research article a cas approach to handle the anisotropic

Research ArticleA CAS Approach to Handle the AnisotropicHookersquos Law for Cancellous Bone and Wood

Sandeep Kumar

Department of Mathematics Government PG Degree College New Tehri Tehri Garhwal Uttarakhand 249 001 India

Correspondence should be addressed to Sandeep Kumar drsandeepkumarmathhotmailcom

Received 29 August 2013 Accepted 1 October 2013 Published 23 March 2014

Academic Editors J Deng and D Sun

Copyright copy 2014 Sandeep Kumar This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The present research entirely relies on the Computer Algebric Systems (CAS) to develop techniques for the data analysis of thesets of elastic constant data measurements In particular this study deals with the development of some appropriate programmingcodes that favor the data analysis of known values of elastic constants for cancellous bone hardwoods and softwood species Moreprecisely a ldquoMathematicardquo code which has an ability to unfold a fourth-order elasticity tensor is discussed Also an effort towardsthe fabrication of an appropriate ldquoMAPLErdquo code has been exposed that can calculate not only the eigenvalues and eigenvectors forcancellous bone hardwoods and softwood species but also computes the nominal average of eigenvectors average eigenvectorsaverage eigenvalues and the average elasticity matrices for these materials Further using such a MAPLE code the histogramscorresponding to average elasticity matrices of 15 hardwood species have been plotted and the graphs for I II III IV V and VIeigenvalues of each hardwood species against their apparent densities are also drawn

1 Introduction

The study of material symmetry of 3-dimensional spaceis of great interest due to having crucial theoretical aswell as practical significance This is because a symmetricalspace includes crystals and all homogeneous fields withoutexceptions electric magnetic gravitational and so forth

The variation of material properties with respect todirection at a stagnant point in a material is called materialsymmetry for instance if the material properties are samein all directions at some fixed point they are called isotropicwhereas if the material properties show variation at the samepoint they are called anisotropic [1]Of course the familiaritywith material symmetries is the best way to categorize thematerials However according to [2] many materials areanisotropic and inhomogeneous due to the varying composi-tion of their constituents In such materials it becomes tick-lish to identify the symmetries ormore particularly the elasticsymmetries The variable composition method to identifymaterialrsquos elastic symmetry becomes complicated and henceto overcome this difficulty an approach is developed by[3] called ldquoaveraging anisotropic elastic constant datardquo Inthis approach the identification of elastic symmetries and

method of variable composition are analyzed separatelyWith the aid of this fabulous approach a sheer volume ofresearch towards material symmetry has been put forward byvarious researchers for example estimation of the effectivetransversely isotropic elastic constants of a material fromknown values of the materialrsquos orthotropic elastic constants[4] anisotropic Hookersquos law for cancellous bone and wood[2] the validation of generalized Hookersquos law for coronaryarteries [5] constitutive relationships of fabric density andelastic properties in cancellous bone architecture [6] analysisof elastic anisotropy of wood material for engineering appli-cations [7] a multidimensional anisotropic strength criterionbased on Kelvinrsquos modes [8] spectral decomposition of a 4th-order covariance tensor application to diffusion tensor MRI[9] a normal distribution of tensor valued random variablesapplications to diffusion tensor MRI [10] and so forth

Concerning computer assisted analysis of bonematerial afew of the articles like directmechanical assessment of elasticsymmetries and properties of trabecular bone architecture[11] relationships between morphology and bone elasticproperties can be accurately quantified using high-resolutioncomputer reconstructions [12] and intrinsic mechanicalproperties of trabecular calcaneus determined by finite

Hindawi Publishing CorporationChinese Journal of EngineeringVolume 2014 Article ID 487314 28 pageshttpdxdoiorg1011552014487314

2 Chinese Journal of Engineering

Table 1 Voigtrsquos mappings for stress (left) and strain tensors (right) respectively

1205901112059022

12059033

12059023

1205901312059012

12059023

1205901212059013

120590111205902212059033120590231205901312059012

1205981112059822

120598331205982312059813

12059812

12059812

12059823

12059813

12059811

1205982212059833212059823212059813212059812

Table 2 Index conversion rule of Voigt

119894 119895 11 22 33 23 or 32 13 or 31 12 or 21119901 1 2 3 4 5 6

Table 3 Voigt to Kelvin and vice versa

120585 = 1 1 1 radic2 radic2 radic2

element models using 3D synchrotron microtomography[13] have explored the subject material up to some greatextent

Now as far as the present proposed research concernedwe turn our concentration towards cancellous bone andwood (soft and hard) According to [4] the type of materialsymmetry exposed by bone tissue is often ambiguous Soin order to attain accuracy we have to measure the elasticconstants of bone specimen which may be represented asorthotropic However the orthotropy of the bone specimenmay or may not be of higher degree in all the directionswith respect to a fixed point in the specimen Thereby for allpractical purposes the bone specimen can be represented astransversely isotropic or even isotropic

Now the question arises iswhat is the need of studying themechanical properties of cancellous bone and woodThe rig-orous answer is that the knowledge about mechanical prop-erties of cancellous bone is essential for the determinationof bone fracture risk in osteoporosis and other pathologicalconditions involving impaired bone strength [6] Also themechanical properties of cancellous bone are determined bythe properties of its bone tissue and its architecture [4 14ndash21] Further the wood is a cellulosic semicrystalline cellu-lar material and the mechanical properties (eg elasticitystrength and rheology etc) are generally higher along thebole of a tree than across the bole [7]

Mechanically clear wood obeys the law of elasticorthotropic material like the bone does In general the woodysubstance also exposes the high toughness and stiffnessproperties and these properties vary according to the typeof wood and the direction in which the woody substance isexamined as the woody substance shows a higher degree ofanisotropy

In addition to deal with anisotropic Hookersquos law oneshould be familiar with the algebra of the 4th-order tensorWe would like to emphasize that the 4th-order tensor algebrais not only involved in the study of material symmetrybut its crucial appearance has been set up in the field ofdiffusion tensor MRI [9 10] visualization and processing of

tensor fields [22] biomechanics [13] tissue mechanics [1]geophysics [23] and so forth

Though an adequate amount of research work has beendedicated towards the algebra of 4th-order tensors thenalso in the following section we shall present a brief digestregarding this issue as this issue hits the present study up tosome great level

2 The Algebra of Fourth-Order Tensor

The splendid word ldquoTensorrdquo has been derived from the Latinword ldquoTensusrdquo which is the past participle of ldquotendererdquo andstands for ldquoStretchrdquo This word was used in anatomy inthe early 1704 to represent muscle and stretches Howeverin Mathematics it existed in 1846 when William RowanHamilton has explored his quaternion Algebra Even thoughthe Hamilton sense regarding tensor did not survive Therecent meaning of tensor is due to Voigt who used the termldquoTensortriplerdquo in crystal elasticity around 1899

Likewise the second-order tensor a fourth-order tensorA (say) is a linear map from LinV to LinV That is it assignsto each second-order tensor A the second-order tensor AUthat is

A Lin (V) 997888rarr Lin (V) U 997891997888rarr AU forallU isin Lin (V)

(1)

where V is a vector space and Lin(V) is a linear space ofdimension 119899

2 and is equivalent to

Lin (V) ≑ 119860 V 997888rarr V 119906 997891997888rarr 119860119906 forall119906 isin V (2)

Also any 4th-order tensor A has the representation

A = 119860119894119895119896119897

119890119894otimes 119890

119895otimes 119890

119896otimes 119890

119897 (3)

where

119860119894119895119896119897

= ⟨119890119894otimes 119890

119895A (119890

119896otimes 119890

119896)⟩ (4)

and the set of all four vectors that is the set of all linear mapsfrom Lin(V) to Lin(V) is a linear space of dimension 119899

4 andis denoted by Lin(V)

The outer (or open) product of the two 4th-order tensorsis given by the composition

(AB)U = A (BU) forallU isin Lin (V) (5)

In abstract index notations we have

(AB)119894119895119896119897

= 119860119894119895119901119902

119861119901119901119902119896119897

(6)

Chinese Journal of Engineering 3

In[2]= SymIndex[2 dim ] =

Apply[ (Sort[List[]] amp) Array[ List Array[dim amp 2]] 2 ]

SymIndex[2 3]

SymIndex[2 6]

In[3]=SymIndex[4] =

Apply[ Flatten[ Sort[ Map[ Sort Partition[List[] 2]]]] amp

Array[ List Array[3 amp 4]] 4]

Algorithm 1

In[4]=MakeName[myexp ] =

ToExpression ( StringJoin Map[ ToString[ ] amp List[ myexp]])

MakeTensor[ mystring 2 dim ] =

Apply[(MakeName[mystring ] amp) SymIndex[2 dim] 2]

MakeTensor[ mystring 4 ] =

Apply[(MakeName[mystring ] amp) SymIndex[4] 4]

In[5]=( stress = MakeTensor[sigma 2 3]) MatrixForm

In[6]=( strain = MakeTensor[epsilon 2 3]) MatrixForm

In[7]=( C4 = MakeTensor[C 4] ) MatrixForm

Algorithm 2

Moreover the transpose of A isin Lin(V) is a 4th-order tensorA119879 and is defined by

⟨UA119879V⟩ = ⟨VAU⟩ forallUV isin Lin (V) (7)

On Lin(V) we can define the inner product as

⟨(119860) B⟩ ≑ tr (A119879B) = 119860119894119895119896119897

119861119894119895119896119897

(8)

The induced norm is

A = [tr (A119879A)]12

(9)

and the metric is

119889119864(AB) = A minus B (10)

Hence with respect to the inner product given by (8) theset 119890

119894otimes 119890

119895otimes 119890

119896otimes 119890

1198971le119894119895119896119897le119899

forms an orthonormal basis ofLin(V)

21 Symmetries of a 4th-Order Tensor In case of a second-order tensor the only known symmetry represents an invari-ance under the mutual rotation of two indices while for the4th-order tensor there are several notions of symmetry thatrepresent invariance under exchanging pair of indices Thusfor the 4th-order tensor generally we have three types ofsymmetries namely major minor and total symmetries andthese notions of symmetries widely play an important role inthe theory of elasticity [24]

211 Major Symmetry of a Fourth-Rank Tensor For A isin

Lin(V) we say that A is symmetric if A = A119879 or incomponent form 119860

119894119895119896119897= 119860

119896119897119894119895and we say that A is skew-

symmetric if A = minusA119879 or 119860119894119895119896119897

= minus119860119896119897119894119895

Moreover we have the decomposition

A =1

2(A + A

119879) +

1

2(A minus A

119879) (11)

Then the set of all symmetric fourth-order tensors is

Sym (V) = A isin Lin (V) | A = A119879 (12)

is a subspace of Lin(V) of dimension 1198992(1198992+ 1)2

In the theory of elasticity this symmetry is often evokedas ldquomajor symmetryrdquo Since the major symmetry representsinvariance under exchanging the pair (119894 119895) and (119896 119897) then ifA isin Sym(V) then exist120582

119894119894=12119899

2 isin R and 119880119894119894=12119899

2 is anorthonormal basis of Lin(V) such that

A =

1198992

sum

119894=1

120582119894119880119894otimes 119880

119894 (13)

The real number 120582119894is called the eigenvalues of A associated

with the eigentensor 119880119894

Such a representation is called the spectral decompositionofAThe trace and the determinant of the fourth-order tensorare defined respectively as

tr (A) =

1198992

sum

119894=1

120582119894 det (A) =

1198992

prod

119894=1

120582119894 (14)


In[8] =indexrule2to1 = 1 1 -gt 1 2 2 -gt 2 3 3 -gt 3 2 3 -gt

4 3 1 -gt 5 1 2 -gt 6 3 2 -gt 4 1 3 -gt

5 2 1 -gt 6

indexrule1to2 = Map[Rule[[[2]] [[1]] ] amp indexrule2to1]

In[9] =Index6[1] = Range[ 6] indexrule1to2

Algorithm 3

In[10]=HookeVto4[ myC ] =

Array[ myC[[ 1 2 indexrule2to1 3 4 indexrule2to1]] amp Array[3 amp 4]]

In[11]=(C2 = MakeTensor[C 2 6]) MatrixForm

In[12]=C2

In[13]=(C2to4 = HookeVto4[C2] ) MatrixForm

In[14]=C2to4

In[15]=Hooke4toV[ myC ] =

Apply[ Part[ myC ] amp

Array[ Join[ 1 indexrule1to2 2 indexrule1to2] amp

Array[6 amp 2]] 2]

In[16]=C2back = Hooke4toV[C2to4] ) MatrixForm

In[17]=(C4to2 = Hooke4toV[C4] ) MatrixForm

In[18]=(C4back = HookeVto4[C4to2]) MatrixForm

In[19]=Hooke2toV[myc2 ] = Table[

Which[

i lt= 3 ampamp j lt= 3 myc2[[i j]]

4 lt= i ampamp j lt= 3 myc2[[i j]] 2 and(12)

i lt= 3 ampamp 4 lt= j myc2[[i j]] 2 and(12)

4 lt= i ampamp 4 lt= j myc2[[i j]] 2

] i 6 j 6]

HookeVto2[mycV ] = Table[

Which[

i lt= 3 ampamp j lt= 3 mycV[[i j]]

4 lt= i ampamp j lt= 3 mycV[[i j]] lowast 2 and(12)

i lt= 3 ampamp 4 lt= j mycV[[i j]] lowast 2 and(12)

4 lt= i ampamp 4 lt= j mycV[[i j]] lowast 2

] i 6 j 6]

Hooke4to2[myC ] = HookeVto2[Hooke4toV[ myC ]]


In[20]=(CV = Hooke2toV[ C2 ] ) MatrixForm

In[21]=(C2back = HookeVto2[ CV ] ) MatrixForm

In[22]=(CV = Hooke4to2[C4]) MatrixForm

In[23]=C4back = Hooke2to4[CV]) MatrixForm

Algorithm 4

212 Minor Symmetry The second type of symmetry iscalled ldquominor symmetryrdquo and is defined by

⟨UAV⟩ = ⟨U119879AV119879⟩ forallUV isin Lin (V) (15)

In component form 119860119894119895119896119897

= 119860119895119894119896119897

= 119860119894119895119897119896

1 le 119894 119895 119896 119897 le 119899Now the invariance under the exchange of first pair of

the indices is called the ldquofirst minor symmetryrdquo and theinvariance under the exchange of second pair of indices iscalled ldquosecond minor symmetryrdquo

The set of all 4th-order tensors that satisfy the minorsymmetry is denoted by

Sym (V) = A isin Lin (V) | A satisfies minor symmetry (16)

213 Total Symmetry A fourth-order tensorA is said to havetotal symmetry if in addition to satisfying the major andminor symmetries it also satisfies

A119906 otimes U otimes V = A119906 otimes U119879 otimes V forallU isin Lin (V) 119906 V isin V

(17)


11 12 1312 22 2313 23 33

11 12 1312 22 2313 23 33

11 12 1312 22 2313 23 33

11 12 1312 22 2313 23 33

11 12 1312 22 2313 23 33

11 12 1312 22 2313 23 33

11 12 1312 22 2313 23 33

11 12 1312 22 2313 23 33

11 12 1312 22 2313 23 33

11 12 1312 22 2313 23 33

11 12 1312 22 2313 23 33

11 12 13

12 22 23

13 23 33

120590ij

Cijkl

120598ij

lowast=

Figure 1 Hookersquos law (reduction process of elastic coefficients) Here still there are 36 components seen in the stiffness table but due to thecomponents for example 119862

2313= 119862

1323 and so forth the counting of 36 components will be reduced up to 21 Also in the above figure the

components having gray background expose symmetry

1

2

3

12

34

56

Column Row

12

34

56

Figure 2 Histogram showing average elasticity matrix of Quipo

2468101214

12

34

56

Column Row

12

34

56

Figure 3 Histogram showing average elasticity matrix of white

246810121416

12

34

56

Column Row

12

34

56

Figure 4 Histogram showing average elasticity matrix of Khaya

or in abstract index notations119860119894119895119896119897

= 119860120590(119894)120590(119895)120590(119895)120590(119897)

for somepermutation 120590 of [1 119899]

The set of the fourth-rank tensors bearing total symmetryis denoted by

Symtotal(V) = A isin Lin (V) | A satisfies total symmetry

(18)

Thus any fourth-order tensor can be decomposed into itstotally symmetric part A119904 and its totally antisymmetric partA119886 that is A = A119904 + A119886

The components of totally symmetric and antisymmetricparts of a fourth-order tensor are described as [25]

119860119904

119894119895119896119897=

1

3(119860

119894119895119896119897+ 119860

119894119896119895119897+ 119860

119894119897119896119895)

119860119886

119894119895119896119897=

1

3(119860

119894119895119896119897minus 119860

119894119896119895119897minus 119860

119894119897119895119896)

(19)


Table 4 The stress strain elasticity and compliance tensorscomponents in Voigtrsquos Kelvinrsquos and MCN patterns

Voigtrsquos notations Kelvinrsquos notations MCN

Stress

12059011

1205901

1

12059022

1205902

2

12059033

1205903

3

12059023

1205904

4

12059013

1205905

5

12059012

1205905

6

Strain

12059811

1205981

1205981

12059822

1205982

1205982

12059833

1205983

1205983

12059823

1205984

1205984

12059813

12059815

1205985

12059812

1205986

1205986

Elasticity

1198621111

11986211

11986211

1198622222

11986222

11986222

1198623333

11986233

11986233

1198621122

11986212

11986212

1198621133

11986213

11986213

1198622233

11986223

11986223

1198622323

11986244

1

211986244

1198621313

11986255

1

211986255

1198621212

11986266

1

211986266

1198621323

11986254

1

211986254

1198621312

11986256

1

211986256

1198621223

11986264

1

411986264

1198622311

11986241

1

radic211986241

1198621311

11986251

1

radic211986251

1198621211

11986261

1

radic211986261

1198622322

11986242

1

radic211986242

1198621322

11986252

1

radic211986252

1198621222

11986262

1

radic211986262

1198622333

11986243

1

radic211986243

1198621333

11986253

1

radic211986253

1198621233

11986263

1

radic211986263

Table 4 Continued


Compliance

1198701111

11987811

11987811

1198702222

11987822

11987822

1198703333

11987833

11987833

1198701122

11987812

11987812

1198701133

11987813

11987813

1198702233

11987823

11987823

1198702323

1

411987844

1

211987844

1198701313

1

411987855

1

211987855

1198701212

1

411987866

1

211987866

1198701323

1

411987854

1

211987854

1198701312

1

411987856

1

211987856

1198701223

1

411987864

1

211987864

1198702311

1

211987841

1

radic211987841

1198701311

1

211987851

1

radic211987851

1198701211

1

211987861

1

radic211987861

1198702322

1

211987842

1

radic211987842

1198701322

1

211987852

1

radic211987852

1198701222

1

211987862

1

radic211987862

1198702333

1

211987843

1

radic211987843

1198701333

1

211987853

1

radic211987853

1198701233

1

211987863

1

radic211987863

24681012141618

12

34

56

Column Row

12

34

56

Figure 5 Histogram showing average elasticity matrix ofMahogany


Table 5 The elastic constants data for hardwoods

S no Species 120588 11986211

11986222

11986233

11986212

11986213

11986223

11986244

11986255

11986266

1 Quipo 01 0045 0251 1075 0027 0033 0025 0226 0118 00782 Quipo 02 0159 0427 3446 0069 0131 0178 0430 0280 01443 White 038 0547 1192 10041 0399 0360 0555 1442 1344 00224 Khaya 044 0631 1381 10725 0389 0520 0662 1800 1196 04205 Mahogany 050 0952 1575 11996 0571 0682 0790 1960 1498 06386 Mahogany 053 0765 1538 13010 0655 0631 0841 1218 0938 03007 S Germ 054 0772 1772 12240 0558 0530 0871 2318 1582 05408 Maple 058 1451 2565 11492 1197 1267 1818 2460 2194 05849 Walnut 059 0927 1760 12432 0707 0936 1312 1922 1400 046010 Birch 062 0898 1623 17173 0671 0714 1075 2346 1816 037211 Y Birch 064 1084 1697 15288 0777 0883 1191 2120 1942 048012 Oak 067 1350 2983 16958 1007 1005 1463 2380 1532 078413 Ash 068 1135 2142 16958 0827 0917 1427 2684 1784 054014 Ash 080 1439 2439 17000 1037 1485 1968 1720 1218 050015 Beech 074 1659 3301 15437 1279 1433 2142 3216 2112 0912Source this data is consulted from [47 48] The number of repetitions of the particular species in this table shows the number of measurements done for thatparticular wood species The units of 120588 (bulk or apparent density) are gcm3 and the units of elastic constants are GPa = 1010 dynescm2

Table 6 The elastic constants data for softwoods

S no Species 120588 11986211

11986222

11986233

11986212

11986213

11986223

11986244

11986255

11986266

1 Blasa 02 0127 0360 6380 0086 0091 0154 0624 0406 00662 Spruce 039 0572 1030 11950 0262 0365 0506 1498 1442 00783 Spruce 043 0594 1106 14055 0346 0476 0686 1442 10 00644 Spruce 044 0443 0775 16286 0192 0321 0442 1234 152 00725 Spruce 050 0755 0963 17221 0333 0549 0548 125 1706 0076 Douglas Fir 045 0929 1173 16095 0409 0539 0539 1767 1766 01767 Douglas Fir 059 1226 1775 17004 0753 0747 0941 2348 1816 01608 Pine 054 0721 1405 16929 0454 0535 0857 3484 1344 0132Source this data is consulted from [47 48] The number of repetitions of the particular species in this table shows the number of measurements done for thatparticular wood species The units of 120588 (bulk or apparent density) are gcm3 and the units of elastic constants are GPa = 1010 dynescm2

Obviously a fourth-order tensor is totally symmetric if A =

A119904 or equivalently A119886 = O where O stands for a fourth-order null tensor Likewise A is antisymmetric if A = A119886 orequivalently A119904 = O

Now it is straightforward to show that if A is totallysymmetric and B is a symmetric tensor then tr(AB) = 0

Reference [25] has also shown that there is an isomor-phism (ie a 1-1 linear map) between totally symmetricfourth-rank tensor and a homogeneous polynomial of degreefour Also if a totally symmetric fourth-order tensor istraceless that is tr(A) = 0 then it is isomorphic toa harmonic polynomial of degree four For this reason atotally symmetric and traceless tensor is called harmonictensor

To meet the objectives of proposed research let us turnour attention to the flattening (sometimes called unfolding)of the fourth-order 3-dimensional tensor withminor symme-tries as it is customary in the 3-dimensional elasticity theory

In the following section we discuss the notion of9-dimensional representation of a 3-dimensional fourth-order tensor and then particularize this representation to

a fourth-order elasticity tensor havingminor symmetries dueto the well-known anisotropic Hookersquos law

3 A 9-Dimensional Exposition ofa 3-Dimensional 4th-Order Tensorand the Anisotropic Hookersquos Law

In accordance with the evolution of tensor theory the algebraof the 3-dimensional fourth- order tensor is still not fullydeveloped For instance the techniques for calculating thelatent roots an latent tensors of a tensor like119862119894119895119896119897 or the com-putation of its inverse (which is called a compliance 119878119894119895119896119897) arestill not widely available There are also some certain psycho-logical constraints that is we are trained to tackle matricesbut not trained to deal with the multiway arrays (sometimescalled matrix of matrices) Though recently Tamra [26ndash29]has developed a tensor toolbox that runs under MATLAB[30] in my view this tool still needs some enhancementsto handle symbolic tensor data Moreover Constantinescuand Korsunsky [31] have developed some crucial packages


Table 7 The elastic constants data for the three specimens of cancellous bone

Elastic constants Specimen 1 Specimen 2 Specimen 311986211

986 7912 698311986212

2147 3609 281611986213

2773 3186 284511986214

minus0162 11 8011986215

minus293 54 minus0711986216

minus0821 minus06 11411986222

935 8529 896711986223

2346 3281 322711986224

minus1204 09 minus162311986225

minus0936 minus25 minus3211986226

0445 minus42 16011986233

5952 14730 916011986234


minus1791 169 minus7211986236

054 06 2411986244

561 3587 348911986245

0798 05 10511986246

minus2159 51 minus9811986255

6032 3448 242611986256


7425 2304 2378Source the data for cancellous bone has been taken from [2 11]

24681012141618

12

34

56

Column Row

12

34

56

Figure 6 Histogram showing average elasticity matrix of S Germ

namely ldquoTensor2Analysismrdquo ldquoIntegrateStrainmrdquo ldquoParamet-ricMeshmrdquo and ldquoVectorAnalysismrdquo which are compatiblewith Mathematica and Mathematica programming usingthese packages enables one to compute symbolic computa-tions regarding elasticity tensors Anyways we shall comeback to these CAS assisted issues in the subsequent sectionswhere we shall demonstrate the proposed objectives usingCAS

5

10

15

20

12

34

56

Column Row

12

34

56

Figure 7 Histogram showing average elasticity matrix of maple

In order to study material symmetries and anisotropicHookersquos law it is customary to transform the 3-dimensional4th-order tensor as a 9 times 9 matrix that is a 9-dimensionalrepresentation of a 3-dimensional fourth-order tensor Forthis purpose there is a basic notion of representinga fourth-order tensor as a second-order tensor [22] Accord-ing to [22] if 1 le 119894 119895 le 119899 then we set

119890120595(119894119895)

= 119890119894otimes 119890

119895 (20)


Table 8 The eigenvalues and eigenvectors for 15 hardwoods species calculated using MAPLE Λ[119894] stand for the eigenvalues of 119894th softwoodspecies and 119881[119894] stand for the corresponding eigenvectors

S no Hardwood species Eigenvalues Λ[119894] Eigenvectors 119881[119894]

1 Quipo

[[[[[[[[[

[

1076866026

004067345638

02534605191

02260000000

01180000000

007800000000

]]]]]]]]]

]

[[[[[[[[[

[

003276733390 09918780499 minus01228992897 00 00 00

003131140851 minus01239237163 minus09917976143 00 00 00

09989724208 minus002865041271 003511774899 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

2 Quipo

[[[[[[[[[

[

3461963876

01399776173

04300584948

04300000000

02800000000

01440000000

]]]]]]]]]

]

[[[[[[[[[

[

004079957804 09756137601 minus02156691559 00 00 00

005942480245 minus02178361212 minus09741745817 00 00 00

09973986606 minus002692981555 006686327745 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

3 White

[[[[[[[[[

[

1009116301

03556701722

1333166815

1442000000

1344000000

002200000000

]]]]]]]]]

]

[[[[[[[[[

[

004028666644 09048796003 minus04237568827 00 00 00

006399313350 minus04255671399 minus09026613361 00 00 00

09971368323 minus0009247686096 007505076253 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

4 Khaya

[[[[[[[[[

[

1080102614

04606834511

1475290392

1800000000

1196000000

04200000000

]]]]]]]]]

]

[[[[[[[[[

[

005368516527 09266208315 minus03721447810 00 00 00

007220768570 minus03753089731 minus09240829102 00 00 00

09959437502 minus002273783078 008705767064 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

5 Mahogany

[[[[[[[[[

[

1210253321

06098853915

1810581412

1960000000

1498000000

06380000000

]]]]]]]]]

]

[[[[[[[[[

[

minus006484967920 minus08666704601 minus04946481887 00 00 00

minus007817061387 04985803653 minus08633116316 00 00 00

minus09948285648 001731853005 01000809391 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

6 Mahogany

[[[[[[[[[

[

1310854783

03895295651

1814922632

1218000000

09380000000

03000000000

]]]]]]]]]

]

[[[[[[[[[

[


minus007547523076 04892979419 minus08688446422 00 00 00

minus09956351187 001099502014 009268127742 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

7 S Germ

[[[[[[[[[

[

1234053410

05212570563

1922208822

2318000000

1582000000

05400000000

]]]]]]]]]

]

[[[[[[[[[

[

004967528691 09161715971 minus03976958243 00 00 00

008463937788 minus04006166011 minus09123280736 00 00 00

09951726186 minus001165943224 009744493938 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

8 Maple

[[[[[[[[[

[

1205449768

06868279555

2766674361

2460000000

2194000000

05840000000

]]]]]]]]]

]

[[[[[[[[[

[


minus02031928413 05304148448 minus08230265872 00 00 00

minus09692575344 001015375725 02458388325 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]


Table 9 In continuation to Table 8


9 Walnut

[[[[[[[[[

[

1267869546

05204134969

1919891015

1922000000

1400000000

04600000000

]]]]]]]]]

]

[[[[[[[[[

[

008621257414 08755641188 minus04753471023 00 00 00

01243595697 minus04828494241 minus08668282006 00 00 00

09884847438 minus001561753067 01505124700 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

10 Birch

[[[[[[[[[

[

3168908680

04747157903

05946755301

2346000000

1816000000

03720000000

]]]]]]]]]

]

[[[[[[[[[

[

03961520086 09160614623 006240981414 00 00 00

06338768023 minus02236797319 minus07403833993 00 00 00

06642768888 minus03328645006 06692812840 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

11 Y Birch

[[[[[[[[[

[

1545414040

05549651499

2059894445

2120000000

1942000000

04800000000

]]]]]]]]]

]

[[[[[[[[[

[


minus008975775227 05593224991 minus08240763851 00 00 00

minus09937798435 0004466103673 01112729817 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

12 Oak

[[[[[[[[[

[

1718666431

08648974218

3239438239

2380000000

1532000000

07840000000

]]]]]]]]]

]

[[[[[[[[[

[

006975010637 09079000793 minus04133429180 00 00 00

01071019008 minus04187725641 minus09017531389 00 00 00

09917984199 minus001862756541 01264472547 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

13 Ash

[[[[[[[[[

[

1715567967

06696042476

2409716064

2684000000

1784000000

05400000000

]]]]]]]]]

]

[[[[[[[[[

[

006190313535 08746549514 minus04807772027 00 00 00

009781749768 minus04846986500 minus08691944279 00 00 00

09932772717 minus0006777439445 01155609239 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

14 Ash

[[[[[[[[[

[

1742363268

07844815431

2669885744

1720000000

1218000000

05000000000

]]]]]]]]]

]

[[[[[[[[[

[


minus01363859604 05177044741 minus08446188165 00 00 00

minus09855542417 001550704374 01686486565 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

15 Beech

[[[[[[[[[

[

1599002755

09558575345

3451114905

3216000000

2112000000

09120000000

]]]]]]]]]

]

[[[[[[[[[

[

01135176976 08848520771 minus04518302069 00 00 00

01764907831 minus04654963442 minus08672739791 00 00 00

09777344911 minus001870707734 02090103088 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

Thus 119890120572otimes 119890

1205731le120572120573le119899

2 is an orthonormal basis of Lin(V) equiv

Lin(V)Therefore any fourth-rank tensor A isin Lin(V) can

be assumed as an 119899-dimensional fourth rank tensor A =

119860119894119895119896119897

119890119894otimes 119890

119895otimes 119890

119896otimes 119890

119897 or as an 119899

2-dimensional second-ordertensor A harr 119860 = 119860

120572120573119890120572otimes 119890

120573 where 119860

120595(119894119895)120595119896119897= 119860

119894119895119896119897

1 le 119894 119895 119896 119897 le 119899

Again we have from (8) that

tr (A119879B) = ⟨AB⟩ = tr (119860119879119861119879) A =1003817100381710038171003817100381711986010038171003817100381710038171003817 (21)

hence the two-way map A harr 119860 is an isometryLet us apply this concept to anisotropic Hookersquos law by

assuming that the anisotropic Hookersquos law given below is


Table 10 The eigenvalues and eigenvectors for 8 softwoods species calculated using MAPLE Λ[119894] stand for the eigenvalues of 119894th softwoodspecies and 119881[119894] stand for the corresponding eigenvectors

S no Softwood species Eigenvalues Λ[119894] Eigenvectors 119881[119894]

1 Blasa

[[[[[[[[[

[

6385324208

009845837744

03832174064

06240000000

04060000000

006600000000

]]]]]]]]]

]

[[[[[[[[[

[

001488818113 09510211778 minus03087669978 00 00 00

002575997614 minus03090635474 minus09506924583 00 00 00

09995572851 minus0006200252135 002909967665 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

2 Spruce

[[[[[[[[[

[

1198583568

04516442734

1114520065

1498000000

1442000000

007800000000

]]]]]]]]]

]

[[[[[[[[[

[


minus004689865645 04044467539 minus09133582749 00 00 00

minus09983543167 001123114838 005623628701 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

3 Spruce

[[[[[[[[[

[

1410927768

04185382987

1227183961

1442000000

10

006400000000

]]]]]]]]]

]

[[[[[[[[[

[


minus005361649666 04423303564 minus08952480817 00 00 00

minus09978936411 0009052295016 006423657043 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

4 Spruce

[[[[[[[[[

[

1630529953

03544288592

08442715844

1234000000

1520000000

007200000000

]]]]]]]]]

]

[[[[[[[[[

[


minus002869707087 04091564350 minus09120128765 00 00 00

minus09993764538 0007032901380 003460119282 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

5 Spruce

[[[[[[[[[

[

1725845208

05092652753

1171282625

1250000000

1706000000

007000000000

]]]]]]]]]

]

[[[[[[[[[

[


minus003428302033 05858146064 minus08097196604 00 00 00

minus09988364173 0007413313465 004765347716 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

6 Douglas Fir

[[[[[[[[[

[

1613459769

06233733771

1439028943

1767000000

1766000000

01760000000

]]]]]]]]]

]

[[[[[[[[[

[


minus003697194691 05922678885 minus08048924305 00 00 00

minus09986533619 0007435777607 005134366998 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

7 Douglas Fir

[[[[[[[[[

[

1710150156

06986859431

2204812526

2348000000

1816000000

01600000000

]]]]]]]]]

]

[[[[[[[[[

[


minus006364825251 05704773676 minus08188433771 00 00 00

minus09967231590 0004703484281 008075160717 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

8 Pine

[[[[[[[[[

[

1699539133

04940019684

1565606760

3484000000

1344000000

01320000000

]]]]]]]]]

]

[[[[[[[[[

[


minus005585205149 04413516031 minus08955943895 00 00 00

minus09978476167 0006195562437 006528207193 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]


Table11Th

eeigenvalues

andeigenvectorsfor3

specim

enso

fcancello

usbo

necalculated

usingMAPL

EΛ[119894]sta

ndforthe

eigenvalueso

f119894th

specim

enof

cancellous

boneand

119881[119894]sta

ndfor

thec

orrespon

ding

eigenvectors

Cancellous

bone

EigenvaluesΛ

[119894]

Eigenvectors

119881[119894]

Specim

en1

[ [ [ [ [ [ [ [

1334869880

minus02071167473

8644210768

6837676373

3768806080

4123724855

] ] ] ] ] ] ] ]

[ [ [ [ [ [ [ [

03316115964

08831238953

02356441950

02242668491

005354786914

003787816350

06475092156

minus01099136965

01556396475

minus06940063479

minus01278953723

02154647417

04800906963

minus02832250652

007648827487

02494278793

06410471445

minus04585742261

minus02737532692

minus006354940263

04518342532

minus006214920576

05836952694

06101669758

minus03706582070

03314035780

minus01276606259

minus06337030234

03639432801

minus04499474345

01671829224

01180163925

minus08330343502

002008124471

03109325968

04087706408

] ] ] ] ] ] ] ]

Specim

en2

[ [ [ [ [ [ [ [

1754634171

7773848354

1550688791

2301790452

3445174379

3589156342

] ] ] ] ] ] ] ]

[ [ [ [ [ [ [ [

minus03057290719

minus03005337771

minus09030341401

001584500766

minus002175914011

minus0003741933916

minus04311831740

minus08021570593

04130146809

00001362645283

minus0003854432451

0005396441669

minus08488209390

05154657137

01152135966

minus0004741957786

002229739182

minus0002068264586

00009402467441

minus0005358584867

0003987518000

minus003989381331

003796849613

minus09984595017

minus001058157817

002100470582

002092543658

0006230946613

minus09987520802

minus003826770492

00009823402498

0006977574167

001484146906

09990475909

0008196718814

minus003958286493

] ] ] ] ] ] ] ]

Specim

en3

[ [ [ [ [ [ [ [

1418542670

5838388363

9959058231

3433818493

3024968125

1757492470

] ] ] ] ] ] ] ]

[ [ [ [ [ [ [ [

03244960223

minus0004831643783

minus08451713783

04062092722

01164524794

minus004239310198

06461152835

minus07125334236

02668138654

004131783689

minus002431040234

003665187401

06621315621

07009969173

02633438279

002836289512

minus0001711614840

0005269111592

minus01962401199

0009159859564

03740839141

08850746598

01904577552

004284654369

minus0008597323304

minus0002526763452

001897677365

01738799329

minus06977939665

minus06945566457

001533190599

minus002803230310

006961922985

minus01374446216

06801869735

minus07159517722

] ] ] ] ] ] ] ]


24681012141618

12

34

56

Column Row

12

34

56

Figure 8 Histogram showing average elasticity matrix of Walnut

12345678

12

34

56

Column Row

12

34

56

Figure 9 Histogram showing average elasticity matrix of Birch

generalized one and also valid in the case where stress andstrain tensors are not necessarily symmetric

The anisotropic Hookersquos law in abstract index notations isoften depicted as

120590119894119895= 119862

119894119895119896119897120598119896119897 or in index free notations 120590 = C120598 (22)

where 119862119894119895119896119897

are the components of an elastic tensor Writingthe stress strain and elastic tensors in usual tensor bases wehave

120590 = 120590119894119895119890119894otimes 119890

119895 120598 = 120598

119896119897119890119896otimes 119890

119897

C = 119862119894119895119896119897

119890119894otimes 119890

119896otimes 119890

119896otimes 119890

119897

(23)

5

10

15

20

12

34

56

Column Row

12

34

56

Figure 10 Histogram showing average elasticity matrix of Y Birch

5

10

15

20

25

12

34

56

Column Row

12

34

56

Figure 11 Histogram showing average elasticity matrix of Oak

Now when working with orthonormal basis one needs tointroduce a new basis which should be composed of the threediagonal elements

E1equiv 119890

1otimes 119890

1

E2equiv 119890

2otimes 119890

2

E3equiv 119890

3otimes 119890

3

(24)

the three symmetric elements

E4equiv

1

radic2(1198901otimes 119890

2+ 119890

2otimes 119890

1)

E5equiv

1


3+ 119890

3otimes 119890

2)

E6equiv

1


1+ 119890

1otimes 119890

3)

(25)


and the three asymmetric elements

E7equiv

1


2minus 119890

2otimes 119890

1)

E8equiv

1


3minus 119890

3otimes 119890

2)

E9equiv

1


1minus 119890

1otimes 119890

3)

(26)

In this new system of bases the components of stress strainand elastic tensors respectively are defined as

120590 = 119878119860E

119860

120598 = 119864119860E

119860

C = 119862119860119861

119890119860otimes 119890

119861

(27)

where all the implicit sums concerning the indices 119860 119861

range from 1 to 9 and 119878 is the compliance tensorThus for Hookersquos law and for eigenstiffness-eigenstrain

equations one can have the following equivalences

120590119894119895= 119862

119894119895119896119897120598119896119897

lArrrArr 119878119860= 119862

119860119861E119861

119862119894119895119896119897

120598119896119897

= 120582120598119894119895lArrrArr 119862

119860119861E119861= 120582119864

119860

(28)

Using elementary algebra we can have the components of thestress and strain tensors in two bases

((((((((

(

1198781

1198782

1198783

1198784

1198785

1198786

1198787

1198788

1198789

))))))))

)

=

((((((((

(

12059011

12059022

12059033

120590(12)

120590(23)

120590(31)

120590[12]

120590[23]

120590[31]

))))))))

)

((((((((

(

1198641

1198642

1198643

1198644

1198645

1198646

1198647

1198648

1198649

))))))))

)

=

((((((((

(

12059811

12059822

12059833

120598(12)

120598(23)

120598(31)

120598[12]

120598[23]

120598[31]

))))))))

)

(29)

where we have used the following notations

120579(119894119895)

equiv1

radic2(120579119894119895+ 120579

119895119894) 120579

[119894119895]equiv

1

radic2(120579119894119895minus 120579

119895119894) (30)

Now in the new basis 119864119860 the new components of stiffness

tensor C are

((((((((

(

11986211

11986212

11986213

11986214

11986215

11986216

11986217

11986218

11986219

11986221

11986222

11986223

11986224

11986225

11986226

11986227

11986228

11986229

11986231

11986232

11986233

11986234

11986235

11986236

11986237

11986238

11986239

11986241

11986242

11986243

11986244

11986245

11986246

11986247

11986248

11986249

11986251

11986252

11986253

11986254

11986255

11986256

11986257

11986258

11986259

11986261

11986262

11986263

11986264

11986265

11986266

11986267

11986268

11986269

11986271

11986272

11986273

11986274

11986275

11986276

11986277

11986278

11986279

11986281

11986282

11986283

11986284

11986285

11986286

11986287

11986288

11986289

11986291

11986292

11986293

11986294

11986295

11986296

11986297

11986298

11986299

))))))))

)

=

1198621111

1198621122

1198621133

1198622211

1198622222

1198623333

1198623311

1198623322

1198623333

11986211(12)

11986211(23)

11986211(31)

11986222(12)

11986222(23)

11986222(31)

11986233(12)

11986233(23)

11986233(31)

11986211[12]

11986211[23]

11986211[31]

11986222[12]

11986222[23]

11986222[31]

11986233[12]

11986233[23]

11986233[31]

119862(12)11

119862(12)22

119862(12)33

119862(23)11

119862(23)22

119862(23)33

119862(31)11

119862(31)22

119862(31)33

119862(1212)

119862(1223)

119862(1231)

119862(2312)

119862(2323)

119862(2331)

119862(3112)

119862(3123)

119862(3131)

119862(12)[12]

119862(12)[23]

119862(12)[31]

119862(23)[12]

119862(23)[23]

119862(23)[31]

119862(31)[12]

119862(31)[23]

119862(31)[31]

119862[12]11

119862[12]22

119862[12]33

119862[23]11

119862[23]22

119862[123]33

119862[21]11

119862[31]22

119862[31]33

119862[12](12)

119862[12](23)

119862[12](21)

119862[23](12)

119862[23](23)

119862[23](21)

119862[31](12)

119862[31](23)

119862[31](21)

119862[1212]

119862[1223]

119862[1231]

119862[2312]

119862[2323]

119862[2331]

119862[3112]

119862[3123]

119862[3131]

(31)

where

119862119894119895(119896119897)

equiv1

radic2(119862

119894119895119896119897+ 119862

119894119895119897119896) 119862

119894119895[119896119897]equiv

1

radic2(119862

119894119895119896119897minus 119862

119894119895119897119896)

119862(119894119895)119896119897

equiv1

radic2(119862

119894119895119896119897+ 119862

119895119894119896119897) 119862

[119894119895]119896119897equiv

1

radic2(119862

119894119895119896119897minus 119862

119895119894119896119897)

119862(119894119895119896119897)

equiv1

2(119862

119894119895119896119897+ 119862

119894119895119897119896+ 119862

119895119894119896119897+ 119862

119895119894119897119896)

119862(119894119895)[119896119897]

equiv1

2(119862

119894119895119896119897minus 119862

119894119895119897119896+ 119862

119895119894119896119897minus 119862

119895119894119897119896)

119862[119894119895](119896119897)

equiv1

2(119862

119894119895119896119897+ 119862

119894119895119897119896minus 119862

119895119894119896119897minus 119862

119895119894119897119896)

119862[119894119895119896119897]

equiv1

2(119862

119894119895119896119897minus 119862

119895119894119896119897minus 119862

119894119895119897119896+ 119862

119895119894119897119896)

(32)


12

34

56

ColumnRow

12

34

56

600005000040000300002000010000

Figure 12 Histogram showing average elasticity matrix of Ash

5

10

15

20

25

12

34

56

Column Row

12

34

56

Figure 13 Histogram showing average elasticity matrix of Beech

In case if we impose symmetries of stress and strain tensorslet us see what happens with generalized Hookersquos law (22)

In generalized Hookersquos law when the stress-strain sym-metries do not affect the stiffness tensor C the number ofcomponents of stiffness tensor in 3-dimensional space isequal to 3

4= 81 Now if we impose the symmetries of stress-

strain tensors that is 120590119894119895= 120590

119895119894and 120598

119896119897= 120598

119897119896 the 119862

119894119895119896119897will be

like 119862119894119895119896119897

= 119862119895119894119896119897

= 119862119894119895119897119896

Moreover imposing symmetrical connection that is

119862119894119895119896119897

= 119862119896119897119894119895

we would have only 21 significant componentsout of 81 Thus if we flatten the stiffness tensor under thenotion of Hookersquos law we will definitely have a 6 times 6 matrixhaving only 21 independent elastic coefficients instead of 9 times

9 matrix having 81 elastic coefficientsHere we depict a figure (see Figure 1) which delineates

the component reduction process for the stiffness tensor

Now with 21 significant independent components thestiffness tensor 119862

119894119895119896119897can be mapped on a symmetric 6 times 6

matrixAs the elasticity of a material is described by a fourth-

order tensor with 21 independent components as shownin (Figure 1) and the mathematical description of elasticitytensor appears in Hookersquos law now how to map this 3-dimensional fourth order tensor on a 6 times 6 matrix

We are fortunate to have a long series of research papersconcerning this issue For instance [2 3 32ndash39] and manymore

The very first approach to map 21 significant componentsof the elasticity tensor on a symmetric 6 times 6 matrix wasintroduced by Voigt [32] and after that various successors ofVoigt have used his fabulous notions for flattening of fourth-order tensors But Lord Kelvin [40] found the Voigt notationsare inadequate from the perspective of tensorial nature andthen introduced his own advanced notations now known asKelvinrsquos mapping More recently [39] have introduced somesophisticated methodology to unfold a fourth-rank tensorcalled ldquoMCNrdquo (Mehrabadi and Cowinrsquos notations)

Let us briefly go through these three notions of tensorflattening one by one

31 The Voigt Six-Dimensional Notations for Unfolding anElasticity Tensor It is well known that the Voigt mappingpreserves the elastic energy density of thematerial and elasticstiffness and is given by

2 sdot 119864energy = 120590119894119895120598119894119895= 120590

119901120598119901 (33)

TheVoigtmapping receives this relation by themapping rules

119901 = 119894120575119894119895+ (1 minus 120575

119894119895) (9 minus 119894 minus 119895)

119902 = 119896120575119896119897+ (1 minus 120575

119896119897) (9 minus 119896 minus 119897)

(34)

Using this rule we have 120590119894119895= 120590

119901 119862

119894119895119896119897= 119862

119901119902 and 120598

119902= (2 minus

120575119896119897)120598119896119897

This Voigt mapping can be visualized as shown in Table 1Thus in accordance with Voigtrsquos mappings Hookersquos law

(22) can be represented in matrix form as follows

(

(

1205901

1205902

1205903

1205904

1205905

1205906

)

)

= (

(

11986211

11986212

11986213

11986214

11986215

11986216

11986221

11986222

11986223

11986224

11986225

11986226

11986231

11986232

11986233

11986234

11986235

11986236

11986241

11986242

11986243

11986244

11986245

11986246

11986251

11986252

11986253

11986254

11986255

11986256

11986261

11986262

11986263

11986264

11986265

11986266

)

)

(

(

1205981

1205982

1205983

1205984

1205985

1205986

)

)

(35)

where the simple index conversion rule of Voigt (see Table 2)is applied

But in the Voigt notations many disadvantages werenoticed For instance

(1) the 120590119894119895and 120598

119896119897are treated differently

(2) the norms of 120590119894119895 120598119896119897 and 119862

119894119895119896119897are not preserved

(3) the entries in all the three Voigt arrays (see (35)) arenot the tensor or the vector components and thus


01 02 03 04 05 06 07 08

2

4

6

8

10

12

14

16

(a)

01

02

03

04

05

06

07

08

09

01 02 03 04 05 06 07 08

(b)

Figure 14The graphics placed in left as well as right positions showing graphs of I and II eigenvalues of all 15 hardwood species against theirapparent densities

01 02 03 04 05 06 07 08

05

1

15

2

25

3

(a)

01 02 03 04 05 06 07 08

05

1

15

2

25

3

(b)

Figure 15 The graphics placed in left as well as right positions showing graphs of III and IV eigenvalues of all 15 hardwood species againsttheir apparent densities

one may not be able to have advantages of tensoralgebra like tensor law of transformation rotation ofcoordinate system and so forth

32 Kelvinrsquos Mapping Rules Likewise Voigtrsquos mapping rulesKelvinrsquosmapping rules also preserve the elastic energy densityof material under the following methodology

119901 = 119894120575119894119895+ (1 minus 120575

119894119895) (9 minus 119894 minus 119895)

119902 = 119896120575119896119897+ (1 minus 120575

119896119897) (9 minus 119896 minus 119897)

(36)

In this way

120590119901= (120575

119894119895+ radic2 (1 minus 120575

119894119895)) 120598

119894119895

120598119902= (120575

119896119897+ radic2 (1 minus 120575

119896119897)) 120598

119896119897

119862119901119902

= (120575119894119895+ radic2 (1 minus 120575

119894119895)) (120575

119896119897+ radic2 (1 minus 120575

119896119897)) 119862

119894119895119896119897

(37)

Naturally Kelvinrsquos mapping preserves the norms of the threetensors and the stress and strain are treated identically Alsothe mappings of stress strain and elasticity tensors have allthe properties of tensor of first- and second-rank tensorsrespectively in 6-dimensional space


But the only disadvantage of this mapping is that thevalues of stiffness components are changed However thereis a simple tool for conversion between Voigtrsquos and Kelvinrsquosnotations For this purpose one just needed a single array (seeTable 3) Thus

120590kelvin

= 120590viogt

120585 120590voigt

=120590kelvin

120585

120598kelvin

= 120598voigt

120585 120598voigt

=120598kelvin

120585

(38)

33 Mehrabadi-Cowin Notations (MCN) To preserve thetensor properties of Hookersquos law during the unfolding offourth-order elasticity tensor [39] have introduced a newnotion which is nothing but a conversion of Voigtrsquos notationsinto Kelvin ones This conversion is done using the followingrelation

119862kelvin

=((

(

1 1 1 radic2 radic2 radic2



radic2 radic2 radic2 2 2 2



))

)

119862voigt

(39)

Exploiting this new notion (35) becomes

(

(

12059011

12059022

12059033

radic212059023

radic212059013

radic212059012

)

)

= (

1198621111 1198621122 1198621133radic21198621123

radic21198621113radic21198621112

1198622211 1198622222 1198622333radic21198622223

radic21198622213radic21198622212

1198623311 1198623322 1198623333radic21198623323

radic21198623313radic21198623312

radic21198622311radic21198622322

radic21198622333 21198622323 21198622313 21198622312

radic21198621311radic21198621322

radic21198621333 21198621323 21198621313 21198621312

radic21198621211radic21198621222

radic21198621233 21198621223 21198621213 21198621212

)

lowast(

(

12059811

12059822

12059833

radic212059823

radic212059813

radic212059812

)

)

(40)

Further to involve the features of tensor in the matrix rep-resentation (40) [39] have evoked that (40) can be rewrittenas

= C (41)

where the new six-dimensional stress and strain vectorsdenoted by and respectively are multiplied by the factors

radic2 Also C is a new six-by-six matrix [39] The matrix formof (40) is given as

(

(

12059011

12059022

12059033

radic212059023

radic212059013

radic212059012

)

)

=((

(

11986211

11986212

11986213

radic211986214

radic211986215

radic211986216

11986212

11986222

11986223

radic211986224

radic211986225

radic211986226

11986213

11986223

11986233

radic211986234

radic211986235

radic211986236

radic211986214

radic211986224

radic211986234

211986244

211986245

211986246

radic211986215

radic211986225

radic211986235

211986245

211986255

211986256

radic211986216

radic211986226

radic211986236

211986246

211986256

211986266

))

)

lowast (

(

12059811

12059822

12059833

radic212059823

radic212059813

radic212059812

)

)

(42)

The representation (42) is further produced in MCN patternlike below

(

(

1

2

3

4

5

6

)

)

=((

(

11986211

11986212

11986213

11986214

11986215

11986216

11986212

11986222

11986223

11986224

11986225

11986226

11986213

11986223

11986233

11986234

11986235

11986236

11986214

11986224

11986234

11986244

11986245

11986246

11986215

11986225

11986235

11986245

11986255

11986256

11986216

11986226

11986236

11986246

11986256

11986266

))

)

lowast (

(

1205981

1205982

1205983

1205984

1205985

1205986

)

)

(43)

To deal with all the above representations of a fourth-orderelasticity tensor as a six-by-six matrix a simple conversiontable has been proposed by [39] which is depicted as inTable 4

Even though the foregoing detailed description is inter-esting and important from the viewpoint of those who arebeginners in the field of mathematical elasticity the modernscenario of the literature of mathematical elasticity nowinvolves the assistance of various computer algebraic systems(CAS) like MATLAB [30] Maple [41] Mathematica [42]wxMaxima [43] and some peculiar packages like TensorToolbox [26ndash29] GRTensor [44] Ricci [45] and so forth tohandle particular problems of the scientific literature

However in the following section we shall delineate aCAS approach for Section 3


4 Unfolding the Anisotropic Hookersquos Lawwith the Aid of lsquolsquoMathematicarsquorsquo

ldquoMathematicardquo is a general computing environment inti-mated with organizing algorithmic visualization and user-friendly interface capabilities Moreover many mathematicalalgorithms encapsulated by ldquoMathematicardquo make the compu-tation easy and fast [42 46]

A fabulous book entitled ldquoElasticity with Mathematicardquohas been produced by [31] This book is equipped with somegreat Mathematica packages namely VectorAnalysismDisplacementm IntegrateStrainm ParametricMeshm andTensor2Analysism

Overall the effort of [31] is greatly appreciated for pro-ducing amasterpiecewithMathematica packages notebooksand worked examples Moreover all the aforementionedpackages notebooks and worked examples are freely avail-able to readers and can be downloaded from the publisherrsquoswebsite httpwwwcambridgeorg9780521842013

We consider the following code that easily meets thepurpose discussed in Section 3

Here kindly note that only the input Mathematica codesare being exposed For considering the entire processingan Electronic (see Appendix A in Supplementary Materialavailable online at httpdxdoiorg1011552014487314) isbeing proposed to readers

First of all install the ldquoTensor2Analysisrdquo package inMathematica using the following command

In [1]= ltlt Tensor2AnalysislsquoWe have loaded the above package like Loading the

package ldquoTensor2Analysismrdquo by setting the following pathSetDirectory[CProgram FilesWolframResearch

Mathematica80AddOnsPackages]Next is concerning the creation of tensor The code in

Algorithm 1 generates the stress and strain tensors using theindex rules specified by [31]

Use theMakeTensor command to create tensorial expres-sions having components with respect to symmetric condi-tions This command combines all the previous commandsto do so (see Algorithm 2)

Now the next code deals with the index rules that is ableto handle Hookersquos tensor conversion from the fourth-orderto the second-order tensor or second-order Voigt form usingindexrule (see Algorithm 3)

Afterwards we have a crucial stage that concerns withthe transformation of Voigtrsquos notations into a fourth-ranktensor and vice versa This stage includes the codes likeHookeVto4 Hooke4toV Also the codes HookeVto2 andHook2toV transform the Voigt notations into the second-order tensor notations and vice versa Finally the code inAlgorithm 4 provides us a splendid form of fourth-orderelasticity tensor as a six-by-six matrix in MCN notations

We have presented here a minimal code of mathematicadeveloped by [31] However a numerical demonstration ofthis code has also been presented by [31] and the reader(s)interested in understanding this code in full are requested torefer to the website already mentioned above

Let us now proceed to Section 5 which in our viewis the soul of the present work In this section we shall

deal with the eigenvalues eigenvectors the nominal averageof eigenvectors and the average eigenvectors and so forthfor different species of softwoods hardwoods and somespecimen of cancellous bone

5 Analysis of Known Values of ElasticConstants of Woods and CancellousBone Using MAPLE

Of course ldquoMAPLErdquo is a sophisticated CAS and producedunder the results of over 30 years of cutting-edge researchand development which assists us in analyzing exploringvisualizing and manipulating almost every mathematicalproblem Having more than 500 functions this CAS offersbroadness depth and performance to handle every phe-nomenon of mathematics In a nutshell it is a CAS that offershigh performance mathematics capabilities with integratednumeric and symbolic computations

However in the present study we attempt to explorehow this CAS handles complicated analysis regarding elasticconstant data

The elastic constant data of our interest for hardwoodsand softwoods as calculated by Hearmon [47 48] aretabulated in Tables 5 and 6

For cancellous bone we have the elastic constants dataavailable for three specimens [2 11] These data are tabulatedin Table 7

Precisely speaking to meet the requirements of proposedresearch a MAPLE code consisting of almost 81 steps hasbeen developed This MAPLE code (in its minimal form) isappended to Appendix A while its entire working mecha-nism is included in an electronic appendix (Appendix B) andcan be accessed from httpdxdoiorg1011552014487314Particularly as far as our intention concerns analysis ofelastic constants data regarding hardwoods softwoods andcancellous bone using MAPLE is consisting of the followingsteps

(i) Computating the eigenvalues and eigenvectors for allthe 15 hardwood species 8 softwood species and 3specimens of cancellous bone using MAPLE

(ii) Computing the nominal averages of eigenvectors andthe average eigenvectors for all the 15 hardwoodspecies

(iii) Computing the average eigenvalues for hardwoodspecies

(iv) Computing the average elasticity matrices for all the15 hardwood species

(v) Plotting the histograms for elasticity matrices of allthe 15 hardwood species

(vi) Plotting the graphs for I II III IV V and VI eigen-values of 15 hardwood species against their apparentdensities (see Figures 14ndash16 also see Figure 17 for acombined view)


01 02 03 04 05 06 07 08

05

1

15

2

25

3

(a)

01 02 03 04 05 06 07 08

05

1

15

2

25

3

(b)

Figure 16 The graphics placed in left as well as right positions showing graphs of V and VI eigenvalues of all 15 hardwood species againsttheir apparent densities

01 02 03 04 05 06 07 08

2

4

6

8

10

12

14

16

Figure 17 A combined view of Figures 14 15 and 16

51 Computing the Eigenvalues and Eigenvectors for Hard-woods Species Softwood Species and Cancellous Bone UsingMAPLE Here we present the outcomes of MAPLE code(which is mentioned in Appendix A) regarding the computa-tion of eigenvalues and eigenvectors of the stiffness matricesC concerning 15 hardwood species (see Table 5)

It is well known that the eigenvalues and eigenvectors of astiffness matrix C or its compliance matrix S are determinedfrom the equations [3]

(C minus ΛI) N = 0 or (S minus1

ΛI) N = 0 (44)

where I is the 6 times 6 identity matrix and N representsthe normalized eigenvectors of C or S Naturally C or Sbeing positive definite therefore there would be six positiveeigenvalues and these values are known as Kelvinrsquos moduliThese Kelvinrsquos moduli are denoted by Λ

119894 119894 = 1 2 6 and

if possible are ordered in such a way that Λ1ge Λ

2ge sdot sdot sdot ge

Λ6gt 0 Also the eigenvalues of S can simply de computed by

taking the inverse of the eigenvalues of CA MAPLE code mentioned in Step 16 of Appendix A

enables us to simultaneously compute the eigenvalues andeigenvectors for all the 15 hardwood species Also the resultsof this MAPLE code are summarized in Tables 8 and 9Further by simply substituting the elasticity constants fromTable 6 in Step 2 to Step 11 of Appendix A and performingStep 16 again one can have the eigenvalues and eigenvectorsfor the 8 softwood speciesThe computed results for softwoodspecies are summarized in Table 10 Similarly substitution ofelastic constants data for cancellous bone from Table 7 intoStep 2 to Step 11 and repeating Step 16 of the MAPLE codeyields the desired results tabulated in Table 11

52 Computing the Average Elasticity Tensors for Woodsand Cancellous Bone Using MAPLE In order to computeaverage elasticity tensors for the hardwoods softwoods andcancellous bone let us go through a brief mathematicaldescription regarding this issue [3] as this notion helpsus developing an appropriate MAPLE code to have desiredresults

Suppose we have 119872 measurements of the elasticconstants data for some particular species or materialC119868 C119868119868 C119872 and we want to construct a tensor CAVG

representing an average elasticity tensor for that particular


material Then to do so we need the following key algebra[3]

(i) The eigenvalues Λ119884119896 119896 = 1 2 6 119884 = 1 2 119872

and the eigenvectors N119884119896 119896 = 1 2 6 119884 =

1 2 119872 for each of the 119884th measurement of theparticular species or material

(ii) The nominal average (NA) NNA119896

of the eigenvectorsN119884119896associated with the particular species The nomi-

nal average is given by

NNA119896

equiv1

119872

119872

sum

119884=1

N119884119896 (45)

(iii) The average NAVG119896

which is obtained by the followingformula

NAVG119896

equiv JNANNA119896

(46)

where JNA stands for the inverse square root of thepositive definite tensor | sum6

119896=1NNA119896

otimes NNA119896

| that is

JNA equiv (

6

sum

119896=1

NNA119896

otimes NNA119896

)

minus12

(47)

(iv) Now we proceed to average the eigenvalue Forthis we need to transform each set of eigenvaluescorresponding to each eigenvector to the averageeigenvector NAVG

119896 that is

ΛAVG119902

equiv1

119872

119872

sum

119884=1

6

sum

119896=1

Λ119884

119896(N119884

119896sdot NAVG

119902)2

(48)

It is obvious that ΛAVG119896

gt 0 for all 119896 = 1 2 6

and ΛAVG119896

= ΛNA119896 where Λ

NA119896

is the nominal averageof eigenvalues of material and is defined by

ΛNA119896

equiv1

119872

119872

sum

119884=1

Λ119884

119896 (49)

(v) Eventually we can now calculate the average elasticitymatrix CAVG in such a way that

CAVG=

6

sum

119896=1

ΛAVG119896

NAVG119896

otimes NAVG119896

(50)

Taking care of the above outlined steps we have developedsome MAPLE codes (See Step 17 to Step 81 of AppendixA) that enable us to calculate the nominal averages averageeigenvectors average eigenvalues and the average elasticitymatrices for each of the 15 hardwood species tabulated inTable 5 In addition to this Appendix A also retains few stepsfor example Steps 21 26 31 36 41 46 51 56 61 66 73 and80 which are particularly developed to plot histograms ofaverage elasticity matrices of each of the hardwood species aswell as graphs between I II III IV V and VI eigenvalues ofall hardwood species and the apparent densities (see Table 5)

We summarize the above claimed consequences of eachof the hardwood species one by one ss follows

521 MAPLE Evaluations for Quipo

(1) The nominal averages of eigenvectors for the twomeasurements (S nos 1 and 2 of Table 5) of Quipoare

[[[[[[[

[

00367834559700000

00453681054800000

0998185540700000

00

00

00

]]]]]]]

]

[[[[[[[

[

0983745905000000

minus0170879918750000

minus00277901141300000

00

00

00

]]]]]]]

]

[[[[[[[

[

minus0169284222800000

minus0982986098000000

00509905132200000

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(51)

(2) The average eigenvectors for Quipo are

[[[[[[[

[

003679134179

004537783172

09983995367

00

00

00

]]]]]]]

]

[[[[[[[

[

09859858256

minus01712690003

minus002785339027

00

00

00

]]]]]]]

]

[[[[[[[

[

minus01697052873

minus09854311019

005111734310

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(52)

(3) The averages eigenvalue for the twomeasurements (Snos 1 and 2 of Table 5) of Quipo is 3339500000

(4) The average elasticity matrix for Quipo is


[[[

[

334725276837330 0000112116752981933 000198542359441849 00 00 00

0000112116752981933 334773746006714 minus0000991802322788501 00 00 00

000198542359441849 minus0000991802322788501 334013593769726 00 00 00

00 00 00 333950000000000 00 00

00 00 00 00 333950000000000 00

00 00 00 00 00 333950000000000

]]]

]

(53)

(5) The histogram of the average elasticity matrix forQuipo is shown in Figure 2

522 MAPLE Evaluations for White

(1) The nominal averages of eigenvectors for the singlemeasurements (S no 3 of Table 5) of White are

[[[[[[[

[

004028666644

006399313350

09971368323

00

00

00

]]]]]]]

]

[[[[[[[

[

09048796003

minus04255671399

minus0009247686096

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04237568827

minus09026613361

007505076253

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(54)

(2) The average eigenvectors for White are

[[[[[[[

[

004028666644

006399313350

09971368323

00

00

00

]]]]]]]

]

[[[[[[[

[

09048795994

minus04255671395

minus0009247686087

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04237568827

minus09026613361

007505076253

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(55)

(3) Theaverage eigenvalue for the singlemeasurement (Sno 3 of Table 5) of White is 1458799998

(4) The average elasticity matrix for White is given as

[[[

[

1458799998 000000001785571215 minus000000001009489597 00 00 00000000001785571215 1458799997 0000000002407020197 00 00 00minus000000001009489597 0000000002407020197 1458799997 00 00 00

00 00 00 1458799998 00 0000 00 00 00 1458799998 0000 00 00 00 00 1458799998

]]]

]

(56)

(5) The histogram of the average elasticity matrix forWhite is shown in Figure 3

523 MAPLE Evaluations for Khaya

(1) The nominal averages of eigenvectors for the singlemeasurements (S no 4 of Table 5) of Khaya are

[[[[[

[

005368516527

007220768570

09959437502

00

00

00

]]]]]

]

[[[[[

[

09266208315

minus03753089731

minus002273783078

00

00

00

]]]]]

]

[[[[[

[

minus03721447810

minus09240829102

008705767064

00

00

00

]]]]]

]

[[[[[

[

00

00

00

00

00

10

]]]]]

]

[[[[[

[

00

00

00

10

00

00

]]]]]

]

[[[[[

[

00

00

00

00

10

00

]]]]]

]

(57)

(2) The average eigenvectors for Khaya are

[[[[[[[[[

[

005368516527

007220768570

09959437502

00

00

00

]]]]]]]]]

]

[[[[[[[[[

[

09266208315

minus03753089731

minus002273783078

00

00

00

]]]]]]]]]

]

[[[[[[[[[

[

minus03721447806

minus09240829093

008705767055

00

00

00

]]]]]]]]]

]

[[[[[[[[[

[

00

00

00

10

00

00

]]]]]]]]]

]

[[[[[[[[[

[

00

00

00

00

10

00

]]]]]]]]]

]

[[[[[[[[[

[

00

00

00

00

00

10

]]]]]]]]]

]

(58)


(3) Theaverage eigenvalue for the singlemeasurement (Sno 4 of Table 5) of Khaya is 1615299998

(4) The average elasticity matrix for Khaya is given as

[[[

[

1615299998 0000000005508173090 minus0000000008722620038 00 00 00

0000000005508173090 1615299996 minus0000000004345156817 00 00 00

minus0000000008722620038 minus0000000004345156817 1615300000 00 00 00

00 00 00 1615299998 00 00

00 00 00 00 1615299998 00

00 00 00 00 00 1615299998

]]]

]

(59)

(5) The histogram of the average elasticity matrix forKhaya is shown in Figure 4

524 MAPLE Evaluations for Mahogany

(1) The nominal averages of eigenvectors for the twomeasurements (S nos 5 and 6 of Table 5) ofMahogany are

[[[[[[[

[

minus00598757013800000

minus00768229223150000

minus0995231841750000

00

00

00

]]]]]]]

]

[[[[[[[

[

minus0869358919900000

0493939153600000

00141567750950000

00

00

00

]]]]]]]

]

[[[[[[[

[

minus0490490276500000

minus0866078136900000

00963811082600000

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(60)

(2) The average eigenvectors for Mahogany are

[[[[[[[[[[[

[

minus005987730132

minus007682497510

minus09952584354

00

00

00

]]]]]]]]]]]

]

[[[[[[[[[[[

[

minus08693926232

04939583026

001415732393

00

00

00

]]]]]]]]]]]

]

[[[[[[[[[[[

[

00

00

00

10

00

00

]]]]]]]]]]]

]

[[[[[[[[[[[

[

00

00

00

00

10

00

]]]]]]]]]]]

]

[[[[[[[[[[[

[

00

00

00

00

00

10

]]]]]]]]]]]

]

(61)

(3) The average eigenvalue for the two measurements (Snos 5 and 6 of Table 5) of Mahogany is 1819400002

(4) The average elasticity matrix forMahogany is given as

[[[

[

1819451173 minus00001438038661 00001358556898 00 00 00minus00001438038661 1819484018 minus00004764690574 00 00 0000001358556898 minus00004764690574 1819454255 00 00 00

00 00 00 1819400002 1819400002 0000 00 00 00 00 0000 00 00 00 00 1819400002

]]]

]

(62)

(5) The histogram of the average elasticity matrix forMahogany is shown in Figure 5

525 MAPLE Evaluations for S Germ

(1) The nominal averages of eigenvectors for the singlemeasurement (S no 7 of Table 5) of S Gem are

[[[[[[[

[

004967528691

008463937788

09951726186

00

00

00

]]]]]]]

]

[[[[[[[

[

09161715971

minus04006166011

minus001165943224

00

00

00

]]]]]]]

]

[[[[[

[

minus03976958243

minus09123280736

009744493938

00

00

00

]]]]]

]

[[[[[

[

00

00

00

10

00

00

]]]]]

]

[[[[[

[

00

00

00

00

10

00

]]]]]

]

[[[[[

[

00

00

00

00

00

10

]]]]]

]

(63)

(2) The average eigenvectors for S Germ are

[[[[[[[

[

004967528696

008463937796

09951726196

00

00

00

]]]]]]]

]

[[[[[[[

[

09161715980

minus04006166015

minus001165943225

00

00

00

]]]]]]]

]


[[[[[[[

[

minus03976958247

minus09123280745

009744493948

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(64)

(3) Theaverage eigenvalue for the singlemeasurement (Sno 7 of Table 5) of S Germ is 1922399998

(4) The average elasticity matrix for S Germ is givenas

[[[

[

1922399998 minus000000001176508811 minus000000001537920006 00 00 00

minus000000001176508811 1922400000 minus0000000007631928036 00 00 00

minus000000001537920006 minus0000000007631928036 1922400000 00 00 00

00 00 00 1922399998 1922399998 00

00 00 00 00 00 00

00 00 00 00 00 1922399998

]]]

]

(65)

(5) The histogram of the average elasticity matrix for SGerm is shown in Figure 6

526 MAPLE Evaluations for maple

(1) The nominal averages of eigenvectors for the singlemeasurement (S no 8 of Table 5) of maple are

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

[[[[[[[

[

minus01387533797

minus02031928413

minus09692575344

00

00

00

]]]]]]]

]

[[[[[[[

[

minus08476774112

05304148448

001015375725

00

00

00

]]]]]]]

]

[[[[[[[

[

minus05120454135

minus08230265872

02458388325

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

(66)

(2) The average eigenvectors for maple are

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

[[[[[[[

[

minus01387533798

minus02031928415

minus09692575354

00

00

00

]]]]]]]

]

[[[[[[[

[

minus08476774112

05304148448

001015375725

00

00

00

]]]]]]]

]

[[[[[[[

[

minus05120454135

minus08230265872

02458388325

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

(67)

(3) Theaverage eigenvalue for the singlemeasurement (Sno 8 of Table 5) of maple is 2074600000

(4) The average elasticity matrix for maple is given as

[[[

[

2074599999 0000000004356660361 000000003402343994 00 00 00

0000000004356660361 2074600004 000000002232269567 00 00 00

000000003402343994 000000002232269567 2074600000 00 00 00

00 00 00 2074600000 2074600000 00

00 00 00 00 00 00

00 00 00 00 00 2074600000

]]]

]

(68)

(5) The histogram of the average elasticity matrix forMAPLE is shown in Figure 7

527 MAPLE Evaluations for Walnut

(1) The nominal averages of eigenvectors for the singlemeasurement (S no 9 of Table 5) of Walnut are

[[[[[[[

[

008621257414

01243595697

09884847438

00

00

00

]]]]]]]

]

[[[[[[[

[

08755641188

minus04828494241

minus001561753067

00

00

00

]]]]]]]

]

[[[[[

[

minus04753471023

minus08668282006

01505124700

00

00

00

]]]]]

]

[[[[[

[

00

00

00

10

00

00

]]]]]

]

[[[[[

[

00

00

00

00

10

00

]]]]]

]

[[[[[

[

00

00

00

00

00

10

]]]]]

]

(69)

(2) The average eigenvectors for Walnut are

[[[[[

[

008621257423

01243595698

09884847448

00

00

00

]]]]]

]

[[[[[

[

08755641188

minus04828494241

minus001561753067

00

00

00

]]]]]

]


[[[[[[[

[

minus04753471018

minus08668281997

01505124698

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(70)

(3) Theaverage eigenvalue for the singlemeasurement (Sno 9 of Table 5) of walnut is 1890099997

(4) The average elasticity matrix for Walnut is givenas

[[[

[

1890099999 000000001209663993 minus000000002532733996 00 00 00000000001209663993 1890099991 0000000001701090138 00 00 00minus000000002532733996 0000000001701090138 1890100001 00 00 00

00 00 00 1890099997 1890099997 0000 00 00 00 00 0000 00 00 00 00 1890099997

]]]

]

(71)

(5) The histogram of the average elasticity matrix forWalnut is shown in Figure 8

528 MAPLE Evaluations for Birch

(1) The nominal averages of eigenvectors for the singlemeasurement (S no 10 of Table 5) of Birch are

[[[[[[[

[

03961520086

06338768023

06642768888

00

00

00

]]]]]]]

]

[[[[[[[

[

09160614623

minus02236797319

minus03328645006

00

00

00

]]]]]]]

]

[[[[[[[

[

006240981414

minus07403833993

06692812840

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(72)

(2) The average eigenvectors for Birch are

[[[[[[[

[

03961520086

06338768023

06642768888

00

00

00

]]]]]]]

]

[[[[[[[

[

09160614614

minus02236797317

minus03328645003

00

00

00

]]]]]]]

]

[[[[[[[

[

006240981426

minus07403834008

06692812853

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(73)

(3) Theaverage eigenvalue for the singlemeasurement (Sno 10 of Table 5) of Birch is 8772299998

(4) The average elasticity matrix for Birch is given as

[[[

[


00 00 00 8772299998 8772299998 0000 00 00 00 00 0000 00 00 00 00 8772299998

]]]

]

(74)

(5) The histogram of the average elasticity matrix forBirch is shown in Figure 9

529 MAPLE Evaluations for Y Birch

(1) The nominal averages of eigenvectors for the singlemeasurement (S no 11 of Table 5) of Y Birch are

[[[[[[[

[

minus006591789198

minus008975775227

minus09937798435

00

00

00

]]]]]]]

]

[[[[[[[

[

minus08289381135

05593224991

0004466103673

00

00

00

]]]]]]]

]

[[[[[

[

minus05554425577

minus08240763851

01112729817

00

00

00

]]]]]

]

[[[[[

[

00

00

00

10

00

00

]]]]]

]

[[[[[

[

00

00

00

00

10

00

]]]]]

]

[[[[[

[

00

00

00

00

00

10

]]]]]

]

(75)(2) The average eigenvectors for Y Birch are

[[[[[[[

[

minus01387533798

minus02031928415

minus09692575354

00

00

00

]]]]]]]

]

[[[[[[[

[

minus08476774112

05304148448

001015375725

00

00

00

]]]]]]]

]


[[[[

[

minus05120454135

minus08230265872

02458388325

00

00

00

]]]]

]

[[[[

[

00

00

00

10

00

00

]]]]

]

[[[[

[

00

00

00

00

10

00

]]]]

]

[[[[

[

00

00

00

00

00

10

]]]]

]

(76)

(3) Theaverage eigenvalue for the singlemeasurement (Sno 11 of Table 5) of Y Birch is 2228057495

(4) The average elasticity matrix for Y Birch is given as

[[[

[

2228057494 0000000004678921127 000000003654014285 00 00 000000000004678921127 2228057499 000000002397389829 00 00 00000000003654014285 000000002397389829 2228057495 00 00 00

00 00 00 2228057495 2228057495 0000 00 00 00 00 0000 00 00 00 00 2228057495

]]]

]

(77)

(5) The histogram of the average elasticity matrix for YBirch is shown in Figure 10

5210 MAPLE Evaluations for Oak

(1) The nominal averages of eigenvectors for the singlemeasurement (S no 12 of Table 5) of Oak are

[[[[[[[

[

006975010637

01071019008

09917984199

00

00

00

]]]]]]]

]

[[[[[[[

[

09079000793

minus04187725641

minus001862756541

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04133429180

minus09017531389

01264472547

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(78)

(2) The average eigenvectors for Oak are

[[[[[[[

[

006975010637

01071019008

09917984199

00

00

00

]]]]]]]

]

[[[[[[[

[

09079000793

minus04187725641

minus001862756541

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04133429184

minus09017531398

01264472548

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(79)

(3) Theaverage eigenvalue for the singlemeasurement (Sno 12 of Table 5) of Oak is 2598699998

(4) The average elasticity matrix for Oak is given as

[[[

[

2598699997 minus000000001889254939 minus0000000003638179937 00 00 00minus000000001889254939 2598700006 0000000008575709980 00 00 00minus0000000003638179937 0000000008575709980 2598699998 00 00 00

00 00 00 2598699998 2598699998 0000 00 00 00 00 0000 00 00 00 00 2598699998

]]]

]

(80)

(5) Thehistogram of the average elasticity matrix for Oakis shown in Figure 11

5211 MAPLE Evaluations for Ash

(1) The nominal averages of eigenvectors for the twomeasurements (S nos 13 and 14 of Table 5) of Ash are

[[[[[

[

minus00192522847250000

minus00192842313600000

000386151499999998

00

00

00

]]]]]

]

[[[[[

[

000961799224999998

00165029120500000

000436480214750000

00

00

00

]]]]]

]

[[[[[[

[

minus0494444050150000

minus0856906622200000

0142104790200000

00

00

00

]]]]]]

]

[[[[[[

[

00

00

00

10

00

00

]]]]]]

]

[[[[[[

[

00

00

00

00

10

00

]]]]]]

]

[[[[[[

[

00

00

00

00

00

10

]]]]]]

]

(81)


(2) The average eigenvectors for Ash are

[[[[[[[[[[

[

minus2541745843

minus2545963537

5098090872

00

00

00

]]]]]]]]]]

]

[[[[[[[[[[

[

2505315863

4298714977

1136953303

00

00

00

]]]]]]]]]]

]

[[[[[[[

[

minus04949599718

minus08578007511

01422530677

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(82)

(3) The average eigenvalue for the two measurements (Snos 13 and 14 of Table 5) of Ash is 2477950014

(4) The average elasticity matrix for Ash is given as

[[[

[

315679169478799 427324383657779 384557503470365 00 00 00427324383657779 618700481877479 889153535276175 00 00 00384557503470365 889153535276175 384768762708609 00 00 00

00 00 00 247795001400000 00 0000 00 00 00 247795001400000 0000 00 00 00 00 247795001400000

]]]

]

(83)

(5) The histogram of the average elasticity matrix for Ashis shown in Figure 12

5212 MAPLE Evaluations for Beech

(1) The nominal averages of eigenvectors for the singlemeasurements (S no 15 of Table 5) of Beech are

[[[[[[[

[

01135176976

01764907831

09777344911

00

00

00

]]]]]]]

]

[[[[[[[

[

08848520771

minus04654963442

minus001870707734

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04518302069

minus08672739791

02090103088

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(84)

(2) The average eigenvectors for Beech are

[[[[[[[

[

01135176977

01764907833

09777344921

00

00

00

]]]]]]]

]

[[[[[[[

[

08848520771

minus04654963442

minus001870707734

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04518302069

minus08672739791

02090103088

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(85)

(3) The average eigenvalue for the single measurements(S no 15 of Table 5) of Beech is 2663699998

(4) The average elasticity matrix for Beech is given as

[[[

[

2663700003 000000004768023095 000000003169803038 00 00 00000000004768023095 2663699992 0000000008310743498 00 00 00000000003169803038 0000000008310743498 2663700001 00 00 00

00 00 00 2663699998 2663699998 0000 00 00 00 00 0000 00 00 00 00 2663699998

]]]

]

(86)

(5) The histogram of the average elasticity matrix forBeech is shown in Figure 13

53 MAPLE Plotted Graphics for I II III IV V and VIEigenvalues of 15 Hardwood Species against Their ApparentDensities This subsection is dedicated to the expositionof the extreme quality of MAPLE which can enable one

to simultaneously sketch up graphics for I II III IV Vand VI eigenvalues of each of the 15 hardwood species(See Tables 8 and 9) against the corresponding apparentdensities 120588 (See Table 5 for apparent density) The MAPLEcode regarding this issue has been mentioned in Step 81 ofAppendix A


6 Conclusions

In this paper we have tried to develop a CAS environmentthat suits best to numerically analyze the theory of anisotropicHookersquos law for different species ofwood and cancellous boneParticularly the two fabulous CAS namely Mathematicaand MAPLE have been used in relation to our proposedresearch work More precisely a Mathematica package ldquoTen-sor2Analysismrdquo has been employed to rectify the issueregarding unfolding the tensorial form of anisotropicHookersquoslaw whereas a MAPLE code consisting of almost 81 steps hasbeen developed to dig out the following numerical analysis

(i) Computation of eigenvalues and eigenvectors for allthe 15 hardwood species 8 softwood species and 3specimens of cancellous bone using MAPLE

(ii) Computation of the nominal averages of eigenvectorsand the average eigenvectors for all the 15 hardwoodspecies

(iii) Computation of the average eigenvalues for all 15hardwood species

(iv) Computation of the average elasticity matrices for allthe 15 hardwood species


(vi) Plotting the graphics for I II III IV V and VIeigenvalues of 15 hardwood species against theirapparent densities

It is also claimed that the developedMAPLE code cannotonly simultaneously tackle the 6 times 6 matrices relevant to thedifferent wood species and cancellous bone specimen butalso bears the capacity of handling 100 times 100 matrix whoseentries vary with respect to some parameter Thus from theviewpoint of author the MAPLE code developed by him isof great interest in those fields where higher order matrixalgebra is required

Disclosure

This is to bring in the kind knowledge of Editor(s) andReader(s) that due to the excessive length of present researchwe have ceased some computations concerning cancellousbone and softwood species These remaining computationswill soon be presented in the next series of this paper

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The author expresses his hearty thanks and gratefulnessto all those scientists whose masterpieces have been con-sulted during the preparation of the present research articleSpecial thanks are due to the developer(s) of Mathematicaand MAPLE who have developed such versatile Computer

Algebraic Systems and provided 24-hour online support forall the scientific community

References

[1] S C Cowin and S B Doty Tissue Mechanics Springer NewYork NY USA 2007

[2] G Yang J Kabel B Van Rietbergen A Odgaard R Huiskesand S C Cowin ldquoAnisotropic Hookersquos law for cancellous boneand woodrdquo Journal of Elasticity vol 53 no 2 pp 125ndash146 1998

[3] S C Cowin andGYang ldquoAveraging anisotropic elastic constantdatardquo Journal of Elasticity vol 46 no 2 pp 151ndash180 1997

[4] Y J Yoon and S C Cowin ldquoEstimation of the effectivetransversely isotropic elastic constants of a material fromknown values of the materialrsquos orthotropic elastic constantsrdquoBiomechan Model Mechanobiol vol 1 pp 83ndash93 2002

[5] C Wang W Zhang and G S Kassab ldquoThe validation ofa generalized Hookersquos law for coronary arteriesrdquo AmericanJournal of Physiology Heart andCirculatory Physiology vol 294no 1 pp H66ndashH73 2008

[6] J Kabel B Van Rietbergen A Odgaard and R Huiskes ldquoCon-stitutive relationships of fabric density and elastic properties incancellous bone architecturerdquo Bone vol 25 no 4 pp 481ndash4861999

[7] C Dinckal ldquoAnalysis of elastic anisotropy of wood materialfor engineering applicationsrdquo Journal of Innovative Research inEngineering and Science vol 2 no 2 pp 67ndash80 2011

[8] Y P Arramon M M Mehrabadi D W Martin and SC Cowin ldquoA multidimensional anisotropic strength criterionbased on Kelvin modesrdquo International Journal of Solids andStructures vol 37 no 21 pp 2915ndash2935 2000

[9] P J Basser and S Pajevic ldquoSpectral decomposition of a 4th-order covariance tensor applications to diffusion tensor MRIrdquoSignal Processing vol 87 no 2 pp 220ndash236 2007

[10] P J Basser and S Pajevic ldquoA normal distribution for tensor-valued random variables applications to diffusion tensor MRIrdquoIEEE Transactions on Medical Imaging vol 22 no 7 pp 785ndash794 2003

[11] B Van Rietbergen A Odgaard J Kabel and RHuiskes ldquoDirectmechanics assessment of elastic symmetries and properties oftrabecular bone architecturerdquo Journal of Biomechanics vol 29no 12 pp 1653ndash1657 1996

[12] B Van Rietbergen A Odgaard J Kabel and R HuiskesldquoRelationships between bone morphology and bone elasticproperties can be accurately quantified using high-resolutioncomputer reconstructionsrdquo Journal of Orthopaedic Researchvol 16 no 1 pp 23ndash28 1998

[13] H Follet F Peyrin E Vidal-Salle A Bonnassie C Rumelhartand P J Meunier ldquoIntrinsicmechanical properties of trabecularcalcaneus determined by finite-element models using 3D syn-chrotron microtomographyrdquo Journal of Biomechanics vol 40no 10 pp 2174ndash2183 2007

[14] D R Carte and W C Hayes ldquoThe compressive behavior ofbone as a two-phase porous structurerdquo Journal of Bone and JointSurgery A vol 59 no 7 pp 954ndash962 1977

[15] F Linde I Hvid and B Pongsoipetch ldquoEnergy absorptiveproperties of human trabecular bone specimens during axialcompressionrdquo Journal of Orthopaedic Research vol 7 no 3 pp432ndash439 1989

[16] J C Rice S C Cowin and J A Bowman ldquoOn the dependenceof the elasticity and strength of cancellous bone on apparentdensityrdquo Journal of Biomechanics vol 21 no 2 pp 155ndash168 1988


[17] R W Goulet S A Goldstein M J Ciarelli J L Kuhn MB Brown and L A Feldkamp ldquoThe relationship between thestructural and orthogonal compressive properties of trabecularbonerdquo Journal of Biomechanics vol 27 no 4 pp 375ndash389 1994

[18] R Hodgskinson and J D Currey ldquoEffect of variation instructure on the Youngrsquos modulus of cancellous bone A com-parison of human and non-human materialrdquo Proceedings of theInstitution ofMechanical EngineersH vol 204 no 2 pp 115ndash1211990

[19] R Hodgskinson and J D Currey ldquoEffects of structural variationon Youngrsquos modulus of non-human cancellous bonerdquo Proceed-ings of the Institution of Mechanical Engineers H vol 204 no 1pp 43ndash52 1990

[20] B D Snyder E J Cheal J A Hipp and W C HayesldquoAnisotropic structure-property relations for trabecular bonerdquoTransactions of the Orthopedic Research Society vol 35 p 2651989

[21] C H Turner S C Cowin J Young Rho R B Ashman andJ C Rice ldquoThe fabric dependence of the orthotropic elasticconstants of cancellous bonerdquo Journal of Biomechanics vol 23no 6 pp 549ndash561 1990

[22] D H Laidlaw and J Weickert Visualization and Processingof Tensor Fields Advances and Perspectives Springer BerlinGermany 2009

[23] J T Browaeys and S Chevrot ldquoDecomposition of the elastictensor and geophysical applicationsrdquo Geophysical Journal Inter-national vol 159 no 2 pp 667ndash678 2004

[24] A E H Love A Treatise on the Mathematical Theory ofElasticity Dover New York NY USA 4th edition 1944

[25] G Backus ldquoA geometric picture of anisotropic elastic tensorsrdquoReviews of Geophysics and Space Physics vol 8 no 3 pp 633ndash671 1970

[26] T G Kolda and B W Bader ldquoTensor decompositions andapplicationsrdquo SIAM Review vol 51 no 3 pp 455ndash500 2009

[27] B W Bader and T G Kolda ldquoAlgorithm 862 MATLAB tensorclasses for fast algorithm prototypingrdquo ACM Transactions onMathematical Software vol 32 no 4 pp 635ndash653 2006

[28] M D Daniel T G Kolda and W P Kegelmeyer ldquoMultilinearalgebra for analyzing data with multiple linkagesrdquo SANDIAReport Sandia National Laboratories Albuquerque NM USA2006

[29] W B Brett and T G Kolda ldquoEffcient MATLAB computationswith sparse and factored tensorsrdquo SANDIA Report SandiaNational Laboratories Albuquerque NM USA 2006

[30] R H Brian L L Ronald and M R Jonathan A Guideto MATLAB for Beginners and Experienced Users CambridgeUniversity Press Cambridge UK 2nd edition 2006

[31] A Constantinescu and A Korsunsky Elasticity with Mathe-matica An Introduction to Continuum Mechanics and LinearElasticity Cambridge University Press Cambridge UK 2007

[32] WVoigt LehrbuchDer Kristallphysik JohnsonReprint LeipzigGermany

[33] J Dellinger D Vasicek and C Sondergeld ldquoKelvin notationfor stabilizing elastic-constant inversionrdquo Revue de lrsquoInstitutFrancais du Petrole vol 53 no 5 pp 709ndash719 1998

[34] B A AuldAcoustic Fields andWaves in Solids vol 1 JohnWileyamp Sons 1973

[35] S C Cowin and M M Mehrabadi ldquoOn the identification ofmaterial symmetry for anisotropic elastic materialsrdquo QuarterlyJournal of Mechanics and Applied Mathematics vol 40 pp 451ndash476 1987

[36] S C Cowin BoneMechanics CRC Press Boca Raton Fla USA1989

[37] S C Cowin and M M Mehrabadi ldquoAnisotropic symmetries oflinear elasticityrdquo Applied Mechanics Reviews vol 48 no 5 pp247ndash285 1995

[38] M M Mehrabadi S C Cowin and J Jaric ldquoSix-dimensionalorthogonal tensorrepresentation of the rotation about an axis inthree dimensionsrdquo International Journal of Solids and Structuresvol 32 no 3-4 pp 439ndash449 1995

[39] M M Mehrabadi and S C Cowin ldquoEigentensors of linearanisotropic elastic materialsrdquo Quarterly Journal of Mechanicsand Applied Mathematics vol 43 no 1 pp 15ndash41 1990

[40] W K Thomson ldquoElements of a mathematical theory of elastic-ityrdquo Philosophical Transactions of the Royal Society of Londonvol 166 pp 481ndash498 1856

[41] YW Frank Physics withMapleThe Computer Algebra Resourcefor Mathematical Methods in Physics Wiley-VCH 2005

[42] R HeikkiMathematica Navigator Mathematics Statistics andGraphics Academic Press 2009

[43] T Viktor ldquoTensor manipulation in GPL maximardquo httpgrten-sorphyqueensucaindexhtml

[44] httpgrtensorphyqueensucaindexhtml[45] J M Lee D Lear J Roth J Coskey Nave and L Ricci A

Mathematica Package For Doing Tensor Calculations in Differ-ential Geometry User Manual Department of MathematicsUniversity of Washington Seattle Wash USA Version 132

[46] httpwwwwolframcom[47] R F S Hearmon ldquoThe elastic constants of anisotropic materi-

alsrdquo Reviews ofModern Physics vol 18 no 3 pp 409ndash440 1946[48] R F S Hearmon ldquoThe elasticity of wood and plywoodrdquo Forest

Products Research Special Report 9 Department of Scientificand Industrial Research HMSO London UK 1948

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



Table 1 Voigtrsquos mappings for stress (left) and strain tensors (right) respectively

1205901112059022

12059033

12059023

1205901312059012

12059023

1205901212059013

120590111205902212059033120590231205901312059012

1205981112059822

120598331205982312059813

12059812

12059812

12059823

12059813

12059811

1205982212059833212059823212059813212059812

Table 2 Index conversion rule of Voigt

119894 119895 11 22 33 23 or 32 13 or 31 12 or 21119901 1 2 3 4 5 6

Table 3 Voigt to Kelvin and vice versa

120585 = 1 1 1 radic2 radic2 radic2

element models using 3D synchrotron microtomography[13] have explored the subject material up to some greatextent

Now as far as the present proposed research concernedwe turn our concentration towards cancellous bone andwood (soft and hard) According to [4] the type of materialsymmetry exposed by bone tissue is often ambiguous Soin order to attain accuracy we have to measure the elasticconstants of bone specimen which may be represented asorthotropic However the orthotropy of the bone specimenmay or may not be of higher degree in all the directionswith respect to a fixed point in the specimen Thereby for allpractical purposes the bone specimen can be represented astransversely isotropic or even isotropic

Now the question arises iswhat is the need of studying themechanical properties of cancellous bone and woodThe rig-orous answer is that the knowledge about mechanical prop-erties of cancellous bone is essential for the determinationof bone fracture risk in osteoporosis and other pathologicalconditions involving impaired bone strength [6] Also themechanical properties of cancellous bone are determined bythe properties of its bone tissue and its architecture [4 14ndash21] Further the wood is a cellulosic semicrystalline cellu-lar material and the mechanical properties (eg elasticitystrength and rheology etc) are generally higher along thebole of a tree than across the bole [7]

Mechanically clear wood obeys the law of elasticorthotropic material like the bone does In general the woodysubstance also exposes the high toughness and stiffnessproperties and these properties vary according to the typeof wood and the direction in which the woody substance isexamined as the woody substance shows a higher degree ofanisotropy

In addition to deal with anisotropic Hookersquos law oneshould be familiar with the algebra of the 4th-order tensorWe would like to emphasize that the 4th-order tensor algebrais not only involved in the study of material symmetrybut its crucial appearance has been set up in the field ofdiffusion tensor MRI [9 10] visualization and processing of

tensor fields [22] biomechanics [13] tissue mechanics [1]geophysics [23] and so forth

Though an adequate amount of research work has beendedicated towards the algebra of 4th-order tensors thenalso in the following section we shall present a brief digestregarding this issue as this issue hits the present study up tosome great level

2 The Algebra of Fourth-Order Tensor

The splendid word ldquoTensorrdquo has been derived from the Latinword ldquoTensusrdquo which is the past participle of ldquotendererdquo andstands for ldquoStretchrdquo This word was used in anatomy inthe early 1704 to represent muscle and stretches Howeverin Mathematics it existed in 1846 when William RowanHamilton has explored his quaternion Algebra Even thoughthe Hamilton sense regarding tensor did not survive Therecent meaning of tensor is due to Voigt who used the termldquoTensortriplerdquo in crystal elasticity around 1899

Likewise the second-order tensor a fourth-order tensorA (say) is a linear map from LinV to LinV That is it assignsto each second-order tensor A the second-order tensor AUthat is

A Lin (V) 997888rarr Lin (V) U 997891997888rarr AU forallU isin Lin (V)

(1)

where V is a vector space and Lin(V) is a linear space ofdimension 119899

2 and is equivalent to

Lin (V) ≑ 119860 V 997888rarr V 119906 997891997888rarr 119860119906 forall119906 isin V (2)

Also any 4th-order tensor A has the representation

A = 119860119894119895119896119897

119890119894otimes 119890

119895otimes 119890

119896otimes 119890

119897 (3)

where

119860119894119895119896119897

= ⟨119890119894otimes 119890

119895A (119890

119896otimes 119890

119896)⟩ (4)

and the set of all four vectors that is the set of all linear mapsfrom Lin(V) to Lin(V) is a linear space of dimension 119899

4 andis denoted by Lin(V)

The outer (or open) product of the two 4th-order tensorsis given by the composition

(AB)U = A (BU) forallU isin Lin (V) (5)

In abstract index notations we have

(AB)119894119895119896119897

= 119860119894119895119901119902

119861119901119901119902119896119897

(6)




SymIndex[2 3]

SymIndex[2 6]

In[3]=SymIndex[4] =



Algorithm 1










Algorithm 2




⟨(119860) B⟩ ≑ tr (A119879B) = 119860119894119895119896119897

119861119894119895119896119897

(8)

The induced norm is

A = [tr (A119879A)]12

(9)

and the metric is

119889119864(AB) = A minus B (10)


119894otimes 119890

119895otimes 119890

119896otimes 119890

1198971le119894119895119896119897le119899





119894119895119896119897= 119860



= minus119860119896119897119894119895


A =1

2(A + A

119879) +

1

2(A minus A

119879) (11)





119894119894=12119899

2 isin R and 119880119894119894=12119899


A =

1198992

sum

119894=1

120582119894119880119894otimes 119880

119894 (13)




tr (A) =

1198992

sum

119894=1

120582119894 det (A) =

1198992

prod

119894=1

120582119894 (14)



4 3 1 -gt 5 1 2 -gt 6 3 2 -gt 4 1 3 -gt

5 2 1 -gt 6



Algorithm 3




In[12]=C2


In[14]=C2to4




Array[6 amp 2]] 2]





Which[





] i 6 j 6]


Which[





] i 6 j 6]







Algorithm 4




= 119860119895119894119896119897

= 119860119894119895119897119896







(17)


11 12 1312 22 2313 23 33

11 12 1312 22 2313 23 33

11 12 1312 22 2313 23 33

11 12 1312 22 2313 23 33

11 12 1312 22 2313 23 33

11 12 1312 22 2313 23 33

11 12 1312 22 2313 23 33

11 12 1312 22 2313 23 33

11 12 1312 22 2313 23 33

11 12 1312 22 2313 23 33

11 12 1312 22 2313 23 33

11 12 13

12 22 23

13 23 33

120590ij

Cijkl

120598ij

lowast=


2313= 119862



1

2

3

12

34

56

Column Row

12

34

56


2468101214

12

34

56

Column Row

12

34

56


246810121416

12

34

56

Column Row

12

34

56



= 119860120590(119894)120590(119895)120590(119895)120590(119897)




(18)



119860119904

119894119895119896119897=

1

3(119860

119894119895119896119897+ 119860

119894119896119895119897+ 119860

119894119897119896119895)

119860119886

119894119895119896119897=

1

3(119860

119894119895119896119897minus 119860

119894119896119895119897minus 119860

119894119897119895119896)

(19)




Stress

12059011

1205901

1

12059022

1205902

2

12059033

1205903

3

12059023

1205904

4

12059013

1205905

5

12059012

1205905

6

Strain

12059811

1205981

1205981

12059822

1205982

1205982

12059833

1205983

1205983

12059823

1205984

1205984

12059813

12059815

1205985

12059812

1205986

1205986

Elasticity

1198621111

11986211

11986211

1198622222

11986222

11986222

1198623333

11986233

11986233

1198621122

11986212

11986212

1198621133

11986213

11986213

1198622233

11986223

11986223

1198622323

11986244

1

211986244

1198621313

11986255

1

211986255

1198621212

11986266

1

211986266

1198621323

11986254

1

211986254

1198621312

11986256

1

211986256

1198621223

11986264

1

411986264

1198622311

11986241

1

radic211986241

1198621311

11986251

1

radic211986251

1198621211

11986261

1

radic211986261

1198622322

11986242

1

radic211986242

1198621322

11986252

1

radic211986252

1198621222

11986262

1

radic211986262

1198622333

11986243

1

radic211986243

1198621333

11986253

1

radic211986253

1198621233

11986263

1

radic211986263

Table 4 Continued


Compliance

1198701111

11987811

11987811

1198702222

11987822

11987822

1198703333

11987833

11987833

1198701122

11987812

11987812

1198701133

11987813

11987813

1198702233

11987823

11987823

1198702323

1

411987844

1

211987844

1198701313

1

411987855

1

211987855

1198701212

1

411987866

1

211987866

1198701323

1

411987854

1

211987854

1198701312

1

411987856

1

211987856

1198701223

1

411987864

1

211987864

1198702311

1

211987841

1

radic211987841

1198701311

1

211987851

1

radic211987851

1198701211

1

211987861

1

radic211987861

1198702322

1

211987842

1

radic211987842

1198701322

1

211987852

1

radic211987852

1198701222

1

211987862

1

radic211987862

1198702333

1

211987843

1

radic211987843

1198701333

1

211987853

1

radic211987853

1198701233

1

211987863

1

radic211987863

24681012141618

12

34

56

Column Row

12

34

56




S no Species 120588 11986211

11986222

11986233

11986212

11986213

11986223

11986244

11986255

11986266



S no Species 120588 11986211

11986222

11986233

11986212

11986213

11986223

11986244

11986255

11986266














986 7912 698311986212

2147 3609 281611986213

2773 3186 284511986214

minus0162 11 8011986215

minus293 54 minus0711986216

minus0821 minus06 11411986222

935 8529 896711986223

2346 3281 322711986224

minus1204 09 minus162311986225


0445 minus42 16011986233

5952 14730 916011986234


minus1791 169 minus7211986236

054 06 2411986244

561 3587 348911986245

0798 05 10511986246

minus2159 51 minus9811986255

6032 3448 242611986256



24681012141618

12

34

56

Column Row

12

34

56



5

10

15

20

12

34

56

Column Row

12

34

56



119890120595(119894119895)

= 119890119894otimes 119890

119895 (20)




1 Quipo

[[[[[[[[[

[

1076866026

004067345638

02534605191

02260000000

01180000000

007800000000

]]]]]]]]]

]

[[[[[[[[[

[

003276733390 09918780499 minus01228992897 00 00 00

003131140851 minus01239237163 minus09917976143 00 00 00

09989724208 minus002865041271 003511774899 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

2 Quipo

[[[[[[[[[

[

3461963876

01399776173

04300584948

04300000000

02800000000

01440000000

]]]]]]]]]

]

[[[[[[[[[

[

004079957804 09756137601 minus02156691559 00 00 00

005942480245 minus02178361212 minus09741745817 00 00 00

09973986606 minus002692981555 006686327745 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

3 White

[[[[[[[[[

[

1009116301

03556701722

1333166815

1442000000

1344000000

002200000000

]]]]]]]]]

]

[[[[[[[[[

[

004028666644 09048796003 minus04237568827 00 00 00

006399313350 minus04255671399 minus09026613361 00 00 00

09971368323 minus0009247686096 007505076253 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

4 Khaya

[[[[[[[[[

[

1080102614

04606834511

1475290392

1800000000

1196000000

04200000000

]]]]]]]]]

]

[[[[[[[[[

[

005368516527 09266208315 minus03721447810 00 00 00

007220768570 minus03753089731 minus09240829102 00 00 00

09959437502 minus002273783078 008705767064 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

5 Mahogany

[[[[[[[[[

[

1210253321

06098853915

1810581412

1960000000

1498000000

06380000000

]]]]]]]]]

]

[[[[[[[[[

[


minus007817061387 04985803653 minus08633116316 00 00 00

minus09948285648 001731853005 01000809391 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

6 Mahogany

[[[[[[[[[

[

1310854783

03895295651

1814922632

1218000000

09380000000

03000000000

]]]]]]]]]

]

[[[[[[[[[

[


minus007547523076 04892979419 minus08688446422 00 00 00

minus09956351187 001099502014 009268127742 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

7 S Germ

[[[[[[[[[

[

1234053410

05212570563

1922208822

2318000000

1582000000

05400000000

]]]]]]]]]

]

[[[[[[[[[

[

004967528691 09161715971 minus03976958243 00 00 00

008463937788 minus04006166011 minus09123280736 00 00 00

09951726186 minus001165943224 009744493938 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

8 Maple

[[[[[[[[[

[

1205449768

06868279555

2766674361

2460000000

2194000000

05840000000

]]]]]]]]]

]

[[[[[[[[[

[


minus02031928413 05304148448 minus08230265872 00 00 00

minus09692575344 001015375725 02458388325 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]




9 Walnut

[[[[[[[[[

[

1267869546

05204134969

1919891015

1922000000

1400000000

04600000000

]]]]]]]]]

]

[[[[[[[[[

[

008621257414 08755641188 minus04753471023 00 00 00

01243595697 minus04828494241 minus08668282006 00 00 00

09884847438 minus001561753067 01505124700 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

10 Birch

[[[[[[[[[

[

3168908680

04747157903

05946755301

2346000000

1816000000

03720000000

]]]]]]]]]

]

[[[[[[[[[

[

03961520086 09160614623 006240981414 00 00 00

06338768023 minus02236797319 minus07403833993 00 00 00

06642768888 minus03328645006 06692812840 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

11 Y Birch

[[[[[[[[[

[

1545414040

05549651499

2059894445

2120000000

1942000000

04800000000

]]]]]]]]]

]

[[[[[[[[[

[


minus008975775227 05593224991 minus08240763851 00 00 00

minus09937798435 0004466103673 01112729817 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

12 Oak

[[[[[[[[[

[

1718666431

08648974218

3239438239

2380000000

1532000000

07840000000

]]]]]]]]]

]

[[[[[[[[[

[

006975010637 09079000793 minus04133429180 00 00 00

01071019008 minus04187725641 minus09017531389 00 00 00

09917984199 minus001862756541 01264472547 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

13 Ash

[[[[[[[[[

[

1715567967

06696042476

2409716064

2684000000

1784000000

05400000000

]]]]]]]]]

]

[[[[[[[[[

[

006190313535 08746549514 minus04807772027 00 00 00

009781749768 minus04846986500 minus08691944279 00 00 00

09932772717 minus0006777439445 01155609239 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

14 Ash

[[[[[[[[[

[

1742363268

07844815431

2669885744

1720000000

1218000000

05000000000

]]]]]]]]]

]

[[[[[[[[[

[


minus01363859604 05177044741 minus08446188165 00 00 00

minus09855542417 001550704374 01686486565 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

15 Beech

[[[[[[[[[

[

1599002755

09558575345

3451114905

3216000000

2112000000

09120000000

]]]]]]]]]

]

[[[[[[[[[

[

01135176976 08848520771 minus04518302069 00 00 00

01764907831 minus04654963442 minus08672739791 00 00 00

09777344911 minus001870707734 02090103088 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

Thus 119890120572otimes 119890

1205731le120572120573le119899




119860119894119895119896119897

119890119894otimes 119890

119895otimes 119890

119896otimes 119890

119897 or as an 119899


120572120573119890120572otimes 119890

120573 where 119860

120595(119894119895)120595119896119897= 119860

119894119895119896119897

1 le 119894 119895 119896 119897 le 119899


tr (A119879B) = ⟨AB⟩ = tr (119860119879119861119879) A =1003817100381710038171003817100381711986010038171003817100381710038171003817 (21)






1 Blasa

[[[[[[[[[

[

6385324208

009845837744

03832174064

06240000000

04060000000

006600000000

]]]]]]]]]

]

[[[[[[[[[

[

001488818113 09510211778 minus03087669978 00 00 00

002575997614 minus03090635474 minus09506924583 00 00 00

09995572851 minus0006200252135 002909967665 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

2 Spruce

[[[[[[[[[

[

1198583568

04516442734

1114520065

1498000000

1442000000

007800000000

]]]]]]]]]

]

[[[[[[[[[

[


minus004689865645 04044467539 minus09133582749 00 00 00

minus09983543167 001123114838 005623628701 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

3 Spruce

[[[[[[[[[

[

1410927768

04185382987

1227183961

1442000000

10

006400000000

]]]]]]]]]

]

[[[[[[[[[

[


minus005361649666 04423303564 minus08952480817 00 00 00

minus09978936411 0009052295016 006423657043 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

4 Spruce

[[[[[[[[[

[

1630529953

03544288592

08442715844

1234000000

1520000000

007200000000

]]]]]]]]]

]

[[[[[[[[[

[


minus002869707087 04091564350 minus09120128765 00 00 00

minus09993764538 0007032901380 003460119282 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

5 Spruce

[[[[[[[[[

[

1725845208

05092652753

1171282625

1250000000

1706000000

007000000000

]]]]]]]]]

]

[[[[[[[[[

[


minus003428302033 05858146064 minus08097196604 00 00 00

minus09988364173 0007413313465 004765347716 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

6 Douglas Fir

[[[[[[[[[

[

1613459769

06233733771

1439028943

1767000000

1766000000

01760000000

]]]]]]]]]

]

[[[[[[[[[

[


minus003697194691 05922678885 minus08048924305 00 00 00

minus09986533619 0007435777607 005134366998 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

7 Douglas Fir

[[[[[[[[[

[

1710150156

06986859431

2204812526

2348000000

1816000000

01600000000

]]]]]]]]]

]

[[[[[[[[[

[


minus006364825251 05704773676 minus08188433771 00 00 00

minus09967231590 0004703484281 008075160717 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

8 Pine

[[[[[[[[[

[

1699539133

04940019684

1565606760

3484000000

1344000000

01320000000

]]]]]]]]]

]

[[[[[[[[[

[


minus005585205149 04413516031 minus08955943895 00 00 00

minus09978476167 0006195562437 006528207193 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]


Table11Th

eeigenvalues

andeigenvectorsfor3

specim

enso

fcancello

usbo

necalculated

usingMAPL

EΛ[119894]sta

ndforthe

eigenvalueso

f119894th

specim

enof

cancellous

boneand

119881[119894]sta

ndfor

thec

orrespon

ding

eigenvectors

Cancellous

bone

EigenvaluesΛ

[119894]

Eigenvectors

119881[119894]

Specim

en1

[ [ [ [ [ [ [ [

1334869880

minus02071167473

8644210768

6837676373

3768806080

4123724855

] ] ] ] ] ] ] ]

[ [ [ [ [ [ [ [

03316115964

08831238953

02356441950

02242668491

005354786914

003787816350

06475092156

minus01099136965

01556396475

minus06940063479

minus01278953723

02154647417

04800906963

minus02832250652

007648827487

02494278793

06410471445

minus04585742261

minus02737532692

minus006354940263

04518342532

minus006214920576

05836952694

06101669758

minus03706582070

03314035780

minus01276606259

minus06337030234

03639432801

minus04499474345

01671829224

01180163925

minus08330343502

002008124471

03109325968

04087706408

] ] ] ] ] ] ] ]

Specim

en2

[ [ [ [ [ [ [ [

1754634171

7773848354

1550688791

2301790452

3445174379

3589156342

] ] ] ] ] ] ] ]

[ [ [ [ [ [ [ [

minus03057290719

minus03005337771

minus09030341401

001584500766

minus002175914011

minus0003741933916

minus04311831740

minus08021570593

04130146809

00001362645283

minus0003854432451

0005396441669

minus08488209390

05154657137

01152135966

minus0004741957786

002229739182

minus0002068264586

00009402467441

minus0005358584867

0003987518000

minus003989381331

003796849613

minus09984595017

minus001058157817

002100470582

002092543658

0006230946613

minus09987520802

minus003826770492

00009823402498

0006977574167

001484146906

09990475909

0008196718814

minus003958286493

] ] ] ] ] ] ] ]

Specim

en3

[ [ [ [ [ [ [ [

1418542670

5838388363

9959058231

3433818493

3024968125

1757492470

] ] ] ] ] ] ] ]

[ [ [ [ [ [ [ [

03244960223

minus0004831643783

minus08451713783

04062092722

01164524794

minus004239310198

06461152835

minus07125334236

02668138654

004131783689

minus002431040234

003665187401

06621315621

07009969173

02633438279

002836289512

minus0001711614840

0005269111592

minus01962401199

0009159859564

03740839141

08850746598

01904577552

004284654369

minus0008597323304

minus0002526763452

001897677365

01738799329

minus06977939665

minus06945566457

001533190599

minus002803230310

006961922985

minus01374446216

06801869735

minus07159517722

] ] ] ] ] ] ] ]


24681012141618

12

34

56

Column Row

12

34

56


12345678

12

34

56

Column Row

12

34

56




120590119894119895= 119862


where 119862119894119895119896119897


120590 = 120590119894119895119890119894otimes 119890

119895 120598 = 120598

119896119897119890119896otimes 119890

119897

C = 119862119894119895119896119897

119890119894otimes 119890

119896otimes 119890

119896otimes 119890

119897

(23)

5

10

15

20

12

34

56

Column Row

12

34

56


5

10

15

20

25

12

34

56

Column Row

12

34

56



E1equiv 119890

1otimes 119890

1

E2equiv 119890

2otimes 119890

2

E3equiv 119890

3otimes 119890

3

(24)


E4equiv

1


2+ 119890

2otimes 119890

1)

E5equiv

1


3+ 119890

3otimes 119890

2)

E6equiv

1


1+ 119890

1otimes 119890

3)

(25)



E7equiv

1


2minus 119890

2otimes 119890

1)

E8equiv

1


3minus 119890

3otimes 119890

2)

E9equiv

1


1minus 119890

1otimes 119890

3)

(26)


120590 = 119878119860E

119860

120598 = 119864119860E

119860

C = 119862119860119861

119890119860otimes 119890

119861

(27)




120590119894119895= 119862

119894119895119896119897120598119896119897

lArrrArr 119878119860= 119862

119860119861E119861

119862119894119895119896119897

120598119896119897

= 120582120598119894119895lArrrArr 119862

119860119861E119861= 120582119864

119860

(28)


((((((((

(

1198781

1198782

1198783

1198784

1198785

1198786

1198787

1198788

1198789

))))))))

)

=

((((((((

(

12059011

12059022

12059033

120590(12)

120590(23)

120590(31)

120590[12]

120590[23]

120590[31]

))))))))

)

((((((((

(

1198641

1198642

1198643

1198644

1198645

1198646

1198647

1198648

1198649

))))))))

)

=

((((((((

(

12059811

12059822

12059833

120598(12)

120598(23)

120598(31)

120598[12]

120598[23]

120598[31]

))))))))

)

(29)


120579(119894119895)

equiv1

radic2(120579119894119895+ 120579

119895119894) 120579

[119894119895]equiv

1

radic2(120579119894119895minus 120579

119895119894) (30)


tensor C are

((((((((

(

11986211

11986212

11986213

11986214

11986215

11986216

11986217

11986218

11986219

11986221

11986222

11986223

11986224

11986225

11986226

11986227

11986228

11986229

11986231

11986232

11986233

11986234

11986235

11986236

11986237

11986238

11986239

11986241

11986242

11986243

11986244

11986245

11986246

11986247

11986248

11986249

11986251

11986252

11986253

11986254

11986255

11986256

11986257

11986258

11986259

11986261

11986262

11986263

11986264

11986265

11986266

11986267

11986268

11986269

11986271

11986272

11986273

11986274

11986275

11986276

11986277

11986278

11986279

11986281

11986282

11986283

11986284

11986285

11986286

11986287

11986288

11986289

11986291

11986292

11986293

11986294

11986295

11986296

11986297

11986298

11986299

))))))))

)

=

1198621111

1198621122

1198621133

1198622211

1198622222

1198623333

1198623311

1198623322

1198623333

11986211(12)

11986211(23)

11986211(31)

11986222(12)

11986222(23)

11986222(31)

11986233(12)

11986233(23)

11986233(31)

11986211[12]

11986211[23]

11986211[31]

11986222[12]

11986222[23]

11986222[31]

11986233[12]

11986233[23]

11986233[31]

119862(12)11

119862(12)22

119862(12)33

119862(23)11

119862(23)22

119862(23)33

119862(31)11

119862(31)22

119862(31)33

119862(1212)

119862(1223)

119862(1231)

119862(2312)

119862(2323)

119862(2331)

119862(3112)

119862(3123)

119862(3131)

119862(12)[12]

119862(12)[23]

119862(12)[31]

119862(23)[12]

119862(23)[23]

119862(23)[31]

119862(31)[12]

119862(31)[23]

119862(31)[31]

119862[12]11

119862[12]22

119862[12]33

119862[23]11

119862[23]22

119862[123]33

119862[21]11

119862[31]22

119862[31]33

119862[12](12)

119862[12](23)

119862[12](21)

119862[23](12)

119862[23](23)

119862[23](21)

119862[31](12)

119862[31](23)

119862[31](21)

119862[1212]

119862[1223]

119862[1231]

119862[2312]

119862[2323]

119862[2331]

119862[3112]

119862[3123]

119862[3131]

(31)

where

119862119894119895(119896119897)

equiv1

radic2(119862

119894119895119896119897+ 119862

119894119895119897119896) 119862

119894119895[119896119897]equiv

1

radic2(119862

119894119895119896119897minus 119862

119894119895119897119896)

119862(119894119895)119896119897

equiv1

radic2(119862

119894119895119896119897+ 119862

119895119894119896119897) 119862

[119894119895]119896119897equiv

1

radic2(119862

119894119895119896119897minus 119862

119895119894119896119897)

119862(119894119895119896119897)

equiv1

2(119862

119894119895119896119897+ 119862

119894119895119897119896+ 119862

119895119894119896119897+ 119862

119895119894119897119896)

119862(119894119895)[119896119897]

equiv1

2(119862

119894119895119896119897minus 119862

119894119895119897119896+ 119862

119895119894119896119897minus 119862

119895119894119897119896)

119862[119894119895](119896119897)

equiv1

2(119862

119894119895119896119897+ 119862

119894119895119897119896minus 119862

119895119894119896119897minus 119862

119895119894119897119896)

119862[119894119895119896119897]

equiv1

2(119862

119894119895119896119897minus 119862

119895119894119896119897minus 119862

119894119895119897119896+ 119862

119895119894119897119896)

(32)


12

34

56

ColumnRow

12

34

56

600005000040000300002000010000


5

10

15

20

25

12

34

56

Column Row

12

34

56






119895119894and 120598

119896119897= 120598

119897119896 the 119862

119894119895119896119897will be

like 119862119894119895119896119897

= 119862119895119894119896119897

= 119862119894119895119897119896


119862119894119895119896119897

= 119862119896119897119894119895












2 sdot 119864energy = 120590119894119895120598119894119895= 120590

119901120598119901 (33)


119901 = 119894120575119894119895+ (1 minus 120575

119894119895) (9 minus 119894 minus 119895)

119902 = 119896120575119896119897+ (1 minus 120575

119896119897) (9 minus 119896 minus 119897)

(34)


119901 119862

119894119895119896119897= 119862

119901119902 and 120598

119902= (2 minus

120575119896119897)120598119896119897



(

(

1205901

1205902

1205903

1205904

1205905

1205906

)

)

= (

(

11986211

11986212

11986213

11986214

11986215

11986216

11986221

11986222

11986223

11986224

11986225

11986226

11986231

11986232

11986233

11986234

11986235

11986236

11986241

11986242

11986243

11986244

11986245

11986246

11986251

11986252

11986253

11986254

11986255

11986256

11986261

11986262

11986263

11986264

11986265

11986266

)

)

(

(

1205981

1205982

1205983

1205984

1205985

1205986

)

)

(35)



(1) the 120590119894119895and 120598


(2) the norms of 120590119894119895 120598119896119897 and 119862




01 02 03 04 05 06 07 08

2

4

6

8

10

12

14

16

(a)

01

02

03

04

05

06

07

08

09

01 02 03 04 05 06 07 08

(b)


01 02 03 04 05 06 07 08

05

1

15

2

25

3

(a)

01 02 03 04 05 06 07 08

05

1

15

2

25

3

(b)




119901 = 119894120575119894119895+ (1 minus 120575

119894119895) (9 minus 119894 minus 119895)

119902 = 119896120575119896119897+ (1 minus 120575

119896119897) (9 minus 119896 minus 119897)

(36)

In this way

120590119901= (120575

119894119895+ radic2 (1 minus 120575

119894119895)) 120598

119894119895

120598119902= (120575

119896119897+ radic2 (1 minus 120575

119896119897)) 120598

119896119897

119862119901119902

= (120575119894119895+ radic2 (1 minus 120575

119894119895)) (120575

119896119897+ radic2 (1 minus 120575

119896119897)) 119862

119894119895119896119897

(37)




120590kelvin

= 120590viogt

120585 120590voigt

=120590kelvin

120585

120598kelvin

= 120598voigt

120585 120598voigt

=120598kelvin

120585

(38)


119862kelvin

=((

(







))

)

119862voigt

(39)


(

(

12059011

12059022

12059033

radic212059023

radic212059013

radic212059012

)

)

= (

1198621111 1198621122 1198621133radic21198621123

radic21198621113radic21198621112

1198622211 1198622222 1198622333radic21198622223

radic21198622213radic21198622212

1198623311 1198623322 1198623333radic21198623323

radic21198623313radic21198623312

radic21198622311radic21198622322

radic21198622333 21198622323 21198622313 21198622312

radic21198621311radic21198621322

radic21198621333 21198621323 21198621313 21198621312

radic21198621211radic21198621222

radic21198621233 21198621223 21198621213 21198621212

)

lowast(

(

12059811

12059822

12059833

radic212059823

radic212059813

radic212059812

)

)

(40)


= C (41)



(

(

12059011

12059022

12059033

radic212059023

radic212059013

radic212059012

)

)

=((

(

11986211

11986212

11986213

radic211986214

radic211986215

radic211986216

11986212

11986222

11986223

radic211986224

radic211986225

radic211986226

11986213

11986223

11986233

radic211986234

radic211986235

radic211986236

radic211986214

radic211986224

radic211986234

211986244

211986245

211986246

radic211986215

radic211986225

radic211986235

211986245

211986255

211986256

radic211986216

radic211986226

radic211986236

211986246

211986256

211986266

))

)

lowast (

(

12059811

12059822

12059833

radic212059823

radic212059813

radic212059812

)

)

(42)


(

(

1

2

3

4

5

6

)

)

=((

(

11986211

11986212

11986213

11986214

11986215

11986216

11986212

11986222

11986223

11986224

11986225

11986226

11986213

11986223

11986233

11986234

11986235

11986236

11986214

11986224

11986234

11986244

11986245

11986246

11986215

11986225

11986235

11986245

11986255

11986256

11986216

11986226

11986236

11986246

11986256

11986266

))

)

lowast (

(

1205981

1205982

1205983

1205984

1205985

1205986

)

)

(43)



































01 02 03 04 05 06 07 08

05

1

15

2

25

3

(a)

01 02 03 04 05 06 07 08

05

1

15

2

25

3

(b)


01 02 03 04 05 06 07 08

2

4

6

8

10

12

14

16





ΛI) N = 0 (44)


119894 119894 = 1 2 6 and

















NNA119896

equiv1

119872

119872

sum

119884=1

N119884119896 (45)



NAVG119896

equiv JNANNA119896

(46)


119896=1NNA119896

otimes NNA119896

| that is

JNA equiv (

6

sum

119896=1

NNA119896

otimes NNA119896

)

minus12

(47)


119896 that is

ΛAVG119902

equiv1

119872

119872

sum

119884=1

6

sum

119896=1

Λ119884

119896(N119884

119896sdot NAVG

119902)2

(48)


gt 0 for all 119896 = 1 2 6

and ΛAVG119896


NA119896


ΛNA119896

equiv1

119872

119872

sum

119884=1

Λ119884

119896 (49)


CAVG=

6

sum

119896=1

ΛAVG119896

NAVG119896

otimes NAVG119896

(50)





[[[[[[[

[

00367834559700000

00453681054800000

0998185540700000

00

00

00

]]]]]]]

]

[[[[[[[

[

0983745905000000

minus0170879918750000

minus00277901141300000

00

00

00

]]]]]]]

]

[[[[[[[

[

minus0169284222800000

minus0982986098000000

00509905132200000

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(51)


[[[[[[[

[

003679134179

004537783172

09983995367

00

00

00

]]]]]]]

]

[[[[[[[

[

09859858256

minus01712690003

minus002785339027

00

00

00

]]]]]]]

]

[[[[[[[

[

minus01697052873

minus09854311019

005111734310

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(52)




[[[

[

334725276837330 0000112116752981933 000198542359441849 00 00 00

0000112116752981933 334773746006714 minus0000991802322788501 00 00 00

000198542359441849 minus0000991802322788501 334013593769726 00 00 00

00 00 00 333950000000000 00 00

00 00 00 00 333950000000000 00

00 00 00 00 00 333950000000000

]]]

]

(53)




[[[[[[[

[

004028666644

006399313350

09971368323

00

00

00

]]]]]]]

]

[[[[[[[

[

09048796003

minus04255671399

minus0009247686096

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04237568827

minus09026613361

007505076253

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(54)


[[[[[[[

[

004028666644

006399313350

09971368323

00

00

00

]]]]]]]

]

[[[[[[[

[

09048795994

minus04255671395

minus0009247686087

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04237568827

minus09026613361

007505076253

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(55)



[[[

[

1458799998 000000001785571215 minus000000001009489597 00 00 00000000001785571215 1458799997 0000000002407020197 00 00 00minus000000001009489597 0000000002407020197 1458799997 00 00 00

00 00 00 1458799998 00 0000 00 00 00 1458799998 0000 00 00 00 00 1458799998

]]]

]

(56)




[[[[[

[

005368516527

007220768570

09959437502

00

00

00

]]]]]

]

[[[[[

[

09266208315

minus03753089731

minus002273783078

00

00

00

]]]]]

]

[[[[[

[

minus03721447810

minus09240829102

008705767064

00

00

00

]]]]]

]

[[[[[

[

00

00

00

00

00

10

]]]]]

]

[[[[[

[

00

00

00

10

00

00

]]]]]

]

[[[[[

[

00

00

00

00

10

00

]]]]]

]

(57)


[[[[[[[[[

[

005368516527

007220768570

09959437502

00

00

00

]]]]]]]]]

]

[[[[[[[[[

[

09266208315

minus03753089731

minus002273783078

00

00

00

]]]]]]]]]

]

[[[[[[[[[

[

minus03721447806

minus09240829093

008705767055

00

00

00

]]]]]]]]]

]

[[[[[[[[[

[

00

00

00

10

00

00

]]]]]]]]]

]

[[[[[[[[[

[

00

00

00

00

10

00

]]]]]]]]]

]

[[[[[[[[[

[

00

00

00

00

00

10

]]]]]]]]]

]

(58)




[[[

[

1615299998 0000000005508173090 minus0000000008722620038 00 00 00

0000000005508173090 1615299996 minus0000000004345156817 00 00 00

minus0000000008722620038 minus0000000004345156817 1615300000 00 00 00

00 00 00 1615299998 00 00

00 00 00 00 1615299998 00

00 00 00 00 00 1615299998

]]]

]

(59)




[[[[[[[

[

minus00598757013800000

minus00768229223150000

minus0995231841750000

00

00

00

]]]]]]]

]

[[[[[[[

[

minus0869358919900000

0493939153600000

00141567750950000

00

00

00

]]]]]]]

]

[[[[[[[

[

minus0490490276500000

minus0866078136900000

00963811082600000

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(60)


[[[[[[[[[[[

[

minus005987730132

minus007682497510

minus09952584354

00

00

00

]]]]]]]]]]]

]

[[[[[[[[[[[

[

minus08693926232

04939583026

001415732393

00

00

00

]]]]]]]]]]]

]

[[[[[[[[[[[

[

00

00

00

10

00

00

]]]]]]]]]]]

]

[[[[[[[[[[[

[

00

00

00

00

10

00

]]]]]]]]]]]

]

[[[[[[[[[[[

[

00

00

00

00

00

10

]]]]]]]]]]]

]

(61)



[[[

[


00 00 00 1819400002 1819400002 0000 00 00 00 00 0000 00 00 00 00 1819400002

]]]

]

(62)




[[[[[[[

[

004967528691

008463937788

09951726186

00

00

00

]]]]]]]

]

[[[[[[[

[

09161715971

minus04006166011

minus001165943224

00

00

00

]]]]]]]

]

[[[[[

[

minus03976958243

minus09123280736

009744493938

00

00

00

]]]]]

]

[[[[[

[

00

00

00

10

00

00

]]]]]

]

[[[[[

[

00

00

00

00

10

00

]]]]]

]

[[[[[

[

00

00

00

00

00

10

]]]]]

]

(63)


[[[[[[[

[

004967528696

008463937796

09951726196

00

00

00

]]]]]]]

]

[[[[[[[

[

09161715980

minus04006166015

minus001165943225

00

00

00

]]]]]]]

]


[[[[[[[

[

minus03976958247

minus09123280745

009744493948

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(64)



[[[

[

1922399998 minus000000001176508811 minus000000001537920006 00 00 00

minus000000001176508811 1922400000 minus0000000007631928036 00 00 00

minus000000001537920006 minus0000000007631928036 1922400000 00 00 00

00 00 00 1922399998 1922399998 00

00 00 00 00 00 00

00 00 00 00 00 1922399998

]]]

]

(65)




[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

[[[[[[[

[

minus01387533797

minus02031928413

minus09692575344

00

00

00

]]]]]]]

]

[[[[[[[

[

minus08476774112

05304148448

001015375725

00

00

00

]]]]]]]

]

[[[[[[[

[

minus05120454135

minus08230265872

02458388325

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

(66)


[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

[[[[[[[

[

minus01387533798

minus02031928415

minus09692575354

00

00

00

]]]]]]]

]

[[[[[[[

[

minus08476774112

05304148448

001015375725

00

00

00

]]]]]]]

]

[[[[[[[

[

minus05120454135

minus08230265872

02458388325

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

(67)



[[[

[

2074599999 0000000004356660361 000000003402343994 00 00 00

0000000004356660361 2074600004 000000002232269567 00 00 00

000000003402343994 000000002232269567 2074600000 00 00 00

00 00 00 2074600000 2074600000 00

00 00 00 00 00 00

00 00 00 00 00 2074600000

]]]

]

(68)




[[[[[[[

[

008621257414

01243595697

09884847438

00

00

00

]]]]]]]

]

[[[[[[[

[

08755641188

minus04828494241

minus001561753067

00

00

00

]]]]]]]

]

[[[[[

[

minus04753471023

minus08668282006

01505124700

00

00

00

]]]]]

]

[[[[[

[

00

00

00

10

00

00

]]]]]

]

[[[[[

[

00

00

00

00

10

00

]]]]]

]

[[[[[

[

00

00

00

00

00

10

]]]]]

]

(69)


[[[[[

[

008621257423

01243595698

09884847448

00

00

00

]]]]]

]

[[[[[

[

08755641188

minus04828494241

minus001561753067

00

00

00

]]]]]

]


[[[[[[[

[

minus04753471018

minus08668281997

01505124698

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(70)



[[[

[

1890099999 000000001209663993 minus000000002532733996 00 00 00000000001209663993 1890099991 0000000001701090138 00 00 00minus000000002532733996 0000000001701090138 1890100001 00 00 00

00 00 00 1890099997 1890099997 0000 00 00 00 00 0000 00 00 00 00 1890099997

]]]

]

(71)




[[[[[[[

[

03961520086

06338768023

06642768888

00

00

00

]]]]]]]

]

[[[[[[[

[

09160614623

minus02236797319

minus03328645006

00

00

00

]]]]]]]

]

[[[[[[[

[

006240981414

minus07403833993

06692812840

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(72)


[[[[[[[

[

03961520086

06338768023

06642768888

00

00

00

]]]]]]]

]

[[[[[[[

[

09160614614

minus02236797317

minus03328645003

00

00

00

]]]]]]]

]

[[[[[[[

[

006240981426

minus07403834008

06692812853

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(73)



[[[

[


00 00 00 8772299998 8772299998 0000 00 00 00 00 0000 00 00 00 00 8772299998

]]]

]

(74)




[[[[[[[

[

minus006591789198

minus008975775227

minus09937798435

00

00

00

]]]]]]]

]

[[[[[[[

[

minus08289381135

05593224991

0004466103673

00

00

00

]]]]]]]

]

[[[[[

[

minus05554425577

minus08240763851

01112729817

00

00

00

]]]]]

]

[[[[[

[

00

00

00

10

00

00

]]]]]

]

[[[[[

[

00

00

00

00

10

00

]]]]]

]

[[[[[

[

00

00

00

00

00

10

]]]]]

]


[[[[[[[

[

minus01387533798

minus02031928415

minus09692575354

00

00

00

]]]]]]]

]

[[[[[[[

[

minus08476774112

05304148448

001015375725

00

00

00

]]]]]]]

]


[[[[

[

minus05120454135

minus08230265872

02458388325

00

00

00

]]]]

]

[[[[

[

00

00

00

10

00

00

]]]]

]

[[[[

[

00

00

00

00

10

00

]]]]

]

[[[[

[

00

00

00

00

00

10

]]]]

]

(76)



[[[

[

2228057494 0000000004678921127 000000003654014285 00 00 000000000004678921127 2228057499 000000002397389829 00 00 00000000003654014285 000000002397389829 2228057495 00 00 00

00 00 00 2228057495 2228057495 0000 00 00 00 00 0000 00 00 00 00 2228057495

]]]

]

(77)




[[[[[[[

[

006975010637

01071019008

09917984199

00

00

00

]]]]]]]

]

[[[[[[[

[

09079000793

minus04187725641

minus001862756541

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04133429180

minus09017531389

01264472547

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(78)


[[[[[[[

[

006975010637

01071019008

09917984199

00

00

00

]]]]]]]

]

[[[[[[[

[

09079000793

minus04187725641

minus001862756541

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04133429184

minus09017531398

01264472548

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(79)



[[[

[


00 00 00 2598699998 2598699998 0000 00 00 00 00 0000 00 00 00 00 2598699998

]]]

]

(80)




[[[[[

[

minus00192522847250000

minus00192842313600000

000386151499999998

00

00

00

]]]]]

]

[[[[[

[

000961799224999998

00165029120500000

000436480214750000

00

00

00

]]]]]

]

[[[[[[

[

minus0494444050150000

minus0856906622200000

0142104790200000

00

00

00

]]]]]]

]

[[[[[[

[

00

00

00

10

00

00

]]]]]]

]

[[[[[[

[

00

00

00

00

10

00

]]]]]]

]

[[[[[[

[

00

00

00

00

00

10

]]]]]]

]

(81)



[[[[[[[[[[

[

minus2541745843

minus2545963537

5098090872

00

00

00

]]]]]]]]]]

]

[[[[[[[[[[

[

2505315863

4298714977

1136953303

00

00

00

]]]]]]]]]]

]

[[[[[[[

[

minus04949599718

minus08578007511

01422530677

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(82)



[[[

[

315679169478799 427324383657779 384557503470365 00 00 00427324383657779 618700481877479 889153535276175 00 00 00384557503470365 889153535276175 384768762708609 00 00 00

00 00 00 247795001400000 00 0000 00 00 00 247795001400000 0000 00 00 00 00 247795001400000

]]]

]

(83)




[[[[[[[

[

01135176976

01764907831

09777344911

00

00

00

]]]]]]]

]

[[[[[[[

[

08848520771

minus04654963442

minus001870707734

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04518302069

minus08672739791

02090103088

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(84)


[[[[[[[

[

01135176977

01764907833

09777344921

00

00

00

]]]]]]]

]

[[[[[[[

[

08848520771

minus04654963442

minus001870707734

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04518302069

minus08672739791

02090103088

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(85)



[[[

[

2663700003 000000004768023095 000000003169803038 00 00 00000000004768023095 2663699992 0000000008310743498 00 00 00000000003169803038 0000000008310743498 2663700001 00 00 00

00 00 00 2663699998 2663699998 0000 00 00 00 00 0000 00 00 00 00 2663699998

]]]

]

(86)





6 Conclusions









Disclosure




Acknowledgments



References




















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












SymIndex[2 3]

SymIndex[2 6]

In[3]=SymIndex[4] =



Algorithm 1










Algorithm 2




⟨(119860) B⟩ ≑ tr (A119879B) = 119860119894119895119896119897

119861119894119895119896119897

(8)

The induced norm is

A = [tr (A119879A)]12

(9)

and the metric is

119889119864(AB) = A minus B (10)


119894otimes 119890

119895otimes 119890

119896otimes 119890

1198971le119894119895119896119897le119899





119894119895119896119897= 119860



= minus119860119896119897119894119895


A =1

2(A + A

119879) +

1

2(A minus A

119879) (11)





119894119894=12119899

2 isin R and 119880119894119894=12119899


A =

1198992

sum

119894=1

120582119894119880119894otimes 119880

119894 (13)




tr (A) =

1198992

sum

119894=1

120582119894 det (A) =

1198992

prod

119894=1

120582119894 (14)



4 3 1 -gt 5 1 2 -gt 6 3 2 -gt 4 1 3 -gt

5 2 1 -gt 6



Algorithm 3




In[12]=C2


In[14]=C2to4




Array[6 amp 2]] 2]





Which[





] i 6 j 6]


Which[





] i 6 j 6]







Algorithm 4




= 119860119895119894119896119897

= 119860119894119895119897119896







(17)


11 12 1312 22 2313 23 33

11 12 1312 22 2313 23 33

11 12 1312 22 2313 23 33

11 12 1312 22 2313 23 33

11 12 1312 22 2313 23 33

11 12 1312 22 2313 23 33

11 12 1312 22 2313 23 33

11 12 1312 22 2313 23 33

11 12 1312 22 2313 23 33

11 12 1312 22 2313 23 33

11 12 1312 22 2313 23 33

11 12 13

12 22 23

13 23 33

120590ij

Cijkl

120598ij

lowast=


2313= 119862



1

2

3

12

34

56

Column Row

12

34

56


2468101214

12

34

56

Column Row

12

34

56


246810121416

12

34

56

Column Row

12

34

56



= 119860120590(119894)120590(119895)120590(119895)120590(119897)




(18)



119860119904

119894119895119896119897=

1

3(119860

119894119895119896119897+ 119860

119894119896119895119897+ 119860

119894119897119896119895)

119860119886

119894119895119896119897=

1

3(119860

119894119895119896119897minus 119860

119894119896119895119897minus 119860

119894119897119895119896)

(19)




Stress

12059011

1205901

1

12059022

1205902

2

12059033

1205903

3

12059023

1205904

4

12059013

1205905

5

12059012

1205905

6

Strain

12059811

1205981

1205981

12059822

1205982

1205982

12059833

1205983

1205983

12059823

1205984

1205984

12059813

12059815

1205985

12059812

1205986

1205986

Elasticity

1198621111

11986211

11986211

1198622222

11986222

11986222

1198623333

11986233

11986233

1198621122

11986212

11986212

1198621133

11986213

11986213

1198622233

11986223

11986223

1198622323

11986244

1

211986244

1198621313

11986255

1

211986255

1198621212

11986266

1

211986266

1198621323

11986254

1

211986254

1198621312

11986256

1

211986256

1198621223

11986264

1

411986264

1198622311

11986241

1

radic211986241

1198621311

11986251

1

radic211986251

1198621211

11986261

1

radic211986261

1198622322

11986242

1

radic211986242

1198621322

11986252

1

radic211986252

1198621222

11986262

1

radic211986262

1198622333

11986243

1

radic211986243

1198621333

11986253

1

radic211986253

1198621233

11986263

1

radic211986263

Table 4 Continued


Compliance

1198701111

11987811

11987811

1198702222

11987822

11987822

1198703333

11987833

11987833

1198701122

11987812

11987812

1198701133

11987813

11987813

1198702233

11987823

11987823

1198702323

1

411987844

1

211987844

1198701313

1

411987855

1

211987855

1198701212

1

411987866

1

211987866

1198701323

1

411987854

1

211987854

1198701312

1

411987856

1

211987856

1198701223

1

411987864

1

211987864

1198702311

1

211987841

1

radic211987841

1198701311

1

211987851

1

radic211987851

1198701211

1

211987861

1

radic211987861

1198702322

1

211987842

1

radic211987842

1198701322

1

211987852

1

radic211987852

1198701222

1

211987862

1

radic211987862

1198702333

1

211987843

1

radic211987843

1198701333

1

211987853

1

radic211987853

1198701233

1

211987863

1

radic211987863

24681012141618

12

34

56

Column Row

12

34

56




S no Species 120588 11986211

11986222

11986233

11986212

11986213

11986223

11986244

11986255

11986266



S no Species 120588 11986211

11986222

11986233

11986212

11986213

11986223

11986244

11986255

11986266














986 7912 698311986212

2147 3609 281611986213

2773 3186 284511986214

minus0162 11 8011986215

minus293 54 minus0711986216

minus0821 minus06 11411986222

935 8529 896711986223

2346 3281 322711986224

minus1204 09 minus162311986225


0445 minus42 16011986233

5952 14730 916011986234


minus1791 169 minus7211986236

054 06 2411986244

561 3587 348911986245

0798 05 10511986246

minus2159 51 minus9811986255

6032 3448 242611986256



24681012141618

12

34

56

Column Row

12

34

56



5

10

15

20

12

34

56

Column Row

12

34

56



119890120595(119894119895)

= 119890119894otimes 119890

119895 (20)




1 Quipo

[[[[[[[[[

[

1076866026

004067345638

02534605191

02260000000

01180000000

007800000000

]]]]]]]]]

]

[[[[[[[[[

[

003276733390 09918780499 minus01228992897 00 00 00

003131140851 minus01239237163 minus09917976143 00 00 00

09989724208 minus002865041271 003511774899 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

2 Quipo

[[[[[[[[[

[

3461963876

01399776173

04300584948

04300000000

02800000000

01440000000

]]]]]]]]]

]

[[[[[[[[[

[

004079957804 09756137601 minus02156691559 00 00 00

005942480245 minus02178361212 minus09741745817 00 00 00

09973986606 minus002692981555 006686327745 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

3 White

[[[[[[[[[

[

1009116301

03556701722

1333166815

1442000000

1344000000

002200000000

]]]]]]]]]

]

[[[[[[[[[

[

004028666644 09048796003 minus04237568827 00 00 00

006399313350 minus04255671399 minus09026613361 00 00 00

09971368323 minus0009247686096 007505076253 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

4 Khaya

[[[[[[[[[

[

1080102614

04606834511

1475290392

1800000000

1196000000

04200000000

]]]]]]]]]

]

[[[[[[[[[

[

005368516527 09266208315 minus03721447810 00 00 00

007220768570 minus03753089731 minus09240829102 00 00 00

09959437502 minus002273783078 008705767064 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

5 Mahogany

[[[[[[[[[

[

1210253321

06098853915

1810581412

1960000000

1498000000

06380000000

]]]]]]]]]

]

[[[[[[[[[

[


minus007817061387 04985803653 minus08633116316 00 00 00

minus09948285648 001731853005 01000809391 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

6 Mahogany

[[[[[[[[[

[

1310854783

03895295651

1814922632

1218000000

09380000000

03000000000

]]]]]]]]]

]

[[[[[[[[[

[


minus007547523076 04892979419 minus08688446422 00 00 00

minus09956351187 001099502014 009268127742 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

7 S Germ

[[[[[[[[[

[

1234053410

05212570563

1922208822

2318000000

1582000000

05400000000

]]]]]]]]]

]

[[[[[[[[[

[

004967528691 09161715971 minus03976958243 00 00 00

008463937788 minus04006166011 minus09123280736 00 00 00

09951726186 minus001165943224 009744493938 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

8 Maple

[[[[[[[[[

[

1205449768

06868279555

2766674361

2460000000

2194000000

05840000000

]]]]]]]]]

]

[[[[[[[[[

[


minus02031928413 05304148448 minus08230265872 00 00 00

minus09692575344 001015375725 02458388325 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]




9 Walnut

[[[[[[[[[

[

1267869546

05204134969

1919891015

1922000000

1400000000

04600000000

]]]]]]]]]

]

[[[[[[[[[

[

008621257414 08755641188 minus04753471023 00 00 00

01243595697 minus04828494241 minus08668282006 00 00 00

09884847438 minus001561753067 01505124700 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

10 Birch

[[[[[[[[[

[

3168908680

04747157903

05946755301

2346000000

1816000000

03720000000

]]]]]]]]]

]

[[[[[[[[[

[

03961520086 09160614623 006240981414 00 00 00

06338768023 minus02236797319 minus07403833993 00 00 00

06642768888 minus03328645006 06692812840 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

11 Y Birch

[[[[[[[[[

[

1545414040

05549651499

2059894445

2120000000

1942000000

04800000000

]]]]]]]]]

]

[[[[[[[[[

[


minus008975775227 05593224991 minus08240763851 00 00 00

minus09937798435 0004466103673 01112729817 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

12 Oak

[[[[[[[[[

[

1718666431

08648974218

3239438239

2380000000

1532000000

07840000000

]]]]]]]]]

]

[[[[[[[[[

[

006975010637 09079000793 minus04133429180 00 00 00

01071019008 minus04187725641 minus09017531389 00 00 00

09917984199 minus001862756541 01264472547 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

13 Ash

[[[[[[[[[

[

1715567967

06696042476

2409716064

2684000000

1784000000

05400000000

]]]]]]]]]

]

[[[[[[[[[

[

006190313535 08746549514 minus04807772027 00 00 00

009781749768 minus04846986500 minus08691944279 00 00 00

09932772717 minus0006777439445 01155609239 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

14 Ash

[[[[[[[[[

[

1742363268

07844815431

2669885744

1720000000

1218000000

05000000000

]]]]]]]]]

]

[[[[[[[[[

[


minus01363859604 05177044741 minus08446188165 00 00 00

minus09855542417 001550704374 01686486565 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

15 Beech

[[[[[[[[[

[

1599002755

09558575345

3451114905

3216000000

2112000000

09120000000

]]]]]]]]]

]

[[[[[[[[[

[

01135176976 08848520771 minus04518302069 00 00 00

01764907831 minus04654963442 minus08672739791 00 00 00

09777344911 minus001870707734 02090103088 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

Thus 119890120572otimes 119890

1205731le120572120573le119899




119860119894119895119896119897

119890119894otimes 119890

119895otimes 119890

119896otimes 119890

119897 or as an 119899


120572120573119890120572otimes 119890

120573 where 119860

120595(119894119895)120595119896119897= 119860

119894119895119896119897

1 le 119894 119895 119896 119897 le 119899


tr (A119879B) = ⟨AB⟩ = tr (119860119879119861119879) A =1003817100381710038171003817100381711986010038171003817100381710038171003817 (21)






1 Blasa

[[[[[[[[[

[

6385324208

009845837744

03832174064

06240000000

04060000000

006600000000

]]]]]]]]]

]

[[[[[[[[[

[

001488818113 09510211778 minus03087669978 00 00 00

002575997614 minus03090635474 minus09506924583 00 00 00

09995572851 minus0006200252135 002909967665 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

2 Spruce

[[[[[[[[[

[

1198583568

04516442734

1114520065

1498000000

1442000000

007800000000

]]]]]]]]]

]

[[[[[[[[[

[


minus004689865645 04044467539 minus09133582749 00 00 00

minus09983543167 001123114838 005623628701 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

3 Spruce

[[[[[[[[[

[

1410927768

04185382987

1227183961

1442000000

10

006400000000

]]]]]]]]]

]

[[[[[[[[[

[


minus005361649666 04423303564 minus08952480817 00 00 00

minus09978936411 0009052295016 006423657043 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

4 Spruce

[[[[[[[[[

[

1630529953

03544288592

08442715844

1234000000

1520000000

007200000000

]]]]]]]]]

]

[[[[[[[[[

[


minus002869707087 04091564350 minus09120128765 00 00 00

minus09993764538 0007032901380 003460119282 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

5 Spruce

[[[[[[[[[

[

1725845208

05092652753

1171282625

1250000000

1706000000

007000000000

]]]]]]]]]

]

[[[[[[[[[

[


minus003428302033 05858146064 minus08097196604 00 00 00

minus09988364173 0007413313465 004765347716 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

6 Douglas Fir

[[[[[[[[[

[

1613459769

06233733771

1439028943

1767000000

1766000000

01760000000

]]]]]]]]]

]

[[[[[[[[[

[


minus003697194691 05922678885 minus08048924305 00 00 00

minus09986533619 0007435777607 005134366998 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

7 Douglas Fir

[[[[[[[[[

[

1710150156

06986859431

2204812526

2348000000

1816000000

01600000000

]]]]]]]]]

]

[[[[[[[[[

[


minus006364825251 05704773676 minus08188433771 00 00 00

minus09967231590 0004703484281 008075160717 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

8 Pine

[[[[[[[[[

[

1699539133

04940019684

1565606760

3484000000

1344000000

01320000000

]]]]]]]]]

]

[[[[[[[[[

[


minus005585205149 04413516031 minus08955943895 00 00 00

minus09978476167 0006195562437 006528207193 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]


Table11Th

eeigenvalues

andeigenvectorsfor3

specim

enso

fcancello

usbo

necalculated

usingMAPL

EΛ[119894]sta

ndforthe

eigenvalueso

f119894th

specim

enof

cancellous

boneand

119881[119894]sta

ndfor

thec

orrespon

ding

eigenvectors

Cancellous

bone

EigenvaluesΛ

[119894]

Eigenvectors

119881[119894]

Specim

en1

[ [ [ [ [ [ [ [

1334869880

minus02071167473

8644210768

6837676373

3768806080

4123724855

] ] ] ] ] ] ] ]

[ [ [ [ [ [ [ [

03316115964

08831238953

02356441950

02242668491

005354786914

003787816350

06475092156

minus01099136965

01556396475

minus06940063479

minus01278953723

02154647417

04800906963

minus02832250652

007648827487

02494278793

06410471445

minus04585742261

minus02737532692

minus006354940263

04518342532

minus006214920576

05836952694

06101669758

minus03706582070

03314035780

minus01276606259

minus06337030234

03639432801

minus04499474345

01671829224

01180163925

minus08330343502

002008124471

03109325968

04087706408

] ] ] ] ] ] ] ]

Specim

en2

[ [ [ [ [ [ [ [

1754634171

7773848354

1550688791

2301790452

3445174379

3589156342

] ] ] ] ] ] ] ]

[ [ [ [ [ [ [ [

minus03057290719

minus03005337771

minus09030341401

001584500766

minus002175914011

minus0003741933916

minus04311831740

minus08021570593

04130146809

00001362645283

minus0003854432451

0005396441669

minus08488209390

05154657137

01152135966

minus0004741957786

002229739182

minus0002068264586

00009402467441

minus0005358584867

0003987518000

minus003989381331

003796849613

minus09984595017

minus001058157817

002100470582

002092543658

0006230946613

minus09987520802

minus003826770492

00009823402498

0006977574167

001484146906

09990475909

0008196718814

minus003958286493

] ] ] ] ] ] ] ]

Specim

en3

[ [ [ [ [ [ [ [

1418542670

5838388363

9959058231

3433818493

3024968125

1757492470

] ] ] ] ] ] ] ]

[ [ [ [ [ [ [ [

03244960223

minus0004831643783

minus08451713783

04062092722

01164524794

minus004239310198

06461152835

minus07125334236

02668138654

004131783689

minus002431040234

003665187401

06621315621

07009969173

02633438279

002836289512

minus0001711614840

0005269111592

minus01962401199

0009159859564

03740839141

08850746598

01904577552

004284654369

minus0008597323304

minus0002526763452

001897677365

01738799329

minus06977939665

minus06945566457

001533190599

minus002803230310

006961922985

minus01374446216

06801869735

minus07159517722

] ] ] ] ] ] ] ]


24681012141618

12

34

56

Column Row

12

34

56


12345678

12

34

56

Column Row

12

34

56




120590119894119895= 119862


where 119862119894119895119896119897


120590 = 120590119894119895119890119894otimes 119890

119895 120598 = 120598

119896119897119890119896otimes 119890

119897

C = 119862119894119895119896119897

119890119894otimes 119890

119896otimes 119890

119896otimes 119890

119897

(23)

5

10

15

20

12

34

56

Column Row

12

34

56


5

10

15

20

25

12

34

56

Column Row

12

34

56



E1equiv 119890

1otimes 119890

1

E2equiv 119890

2otimes 119890

2

E3equiv 119890

3otimes 119890

3

(24)


E4equiv

1


2+ 119890

2otimes 119890

1)

E5equiv

1


3+ 119890

3otimes 119890

2)

E6equiv

1


1+ 119890

1otimes 119890

3)

(25)



E7equiv

1


2minus 119890

2otimes 119890

1)

E8equiv

1


3minus 119890

3otimes 119890

2)

E9equiv

1


1minus 119890

1otimes 119890

3)

(26)


120590 = 119878119860E

119860

120598 = 119864119860E

119860

C = 119862119860119861

119890119860otimes 119890

119861

(27)




120590119894119895= 119862

119894119895119896119897120598119896119897

lArrrArr 119878119860= 119862

119860119861E119861

119862119894119895119896119897

120598119896119897

= 120582120598119894119895lArrrArr 119862

119860119861E119861= 120582119864

119860

(28)


((((((((

(

1198781

1198782

1198783

1198784

1198785

1198786

1198787

1198788

1198789

))))))))

)

=

((((((((

(

12059011

12059022

12059033

120590(12)

120590(23)

120590(31)

120590[12]

120590[23]

120590[31]

))))))))

)

((((((((

(

1198641

1198642

1198643

1198644

1198645

1198646

1198647

1198648

1198649

))))))))

)

=

((((((((

(

12059811

12059822

12059833

120598(12)

120598(23)

120598(31)

120598[12]

120598[23]

120598[31]

))))))))

)

(29)


120579(119894119895)

equiv1

radic2(120579119894119895+ 120579

119895119894) 120579

[119894119895]equiv

1

radic2(120579119894119895minus 120579

119895119894) (30)


tensor C are

((((((((

(

11986211

11986212

11986213

11986214

11986215

11986216

11986217

11986218

11986219

11986221

11986222

11986223

11986224

11986225

11986226

11986227

11986228

11986229

11986231

11986232

11986233

11986234

11986235

11986236

11986237

11986238

11986239

11986241

11986242

11986243

11986244

11986245

11986246

11986247

11986248

11986249

11986251

11986252

11986253

11986254

11986255

11986256

11986257

11986258

11986259

11986261

11986262

11986263

11986264

11986265

11986266

11986267

11986268

11986269

11986271

11986272

11986273

11986274

11986275

11986276

11986277

11986278

11986279

11986281

11986282

11986283

11986284

11986285

11986286

11986287

11986288

11986289

11986291

11986292

11986293

11986294

11986295

11986296

11986297

11986298

11986299

))))))))

)

=

1198621111

1198621122

1198621133

1198622211

1198622222

1198623333

1198623311

1198623322

1198623333

11986211(12)

11986211(23)

11986211(31)

11986222(12)

11986222(23)

11986222(31)

11986233(12)

11986233(23)

11986233(31)

11986211[12]

11986211[23]

11986211[31]

11986222[12]

11986222[23]

11986222[31]

11986233[12]

11986233[23]

11986233[31]

119862(12)11

119862(12)22

119862(12)33

119862(23)11

119862(23)22

119862(23)33

119862(31)11

119862(31)22

119862(31)33

119862(1212)

119862(1223)

119862(1231)

119862(2312)

119862(2323)

119862(2331)

119862(3112)

119862(3123)

119862(3131)

119862(12)[12]

119862(12)[23]

119862(12)[31]

119862(23)[12]

119862(23)[23]

119862(23)[31]

119862(31)[12]

119862(31)[23]

119862(31)[31]

119862[12]11

119862[12]22

119862[12]33

119862[23]11

119862[23]22

119862[123]33

119862[21]11

119862[31]22

119862[31]33

119862[12](12)

119862[12](23)

119862[12](21)

119862[23](12)

119862[23](23)

119862[23](21)

119862[31](12)

119862[31](23)

119862[31](21)

119862[1212]

119862[1223]

119862[1231]

119862[2312]

119862[2323]

119862[2331]

119862[3112]

119862[3123]

119862[3131]

(31)

where

119862119894119895(119896119897)

equiv1

radic2(119862

119894119895119896119897+ 119862

119894119895119897119896) 119862

119894119895[119896119897]equiv

1

radic2(119862

119894119895119896119897minus 119862

119894119895119897119896)

119862(119894119895)119896119897

equiv1

radic2(119862

119894119895119896119897+ 119862

119895119894119896119897) 119862

[119894119895]119896119897equiv

1

radic2(119862

119894119895119896119897minus 119862

119895119894119896119897)

119862(119894119895119896119897)

equiv1

2(119862

119894119895119896119897+ 119862

119894119895119897119896+ 119862

119895119894119896119897+ 119862

119895119894119897119896)

119862(119894119895)[119896119897]

equiv1

2(119862

119894119895119896119897minus 119862

119894119895119897119896+ 119862

119895119894119896119897minus 119862

119895119894119897119896)

119862[119894119895](119896119897)

equiv1

2(119862

119894119895119896119897+ 119862

119894119895119897119896minus 119862

119895119894119896119897minus 119862

119895119894119897119896)

119862[119894119895119896119897]

equiv1

2(119862

119894119895119896119897minus 119862

119895119894119896119897minus 119862

119894119895119897119896+ 119862

119895119894119897119896)

(32)


12

34

56

ColumnRow

12

34

56

600005000040000300002000010000


5

10

15

20

25

12

34

56

Column Row

12

34

56






119895119894and 120598

119896119897= 120598

119897119896 the 119862

119894119895119896119897will be

like 119862119894119895119896119897

= 119862119895119894119896119897

= 119862119894119895119897119896


119862119894119895119896119897

= 119862119896119897119894119895












2 sdot 119864energy = 120590119894119895120598119894119895= 120590

119901120598119901 (33)


119901 = 119894120575119894119895+ (1 minus 120575

119894119895) (9 minus 119894 minus 119895)

119902 = 119896120575119896119897+ (1 minus 120575

119896119897) (9 minus 119896 minus 119897)

(34)


119901 119862

119894119895119896119897= 119862

119901119902 and 120598

119902= (2 minus

120575119896119897)120598119896119897



(

(

1205901

1205902

1205903

1205904

1205905

1205906

)

)

= (

(

11986211

11986212

11986213

11986214

11986215

11986216

11986221

11986222

11986223

11986224

11986225

11986226

11986231

11986232

11986233

11986234

11986235

11986236

11986241

11986242

11986243

11986244

11986245

11986246

11986251

11986252

11986253

11986254

11986255

11986256

11986261

11986262

11986263

11986264

11986265

11986266

)

)

(

(

1205981

1205982

1205983

1205984

1205985

1205986

)

)

(35)



(1) the 120590119894119895and 120598


(2) the norms of 120590119894119895 120598119896119897 and 119862




01 02 03 04 05 06 07 08

2

4

6

8

10

12

14

16

(a)

01

02

03

04

05

06

07

08

09

01 02 03 04 05 06 07 08

(b)


01 02 03 04 05 06 07 08

05

1

15

2

25

3

(a)

01 02 03 04 05 06 07 08

05

1

15

2

25

3

(b)




119901 = 119894120575119894119895+ (1 minus 120575

119894119895) (9 minus 119894 minus 119895)

119902 = 119896120575119896119897+ (1 minus 120575

119896119897) (9 minus 119896 minus 119897)

(36)

In this way

120590119901= (120575

119894119895+ radic2 (1 minus 120575

119894119895)) 120598

119894119895

120598119902= (120575

119896119897+ radic2 (1 minus 120575

119896119897)) 120598

119896119897

119862119901119902

= (120575119894119895+ radic2 (1 minus 120575

119894119895)) (120575

119896119897+ radic2 (1 minus 120575

119896119897)) 119862

119894119895119896119897

(37)




120590kelvin

= 120590viogt

120585 120590voigt

=120590kelvin

120585

120598kelvin

= 120598voigt

120585 120598voigt

=120598kelvin

120585

(38)


119862kelvin

=((

(







))

)

119862voigt

(39)


(

(

12059011

12059022

12059033

radic212059023

radic212059013

radic212059012

)

)

= (

1198621111 1198621122 1198621133radic21198621123

radic21198621113radic21198621112

1198622211 1198622222 1198622333radic21198622223

radic21198622213radic21198622212

1198623311 1198623322 1198623333radic21198623323

radic21198623313radic21198623312

radic21198622311radic21198622322

radic21198622333 21198622323 21198622313 21198622312

radic21198621311radic21198621322

radic21198621333 21198621323 21198621313 21198621312

radic21198621211radic21198621222

radic21198621233 21198621223 21198621213 21198621212

)

lowast(

(

12059811

12059822

12059833

radic212059823

radic212059813

radic212059812

)

)

(40)


= C (41)



(

(

12059011

12059022

12059033

radic212059023

radic212059013

radic212059012

)

)

=((

(

11986211

11986212

11986213

radic211986214

radic211986215

radic211986216

11986212

11986222

11986223

radic211986224

radic211986225

radic211986226

11986213

11986223

11986233

radic211986234

radic211986235

radic211986236

radic211986214

radic211986224

radic211986234

211986244

211986245

211986246

radic211986215

radic211986225

radic211986235

211986245

211986255

211986256

radic211986216

radic211986226

radic211986236

211986246

211986256

211986266

))

)

lowast (

(

12059811

12059822

12059833

radic212059823

radic212059813

radic212059812

)

)

(42)


(

(

1

2

3

4

5

6

)

)

=((

(

11986211

11986212

11986213

11986214

11986215

11986216

11986212

11986222

11986223

11986224

11986225

11986226

11986213

11986223

11986233

11986234

11986235

11986236

11986214

11986224

11986234

11986244

11986245

11986246

11986215

11986225

11986235

11986245

11986255

11986256

11986216

11986226

11986236

11986246

11986256

11986266

))

)

lowast (

(

1205981

1205982

1205983

1205984

1205985

1205986

)

)

(43)



































01 02 03 04 05 06 07 08

05

1

15

2

25

3

(a)

01 02 03 04 05 06 07 08

05

1

15

2

25

3

(b)


01 02 03 04 05 06 07 08

2

4

6

8

10

12

14

16





ΛI) N = 0 (44)


119894 119894 = 1 2 6 and

















NNA119896

equiv1

119872

119872

sum

119884=1

N119884119896 (45)



NAVG119896

equiv JNANNA119896

(46)


119896=1NNA119896

otimes NNA119896

| that is

JNA equiv (

6

sum

119896=1

NNA119896

otimes NNA119896

)

minus12

(47)


119896 that is

ΛAVG119902

equiv1

119872

119872

sum

119884=1

6

sum

119896=1

Λ119884

119896(N119884

119896sdot NAVG

119902)2

(48)


gt 0 for all 119896 = 1 2 6

and ΛAVG119896


NA119896


ΛNA119896

equiv1

119872

119872

sum

119884=1

Λ119884

119896 (49)


CAVG=

6

sum

119896=1

ΛAVG119896

NAVG119896

otimes NAVG119896

(50)





[[[[[[[

[

00367834559700000

00453681054800000

0998185540700000

00

00

00

]]]]]]]

]

[[[[[[[

[

0983745905000000

minus0170879918750000

minus00277901141300000

00

00

00

]]]]]]]

]

[[[[[[[

[

minus0169284222800000

minus0982986098000000

00509905132200000

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

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]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(51)


[[[[[[[

[

003679134179

004537783172

09983995367

00

00

00

]]]]]]]

]

[[[[[[[

[

09859858256

minus01712690003

minus002785339027

00

00

00

]]]]]]]

]

[[[[[[[

[

minus01697052873

minus09854311019

005111734310

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

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]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(52)




[[[

[

334725276837330 0000112116752981933 000198542359441849 00 00 00

0000112116752981933 334773746006714 minus0000991802322788501 00 00 00

000198542359441849 minus0000991802322788501 334013593769726 00 00 00

00 00 00 333950000000000 00 00

00 00 00 00 333950000000000 00

00 00 00 00 00 333950000000000

]]]

]

(53)




[[[[[[[

[

004028666644

006399313350

09971368323

00

00

00

]]]]]]]

]

[[[[[[[

[

09048796003

minus04255671399

minus0009247686096

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04237568827

minus09026613361

007505076253

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(54)


[[[[[[[

[

004028666644

006399313350

09971368323

00

00

00

]]]]]]]

]

[[[[[[[

[

09048795994

minus04255671395

minus0009247686087

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04237568827

minus09026613361

007505076253

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(55)



[[[

[

1458799998 000000001785571215 minus000000001009489597 00 00 00000000001785571215 1458799997 0000000002407020197 00 00 00minus000000001009489597 0000000002407020197 1458799997 00 00 00

00 00 00 1458799998 00 0000 00 00 00 1458799998 0000 00 00 00 00 1458799998

]]]

]

(56)




[[[[[

[

005368516527

007220768570

09959437502

00

00

00

]]]]]

]

[[[[[

[

09266208315

minus03753089731

minus002273783078

00

00

00

]]]]]

]

[[[[[

[

minus03721447810

minus09240829102

008705767064

00

00

00

]]]]]

]

[[[[[

[

00

00

00

00

00

10

]]]]]

]

[[[[[

[

00

00

00

10

00

00

]]]]]

]

[[[[[

[

00

00

00

00

10

00

]]]]]

]

(57)


[[[[[[[[[

[

005368516527

007220768570

09959437502

00

00

00

]]]]]]]]]

]

[[[[[[[[[

[

09266208315

minus03753089731

minus002273783078

00

00

00

]]]]]]]]]

]

[[[[[[[[[

[

minus03721447806

minus09240829093

008705767055

00

00

00

]]]]]]]]]

]

[[[[[[[[[

[

00

00

00

10

00

00

]]]]]]]]]

]

[[[[[[[[[

[

00

00

00

00

10

00

]]]]]]]]]

]

[[[[[[[[[

[

00

00

00

00

00

10

]]]]]]]]]

]

(58)




[[[

[

1615299998 0000000005508173090 minus0000000008722620038 00 00 00

0000000005508173090 1615299996 minus0000000004345156817 00 00 00

minus0000000008722620038 minus0000000004345156817 1615300000 00 00 00

00 00 00 1615299998 00 00

00 00 00 00 1615299998 00

00 00 00 00 00 1615299998

]]]

]

(59)




[[[[[[[

[

minus00598757013800000

minus00768229223150000

minus0995231841750000

00

00

00

]]]]]]]

]

[[[[[[[

[

minus0869358919900000

0493939153600000

00141567750950000

00

00

00

]]]]]]]

]

[[[[[[[

[

minus0490490276500000

minus0866078136900000

00963811082600000

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(60)


[[[[[[[[[[[

[

minus005987730132

minus007682497510

minus09952584354

00

00

00

]]]]]]]]]]]

]

[[[[[[[[[[[

[

minus08693926232

04939583026

001415732393

00

00

00

]]]]]]]]]]]

]

[[[[[[[[[[[

[

00

00

00

10

00

00

]]]]]]]]]]]

]

[[[[[[[[[[[

[

00

00

00

00

10

00

]]]]]]]]]]]

]

[[[[[[[[[[[

[

00

00

00

00

00

10

]]]]]]]]]]]

]

(61)



[[[

[


00 00 00 1819400002 1819400002 0000 00 00 00 00 0000 00 00 00 00 1819400002

]]]

]

(62)




[[[[[[[

[

004967528691

008463937788

09951726186

00

00

00

]]]]]]]

]

[[[[[[[

[

09161715971

minus04006166011

minus001165943224

00

00

00

]]]]]]]

]

[[[[[

[

minus03976958243

minus09123280736

009744493938

00

00

00

]]]]]

]

[[[[[

[

00

00

00

10

00

00

]]]]]

]

[[[[[

[

00

00

00

00

10

00

]]]]]

]

[[[[[

[

00

00

00

00

00

10

]]]]]

]

(63)


[[[[[[[

[

004967528696

008463937796

09951726196

00

00

00

]]]]]]]

]

[[[[[[[

[

09161715980

minus04006166015

minus001165943225

00

00

00

]]]]]]]

]


[[[[[[[

[

minus03976958247

minus09123280745

009744493948

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(64)



[[[

[

1922399998 minus000000001176508811 minus000000001537920006 00 00 00

minus000000001176508811 1922400000 minus0000000007631928036 00 00 00

minus000000001537920006 minus0000000007631928036 1922400000 00 00 00

00 00 00 1922399998 1922399998 00

00 00 00 00 00 00

00 00 00 00 00 1922399998

]]]

]

(65)




[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

[[[[[[[

[

minus01387533797

minus02031928413

minus09692575344

00

00

00

]]]]]]]

]

[[[[[[[

[

minus08476774112

05304148448

001015375725

00

00

00

]]]]]]]

]

[[[[[[[

[

minus05120454135

minus08230265872

02458388325

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

(66)


[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

[[[[[[[

[

minus01387533798

minus02031928415

minus09692575354

00

00

00

]]]]]]]

]

[[[[[[[

[

minus08476774112

05304148448

001015375725

00

00

00

]]]]]]]

]

[[[[[[[

[

minus05120454135

minus08230265872

02458388325

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

(67)



[[[

[

2074599999 0000000004356660361 000000003402343994 00 00 00

0000000004356660361 2074600004 000000002232269567 00 00 00

000000003402343994 000000002232269567 2074600000 00 00 00

00 00 00 2074600000 2074600000 00

00 00 00 00 00 00

00 00 00 00 00 2074600000

]]]

]

(68)




[[[[[[[

[

008621257414

01243595697

09884847438

00

00

00

]]]]]]]

]

[[[[[[[

[

08755641188

minus04828494241

minus001561753067

00

00

00

]]]]]]]

]

[[[[[

[

minus04753471023

minus08668282006

01505124700

00

00

00

]]]]]

]

[[[[[

[

00

00

00

10

00

00

]]]]]

]

[[[[[

[

00

00

00

00

10

00

]]]]]

]

[[[[[

[

00

00

00

00

00

10

]]]]]

]

(69)


[[[[[

[

008621257423

01243595698

09884847448

00

00

00

]]]]]

]

[[[[[

[

08755641188

minus04828494241

minus001561753067

00

00

00

]]]]]

]


[[[[[[[

[

minus04753471018

minus08668281997

01505124698

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(70)



[[[

[

1890099999 000000001209663993 minus000000002532733996 00 00 00000000001209663993 1890099991 0000000001701090138 00 00 00minus000000002532733996 0000000001701090138 1890100001 00 00 00

00 00 00 1890099997 1890099997 0000 00 00 00 00 0000 00 00 00 00 1890099997

]]]

]

(71)




[[[[[[[

[

03961520086

06338768023

06642768888

00

00

00

]]]]]]]

]

[[[[[[[

[

09160614623

minus02236797319

minus03328645006

00

00

00

]]]]]]]

]

[[[[[[[

[

006240981414

minus07403833993

06692812840

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(72)


[[[[[[[

[

03961520086

06338768023

06642768888

00

00

00

]]]]]]]

]

[[[[[[[

[

09160614614

minus02236797317

minus03328645003

00

00

00

]]]]]]]

]

[[[[[[[

[

006240981426

minus07403834008

06692812853

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(73)



[[[

[


00 00 00 8772299998 8772299998 0000 00 00 00 00 0000 00 00 00 00 8772299998

]]]

]

(74)




[[[[[[[

[

minus006591789198

minus008975775227

minus09937798435

00

00

00

]]]]]]]

]

[[[[[[[

[

minus08289381135

05593224991

0004466103673

00

00

00

]]]]]]]

]

[[[[[

[

minus05554425577

minus08240763851

01112729817

00

00

00

]]]]]

]

[[[[[

[

00

00

00

10

00

00

]]]]]

]

[[[[[

[

00

00

00

00

10

00

]]]]]

]

[[[[[

[

00

00

00

00

00

10

]]]]]

]


[[[[[[[

[

minus01387533798

minus02031928415

minus09692575354

00

00

00

]]]]]]]

]

[[[[[[[

[

minus08476774112

05304148448

001015375725

00

00

00

]]]]]]]

]


[[[[

[

minus05120454135

minus08230265872

02458388325

00

00

00

]]]]

]

[[[[

[

00

00

00

10

00

00

]]]]

]

[[[[

[

00

00

00

00

10

00

]]]]

]

[[[[

[

00

00

00

00

00

10

]]]]

]

(76)



[[[

[

2228057494 0000000004678921127 000000003654014285 00 00 000000000004678921127 2228057499 000000002397389829 00 00 00000000003654014285 000000002397389829 2228057495 00 00 00

00 00 00 2228057495 2228057495 0000 00 00 00 00 0000 00 00 00 00 2228057495

]]]

]

(77)




[[[[[[[

[

006975010637

01071019008

09917984199

00

00

00

]]]]]]]

]

[[[[[[[

[

09079000793

minus04187725641

minus001862756541

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04133429180

minus09017531389

01264472547

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(78)


[[[[[[[

[

006975010637

01071019008

09917984199

00

00

00

]]]]]]]

]

[[[[[[[

[

09079000793

minus04187725641

minus001862756541

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04133429184

minus09017531398

01264472548

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(79)



[[[

[


00 00 00 2598699998 2598699998 0000 00 00 00 00 0000 00 00 00 00 2598699998

]]]

]

(80)




[[[[[

[

minus00192522847250000

minus00192842313600000

000386151499999998

00

00

00

]]]]]

]

[[[[[

[

000961799224999998

00165029120500000

000436480214750000

00

00

00

]]]]]

]

[[[[[[

[

minus0494444050150000

minus0856906622200000

0142104790200000

00

00

00

]]]]]]

]

[[[[[[

[

00

00

00

10

00

00

]]]]]]

]

[[[[[[

[

00

00

00

00

10

00

]]]]]]

]

[[[[[[

[

00

00

00

00

00

10

]]]]]]

]

(81)



[[[[[[[[[[

[

minus2541745843

minus2545963537

5098090872

00

00

00

]]]]]]]]]]

]

[[[[[[[[[[

[

2505315863

4298714977

1136953303

00

00

00

]]]]]]]]]]

]

[[[[[[[

[

minus04949599718

minus08578007511

01422530677

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(82)



[[[

[

315679169478799 427324383657779 384557503470365 00 00 00427324383657779 618700481877479 889153535276175 00 00 00384557503470365 889153535276175 384768762708609 00 00 00

00 00 00 247795001400000 00 0000 00 00 00 247795001400000 0000 00 00 00 00 247795001400000

]]]

]

(83)




[[[[[[[

[

01135176976

01764907831

09777344911

00

00

00

]]]]]]]

]

[[[[[[[

[

08848520771

minus04654963442

minus001870707734

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04518302069

minus08672739791

02090103088

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(84)


[[[[[[[

[

01135176977

01764907833

09777344921

00

00

00

]]]]]]]

]

[[[[[[[

[

08848520771

minus04654963442

minus001870707734

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04518302069

minus08672739791

02090103088

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(85)



[[[

[

2663700003 000000004768023095 000000003169803038 00 00 00000000004768023095 2663699992 0000000008310743498 00 00 00000000003169803038 0000000008310743498 2663700001 00 00 00

00 00 00 2663699998 2663699998 0000 00 00 00 00 0000 00 00 00 00 2663699998

]]]

]

(86)





6 Conclusions









Disclosure




Acknowledgments



References




















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











4 3 1 -gt 5 1 2 -gt 6 3 2 -gt 4 1 3 -gt

5 2 1 -gt 6



Algorithm 3




In[12]=C2


In[14]=C2to4




Array[6 amp 2]] 2]





Which[





] i 6 j 6]


Which[





] i 6 j 6]







Algorithm 4




= 119860119895119894119896119897

= 119860119894119895119897119896







(17)


11 12 1312 22 2313 23 33

11 12 1312 22 2313 23 33

11 12 1312 22 2313 23 33

11 12 1312 22 2313 23 33

11 12 1312 22 2313 23 33

11 12 1312 22 2313 23 33

11 12 1312 22 2313 23 33

11 12 1312 22 2313 23 33

11 12 1312 22 2313 23 33

11 12 1312 22 2313 23 33

11 12 1312 22 2313 23 33

11 12 13

12 22 23

13 23 33

120590ij

Cijkl

120598ij

lowast=


2313= 119862



1

2

3

12

34

56

Column Row

12

34

56


2468101214

12

34

56

Column Row

12

34

56


246810121416

12

34

56

Column Row

12

34

56



= 119860120590(119894)120590(119895)120590(119895)120590(119897)




(18)



119860119904

119894119895119896119897=

1

3(119860

119894119895119896119897+ 119860

119894119896119895119897+ 119860

119894119897119896119895)

119860119886

119894119895119896119897=

1

3(119860

119894119895119896119897minus 119860

119894119896119895119897minus 119860

119894119897119895119896)

(19)




Stress

12059011

1205901

1

12059022

1205902

2

12059033

1205903

3

12059023

1205904

4

12059013

1205905

5

12059012

1205905

6

Strain

12059811

1205981

1205981

12059822

1205982

1205982

12059833

1205983

1205983

12059823

1205984

1205984

12059813

12059815

1205985

12059812

1205986

1205986

Elasticity

1198621111

11986211

11986211

1198622222

11986222

11986222

1198623333

11986233

11986233

1198621122

11986212

11986212

1198621133

11986213

11986213

1198622233

11986223

11986223

1198622323

11986244

1

211986244

1198621313

11986255

1

211986255

1198621212

11986266

1

211986266

1198621323

11986254

1

211986254

1198621312

11986256

1

211986256

1198621223

11986264

1

411986264

1198622311

11986241

1

radic211986241

1198621311

11986251

1

radic211986251

1198621211

11986261

1

radic211986261

1198622322

11986242

1

radic211986242

1198621322

11986252

1

radic211986252

1198621222

11986262

1

radic211986262

1198622333

11986243

1

radic211986243

1198621333

11986253

1

radic211986253

1198621233

11986263

1

radic211986263

Table 4 Continued


Compliance

1198701111

11987811

11987811

1198702222

11987822

11987822

1198703333

11987833

11987833

1198701122

11987812

11987812

1198701133

11987813

11987813

1198702233

11987823

11987823

1198702323

1

411987844

1

211987844

1198701313

1

411987855

1

211987855

1198701212

1

411987866

1

211987866

1198701323

1

411987854

1

211987854

1198701312

1

411987856

1

211987856

1198701223

1

411987864

1

211987864

1198702311

1

211987841

1

radic211987841

1198701311

1

211987851

1

radic211987851

1198701211

1

211987861

1

radic211987861

1198702322

1

211987842

1

radic211987842

1198701322

1

211987852

1

radic211987852

1198701222

1

211987862

1

radic211987862

1198702333

1

211987843

1

radic211987843

1198701333

1

211987853

1

radic211987853

1198701233

1

211987863

1

radic211987863

24681012141618

12

34

56

Column Row

12

34

56




S no Species 120588 11986211

11986222

11986233

11986212

11986213

11986223

11986244

11986255

11986266



S no Species 120588 11986211

11986222

11986233

11986212

11986213

11986223

11986244

11986255

11986266














986 7912 698311986212

2147 3609 281611986213

2773 3186 284511986214

minus0162 11 8011986215

minus293 54 minus0711986216

minus0821 minus06 11411986222

935 8529 896711986223

2346 3281 322711986224

minus1204 09 minus162311986225


0445 minus42 16011986233

5952 14730 916011986234


minus1791 169 minus7211986236

054 06 2411986244

561 3587 348911986245

0798 05 10511986246

minus2159 51 minus9811986255

6032 3448 242611986256



24681012141618

12

34

56

Column Row

12

34

56



5

10

15

20

12

34

56

Column Row

12

34

56



119890120595(119894119895)

= 119890119894otimes 119890

119895 (20)




1 Quipo

[[[[[[[[[

[

1076866026

004067345638

02534605191

02260000000

01180000000

007800000000

]]]]]]]]]

]

[[[[[[[[[

[

003276733390 09918780499 minus01228992897 00 00 00

003131140851 minus01239237163 minus09917976143 00 00 00

09989724208 minus002865041271 003511774899 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

2 Quipo

[[[[[[[[[

[

3461963876

01399776173

04300584948

04300000000

02800000000

01440000000

]]]]]]]]]

]

[[[[[[[[[

[

004079957804 09756137601 minus02156691559 00 00 00

005942480245 minus02178361212 minus09741745817 00 00 00

09973986606 minus002692981555 006686327745 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

3 White

[[[[[[[[[

[

1009116301

03556701722

1333166815

1442000000

1344000000

002200000000

]]]]]]]]]

]

[[[[[[[[[

[

004028666644 09048796003 minus04237568827 00 00 00

006399313350 minus04255671399 minus09026613361 00 00 00

09971368323 minus0009247686096 007505076253 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

4 Khaya

[[[[[[[[[

[

1080102614

04606834511

1475290392

1800000000

1196000000

04200000000

]]]]]]]]]

]

[[[[[[[[[

[

005368516527 09266208315 minus03721447810 00 00 00

007220768570 minus03753089731 minus09240829102 00 00 00

09959437502 minus002273783078 008705767064 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

5 Mahogany

[[[[[[[[[

[

1210253321

06098853915

1810581412

1960000000

1498000000

06380000000

]]]]]]]]]

]

[[[[[[[[[

[


minus007817061387 04985803653 minus08633116316 00 00 00

minus09948285648 001731853005 01000809391 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

6 Mahogany

[[[[[[[[[

[

1310854783

03895295651

1814922632

1218000000

09380000000

03000000000

]]]]]]]]]

]

[[[[[[[[[

[


minus007547523076 04892979419 minus08688446422 00 00 00

minus09956351187 001099502014 009268127742 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

7 S Germ

[[[[[[[[[

[

1234053410

05212570563

1922208822

2318000000

1582000000

05400000000

]]]]]]]]]

]

[[[[[[[[[

[

004967528691 09161715971 minus03976958243 00 00 00

008463937788 minus04006166011 minus09123280736 00 00 00

09951726186 minus001165943224 009744493938 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

8 Maple

[[[[[[[[[

[

1205449768

06868279555

2766674361

2460000000

2194000000

05840000000

]]]]]]]]]

]

[[[[[[[[[

[


minus02031928413 05304148448 minus08230265872 00 00 00

minus09692575344 001015375725 02458388325 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]




9 Walnut

[[[[[[[[[

[

1267869546

05204134969

1919891015

1922000000

1400000000

04600000000

]]]]]]]]]

]

[[[[[[[[[

[

008621257414 08755641188 minus04753471023 00 00 00

01243595697 minus04828494241 minus08668282006 00 00 00

09884847438 minus001561753067 01505124700 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

10 Birch

[[[[[[[[[

[

3168908680

04747157903

05946755301

2346000000

1816000000

03720000000

]]]]]]]]]

]

[[[[[[[[[

[

03961520086 09160614623 006240981414 00 00 00

06338768023 minus02236797319 minus07403833993 00 00 00

06642768888 minus03328645006 06692812840 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

11 Y Birch

[[[[[[[[[

[

1545414040

05549651499

2059894445

2120000000

1942000000

04800000000

]]]]]]]]]

]

[[[[[[[[[

[


minus008975775227 05593224991 minus08240763851 00 00 00

minus09937798435 0004466103673 01112729817 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

12 Oak

[[[[[[[[[

[

1718666431

08648974218

3239438239

2380000000

1532000000

07840000000

]]]]]]]]]

]

[[[[[[[[[

[

006975010637 09079000793 minus04133429180 00 00 00

01071019008 minus04187725641 minus09017531389 00 00 00

09917984199 minus001862756541 01264472547 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

13 Ash

[[[[[[[[[

[

1715567967

06696042476

2409716064

2684000000

1784000000

05400000000

]]]]]]]]]

]

[[[[[[[[[

[

006190313535 08746549514 minus04807772027 00 00 00

009781749768 minus04846986500 minus08691944279 00 00 00

09932772717 minus0006777439445 01155609239 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

14 Ash

[[[[[[[[[

[

1742363268

07844815431

2669885744

1720000000

1218000000

05000000000

]]]]]]]]]

]

[[[[[[[[[

[


minus01363859604 05177044741 minus08446188165 00 00 00

minus09855542417 001550704374 01686486565 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

15 Beech

[[[[[[[[[

[

1599002755

09558575345

3451114905

3216000000

2112000000

09120000000

]]]]]]]]]

]

[[[[[[[[[

[

01135176976 08848520771 minus04518302069 00 00 00

01764907831 minus04654963442 minus08672739791 00 00 00

09777344911 minus001870707734 02090103088 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

Thus 119890120572otimes 119890

1205731le120572120573le119899




119860119894119895119896119897

119890119894otimes 119890

119895otimes 119890

119896otimes 119890

119897 or as an 119899


120572120573119890120572otimes 119890

120573 where 119860

120595(119894119895)120595119896119897= 119860

119894119895119896119897

1 le 119894 119895 119896 119897 le 119899


tr (A119879B) = ⟨AB⟩ = tr (119860119879119861119879) A =1003817100381710038171003817100381711986010038171003817100381710038171003817 (21)






1 Blasa

[[[[[[[[[

[

6385324208

009845837744

03832174064

06240000000

04060000000

006600000000

]]]]]]]]]

]

[[[[[[[[[

[

001488818113 09510211778 minus03087669978 00 00 00

002575997614 minus03090635474 minus09506924583 00 00 00

09995572851 minus0006200252135 002909967665 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

2 Spruce

[[[[[[[[[

[

1198583568

04516442734

1114520065

1498000000

1442000000

007800000000

]]]]]]]]]

]

[[[[[[[[[

[


minus004689865645 04044467539 minus09133582749 00 00 00

minus09983543167 001123114838 005623628701 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

3 Spruce

[[[[[[[[[

[

1410927768

04185382987

1227183961

1442000000

10

006400000000

]]]]]]]]]

]

[[[[[[[[[

[


minus005361649666 04423303564 minus08952480817 00 00 00

minus09978936411 0009052295016 006423657043 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

4 Spruce

[[[[[[[[[

[

1630529953

03544288592

08442715844

1234000000

1520000000

007200000000

]]]]]]]]]

]

[[[[[[[[[

[


minus002869707087 04091564350 minus09120128765 00 00 00

minus09993764538 0007032901380 003460119282 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

5 Spruce

[[[[[[[[[

[

1725845208

05092652753

1171282625

1250000000

1706000000

007000000000

]]]]]]]]]

]

[[[[[[[[[

[


minus003428302033 05858146064 minus08097196604 00 00 00

minus09988364173 0007413313465 004765347716 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

6 Douglas Fir

[[[[[[[[[

[

1613459769

06233733771

1439028943

1767000000

1766000000

01760000000

]]]]]]]]]

]

[[[[[[[[[

[


minus003697194691 05922678885 minus08048924305 00 00 00

minus09986533619 0007435777607 005134366998 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

7 Douglas Fir

[[[[[[[[[

[

1710150156

06986859431

2204812526

2348000000

1816000000

01600000000

]]]]]]]]]

]

[[[[[[[[[

[


minus006364825251 05704773676 minus08188433771 00 00 00

minus09967231590 0004703484281 008075160717 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

8 Pine

[[[[[[[[[

[

1699539133

04940019684

1565606760

3484000000

1344000000

01320000000

]]]]]]]]]

]

[[[[[[[[[

[


minus005585205149 04413516031 minus08955943895 00 00 00

minus09978476167 0006195562437 006528207193 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]


Table11Th

eeigenvalues

andeigenvectorsfor3

specim

enso

fcancello

usbo

necalculated

usingMAPL

EΛ[119894]sta

ndforthe

eigenvalueso

f119894th

specim

enof

cancellous

boneand

119881[119894]sta

ndfor

thec

orrespon

ding

eigenvectors

Cancellous

bone

EigenvaluesΛ

[119894]

Eigenvectors

119881[119894]

Specim

en1

[ [ [ [ [ [ [ [

1334869880

minus02071167473

8644210768

6837676373

3768806080

4123724855

] ] ] ] ] ] ] ]

[ [ [ [ [ [ [ [

03316115964

08831238953

02356441950

02242668491

005354786914

003787816350

06475092156

minus01099136965

01556396475

minus06940063479

minus01278953723

02154647417

04800906963

minus02832250652

007648827487

02494278793

06410471445

minus04585742261

minus02737532692

minus006354940263

04518342532

minus006214920576

05836952694

06101669758

minus03706582070

03314035780

minus01276606259

minus06337030234

03639432801

minus04499474345

01671829224

01180163925

minus08330343502

002008124471

03109325968

04087706408

] ] ] ] ] ] ] ]

Specim

en2

[ [ [ [ [ [ [ [

1754634171

7773848354

1550688791

2301790452

3445174379

3589156342

] ] ] ] ] ] ] ]

[ [ [ [ [ [ [ [

minus03057290719

minus03005337771

minus09030341401

001584500766

minus002175914011

minus0003741933916

minus04311831740

minus08021570593

04130146809

00001362645283

minus0003854432451

0005396441669

minus08488209390

05154657137

01152135966

minus0004741957786

002229739182

minus0002068264586

00009402467441

minus0005358584867

0003987518000

minus003989381331

003796849613

minus09984595017

minus001058157817

002100470582

002092543658

0006230946613

minus09987520802

minus003826770492

00009823402498

0006977574167

001484146906

09990475909

0008196718814

minus003958286493

] ] ] ] ] ] ] ]

Specim

en3

[ [ [ [ [ [ [ [

1418542670

5838388363

9959058231

3433818493

3024968125

1757492470

] ] ] ] ] ] ] ]

[ [ [ [ [ [ [ [

03244960223

minus0004831643783

minus08451713783

04062092722

01164524794

minus004239310198

06461152835

minus07125334236

02668138654

004131783689

minus002431040234

003665187401

06621315621

07009969173

02633438279

002836289512

minus0001711614840

0005269111592

minus01962401199

0009159859564

03740839141

08850746598

01904577552

004284654369

minus0008597323304

minus0002526763452

001897677365

01738799329

minus06977939665

minus06945566457

001533190599

minus002803230310

006961922985

minus01374446216

06801869735

minus07159517722

] ] ] ] ] ] ] ]


24681012141618

12

34

56

Column Row

12

34

56


12345678

12

34

56

Column Row

12

34

56




120590119894119895= 119862


where 119862119894119895119896119897


120590 = 120590119894119895119890119894otimes 119890

119895 120598 = 120598

119896119897119890119896otimes 119890

119897

C = 119862119894119895119896119897

119890119894otimes 119890

119896otimes 119890

119896otimes 119890

119897

(23)

5

10

15

20

12

34

56

Column Row

12

34

56


5

10

15

20

25

12

34

56

Column Row

12

34

56



E1equiv 119890

1otimes 119890

1

E2equiv 119890

2otimes 119890

2

E3equiv 119890

3otimes 119890

3

(24)


E4equiv

1


2+ 119890

2otimes 119890

1)

E5equiv

1


3+ 119890

3otimes 119890

2)

E6equiv

1


1+ 119890

1otimes 119890

3)

(25)



E7equiv

1


2minus 119890

2otimes 119890

1)

E8equiv

1


3minus 119890

3otimes 119890

2)

E9equiv

1


1minus 119890

1otimes 119890

3)

(26)


120590 = 119878119860E

119860

120598 = 119864119860E

119860

C = 119862119860119861

119890119860otimes 119890

119861

(27)




120590119894119895= 119862

119894119895119896119897120598119896119897

lArrrArr 119878119860= 119862

119860119861E119861

119862119894119895119896119897

120598119896119897

= 120582120598119894119895lArrrArr 119862

119860119861E119861= 120582119864

119860

(28)


((((((((

(

1198781

1198782

1198783

1198784

1198785

1198786

1198787

1198788

1198789

))))))))

)

=

((((((((

(

12059011

12059022

12059033

120590(12)

120590(23)

120590(31)

120590[12]

120590[23]

120590[31]

))))))))

)

((((((((

(

1198641

1198642

1198643

1198644

1198645

1198646

1198647

1198648

1198649

))))))))

)

=

((((((((

(

12059811

12059822

12059833

120598(12)

120598(23)

120598(31)

120598[12]

120598[23]

120598[31]

))))))))

)

(29)


120579(119894119895)

equiv1

radic2(120579119894119895+ 120579

119895119894) 120579

[119894119895]equiv

1

radic2(120579119894119895minus 120579

119895119894) (30)


tensor C are

((((((((

(

11986211

11986212

11986213

11986214

11986215

11986216

11986217

11986218

11986219

11986221

11986222

11986223

11986224

11986225

11986226

11986227

11986228

11986229

11986231

11986232

11986233

11986234

11986235

11986236

11986237

11986238

11986239

11986241

11986242

11986243

11986244

11986245

11986246

11986247

11986248

11986249

11986251

11986252

11986253

11986254

11986255

11986256

11986257

11986258

11986259

11986261

11986262

11986263

11986264

11986265

11986266

11986267

11986268

11986269

11986271

11986272

11986273

11986274

11986275

11986276

11986277

11986278

11986279

11986281

11986282

11986283

11986284

11986285

11986286

11986287

11986288

11986289

11986291

11986292

11986293

11986294

11986295

11986296

11986297

11986298

11986299

))))))))

)

=

1198621111

1198621122

1198621133

1198622211

1198622222

1198623333

1198623311

1198623322

1198623333

11986211(12)

11986211(23)

11986211(31)

11986222(12)

11986222(23)

11986222(31)

11986233(12)

11986233(23)

11986233(31)

11986211[12]

11986211[23]

11986211[31]

11986222[12]

11986222[23]

11986222[31]

11986233[12]

11986233[23]

11986233[31]

119862(12)11

119862(12)22

119862(12)33

119862(23)11

119862(23)22

119862(23)33

119862(31)11

119862(31)22

119862(31)33

119862(1212)

119862(1223)

119862(1231)

119862(2312)

119862(2323)

119862(2331)

119862(3112)

119862(3123)

119862(3131)

119862(12)[12]

119862(12)[23]

119862(12)[31]

119862(23)[12]

119862(23)[23]

119862(23)[31]

119862(31)[12]

119862(31)[23]

119862(31)[31]

119862[12]11

119862[12]22

119862[12]33

119862[23]11

119862[23]22

119862[123]33

119862[21]11

119862[31]22

119862[31]33

119862[12](12)

119862[12](23)

119862[12](21)

119862[23](12)

119862[23](23)

119862[23](21)

119862[31](12)

119862[31](23)

119862[31](21)

119862[1212]

119862[1223]

119862[1231]

119862[2312]

119862[2323]

119862[2331]

119862[3112]

119862[3123]

119862[3131]

(31)

where

119862119894119895(119896119897)

equiv1

radic2(119862

119894119895119896119897+ 119862

119894119895119897119896) 119862

119894119895[119896119897]equiv

1

radic2(119862

119894119895119896119897minus 119862

119894119895119897119896)

119862(119894119895)119896119897

equiv1

radic2(119862

119894119895119896119897+ 119862

119895119894119896119897) 119862

[119894119895]119896119897equiv

1

radic2(119862

119894119895119896119897minus 119862

119895119894119896119897)

119862(119894119895119896119897)

equiv1

2(119862

119894119895119896119897+ 119862

119894119895119897119896+ 119862

119895119894119896119897+ 119862

119895119894119897119896)

119862(119894119895)[119896119897]

equiv1

2(119862

119894119895119896119897minus 119862

119894119895119897119896+ 119862

119895119894119896119897minus 119862

119895119894119897119896)

119862[119894119895](119896119897)

equiv1

2(119862

119894119895119896119897+ 119862

119894119895119897119896minus 119862

119895119894119896119897minus 119862

119895119894119897119896)

119862[119894119895119896119897]

equiv1

2(119862

119894119895119896119897minus 119862

119895119894119896119897minus 119862

119894119895119897119896+ 119862

119895119894119897119896)

(32)


12

34

56

ColumnRow

12

34

56

600005000040000300002000010000


5

10

15

20

25

12

34

56

Column Row

12

34

56






119895119894and 120598

119896119897= 120598

119897119896 the 119862

119894119895119896119897will be

like 119862119894119895119896119897

= 119862119895119894119896119897

= 119862119894119895119897119896


119862119894119895119896119897

= 119862119896119897119894119895












2 sdot 119864energy = 120590119894119895120598119894119895= 120590

119901120598119901 (33)


119901 = 119894120575119894119895+ (1 minus 120575

119894119895) (9 minus 119894 minus 119895)

119902 = 119896120575119896119897+ (1 minus 120575

119896119897) (9 minus 119896 minus 119897)

(34)


119901 119862

119894119895119896119897= 119862

119901119902 and 120598

119902= (2 minus

120575119896119897)120598119896119897



(

(

1205901

1205902

1205903

1205904

1205905

1205906

)

)

= (

(

11986211

11986212

11986213

11986214

11986215

11986216

11986221

11986222

11986223

11986224

11986225

11986226

11986231

11986232

11986233

11986234

11986235

11986236

11986241

11986242

11986243

11986244

11986245

11986246

11986251

11986252

11986253

11986254

11986255

11986256

11986261

11986262

11986263

11986264

11986265

11986266

)

)

(

(

1205981

1205982

1205983

1205984

1205985

1205986

)

)

(35)



(1) the 120590119894119895and 120598


(2) the norms of 120590119894119895 120598119896119897 and 119862




01 02 03 04 05 06 07 08

2

4

6

8

10

12

14

16

(a)

01

02

03

04

05

06

07

08

09

01 02 03 04 05 06 07 08

(b)


01 02 03 04 05 06 07 08

05

1

15

2

25

3

(a)

01 02 03 04 05 06 07 08

05

1

15

2

25

3

(b)




119901 = 119894120575119894119895+ (1 minus 120575

119894119895) (9 minus 119894 minus 119895)

119902 = 119896120575119896119897+ (1 minus 120575

119896119897) (9 minus 119896 minus 119897)

(36)

In this way

120590119901= (120575

119894119895+ radic2 (1 minus 120575

119894119895)) 120598

119894119895

120598119902= (120575

119896119897+ radic2 (1 minus 120575

119896119897)) 120598

119896119897

119862119901119902

= (120575119894119895+ radic2 (1 minus 120575

119894119895)) (120575

119896119897+ radic2 (1 minus 120575

119896119897)) 119862

119894119895119896119897

(37)




120590kelvin

= 120590viogt

120585 120590voigt

=120590kelvin

120585

120598kelvin

= 120598voigt

120585 120598voigt

=120598kelvin

120585

(38)


119862kelvin

=((

(







))

)

119862voigt

(39)


(

(

12059011

12059022

12059033

radic212059023

radic212059013

radic212059012

)

)

= (

1198621111 1198621122 1198621133radic21198621123

radic21198621113radic21198621112

1198622211 1198622222 1198622333radic21198622223

radic21198622213radic21198622212

1198623311 1198623322 1198623333radic21198623323

radic21198623313radic21198623312

radic21198622311radic21198622322

radic21198622333 21198622323 21198622313 21198622312

radic21198621311radic21198621322

radic21198621333 21198621323 21198621313 21198621312

radic21198621211radic21198621222

radic21198621233 21198621223 21198621213 21198621212

)

lowast(

(

12059811

12059822

12059833

radic212059823

radic212059813

radic212059812

)

)

(40)


= C (41)



(

(

12059011

12059022

12059033

radic212059023

radic212059013

radic212059012

)

)

=((

(

11986211

11986212

11986213

radic211986214

radic211986215

radic211986216

11986212

11986222

11986223

radic211986224

radic211986225

radic211986226

11986213

11986223

11986233

radic211986234

radic211986235

radic211986236

radic211986214

radic211986224

radic211986234

211986244

211986245

211986246

radic211986215

radic211986225

radic211986235

211986245

211986255

211986256

radic211986216

radic211986226

radic211986236

211986246

211986256

211986266

))

)

lowast (

(

12059811

12059822

12059833

radic212059823

radic212059813

radic212059812

)

)

(42)


(

(

1

2

3

4

5

6

)

)

=((

(

11986211

11986212

11986213

11986214

11986215

11986216

11986212

11986222

11986223

11986224

11986225

11986226

11986213

11986223

11986233

11986234

11986235

11986236

11986214

11986224

11986234

11986244

11986245

11986246

11986215

11986225

11986235

11986245

11986255

11986256

11986216

11986226

11986236

11986246

11986256

11986266

))

)

lowast (

(

1205981

1205982

1205983

1205984

1205985

1205986

)

)

(43)



































01 02 03 04 05 06 07 08

05

1

15

2

25

3

(a)

01 02 03 04 05 06 07 08

05

1

15

2

25

3

(b)


01 02 03 04 05 06 07 08

2

4

6

8

10

12

14

16





ΛI) N = 0 (44)


119894 119894 = 1 2 6 and

















NNA119896

equiv1

119872

119872

sum

119884=1

N119884119896 (45)



NAVG119896

equiv JNANNA119896

(46)


119896=1NNA119896

otimes NNA119896

| that is

JNA equiv (

6

sum

119896=1

NNA119896

otimes NNA119896

)

minus12

(47)


119896 that is

ΛAVG119902

equiv1

119872

119872

sum

119884=1

6

sum

119896=1

Λ119884

119896(N119884

119896sdot NAVG

119902)2

(48)


gt 0 for all 119896 = 1 2 6

and ΛAVG119896


NA119896


ΛNA119896

equiv1

119872

119872

sum

119884=1

Λ119884

119896 (49)


CAVG=

6

sum

119896=1

ΛAVG119896

NAVG119896

otimes NAVG119896

(50)





[[[[[[[

[

00367834559700000

00453681054800000

0998185540700000

00

00

00

]]]]]]]

]

[[[[[[[

[

0983745905000000

minus0170879918750000

minus00277901141300000

00

00

00

]]]]]]]

]

[[[[[[[

[

minus0169284222800000

minus0982986098000000

00509905132200000

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(51)


[[[[[[[

[

003679134179

004537783172

09983995367

00

00

00

]]]]]]]

]

[[[[[[[

[

09859858256

minus01712690003

minus002785339027

00

00

00

]]]]]]]

]

[[[[[[[

[

minus01697052873

minus09854311019

005111734310

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(52)




[[[

[

334725276837330 0000112116752981933 000198542359441849 00 00 00

0000112116752981933 334773746006714 minus0000991802322788501 00 00 00

000198542359441849 minus0000991802322788501 334013593769726 00 00 00

00 00 00 333950000000000 00 00

00 00 00 00 333950000000000 00

00 00 00 00 00 333950000000000

]]]

]

(53)




[[[[[[[

[

004028666644

006399313350

09971368323

00

00

00

]]]]]]]

]

[[[[[[[

[

09048796003

minus04255671399

minus0009247686096

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04237568827

minus09026613361

007505076253

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(54)


[[[[[[[

[

004028666644

006399313350

09971368323

00

00

00

]]]]]]]

]

[[[[[[[

[

09048795994

minus04255671395

minus0009247686087

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04237568827

minus09026613361

007505076253

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(55)



[[[

[

1458799998 000000001785571215 minus000000001009489597 00 00 00000000001785571215 1458799997 0000000002407020197 00 00 00minus000000001009489597 0000000002407020197 1458799997 00 00 00

00 00 00 1458799998 00 0000 00 00 00 1458799998 0000 00 00 00 00 1458799998

]]]

]

(56)




[[[[[

[

005368516527

007220768570

09959437502

00

00

00

]]]]]

]

[[[[[

[

09266208315

minus03753089731

minus002273783078

00

00

00

]]]]]

]

[[[[[

[

minus03721447810

minus09240829102

008705767064

00

00

00

]]]]]

]

[[[[[

[

00

00

00

00

00

10

]]]]]

]

[[[[[

[

00

00

00

10

00

00

]]]]]

]

[[[[[

[

00

00

00

00

10

00

]]]]]

]

(57)


[[[[[[[[[

[

005368516527

007220768570

09959437502

00

00

00

]]]]]]]]]

]

[[[[[[[[[

[

09266208315

minus03753089731

minus002273783078

00

00

00

]]]]]]]]]

]

[[[[[[[[[

[

minus03721447806

minus09240829093

008705767055

00

00

00

]]]]]]]]]

]

[[[[[[[[[

[

00

00

00

10

00

00

]]]]]]]]]

]

[[[[[[[[[

[

00

00

00

00

10

00

]]]]]]]]]

]

[[[[[[[[[

[

00

00

00

00

00

10

]]]]]]]]]

]

(58)




[[[

[

1615299998 0000000005508173090 minus0000000008722620038 00 00 00

0000000005508173090 1615299996 minus0000000004345156817 00 00 00

minus0000000008722620038 minus0000000004345156817 1615300000 00 00 00

00 00 00 1615299998 00 00

00 00 00 00 1615299998 00

00 00 00 00 00 1615299998

]]]

]

(59)




[[[[[[[

[

minus00598757013800000

minus00768229223150000

minus0995231841750000

00

00

00

]]]]]]]

]

[[[[[[[

[

minus0869358919900000

0493939153600000

00141567750950000

00

00

00

]]]]]]]

]

[[[[[[[

[

minus0490490276500000

minus0866078136900000

00963811082600000

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(60)


[[[[[[[[[[[

[

minus005987730132

minus007682497510

minus09952584354

00

00

00

]]]]]]]]]]]

]

[[[[[[[[[[[

[

minus08693926232

04939583026

001415732393

00

00

00

]]]]]]]]]]]

]

[[[[[[[[[[[

[

00

00

00

10

00

00

]]]]]]]]]]]

]

[[[[[[[[[[[

[

00

00

00

00

10

00

]]]]]]]]]]]

]

[[[[[[[[[[[

[

00

00

00

00

00

10

]]]]]]]]]]]

]

(61)



[[[

[


00 00 00 1819400002 1819400002 0000 00 00 00 00 0000 00 00 00 00 1819400002

]]]

]

(62)




[[[[[[[

[

004967528691

008463937788

09951726186

00

00

00

]]]]]]]

]

[[[[[[[

[

09161715971

minus04006166011

minus001165943224

00

00

00

]]]]]]]

]

[[[[[

[

minus03976958243

minus09123280736

009744493938

00

00

00

]]]]]

]

[[[[[

[

00

00

00

10

00

00

]]]]]

]

[[[[[

[

00

00

00

00

10

00

]]]]]

]

[[[[[

[

00

00

00

00

00

10

]]]]]

]

(63)


[[[[[[[

[

004967528696

008463937796

09951726196

00

00

00

]]]]]]]

]

[[[[[[[

[

09161715980

minus04006166015

minus001165943225

00

00

00

]]]]]]]

]


[[[[[[[

[

minus03976958247

minus09123280745

009744493948

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(64)



[[[

[

1922399998 minus000000001176508811 minus000000001537920006 00 00 00

minus000000001176508811 1922400000 minus0000000007631928036 00 00 00

minus000000001537920006 minus0000000007631928036 1922400000 00 00 00

00 00 00 1922399998 1922399998 00

00 00 00 00 00 00

00 00 00 00 00 1922399998

]]]

]

(65)




[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

[[[[[[[

[

minus01387533797

minus02031928413

minus09692575344

00

00

00

]]]]]]]

]

[[[[[[[

[

minus08476774112

05304148448

001015375725

00

00

00

]]]]]]]

]

[[[[[[[

[

minus05120454135

minus08230265872

02458388325

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

(66)


[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

[[[[[[[

[

minus01387533798

minus02031928415

minus09692575354

00

00

00

]]]]]]]

]

[[[[[[[

[

minus08476774112

05304148448

001015375725

00

00

00

]]]]]]]

]

[[[[[[[

[

minus05120454135

minus08230265872

02458388325

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

(67)



[[[

[

2074599999 0000000004356660361 000000003402343994 00 00 00

0000000004356660361 2074600004 000000002232269567 00 00 00

000000003402343994 000000002232269567 2074600000 00 00 00

00 00 00 2074600000 2074600000 00

00 00 00 00 00 00

00 00 00 00 00 2074600000

]]]

]

(68)




[[[[[[[

[

008621257414

01243595697

09884847438

00

00

00

]]]]]]]

]

[[[[[[[

[

08755641188

minus04828494241

minus001561753067

00

00

00

]]]]]]]

]

[[[[[

[

minus04753471023

minus08668282006

01505124700

00

00

00

]]]]]

]

[[[[[

[

00

00

00

10

00

00

]]]]]

]

[[[[[

[

00

00

00

00

10

00

]]]]]

]

[[[[[

[

00

00

00

00

00

10

]]]]]

]

(69)


[[[[[

[

008621257423

01243595698

09884847448

00

00

00

]]]]]

]

[[[[[

[

08755641188

minus04828494241

minus001561753067

00

00

00

]]]]]

]


[[[[[[[

[

minus04753471018

minus08668281997

01505124698

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(70)



[[[

[

1890099999 000000001209663993 minus000000002532733996 00 00 00000000001209663993 1890099991 0000000001701090138 00 00 00minus000000002532733996 0000000001701090138 1890100001 00 00 00

00 00 00 1890099997 1890099997 0000 00 00 00 00 0000 00 00 00 00 1890099997

]]]

]

(71)




[[[[[[[

[

03961520086

06338768023

06642768888

00

00

00

]]]]]]]

]

[[[[[[[

[

09160614623

minus02236797319

minus03328645006

00

00

00

]]]]]]]

]

[[[[[[[

[

006240981414

minus07403833993

06692812840

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(72)


[[[[[[[

[

03961520086

06338768023

06642768888

00

00

00

]]]]]]]

]

[[[[[[[

[

09160614614

minus02236797317

minus03328645003

00

00

00

]]]]]]]

]

[[[[[[[

[

006240981426

minus07403834008

06692812853

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(73)



[[[

[


00 00 00 8772299998 8772299998 0000 00 00 00 00 0000 00 00 00 00 8772299998

]]]

]

(74)




[[[[[[[

[

minus006591789198

minus008975775227

minus09937798435

00

00

00

]]]]]]]

]

[[[[[[[

[

minus08289381135

05593224991

0004466103673

00

00

00

]]]]]]]

]

[[[[[

[

minus05554425577

minus08240763851

01112729817

00

00

00

]]]]]

]

[[[[[

[

00

00

00

10

00

00

]]]]]

]

[[[[[

[

00

00

00

00

10

00

]]]]]

]

[[[[[

[

00

00

00

00

00

10

]]]]]

]


[[[[[[[

[

minus01387533798

minus02031928415

minus09692575354

00

00

00

]]]]]]]

]

[[[[[[[

[

minus08476774112

05304148448

001015375725

00

00

00

]]]]]]]

]


[[[[

[

minus05120454135

minus08230265872

02458388325

00

00

00

]]]]

]

[[[[

[

00

00

00

10

00

00

]]]]

]

[[[[

[

00

00

00

00

10

00

]]]]

]

[[[[

[

00

00

00

00

00

10

]]]]

]

(76)



[[[

[

2228057494 0000000004678921127 000000003654014285 00 00 000000000004678921127 2228057499 000000002397389829 00 00 00000000003654014285 000000002397389829 2228057495 00 00 00

00 00 00 2228057495 2228057495 0000 00 00 00 00 0000 00 00 00 00 2228057495

]]]

]

(77)




[[[[[[[

[

006975010637

01071019008

09917984199

00

00

00

]]]]]]]

]

[[[[[[[

[

09079000793

minus04187725641

minus001862756541

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04133429180

minus09017531389

01264472547

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(78)


[[[[[[[

[

006975010637

01071019008

09917984199

00

00

00

]]]]]]]

]

[[[[[[[

[

09079000793

minus04187725641

minus001862756541

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04133429184

minus09017531398

01264472548

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(79)



[[[

[


00 00 00 2598699998 2598699998 0000 00 00 00 00 0000 00 00 00 00 2598699998

]]]

]

(80)




[[[[[

[

minus00192522847250000

minus00192842313600000

000386151499999998

00

00

00

]]]]]

]

[[[[[

[

000961799224999998

00165029120500000

000436480214750000

00

00

00

]]]]]

]

[[[[[[

[

minus0494444050150000

minus0856906622200000

0142104790200000

00

00

00

]]]]]]

]

[[[[[[

[

00

00

00

10

00

00

]]]]]]

]

[[[[[[

[

00

00

00

00

10

00

]]]]]]

]

[[[[[[

[

00

00

00

00

00

10

]]]]]]

]

(81)



[[[[[[[[[[

[

minus2541745843

minus2545963537

5098090872

00

00

00

]]]]]]]]]]

]

[[[[[[[[[[

[

2505315863

4298714977

1136953303

00

00

00

]]]]]]]]]]

]

[[[[[[[

[

minus04949599718

minus08578007511

01422530677

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(82)



[[[

[

315679169478799 427324383657779 384557503470365 00 00 00427324383657779 618700481877479 889153535276175 00 00 00384557503470365 889153535276175 384768762708609 00 00 00

00 00 00 247795001400000 00 0000 00 00 00 247795001400000 0000 00 00 00 00 247795001400000

]]]

]

(83)




[[[[[[[

[

01135176976

01764907831

09777344911

00

00

00

]]]]]]]

]

[[[[[[[

[

08848520771

minus04654963442

minus001870707734

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04518302069

minus08672739791

02090103088

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(84)


[[[[[[[

[

01135176977

01764907833

09777344921

00

00

00

]]]]]]]

]

[[[[[[[

[

08848520771

minus04654963442

minus001870707734

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04518302069

minus08672739791

02090103088

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(85)



[[[

[

2663700003 000000004768023095 000000003169803038 00 00 00000000004768023095 2663699992 0000000008310743498 00 00 00000000003169803038 0000000008310743498 2663700001 00 00 00

00 00 00 2663699998 2663699998 0000 00 00 00 00 0000 00 00 00 00 2663699998

]]]

]

(86)





6 Conclusions









Disclosure




Acknowledgments



References




















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










11 12 1312 22 2313 23 33

11 12 1312 22 2313 23 33

11 12 1312 22 2313 23 33

11 12 1312 22 2313 23 33

11 12 1312 22 2313 23 33

11 12 1312 22 2313 23 33

11 12 1312 22 2313 23 33

11 12 1312 22 2313 23 33

11 12 1312 22 2313 23 33

11 12 1312 22 2313 23 33

11 12 1312 22 2313 23 33

11 12 13

12 22 23

13 23 33

120590ij

Cijkl

120598ij

lowast=


2313= 119862



1

2

3

12

34

56

Column Row

12

34

56


2468101214

12

34

56

Column Row

12

34

56


246810121416

12

34

56

Column Row

12

34

56



= 119860120590(119894)120590(119895)120590(119895)120590(119897)




(18)



119860119904

119894119895119896119897=

1

3(119860

119894119895119896119897+ 119860

119894119896119895119897+ 119860

119894119897119896119895)

119860119886

119894119895119896119897=

1

3(119860

119894119895119896119897minus 119860

119894119896119895119897minus 119860

119894119897119895119896)

(19)




Stress

12059011

1205901

1

12059022

1205902

2

12059033

1205903

3

12059023

1205904

4

12059013

1205905

5

12059012

1205905

6

Strain

12059811

1205981

1205981

12059822

1205982

1205982

12059833

1205983

1205983

12059823

1205984

1205984

12059813

12059815

1205985

12059812

1205986

1205986

Elasticity

1198621111

11986211

11986211

1198622222

11986222

11986222

1198623333

11986233

11986233

1198621122

11986212

11986212

1198621133

11986213

11986213

1198622233

11986223

11986223

1198622323

11986244

1

211986244

1198621313

11986255

1

211986255

1198621212

11986266

1

211986266

1198621323

11986254

1

211986254

1198621312

11986256

1

211986256

1198621223

11986264

1

411986264

1198622311

11986241

1

radic211986241

1198621311

11986251

1

radic211986251

1198621211

11986261

1

radic211986261

1198622322

11986242

1

radic211986242

1198621322

11986252

1

radic211986252

1198621222

11986262

1

radic211986262

1198622333

11986243

1

radic211986243

1198621333

11986253

1

radic211986253

1198621233

11986263

1

radic211986263

Table 4 Continued


Compliance

1198701111

11987811

11987811

1198702222

11987822

11987822

1198703333

11987833

11987833

1198701122

11987812

11987812

1198701133

11987813

11987813

1198702233

11987823

11987823

1198702323

1

411987844

1

211987844

1198701313

1

411987855

1

211987855

1198701212

1

411987866

1

211987866

1198701323

1

411987854

1

211987854

1198701312

1

411987856

1

211987856

1198701223

1

411987864

1

211987864

1198702311

1

211987841

1

radic211987841

1198701311

1

211987851

1

radic211987851

1198701211

1

211987861

1

radic211987861

1198702322

1

211987842

1

radic211987842

1198701322

1

211987852

1

radic211987852

1198701222

1

211987862

1

radic211987862

1198702333

1

211987843

1

radic211987843

1198701333

1

211987853

1

radic211987853

1198701233

1

211987863

1

radic211987863

24681012141618

12

34

56

Column Row

12

34

56




S no Species 120588 11986211

11986222

11986233

11986212

11986213

11986223

11986244

11986255

11986266



S no Species 120588 11986211

11986222

11986233

11986212

11986213

11986223

11986244

11986255

11986266














986 7912 698311986212

2147 3609 281611986213

2773 3186 284511986214

minus0162 11 8011986215

minus293 54 minus0711986216

minus0821 minus06 11411986222

935 8529 896711986223

2346 3281 322711986224

minus1204 09 minus162311986225


0445 minus42 16011986233

5952 14730 916011986234


minus1791 169 minus7211986236

054 06 2411986244

561 3587 348911986245

0798 05 10511986246

minus2159 51 minus9811986255

6032 3448 242611986256



24681012141618

12

34

56

Column Row

12

34

56



5

10

15

20

12

34

56

Column Row

12

34

56



119890120595(119894119895)

= 119890119894otimes 119890

119895 (20)




1 Quipo

[[[[[[[[[

[

1076866026

004067345638

02534605191

02260000000

01180000000

007800000000

]]]]]]]]]

]

[[[[[[[[[

[

003276733390 09918780499 minus01228992897 00 00 00

003131140851 minus01239237163 minus09917976143 00 00 00

09989724208 minus002865041271 003511774899 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

2 Quipo

[[[[[[[[[

[

3461963876

01399776173

04300584948

04300000000

02800000000

01440000000

]]]]]]]]]

]

[[[[[[[[[

[

004079957804 09756137601 minus02156691559 00 00 00

005942480245 minus02178361212 minus09741745817 00 00 00

09973986606 minus002692981555 006686327745 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

3 White

[[[[[[[[[

[

1009116301

03556701722

1333166815

1442000000

1344000000

002200000000

]]]]]]]]]

]

[[[[[[[[[

[

004028666644 09048796003 minus04237568827 00 00 00

006399313350 minus04255671399 minus09026613361 00 00 00

09971368323 minus0009247686096 007505076253 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

4 Khaya

[[[[[[[[[

[

1080102614

04606834511

1475290392

1800000000

1196000000

04200000000

]]]]]]]]]

]

[[[[[[[[[

[

005368516527 09266208315 minus03721447810 00 00 00

007220768570 minus03753089731 minus09240829102 00 00 00

09959437502 minus002273783078 008705767064 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

5 Mahogany

[[[[[[[[[

[

1210253321

06098853915

1810581412

1960000000

1498000000

06380000000

]]]]]]]]]

]

[[[[[[[[[

[


minus007817061387 04985803653 minus08633116316 00 00 00

minus09948285648 001731853005 01000809391 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

6 Mahogany

[[[[[[[[[

[

1310854783

03895295651

1814922632

1218000000

09380000000

03000000000

]]]]]]]]]

]

[[[[[[[[[

[


minus007547523076 04892979419 minus08688446422 00 00 00

minus09956351187 001099502014 009268127742 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

7 S Germ

[[[[[[[[[

[

1234053410

05212570563

1922208822

2318000000

1582000000

05400000000

]]]]]]]]]

]

[[[[[[[[[

[

004967528691 09161715971 minus03976958243 00 00 00

008463937788 minus04006166011 minus09123280736 00 00 00

09951726186 minus001165943224 009744493938 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

8 Maple

[[[[[[[[[

[

1205449768

06868279555

2766674361

2460000000

2194000000

05840000000

]]]]]]]]]

]

[[[[[[[[[

[


minus02031928413 05304148448 minus08230265872 00 00 00

minus09692575344 001015375725 02458388325 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]




9 Walnut

[[[[[[[[[

[

1267869546

05204134969

1919891015

1922000000

1400000000

04600000000

]]]]]]]]]

]

[[[[[[[[[

[

008621257414 08755641188 minus04753471023 00 00 00

01243595697 minus04828494241 minus08668282006 00 00 00

09884847438 minus001561753067 01505124700 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

10 Birch

[[[[[[[[[

[

3168908680

04747157903

05946755301

2346000000

1816000000

03720000000

]]]]]]]]]

]

[[[[[[[[[

[

03961520086 09160614623 006240981414 00 00 00

06338768023 minus02236797319 minus07403833993 00 00 00

06642768888 minus03328645006 06692812840 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

11 Y Birch

[[[[[[[[[

[

1545414040

05549651499

2059894445

2120000000

1942000000

04800000000

]]]]]]]]]

]

[[[[[[[[[

[


minus008975775227 05593224991 minus08240763851 00 00 00

minus09937798435 0004466103673 01112729817 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

12 Oak

[[[[[[[[[

[

1718666431

08648974218

3239438239

2380000000

1532000000

07840000000

]]]]]]]]]

]

[[[[[[[[[

[

006975010637 09079000793 minus04133429180 00 00 00

01071019008 minus04187725641 minus09017531389 00 00 00

09917984199 minus001862756541 01264472547 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

13 Ash

[[[[[[[[[

[

1715567967

06696042476

2409716064

2684000000

1784000000

05400000000

]]]]]]]]]

]

[[[[[[[[[

[

006190313535 08746549514 minus04807772027 00 00 00

009781749768 minus04846986500 minus08691944279 00 00 00

09932772717 minus0006777439445 01155609239 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

14 Ash

[[[[[[[[[

[

1742363268

07844815431

2669885744

1720000000

1218000000

05000000000

]]]]]]]]]

]

[[[[[[[[[

[


minus01363859604 05177044741 minus08446188165 00 00 00

minus09855542417 001550704374 01686486565 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

15 Beech

[[[[[[[[[

[

1599002755

09558575345

3451114905

3216000000

2112000000

09120000000

]]]]]]]]]

]

[[[[[[[[[

[

01135176976 08848520771 minus04518302069 00 00 00

01764907831 minus04654963442 minus08672739791 00 00 00

09777344911 minus001870707734 02090103088 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

Thus 119890120572otimes 119890

1205731le120572120573le119899




119860119894119895119896119897

119890119894otimes 119890

119895otimes 119890

119896otimes 119890

119897 or as an 119899


120572120573119890120572otimes 119890

120573 where 119860

120595(119894119895)120595119896119897= 119860

119894119895119896119897

1 le 119894 119895 119896 119897 le 119899


tr (A119879B) = ⟨AB⟩ = tr (119860119879119861119879) A =1003817100381710038171003817100381711986010038171003817100381710038171003817 (21)






1 Blasa

[[[[[[[[[

[

6385324208

009845837744

03832174064

06240000000

04060000000

006600000000

]]]]]]]]]

]

[[[[[[[[[

[

001488818113 09510211778 minus03087669978 00 00 00

002575997614 minus03090635474 minus09506924583 00 00 00

09995572851 minus0006200252135 002909967665 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

2 Spruce

[[[[[[[[[

[

1198583568

04516442734

1114520065

1498000000

1442000000

007800000000

]]]]]]]]]

]

[[[[[[[[[

[


minus004689865645 04044467539 minus09133582749 00 00 00

minus09983543167 001123114838 005623628701 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

3 Spruce

[[[[[[[[[

[

1410927768

04185382987

1227183961

1442000000

10

006400000000

]]]]]]]]]

]

[[[[[[[[[

[


minus005361649666 04423303564 minus08952480817 00 00 00

minus09978936411 0009052295016 006423657043 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

4 Spruce

[[[[[[[[[

[

1630529953

03544288592

08442715844

1234000000

1520000000

007200000000

]]]]]]]]]

]

[[[[[[[[[

[


minus002869707087 04091564350 minus09120128765 00 00 00

minus09993764538 0007032901380 003460119282 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

5 Spruce

[[[[[[[[[

[

1725845208

05092652753

1171282625

1250000000

1706000000

007000000000

]]]]]]]]]

]

[[[[[[[[[

[


minus003428302033 05858146064 minus08097196604 00 00 00

minus09988364173 0007413313465 004765347716 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

6 Douglas Fir

[[[[[[[[[

[

1613459769

06233733771

1439028943

1767000000

1766000000

01760000000

]]]]]]]]]

]

[[[[[[[[[

[


minus003697194691 05922678885 minus08048924305 00 00 00

minus09986533619 0007435777607 005134366998 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

7 Douglas Fir

[[[[[[[[[

[

1710150156

06986859431

2204812526

2348000000

1816000000

01600000000

]]]]]]]]]

]

[[[[[[[[[

[


minus006364825251 05704773676 minus08188433771 00 00 00

minus09967231590 0004703484281 008075160717 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

8 Pine

[[[[[[[[[

[

1699539133

04940019684

1565606760

3484000000

1344000000

01320000000

]]]]]]]]]

]

[[[[[[[[[

[


minus005585205149 04413516031 minus08955943895 00 00 00

minus09978476167 0006195562437 006528207193 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]


Table11Th

eeigenvalues

andeigenvectorsfor3

specim

enso

fcancello

usbo

necalculated

usingMAPL

EΛ[119894]sta

ndforthe

eigenvalueso

f119894th

specim

enof

cancellous

boneand

119881[119894]sta

ndfor

thec

orrespon

ding

eigenvectors

Cancellous

bone

EigenvaluesΛ

[119894]

Eigenvectors

119881[119894]

Specim

en1

[ [ [ [ [ [ [ [

1334869880

minus02071167473

8644210768

6837676373

3768806080

4123724855

] ] ] ] ] ] ] ]

[ [ [ [ [ [ [ [

03316115964

08831238953

02356441950

02242668491

005354786914

003787816350

06475092156

minus01099136965

01556396475

minus06940063479

minus01278953723

02154647417

04800906963

minus02832250652

007648827487

02494278793

06410471445

minus04585742261

minus02737532692

minus006354940263

04518342532

minus006214920576

05836952694

06101669758

minus03706582070

03314035780

minus01276606259

minus06337030234

03639432801

minus04499474345

01671829224

01180163925

minus08330343502

002008124471

03109325968

04087706408

] ] ] ] ] ] ] ]

Specim

en2

[ [ [ [ [ [ [ [

1754634171

7773848354

1550688791

2301790452

3445174379

3589156342

] ] ] ] ] ] ] ]

[ [ [ [ [ [ [ [

minus03057290719

minus03005337771

minus09030341401

001584500766

minus002175914011

minus0003741933916

minus04311831740

minus08021570593

04130146809

00001362645283

minus0003854432451

0005396441669

minus08488209390

05154657137

01152135966

minus0004741957786

002229739182

minus0002068264586

00009402467441

minus0005358584867

0003987518000

minus003989381331

003796849613

minus09984595017

minus001058157817

002100470582

002092543658

0006230946613

minus09987520802

minus003826770492

00009823402498

0006977574167

001484146906

09990475909

0008196718814

minus003958286493

] ] ] ] ] ] ] ]

Specim

en3

[ [ [ [ [ [ [ [

1418542670

5838388363

9959058231

3433818493

3024968125

1757492470

] ] ] ] ] ] ] ]

[ [ [ [ [ [ [ [

03244960223

minus0004831643783

minus08451713783

04062092722

01164524794

minus004239310198

06461152835

minus07125334236

02668138654

004131783689

minus002431040234

003665187401

06621315621

07009969173

02633438279

002836289512

minus0001711614840

0005269111592

minus01962401199

0009159859564

03740839141

08850746598

01904577552

004284654369

minus0008597323304

minus0002526763452

001897677365

01738799329

minus06977939665

minus06945566457

001533190599

minus002803230310

006961922985

minus01374446216

06801869735

minus07159517722

] ] ] ] ] ] ] ]


24681012141618

12

34

56

Column Row

12

34

56


12345678

12

34

56

Column Row

12

34

56




120590119894119895= 119862


where 119862119894119895119896119897


120590 = 120590119894119895119890119894otimes 119890

119895 120598 = 120598

119896119897119890119896otimes 119890

119897

C = 119862119894119895119896119897

119890119894otimes 119890

119896otimes 119890

119896otimes 119890

119897

(23)

5

10

15

20

12

34

56

Column Row

12

34

56


5

10

15

20

25

12

34

56

Column Row

12

34

56



E1equiv 119890

1otimes 119890

1

E2equiv 119890

2otimes 119890

2

E3equiv 119890

3otimes 119890

3

(24)


E4equiv

1


2+ 119890

2otimes 119890

1)

E5equiv

1


3+ 119890

3otimes 119890

2)

E6equiv

1


1+ 119890

1otimes 119890

3)

(25)



E7equiv

1


2minus 119890

2otimes 119890

1)

E8equiv

1


3minus 119890

3otimes 119890

2)

E9equiv

1


1minus 119890

1otimes 119890

3)

(26)


120590 = 119878119860E

119860

120598 = 119864119860E

119860

C = 119862119860119861

119890119860otimes 119890

119861

(27)




120590119894119895= 119862

119894119895119896119897120598119896119897

lArrrArr 119878119860= 119862

119860119861E119861

119862119894119895119896119897

120598119896119897

= 120582120598119894119895lArrrArr 119862

119860119861E119861= 120582119864

119860

(28)


((((((((

(

1198781

1198782

1198783

1198784

1198785

1198786

1198787

1198788

1198789

))))))))

)

=

((((((((

(

12059011

12059022

12059033

120590(12)

120590(23)

120590(31)

120590[12]

120590[23]

120590[31]

))))))))

)

((((((((

(

1198641

1198642

1198643

1198644

1198645

1198646

1198647

1198648

1198649

))))))))

)

=

((((((((

(

12059811

12059822

12059833

120598(12)

120598(23)

120598(31)

120598[12]

120598[23]

120598[31]

))))))))

)

(29)


120579(119894119895)

equiv1

radic2(120579119894119895+ 120579

119895119894) 120579

[119894119895]equiv

1

radic2(120579119894119895minus 120579

119895119894) (30)


tensor C are

((((((((

(

11986211

11986212

11986213

11986214

11986215

11986216

11986217

11986218

11986219

11986221

11986222

11986223

11986224

11986225

11986226

11986227

11986228

11986229

11986231

11986232

11986233

11986234

11986235

11986236

11986237

11986238

11986239

11986241

11986242

11986243

11986244

11986245

11986246

11986247

11986248

11986249

11986251

11986252

11986253

11986254

11986255

11986256

11986257

11986258

11986259

11986261

11986262

11986263

11986264

11986265

11986266

11986267

11986268

11986269

11986271

11986272

11986273

11986274

11986275

11986276

11986277

11986278

11986279

11986281

11986282

11986283

11986284

11986285

11986286

11986287

11986288

11986289

11986291

11986292

11986293

11986294

11986295

11986296

11986297

11986298

11986299

))))))))

)

=

1198621111

1198621122

1198621133

1198622211

1198622222

1198623333

1198623311

1198623322

1198623333

11986211(12)

11986211(23)

11986211(31)

11986222(12)

11986222(23)

11986222(31)

11986233(12)

11986233(23)

11986233(31)

11986211[12]

11986211[23]

11986211[31]

11986222[12]

11986222[23]

11986222[31]

11986233[12]

11986233[23]

11986233[31]

119862(12)11

119862(12)22

119862(12)33

119862(23)11

119862(23)22

119862(23)33

119862(31)11

119862(31)22

119862(31)33

119862(1212)

119862(1223)

119862(1231)

119862(2312)

119862(2323)

119862(2331)

119862(3112)

119862(3123)

119862(3131)

119862(12)[12]

119862(12)[23]

119862(12)[31]

119862(23)[12]

119862(23)[23]

119862(23)[31]

119862(31)[12]

119862(31)[23]

119862(31)[31]

119862[12]11

119862[12]22

119862[12]33

119862[23]11

119862[23]22

119862[123]33

119862[21]11

119862[31]22

119862[31]33

119862[12](12)

119862[12](23)

119862[12](21)

119862[23](12)

119862[23](23)

119862[23](21)

119862[31](12)

119862[31](23)

119862[31](21)

119862[1212]

119862[1223]

119862[1231]

119862[2312]

119862[2323]

119862[2331]

119862[3112]

119862[3123]

119862[3131]

(31)

where

119862119894119895(119896119897)

equiv1

radic2(119862

119894119895119896119897+ 119862

119894119895119897119896) 119862

119894119895[119896119897]equiv

1

radic2(119862

119894119895119896119897minus 119862

119894119895119897119896)

119862(119894119895)119896119897

equiv1

radic2(119862

119894119895119896119897+ 119862

119895119894119896119897) 119862

[119894119895]119896119897equiv

1

radic2(119862

119894119895119896119897minus 119862

119895119894119896119897)

119862(119894119895119896119897)

equiv1

2(119862

119894119895119896119897+ 119862

119894119895119897119896+ 119862

119895119894119896119897+ 119862

119895119894119897119896)

119862(119894119895)[119896119897]

equiv1

2(119862

119894119895119896119897minus 119862

119894119895119897119896+ 119862

119895119894119896119897minus 119862

119895119894119897119896)

119862[119894119895](119896119897)

equiv1

2(119862

119894119895119896119897+ 119862

119894119895119897119896minus 119862

119895119894119896119897minus 119862

119895119894119897119896)

119862[119894119895119896119897]

equiv1

2(119862

119894119895119896119897minus 119862

119895119894119896119897minus 119862

119894119895119897119896+ 119862

119895119894119897119896)

(32)


12

34

56

ColumnRow

12

34

56

600005000040000300002000010000


5

10

15

20

25

12

34

56

Column Row

12

34

56






119895119894and 120598

119896119897= 120598

119897119896 the 119862

119894119895119896119897will be

like 119862119894119895119896119897

= 119862119895119894119896119897

= 119862119894119895119897119896


119862119894119895119896119897

= 119862119896119897119894119895












2 sdot 119864energy = 120590119894119895120598119894119895= 120590

119901120598119901 (33)


119901 = 119894120575119894119895+ (1 minus 120575

119894119895) (9 minus 119894 minus 119895)

119902 = 119896120575119896119897+ (1 minus 120575

119896119897) (9 minus 119896 minus 119897)

(34)


119901 119862

119894119895119896119897= 119862

119901119902 and 120598

119902= (2 minus

120575119896119897)120598119896119897



(

(

1205901

1205902

1205903

1205904

1205905

1205906

)

)

= (

(

11986211

11986212

11986213

11986214

11986215

11986216

11986221

11986222

11986223

11986224

11986225

11986226

11986231

11986232

11986233

11986234

11986235

11986236

11986241

11986242

11986243

11986244

11986245

11986246

11986251

11986252

11986253

11986254

11986255

11986256

11986261

11986262

11986263

11986264

11986265

11986266

)

)

(

(

1205981

1205982

1205983

1205984

1205985

1205986

)

)

(35)



(1) the 120590119894119895and 120598


(2) the norms of 120590119894119895 120598119896119897 and 119862




01 02 03 04 05 06 07 08

2

4

6

8

10

12

14

16

(a)

01

02

03

04

05

06

07

08

09

01 02 03 04 05 06 07 08

(b)


01 02 03 04 05 06 07 08

05

1

15

2

25

3

(a)

01 02 03 04 05 06 07 08

05

1

15

2

25

3

(b)




119901 = 119894120575119894119895+ (1 minus 120575

119894119895) (9 minus 119894 minus 119895)

119902 = 119896120575119896119897+ (1 minus 120575

119896119897) (9 minus 119896 minus 119897)

(36)

In this way

120590119901= (120575

119894119895+ radic2 (1 minus 120575

119894119895)) 120598

119894119895

120598119902= (120575

119896119897+ radic2 (1 minus 120575

119896119897)) 120598

119896119897

119862119901119902

= (120575119894119895+ radic2 (1 minus 120575

119894119895)) (120575

119896119897+ radic2 (1 minus 120575

119896119897)) 119862

119894119895119896119897

(37)




120590kelvin

= 120590viogt

120585 120590voigt

=120590kelvin

120585

120598kelvin

= 120598voigt

120585 120598voigt

=120598kelvin

120585

(38)


119862kelvin

=((

(







))

)

119862voigt

(39)


(

(

12059011

12059022

12059033

radic212059023

radic212059013

radic212059012

)

)

= (

1198621111 1198621122 1198621133radic21198621123

radic21198621113radic21198621112

1198622211 1198622222 1198622333radic21198622223

radic21198622213radic21198622212

1198623311 1198623322 1198623333radic21198623323

radic21198623313radic21198623312

radic21198622311radic21198622322

radic21198622333 21198622323 21198622313 21198622312

radic21198621311radic21198621322

radic21198621333 21198621323 21198621313 21198621312

radic21198621211radic21198621222

radic21198621233 21198621223 21198621213 21198621212

)

lowast(

(

12059811

12059822

12059833

radic212059823

radic212059813

radic212059812

)

)

(40)


= C (41)



(

(

12059011

12059022

12059033

radic212059023

radic212059013

radic212059012

)

)

=((

(

11986211

11986212

11986213

radic211986214

radic211986215

radic211986216

11986212

11986222

11986223

radic211986224

radic211986225

radic211986226

11986213

11986223

11986233

radic211986234

radic211986235

radic211986236

radic211986214

radic211986224

radic211986234

211986244

211986245

211986246

radic211986215

radic211986225

radic211986235

211986245

211986255

211986256

radic211986216

radic211986226

radic211986236

211986246

211986256

211986266

))

)

lowast (

(

12059811

12059822

12059833

radic212059823

radic212059813

radic212059812

)

)

(42)


(

(

1

2

3

4

5

6

)

)

=((

(

11986211

11986212

11986213

11986214

11986215

11986216

11986212

11986222

11986223

11986224

11986225

11986226

11986213

11986223

11986233

11986234

11986235

11986236

11986214

11986224

11986234

11986244

11986245

11986246

11986215

11986225

11986235

11986245

11986255

11986256

11986216

11986226

11986236

11986246

11986256

11986266

))

)

lowast (

(

1205981

1205982

1205983

1205984

1205985

1205986

)

)

(43)



































01 02 03 04 05 06 07 08

05

1

15

2

25

3

(a)

01 02 03 04 05 06 07 08

05

1

15

2

25

3

(b)


01 02 03 04 05 06 07 08

2

4

6

8

10

12

14

16





ΛI) N = 0 (44)


119894 119894 = 1 2 6 and

















NNA119896

equiv1

119872

119872

sum

119884=1

N119884119896 (45)



NAVG119896

equiv JNANNA119896

(46)


119896=1NNA119896

otimes NNA119896

| that is

JNA equiv (

6

sum

119896=1

NNA119896

otimes NNA119896

)

minus12

(47)


119896 that is

ΛAVG119902

equiv1

119872

119872

sum

119884=1

6

sum

119896=1

Λ119884

119896(N119884

119896sdot NAVG

119902)2

(48)


gt 0 for all 119896 = 1 2 6

and ΛAVG119896


NA119896


ΛNA119896

equiv1

119872

119872

sum

119884=1

Λ119884

119896 (49)


CAVG=

6

sum

119896=1

ΛAVG119896

NAVG119896

otimes NAVG119896

(50)





[[[[[[[

[

00367834559700000

00453681054800000

0998185540700000

00

00

00

]]]]]]]

]

[[[[[[[

[

0983745905000000

minus0170879918750000

minus00277901141300000

00

00

00

]]]]]]]

]

[[[[[[[

[

minus0169284222800000

minus0982986098000000

00509905132200000

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(51)


[[[[[[[

[

003679134179

004537783172

09983995367

00

00

00

]]]]]]]

]

[[[[[[[

[

09859858256

minus01712690003

minus002785339027

00

00

00

]]]]]]]

]

[[[[[[[

[

minus01697052873

minus09854311019

005111734310

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(52)




[[[

[

334725276837330 0000112116752981933 000198542359441849 00 00 00

0000112116752981933 334773746006714 minus0000991802322788501 00 00 00

000198542359441849 minus0000991802322788501 334013593769726 00 00 00

00 00 00 333950000000000 00 00

00 00 00 00 333950000000000 00

00 00 00 00 00 333950000000000

]]]

]

(53)




[[[[[[[

[

004028666644

006399313350

09971368323

00

00

00

]]]]]]]

]

[[[[[[[

[

09048796003

minus04255671399

minus0009247686096

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04237568827

minus09026613361

007505076253

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(54)


[[[[[[[

[

004028666644

006399313350

09971368323

00

00

00

]]]]]]]

]

[[[[[[[

[

09048795994

minus04255671395

minus0009247686087

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04237568827

minus09026613361

007505076253

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(55)



[[[

[

1458799998 000000001785571215 minus000000001009489597 00 00 00000000001785571215 1458799997 0000000002407020197 00 00 00minus000000001009489597 0000000002407020197 1458799997 00 00 00

00 00 00 1458799998 00 0000 00 00 00 1458799998 0000 00 00 00 00 1458799998

]]]

]

(56)




[[[[[

[

005368516527

007220768570

09959437502

00

00

00

]]]]]

]

[[[[[

[

09266208315

minus03753089731

minus002273783078

00

00

00

]]]]]

]

[[[[[

[

minus03721447810

minus09240829102

008705767064

00

00

00

]]]]]

]

[[[[[

[

00

00

00

00

00

10

]]]]]

]

[[[[[

[

00

00

00

10

00

00

]]]]]

]

[[[[[

[

00

00

00

00

10

00

]]]]]

]

(57)


[[[[[[[[[

[

005368516527

007220768570

09959437502

00

00

00

]]]]]]]]]

]

[[[[[[[[[

[

09266208315

minus03753089731

minus002273783078

00

00

00

]]]]]]]]]

]

[[[[[[[[[

[

minus03721447806

minus09240829093

008705767055

00

00

00

]]]]]]]]]

]

[[[[[[[[[

[

00

00

00

10

00

00

]]]]]]]]]

]

[[[[[[[[[

[

00

00

00

00

10

00

]]]]]]]]]

]

[[[[[[[[[

[

00

00

00

00

00

10

]]]]]]]]]

]

(58)




[[[

[

1615299998 0000000005508173090 minus0000000008722620038 00 00 00

0000000005508173090 1615299996 minus0000000004345156817 00 00 00

minus0000000008722620038 minus0000000004345156817 1615300000 00 00 00

00 00 00 1615299998 00 00

00 00 00 00 1615299998 00

00 00 00 00 00 1615299998

]]]

]

(59)




[[[[[[[

[

minus00598757013800000

minus00768229223150000

minus0995231841750000

00

00

00

]]]]]]]

]

[[[[[[[

[

minus0869358919900000

0493939153600000

00141567750950000

00

00

00

]]]]]]]

]

[[[[[[[

[

minus0490490276500000

minus0866078136900000

00963811082600000

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(60)


[[[[[[[[[[[

[

minus005987730132

minus007682497510

minus09952584354

00

00

00

]]]]]]]]]]]

]

[[[[[[[[[[[

[

minus08693926232

04939583026

001415732393

00

00

00

]]]]]]]]]]]

]

[[[[[[[[[[[

[

00

00

00

10

00

00

]]]]]]]]]]]

]

[[[[[[[[[[[

[

00

00

00

00

10

00

]]]]]]]]]]]

]

[[[[[[[[[[[

[

00

00

00

00

00

10

]]]]]]]]]]]

]

(61)



[[[

[


00 00 00 1819400002 1819400002 0000 00 00 00 00 0000 00 00 00 00 1819400002

]]]

]

(62)




[[[[[[[

[

004967528691

008463937788

09951726186

00

00

00

]]]]]]]

]

[[[[[[[

[

09161715971

minus04006166011

minus001165943224

00

00

00

]]]]]]]

]

[[[[[

[

minus03976958243

minus09123280736

009744493938

00

00

00

]]]]]

]

[[[[[

[

00

00

00

10

00

00

]]]]]

]

[[[[[

[

00

00

00

00

10

00

]]]]]

]

[[[[[

[

00

00

00

00

00

10

]]]]]

]

(63)


[[[[[[[

[

004967528696

008463937796

09951726196

00

00

00

]]]]]]]

]

[[[[[[[

[

09161715980

minus04006166015

minus001165943225

00

00

00

]]]]]]]

]


[[[[[[[

[

minus03976958247

minus09123280745

009744493948

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(64)



[[[

[

1922399998 minus000000001176508811 minus000000001537920006 00 00 00

minus000000001176508811 1922400000 minus0000000007631928036 00 00 00

minus000000001537920006 minus0000000007631928036 1922400000 00 00 00

00 00 00 1922399998 1922399998 00

00 00 00 00 00 00

00 00 00 00 00 1922399998

]]]

]

(65)




[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

[[[[[[[

[

minus01387533797

minus02031928413

minus09692575344

00

00

00

]]]]]]]

]

[[[[[[[

[

minus08476774112

05304148448

001015375725

00

00

00

]]]]]]]

]

[[[[[[[

[

minus05120454135

minus08230265872

02458388325

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

(66)


[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

[[[[[[[

[

minus01387533798

minus02031928415

minus09692575354

00

00

00

]]]]]]]

]

[[[[[[[

[

minus08476774112

05304148448

001015375725

00

00

00

]]]]]]]

]

[[[[[[[

[

minus05120454135

minus08230265872

02458388325

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

(67)



[[[

[

2074599999 0000000004356660361 000000003402343994 00 00 00

0000000004356660361 2074600004 000000002232269567 00 00 00

000000003402343994 000000002232269567 2074600000 00 00 00

00 00 00 2074600000 2074600000 00

00 00 00 00 00 00

00 00 00 00 00 2074600000

]]]

]

(68)




[[[[[[[

[

008621257414

01243595697

09884847438

00

00

00

]]]]]]]

]

[[[[[[[

[

08755641188

minus04828494241

minus001561753067

00

00

00

]]]]]]]

]

[[[[[

[

minus04753471023

minus08668282006

01505124700

00

00

00

]]]]]

]

[[[[[

[

00

00

00

10

00

00

]]]]]

]

[[[[[

[

00

00

00

00

10

00

]]]]]

]

[[[[[

[

00

00

00

00

00

10

]]]]]

]

(69)


[[[[[

[

008621257423

01243595698

09884847448

00

00

00

]]]]]

]

[[[[[

[

08755641188

minus04828494241

minus001561753067

00

00

00

]]]]]

]


[[[[[[[

[

minus04753471018

minus08668281997

01505124698

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(70)



[[[

[

1890099999 000000001209663993 minus000000002532733996 00 00 00000000001209663993 1890099991 0000000001701090138 00 00 00minus000000002532733996 0000000001701090138 1890100001 00 00 00

00 00 00 1890099997 1890099997 0000 00 00 00 00 0000 00 00 00 00 1890099997

]]]

]

(71)




[[[[[[[

[

03961520086

06338768023

06642768888

00

00

00

]]]]]]]

]

[[[[[[[

[

09160614623

minus02236797319

minus03328645006

00

00

00

]]]]]]]

]

[[[[[[[

[

006240981414

minus07403833993

06692812840

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(72)


[[[[[[[

[

03961520086

06338768023

06642768888

00

00

00

]]]]]]]

]

[[[[[[[

[

09160614614

minus02236797317

minus03328645003

00

00

00

]]]]]]]

]

[[[[[[[

[

006240981426

minus07403834008

06692812853

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(73)



[[[

[


00 00 00 8772299998 8772299998 0000 00 00 00 00 0000 00 00 00 00 8772299998

]]]

]

(74)




[[[[[[[

[

minus006591789198

minus008975775227

minus09937798435

00

00

00

]]]]]]]

]

[[[[[[[

[

minus08289381135

05593224991

0004466103673

00

00

00

]]]]]]]

]

[[[[[

[

minus05554425577

minus08240763851

01112729817

00

00

00

]]]]]

]

[[[[[

[

00

00

00

10

00

00

]]]]]

]

[[[[[

[

00

00

00

00

10

00

]]]]]

]

[[[[[

[

00

00

00

00

00

10

]]]]]

]


[[[[[[[

[

minus01387533798

minus02031928415

minus09692575354

00

00

00

]]]]]]]

]

[[[[[[[

[

minus08476774112

05304148448

001015375725

00

00

00

]]]]]]]

]


[[[[

[

minus05120454135

minus08230265872

02458388325

00

00

00

]]]]

]

[[[[

[

00

00

00

10

00

00

]]]]

]

[[[[

[

00

00

00

00

10

00

]]]]

]

[[[[

[

00

00

00

00

00

10

]]]]

]

(76)



[[[

[

2228057494 0000000004678921127 000000003654014285 00 00 000000000004678921127 2228057499 000000002397389829 00 00 00000000003654014285 000000002397389829 2228057495 00 00 00

00 00 00 2228057495 2228057495 0000 00 00 00 00 0000 00 00 00 00 2228057495

]]]

]

(77)




[[[[[[[

[

006975010637

01071019008

09917984199

00

00

00

]]]]]]]

]

[[[[[[[

[

09079000793

minus04187725641

minus001862756541

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04133429180

minus09017531389

01264472547

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(78)


[[[[[[[

[

006975010637

01071019008

09917984199

00

00

00

]]]]]]]

]

[[[[[[[

[

09079000793

minus04187725641

minus001862756541

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04133429184

minus09017531398

01264472548

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(79)



[[[

[


00 00 00 2598699998 2598699998 0000 00 00 00 00 0000 00 00 00 00 2598699998

]]]

]

(80)




[[[[[

[

minus00192522847250000

minus00192842313600000

000386151499999998

00

00

00

]]]]]

]

[[[[[

[

000961799224999998

00165029120500000

000436480214750000

00

00

00

]]]]]

]

[[[[[[

[

minus0494444050150000

minus0856906622200000

0142104790200000

00

00

00

]]]]]]

]

[[[[[[

[

00

00

00

10

00

00

]]]]]]

]

[[[[[[

[

00

00

00

00

10

00

]]]]]]

]

[[[[[[

[

00

00

00

00

00

10

]]]]]]

]

(81)



[[[[[[[[[[

[

minus2541745843

minus2545963537

5098090872

00

00

00

]]]]]]]]]]

]

[[[[[[[[[[

[

2505315863

4298714977

1136953303

00

00

00

]]]]]]]]]]

]

[[[[[[[

[

minus04949599718

minus08578007511

01422530677

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(82)



[[[

[

315679169478799 427324383657779 384557503470365 00 00 00427324383657779 618700481877479 889153535276175 00 00 00384557503470365 889153535276175 384768762708609 00 00 00

00 00 00 247795001400000 00 0000 00 00 00 247795001400000 0000 00 00 00 00 247795001400000

]]]

]

(83)




[[[[[[[

[

01135176976

01764907831

09777344911

00

00

00

]]]]]]]

]

[[[[[[[

[

08848520771

minus04654963442

minus001870707734

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04518302069

minus08672739791

02090103088

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(84)


[[[[[[[

[

01135176977

01764907833

09777344921

00

00

00

]]]]]]]

]

[[[[[[[

[

08848520771

minus04654963442

minus001870707734

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04518302069

minus08672739791

02090103088

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(85)



[[[

[

2663700003 000000004768023095 000000003169803038 00 00 00000000004768023095 2663699992 0000000008310743498 00 00 00000000003169803038 0000000008310743498 2663700001 00 00 00

00 00 00 2663699998 2663699998 0000 00 00 00 00 0000 00 00 00 00 2663699998

]]]

]

(86)





6 Conclusions









Disclosure




Acknowledgments



References




















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












Stress

12059011

1205901

1

12059022

1205902

2

12059033

1205903

3

12059023

1205904

4

12059013

1205905

5

12059012

1205905

6

Strain

12059811

1205981

1205981

12059822

1205982

1205982

12059833

1205983

1205983

12059823

1205984

1205984

12059813

12059815

1205985

12059812

1205986

1205986

Elasticity

1198621111

11986211

11986211

1198622222

11986222

11986222

1198623333

11986233

11986233

1198621122

11986212

11986212

1198621133

11986213

11986213

1198622233

11986223

11986223

1198622323

11986244

1

211986244

1198621313

11986255

1

211986255

1198621212

11986266

1

211986266

1198621323

11986254

1

211986254

1198621312

11986256

1

211986256

1198621223

11986264

1

411986264

1198622311

11986241

1

radic211986241

1198621311

11986251

1

radic211986251

1198621211

11986261

1

radic211986261

1198622322

11986242

1

radic211986242

1198621322

11986252

1

radic211986252

1198621222

11986262

1

radic211986262

1198622333

11986243

1

radic211986243

1198621333

11986253

1

radic211986253

1198621233

11986263

1

radic211986263

Table 4 Continued


Compliance

1198701111

11987811

11987811

1198702222

11987822

11987822

1198703333

11987833

11987833

1198701122

11987812

11987812

1198701133

11987813

11987813

1198702233

11987823

11987823

1198702323

1

411987844

1

211987844

1198701313

1

411987855

1

211987855

1198701212

1

411987866

1

211987866

1198701323

1

411987854

1

211987854

1198701312

1

411987856

1

211987856

1198701223

1

411987864

1

211987864

1198702311

1

211987841

1

radic211987841

1198701311

1

211987851

1

radic211987851

1198701211

1

211987861

1

radic211987861

1198702322

1

211987842

1

radic211987842

1198701322

1

211987852

1

radic211987852

1198701222

1

211987862

1

radic211987862

1198702333

1

211987843

1

radic211987843

1198701333

1

211987853

1

radic211987853

1198701233

1

211987863

1

radic211987863

24681012141618

12

34

56

Column Row

12

34

56




S no Species 120588 11986211

11986222

11986233

11986212

11986213

11986223

11986244

11986255

11986266



S no Species 120588 11986211

11986222

11986233

11986212

11986213

11986223

11986244

11986255

11986266














986 7912 698311986212

2147 3609 281611986213

2773 3186 284511986214

minus0162 11 8011986215

minus293 54 minus0711986216

minus0821 minus06 11411986222

935 8529 896711986223

2346 3281 322711986224

minus1204 09 minus162311986225


0445 minus42 16011986233

5952 14730 916011986234


minus1791 169 minus7211986236

054 06 2411986244

561 3587 348911986245

0798 05 10511986246

minus2159 51 minus9811986255

6032 3448 242611986256



24681012141618

12

34

56

Column Row

12

34

56



5

10

15

20

12

34

56

Column Row

12

34

56



119890120595(119894119895)

= 119890119894otimes 119890

119895 (20)




1 Quipo

[[[[[[[[[

[

1076866026

004067345638

02534605191

02260000000

01180000000

007800000000

]]]]]]]]]

]

[[[[[[[[[

[

003276733390 09918780499 minus01228992897 00 00 00

003131140851 minus01239237163 minus09917976143 00 00 00

09989724208 minus002865041271 003511774899 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

2 Quipo

[[[[[[[[[

[

3461963876

01399776173

04300584948

04300000000

02800000000

01440000000

]]]]]]]]]

]

[[[[[[[[[

[

004079957804 09756137601 minus02156691559 00 00 00

005942480245 minus02178361212 minus09741745817 00 00 00

09973986606 minus002692981555 006686327745 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

3 White

[[[[[[[[[

[

1009116301

03556701722

1333166815

1442000000

1344000000

002200000000

]]]]]]]]]

]

[[[[[[[[[

[

004028666644 09048796003 minus04237568827 00 00 00

006399313350 minus04255671399 minus09026613361 00 00 00

09971368323 minus0009247686096 007505076253 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

4 Khaya

[[[[[[[[[

[

1080102614

04606834511

1475290392

1800000000

1196000000

04200000000

]]]]]]]]]

]

[[[[[[[[[

[

005368516527 09266208315 minus03721447810 00 00 00

007220768570 minus03753089731 minus09240829102 00 00 00

09959437502 minus002273783078 008705767064 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

5 Mahogany

[[[[[[[[[

[

1210253321

06098853915

1810581412

1960000000

1498000000

06380000000

]]]]]]]]]

]

[[[[[[[[[

[


minus007817061387 04985803653 minus08633116316 00 00 00

minus09948285648 001731853005 01000809391 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

6 Mahogany

[[[[[[[[[

[

1310854783

03895295651

1814922632

1218000000

09380000000

03000000000

]]]]]]]]]

]

[[[[[[[[[

[


minus007547523076 04892979419 minus08688446422 00 00 00

minus09956351187 001099502014 009268127742 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

7 S Germ

[[[[[[[[[

[

1234053410

05212570563

1922208822

2318000000

1582000000

05400000000

]]]]]]]]]

]

[[[[[[[[[

[

004967528691 09161715971 minus03976958243 00 00 00

008463937788 minus04006166011 minus09123280736 00 00 00

09951726186 minus001165943224 009744493938 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

8 Maple

[[[[[[[[[

[

1205449768

06868279555

2766674361

2460000000

2194000000

05840000000

]]]]]]]]]

]

[[[[[[[[[

[


minus02031928413 05304148448 minus08230265872 00 00 00

minus09692575344 001015375725 02458388325 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]




9 Walnut

[[[[[[[[[

[

1267869546

05204134969

1919891015

1922000000

1400000000

04600000000

]]]]]]]]]

]

[[[[[[[[[

[

008621257414 08755641188 minus04753471023 00 00 00

01243595697 minus04828494241 minus08668282006 00 00 00

09884847438 minus001561753067 01505124700 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

10 Birch

[[[[[[[[[

[

3168908680

04747157903

05946755301

2346000000

1816000000

03720000000

]]]]]]]]]

]

[[[[[[[[[

[

03961520086 09160614623 006240981414 00 00 00

06338768023 minus02236797319 minus07403833993 00 00 00

06642768888 minus03328645006 06692812840 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

11 Y Birch

[[[[[[[[[

[

1545414040

05549651499

2059894445

2120000000

1942000000

04800000000

]]]]]]]]]

]

[[[[[[[[[

[


minus008975775227 05593224991 minus08240763851 00 00 00

minus09937798435 0004466103673 01112729817 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

12 Oak

[[[[[[[[[

[

1718666431

08648974218

3239438239

2380000000

1532000000

07840000000

]]]]]]]]]

]

[[[[[[[[[

[

006975010637 09079000793 minus04133429180 00 00 00

01071019008 minus04187725641 minus09017531389 00 00 00

09917984199 minus001862756541 01264472547 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

13 Ash

[[[[[[[[[

[

1715567967

06696042476

2409716064

2684000000

1784000000

05400000000

]]]]]]]]]

]

[[[[[[[[[

[

006190313535 08746549514 minus04807772027 00 00 00

009781749768 minus04846986500 minus08691944279 00 00 00

09932772717 minus0006777439445 01155609239 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

14 Ash

[[[[[[[[[

[

1742363268

07844815431

2669885744

1720000000

1218000000

05000000000

]]]]]]]]]

]

[[[[[[[[[

[


minus01363859604 05177044741 minus08446188165 00 00 00

minus09855542417 001550704374 01686486565 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

15 Beech

[[[[[[[[[

[

1599002755

09558575345

3451114905

3216000000

2112000000

09120000000

]]]]]]]]]

]

[[[[[[[[[

[

01135176976 08848520771 minus04518302069 00 00 00

01764907831 minus04654963442 minus08672739791 00 00 00

09777344911 minus001870707734 02090103088 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

Thus 119890120572otimes 119890

1205731le120572120573le119899




119860119894119895119896119897

119890119894otimes 119890

119895otimes 119890

119896otimes 119890

119897 or as an 119899


120572120573119890120572otimes 119890

120573 where 119860

120595(119894119895)120595119896119897= 119860

119894119895119896119897

1 le 119894 119895 119896 119897 le 119899


tr (A119879B) = ⟨AB⟩ = tr (119860119879119861119879) A =1003817100381710038171003817100381711986010038171003817100381710038171003817 (21)






1 Blasa

[[[[[[[[[

[

6385324208

009845837744

03832174064

06240000000

04060000000

006600000000

]]]]]]]]]

]

[[[[[[[[[

[

001488818113 09510211778 minus03087669978 00 00 00

002575997614 minus03090635474 minus09506924583 00 00 00

09995572851 minus0006200252135 002909967665 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

2 Spruce

[[[[[[[[[

[

1198583568

04516442734

1114520065

1498000000

1442000000

007800000000

]]]]]]]]]

]

[[[[[[[[[

[


minus004689865645 04044467539 minus09133582749 00 00 00

minus09983543167 001123114838 005623628701 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

3 Spruce

[[[[[[[[[

[

1410927768

04185382987

1227183961

1442000000

10

006400000000

]]]]]]]]]

]

[[[[[[[[[

[


minus005361649666 04423303564 minus08952480817 00 00 00

minus09978936411 0009052295016 006423657043 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

4 Spruce

[[[[[[[[[

[

1630529953

03544288592

08442715844

1234000000

1520000000

007200000000

]]]]]]]]]

]

[[[[[[[[[

[


minus002869707087 04091564350 minus09120128765 00 00 00

minus09993764538 0007032901380 003460119282 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

5 Spruce

[[[[[[[[[

[

1725845208

05092652753

1171282625

1250000000

1706000000

007000000000

]]]]]]]]]

]

[[[[[[[[[

[


minus003428302033 05858146064 minus08097196604 00 00 00

minus09988364173 0007413313465 004765347716 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

6 Douglas Fir

[[[[[[[[[

[

1613459769

06233733771

1439028943

1767000000

1766000000

01760000000

]]]]]]]]]

]

[[[[[[[[[

[


minus003697194691 05922678885 minus08048924305 00 00 00

minus09986533619 0007435777607 005134366998 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

7 Douglas Fir

[[[[[[[[[

[

1710150156

06986859431

2204812526

2348000000

1816000000

01600000000

]]]]]]]]]

]

[[[[[[[[[

[


minus006364825251 05704773676 minus08188433771 00 00 00

minus09967231590 0004703484281 008075160717 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

8 Pine

[[[[[[[[[

[

1699539133

04940019684

1565606760

3484000000

1344000000

01320000000

]]]]]]]]]

]

[[[[[[[[[

[


minus005585205149 04413516031 minus08955943895 00 00 00

minus09978476167 0006195562437 006528207193 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]


Table11Th

eeigenvalues

andeigenvectorsfor3

specim

enso

fcancello

usbo

necalculated

usingMAPL

EΛ[119894]sta

ndforthe

eigenvalueso

f119894th

specim

enof

cancellous

boneand

119881[119894]sta

ndfor

thec

orrespon

ding

eigenvectors

Cancellous

bone

EigenvaluesΛ

[119894]

Eigenvectors

119881[119894]

Specim

en1

[ [ [ [ [ [ [ [

1334869880

minus02071167473

8644210768

6837676373

3768806080

4123724855

] ] ] ] ] ] ] ]

[ [ [ [ [ [ [ [

03316115964

08831238953

02356441950

02242668491

005354786914

003787816350

06475092156

minus01099136965

01556396475

minus06940063479

minus01278953723

02154647417

04800906963

minus02832250652

007648827487

02494278793

06410471445

minus04585742261

minus02737532692

minus006354940263

04518342532

minus006214920576

05836952694

06101669758

minus03706582070

03314035780

minus01276606259

minus06337030234

03639432801

minus04499474345

01671829224

01180163925

minus08330343502

002008124471

03109325968

04087706408

] ] ] ] ] ] ] ]

Specim

en2

[ [ [ [ [ [ [ [

1754634171

7773848354

1550688791

2301790452

3445174379

3589156342

] ] ] ] ] ] ] ]

[ [ [ [ [ [ [ [

minus03057290719

minus03005337771

minus09030341401

001584500766

minus002175914011

minus0003741933916

minus04311831740

minus08021570593

04130146809

00001362645283

minus0003854432451

0005396441669

minus08488209390

05154657137

01152135966

minus0004741957786

002229739182

minus0002068264586

00009402467441

minus0005358584867

0003987518000

minus003989381331

003796849613

minus09984595017

minus001058157817

002100470582

002092543658

0006230946613

minus09987520802

minus003826770492

00009823402498

0006977574167

001484146906

09990475909

0008196718814

minus003958286493

] ] ] ] ] ] ] ]

Specim

en3

[ [ [ [ [ [ [ [

1418542670

5838388363

9959058231

3433818493

3024968125

1757492470

] ] ] ] ] ] ] ]

[ [ [ [ [ [ [ [

03244960223

minus0004831643783

minus08451713783

04062092722

01164524794

minus004239310198

06461152835

minus07125334236

02668138654

004131783689

minus002431040234

003665187401

06621315621

07009969173

02633438279

002836289512

minus0001711614840

0005269111592

minus01962401199

0009159859564

03740839141

08850746598

01904577552

004284654369

minus0008597323304

minus0002526763452

001897677365

01738799329

minus06977939665

minus06945566457

001533190599

minus002803230310

006961922985

minus01374446216

06801869735

minus07159517722

] ] ] ] ] ] ] ]


24681012141618

12

34

56

Column Row

12

34

56


12345678

12

34

56

Column Row

12

34

56




120590119894119895= 119862


where 119862119894119895119896119897


120590 = 120590119894119895119890119894otimes 119890

119895 120598 = 120598

119896119897119890119896otimes 119890

119897

C = 119862119894119895119896119897

119890119894otimes 119890

119896otimes 119890

119896otimes 119890

119897

(23)

5

10

15

20

12

34

56

Column Row

12

34

56


5

10

15

20

25

12

34

56

Column Row

12

34

56



E1equiv 119890

1otimes 119890

1

E2equiv 119890

2otimes 119890

2

E3equiv 119890

3otimes 119890

3

(24)


E4equiv

1


2+ 119890

2otimes 119890

1)

E5equiv

1


3+ 119890

3otimes 119890

2)

E6equiv

1


1+ 119890

1otimes 119890

3)

(25)



E7equiv

1


2minus 119890

2otimes 119890

1)

E8equiv

1


3minus 119890

3otimes 119890

2)

E9equiv

1


1minus 119890

1otimes 119890

3)

(26)


120590 = 119878119860E

119860

120598 = 119864119860E

119860

C = 119862119860119861

119890119860otimes 119890

119861

(27)




120590119894119895= 119862

119894119895119896119897120598119896119897

lArrrArr 119878119860= 119862

119860119861E119861

119862119894119895119896119897

120598119896119897

= 120582120598119894119895lArrrArr 119862

119860119861E119861= 120582119864

119860

(28)


((((((((

(

1198781

1198782

1198783

1198784

1198785

1198786

1198787

1198788

1198789

))))))))

)

=

((((((((

(

12059011

12059022

12059033

120590(12)

120590(23)

120590(31)

120590[12]

120590[23]

120590[31]

))))))))

)

((((((((

(

1198641

1198642

1198643

1198644

1198645

1198646

1198647

1198648

1198649

))))))))

)

=

((((((((

(

12059811

12059822

12059833

120598(12)

120598(23)

120598(31)

120598[12]

120598[23]

120598[31]

))))))))

)

(29)


120579(119894119895)

equiv1

radic2(120579119894119895+ 120579

119895119894) 120579

[119894119895]equiv

1

radic2(120579119894119895minus 120579

119895119894) (30)


tensor C are

((((((((

(

11986211

11986212

11986213

11986214

11986215

11986216

11986217

11986218

11986219

11986221

11986222

11986223

11986224

11986225

11986226

11986227

11986228

11986229

11986231

11986232

11986233

11986234

11986235

11986236

11986237

11986238

11986239

11986241

11986242

11986243

11986244

11986245

11986246

11986247

11986248

11986249

11986251

11986252

11986253

11986254

11986255

11986256

11986257

11986258

11986259

11986261

11986262

11986263

11986264

11986265

11986266

11986267

11986268

11986269

11986271

11986272

11986273

11986274

11986275

11986276

11986277

11986278

11986279

11986281

11986282

11986283

11986284

11986285

11986286

11986287

11986288

11986289

11986291

11986292

11986293

11986294

11986295

11986296

11986297

11986298

11986299

))))))))

)

=

1198621111

1198621122

1198621133

1198622211

1198622222

1198623333

1198623311

1198623322

1198623333

11986211(12)

11986211(23)

11986211(31)

11986222(12)

11986222(23)

11986222(31)

11986233(12)

11986233(23)

11986233(31)

11986211[12]

11986211[23]

11986211[31]

11986222[12]

11986222[23]

11986222[31]

11986233[12]

11986233[23]

11986233[31]

119862(12)11

119862(12)22

119862(12)33

119862(23)11

119862(23)22

119862(23)33

119862(31)11

119862(31)22

119862(31)33

119862(1212)

119862(1223)

119862(1231)

119862(2312)

119862(2323)

119862(2331)

119862(3112)

119862(3123)

119862(3131)

119862(12)[12]

119862(12)[23]

119862(12)[31]

119862(23)[12]

119862(23)[23]

119862(23)[31]

119862(31)[12]

119862(31)[23]

119862(31)[31]

119862[12]11

119862[12]22

119862[12]33

119862[23]11

119862[23]22

119862[123]33

119862[21]11

119862[31]22

119862[31]33

119862[12](12)

119862[12](23)

119862[12](21)

119862[23](12)

119862[23](23)

119862[23](21)

119862[31](12)

119862[31](23)

119862[31](21)

119862[1212]

119862[1223]

119862[1231]

119862[2312]

119862[2323]

119862[2331]

119862[3112]

119862[3123]

119862[3131]

(31)

where

119862119894119895(119896119897)

equiv1

radic2(119862

119894119895119896119897+ 119862

119894119895119897119896) 119862

119894119895[119896119897]equiv

1

radic2(119862

119894119895119896119897minus 119862

119894119895119897119896)

119862(119894119895)119896119897

equiv1

radic2(119862

119894119895119896119897+ 119862

119895119894119896119897) 119862

[119894119895]119896119897equiv

1

radic2(119862

119894119895119896119897minus 119862

119895119894119896119897)

119862(119894119895119896119897)

equiv1

2(119862

119894119895119896119897+ 119862

119894119895119897119896+ 119862

119895119894119896119897+ 119862

119895119894119897119896)

119862(119894119895)[119896119897]

equiv1

2(119862

119894119895119896119897minus 119862

119894119895119897119896+ 119862

119895119894119896119897minus 119862

119895119894119897119896)

119862[119894119895](119896119897)

equiv1

2(119862

119894119895119896119897+ 119862

119894119895119897119896minus 119862

119895119894119896119897minus 119862

119895119894119897119896)

119862[119894119895119896119897]

equiv1

2(119862

119894119895119896119897minus 119862

119895119894119896119897minus 119862

119894119895119897119896+ 119862

119895119894119897119896)

(32)


12

34

56

ColumnRow

12

34

56

600005000040000300002000010000


5

10

15

20

25

12

34

56

Column Row

12

34

56






119895119894and 120598

119896119897= 120598

119897119896 the 119862

119894119895119896119897will be

like 119862119894119895119896119897

= 119862119895119894119896119897

= 119862119894119895119897119896


119862119894119895119896119897

= 119862119896119897119894119895












2 sdot 119864energy = 120590119894119895120598119894119895= 120590

119901120598119901 (33)


119901 = 119894120575119894119895+ (1 minus 120575

119894119895) (9 minus 119894 minus 119895)

119902 = 119896120575119896119897+ (1 minus 120575

119896119897) (9 minus 119896 minus 119897)

(34)


119901 119862

119894119895119896119897= 119862

119901119902 and 120598

119902= (2 minus

120575119896119897)120598119896119897



(

(

1205901

1205902

1205903

1205904

1205905

1205906

)

)

= (

(

11986211

11986212

11986213

11986214

11986215

11986216

11986221

11986222

11986223

11986224

11986225

11986226

11986231

11986232

11986233

11986234

11986235

11986236

11986241

11986242

11986243

11986244

11986245

11986246

11986251

11986252

11986253

11986254

11986255

11986256

11986261

11986262

11986263

11986264

11986265

11986266

)

)

(

(

1205981

1205982

1205983

1205984

1205985

1205986

)

)

(35)



(1) the 120590119894119895and 120598


(2) the norms of 120590119894119895 120598119896119897 and 119862




01 02 03 04 05 06 07 08

2

4

6

8

10

12

14

16

(a)

01

02

03

04

05

06

07

08

09

01 02 03 04 05 06 07 08

(b)


01 02 03 04 05 06 07 08

05

1

15

2

25

3

(a)

01 02 03 04 05 06 07 08

05

1

15

2

25

3

(b)




119901 = 119894120575119894119895+ (1 minus 120575

119894119895) (9 minus 119894 minus 119895)

119902 = 119896120575119896119897+ (1 minus 120575

119896119897) (9 minus 119896 minus 119897)

(36)

In this way

120590119901= (120575

119894119895+ radic2 (1 minus 120575

119894119895)) 120598

119894119895

120598119902= (120575

119896119897+ radic2 (1 minus 120575

119896119897)) 120598

119896119897

119862119901119902

= (120575119894119895+ radic2 (1 minus 120575

119894119895)) (120575

119896119897+ radic2 (1 minus 120575

119896119897)) 119862

119894119895119896119897

(37)




120590kelvin

= 120590viogt

120585 120590voigt

=120590kelvin

120585

120598kelvin

= 120598voigt

120585 120598voigt

=120598kelvin

120585

(38)


119862kelvin

=((

(







))

)

119862voigt

(39)


(

(

12059011

12059022

12059033

radic212059023

radic212059013

radic212059012

)

)

= (

1198621111 1198621122 1198621133radic21198621123

radic21198621113radic21198621112

1198622211 1198622222 1198622333radic21198622223

radic21198622213radic21198622212

1198623311 1198623322 1198623333radic21198623323

radic21198623313radic21198623312

radic21198622311radic21198622322

radic21198622333 21198622323 21198622313 21198622312

radic21198621311radic21198621322

radic21198621333 21198621323 21198621313 21198621312

radic21198621211radic21198621222

radic21198621233 21198621223 21198621213 21198621212

)

lowast(

(

12059811

12059822

12059833

radic212059823

radic212059813

radic212059812

)

)

(40)


= C (41)



(

(

12059011

12059022

12059033

radic212059023

radic212059013

radic212059012

)

)

=((

(

11986211

11986212

11986213

radic211986214

radic211986215

radic211986216

11986212

11986222

11986223

radic211986224

radic211986225

radic211986226

11986213

11986223

11986233

radic211986234

radic211986235

radic211986236

radic211986214

radic211986224

radic211986234

211986244

211986245

211986246

radic211986215

radic211986225

radic211986235

211986245

211986255

211986256

radic211986216

radic211986226

radic211986236

211986246

211986256

211986266

))

)

lowast (

(

12059811

12059822

12059833

radic212059823

radic212059813

radic212059812

)

)

(42)


(

(

1

2

3

4

5

6

)

)

=((

(

11986211

11986212

11986213

11986214

11986215

11986216

11986212

11986222

11986223

11986224

11986225

11986226

11986213

11986223

11986233

11986234

11986235

11986236

11986214

11986224

11986234

11986244

11986245

11986246

11986215

11986225

11986235

11986245

11986255

11986256

11986216

11986226

11986236

11986246

11986256

11986266

))

)

lowast (

(

1205981

1205982

1205983

1205984

1205985

1205986

)

)

(43)



































01 02 03 04 05 06 07 08

05

1

15

2

25

3

(a)

01 02 03 04 05 06 07 08

05

1

15

2

25

3

(b)


01 02 03 04 05 06 07 08

2

4

6

8

10

12

14

16





ΛI) N = 0 (44)


119894 119894 = 1 2 6 and

















NNA119896

equiv1

119872

119872

sum

119884=1

N119884119896 (45)



NAVG119896

equiv JNANNA119896

(46)


119896=1NNA119896

otimes NNA119896

| that is

JNA equiv (

6

sum

119896=1

NNA119896

otimes NNA119896

)

minus12

(47)


119896 that is

ΛAVG119902

equiv1

119872

119872

sum

119884=1

6

sum

119896=1

Λ119884

119896(N119884

119896sdot NAVG

119902)2

(48)


gt 0 for all 119896 = 1 2 6

and ΛAVG119896


NA119896


ΛNA119896

equiv1

119872

119872

sum

119884=1

Λ119884

119896 (49)


CAVG=

6

sum

119896=1

ΛAVG119896

NAVG119896

otimes NAVG119896

(50)





[[[[[[[

[

00367834559700000

00453681054800000

0998185540700000

00

00

00

]]]]]]]

]

[[[[[[[

[

0983745905000000

minus0170879918750000

minus00277901141300000

00

00

00

]]]]]]]

]

[[[[[[[

[

minus0169284222800000

minus0982986098000000

00509905132200000

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(51)


[[[[[[[

[

003679134179

004537783172

09983995367

00

00

00

]]]]]]]

]

[[[[[[[

[

09859858256

minus01712690003

minus002785339027

00

00

00

]]]]]]]

]

[[[[[[[

[

minus01697052873

minus09854311019

005111734310

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(52)




[[[

[

334725276837330 0000112116752981933 000198542359441849 00 00 00

0000112116752981933 334773746006714 minus0000991802322788501 00 00 00

000198542359441849 minus0000991802322788501 334013593769726 00 00 00

00 00 00 333950000000000 00 00

00 00 00 00 333950000000000 00

00 00 00 00 00 333950000000000

]]]

]

(53)




[[[[[[[

[

004028666644

006399313350

09971368323

00

00

00

]]]]]]]

]

[[[[[[[

[

09048796003

minus04255671399

minus0009247686096

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04237568827

minus09026613361

007505076253

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(54)


[[[[[[[

[

004028666644

006399313350

09971368323

00

00

00

]]]]]]]

]

[[[[[[[

[

09048795994

minus04255671395

minus0009247686087

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04237568827

minus09026613361

007505076253

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(55)



[[[

[

1458799998 000000001785571215 minus000000001009489597 00 00 00000000001785571215 1458799997 0000000002407020197 00 00 00minus000000001009489597 0000000002407020197 1458799997 00 00 00

00 00 00 1458799998 00 0000 00 00 00 1458799998 0000 00 00 00 00 1458799998

]]]

]

(56)




[[[[[

[

005368516527

007220768570

09959437502

00

00

00

]]]]]

]

[[[[[

[

09266208315

minus03753089731

minus002273783078

00

00

00

]]]]]

]

[[[[[

[

minus03721447810

minus09240829102

008705767064

00

00

00

]]]]]

]

[[[[[

[

00

00

00

00

00

10

]]]]]

]

[[[[[

[

00

00

00

10

00

00

]]]]]

]

[[[[[

[

00

00

00

00

10

00

]]]]]

]

(57)


[[[[[[[[[

[

005368516527

007220768570

09959437502

00

00

00

]]]]]]]]]

]

[[[[[[[[[

[

09266208315

minus03753089731

minus002273783078

00

00

00

]]]]]]]]]

]

[[[[[[[[[

[

minus03721447806

minus09240829093

008705767055

00

00

00

]]]]]]]]]

]

[[[[[[[[[

[

00

00

00

10

00

00

]]]]]]]]]

]

[[[[[[[[[

[

00

00

00

00

10

00

]]]]]]]]]

]

[[[[[[[[[

[

00

00

00

00

00

10

]]]]]]]]]

]

(58)




[[[

[

1615299998 0000000005508173090 minus0000000008722620038 00 00 00

0000000005508173090 1615299996 minus0000000004345156817 00 00 00

minus0000000008722620038 minus0000000004345156817 1615300000 00 00 00

00 00 00 1615299998 00 00

00 00 00 00 1615299998 00

00 00 00 00 00 1615299998

]]]

]

(59)




[[[[[[[

[

minus00598757013800000

minus00768229223150000

minus0995231841750000

00

00

00

]]]]]]]

]

[[[[[[[

[

minus0869358919900000

0493939153600000

00141567750950000

00

00

00

]]]]]]]

]

[[[[[[[

[

minus0490490276500000

minus0866078136900000

00963811082600000

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(60)


[[[[[[[[[[[

[

minus005987730132

minus007682497510

minus09952584354

00

00

00

]]]]]]]]]]]

]

[[[[[[[[[[[

[

minus08693926232

04939583026

001415732393

00

00

00

]]]]]]]]]]]

]

[[[[[[[[[[[

[

00

00

00

10

00

00

]]]]]]]]]]]

]

[[[[[[[[[[[

[

00

00

00

00

10

00

]]]]]]]]]]]

]

[[[[[[[[[[[

[

00

00

00

00

00

10

]]]]]]]]]]]

]

(61)



[[[

[


00 00 00 1819400002 1819400002 0000 00 00 00 00 0000 00 00 00 00 1819400002

]]]

]

(62)




[[[[[[[

[

004967528691

008463937788

09951726186

00

00

00

]]]]]]]

]

[[[[[[[

[

09161715971

minus04006166011

minus001165943224

00

00

00

]]]]]]]

]

[[[[[

[

minus03976958243

minus09123280736

009744493938

00

00

00

]]]]]

]

[[[[[

[

00

00

00

10

00

00

]]]]]

]

[[[[[

[

00

00

00

00

10

00

]]]]]

]

[[[[[

[

00

00

00

00

00

10

]]]]]

]

(63)


[[[[[[[

[

004967528696

008463937796

09951726196

00

00

00

]]]]]]]

]

[[[[[[[

[

09161715980

minus04006166015

minus001165943225

00

00

00

]]]]]]]

]


[[[[[[[

[

minus03976958247

minus09123280745

009744493948

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(64)



[[[

[

1922399998 minus000000001176508811 minus000000001537920006 00 00 00

minus000000001176508811 1922400000 minus0000000007631928036 00 00 00

minus000000001537920006 minus0000000007631928036 1922400000 00 00 00

00 00 00 1922399998 1922399998 00

00 00 00 00 00 00

00 00 00 00 00 1922399998

]]]

]

(65)




[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

[[[[[[[

[

minus01387533797

minus02031928413

minus09692575344

00

00

00

]]]]]]]

]

[[[[[[[

[

minus08476774112

05304148448

001015375725

00

00

00

]]]]]]]

]

[[[[[[[

[

minus05120454135

minus08230265872

02458388325

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

(66)


[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

[[[[[[[

[

minus01387533798

minus02031928415

minus09692575354

00

00

00

]]]]]]]

]

[[[[[[[

[

minus08476774112

05304148448

001015375725

00

00

00

]]]]]]]

]

[[[[[[[

[

minus05120454135

minus08230265872

02458388325

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

(67)



[[[

[

2074599999 0000000004356660361 000000003402343994 00 00 00

0000000004356660361 2074600004 000000002232269567 00 00 00

000000003402343994 000000002232269567 2074600000 00 00 00

00 00 00 2074600000 2074600000 00

00 00 00 00 00 00

00 00 00 00 00 2074600000

]]]

]

(68)




[[[[[[[

[

008621257414

01243595697

09884847438

00

00

00

]]]]]]]

]

[[[[[[[

[

08755641188

minus04828494241

minus001561753067

00

00

00

]]]]]]]

]

[[[[[

[

minus04753471023

minus08668282006

01505124700

00

00

00

]]]]]

]

[[[[[

[

00

00

00

10

00

00

]]]]]

]

[[[[[

[

00

00

00

00

10

00

]]]]]

]

[[[[[

[

00

00

00

00

00

10

]]]]]

]

(69)


[[[[[

[

008621257423

01243595698

09884847448

00

00

00

]]]]]

]

[[[[[

[

08755641188

minus04828494241

minus001561753067

00

00

00

]]]]]

]


[[[[[[[

[

minus04753471018

minus08668281997

01505124698

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(70)



[[[

[

1890099999 000000001209663993 minus000000002532733996 00 00 00000000001209663993 1890099991 0000000001701090138 00 00 00minus000000002532733996 0000000001701090138 1890100001 00 00 00

00 00 00 1890099997 1890099997 0000 00 00 00 00 0000 00 00 00 00 1890099997

]]]

]

(71)




[[[[[[[

[

03961520086

06338768023

06642768888

00

00

00

]]]]]]]

]

[[[[[[[

[

09160614623

minus02236797319

minus03328645006

00

00

00

]]]]]]]

]

[[[[[[[

[

006240981414

minus07403833993

06692812840

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(72)


[[[[[[[

[

03961520086

06338768023

06642768888

00

00

00

]]]]]]]

]

[[[[[[[

[

09160614614

minus02236797317

minus03328645003

00

00

00

]]]]]]]

]

[[[[[[[

[

006240981426

minus07403834008

06692812853

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(73)



[[[

[


00 00 00 8772299998 8772299998 0000 00 00 00 00 0000 00 00 00 00 8772299998

]]]

]

(74)




[[[[[[[

[

minus006591789198

minus008975775227

minus09937798435

00

00

00

]]]]]]]

]

[[[[[[[

[

minus08289381135

05593224991

0004466103673

00

00

00

]]]]]]]

]

[[[[[

[

minus05554425577

minus08240763851

01112729817

00

00

00

]]]]]

]

[[[[[

[

00

00

00

10

00

00

]]]]]

]

[[[[[

[

00

00

00

00

10

00

]]]]]

]

[[[[[

[

00

00

00

00

00

10

]]]]]

]


[[[[[[[

[

minus01387533798

minus02031928415

minus09692575354

00

00

00

]]]]]]]

]

[[[[[[[

[

minus08476774112

05304148448

001015375725

00

00

00

]]]]]]]

]


[[[[

[

minus05120454135

minus08230265872

02458388325

00

00

00

]]]]

]

[[[[

[

00

00

00

10

00

00

]]]]

]

[[[[

[

00

00

00

00

10

00

]]]]

]

[[[[

[

00

00

00

00

00

10

]]]]

]

(76)



[[[

[

2228057494 0000000004678921127 000000003654014285 00 00 000000000004678921127 2228057499 000000002397389829 00 00 00000000003654014285 000000002397389829 2228057495 00 00 00

00 00 00 2228057495 2228057495 0000 00 00 00 00 0000 00 00 00 00 2228057495

]]]

]

(77)




[[[[[[[

[

006975010637

01071019008

09917984199

00

00

00

]]]]]]]

]

[[[[[[[

[

09079000793

minus04187725641

minus001862756541

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04133429180

minus09017531389

01264472547

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(78)


[[[[[[[

[

006975010637

01071019008

09917984199

00

00

00

]]]]]]]

]

[[[[[[[

[

09079000793

minus04187725641

minus001862756541

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04133429184

minus09017531398

01264472548

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(79)



[[[

[


00 00 00 2598699998 2598699998 0000 00 00 00 00 0000 00 00 00 00 2598699998

]]]

]

(80)




[[[[[

[

minus00192522847250000

minus00192842313600000

000386151499999998

00

00

00

]]]]]

]

[[[[[

[

000961799224999998

00165029120500000

000436480214750000

00

00

00

]]]]]

]

[[[[[[

[

minus0494444050150000

minus0856906622200000

0142104790200000

00

00

00

]]]]]]

]

[[[[[[

[

00

00

00

10

00

00

]]]]]]

]

[[[[[[

[

00

00

00

00

10

00

]]]]]]

]

[[[[[[

[

00

00

00

00

00

10

]]]]]]

]

(81)



[[[[[[[[[[

[

minus2541745843

minus2545963537

5098090872

00

00

00

]]]]]]]]]]

]

[[[[[[[[[[

[

2505315863

4298714977

1136953303

00

00

00

]]]]]]]]]]

]

[[[[[[[

[

minus04949599718

minus08578007511

01422530677

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(82)



[[[

[

315679169478799 427324383657779 384557503470365 00 00 00427324383657779 618700481877479 889153535276175 00 00 00384557503470365 889153535276175 384768762708609 00 00 00

00 00 00 247795001400000 00 0000 00 00 00 247795001400000 0000 00 00 00 00 247795001400000

]]]

]

(83)




[[[[[[[

[

01135176976

01764907831

09777344911

00

00

00

]]]]]]]

]

[[[[[[[

[

08848520771

minus04654963442

minus001870707734

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04518302069

minus08672739791

02090103088

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(84)


[[[[[[[

[

01135176977

01764907833

09777344921

00

00

00

]]]]]]]

]

[[[[[[[

[

08848520771

minus04654963442

minus001870707734

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04518302069

minus08672739791

02090103088

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(85)



[[[

[

2663700003 000000004768023095 000000003169803038 00 00 00000000004768023095 2663699992 0000000008310743498 00 00 00000000003169803038 0000000008310743498 2663700001 00 00 00

00 00 00 2663699998 2663699998 0000 00 00 00 00 0000 00 00 00 00 2663699998

]]]

]

(86)





6 Conclusions









Disclosure




Acknowledgments



References




















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











S no Species 120588 11986211

11986222

11986233

11986212

11986213

11986223

11986244

11986255

11986266



S no Species 120588 11986211

11986222

11986233

11986212

11986213

11986223

11986244

11986255

11986266














986 7912 698311986212

2147 3609 281611986213

2773 3186 284511986214

minus0162 11 8011986215

minus293 54 minus0711986216

minus0821 minus06 11411986222

935 8529 896711986223

2346 3281 322711986224

minus1204 09 minus162311986225


0445 minus42 16011986233

5952 14730 916011986234


minus1791 169 minus7211986236

054 06 2411986244

561 3587 348911986245

0798 05 10511986246

minus2159 51 minus9811986255

6032 3448 242611986256



24681012141618

12

34

56

Column Row

12

34

56



5

10

15

20

12

34

56

Column Row

12

34

56



119890120595(119894119895)

= 119890119894otimes 119890

119895 (20)




1 Quipo

[[[[[[[[[

[

1076866026

004067345638

02534605191

02260000000

01180000000

007800000000

]]]]]]]]]

]

[[[[[[[[[

[

003276733390 09918780499 minus01228992897 00 00 00

003131140851 minus01239237163 minus09917976143 00 00 00

09989724208 minus002865041271 003511774899 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

2 Quipo

[[[[[[[[[

[

3461963876

01399776173

04300584948

04300000000

02800000000

01440000000

]]]]]]]]]

]

[[[[[[[[[

[

004079957804 09756137601 minus02156691559 00 00 00

005942480245 minus02178361212 minus09741745817 00 00 00

09973986606 minus002692981555 006686327745 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

3 White

[[[[[[[[[

[

1009116301

03556701722

1333166815

1442000000

1344000000

002200000000

]]]]]]]]]

]

[[[[[[[[[

[

004028666644 09048796003 minus04237568827 00 00 00

006399313350 minus04255671399 minus09026613361 00 00 00

09971368323 minus0009247686096 007505076253 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

4 Khaya

[[[[[[[[[

[

1080102614

04606834511

1475290392

1800000000

1196000000

04200000000

]]]]]]]]]

]

[[[[[[[[[

[

005368516527 09266208315 minus03721447810 00 00 00

007220768570 minus03753089731 minus09240829102 00 00 00

09959437502 minus002273783078 008705767064 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

5 Mahogany

[[[[[[[[[

[

1210253321

06098853915

1810581412

1960000000

1498000000

06380000000

]]]]]]]]]

]

[[[[[[[[[

[


minus007817061387 04985803653 minus08633116316 00 00 00

minus09948285648 001731853005 01000809391 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

6 Mahogany

[[[[[[[[[

[

1310854783

03895295651

1814922632

1218000000

09380000000

03000000000

]]]]]]]]]

]

[[[[[[[[[

[


minus007547523076 04892979419 minus08688446422 00 00 00

minus09956351187 001099502014 009268127742 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

7 S Germ

[[[[[[[[[

[

1234053410

05212570563

1922208822

2318000000

1582000000

05400000000

]]]]]]]]]

]

[[[[[[[[[

[

004967528691 09161715971 minus03976958243 00 00 00

008463937788 minus04006166011 minus09123280736 00 00 00

09951726186 minus001165943224 009744493938 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

8 Maple

[[[[[[[[[

[

1205449768

06868279555

2766674361

2460000000

2194000000

05840000000

]]]]]]]]]

]

[[[[[[[[[

[


minus02031928413 05304148448 minus08230265872 00 00 00

minus09692575344 001015375725 02458388325 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]




9 Walnut

[[[[[[[[[

[

1267869546

05204134969

1919891015

1922000000

1400000000

04600000000

]]]]]]]]]

]

[[[[[[[[[

[

008621257414 08755641188 minus04753471023 00 00 00

01243595697 minus04828494241 minus08668282006 00 00 00

09884847438 minus001561753067 01505124700 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

10 Birch

[[[[[[[[[

[

3168908680

04747157903

05946755301

2346000000

1816000000

03720000000

]]]]]]]]]

]

[[[[[[[[[

[

03961520086 09160614623 006240981414 00 00 00

06338768023 minus02236797319 minus07403833993 00 00 00

06642768888 minus03328645006 06692812840 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

11 Y Birch

[[[[[[[[[

[

1545414040

05549651499

2059894445

2120000000

1942000000

04800000000

]]]]]]]]]

]

[[[[[[[[[

[


minus008975775227 05593224991 minus08240763851 00 00 00

minus09937798435 0004466103673 01112729817 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

12 Oak

[[[[[[[[[

[

1718666431

08648974218

3239438239

2380000000

1532000000

07840000000

]]]]]]]]]

]

[[[[[[[[[

[

006975010637 09079000793 minus04133429180 00 00 00

01071019008 minus04187725641 minus09017531389 00 00 00

09917984199 minus001862756541 01264472547 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

13 Ash

[[[[[[[[[

[

1715567967

06696042476

2409716064

2684000000

1784000000

05400000000

]]]]]]]]]

]

[[[[[[[[[

[

006190313535 08746549514 minus04807772027 00 00 00

009781749768 minus04846986500 minus08691944279 00 00 00

09932772717 minus0006777439445 01155609239 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

14 Ash

[[[[[[[[[

[

1742363268

07844815431

2669885744

1720000000

1218000000

05000000000

]]]]]]]]]

]

[[[[[[[[[

[


minus01363859604 05177044741 minus08446188165 00 00 00

minus09855542417 001550704374 01686486565 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

15 Beech

[[[[[[[[[

[

1599002755

09558575345

3451114905

3216000000

2112000000

09120000000

]]]]]]]]]

]

[[[[[[[[[

[

01135176976 08848520771 minus04518302069 00 00 00

01764907831 minus04654963442 minus08672739791 00 00 00

09777344911 minus001870707734 02090103088 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

Thus 119890120572otimes 119890

1205731le120572120573le119899




119860119894119895119896119897

119890119894otimes 119890

119895otimes 119890

119896otimes 119890

119897 or as an 119899


120572120573119890120572otimes 119890

120573 where 119860

120595(119894119895)120595119896119897= 119860

119894119895119896119897

1 le 119894 119895 119896 119897 le 119899


tr (A119879B) = ⟨AB⟩ = tr (119860119879119861119879) A =1003817100381710038171003817100381711986010038171003817100381710038171003817 (21)






1 Blasa

[[[[[[[[[

[

6385324208

009845837744

03832174064

06240000000

04060000000

006600000000

]]]]]]]]]

]

[[[[[[[[[

[

001488818113 09510211778 minus03087669978 00 00 00

002575997614 minus03090635474 minus09506924583 00 00 00

09995572851 minus0006200252135 002909967665 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

2 Spruce

[[[[[[[[[

[

1198583568

04516442734

1114520065

1498000000

1442000000

007800000000

]]]]]]]]]

]

[[[[[[[[[

[


minus004689865645 04044467539 minus09133582749 00 00 00

minus09983543167 001123114838 005623628701 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

3 Spruce

[[[[[[[[[

[

1410927768

04185382987

1227183961

1442000000

10

006400000000

]]]]]]]]]

]

[[[[[[[[[

[


minus005361649666 04423303564 minus08952480817 00 00 00

minus09978936411 0009052295016 006423657043 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

4 Spruce

[[[[[[[[[

[

1630529953

03544288592

08442715844

1234000000

1520000000

007200000000

]]]]]]]]]

]

[[[[[[[[[

[


minus002869707087 04091564350 minus09120128765 00 00 00

minus09993764538 0007032901380 003460119282 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

5 Spruce

[[[[[[[[[

[

1725845208

05092652753

1171282625

1250000000

1706000000

007000000000

]]]]]]]]]

]

[[[[[[[[[

[


minus003428302033 05858146064 minus08097196604 00 00 00

minus09988364173 0007413313465 004765347716 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

6 Douglas Fir

[[[[[[[[[

[

1613459769

06233733771

1439028943

1767000000

1766000000

01760000000

]]]]]]]]]

]

[[[[[[[[[

[


minus003697194691 05922678885 minus08048924305 00 00 00

minus09986533619 0007435777607 005134366998 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

7 Douglas Fir

[[[[[[[[[

[

1710150156

06986859431

2204812526

2348000000

1816000000

01600000000

]]]]]]]]]

]

[[[[[[[[[

[


minus006364825251 05704773676 minus08188433771 00 00 00

minus09967231590 0004703484281 008075160717 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

8 Pine

[[[[[[[[[

[

1699539133

04940019684

1565606760

3484000000

1344000000

01320000000

]]]]]]]]]

]

[[[[[[[[[

[


minus005585205149 04413516031 minus08955943895 00 00 00

minus09978476167 0006195562437 006528207193 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]


Table11Th

eeigenvalues

andeigenvectorsfor3

specim

enso

fcancello

usbo

necalculated

usingMAPL

EΛ[119894]sta

ndforthe

eigenvalueso

f119894th

specim

enof

cancellous

boneand

119881[119894]sta

ndfor

thec

orrespon

ding

eigenvectors

Cancellous

bone

EigenvaluesΛ

[119894]

Eigenvectors

119881[119894]

Specim

en1

[ [ [ [ [ [ [ [

1334869880

minus02071167473

8644210768

6837676373

3768806080

4123724855

] ] ] ] ] ] ] ]

[ [ [ [ [ [ [ [

03316115964

08831238953

02356441950

02242668491

005354786914

003787816350

06475092156

minus01099136965

01556396475

minus06940063479

minus01278953723

02154647417

04800906963

minus02832250652

007648827487

02494278793

06410471445

minus04585742261

minus02737532692

minus006354940263

04518342532

minus006214920576

05836952694

06101669758

minus03706582070

03314035780

minus01276606259

minus06337030234

03639432801

minus04499474345

01671829224

01180163925

minus08330343502

002008124471

03109325968

04087706408

] ] ] ] ] ] ] ]

Specim

en2

[ [ [ [ [ [ [ [

1754634171

7773848354

1550688791

2301790452

3445174379

3589156342

] ] ] ] ] ] ] ]

[ [ [ [ [ [ [ [

minus03057290719

minus03005337771

minus09030341401

001584500766

minus002175914011

minus0003741933916

minus04311831740

minus08021570593

04130146809

00001362645283

minus0003854432451

0005396441669

minus08488209390

05154657137

01152135966

minus0004741957786

002229739182

minus0002068264586

00009402467441

minus0005358584867

0003987518000

minus003989381331

003796849613

minus09984595017

minus001058157817

002100470582

002092543658

0006230946613

minus09987520802

minus003826770492

00009823402498

0006977574167

001484146906

09990475909

0008196718814

minus003958286493

] ] ] ] ] ] ] ]

Specim

en3

[ [ [ [ [ [ [ [

1418542670

5838388363

9959058231

3433818493

3024968125

1757492470

] ] ] ] ] ] ] ]

[ [ [ [ [ [ [ [

03244960223

minus0004831643783

minus08451713783

04062092722

01164524794

minus004239310198

06461152835

minus07125334236

02668138654

004131783689

minus002431040234

003665187401

06621315621

07009969173

02633438279

002836289512

minus0001711614840

0005269111592

minus01962401199

0009159859564

03740839141

08850746598

01904577552

004284654369

minus0008597323304

minus0002526763452

001897677365

01738799329

minus06977939665

minus06945566457

001533190599

minus002803230310

006961922985

minus01374446216

06801869735

minus07159517722

] ] ] ] ] ] ] ]


24681012141618

12

34

56

Column Row

12

34

56


12345678

12

34

56

Column Row

12

34

56




120590119894119895= 119862


where 119862119894119895119896119897


120590 = 120590119894119895119890119894otimes 119890

119895 120598 = 120598

119896119897119890119896otimes 119890

119897

C = 119862119894119895119896119897

119890119894otimes 119890

119896otimes 119890

119896otimes 119890

119897

(23)

5

10

15

20

12

34

56

Column Row

12

34

56


5

10

15

20

25

12

34

56

Column Row

12

34

56



E1equiv 119890

1otimes 119890

1

E2equiv 119890

2otimes 119890

2

E3equiv 119890

3otimes 119890

3

(24)


E4equiv

1


2+ 119890

2otimes 119890

1)

E5equiv

1


3+ 119890

3otimes 119890

2)

E6equiv

1


1+ 119890

1otimes 119890

3)

(25)



E7equiv

1


2minus 119890

2otimes 119890

1)

E8equiv

1


3minus 119890

3otimes 119890

2)

E9equiv

1


1minus 119890

1otimes 119890

3)

(26)


120590 = 119878119860E

119860

120598 = 119864119860E

119860

C = 119862119860119861

119890119860otimes 119890

119861

(27)




120590119894119895= 119862

119894119895119896119897120598119896119897

lArrrArr 119878119860= 119862

119860119861E119861

119862119894119895119896119897

120598119896119897

= 120582120598119894119895lArrrArr 119862

119860119861E119861= 120582119864

119860

(28)


((((((((

(

1198781

1198782

1198783

1198784

1198785

1198786

1198787

1198788

1198789

))))))))

)

=

((((((((

(

12059011

12059022

12059033

120590(12)

120590(23)

120590(31)

120590[12]

120590[23]

120590[31]

))))))))

)

((((((((

(

1198641

1198642

1198643

1198644

1198645

1198646

1198647

1198648

1198649

))))))))

)

=

((((((((

(

12059811

12059822

12059833

120598(12)

120598(23)

120598(31)

120598[12]

120598[23]

120598[31]

))))))))

)

(29)


120579(119894119895)

equiv1

radic2(120579119894119895+ 120579

119895119894) 120579

[119894119895]equiv

1

radic2(120579119894119895minus 120579

119895119894) (30)


tensor C are

((((((((

(

11986211

11986212

11986213

11986214

11986215

11986216

11986217

11986218

11986219

11986221

11986222

11986223

11986224

11986225

11986226

11986227

11986228

11986229

11986231

11986232

11986233

11986234

11986235

11986236

11986237

11986238

11986239

11986241

11986242

11986243

11986244

11986245

11986246

11986247

11986248

11986249

11986251

11986252

11986253

11986254

11986255

11986256

11986257

11986258

11986259

11986261

11986262

11986263

11986264

11986265

11986266

11986267

11986268

11986269

11986271

11986272

11986273

11986274

11986275

11986276

11986277

11986278

11986279

11986281

11986282

11986283

11986284

11986285

11986286

11986287

11986288

11986289

11986291

11986292

11986293

11986294

11986295

11986296

11986297

11986298

11986299

))))))))

)

=

1198621111

1198621122

1198621133

1198622211

1198622222

1198623333

1198623311

1198623322

1198623333

11986211(12)

11986211(23)

11986211(31)

11986222(12)

11986222(23)

11986222(31)

11986233(12)

11986233(23)

11986233(31)

11986211[12]

11986211[23]

11986211[31]

11986222[12]

11986222[23]

11986222[31]

11986233[12]

11986233[23]

11986233[31]

119862(12)11

119862(12)22

119862(12)33

119862(23)11

119862(23)22

119862(23)33

119862(31)11

119862(31)22

119862(31)33

119862(1212)

119862(1223)

119862(1231)

119862(2312)

119862(2323)

119862(2331)

119862(3112)

119862(3123)

119862(3131)

119862(12)[12]

119862(12)[23]

119862(12)[31]

119862(23)[12]

119862(23)[23]

119862(23)[31]

119862(31)[12]

119862(31)[23]

119862(31)[31]

119862[12]11

119862[12]22

119862[12]33

119862[23]11

119862[23]22

119862[123]33

119862[21]11

119862[31]22

119862[31]33

119862[12](12)

119862[12](23)

119862[12](21)

119862[23](12)

119862[23](23)

119862[23](21)

119862[31](12)

119862[31](23)

119862[31](21)

119862[1212]

119862[1223]

119862[1231]

119862[2312]

119862[2323]

119862[2331]

119862[3112]

119862[3123]

119862[3131]

(31)

where

119862119894119895(119896119897)

equiv1

radic2(119862

119894119895119896119897+ 119862

119894119895119897119896) 119862

119894119895[119896119897]equiv

1

radic2(119862

119894119895119896119897minus 119862

119894119895119897119896)

119862(119894119895)119896119897

equiv1

radic2(119862

119894119895119896119897+ 119862

119895119894119896119897) 119862

[119894119895]119896119897equiv

1

radic2(119862

119894119895119896119897minus 119862

119895119894119896119897)

119862(119894119895119896119897)

equiv1

2(119862

119894119895119896119897+ 119862

119894119895119897119896+ 119862

119895119894119896119897+ 119862

119895119894119897119896)

119862(119894119895)[119896119897]

equiv1

2(119862

119894119895119896119897minus 119862

119894119895119897119896+ 119862

119895119894119896119897minus 119862

119895119894119897119896)

119862[119894119895](119896119897)

equiv1

2(119862

119894119895119896119897+ 119862

119894119895119897119896minus 119862

119895119894119896119897minus 119862

119895119894119897119896)

119862[119894119895119896119897]

equiv1

2(119862

119894119895119896119897minus 119862

119895119894119896119897minus 119862

119894119895119897119896+ 119862

119895119894119897119896)

(32)


12

34

56

ColumnRow

12

34

56

600005000040000300002000010000


5

10

15

20

25

12

34

56

Column Row

12

34

56






119895119894and 120598

119896119897= 120598

119897119896 the 119862

119894119895119896119897will be

like 119862119894119895119896119897

= 119862119895119894119896119897

= 119862119894119895119897119896


119862119894119895119896119897

= 119862119896119897119894119895












2 sdot 119864energy = 120590119894119895120598119894119895= 120590

119901120598119901 (33)


119901 = 119894120575119894119895+ (1 minus 120575

119894119895) (9 minus 119894 minus 119895)

119902 = 119896120575119896119897+ (1 minus 120575

119896119897) (9 minus 119896 minus 119897)

(34)


119901 119862

119894119895119896119897= 119862

119901119902 and 120598

119902= (2 minus

120575119896119897)120598119896119897



(

(

1205901

1205902

1205903

1205904

1205905

1205906

)

)

= (

(

11986211

11986212

11986213

11986214

11986215

11986216

11986221

11986222

11986223

11986224

11986225

11986226

11986231

11986232

11986233

11986234

11986235

11986236

11986241

11986242

11986243

11986244

11986245

11986246

11986251

11986252

11986253

11986254

11986255

11986256

11986261

11986262

11986263

11986264

11986265

11986266

)

)

(

(

1205981

1205982

1205983

1205984

1205985

1205986

)

)

(35)



(1) the 120590119894119895and 120598


(2) the norms of 120590119894119895 120598119896119897 and 119862




01 02 03 04 05 06 07 08

2

4

6

8

10

12

14

16

(a)

01

02

03

04

05

06

07

08

09

01 02 03 04 05 06 07 08

(b)


01 02 03 04 05 06 07 08

05

1

15

2

25

3

(a)

01 02 03 04 05 06 07 08

05

1

15

2

25

3

(b)




119901 = 119894120575119894119895+ (1 minus 120575

119894119895) (9 minus 119894 minus 119895)

119902 = 119896120575119896119897+ (1 minus 120575

119896119897) (9 minus 119896 minus 119897)

(36)

In this way

120590119901= (120575

119894119895+ radic2 (1 minus 120575

119894119895)) 120598

119894119895

120598119902= (120575

119896119897+ radic2 (1 minus 120575

119896119897)) 120598

119896119897

119862119901119902

= (120575119894119895+ radic2 (1 minus 120575

119894119895)) (120575

119896119897+ radic2 (1 minus 120575

119896119897)) 119862

119894119895119896119897

(37)




120590kelvin

= 120590viogt

120585 120590voigt

=120590kelvin

120585

120598kelvin

= 120598voigt

120585 120598voigt

=120598kelvin

120585

(38)


119862kelvin

=((

(







))

)

119862voigt

(39)


(

(

12059011

12059022

12059033

radic212059023

radic212059013

radic212059012

)

)

= (

1198621111 1198621122 1198621133radic21198621123

radic21198621113radic21198621112

1198622211 1198622222 1198622333radic21198622223

radic21198622213radic21198622212

1198623311 1198623322 1198623333radic21198623323

radic21198623313radic21198623312

radic21198622311radic21198622322

radic21198622333 21198622323 21198622313 21198622312

radic21198621311radic21198621322

radic21198621333 21198621323 21198621313 21198621312

radic21198621211radic21198621222

radic21198621233 21198621223 21198621213 21198621212

)

lowast(

(

12059811

12059822

12059833

radic212059823

radic212059813

radic212059812

)

)

(40)


= C (41)



(

(

12059011

12059022

12059033

radic212059023

radic212059013

radic212059012

)

)

=((

(

11986211

11986212

11986213

radic211986214

radic211986215

radic211986216

11986212

11986222

11986223

radic211986224

radic211986225

radic211986226

11986213

11986223

11986233

radic211986234

radic211986235

radic211986236

radic211986214

radic211986224

radic211986234

211986244

211986245

211986246

radic211986215

radic211986225

radic211986235

211986245

211986255

211986256

radic211986216

radic211986226

radic211986236

211986246

211986256

211986266

))

)

lowast (

(

12059811

12059822

12059833

radic212059823

radic212059813

radic212059812

)

)

(42)


(

(

1

2

3

4

5

6

)

)

=((

(

11986211

11986212

11986213

11986214

11986215

11986216

11986212

11986222

11986223

11986224

11986225

11986226

11986213

11986223

11986233

11986234

11986235

11986236

11986214

11986224

11986234

11986244

11986245

11986246

11986215

11986225

11986235

11986245

11986255

11986256

11986216

11986226

11986236

11986246

11986256

11986266

))

)

lowast (

(

1205981

1205982

1205983

1205984

1205985

1205986

)

)

(43)



































01 02 03 04 05 06 07 08

05

1

15

2

25

3

(a)

01 02 03 04 05 06 07 08

05

1

15

2

25

3

(b)


01 02 03 04 05 06 07 08

2

4

6

8

10

12

14

16





ΛI) N = 0 (44)


119894 119894 = 1 2 6 and

















NNA119896

equiv1

119872

119872

sum

119884=1

N119884119896 (45)



NAVG119896

equiv JNANNA119896

(46)


119896=1NNA119896

otimes NNA119896

| that is

JNA equiv (

6

sum

119896=1

NNA119896

otimes NNA119896

)

minus12

(47)


119896 that is

ΛAVG119902

equiv1

119872

119872

sum

119884=1

6

sum

119896=1

Λ119884

119896(N119884

119896sdot NAVG

119902)2

(48)


gt 0 for all 119896 = 1 2 6

and ΛAVG119896


NA119896


ΛNA119896

equiv1

119872

119872

sum

119884=1

Λ119884

119896 (49)


CAVG=

6

sum

119896=1

ΛAVG119896

NAVG119896

otimes NAVG119896

(50)





[[[[[[[

[

00367834559700000

00453681054800000

0998185540700000

00

00

00

]]]]]]]

]

[[[[[[[

[

0983745905000000

minus0170879918750000

minus00277901141300000

00

00

00

]]]]]]]

]

[[[[[[[

[

minus0169284222800000

minus0982986098000000

00509905132200000

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(51)


[[[[[[[

[

003679134179

004537783172

09983995367

00

00

00

]]]]]]]

]

[[[[[[[

[

09859858256

minus01712690003

minus002785339027

00

00

00

]]]]]]]

]

[[[[[[[

[

minus01697052873

minus09854311019

005111734310

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(52)




[[[

[

334725276837330 0000112116752981933 000198542359441849 00 00 00

0000112116752981933 334773746006714 minus0000991802322788501 00 00 00

000198542359441849 minus0000991802322788501 334013593769726 00 00 00

00 00 00 333950000000000 00 00

00 00 00 00 333950000000000 00

00 00 00 00 00 333950000000000

]]]

]

(53)




[[[[[[[

[

004028666644

006399313350

09971368323

00

00

00

]]]]]]]

]

[[[[[[[

[

09048796003

minus04255671399

minus0009247686096

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04237568827

minus09026613361

007505076253

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(54)


[[[[[[[

[

004028666644

006399313350

09971368323

00

00

00

]]]]]]]

]

[[[[[[[

[

09048795994

minus04255671395

minus0009247686087

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04237568827

minus09026613361

007505076253

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(55)



[[[

[

1458799998 000000001785571215 minus000000001009489597 00 00 00000000001785571215 1458799997 0000000002407020197 00 00 00minus000000001009489597 0000000002407020197 1458799997 00 00 00

00 00 00 1458799998 00 0000 00 00 00 1458799998 0000 00 00 00 00 1458799998

]]]

]

(56)




[[[[[

[

005368516527

007220768570

09959437502

00

00

00

]]]]]

]

[[[[[

[

09266208315

minus03753089731

minus002273783078

00

00

00

]]]]]

]

[[[[[

[

minus03721447810

minus09240829102

008705767064

00

00

00

]]]]]

]

[[[[[

[

00

00

00

00

00

10

]]]]]

]

[[[[[

[

00

00

00

10

00

00

]]]]]

]

[[[[[

[

00

00

00

00

10

00

]]]]]

]

(57)


[[[[[[[[[

[

005368516527

007220768570

09959437502

00

00

00

]]]]]]]]]

]

[[[[[[[[[

[

09266208315

minus03753089731

minus002273783078

00

00

00

]]]]]]]]]

]

[[[[[[[[[

[

minus03721447806

minus09240829093

008705767055

00

00

00

]]]]]]]]]

]

[[[[[[[[[

[

00

00

00

10

00

00

]]]]]]]]]

]

[[[[[[[[[

[

00

00

00

00

10

00

]]]]]]]]]

]

[[[[[[[[[

[

00

00

00

00

00

10

]]]]]]]]]

]

(58)




[[[

[

1615299998 0000000005508173090 minus0000000008722620038 00 00 00

0000000005508173090 1615299996 minus0000000004345156817 00 00 00

minus0000000008722620038 minus0000000004345156817 1615300000 00 00 00

00 00 00 1615299998 00 00

00 00 00 00 1615299998 00

00 00 00 00 00 1615299998

]]]

]

(59)




[[[[[[[

[

minus00598757013800000

minus00768229223150000

minus0995231841750000

00

00

00

]]]]]]]

]

[[[[[[[

[

minus0869358919900000

0493939153600000

00141567750950000

00

00

00

]]]]]]]

]

[[[[[[[

[

minus0490490276500000

minus0866078136900000

00963811082600000

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(60)


[[[[[[[[[[[

[

minus005987730132

minus007682497510

minus09952584354

00

00

00

]]]]]]]]]]]

]

[[[[[[[[[[[

[

minus08693926232

04939583026

001415732393

00

00

00

]]]]]]]]]]]

]

[[[[[[[[[[[

[

00

00

00

10

00

00

]]]]]]]]]]]

]

[[[[[[[[[[[

[

00

00

00

00

10

00

]]]]]]]]]]]

]

[[[[[[[[[[[

[

00

00

00

00

00

10

]]]]]]]]]]]

]

(61)



[[[

[


00 00 00 1819400002 1819400002 0000 00 00 00 00 0000 00 00 00 00 1819400002

]]]

]

(62)




[[[[[[[

[

004967528691

008463937788

09951726186

00

00

00

]]]]]]]

]

[[[[[[[

[

09161715971

minus04006166011

minus001165943224

00

00

00

]]]]]]]

]

[[[[[

[

minus03976958243

minus09123280736

009744493938

00

00

00

]]]]]

]

[[[[[

[

00

00

00

10

00

00

]]]]]

]

[[[[[

[

00

00

00

00

10

00

]]]]]

]

[[[[[

[

00

00

00

00

00

10

]]]]]

]

(63)


[[[[[[[

[

004967528696

008463937796

09951726196

00

00

00

]]]]]]]

]

[[[[[[[

[

09161715980

minus04006166015

minus001165943225

00

00

00

]]]]]]]

]


[[[[[[[

[

minus03976958247

minus09123280745

009744493948

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(64)



[[[

[

1922399998 minus000000001176508811 minus000000001537920006 00 00 00

minus000000001176508811 1922400000 minus0000000007631928036 00 00 00

minus000000001537920006 minus0000000007631928036 1922400000 00 00 00

00 00 00 1922399998 1922399998 00

00 00 00 00 00 00

00 00 00 00 00 1922399998

]]]

]

(65)




[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

[[[[[[[

[

minus01387533797

minus02031928413

minus09692575344

00

00

00

]]]]]]]

]

[[[[[[[

[

minus08476774112

05304148448

001015375725

00

00

00

]]]]]]]

]

[[[[[[[

[

minus05120454135

minus08230265872

02458388325

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

(66)


[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

[[[[[[[

[

minus01387533798

minus02031928415

minus09692575354

00

00

00

]]]]]]]

]

[[[[[[[

[

minus08476774112

05304148448

001015375725

00

00

00

]]]]]]]

]

[[[[[[[

[

minus05120454135

minus08230265872

02458388325

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

(67)



[[[

[

2074599999 0000000004356660361 000000003402343994 00 00 00

0000000004356660361 2074600004 000000002232269567 00 00 00

000000003402343994 000000002232269567 2074600000 00 00 00

00 00 00 2074600000 2074600000 00

00 00 00 00 00 00

00 00 00 00 00 2074600000

]]]

]

(68)




[[[[[[[

[

008621257414

01243595697

09884847438

00

00

00

]]]]]]]

]

[[[[[[[

[

08755641188

minus04828494241

minus001561753067

00

00

00

]]]]]]]

]

[[[[[

[

minus04753471023

minus08668282006

01505124700

00

00

00

]]]]]

]

[[[[[

[

00

00

00

10

00

00

]]]]]

]

[[[[[

[

00

00

00

00

10

00

]]]]]

]

[[[[[

[

00

00

00

00

00

10

]]]]]

]

(69)


[[[[[

[

008621257423

01243595698

09884847448

00

00

00

]]]]]

]

[[[[[

[

08755641188

minus04828494241

minus001561753067

00

00

00

]]]]]

]


[[[[[[[

[

minus04753471018

minus08668281997

01505124698

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(70)



[[[

[

1890099999 000000001209663993 minus000000002532733996 00 00 00000000001209663993 1890099991 0000000001701090138 00 00 00minus000000002532733996 0000000001701090138 1890100001 00 00 00

00 00 00 1890099997 1890099997 0000 00 00 00 00 0000 00 00 00 00 1890099997

]]]

]

(71)




[[[[[[[

[

03961520086

06338768023

06642768888

00

00

00

]]]]]]]

]

[[[[[[[

[

09160614623

minus02236797319

minus03328645006

00

00

00

]]]]]]]

]

[[[[[[[

[

006240981414

minus07403833993

06692812840

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(72)


[[[[[[[

[

03961520086

06338768023

06642768888

00

00

00

]]]]]]]

]

[[[[[[[

[

09160614614

minus02236797317

minus03328645003

00

00

00

]]]]]]]

]

[[[[[[[

[

006240981426

minus07403834008

06692812853

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(73)



[[[

[


00 00 00 8772299998 8772299998 0000 00 00 00 00 0000 00 00 00 00 8772299998

]]]

]

(74)




[[[[[[[

[

minus006591789198

minus008975775227

minus09937798435

00

00

00

]]]]]]]

]

[[[[[[[

[

minus08289381135

05593224991

0004466103673

00

00

00

]]]]]]]

]

[[[[[

[

minus05554425577

minus08240763851

01112729817

00

00

00

]]]]]

]

[[[[[

[

00

00

00

10

00

00

]]]]]

]

[[[[[

[

00

00

00

00

10

00

]]]]]

]

[[[[[

[

00

00

00

00

00

10

]]]]]

]


[[[[[[[

[

minus01387533798

minus02031928415

minus09692575354

00

00

00

]]]]]]]

]

[[[[[[[

[

minus08476774112

05304148448

001015375725

00

00

00

]]]]]]]

]


[[[[

[

minus05120454135

minus08230265872

02458388325

00

00

00

]]]]

]

[[[[

[

00

00

00

10

00

00

]]]]

]

[[[[

[

00

00

00

00

10

00

]]]]

]

[[[[

[

00

00

00

00

00

10

]]]]

]

(76)



[[[

[

2228057494 0000000004678921127 000000003654014285 00 00 000000000004678921127 2228057499 000000002397389829 00 00 00000000003654014285 000000002397389829 2228057495 00 00 00

00 00 00 2228057495 2228057495 0000 00 00 00 00 0000 00 00 00 00 2228057495

]]]

]

(77)




[[[[[[[

[

006975010637

01071019008

09917984199

00

00

00

]]]]]]]

]

[[[[[[[

[

09079000793

minus04187725641

minus001862756541

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04133429180

minus09017531389

01264472547

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(78)


[[[[[[[

[

006975010637

01071019008

09917984199

00

00

00

]]]]]]]

]

[[[[[[[

[

09079000793

minus04187725641

minus001862756541

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04133429184

minus09017531398

01264472548

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(79)



[[[

[


00 00 00 2598699998 2598699998 0000 00 00 00 00 0000 00 00 00 00 2598699998

]]]

]

(80)




[[[[[

[

minus00192522847250000

minus00192842313600000

000386151499999998

00

00

00

]]]]]

]

[[[[[

[

000961799224999998

00165029120500000

000436480214750000

00

00

00

]]]]]

]

[[[[[[

[

minus0494444050150000

minus0856906622200000

0142104790200000

00

00

00

]]]]]]

]

[[[[[[

[

00

00

00

10

00

00

]]]]]]

]

[[[[[[

[

00

00

00

00

10

00

]]]]]]

]

[[[[[[

[

00

00

00

00

00

10

]]]]]]

]

(81)



[[[[[[[[[[

[

minus2541745843

minus2545963537

5098090872

00

00

00

]]]]]]]]]]

]

[[[[[[[[[[

[

2505315863

4298714977

1136953303

00

00

00

]]]]]]]]]]

]

[[[[[[[

[

minus04949599718

minus08578007511

01422530677

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(82)



[[[

[

315679169478799 427324383657779 384557503470365 00 00 00427324383657779 618700481877479 889153535276175 00 00 00384557503470365 889153535276175 384768762708609 00 00 00

00 00 00 247795001400000 00 0000 00 00 00 247795001400000 0000 00 00 00 00 247795001400000

]]]

]

(83)




[[[[[[[

[

01135176976

01764907831

09777344911

00

00

00

]]]]]]]

]

[[[[[[[

[

08848520771

minus04654963442

minus001870707734

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04518302069

minus08672739791

02090103088

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(84)


[[[[[[[

[

01135176977

01764907833

09777344921

00

00

00

]]]]]]]

]

[[[[[[[

[

08848520771

minus04654963442

minus001870707734

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04518302069

minus08672739791

02090103088

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(85)



[[[

[

2663700003 000000004768023095 000000003169803038 00 00 00000000004768023095 2663699992 0000000008310743498 00 00 00000000003169803038 0000000008310743498 2663700001 00 00 00

00 00 00 2663699998 2663699998 0000 00 00 00 00 0000 00 00 00 00 2663699998

]]]

]

(86)





6 Conclusions









Disclosure




Acknowledgments



References




















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












986 7912 698311986212

2147 3609 281611986213

2773 3186 284511986214

minus0162 11 8011986215

minus293 54 minus0711986216

minus0821 minus06 11411986222

935 8529 896711986223

2346 3281 322711986224

minus1204 09 minus162311986225


0445 minus42 16011986233

5952 14730 916011986234


minus1791 169 minus7211986236

054 06 2411986244

561 3587 348911986245

0798 05 10511986246

minus2159 51 minus9811986255

6032 3448 242611986256



24681012141618

12

34

56

Column Row

12

34

56



5

10

15

20

12

34

56

Column Row

12

34

56



119890120595(119894119895)

= 119890119894otimes 119890

119895 (20)




1 Quipo

[[[[[[[[[

[

1076866026

004067345638

02534605191

02260000000

01180000000

007800000000

]]]]]]]]]

]

[[[[[[[[[

[

003276733390 09918780499 minus01228992897 00 00 00

003131140851 minus01239237163 minus09917976143 00 00 00

09989724208 minus002865041271 003511774899 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

2 Quipo

[[[[[[[[[

[

3461963876

01399776173

04300584948

04300000000

02800000000

01440000000

]]]]]]]]]

]

[[[[[[[[[

[

004079957804 09756137601 minus02156691559 00 00 00

005942480245 minus02178361212 minus09741745817 00 00 00

09973986606 minus002692981555 006686327745 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

3 White

[[[[[[[[[

[

1009116301

03556701722

1333166815

1442000000

1344000000

002200000000

]]]]]]]]]

]

[[[[[[[[[

[

004028666644 09048796003 minus04237568827 00 00 00

006399313350 minus04255671399 minus09026613361 00 00 00

09971368323 minus0009247686096 007505076253 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

4 Khaya

[[[[[[[[[

[

1080102614

04606834511

1475290392

1800000000

1196000000

04200000000

]]]]]]]]]

]

[[[[[[[[[

[

005368516527 09266208315 minus03721447810 00 00 00

007220768570 minus03753089731 minus09240829102 00 00 00

09959437502 minus002273783078 008705767064 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

5 Mahogany

[[[[[[[[[

[

1210253321

06098853915

1810581412

1960000000

1498000000

06380000000

]]]]]]]]]

]

[[[[[[[[[

[


minus007817061387 04985803653 minus08633116316 00 00 00

minus09948285648 001731853005 01000809391 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

6 Mahogany

[[[[[[[[[

[

1310854783

03895295651

1814922632

1218000000

09380000000

03000000000

]]]]]]]]]

]

[[[[[[[[[

[


minus007547523076 04892979419 minus08688446422 00 00 00

minus09956351187 001099502014 009268127742 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

7 S Germ

[[[[[[[[[

[

1234053410

05212570563

1922208822

2318000000

1582000000

05400000000

]]]]]]]]]

]

[[[[[[[[[

[

004967528691 09161715971 minus03976958243 00 00 00

008463937788 minus04006166011 minus09123280736 00 00 00

09951726186 minus001165943224 009744493938 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

8 Maple

[[[[[[[[[

[

1205449768

06868279555

2766674361

2460000000

2194000000

05840000000

]]]]]]]]]

]

[[[[[[[[[

[


minus02031928413 05304148448 minus08230265872 00 00 00

minus09692575344 001015375725 02458388325 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]




9 Walnut

[[[[[[[[[

[

1267869546

05204134969

1919891015

1922000000

1400000000

04600000000

]]]]]]]]]

]

[[[[[[[[[

[

008621257414 08755641188 minus04753471023 00 00 00

01243595697 minus04828494241 minus08668282006 00 00 00

09884847438 minus001561753067 01505124700 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

10 Birch

[[[[[[[[[

[

3168908680

04747157903

05946755301

2346000000

1816000000

03720000000

]]]]]]]]]

]

[[[[[[[[[

[

03961520086 09160614623 006240981414 00 00 00

06338768023 minus02236797319 minus07403833993 00 00 00

06642768888 minus03328645006 06692812840 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

11 Y Birch

[[[[[[[[[

[

1545414040

05549651499

2059894445

2120000000

1942000000

04800000000

]]]]]]]]]

]

[[[[[[[[[

[


minus008975775227 05593224991 minus08240763851 00 00 00

minus09937798435 0004466103673 01112729817 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

12 Oak

[[[[[[[[[

[

1718666431

08648974218

3239438239

2380000000

1532000000

07840000000

]]]]]]]]]

]

[[[[[[[[[

[

006975010637 09079000793 minus04133429180 00 00 00

01071019008 minus04187725641 minus09017531389 00 00 00

09917984199 minus001862756541 01264472547 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

13 Ash

[[[[[[[[[

[

1715567967

06696042476

2409716064

2684000000

1784000000

05400000000

]]]]]]]]]

]

[[[[[[[[[

[

006190313535 08746549514 minus04807772027 00 00 00

009781749768 minus04846986500 minus08691944279 00 00 00

09932772717 minus0006777439445 01155609239 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

14 Ash

[[[[[[[[[

[

1742363268

07844815431

2669885744

1720000000

1218000000

05000000000

]]]]]]]]]

]

[[[[[[[[[

[


minus01363859604 05177044741 minus08446188165 00 00 00

minus09855542417 001550704374 01686486565 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

15 Beech

[[[[[[[[[

[

1599002755

09558575345

3451114905

3216000000

2112000000

09120000000

]]]]]]]]]

]

[[[[[[[[[

[

01135176976 08848520771 minus04518302069 00 00 00

01764907831 minus04654963442 minus08672739791 00 00 00

09777344911 minus001870707734 02090103088 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

Thus 119890120572otimes 119890

1205731le120572120573le119899




119860119894119895119896119897

119890119894otimes 119890

119895otimes 119890

119896otimes 119890

119897 or as an 119899


120572120573119890120572otimes 119890

120573 where 119860

120595(119894119895)120595119896119897= 119860

119894119895119896119897

1 le 119894 119895 119896 119897 le 119899


tr (A119879B) = ⟨AB⟩ = tr (119860119879119861119879) A =1003817100381710038171003817100381711986010038171003817100381710038171003817 (21)






1 Blasa

[[[[[[[[[

[

6385324208

009845837744

03832174064

06240000000

04060000000

006600000000

]]]]]]]]]

]

[[[[[[[[[

[

001488818113 09510211778 minus03087669978 00 00 00

002575997614 minus03090635474 minus09506924583 00 00 00

09995572851 minus0006200252135 002909967665 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

2 Spruce

[[[[[[[[[

[

1198583568

04516442734

1114520065

1498000000

1442000000

007800000000

]]]]]]]]]

]

[[[[[[[[[

[


minus004689865645 04044467539 minus09133582749 00 00 00

minus09983543167 001123114838 005623628701 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

3 Spruce

[[[[[[[[[

[

1410927768

04185382987

1227183961

1442000000

10

006400000000

]]]]]]]]]

]

[[[[[[[[[

[


minus005361649666 04423303564 minus08952480817 00 00 00

minus09978936411 0009052295016 006423657043 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

4 Spruce

[[[[[[[[[

[

1630529953

03544288592

08442715844

1234000000

1520000000

007200000000

]]]]]]]]]

]

[[[[[[[[[

[


minus002869707087 04091564350 minus09120128765 00 00 00

minus09993764538 0007032901380 003460119282 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

5 Spruce

[[[[[[[[[

[

1725845208

05092652753

1171282625

1250000000

1706000000

007000000000

]]]]]]]]]

]

[[[[[[[[[

[


minus003428302033 05858146064 minus08097196604 00 00 00

minus09988364173 0007413313465 004765347716 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

6 Douglas Fir

[[[[[[[[[

[

1613459769

06233733771

1439028943

1767000000

1766000000

01760000000

]]]]]]]]]

]

[[[[[[[[[

[


minus003697194691 05922678885 minus08048924305 00 00 00

minus09986533619 0007435777607 005134366998 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

7 Douglas Fir

[[[[[[[[[

[

1710150156

06986859431

2204812526

2348000000

1816000000

01600000000

]]]]]]]]]

]

[[[[[[[[[

[


minus006364825251 05704773676 minus08188433771 00 00 00

minus09967231590 0004703484281 008075160717 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

8 Pine

[[[[[[[[[

[

1699539133

04940019684

1565606760

3484000000

1344000000

01320000000

]]]]]]]]]

]

[[[[[[[[[

[


minus005585205149 04413516031 minus08955943895 00 00 00

minus09978476167 0006195562437 006528207193 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]


Table11Th

eeigenvalues

andeigenvectorsfor3

specim

enso

fcancello

usbo

necalculated

usingMAPL

EΛ[119894]sta

ndforthe

eigenvalueso

f119894th

specim

enof

cancellous

boneand

119881[119894]sta

ndfor

thec

orrespon

ding

eigenvectors

Cancellous

bone

EigenvaluesΛ

[119894]

Eigenvectors

119881[119894]

Specim

en1

[ [ [ [ [ [ [ [

1334869880

minus02071167473

8644210768

6837676373

3768806080

4123724855

] ] ] ] ] ] ] ]

[ [ [ [ [ [ [ [

03316115964

08831238953

02356441950

02242668491

005354786914

003787816350

06475092156

minus01099136965

01556396475

minus06940063479

minus01278953723

02154647417

04800906963

minus02832250652

007648827487

02494278793

06410471445

minus04585742261

minus02737532692

minus006354940263

04518342532

minus006214920576

05836952694

06101669758

minus03706582070

03314035780

minus01276606259

minus06337030234

03639432801

minus04499474345

01671829224

01180163925

minus08330343502

002008124471

03109325968

04087706408

] ] ] ] ] ] ] ]

Specim

en2

[ [ [ [ [ [ [ [

1754634171

7773848354

1550688791

2301790452

3445174379

3589156342

] ] ] ] ] ] ] ]

[ [ [ [ [ [ [ [

minus03057290719

minus03005337771

minus09030341401

001584500766

minus002175914011

minus0003741933916

minus04311831740

minus08021570593

04130146809

00001362645283

minus0003854432451

0005396441669

minus08488209390

05154657137

01152135966

minus0004741957786

002229739182

minus0002068264586

00009402467441

minus0005358584867

0003987518000

minus003989381331

003796849613

minus09984595017

minus001058157817

002100470582

002092543658

0006230946613

minus09987520802

minus003826770492

00009823402498

0006977574167

001484146906

09990475909

0008196718814

minus003958286493

] ] ] ] ] ] ] ]

Specim

en3

[ [ [ [ [ [ [ [

1418542670

5838388363

9959058231

3433818493

3024968125

1757492470

] ] ] ] ] ] ] ]

[ [ [ [ [ [ [ [

03244960223

minus0004831643783

minus08451713783

04062092722

01164524794

minus004239310198

06461152835

minus07125334236

02668138654

004131783689

minus002431040234

003665187401

06621315621

07009969173

02633438279

002836289512

minus0001711614840

0005269111592

minus01962401199

0009159859564

03740839141

08850746598

01904577552

004284654369

minus0008597323304

minus0002526763452

001897677365

01738799329

minus06977939665

minus06945566457

001533190599

minus002803230310

006961922985

minus01374446216

06801869735

minus07159517722

] ] ] ] ] ] ] ]


24681012141618

12

34

56

Column Row

12

34

56


12345678

12

34

56

Column Row

12

34

56




120590119894119895= 119862


where 119862119894119895119896119897


120590 = 120590119894119895119890119894otimes 119890

119895 120598 = 120598

119896119897119890119896otimes 119890

119897

C = 119862119894119895119896119897

119890119894otimes 119890

119896otimes 119890

119896otimes 119890

119897

(23)

5

10

15

20

12

34

56

Column Row

12

34

56


5

10

15

20

25

12

34

56

Column Row

12

34

56



E1equiv 119890

1otimes 119890

1

E2equiv 119890

2otimes 119890

2

E3equiv 119890

3otimes 119890

3

(24)


E4equiv

1


2+ 119890

2otimes 119890

1)

E5equiv

1


3+ 119890

3otimes 119890

2)

E6equiv

1


1+ 119890

1otimes 119890

3)

(25)



E7equiv

1


2minus 119890

2otimes 119890

1)

E8equiv

1


3minus 119890

3otimes 119890

2)

E9equiv

1


1minus 119890

1otimes 119890

3)

(26)


120590 = 119878119860E

119860

120598 = 119864119860E

119860

C = 119862119860119861

119890119860otimes 119890

119861

(27)




120590119894119895= 119862

119894119895119896119897120598119896119897

lArrrArr 119878119860= 119862

119860119861E119861

119862119894119895119896119897

120598119896119897

= 120582120598119894119895lArrrArr 119862

119860119861E119861= 120582119864

119860

(28)


((((((((

(

1198781

1198782

1198783

1198784

1198785

1198786

1198787

1198788

1198789

))))))))

)

=

((((((((

(

12059011

12059022

12059033

120590(12)

120590(23)

120590(31)

120590[12]

120590[23]

120590[31]

))))))))

)

((((((((

(

1198641

1198642

1198643

1198644

1198645

1198646

1198647

1198648

1198649

))))))))

)

=

((((((((

(

12059811

12059822

12059833

120598(12)

120598(23)

120598(31)

120598[12]

120598[23]

120598[31]

))))))))

)

(29)


120579(119894119895)

equiv1

radic2(120579119894119895+ 120579

119895119894) 120579

[119894119895]equiv

1

radic2(120579119894119895minus 120579

119895119894) (30)


tensor C are

((((((((

(

11986211

11986212

11986213

11986214

11986215

11986216

11986217

11986218

11986219

11986221

11986222

11986223

11986224

11986225

11986226

11986227

11986228

11986229

11986231

11986232

11986233

11986234

11986235

11986236

11986237

11986238

11986239

11986241

11986242

11986243

11986244

11986245

11986246

11986247

11986248

11986249

11986251

11986252

11986253

11986254

11986255

11986256

11986257

11986258

11986259

11986261

11986262

11986263

11986264

11986265

11986266

11986267

11986268

11986269

11986271

11986272

11986273

11986274

11986275

11986276

11986277

11986278

11986279

11986281

11986282

11986283

11986284

11986285

11986286

11986287

11986288

11986289

11986291

11986292

11986293

11986294

11986295

11986296

11986297

11986298

11986299

))))))))

)

=

1198621111

1198621122

1198621133

1198622211

1198622222

1198623333

1198623311

1198623322

1198623333

11986211(12)

11986211(23)

11986211(31)

11986222(12)

11986222(23)

11986222(31)

11986233(12)

11986233(23)

11986233(31)

11986211[12]

11986211[23]

11986211[31]

11986222[12]

11986222[23]

11986222[31]

11986233[12]

11986233[23]

11986233[31]

119862(12)11

119862(12)22

119862(12)33

119862(23)11

119862(23)22

119862(23)33

119862(31)11

119862(31)22

119862(31)33

119862(1212)

119862(1223)

119862(1231)

119862(2312)

119862(2323)

119862(2331)

119862(3112)

119862(3123)

119862(3131)

119862(12)[12]

119862(12)[23]

119862(12)[31]

119862(23)[12]

119862(23)[23]

119862(23)[31]

119862(31)[12]

119862(31)[23]

119862(31)[31]

119862[12]11

119862[12]22

119862[12]33

119862[23]11

119862[23]22

119862[123]33

119862[21]11

119862[31]22

119862[31]33

119862[12](12)

119862[12](23)

119862[12](21)

119862[23](12)

119862[23](23)

119862[23](21)

119862[31](12)

119862[31](23)

119862[31](21)

119862[1212]

119862[1223]

119862[1231]

119862[2312]

119862[2323]

119862[2331]

119862[3112]

119862[3123]

119862[3131]

(31)

where

119862119894119895(119896119897)

equiv1

radic2(119862

119894119895119896119897+ 119862

119894119895119897119896) 119862

119894119895[119896119897]equiv

1

radic2(119862

119894119895119896119897minus 119862

119894119895119897119896)

119862(119894119895)119896119897

equiv1

radic2(119862

119894119895119896119897+ 119862

119895119894119896119897) 119862

[119894119895]119896119897equiv

1

radic2(119862

119894119895119896119897minus 119862

119895119894119896119897)

119862(119894119895119896119897)

equiv1

2(119862

119894119895119896119897+ 119862

119894119895119897119896+ 119862

119895119894119896119897+ 119862

119895119894119897119896)

119862(119894119895)[119896119897]

equiv1

2(119862

119894119895119896119897minus 119862

119894119895119897119896+ 119862

119895119894119896119897minus 119862

119895119894119897119896)

119862[119894119895](119896119897)

equiv1

2(119862

119894119895119896119897+ 119862

119894119895119897119896minus 119862

119895119894119896119897minus 119862

119895119894119897119896)

119862[119894119895119896119897]

equiv1

2(119862

119894119895119896119897minus 119862

119895119894119896119897minus 119862

119894119895119897119896+ 119862

119895119894119897119896)

(32)


12

34

56

ColumnRow

12

34

56

600005000040000300002000010000


5

10

15

20

25

12

34

56

Column Row

12

34

56






119895119894and 120598

119896119897= 120598

119897119896 the 119862

119894119895119896119897will be

like 119862119894119895119896119897

= 119862119895119894119896119897

= 119862119894119895119897119896


119862119894119895119896119897

= 119862119896119897119894119895












2 sdot 119864energy = 120590119894119895120598119894119895= 120590

119901120598119901 (33)


119901 = 119894120575119894119895+ (1 minus 120575

119894119895) (9 minus 119894 minus 119895)

119902 = 119896120575119896119897+ (1 minus 120575

119896119897) (9 minus 119896 minus 119897)

(34)


119901 119862

119894119895119896119897= 119862

119901119902 and 120598

119902= (2 minus

120575119896119897)120598119896119897



(

(

1205901

1205902

1205903

1205904

1205905

1205906

)

)

= (

(

11986211

11986212

11986213

11986214

11986215

11986216

11986221

11986222

11986223

11986224

11986225

11986226

11986231

11986232

11986233

11986234

11986235

11986236

11986241

11986242

11986243

11986244

11986245

11986246

11986251

11986252

11986253

11986254

11986255

11986256

11986261

11986262

11986263

11986264

11986265

11986266

)

)

(

(

1205981

1205982

1205983

1205984

1205985

1205986

)

)

(35)



(1) the 120590119894119895and 120598


(2) the norms of 120590119894119895 120598119896119897 and 119862




01 02 03 04 05 06 07 08

2

4

6

8

10

12

14

16

(a)

01

02

03

04

05

06

07

08

09

01 02 03 04 05 06 07 08

(b)


01 02 03 04 05 06 07 08

05

1

15

2

25

3

(a)

01 02 03 04 05 06 07 08

05

1

15

2

25

3

(b)




119901 = 119894120575119894119895+ (1 minus 120575

119894119895) (9 minus 119894 minus 119895)

119902 = 119896120575119896119897+ (1 minus 120575

119896119897) (9 minus 119896 minus 119897)

(36)

In this way

120590119901= (120575

119894119895+ radic2 (1 minus 120575

119894119895)) 120598

119894119895

120598119902= (120575

119896119897+ radic2 (1 minus 120575

119896119897)) 120598

119896119897

119862119901119902

= (120575119894119895+ radic2 (1 minus 120575

119894119895)) (120575

119896119897+ radic2 (1 minus 120575

119896119897)) 119862

119894119895119896119897

(37)




120590kelvin

= 120590viogt

120585 120590voigt

=120590kelvin

120585

120598kelvin

= 120598voigt

120585 120598voigt

=120598kelvin

120585

(38)


119862kelvin

=((

(







))

)

119862voigt

(39)


(

(

12059011

12059022

12059033

radic212059023

radic212059013

radic212059012

)

)

= (

1198621111 1198621122 1198621133radic21198621123

radic21198621113radic21198621112

1198622211 1198622222 1198622333radic21198622223

radic21198622213radic21198622212

1198623311 1198623322 1198623333radic21198623323

radic21198623313radic21198623312

radic21198622311radic21198622322

radic21198622333 21198622323 21198622313 21198622312

radic21198621311radic21198621322

radic21198621333 21198621323 21198621313 21198621312

radic21198621211radic21198621222

radic21198621233 21198621223 21198621213 21198621212

)

lowast(

(

12059811

12059822

12059833

radic212059823

radic212059813

radic212059812

)

)

(40)


= C (41)



(

(

12059011

12059022

12059033

radic212059023

radic212059013

radic212059012

)

)

=((

(

11986211

11986212

11986213

radic211986214

radic211986215

radic211986216

11986212

11986222

11986223

radic211986224

radic211986225

radic211986226

11986213

11986223

11986233

radic211986234

radic211986235

radic211986236

radic211986214

radic211986224

radic211986234

211986244

211986245

211986246

radic211986215

radic211986225

radic211986235

211986245

211986255

211986256

radic211986216

radic211986226

radic211986236

211986246

211986256

211986266

))

)

lowast (

(

12059811

12059822

12059833

radic212059823

radic212059813

radic212059812

)

)

(42)


(

(

1

2

3

4

5

6

)

)

=((

(

11986211

11986212

11986213

11986214

11986215

11986216

11986212

11986222

11986223

11986224

11986225

11986226

11986213

11986223

11986233

11986234

11986235

11986236

11986214

11986224

11986234

11986244

11986245

11986246

11986215

11986225

11986235

11986245

11986255

11986256

11986216

11986226

11986236

11986246

11986256

11986266

))

)

lowast (

(

1205981

1205982

1205983

1205984

1205985

1205986

)

)

(43)



































01 02 03 04 05 06 07 08

05

1

15

2

25

3

(a)

01 02 03 04 05 06 07 08

05

1

15

2

25

3

(b)


01 02 03 04 05 06 07 08

2

4

6

8

10

12

14

16





ΛI) N = 0 (44)


119894 119894 = 1 2 6 and

















NNA119896

equiv1

119872

119872

sum

119884=1

N119884119896 (45)



NAVG119896

equiv JNANNA119896

(46)


119896=1NNA119896

otimes NNA119896

| that is

JNA equiv (

6

sum

119896=1

NNA119896

otimes NNA119896

)

minus12

(47)


119896 that is

ΛAVG119902

equiv1

119872

119872

sum

119884=1

6

sum

119896=1

Λ119884

119896(N119884

119896sdot NAVG

119902)2

(48)


gt 0 for all 119896 = 1 2 6

and ΛAVG119896


NA119896


ΛNA119896

equiv1

119872

119872

sum

119884=1

Λ119884

119896 (49)


CAVG=

6

sum

119896=1

ΛAVG119896

NAVG119896

otimes NAVG119896

(50)





[[[[[[[

[

00367834559700000

00453681054800000

0998185540700000

00

00

00

]]]]]]]

]

[[[[[[[

[

0983745905000000

minus0170879918750000

minus00277901141300000

00

00

00

]]]]]]]

]

[[[[[[[

[

minus0169284222800000

minus0982986098000000

00509905132200000

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(51)


[[[[[[[

[

003679134179

004537783172

09983995367

00

00

00

]]]]]]]

]

[[[[[[[

[

09859858256

minus01712690003

minus002785339027

00

00

00

]]]]]]]

]

[[[[[[[

[

minus01697052873

minus09854311019

005111734310

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(52)




[[[

[

334725276837330 0000112116752981933 000198542359441849 00 00 00

0000112116752981933 334773746006714 minus0000991802322788501 00 00 00

000198542359441849 minus0000991802322788501 334013593769726 00 00 00

00 00 00 333950000000000 00 00

00 00 00 00 333950000000000 00

00 00 00 00 00 333950000000000

]]]

]

(53)




[[[[[[[

[

004028666644

006399313350

09971368323

00

00

00

]]]]]]]

]

[[[[[[[

[

09048796003

minus04255671399

minus0009247686096

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04237568827

minus09026613361

007505076253

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(54)


[[[[[[[

[

004028666644

006399313350

09971368323

00

00

00

]]]]]]]

]

[[[[[[[

[

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00

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]

[[[[[[[

[

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minus09026613361

007505076253

00

00

00

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]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(55)



[[[

[

1458799998 000000001785571215 minus000000001009489597 00 00 00000000001785571215 1458799997 0000000002407020197 00 00 00minus000000001009489597 0000000002407020197 1458799997 00 00 00

00 00 00 1458799998 00 0000 00 00 00 1458799998 0000 00 00 00 00 1458799998

]]]

]

(56)




[[[[[

[

005368516527

007220768570

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00

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]

[[[[[

[

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[[[[[

[

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008705767064

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]

[[[[[

[

00

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]

[[[[[

[

00

00

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]

[[[[[

[

00

00

00

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10

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]

(57)


[[[[[[[[[

[

005368516527

007220768570

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00

00

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]

[[[[[[[[[

[

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00

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]

[[[[[[[[[

[

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008705767055

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00

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]

[[[[[[[[[

[

00

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]

[[[[[[[[[

[

00

00

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10

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]

[[[[[[[[[

[

00

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00

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10

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]

(58)




[[[

[

1615299998 0000000005508173090 minus0000000008722620038 00 00 00

0000000005508173090 1615299996 minus0000000004345156817 00 00 00

minus0000000008722620038 minus0000000004345156817 1615300000 00 00 00

00 00 00 1615299998 00 00

00 00 00 00 1615299998 00

00 00 00 00 00 1615299998

]]]

]

(59)




[[[[[[[

[

minus00598757013800000

minus00768229223150000

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00

00

00

]]]]]]]

]

[[[[[[[

[

minus0869358919900000

0493939153600000

00141567750950000

00

00

00

]]]]]]]

]

[[[[[[[

[

minus0490490276500000

minus0866078136900000

00963811082600000

00

00

00

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]

[[[[[[[

[

00

00

00

10

00

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]

[[[[[[[

[

00

00

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]

[[[[[[[

[

00

00

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00

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10

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]

(60)


[[[[[[[[[[[

[

minus005987730132

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00

00

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]

[[[[[[[[[[[

[

minus08693926232

04939583026

001415732393

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]

[[[[[[[[[[[

[

00

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]

[[[[[[[[[[[

[

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]

[[[[[[[[[[[

[

00

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]

(61)



[[[

[


00 00 00 1819400002 1819400002 0000 00 00 00 00 0000 00 00 00 00 1819400002

]]]

]

(62)




[[[[[[[

[

004967528691

008463937788

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00

00

00

]]]]]]]

]

[[[[[[[

[

09161715971

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00

00

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]

[[[[[

[

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009744493938

00

00

00

]]]]]

]

[[[[[

[

00

00

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10

00

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]]]]]

]

[[[[[

[

00

00

00

00

10

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]]]]]

]

[[[[[

[

00

00

00

00

00

10

]]]]]

]

(63)


[[[[[[[

[

004967528696

008463937796

09951726196

00

00

00

]]]]]]]

]

[[[[[[[

[

09161715980

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00

00

00

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]


[[[[[[[

[

minus03976958247

minus09123280745

009744493948

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(64)



[[[

[

1922399998 minus000000001176508811 minus000000001537920006 00 00 00

minus000000001176508811 1922400000 minus0000000007631928036 00 00 00

minus000000001537920006 minus0000000007631928036 1922400000 00 00 00

00 00 00 1922399998 1922399998 00

00 00 00 00 00 00

00 00 00 00 00 1922399998

]]]

]

(65)




[[[[[[[

[

00

00

00

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10

]]]]]]]

]

[[[[[[[

[

minus01387533797

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00

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]

[[[[[[[

[

minus08476774112

05304148448

001015375725

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00

00

]]]]]]]

]

[[[[[[[

[

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02458388325

00

00

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]

[[[[[[[

[

00

00

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10

00

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]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

(66)


[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

[[[[[[[

[

minus01387533798

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00

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]

[[[[[[[

[

minus08476774112

05304148448

001015375725

00

00

00

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]

[[[[[[[

[

minus05120454135

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02458388325

00

00

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]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

(67)



[[[

[

2074599999 0000000004356660361 000000003402343994 00 00 00

0000000004356660361 2074600004 000000002232269567 00 00 00

000000003402343994 000000002232269567 2074600000 00 00 00

00 00 00 2074600000 2074600000 00

00 00 00 00 00 00

00 00 00 00 00 2074600000

]]]

]

(68)




[[[[[[[

[

008621257414

01243595697

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00

00

00

]]]]]]]

]

[[[[[[[

[

08755641188

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00

00

00

]]]]]]]

]

[[[[[

[

minus04753471023

minus08668282006

01505124700

00

00

00

]]]]]

]

[[[[[

[

00

00

00

10

00

00

]]]]]

]

[[[[[

[

00

00

00

00

10

00

]]]]]

]

[[[[[

[

00

00

00

00

00

10

]]]]]

]

(69)


[[[[[

[

008621257423

01243595698

09884847448

00

00

00

]]]]]

]

[[[[[

[

08755641188

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minus001561753067

00

00

00

]]]]]

]


[[[[[[[

[

minus04753471018

minus08668281997

01505124698

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(70)



[[[

[

1890099999 000000001209663993 minus000000002532733996 00 00 00000000001209663993 1890099991 0000000001701090138 00 00 00minus000000002532733996 0000000001701090138 1890100001 00 00 00

00 00 00 1890099997 1890099997 0000 00 00 00 00 0000 00 00 00 00 1890099997

]]]

]

(71)




[[[[[[[

[

03961520086

06338768023

06642768888

00

00

00

]]]]]]]

]

[[[[[[[

[

09160614623

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minus03328645006

00

00

00

]]]]]]]

]

[[[[[[[

[

006240981414

minus07403833993

06692812840

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(72)


[[[[[[[

[

03961520086

06338768023

06642768888

00

00

00

]]]]]]]

]

[[[[[[[

[

09160614614

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minus03328645003

00

00

00

]]]]]]]

]

[[[[[[[

[

006240981426

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06692812853

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(73)



[[[

[


00 00 00 8772299998 8772299998 0000 00 00 00 00 0000 00 00 00 00 8772299998

]]]

]

(74)




[[[[[[[

[

minus006591789198

minus008975775227

minus09937798435

00

00

00

]]]]]]]

]

[[[[[[[

[

minus08289381135

05593224991

0004466103673

00

00

00

]]]]]]]

]

[[[[[

[

minus05554425577

minus08240763851

01112729817

00

00

00

]]]]]

]

[[[[[

[

00

00

00

10

00

00

]]]]]

]

[[[[[

[

00

00

00

00

10

00

]]]]]

]

[[[[[

[

00

00

00

00

00

10

]]]]]

]


[[[[[[[

[

minus01387533798

minus02031928415

minus09692575354

00

00

00

]]]]]]]

]

[[[[[[[

[

minus08476774112

05304148448

001015375725

00

00

00

]]]]]]]

]


[[[[

[

minus05120454135

minus08230265872

02458388325

00

00

00

]]]]

]

[[[[

[

00

00

00

10

00

00

]]]]

]

[[[[

[

00

00

00

00

10

00

]]]]

]

[[[[

[

00

00

00

00

00

10

]]]]

]

(76)



[[[

[

2228057494 0000000004678921127 000000003654014285 00 00 000000000004678921127 2228057499 000000002397389829 00 00 00000000003654014285 000000002397389829 2228057495 00 00 00

00 00 00 2228057495 2228057495 0000 00 00 00 00 0000 00 00 00 00 2228057495

]]]

]

(77)




[[[[[[[

[

006975010637

01071019008

09917984199

00

00

00

]]]]]]]

]

[[[[[[[

[

09079000793

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minus001862756541

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04133429180

minus09017531389

01264472547

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(78)


[[[[[[[

[

006975010637

01071019008

09917984199

00

00

00

]]]]]]]

]

[[[[[[[

[

09079000793

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minus001862756541

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04133429184

minus09017531398

01264472548

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(79)



[[[

[


00 00 00 2598699998 2598699998 0000 00 00 00 00 0000 00 00 00 00 2598699998

]]]

]

(80)




[[[[[

[

minus00192522847250000

minus00192842313600000

000386151499999998

00

00

00

]]]]]

]

[[[[[

[

000961799224999998

00165029120500000

000436480214750000

00

00

00

]]]]]

]

[[[[[[

[

minus0494444050150000

minus0856906622200000

0142104790200000

00

00

00

]]]]]]

]

[[[[[[

[

00

00

00

10

00

00

]]]]]]

]

[[[[[[

[

00

00

00

00

10

00

]]]]]]

]

[[[[[[

[

00

00

00

00

00

10

]]]]]]

]

(81)



[[[[[[[[[[

[

minus2541745843

minus2545963537

5098090872

00

00

00

]]]]]]]]]]

]

[[[[[[[[[[

[

2505315863

4298714977

1136953303

00

00

00

]]]]]]]]]]

]

[[[[[[[

[

minus04949599718

minus08578007511

01422530677

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(82)



[[[

[

315679169478799 427324383657779 384557503470365 00 00 00427324383657779 618700481877479 889153535276175 00 00 00384557503470365 889153535276175 384768762708609 00 00 00

00 00 00 247795001400000 00 0000 00 00 00 247795001400000 0000 00 00 00 00 247795001400000

]]]

]

(83)




[[[[[[[

[

01135176976

01764907831

09777344911

00

00

00

]]]]]]]

]

[[[[[[[

[

08848520771

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minus001870707734

00

00

00

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]

[[[[[[[

[

minus04518302069

minus08672739791

02090103088

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

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]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(84)


[[[[[[[

[

01135176977

01764907833

09777344921

00

00

00

]]]]]]]

]

[[[[[[[

[

08848520771

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minus001870707734

00

00

00

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]

[[[[[[[

[

minus04518302069

minus08672739791

02090103088

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(85)



[[[

[

2663700003 000000004768023095 000000003169803038 00 00 00000000004768023095 2663699992 0000000008310743498 00 00 00000000003169803038 0000000008310743498 2663700001 00 00 00

00 00 00 2663699998 2663699998 0000 00 00 00 00 0000 00 00 00 00 2663699998

]]]

]

(86)





6 Conclusions









Disclosure




Acknowledgments



References




















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












1 Quipo

[[[[[[[[[

[

1076866026

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]

[[[[[[[[[

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003276733390 09918780499 minus01228992897 00 00 00

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00 00 00 10 00 00

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2 Quipo

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09973986606 minus002692981555 006686327745 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

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]

3 White

[[[[[[[[[

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[[[[[[[[[

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006399313350 minus04255671399 minus09026613361 00 00 00

09971368323 minus0009247686096 007505076253 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

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]

4 Khaya

[[[[[[[[[

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1475290392

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]

[[[[[[[[[

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007220768570 minus03753089731 minus09240829102 00 00 00

09959437502 minus002273783078 008705767064 00 00 00

00 00 00 10 00 00

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]

5 Mahogany

[[[[[[[[[

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1810581412

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1498000000

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]

[[[[[[[[[

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minus09948285648 001731853005 01000809391 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

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]

6 Mahogany

[[[[[[[[[

[

1310854783

03895295651

1814922632

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]

[[[[[[[[[

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minus09956351187 001099502014 009268127742 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

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]

7 S Germ

[[[[[[[[[

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1234053410

05212570563

1922208822

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1582000000

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]

[[[[[[[[[

[

004967528691 09161715971 minus03976958243 00 00 00

008463937788 minus04006166011 minus09123280736 00 00 00

09951726186 minus001165943224 009744493938 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

8 Maple

[[[[[[[[[

[

1205449768

06868279555

2766674361

2460000000

2194000000

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]

[[[[[[[[[

[


minus02031928413 05304148448 minus08230265872 00 00 00

minus09692575344 001015375725 02458388325 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]




9 Walnut

[[[[[[[[[

[

1267869546

05204134969

1919891015

1922000000

1400000000

04600000000

]]]]]]]]]

]

[[[[[[[[[

[

008621257414 08755641188 minus04753471023 00 00 00

01243595697 minus04828494241 minus08668282006 00 00 00

09884847438 minus001561753067 01505124700 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

10 Birch

[[[[[[[[[

[

3168908680

04747157903

05946755301

2346000000

1816000000

03720000000

]]]]]]]]]

]

[[[[[[[[[

[

03961520086 09160614623 006240981414 00 00 00

06338768023 minus02236797319 minus07403833993 00 00 00

06642768888 minus03328645006 06692812840 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

11 Y Birch

[[[[[[[[[

[

1545414040

05549651499

2059894445

2120000000

1942000000

04800000000

]]]]]]]]]

]

[[[[[[[[[

[


minus008975775227 05593224991 minus08240763851 00 00 00

minus09937798435 0004466103673 01112729817 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

12 Oak

[[[[[[[[[

[

1718666431

08648974218

3239438239

2380000000

1532000000

07840000000

]]]]]]]]]

]

[[[[[[[[[

[

006975010637 09079000793 minus04133429180 00 00 00

01071019008 minus04187725641 minus09017531389 00 00 00

09917984199 minus001862756541 01264472547 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

13 Ash

[[[[[[[[[

[

1715567967

06696042476

2409716064

2684000000

1784000000

05400000000

]]]]]]]]]

]

[[[[[[[[[

[

006190313535 08746549514 minus04807772027 00 00 00

009781749768 minus04846986500 minus08691944279 00 00 00

09932772717 minus0006777439445 01155609239 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

14 Ash

[[[[[[[[[

[

1742363268

07844815431

2669885744

1720000000

1218000000

05000000000

]]]]]]]]]

]

[[[[[[[[[

[


minus01363859604 05177044741 minus08446188165 00 00 00

minus09855542417 001550704374 01686486565 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

15 Beech

[[[[[[[[[

[

1599002755

09558575345

3451114905

3216000000

2112000000

09120000000

]]]]]]]]]

]

[[[[[[[[[

[

01135176976 08848520771 minus04518302069 00 00 00

01764907831 minus04654963442 minus08672739791 00 00 00

09777344911 minus001870707734 02090103088 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

Thus 119890120572otimes 119890

1205731le120572120573le119899




119860119894119895119896119897

119890119894otimes 119890

119895otimes 119890

119896otimes 119890

119897 or as an 119899


120572120573119890120572otimes 119890

120573 where 119860

120595(119894119895)120595119896119897= 119860

119894119895119896119897

1 le 119894 119895 119896 119897 le 119899


tr (A119879B) = ⟨AB⟩ = tr (119860119879119861119879) A =1003817100381710038171003817100381711986010038171003817100381710038171003817 (21)






1 Blasa

[[[[[[[[[

[

6385324208

009845837744

03832174064

06240000000

04060000000

006600000000

]]]]]]]]]

]

[[[[[[[[[

[

001488818113 09510211778 minus03087669978 00 00 00

002575997614 minus03090635474 minus09506924583 00 00 00

09995572851 minus0006200252135 002909967665 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

2 Spruce

[[[[[[[[[

[

1198583568

04516442734

1114520065

1498000000

1442000000

007800000000

]]]]]]]]]

]

[[[[[[[[[

[


minus004689865645 04044467539 minus09133582749 00 00 00

minus09983543167 001123114838 005623628701 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

3 Spruce

[[[[[[[[[

[

1410927768

04185382987

1227183961

1442000000

10

006400000000

]]]]]]]]]

]

[[[[[[[[[

[


minus005361649666 04423303564 minus08952480817 00 00 00

minus09978936411 0009052295016 006423657043 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

4 Spruce

[[[[[[[[[

[

1630529953

03544288592

08442715844

1234000000

1520000000

007200000000

]]]]]]]]]

]

[[[[[[[[[

[


minus002869707087 04091564350 minus09120128765 00 00 00

minus09993764538 0007032901380 003460119282 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

5 Spruce

[[[[[[[[[

[

1725845208

05092652753

1171282625

1250000000

1706000000

007000000000

]]]]]]]]]

]

[[[[[[[[[

[


minus003428302033 05858146064 minus08097196604 00 00 00

minus09988364173 0007413313465 004765347716 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

6 Douglas Fir

[[[[[[[[[

[

1613459769

06233733771

1439028943

1767000000

1766000000

01760000000

]]]]]]]]]

]

[[[[[[[[[

[


minus003697194691 05922678885 minus08048924305 00 00 00

minus09986533619 0007435777607 005134366998 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

7 Douglas Fir

[[[[[[[[[

[

1710150156

06986859431

2204812526

2348000000

1816000000

01600000000

]]]]]]]]]

]

[[[[[[[[[

[


minus006364825251 05704773676 minus08188433771 00 00 00

minus09967231590 0004703484281 008075160717 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

8 Pine

[[[[[[[[[

[

1699539133

04940019684

1565606760

3484000000

1344000000

01320000000

]]]]]]]]]

]

[[[[[[[[[

[


minus005585205149 04413516031 minus08955943895 00 00 00

minus09978476167 0006195562437 006528207193 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]


Table11Th

eeigenvalues

andeigenvectorsfor3

specim

enso

fcancello

usbo

necalculated

usingMAPL

EΛ[119894]sta

ndforthe

eigenvalueso

f119894th

specim

enof

cancellous

boneand

119881[119894]sta

ndfor

thec

orrespon

ding

eigenvectors

Cancellous

bone

EigenvaluesΛ

[119894]

Eigenvectors

119881[119894]

Specim

en1

[ [ [ [ [ [ [ [

1334869880

minus02071167473

8644210768

6837676373

3768806080

4123724855

] ] ] ] ] ] ] ]

[ [ [ [ [ [ [ [

03316115964

08831238953

02356441950

02242668491

005354786914

003787816350

06475092156

minus01099136965

01556396475

minus06940063479

minus01278953723

02154647417

04800906963

minus02832250652

007648827487

02494278793

06410471445

minus04585742261

minus02737532692

minus006354940263

04518342532

minus006214920576

05836952694

06101669758

minus03706582070

03314035780

minus01276606259

minus06337030234

03639432801

minus04499474345

01671829224

01180163925

minus08330343502

002008124471

03109325968

04087706408

] ] ] ] ] ] ] ]

Specim

en2

[ [ [ [ [ [ [ [

1754634171

7773848354

1550688791

2301790452

3445174379

3589156342

] ] ] ] ] ] ] ]

[ [ [ [ [ [ [ [

minus03057290719

minus03005337771

minus09030341401

001584500766

minus002175914011

minus0003741933916

minus04311831740

minus08021570593

04130146809

00001362645283

minus0003854432451

0005396441669

minus08488209390

05154657137

01152135966

minus0004741957786

002229739182

minus0002068264586

00009402467441

minus0005358584867

0003987518000

minus003989381331

003796849613

minus09984595017

minus001058157817

002100470582

002092543658

0006230946613

minus09987520802

minus003826770492

00009823402498

0006977574167

001484146906

09990475909

0008196718814

minus003958286493

] ] ] ] ] ] ] ]

Specim

en3

[ [ [ [ [ [ [ [

1418542670

5838388363

9959058231

3433818493

3024968125

1757492470

] ] ] ] ] ] ] ]

[ [ [ [ [ [ [ [

03244960223

minus0004831643783

minus08451713783

04062092722

01164524794

minus004239310198

06461152835

minus07125334236

02668138654

004131783689

minus002431040234

003665187401

06621315621

07009969173

02633438279

002836289512

minus0001711614840

0005269111592

minus01962401199

0009159859564

03740839141

08850746598

01904577552

004284654369

minus0008597323304

minus0002526763452

001897677365

01738799329

minus06977939665

minus06945566457

001533190599

minus002803230310

006961922985

minus01374446216

06801869735

minus07159517722

] ] ] ] ] ] ] ]


24681012141618

12

34

56

Column Row

12

34

56


12345678

12

34

56

Column Row

12

34

56




120590119894119895= 119862


where 119862119894119895119896119897


120590 = 120590119894119895119890119894otimes 119890

119895 120598 = 120598

119896119897119890119896otimes 119890

119897

C = 119862119894119895119896119897

119890119894otimes 119890

119896otimes 119890

119896otimes 119890

119897

(23)

5

10

15

20

12

34

56

Column Row

12

34

56


5

10

15

20

25

12

34

56

Column Row

12

34

56



E1equiv 119890

1otimes 119890

1

E2equiv 119890

2otimes 119890

2

E3equiv 119890

3otimes 119890

3

(24)


E4equiv

1


2+ 119890

2otimes 119890

1)

E5equiv

1


3+ 119890

3otimes 119890

2)

E6equiv

1


1+ 119890

1otimes 119890

3)

(25)



E7equiv

1


2minus 119890

2otimes 119890

1)

E8equiv

1


3minus 119890

3otimes 119890

2)

E9equiv

1


1minus 119890

1otimes 119890

3)

(26)


120590 = 119878119860E

119860

120598 = 119864119860E

119860

C = 119862119860119861

119890119860otimes 119890

119861

(27)




120590119894119895= 119862

119894119895119896119897120598119896119897

lArrrArr 119878119860= 119862

119860119861E119861

119862119894119895119896119897

120598119896119897

= 120582120598119894119895lArrrArr 119862

119860119861E119861= 120582119864

119860

(28)


((((((((

(

1198781

1198782

1198783

1198784

1198785

1198786

1198787

1198788

1198789

))))))))

)

=

((((((((

(

12059011

12059022

12059033

120590(12)

120590(23)

120590(31)

120590[12]

120590[23]

120590[31]

))))))))

)

((((((((

(

1198641

1198642

1198643

1198644

1198645

1198646

1198647

1198648

1198649

))))))))

)

=

((((((((

(

12059811

12059822

12059833

120598(12)

120598(23)

120598(31)

120598[12]

120598[23]

120598[31]

))))))))

)

(29)


120579(119894119895)

equiv1

radic2(120579119894119895+ 120579

119895119894) 120579

[119894119895]equiv

1

radic2(120579119894119895minus 120579

119895119894) (30)


tensor C are

((((((((

(

11986211

11986212

11986213

11986214

11986215

11986216

11986217

11986218

11986219

11986221

11986222

11986223

11986224

11986225

11986226

11986227

11986228

11986229

11986231

11986232

11986233

11986234

11986235

11986236

11986237

11986238

11986239

11986241

11986242

11986243

11986244

11986245

11986246

11986247

11986248

11986249

11986251

11986252

11986253

11986254

11986255

11986256

11986257

11986258

11986259

11986261

11986262

11986263

11986264

11986265

11986266

11986267

11986268

11986269

11986271

11986272

11986273

11986274

11986275

11986276

11986277

11986278

11986279

11986281

11986282

11986283

11986284

11986285

11986286

11986287

11986288

11986289

11986291

11986292

11986293

11986294

11986295

11986296

11986297

11986298

11986299

))))))))

)

=

1198621111

1198621122

1198621133

1198622211

1198622222

1198623333

1198623311

1198623322

1198623333

11986211(12)

11986211(23)

11986211(31)

11986222(12)

11986222(23)

11986222(31)

11986233(12)

11986233(23)

11986233(31)

11986211[12]

11986211[23]

11986211[31]

11986222[12]

11986222[23]

11986222[31]

11986233[12]

11986233[23]

11986233[31]

119862(12)11

119862(12)22

119862(12)33

119862(23)11

119862(23)22

119862(23)33

119862(31)11

119862(31)22

119862(31)33

119862(1212)

119862(1223)

119862(1231)

119862(2312)

119862(2323)

119862(2331)

119862(3112)

119862(3123)

119862(3131)

119862(12)[12]

119862(12)[23]

119862(12)[31]

119862(23)[12]

119862(23)[23]

119862(23)[31]

119862(31)[12]

119862(31)[23]

119862(31)[31]

119862[12]11

119862[12]22

119862[12]33

119862[23]11

119862[23]22

119862[123]33

119862[21]11

119862[31]22

119862[31]33

119862[12](12)

119862[12](23)

119862[12](21)

119862[23](12)

119862[23](23)

119862[23](21)

119862[31](12)

119862[31](23)

119862[31](21)

119862[1212]

119862[1223]

119862[1231]

119862[2312]

119862[2323]

119862[2331]

119862[3112]

119862[3123]

119862[3131]

(31)

where

119862119894119895(119896119897)

equiv1

radic2(119862

119894119895119896119897+ 119862

119894119895119897119896) 119862

119894119895[119896119897]equiv

1

radic2(119862

119894119895119896119897minus 119862

119894119895119897119896)

119862(119894119895)119896119897

equiv1

radic2(119862

119894119895119896119897+ 119862

119895119894119896119897) 119862

[119894119895]119896119897equiv

1

radic2(119862

119894119895119896119897minus 119862

119895119894119896119897)

119862(119894119895119896119897)

equiv1

2(119862

119894119895119896119897+ 119862

119894119895119897119896+ 119862

119895119894119896119897+ 119862

119895119894119897119896)

119862(119894119895)[119896119897]

equiv1

2(119862

119894119895119896119897minus 119862

119894119895119897119896+ 119862

119895119894119896119897minus 119862

119895119894119897119896)

119862[119894119895](119896119897)

equiv1

2(119862

119894119895119896119897+ 119862

119894119895119897119896minus 119862

119895119894119896119897minus 119862

119895119894119897119896)

119862[119894119895119896119897]

equiv1

2(119862

119894119895119896119897minus 119862

119895119894119896119897minus 119862

119894119895119897119896+ 119862

119895119894119897119896)

(32)


12

34

56

ColumnRow

12

34

56

600005000040000300002000010000


5

10

15

20

25

12

34

56

Column Row

12

34

56






119895119894and 120598

119896119897= 120598

119897119896 the 119862

119894119895119896119897will be

like 119862119894119895119896119897

= 119862119895119894119896119897

= 119862119894119895119897119896


119862119894119895119896119897

= 119862119896119897119894119895












2 sdot 119864energy = 120590119894119895120598119894119895= 120590

119901120598119901 (33)


119901 = 119894120575119894119895+ (1 minus 120575

119894119895) (9 minus 119894 minus 119895)

119902 = 119896120575119896119897+ (1 minus 120575

119896119897) (9 minus 119896 minus 119897)

(34)


119901 119862

119894119895119896119897= 119862

119901119902 and 120598

119902= (2 minus

120575119896119897)120598119896119897



(

(

1205901

1205902

1205903

1205904

1205905

1205906

)

)

= (

(

11986211

11986212

11986213

11986214

11986215

11986216

11986221

11986222

11986223

11986224

11986225

11986226

11986231

11986232

11986233

11986234

11986235

11986236

11986241

11986242

11986243

11986244

11986245

11986246

11986251

11986252

11986253

11986254

11986255

11986256

11986261

11986262

11986263

11986264

11986265

11986266

)

)

(

(

1205981

1205982

1205983

1205984

1205985

1205986

)

)

(35)



(1) the 120590119894119895and 120598


(2) the norms of 120590119894119895 120598119896119897 and 119862




01 02 03 04 05 06 07 08

2

4

6

8

10

12

14

16

(a)

01

02

03

04

05

06

07

08

09

01 02 03 04 05 06 07 08

(b)


01 02 03 04 05 06 07 08

05

1

15

2

25

3

(a)

01 02 03 04 05 06 07 08

05

1

15

2

25

3

(b)




119901 = 119894120575119894119895+ (1 minus 120575

119894119895) (9 minus 119894 minus 119895)

119902 = 119896120575119896119897+ (1 minus 120575

119896119897) (9 minus 119896 minus 119897)

(36)

In this way

120590119901= (120575

119894119895+ radic2 (1 minus 120575

119894119895)) 120598

119894119895

120598119902= (120575

119896119897+ radic2 (1 minus 120575

119896119897)) 120598

119896119897

119862119901119902

= (120575119894119895+ radic2 (1 minus 120575

119894119895)) (120575

119896119897+ radic2 (1 minus 120575

119896119897)) 119862

119894119895119896119897

(37)




120590kelvin

= 120590viogt

120585 120590voigt

=120590kelvin

120585

120598kelvin

= 120598voigt

120585 120598voigt

=120598kelvin

120585

(38)


119862kelvin

=((

(







))

)

119862voigt

(39)


(

(

12059011

12059022

12059033

radic212059023

radic212059013

radic212059012

)

)

= (

1198621111 1198621122 1198621133radic21198621123

radic21198621113radic21198621112

1198622211 1198622222 1198622333radic21198622223

radic21198622213radic21198622212

1198623311 1198623322 1198623333radic21198623323

radic21198623313radic21198623312

radic21198622311radic21198622322

radic21198622333 21198622323 21198622313 21198622312

radic21198621311radic21198621322

radic21198621333 21198621323 21198621313 21198621312

radic21198621211radic21198621222

radic21198621233 21198621223 21198621213 21198621212

)

lowast(

(

12059811

12059822

12059833

radic212059823

radic212059813

radic212059812

)

)

(40)


= C (41)



(

(

12059011

12059022

12059033

radic212059023

radic212059013

radic212059012

)

)

=((

(

11986211

11986212

11986213

radic211986214

radic211986215

radic211986216

11986212

11986222

11986223

radic211986224

radic211986225

radic211986226

11986213

11986223

11986233

radic211986234

radic211986235

radic211986236

radic211986214

radic211986224

radic211986234

211986244

211986245

211986246

radic211986215

radic211986225

radic211986235

211986245

211986255

211986256

radic211986216

radic211986226

radic211986236

211986246

211986256

211986266

))

)

lowast (

(

12059811

12059822

12059833

radic212059823

radic212059813

radic212059812

)

)

(42)


(

(

1

2

3

4

5

6

)

)

=((

(

11986211

11986212

11986213

11986214

11986215

11986216

11986212

11986222

11986223

11986224

11986225

11986226

11986213

11986223

11986233

11986234

11986235

11986236

11986214

11986224

11986234

11986244

11986245

11986246

11986215

11986225

11986235

11986245

11986255

11986256

11986216

11986226

11986236

11986246

11986256

11986266

))

)

lowast (

(

1205981

1205982

1205983

1205984

1205985

1205986

)

)

(43)



































01 02 03 04 05 06 07 08

05

1

15

2

25

3

(a)

01 02 03 04 05 06 07 08

05

1

15

2

25

3

(b)


01 02 03 04 05 06 07 08

2

4

6

8

10

12

14

16





ΛI) N = 0 (44)


119894 119894 = 1 2 6 and

















NNA119896

equiv1

119872

119872

sum

119884=1

N119884119896 (45)



NAVG119896

equiv JNANNA119896

(46)


119896=1NNA119896

otimes NNA119896

| that is

JNA equiv (

6

sum

119896=1

NNA119896

otimes NNA119896

)

minus12

(47)


119896 that is

ΛAVG119902

equiv1

119872

119872

sum

119884=1

6

sum

119896=1

Λ119884

119896(N119884

119896sdot NAVG

119902)2

(48)


gt 0 for all 119896 = 1 2 6

and ΛAVG119896


NA119896


ΛNA119896

equiv1

119872

119872

sum

119884=1

Λ119884

119896 (49)


CAVG=

6

sum

119896=1

ΛAVG119896

NAVG119896

otimes NAVG119896

(50)





[[[[[[[

[

00367834559700000

00453681054800000

0998185540700000

00

00

00

]]]]]]]

]

[[[[[[[

[

0983745905000000

minus0170879918750000

minus00277901141300000

00

00

00

]]]]]]]

]

[[[[[[[

[

minus0169284222800000

minus0982986098000000

00509905132200000

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(51)


[[[[[[[

[

003679134179

004537783172

09983995367

00

00

00

]]]]]]]

]

[[[[[[[

[

09859858256

minus01712690003

minus002785339027

00

00

00

]]]]]]]

]

[[[[[[[

[

minus01697052873

minus09854311019

005111734310

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(52)




[[[

[

334725276837330 0000112116752981933 000198542359441849 00 00 00

0000112116752981933 334773746006714 minus0000991802322788501 00 00 00

000198542359441849 minus0000991802322788501 334013593769726 00 00 00

00 00 00 333950000000000 00 00

00 00 00 00 333950000000000 00

00 00 00 00 00 333950000000000

]]]

]

(53)




[[[[[[[

[

004028666644

006399313350

09971368323

00

00

00

]]]]]]]

]

[[[[[[[

[

09048796003

minus04255671399

minus0009247686096

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04237568827

minus09026613361

007505076253

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(54)


[[[[[[[

[

004028666644

006399313350

09971368323

00

00

00

]]]]]]]

]

[[[[[[[

[

09048795994

minus04255671395

minus0009247686087

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04237568827

minus09026613361

007505076253

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(55)



[[[

[

1458799998 000000001785571215 minus000000001009489597 00 00 00000000001785571215 1458799997 0000000002407020197 00 00 00minus000000001009489597 0000000002407020197 1458799997 00 00 00

00 00 00 1458799998 00 0000 00 00 00 1458799998 0000 00 00 00 00 1458799998

]]]

]

(56)




[[[[[

[

005368516527

007220768570

09959437502

00

00

00

]]]]]

]

[[[[[

[

09266208315

minus03753089731

minus002273783078

00

00

00

]]]]]

]

[[[[[

[

minus03721447810

minus09240829102

008705767064

00

00

00

]]]]]

]

[[[[[

[

00

00

00

00

00

10

]]]]]

]

[[[[[

[

00

00

00

10

00

00

]]]]]

]

[[[[[

[

00

00

00

00

10

00

]]]]]

]

(57)


[[[[[[[[[

[

005368516527

007220768570

09959437502

00

00

00

]]]]]]]]]

]

[[[[[[[[[

[

09266208315

minus03753089731

minus002273783078

00

00

00

]]]]]]]]]

]

[[[[[[[[[

[

minus03721447806

minus09240829093

008705767055

00

00

00

]]]]]]]]]

]

[[[[[[[[[

[

00

00

00

10

00

00

]]]]]]]]]

]

[[[[[[[[[

[

00

00

00

00

10

00

]]]]]]]]]

]

[[[[[[[[[

[

00

00

00

00

00

10

]]]]]]]]]

]

(58)




[[[

[

1615299998 0000000005508173090 minus0000000008722620038 00 00 00

0000000005508173090 1615299996 minus0000000004345156817 00 00 00

minus0000000008722620038 minus0000000004345156817 1615300000 00 00 00

00 00 00 1615299998 00 00

00 00 00 00 1615299998 00

00 00 00 00 00 1615299998

]]]

]

(59)




[[[[[[[

[

minus00598757013800000

minus00768229223150000

minus0995231841750000

00

00

00

]]]]]]]

]

[[[[[[[

[

minus0869358919900000

0493939153600000

00141567750950000

00

00

00

]]]]]]]

]

[[[[[[[

[

minus0490490276500000

minus0866078136900000

00963811082600000

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(60)


[[[[[[[[[[[

[

minus005987730132

minus007682497510

minus09952584354

00

00

00

]]]]]]]]]]]

]

[[[[[[[[[[[

[

minus08693926232

04939583026

001415732393

00

00

00

]]]]]]]]]]]

]

[[[[[[[[[[[

[

00

00

00

10

00

00

]]]]]]]]]]]

]

[[[[[[[[[[[

[

00

00

00

00

10

00

]]]]]]]]]]]

]

[[[[[[[[[[[

[

00

00

00

00

00

10

]]]]]]]]]]]

]

(61)



[[[

[


00 00 00 1819400002 1819400002 0000 00 00 00 00 0000 00 00 00 00 1819400002

]]]

]

(62)




[[[[[[[

[

004967528691

008463937788

09951726186

00

00

00

]]]]]]]

]

[[[[[[[

[

09161715971

minus04006166011

minus001165943224

00

00

00

]]]]]]]

]

[[[[[

[

minus03976958243

minus09123280736

009744493938

00

00

00

]]]]]

]

[[[[[

[

00

00

00

10

00

00

]]]]]

]

[[[[[

[

00

00

00

00

10

00

]]]]]

]

[[[[[

[

00

00

00

00

00

10

]]]]]

]

(63)


[[[[[[[

[

004967528696

008463937796

09951726196

00

00

00

]]]]]]]

]

[[[[[[[

[

09161715980

minus04006166015

minus001165943225

00

00

00

]]]]]]]

]


[[[[[[[

[

minus03976958247

minus09123280745

009744493948

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(64)



[[[

[

1922399998 minus000000001176508811 minus000000001537920006 00 00 00

minus000000001176508811 1922400000 minus0000000007631928036 00 00 00

minus000000001537920006 minus0000000007631928036 1922400000 00 00 00

00 00 00 1922399998 1922399998 00

00 00 00 00 00 00

00 00 00 00 00 1922399998

]]]

]

(65)




[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

[[[[[[[

[

minus01387533797

minus02031928413

minus09692575344

00

00

00

]]]]]]]

]

[[[[[[[

[

minus08476774112

05304148448

001015375725

00

00

00

]]]]]]]

]

[[[[[[[

[

minus05120454135

minus08230265872

02458388325

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

(66)


[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

[[[[[[[

[

minus01387533798

minus02031928415

minus09692575354

00

00

00

]]]]]]]

]

[[[[[[[

[

minus08476774112

05304148448

001015375725

00

00

00

]]]]]]]

]

[[[[[[[

[

minus05120454135

minus08230265872

02458388325

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

(67)



[[[

[

2074599999 0000000004356660361 000000003402343994 00 00 00

0000000004356660361 2074600004 000000002232269567 00 00 00

000000003402343994 000000002232269567 2074600000 00 00 00

00 00 00 2074600000 2074600000 00

00 00 00 00 00 00

00 00 00 00 00 2074600000

]]]

]

(68)




[[[[[[[

[

008621257414

01243595697

09884847438

00

00

00

]]]]]]]

]

[[[[[[[

[

08755641188

minus04828494241

minus001561753067

00

00

00

]]]]]]]

]

[[[[[

[

minus04753471023

minus08668282006

01505124700

00

00

00

]]]]]

]

[[[[[

[

00

00

00

10

00

00

]]]]]

]

[[[[[

[

00

00

00

00

10

00

]]]]]

]

[[[[[

[

00

00

00

00

00

10

]]]]]

]

(69)


[[[[[

[

008621257423

01243595698

09884847448

00

00

00

]]]]]

]

[[[[[

[

08755641188

minus04828494241

minus001561753067

00

00

00

]]]]]

]


[[[[[[[

[

minus04753471018

minus08668281997

01505124698

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(70)



[[[

[

1890099999 000000001209663993 minus000000002532733996 00 00 00000000001209663993 1890099991 0000000001701090138 00 00 00minus000000002532733996 0000000001701090138 1890100001 00 00 00

00 00 00 1890099997 1890099997 0000 00 00 00 00 0000 00 00 00 00 1890099997

]]]

]

(71)




[[[[[[[

[

03961520086

06338768023

06642768888

00

00

00

]]]]]]]

]

[[[[[[[

[

09160614623

minus02236797319

minus03328645006

00

00

00

]]]]]]]

]

[[[[[[[

[

006240981414

minus07403833993

06692812840

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(72)


[[[[[[[

[

03961520086

06338768023

06642768888

00

00

00

]]]]]]]

]

[[[[[[[

[

09160614614

minus02236797317

minus03328645003

00

00

00

]]]]]]]

]

[[[[[[[

[

006240981426

minus07403834008

06692812853

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(73)



[[[

[


00 00 00 8772299998 8772299998 0000 00 00 00 00 0000 00 00 00 00 8772299998

]]]

]

(74)




[[[[[[[

[

minus006591789198

minus008975775227

minus09937798435

00

00

00

]]]]]]]

]

[[[[[[[

[

minus08289381135

05593224991

0004466103673

00

00

00

]]]]]]]

]

[[[[[

[

minus05554425577

minus08240763851

01112729817

00

00

00

]]]]]

]

[[[[[

[

00

00

00

10

00

00

]]]]]

]

[[[[[

[

00

00

00

00

10

00

]]]]]

]

[[[[[

[

00

00

00

00

00

10

]]]]]

]


[[[[[[[

[

minus01387533798

minus02031928415

minus09692575354

00

00

00

]]]]]]]

]

[[[[[[[

[

minus08476774112

05304148448

001015375725

00

00

00

]]]]]]]

]


[[[[

[

minus05120454135

minus08230265872

02458388325

00

00

00

]]]]

]

[[[[

[

00

00

00

10

00

00

]]]]

]

[[[[

[

00

00

00

00

10

00

]]]]

]

[[[[

[

00

00

00

00

00

10

]]]]

]

(76)



[[[

[

2228057494 0000000004678921127 000000003654014285 00 00 000000000004678921127 2228057499 000000002397389829 00 00 00000000003654014285 000000002397389829 2228057495 00 00 00

00 00 00 2228057495 2228057495 0000 00 00 00 00 0000 00 00 00 00 2228057495

]]]

]

(77)




[[[[[[[

[

006975010637

01071019008

09917984199

00

00

00

]]]]]]]

]

[[[[[[[

[

09079000793

minus04187725641

minus001862756541

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04133429180

minus09017531389

01264472547

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(78)


[[[[[[[

[

006975010637

01071019008

09917984199

00

00

00

]]]]]]]

]

[[[[[[[

[

09079000793

minus04187725641

minus001862756541

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04133429184

minus09017531398

01264472548

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(79)



[[[

[


00 00 00 2598699998 2598699998 0000 00 00 00 00 0000 00 00 00 00 2598699998

]]]

]

(80)




[[[[[

[

minus00192522847250000

minus00192842313600000

000386151499999998

00

00

00

]]]]]

]

[[[[[

[

000961799224999998

00165029120500000

000436480214750000

00

00

00

]]]]]

]

[[[[[[

[

minus0494444050150000

minus0856906622200000

0142104790200000

00

00

00

]]]]]]

]

[[[[[[

[

00

00

00

10

00

00

]]]]]]

]

[[[[[[

[

00

00

00

00

10

00

]]]]]]

]

[[[[[[

[

00

00

00

00

00

10

]]]]]]

]

(81)



[[[[[[[[[[

[

minus2541745843

minus2545963537

5098090872

00

00

00

]]]]]]]]]]

]

[[[[[[[[[[

[

2505315863

4298714977

1136953303

00

00

00

]]]]]]]]]]

]

[[[[[[[

[

minus04949599718

minus08578007511

01422530677

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(82)



[[[

[

315679169478799 427324383657779 384557503470365 00 00 00427324383657779 618700481877479 889153535276175 00 00 00384557503470365 889153535276175 384768762708609 00 00 00

00 00 00 247795001400000 00 0000 00 00 00 247795001400000 0000 00 00 00 00 247795001400000

]]]

]

(83)




[[[[[[[

[

01135176976

01764907831

09777344911

00

00

00

]]]]]]]

]

[[[[[[[

[

08848520771

minus04654963442

minus001870707734

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04518302069

minus08672739791

02090103088

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(84)


[[[[[[[

[

01135176977

01764907833

09777344921

00

00

00

]]]]]]]

]

[[[[[[[

[

08848520771

minus04654963442

minus001870707734

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04518302069

minus08672739791

02090103088

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(85)



[[[

[

2663700003 000000004768023095 000000003169803038 00 00 00000000004768023095 2663699992 0000000008310743498 00 00 00000000003169803038 0000000008310743498 2663700001 00 00 00

00 00 00 2663699998 2663699998 0000 00 00 00 00 0000 00 00 00 00 2663699998

]]]

]

(86)





6 Conclusions









Disclosure




Acknowledgments



References




















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












9 Walnut

[[[[[[[[[

[

1267869546

05204134969

1919891015

1922000000

1400000000

04600000000

]]]]]]]]]

]

[[[[[[[[[

[

008621257414 08755641188 minus04753471023 00 00 00

01243595697 minus04828494241 minus08668282006 00 00 00

09884847438 minus001561753067 01505124700 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

10 Birch

[[[[[[[[[

[

3168908680

04747157903

05946755301

2346000000

1816000000

03720000000

]]]]]]]]]

]

[[[[[[[[[

[

03961520086 09160614623 006240981414 00 00 00

06338768023 minus02236797319 minus07403833993 00 00 00

06642768888 minus03328645006 06692812840 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

11 Y Birch

[[[[[[[[[

[

1545414040

05549651499

2059894445

2120000000

1942000000

04800000000

]]]]]]]]]

]

[[[[[[[[[

[


minus008975775227 05593224991 minus08240763851 00 00 00

minus09937798435 0004466103673 01112729817 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

12 Oak

[[[[[[[[[

[

1718666431

08648974218

3239438239

2380000000

1532000000

07840000000

]]]]]]]]]

]

[[[[[[[[[

[

006975010637 09079000793 minus04133429180 00 00 00

01071019008 minus04187725641 minus09017531389 00 00 00

09917984199 minus001862756541 01264472547 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

13 Ash

[[[[[[[[[

[

1715567967

06696042476

2409716064

2684000000

1784000000

05400000000

]]]]]]]]]

]

[[[[[[[[[

[

006190313535 08746549514 minus04807772027 00 00 00

009781749768 minus04846986500 minus08691944279 00 00 00

09932772717 minus0006777439445 01155609239 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

14 Ash

[[[[[[[[[

[

1742363268

07844815431

2669885744

1720000000

1218000000

05000000000

]]]]]]]]]

]

[[[[[[[[[

[


minus01363859604 05177044741 minus08446188165 00 00 00

minus09855542417 001550704374 01686486565 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

15 Beech

[[[[[[[[[

[

1599002755

09558575345

3451114905

3216000000

2112000000

09120000000

]]]]]]]]]

]

[[[[[[[[[

[

01135176976 08848520771 minus04518302069 00 00 00

01764907831 minus04654963442 minus08672739791 00 00 00

09777344911 minus001870707734 02090103088 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

Thus 119890120572otimes 119890

1205731le120572120573le119899




119860119894119895119896119897

119890119894otimes 119890

119895otimes 119890

119896otimes 119890

119897 or as an 119899


120572120573119890120572otimes 119890

120573 where 119860

120595(119894119895)120595119896119897= 119860

119894119895119896119897

1 le 119894 119895 119896 119897 le 119899


tr (A119879B) = ⟨AB⟩ = tr (119860119879119861119879) A =1003817100381710038171003817100381711986010038171003817100381710038171003817 (21)






1 Blasa

[[[[[[[[[

[

6385324208

009845837744

03832174064

06240000000

04060000000

006600000000

]]]]]]]]]

]

[[[[[[[[[

[

001488818113 09510211778 minus03087669978 00 00 00

002575997614 minus03090635474 minus09506924583 00 00 00

09995572851 minus0006200252135 002909967665 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

2 Spruce

[[[[[[[[[

[

1198583568

04516442734

1114520065

1498000000

1442000000

007800000000

]]]]]]]]]

]

[[[[[[[[[

[


minus004689865645 04044467539 minus09133582749 00 00 00

minus09983543167 001123114838 005623628701 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

3 Spruce

[[[[[[[[[

[

1410927768

04185382987

1227183961

1442000000

10

006400000000

]]]]]]]]]

]

[[[[[[[[[

[


minus005361649666 04423303564 minus08952480817 00 00 00

minus09978936411 0009052295016 006423657043 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

4 Spruce

[[[[[[[[[

[

1630529953

03544288592

08442715844

1234000000

1520000000

007200000000

]]]]]]]]]

]

[[[[[[[[[

[


minus002869707087 04091564350 minus09120128765 00 00 00

minus09993764538 0007032901380 003460119282 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

5 Spruce

[[[[[[[[[

[

1725845208

05092652753

1171282625

1250000000

1706000000

007000000000

]]]]]]]]]

]

[[[[[[[[[

[


minus003428302033 05858146064 minus08097196604 00 00 00

minus09988364173 0007413313465 004765347716 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

6 Douglas Fir

[[[[[[[[[

[

1613459769

06233733771

1439028943

1767000000

1766000000

01760000000

]]]]]]]]]

]

[[[[[[[[[

[


minus003697194691 05922678885 minus08048924305 00 00 00

minus09986533619 0007435777607 005134366998 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

7 Douglas Fir

[[[[[[[[[

[

1710150156

06986859431

2204812526

2348000000

1816000000

01600000000

]]]]]]]]]

]

[[[[[[[[[

[


minus006364825251 05704773676 minus08188433771 00 00 00

minus09967231590 0004703484281 008075160717 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

8 Pine

[[[[[[[[[

[

1699539133

04940019684

1565606760

3484000000

1344000000

01320000000

]]]]]]]]]

]

[[[[[[[[[

[


minus005585205149 04413516031 minus08955943895 00 00 00

minus09978476167 0006195562437 006528207193 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]


Table11Th

eeigenvalues

andeigenvectorsfor3

specim

enso

fcancello

usbo

necalculated

usingMAPL

EΛ[119894]sta

ndforthe

eigenvalueso

f119894th

specim

enof

cancellous

boneand

119881[119894]sta

ndfor

thec

orrespon

ding

eigenvectors

Cancellous

bone

EigenvaluesΛ

[119894]

Eigenvectors

119881[119894]

Specim

en1

[ [ [ [ [ [ [ [

1334869880

minus02071167473

8644210768

6837676373

3768806080

4123724855

] ] ] ] ] ] ] ]

[ [ [ [ [ [ [ [

03316115964

08831238953

02356441950

02242668491

005354786914

003787816350

06475092156

minus01099136965

01556396475

minus06940063479

minus01278953723

02154647417

04800906963

minus02832250652

007648827487

02494278793

06410471445

minus04585742261

minus02737532692

minus006354940263

04518342532

minus006214920576

05836952694

06101669758

minus03706582070

03314035780

minus01276606259

minus06337030234

03639432801

minus04499474345

01671829224

01180163925

minus08330343502

002008124471

03109325968

04087706408

] ] ] ] ] ] ] ]

Specim

en2

[ [ [ [ [ [ [ [

1754634171

7773848354

1550688791

2301790452

3445174379

3589156342

] ] ] ] ] ] ] ]

[ [ [ [ [ [ [ [

minus03057290719

minus03005337771

minus09030341401

001584500766

minus002175914011

minus0003741933916

minus04311831740

minus08021570593

04130146809

00001362645283

minus0003854432451

0005396441669

minus08488209390

05154657137

01152135966

minus0004741957786

002229739182

minus0002068264586

00009402467441

minus0005358584867

0003987518000

minus003989381331

003796849613

minus09984595017

minus001058157817

002100470582

002092543658

0006230946613

minus09987520802

minus003826770492

00009823402498

0006977574167

001484146906

09990475909

0008196718814

minus003958286493

] ] ] ] ] ] ] ]

Specim

en3

[ [ [ [ [ [ [ [

1418542670

5838388363

9959058231

3433818493

3024968125

1757492470

] ] ] ] ] ] ] ]

[ [ [ [ [ [ [ [

03244960223

minus0004831643783

minus08451713783

04062092722

01164524794

minus004239310198

06461152835

minus07125334236

02668138654

004131783689

minus002431040234

003665187401

06621315621

07009969173

02633438279

002836289512

minus0001711614840

0005269111592

minus01962401199

0009159859564

03740839141

08850746598

01904577552

004284654369

minus0008597323304

minus0002526763452

001897677365

01738799329

minus06977939665

minus06945566457

001533190599

minus002803230310

006961922985

minus01374446216

06801869735

minus07159517722

] ] ] ] ] ] ] ]


24681012141618

12

34

56

Column Row

12

34

56


12345678

12

34

56

Column Row

12

34

56




120590119894119895= 119862


where 119862119894119895119896119897


120590 = 120590119894119895119890119894otimes 119890

119895 120598 = 120598

119896119897119890119896otimes 119890

119897

C = 119862119894119895119896119897

119890119894otimes 119890

119896otimes 119890

119896otimes 119890

119897

(23)

5

10

15

20

12

34

56

Column Row

12

34

56


5

10

15

20

25

12

34

56

Column Row

12

34

56



E1equiv 119890

1otimes 119890

1

E2equiv 119890

2otimes 119890

2

E3equiv 119890

3otimes 119890

3

(24)


E4equiv

1


2+ 119890

2otimes 119890

1)

E5equiv

1


3+ 119890

3otimes 119890

2)

E6equiv

1


1+ 119890

1otimes 119890

3)

(25)



E7equiv

1


2minus 119890

2otimes 119890

1)

E8equiv

1


3minus 119890

3otimes 119890

2)

E9equiv

1


1minus 119890

1otimes 119890

3)

(26)


120590 = 119878119860E

119860

120598 = 119864119860E

119860

C = 119862119860119861

119890119860otimes 119890

119861

(27)




120590119894119895= 119862

119894119895119896119897120598119896119897

lArrrArr 119878119860= 119862

119860119861E119861

119862119894119895119896119897

120598119896119897

= 120582120598119894119895lArrrArr 119862

119860119861E119861= 120582119864

119860

(28)


((((((((

(

1198781

1198782

1198783

1198784

1198785

1198786

1198787

1198788

1198789

))))))))

)

=

((((((((

(

12059011

12059022

12059033

120590(12)

120590(23)

120590(31)

120590[12]

120590[23]

120590[31]

))))))))

)

((((((((

(

1198641

1198642

1198643

1198644

1198645

1198646

1198647

1198648

1198649

))))))))

)

=

((((((((

(

12059811

12059822

12059833

120598(12)

120598(23)

120598(31)

120598[12]

120598[23]

120598[31]

))))))))

)

(29)


120579(119894119895)

equiv1

radic2(120579119894119895+ 120579

119895119894) 120579

[119894119895]equiv

1

radic2(120579119894119895minus 120579

119895119894) (30)


tensor C are

((((((((

(

11986211

11986212

11986213

11986214

11986215

11986216

11986217

11986218

11986219

11986221

11986222

11986223

11986224

11986225

11986226

11986227

11986228

11986229

11986231

11986232

11986233

11986234

11986235

11986236

11986237

11986238

11986239

11986241

11986242

11986243

11986244

11986245

11986246

11986247

11986248

11986249

11986251

11986252

11986253

11986254

11986255

11986256

11986257

11986258

11986259

11986261

11986262

11986263

11986264

11986265

11986266

11986267

11986268

11986269

11986271

11986272

11986273

11986274

11986275

11986276

11986277

11986278

11986279

11986281

11986282

11986283

11986284

11986285

11986286

11986287

11986288

11986289

11986291

11986292

11986293

11986294

11986295

11986296

11986297

11986298

11986299

))))))))

)

=

1198621111

1198621122

1198621133

1198622211

1198622222

1198623333

1198623311

1198623322

1198623333

11986211(12)

11986211(23)

11986211(31)

11986222(12)

11986222(23)

11986222(31)

11986233(12)

11986233(23)

11986233(31)

11986211[12]

11986211[23]

11986211[31]

11986222[12]

11986222[23]

11986222[31]

11986233[12]

11986233[23]

11986233[31]

119862(12)11

119862(12)22

119862(12)33

119862(23)11

119862(23)22

119862(23)33

119862(31)11

119862(31)22

119862(31)33

119862(1212)

119862(1223)

119862(1231)

119862(2312)

119862(2323)

119862(2331)

119862(3112)

119862(3123)

119862(3131)

119862(12)[12]

119862(12)[23]

119862(12)[31]

119862(23)[12]

119862(23)[23]

119862(23)[31]

119862(31)[12]

119862(31)[23]

119862(31)[31]

119862[12]11

119862[12]22

119862[12]33

119862[23]11

119862[23]22

119862[123]33

119862[21]11

119862[31]22

119862[31]33

119862[12](12)

119862[12](23)

119862[12](21)

119862[23](12)

119862[23](23)

119862[23](21)

119862[31](12)

119862[31](23)

119862[31](21)

119862[1212]

119862[1223]

119862[1231]

119862[2312]

119862[2323]

119862[2331]

119862[3112]

119862[3123]

119862[3131]

(31)

where

119862119894119895(119896119897)

equiv1

radic2(119862

119894119895119896119897+ 119862

119894119895119897119896) 119862

119894119895[119896119897]equiv

1

radic2(119862

119894119895119896119897minus 119862

119894119895119897119896)

119862(119894119895)119896119897

equiv1

radic2(119862

119894119895119896119897+ 119862

119895119894119896119897) 119862

[119894119895]119896119897equiv

1

radic2(119862

119894119895119896119897minus 119862

119895119894119896119897)

119862(119894119895119896119897)

equiv1

2(119862

119894119895119896119897+ 119862

119894119895119897119896+ 119862

119895119894119896119897+ 119862

119895119894119897119896)

119862(119894119895)[119896119897]

equiv1

2(119862

119894119895119896119897minus 119862

119894119895119897119896+ 119862

119895119894119896119897minus 119862

119895119894119897119896)

119862[119894119895](119896119897)

equiv1

2(119862

119894119895119896119897+ 119862

119894119895119897119896minus 119862

119895119894119896119897minus 119862

119895119894119897119896)

119862[119894119895119896119897]

equiv1

2(119862

119894119895119896119897minus 119862

119895119894119896119897minus 119862

119894119895119897119896+ 119862

119895119894119897119896)

(32)


12

34

56

ColumnRow

12

34

56

600005000040000300002000010000


5

10

15

20

25

12

34

56

Column Row

12

34

56






119895119894and 120598

119896119897= 120598

119897119896 the 119862

119894119895119896119897will be

like 119862119894119895119896119897

= 119862119895119894119896119897

= 119862119894119895119897119896


119862119894119895119896119897

= 119862119896119897119894119895












2 sdot 119864energy = 120590119894119895120598119894119895= 120590

119901120598119901 (33)


119901 = 119894120575119894119895+ (1 minus 120575

119894119895) (9 minus 119894 minus 119895)

119902 = 119896120575119896119897+ (1 minus 120575

119896119897) (9 minus 119896 minus 119897)

(34)


119901 119862

119894119895119896119897= 119862

119901119902 and 120598

119902= (2 minus

120575119896119897)120598119896119897



(

(

1205901

1205902

1205903

1205904

1205905

1205906

)

)

= (

(

11986211

11986212

11986213

11986214

11986215

11986216

11986221

11986222

11986223

11986224

11986225

11986226

11986231

11986232

11986233

11986234

11986235

11986236

11986241

11986242

11986243

11986244

11986245

11986246

11986251

11986252

11986253

11986254

11986255

11986256

11986261

11986262

11986263

11986264

11986265

11986266

)

)

(

(

1205981

1205982

1205983

1205984

1205985

1205986

)

)

(35)



(1) the 120590119894119895and 120598


(2) the norms of 120590119894119895 120598119896119897 and 119862




01 02 03 04 05 06 07 08

2

4

6

8

10

12

14

16

(a)

01

02

03

04

05

06

07

08

09

01 02 03 04 05 06 07 08

(b)


01 02 03 04 05 06 07 08

05

1

15

2

25

3

(a)

01 02 03 04 05 06 07 08

05

1

15

2

25

3

(b)




119901 = 119894120575119894119895+ (1 minus 120575

119894119895) (9 minus 119894 minus 119895)

119902 = 119896120575119896119897+ (1 minus 120575

119896119897) (9 minus 119896 minus 119897)

(36)

In this way

120590119901= (120575

119894119895+ radic2 (1 minus 120575

119894119895)) 120598

119894119895

120598119902= (120575

119896119897+ radic2 (1 minus 120575

119896119897)) 120598

119896119897

119862119901119902

= (120575119894119895+ radic2 (1 minus 120575

119894119895)) (120575

119896119897+ radic2 (1 minus 120575

119896119897)) 119862

119894119895119896119897

(37)




120590kelvin

= 120590viogt

120585 120590voigt

=120590kelvin

120585

120598kelvin

= 120598voigt

120585 120598voigt

=120598kelvin

120585

(38)


119862kelvin

=((

(







))

)

119862voigt

(39)


(

(

12059011

12059022

12059033

radic212059023

radic212059013

radic212059012

)

)

= (

1198621111 1198621122 1198621133radic21198621123

radic21198621113radic21198621112

1198622211 1198622222 1198622333radic21198622223

radic21198622213radic21198622212

1198623311 1198623322 1198623333radic21198623323

radic21198623313radic21198623312

radic21198622311radic21198622322

radic21198622333 21198622323 21198622313 21198622312

radic21198621311radic21198621322

radic21198621333 21198621323 21198621313 21198621312

radic21198621211radic21198621222

radic21198621233 21198621223 21198621213 21198621212

)

lowast(

(

12059811

12059822

12059833

radic212059823

radic212059813

radic212059812

)

)

(40)


= C (41)



(

(

12059011

12059022

12059033

radic212059023

radic212059013

radic212059012

)

)

=((

(

11986211

11986212

11986213

radic211986214

radic211986215

radic211986216

11986212

11986222

11986223

radic211986224

radic211986225

radic211986226

11986213

11986223

11986233

radic211986234

radic211986235

radic211986236

radic211986214

radic211986224

radic211986234

211986244

211986245

211986246

radic211986215

radic211986225

radic211986235

211986245

211986255

211986256

radic211986216

radic211986226

radic211986236

211986246

211986256

211986266

))

)

lowast (

(

12059811

12059822

12059833

radic212059823

radic212059813

radic212059812

)

)

(42)


(

(

1

2

3

4

5

6

)

)

=((

(

11986211

11986212

11986213

11986214

11986215

11986216

11986212

11986222

11986223

11986224

11986225

11986226

11986213

11986223

11986233

11986234

11986235

11986236

11986214

11986224

11986234

11986244

11986245

11986246

11986215

11986225

11986235

11986245

11986255

11986256

11986216

11986226

11986236

11986246

11986256

11986266

))

)

lowast (

(

1205981

1205982

1205983

1205984

1205985

1205986

)

)

(43)



































01 02 03 04 05 06 07 08

05

1

15

2

25

3

(a)

01 02 03 04 05 06 07 08

05

1

15

2

25

3

(b)


01 02 03 04 05 06 07 08

2

4

6

8

10

12

14

16





ΛI) N = 0 (44)


119894 119894 = 1 2 6 and

















NNA119896

equiv1

119872

119872

sum

119884=1

N119884119896 (45)



NAVG119896

equiv JNANNA119896

(46)


119896=1NNA119896

otimes NNA119896

| that is

JNA equiv (

6

sum

119896=1

NNA119896

otimes NNA119896

)

minus12

(47)


119896 that is

ΛAVG119902

equiv1

119872

119872

sum

119884=1

6

sum

119896=1

Λ119884

119896(N119884

119896sdot NAVG

119902)2

(48)


gt 0 for all 119896 = 1 2 6

and ΛAVG119896


NA119896


ΛNA119896

equiv1

119872

119872

sum

119884=1

Λ119884

119896 (49)


CAVG=

6

sum

119896=1

ΛAVG119896

NAVG119896

otimes NAVG119896

(50)





[[[[[[[

[

00367834559700000

00453681054800000

0998185540700000

00

00

00

]]]]]]]

]

[[[[[[[

[

0983745905000000

minus0170879918750000

minus00277901141300000

00

00

00

]]]]]]]

]

[[[[[[[

[

minus0169284222800000

minus0982986098000000

00509905132200000

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(51)


[[[[[[[

[

003679134179

004537783172

09983995367

00

00

00

]]]]]]]

]

[[[[[[[

[

09859858256

minus01712690003

minus002785339027

00

00

00

]]]]]]]

]

[[[[[[[

[

minus01697052873

minus09854311019

005111734310

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(52)




[[[

[

334725276837330 0000112116752981933 000198542359441849 00 00 00

0000112116752981933 334773746006714 minus0000991802322788501 00 00 00

000198542359441849 minus0000991802322788501 334013593769726 00 00 00

00 00 00 333950000000000 00 00

00 00 00 00 333950000000000 00

00 00 00 00 00 333950000000000

]]]

]

(53)




[[[[[[[

[

004028666644

006399313350

09971368323

00

00

00

]]]]]]]

]

[[[[[[[

[

09048796003

minus04255671399

minus0009247686096

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04237568827

minus09026613361

007505076253

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(54)


[[[[[[[

[

004028666644

006399313350

09971368323

00

00

00

]]]]]]]

]

[[[[[[[

[

09048795994

minus04255671395

minus0009247686087

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04237568827

minus09026613361

007505076253

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(55)



[[[

[

1458799998 000000001785571215 minus000000001009489597 00 00 00000000001785571215 1458799997 0000000002407020197 00 00 00minus000000001009489597 0000000002407020197 1458799997 00 00 00

00 00 00 1458799998 00 0000 00 00 00 1458799998 0000 00 00 00 00 1458799998

]]]

]

(56)




[[[[[

[

005368516527

007220768570

09959437502

00

00

00

]]]]]

]

[[[[[

[

09266208315

minus03753089731

minus002273783078

00

00

00

]]]]]

]

[[[[[

[

minus03721447810

minus09240829102

008705767064

00

00

00

]]]]]

]

[[[[[

[

00

00

00

00

00

10

]]]]]

]

[[[[[

[

00

00

00

10

00

00

]]]]]

]

[[[[[

[

00

00

00

00

10

00

]]]]]

]

(57)


[[[[[[[[[

[

005368516527

007220768570

09959437502

00

00

00

]]]]]]]]]

]

[[[[[[[[[

[

09266208315

minus03753089731

minus002273783078

00

00

00

]]]]]]]]]

]

[[[[[[[[[

[

minus03721447806

minus09240829093

008705767055

00

00

00

]]]]]]]]]

]

[[[[[[[[[

[

00

00

00

10

00

00

]]]]]]]]]

]

[[[[[[[[[

[

00

00

00

00

10

00

]]]]]]]]]

]

[[[[[[[[[

[

00

00

00

00

00

10

]]]]]]]]]

]

(58)




[[[

[

1615299998 0000000005508173090 minus0000000008722620038 00 00 00

0000000005508173090 1615299996 minus0000000004345156817 00 00 00

minus0000000008722620038 minus0000000004345156817 1615300000 00 00 00

00 00 00 1615299998 00 00

00 00 00 00 1615299998 00

00 00 00 00 00 1615299998

]]]

]

(59)




[[[[[[[

[

minus00598757013800000

minus00768229223150000

minus0995231841750000

00

00

00

]]]]]]]

]

[[[[[[[

[

minus0869358919900000

0493939153600000

00141567750950000

00

00

00

]]]]]]]

]

[[[[[[[

[

minus0490490276500000

minus0866078136900000

00963811082600000

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(60)


[[[[[[[[[[[

[

minus005987730132

minus007682497510

minus09952584354

00

00

00

]]]]]]]]]]]

]

[[[[[[[[[[[

[

minus08693926232

04939583026

001415732393

00

00

00

]]]]]]]]]]]

]

[[[[[[[[[[[

[

00

00

00

10

00

00

]]]]]]]]]]]

]

[[[[[[[[[[[

[

00

00

00

00

10

00

]]]]]]]]]]]

]

[[[[[[[[[[[

[

00

00

00

00

00

10

]]]]]]]]]]]

]

(61)



[[[

[


00 00 00 1819400002 1819400002 0000 00 00 00 00 0000 00 00 00 00 1819400002

]]]

]

(62)




[[[[[[[

[

004967528691

008463937788

09951726186

00

00

00

]]]]]]]

]

[[[[[[[

[

09161715971

minus04006166011

minus001165943224

00

00

00

]]]]]]]

]

[[[[[

[

minus03976958243

minus09123280736

009744493938

00

00

00

]]]]]

]

[[[[[

[

00

00

00

10

00

00

]]]]]

]

[[[[[

[

00

00

00

00

10

00

]]]]]

]

[[[[[

[

00

00

00

00

00

10

]]]]]

]

(63)


[[[[[[[

[

004967528696

008463937796

09951726196

00

00

00

]]]]]]]

]

[[[[[[[

[

09161715980

minus04006166015

minus001165943225

00

00

00

]]]]]]]

]


[[[[[[[

[

minus03976958247

minus09123280745

009744493948

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(64)



[[[

[

1922399998 minus000000001176508811 minus000000001537920006 00 00 00

minus000000001176508811 1922400000 minus0000000007631928036 00 00 00

minus000000001537920006 minus0000000007631928036 1922400000 00 00 00

00 00 00 1922399998 1922399998 00

00 00 00 00 00 00

00 00 00 00 00 1922399998

]]]

]

(65)




[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

[[[[[[[

[

minus01387533797

minus02031928413

minus09692575344

00

00

00

]]]]]]]

]

[[[[[[[

[

minus08476774112

05304148448

001015375725

00

00

00

]]]]]]]

]

[[[[[[[

[

minus05120454135

minus08230265872

02458388325

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

(66)


[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

[[[[[[[

[

minus01387533798

minus02031928415

minus09692575354

00

00

00

]]]]]]]

]

[[[[[[[

[

minus08476774112

05304148448

001015375725

00

00

00

]]]]]]]

]

[[[[[[[

[

minus05120454135

minus08230265872

02458388325

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

(67)



[[[

[

2074599999 0000000004356660361 000000003402343994 00 00 00

0000000004356660361 2074600004 000000002232269567 00 00 00

000000003402343994 000000002232269567 2074600000 00 00 00

00 00 00 2074600000 2074600000 00

00 00 00 00 00 00

00 00 00 00 00 2074600000

]]]

]

(68)




[[[[[[[

[

008621257414

01243595697

09884847438

00

00

00

]]]]]]]

]

[[[[[[[

[

08755641188

minus04828494241

minus001561753067

00

00

00

]]]]]]]

]

[[[[[

[

minus04753471023

minus08668282006

01505124700

00

00

00

]]]]]

]

[[[[[

[

00

00

00

10

00

00

]]]]]

]

[[[[[

[

00

00

00

00

10

00

]]]]]

]

[[[[[

[

00

00

00

00

00

10

]]]]]

]

(69)


[[[[[

[

008621257423

01243595698

09884847448

00

00

00

]]]]]

]

[[[[[

[

08755641188

minus04828494241

minus001561753067

00

00

00

]]]]]

]


[[[[[[[

[

minus04753471018

minus08668281997

01505124698

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(70)



[[[

[

1890099999 000000001209663993 minus000000002532733996 00 00 00000000001209663993 1890099991 0000000001701090138 00 00 00minus000000002532733996 0000000001701090138 1890100001 00 00 00

00 00 00 1890099997 1890099997 0000 00 00 00 00 0000 00 00 00 00 1890099997

]]]

]

(71)




[[[[[[[

[

03961520086

06338768023

06642768888

00

00

00

]]]]]]]

]

[[[[[[[

[

09160614623

minus02236797319

minus03328645006

00

00

00

]]]]]]]

]

[[[[[[[

[

006240981414

minus07403833993

06692812840

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(72)


[[[[[[[

[

03961520086

06338768023

06642768888

00

00

00

]]]]]]]

]

[[[[[[[

[

09160614614

minus02236797317

minus03328645003

00

00

00

]]]]]]]

]

[[[[[[[

[

006240981426

minus07403834008

06692812853

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(73)



[[[

[


00 00 00 8772299998 8772299998 0000 00 00 00 00 0000 00 00 00 00 8772299998

]]]

]

(74)




[[[[[[[

[

minus006591789198

minus008975775227

minus09937798435

00

00

00

]]]]]]]

]

[[[[[[[

[

minus08289381135

05593224991

0004466103673

00

00

00

]]]]]]]

]

[[[[[

[

minus05554425577

minus08240763851

01112729817

00

00

00

]]]]]

]

[[[[[

[

00

00

00

10

00

00

]]]]]

]

[[[[[

[

00

00

00

00

10

00

]]]]]

]

[[[[[

[

00

00

00

00

00

10

]]]]]

]


[[[[[[[

[

minus01387533798

minus02031928415

minus09692575354

00

00

00

]]]]]]]

]

[[[[[[[

[

minus08476774112

05304148448

001015375725

00

00

00

]]]]]]]

]


[[[[

[

minus05120454135

minus08230265872

02458388325

00

00

00

]]]]

]

[[[[

[

00

00

00

10

00

00

]]]]

]

[[[[

[

00

00

00

00

10

00

]]]]

]

[[[[

[

00

00

00

00

00

10

]]]]

]

(76)



[[[

[

2228057494 0000000004678921127 000000003654014285 00 00 000000000004678921127 2228057499 000000002397389829 00 00 00000000003654014285 000000002397389829 2228057495 00 00 00

00 00 00 2228057495 2228057495 0000 00 00 00 00 0000 00 00 00 00 2228057495

]]]

]

(77)




[[[[[[[

[

006975010637

01071019008

09917984199

00

00

00

]]]]]]]

]

[[[[[[[

[

09079000793

minus04187725641

minus001862756541

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04133429180

minus09017531389

01264472547

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(78)


[[[[[[[

[

006975010637

01071019008

09917984199

00

00

00

]]]]]]]

]

[[[[[[[

[

09079000793

minus04187725641

minus001862756541

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04133429184

minus09017531398

01264472548

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(79)



[[[

[


00 00 00 2598699998 2598699998 0000 00 00 00 00 0000 00 00 00 00 2598699998

]]]

]

(80)




[[[[[

[

minus00192522847250000

minus00192842313600000

000386151499999998

00

00

00

]]]]]

]

[[[[[

[

000961799224999998

00165029120500000

000436480214750000

00

00

00

]]]]]

]

[[[[[[

[

minus0494444050150000

minus0856906622200000

0142104790200000

00

00

00

]]]]]]

]

[[[[[[

[

00

00

00

10

00

00

]]]]]]

]

[[[[[[

[

00

00

00

00

10

00

]]]]]]

]

[[[[[[

[

00

00

00

00

00

10

]]]]]]

]

(81)



[[[[[[[[[[

[

minus2541745843

minus2545963537

5098090872

00

00

00

]]]]]]]]]]

]

[[[[[[[[[[

[

2505315863

4298714977

1136953303

00

00

00

]]]]]]]]]]

]

[[[[[[[

[

minus04949599718

minus08578007511

01422530677

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(82)



[[[

[

315679169478799 427324383657779 384557503470365 00 00 00427324383657779 618700481877479 889153535276175 00 00 00384557503470365 889153535276175 384768762708609 00 00 00

00 00 00 247795001400000 00 0000 00 00 00 247795001400000 0000 00 00 00 00 247795001400000

]]]

]

(83)




[[[[[[[

[

01135176976

01764907831

09777344911

00

00

00

]]]]]]]

]

[[[[[[[

[

08848520771

minus04654963442

minus001870707734

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04518302069

minus08672739791

02090103088

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(84)


[[[[[[[

[

01135176977

01764907833

09777344921

00

00

00

]]]]]]]

]

[[[[[[[

[

08848520771

minus04654963442

minus001870707734

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04518302069

minus08672739791

02090103088

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(85)



[[[

[

2663700003 000000004768023095 000000003169803038 00 00 00000000004768023095 2663699992 0000000008310743498 00 00 00000000003169803038 0000000008310743498 2663700001 00 00 00

00 00 00 2663699998 2663699998 0000 00 00 00 00 0000 00 00 00 00 2663699998

]]]

]

(86)





6 Conclusions









Disclosure




Acknowledgments



References




















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












1 Blasa

[[[[[[[[[

[

6385324208

009845837744

03832174064

06240000000

04060000000

006600000000

]]]]]]]]]

]

[[[[[[[[[

[

001488818113 09510211778 minus03087669978 00 00 00

002575997614 minus03090635474 minus09506924583 00 00 00

09995572851 minus0006200252135 002909967665 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

2 Spruce

[[[[[[[[[

[

1198583568

04516442734

1114520065

1498000000

1442000000

007800000000

]]]]]]]]]

]

[[[[[[[[[

[


minus004689865645 04044467539 minus09133582749 00 00 00

minus09983543167 001123114838 005623628701 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

3 Spruce

[[[[[[[[[

[

1410927768

04185382987

1227183961

1442000000

10

006400000000

]]]]]]]]]

]

[[[[[[[[[

[


minus005361649666 04423303564 minus08952480817 00 00 00

minus09978936411 0009052295016 006423657043 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

4 Spruce

[[[[[[[[[

[

1630529953

03544288592

08442715844

1234000000

1520000000

007200000000

]]]]]]]]]

]

[[[[[[[[[

[


minus002869707087 04091564350 minus09120128765 00 00 00

minus09993764538 0007032901380 003460119282 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

5 Spruce

[[[[[[[[[

[

1725845208

05092652753

1171282625

1250000000

1706000000

007000000000

]]]]]]]]]

]

[[[[[[[[[

[


minus003428302033 05858146064 minus08097196604 00 00 00

minus09988364173 0007413313465 004765347716 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

6 Douglas Fir

[[[[[[[[[

[

1613459769

06233733771

1439028943

1767000000

1766000000

01760000000

]]]]]]]]]

]

[[[[[[[[[

[


minus003697194691 05922678885 minus08048924305 00 00 00

minus09986533619 0007435777607 005134366998 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

7 Douglas Fir

[[[[[[[[[

[

1710150156

06986859431

2204812526

2348000000

1816000000

01600000000

]]]]]]]]]

]

[[[[[[[[[

[


minus006364825251 05704773676 minus08188433771 00 00 00

minus09967231590 0004703484281 008075160717 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]

8 Pine

[[[[[[[[[

[

1699539133

04940019684

1565606760

3484000000

1344000000

01320000000

]]]]]]]]]

]

[[[[[[[[[

[


minus005585205149 04413516031 minus08955943895 00 00 00

minus09978476167 0006195562437 006528207193 00 00 00

00 00 00 10 00 00

00 00 00 00 10 00

00 00 00 00 00 10

]]]]]]]]]

]


Table11Th

eeigenvalues

andeigenvectorsfor3

specim

enso

fcancello

usbo

necalculated

usingMAPL

EΛ[119894]sta

ndforthe

eigenvalueso

f119894th

specim

enof

cancellous

boneand

119881[119894]sta

ndfor

thec

orrespon

ding

eigenvectors

Cancellous

bone

EigenvaluesΛ

[119894]

Eigenvectors

119881[119894]

Specim

en1

[ [ [ [ [ [ [ [

1334869880

minus02071167473

8644210768

6837676373

3768806080

4123724855

] ] ] ] ] ] ] ]

[ [ [ [ [ [ [ [

03316115964

08831238953

02356441950

02242668491

005354786914

003787816350

06475092156

minus01099136965

01556396475

minus06940063479

minus01278953723

02154647417

04800906963

minus02832250652

007648827487

02494278793

06410471445

minus04585742261

minus02737532692

minus006354940263

04518342532

minus006214920576

05836952694

06101669758

minus03706582070

03314035780

minus01276606259

minus06337030234

03639432801

minus04499474345

01671829224

01180163925

minus08330343502

002008124471

03109325968

04087706408

] ] ] ] ] ] ] ]

Specim

en2

[ [ [ [ [ [ [ [

1754634171

7773848354

1550688791

2301790452

3445174379

3589156342

] ] ] ] ] ] ] ]

[ [ [ [ [ [ [ [

minus03057290719

minus03005337771

minus09030341401

001584500766

minus002175914011

minus0003741933916

minus04311831740

minus08021570593

04130146809

00001362645283

minus0003854432451

0005396441669

minus08488209390

05154657137

01152135966

minus0004741957786

002229739182

minus0002068264586

00009402467441

minus0005358584867

0003987518000

minus003989381331

003796849613

minus09984595017

minus001058157817

002100470582

002092543658

0006230946613

minus09987520802

minus003826770492

00009823402498

0006977574167

001484146906

09990475909

0008196718814

minus003958286493

] ] ] ] ] ] ] ]

Specim

en3

[ [ [ [ [ [ [ [

1418542670

5838388363

9959058231

3433818493

3024968125

1757492470

] ] ] ] ] ] ] ]

[ [ [ [ [ [ [ [

03244960223

minus0004831643783

minus08451713783

04062092722

01164524794

minus004239310198

06461152835

minus07125334236

02668138654

004131783689

minus002431040234

003665187401

06621315621

07009969173

02633438279

002836289512

minus0001711614840

0005269111592

minus01962401199

0009159859564

03740839141

08850746598

01904577552

004284654369

minus0008597323304

minus0002526763452

001897677365

01738799329

minus06977939665

minus06945566457

001533190599

minus002803230310

006961922985

minus01374446216

06801869735

minus07159517722

] ] ] ] ] ] ] ]


24681012141618

12

34

56

Column Row

12

34

56


12345678

12

34

56

Column Row

12

34

56




120590119894119895= 119862


where 119862119894119895119896119897


120590 = 120590119894119895119890119894otimes 119890

119895 120598 = 120598

119896119897119890119896otimes 119890

119897

C = 119862119894119895119896119897

119890119894otimes 119890

119896otimes 119890

119896otimes 119890

119897

(23)

5

10

15

20

12

34

56

Column Row

12

34

56


5

10

15

20

25

12

34

56

Column Row

12

34

56



E1equiv 119890

1otimes 119890

1

E2equiv 119890

2otimes 119890

2

E3equiv 119890

3otimes 119890

3

(24)


E4equiv

1


2+ 119890

2otimes 119890

1)

E5equiv

1


3+ 119890

3otimes 119890

2)

E6equiv

1


1+ 119890

1otimes 119890

3)

(25)



E7equiv

1


2minus 119890

2otimes 119890

1)

E8equiv

1


3minus 119890

3otimes 119890

2)

E9equiv

1


1minus 119890

1otimes 119890

3)

(26)


120590 = 119878119860E

119860

120598 = 119864119860E

119860

C = 119862119860119861

119890119860otimes 119890

119861

(27)




120590119894119895= 119862

119894119895119896119897120598119896119897

lArrrArr 119878119860= 119862

119860119861E119861

119862119894119895119896119897

120598119896119897

= 120582120598119894119895lArrrArr 119862

119860119861E119861= 120582119864

119860

(28)


((((((((

(

1198781

1198782

1198783

1198784

1198785

1198786

1198787

1198788

1198789

))))))))

)

=

((((((((

(

12059011

12059022

12059033

120590(12)

120590(23)

120590(31)

120590[12]

120590[23]

120590[31]

))))))))

)

((((((((

(

1198641

1198642

1198643

1198644

1198645

1198646

1198647

1198648

1198649

))))))))

)

=

((((((((

(

12059811

12059822

12059833

120598(12)

120598(23)

120598(31)

120598[12]

120598[23]

120598[31]

))))))))

)

(29)


120579(119894119895)

equiv1

radic2(120579119894119895+ 120579

119895119894) 120579

[119894119895]equiv

1

radic2(120579119894119895minus 120579

119895119894) (30)


tensor C are

((((((((

(

11986211

11986212

11986213

11986214

11986215

11986216

11986217

11986218

11986219

11986221

11986222

11986223

11986224

11986225

11986226

11986227

11986228

11986229

11986231

11986232

11986233

11986234

11986235

11986236

11986237

11986238

11986239

11986241

11986242

11986243

11986244

11986245

11986246

11986247

11986248

11986249

11986251

11986252

11986253

11986254

11986255

11986256

11986257

11986258

11986259

11986261

11986262

11986263

11986264

11986265

11986266

11986267

11986268

11986269

11986271

11986272

11986273

11986274

11986275

11986276

11986277

11986278

11986279

11986281

11986282

11986283

11986284

11986285

11986286

11986287

11986288

11986289

11986291

11986292

11986293

11986294

11986295

11986296

11986297

11986298

11986299

))))))))

)

=

1198621111

1198621122

1198621133

1198622211

1198622222

1198623333

1198623311

1198623322

1198623333

11986211(12)

11986211(23)

11986211(31)

11986222(12)

11986222(23)

11986222(31)

11986233(12)

11986233(23)

11986233(31)

11986211[12]

11986211[23]

11986211[31]

11986222[12]

11986222[23]

11986222[31]

11986233[12]

11986233[23]

11986233[31]

119862(12)11

119862(12)22

119862(12)33

119862(23)11

119862(23)22

119862(23)33

119862(31)11

119862(31)22

119862(31)33

119862(1212)

119862(1223)

119862(1231)

119862(2312)

119862(2323)

119862(2331)

119862(3112)

119862(3123)

119862(3131)

119862(12)[12]

119862(12)[23]

119862(12)[31]

119862(23)[12]

119862(23)[23]

119862(23)[31]

119862(31)[12]

119862(31)[23]

119862(31)[31]

119862[12]11

119862[12]22

119862[12]33

119862[23]11

119862[23]22

119862[123]33

119862[21]11

119862[31]22

119862[31]33

119862[12](12)

119862[12](23)

119862[12](21)

119862[23](12)

119862[23](23)

119862[23](21)

119862[31](12)

119862[31](23)

119862[31](21)

119862[1212]

119862[1223]

119862[1231]

119862[2312]

119862[2323]

119862[2331]

119862[3112]

119862[3123]

119862[3131]

(31)

where

119862119894119895(119896119897)

equiv1

radic2(119862

119894119895119896119897+ 119862

119894119895119897119896) 119862

119894119895[119896119897]equiv

1

radic2(119862

119894119895119896119897minus 119862

119894119895119897119896)

119862(119894119895)119896119897

equiv1

radic2(119862

119894119895119896119897+ 119862

119895119894119896119897) 119862

[119894119895]119896119897equiv

1

radic2(119862

119894119895119896119897minus 119862

119895119894119896119897)

119862(119894119895119896119897)

equiv1

2(119862

119894119895119896119897+ 119862

119894119895119897119896+ 119862

119895119894119896119897+ 119862

119895119894119897119896)

119862(119894119895)[119896119897]

equiv1

2(119862

119894119895119896119897minus 119862

119894119895119897119896+ 119862

119895119894119896119897minus 119862

119895119894119897119896)

119862[119894119895](119896119897)

equiv1

2(119862

119894119895119896119897+ 119862

119894119895119897119896minus 119862

119895119894119896119897minus 119862

119895119894119897119896)

119862[119894119895119896119897]

equiv1

2(119862

119894119895119896119897minus 119862

119895119894119896119897minus 119862

119894119895119897119896+ 119862

119895119894119897119896)

(32)


12

34

56

ColumnRow

12

34

56

600005000040000300002000010000


5

10

15

20

25

12

34

56

Column Row

12

34

56






119895119894and 120598

119896119897= 120598

119897119896 the 119862

119894119895119896119897will be

like 119862119894119895119896119897

= 119862119895119894119896119897

= 119862119894119895119897119896


119862119894119895119896119897

= 119862119896119897119894119895












2 sdot 119864energy = 120590119894119895120598119894119895= 120590

119901120598119901 (33)


119901 = 119894120575119894119895+ (1 minus 120575

119894119895) (9 minus 119894 minus 119895)

119902 = 119896120575119896119897+ (1 minus 120575

119896119897) (9 minus 119896 minus 119897)

(34)


119901 119862

119894119895119896119897= 119862

119901119902 and 120598

119902= (2 minus

120575119896119897)120598119896119897



(

(

1205901

1205902

1205903

1205904

1205905

1205906

)

)

= (

(

11986211

11986212

11986213

11986214

11986215

11986216

11986221

11986222

11986223

11986224

11986225

11986226

11986231

11986232

11986233

11986234

11986235

11986236

11986241

11986242

11986243

11986244

11986245

11986246

11986251

11986252

11986253

11986254

11986255

11986256

11986261

11986262

11986263

11986264

11986265

11986266

)

)

(

(

1205981

1205982

1205983

1205984

1205985

1205986

)

)

(35)



(1) the 120590119894119895and 120598


(2) the norms of 120590119894119895 120598119896119897 and 119862




01 02 03 04 05 06 07 08

2

4

6

8

10

12

14

16

(a)

01

02

03

04

05

06

07

08

09

01 02 03 04 05 06 07 08

(b)


01 02 03 04 05 06 07 08

05

1

15

2

25

3

(a)

01 02 03 04 05 06 07 08

05

1

15

2

25

3

(b)




119901 = 119894120575119894119895+ (1 minus 120575

119894119895) (9 minus 119894 minus 119895)

119902 = 119896120575119896119897+ (1 minus 120575

119896119897) (9 minus 119896 minus 119897)

(36)

In this way

120590119901= (120575

119894119895+ radic2 (1 minus 120575

119894119895)) 120598

119894119895

120598119902= (120575

119896119897+ radic2 (1 minus 120575

119896119897)) 120598

119896119897

119862119901119902

= (120575119894119895+ radic2 (1 minus 120575

119894119895)) (120575

119896119897+ radic2 (1 minus 120575

119896119897)) 119862

119894119895119896119897

(37)




120590kelvin

= 120590viogt

120585 120590voigt

=120590kelvin

120585

120598kelvin

= 120598voigt

120585 120598voigt

=120598kelvin

120585

(38)


119862kelvin

=((

(







))

)

119862voigt

(39)


(

(

12059011

12059022

12059033

radic212059023

radic212059013

radic212059012

)

)

= (

1198621111 1198621122 1198621133radic21198621123

radic21198621113radic21198621112

1198622211 1198622222 1198622333radic21198622223

radic21198622213radic21198622212

1198623311 1198623322 1198623333radic21198623323

radic21198623313radic21198623312

radic21198622311radic21198622322

radic21198622333 21198622323 21198622313 21198622312

radic21198621311radic21198621322

radic21198621333 21198621323 21198621313 21198621312

radic21198621211radic21198621222

radic21198621233 21198621223 21198621213 21198621212

)

lowast(

(

12059811

12059822

12059833

radic212059823

radic212059813

radic212059812

)

)

(40)


= C (41)



(

(

12059011

12059022

12059033

radic212059023

radic212059013

radic212059012

)

)

=((

(

11986211

11986212

11986213

radic211986214

radic211986215

radic211986216

11986212

11986222

11986223

radic211986224

radic211986225

radic211986226

11986213

11986223

11986233

radic211986234

radic211986235

radic211986236

radic211986214

radic211986224

radic211986234

211986244

211986245

211986246

radic211986215

radic211986225

radic211986235

211986245

211986255

211986256

radic211986216

radic211986226

radic211986236

211986246

211986256

211986266

))

)

lowast (

(

12059811

12059822

12059833

radic212059823

radic212059813

radic212059812

)

)

(42)


(

(

1

2

3

4

5

6

)

)

=((

(

11986211

11986212

11986213

11986214

11986215

11986216

11986212

11986222

11986223

11986224

11986225

11986226

11986213

11986223

11986233

11986234

11986235

11986236

11986214

11986224

11986234

11986244

11986245

11986246

11986215

11986225

11986235

11986245

11986255

11986256

11986216

11986226

11986236

11986246

11986256

11986266

))

)

lowast (

(

1205981

1205982

1205983

1205984

1205985

1205986

)

)

(43)



































01 02 03 04 05 06 07 08

05

1

15

2

25

3

(a)

01 02 03 04 05 06 07 08

05

1

15

2

25

3

(b)


01 02 03 04 05 06 07 08

2

4

6

8

10

12

14

16





ΛI) N = 0 (44)


119894 119894 = 1 2 6 and

















NNA119896

equiv1

119872

119872

sum

119884=1

N119884119896 (45)



NAVG119896

equiv JNANNA119896

(46)


119896=1NNA119896

otimes NNA119896

| that is

JNA equiv (

6

sum

119896=1

NNA119896

otimes NNA119896

)

minus12

(47)


119896 that is

ΛAVG119902

equiv1

119872

119872

sum

119884=1

6

sum

119896=1

Λ119884

119896(N119884

119896sdot NAVG

119902)2

(48)


gt 0 for all 119896 = 1 2 6

and ΛAVG119896


NA119896


ΛNA119896

equiv1

119872

119872

sum

119884=1

Λ119884

119896 (49)


CAVG=

6

sum

119896=1

ΛAVG119896

NAVG119896

otimes NAVG119896

(50)





[[[[[[[

[

00367834559700000

00453681054800000

0998185540700000

00

00

00

]]]]]]]

]

[[[[[[[

[

0983745905000000

minus0170879918750000

minus00277901141300000

00

00

00

]]]]]]]

]

[[[[[[[

[

minus0169284222800000

minus0982986098000000

00509905132200000

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(51)


[[[[[[[

[

003679134179

004537783172

09983995367

00

00

00

]]]]]]]

]

[[[[[[[

[

09859858256

minus01712690003

minus002785339027

00

00

00

]]]]]]]

]

[[[[[[[

[

minus01697052873

minus09854311019

005111734310

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(52)




[[[

[

334725276837330 0000112116752981933 000198542359441849 00 00 00

0000112116752981933 334773746006714 minus0000991802322788501 00 00 00

000198542359441849 minus0000991802322788501 334013593769726 00 00 00

00 00 00 333950000000000 00 00

00 00 00 00 333950000000000 00

00 00 00 00 00 333950000000000

]]]

]

(53)




[[[[[[[

[

004028666644

006399313350

09971368323

00

00

00

]]]]]]]

]

[[[[[[[

[

09048796003

minus04255671399

minus0009247686096

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04237568827

minus09026613361

007505076253

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(54)


[[[[[[[

[

004028666644

006399313350

09971368323

00

00

00

]]]]]]]

]

[[[[[[[

[

09048795994

minus04255671395

minus0009247686087

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04237568827

minus09026613361

007505076253

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(55)



[[[

[

1458799998 000000001785571215 minus000000001009489597 00 00 00000000001785571215 1458799997 0000000002407020197 00 00 00minus000000001009489597 0000000002407020197 1458799997 00 00 00

00 00 00 1458799998 00 0000 00 00 00 1458799998 0000 00 00 00 00 1458799998

]]]

]

(56)




[[[[[

[

005368516527

007220768570

09959437502

00

00

00

]]]]]

]

[[[[[

[

09266208315

minus03753089731

minus002273783078

00

00

00

]]]]]

]

[[[[[

[

minus03721447810

minus09240829102

008705767064

00

00

00

]]]]]

]

[[[[[

[

00

00

00

00

00

10

]]]]]

]

[[[[[

[

00

00

00

10

00

00

]]]]]

]

[[[[[

[

00

00

00

00

10

00

]]]]]

]

(57)


[[[[[[[[[

[

005368516527

007220768570

09959437502

00

00

00

]]]]]]]]]

]

[[[[[[[[[

[

09266208315

minus03753089731

minus002273783078

00

00

00

]]]]]]]]]

]

[[[[[[[[[

[

minus03721447806

minus09240829093

008705767055

00

00

00

]]]]]]]]]

]

[[[[[[[[[

[

00

00

00

10

00

00

]]]]]]]]]

]

[[[[[[[[[

[

00

00

00

00

10

00

]]]]]]]]]

]

[[[[[[[[[

[

00

00

00

00

00

10

]]]]]]]]]

]

(58)




[[[

[

1615299998 0000000005508173090 minus0000000008722620038 00 00 00

0000000005508173090 1615299996 minus0000000004345156817 00 00 00

minus0000000008722620038 minus0000000004345156817 1615300000 00 00 00

00 00 00 1615299998 00 00

00 00 00 00 1615299998 00

00 00 00 00 00 1615299998

]]]

]

(59)




[[[[[[[

[

minus00598757013800000

minus00768229223150000

minus0995231841750000

00

00

00

]]]]]]]

]

[[[[[[[

[

minus0869358919900000

0493939153600000

00141567750950000

00

00

00

]]]]]]]

]

[[[[[[[

[

minus0490490276500000

minus0866078136900000

00963811082600000

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(60)


[[[[[[[[[[[

[

minus005987730132

minus007682497510

minus09952584354

00

00

00

]]]]]]]]]]]

]

[[[[[[[[[[[

[

minus08693926232

04939583026

001415732393

00

00

00

]]]]]]]]]]]

]

[[[[[[[[[[[

[

00

00

00

10

00

00

]]]]]]]]]]]

]

[[[[[[[[[[[

[

00

00

00

00

10

00

]]]]]]]]]]]

]

[[[[[[[[[[[

[

00

00

00

00

00

10

]]]]]]]]]]]

]

(61)



[[[

[


00 00 00 1819400002 1819400002 0000 00 00 00 00 0000 00 00 00 00 1819400002

]]]

]

(62)




[[[[[[[

[

004967528691

008463937788

09951726186

00

00

00

]]]]]]]

]

[[[[[[[

[

09161715971

minus04006166011

minus001165943224

00

00

00

]]]]]]]

]

[[[[[

[

minus03976958243

minus09123280736

009744493938

00

00

00

]]]]]

]

[[[[[

[

00

00

00

10

00

00

]]]]]

]

[[[[[

[

00

00

00

00

10

00

]]]]]

]

[[[[[

[

00

00

00

00

00

10

]]]]]

]

(63)


[[[[[[[

[

004967528696

008463937796

09951726196

00

00

00

]]]]]]]

]

[[[[[[[

[

09161715980

minus04006166015

minus001165943225

00

00

00

]]]]]]]

]


[[[[[[[

[

minus03976958247

minus09123280745

009744493948

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(64)



[[[

[

1922399998 minus000000001176508811 minus000000001537920006 00 00 00

minus000000001176508811 1922400000 minus0000000007631928036 00 00 00

minus000000001537920006 minus0000000007631928036 1922400000 00 00 00

00 00 00 1922399998 1922399998 00

00 00 00 00 00 00

00 00 00 00 00 1922399998

]]]

]

(65)




[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

[[[[[[[

[

minus01387533797

minus02031928413

minus09692575344

00

00

00

]]]]]]]

]

[[[[[[[

[

minus08476774112

05304148448

001015375725

00

00

00

]]]]]]]

]

[[[[[[[

[

minus05120454135

minus08230265872

02458388325

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

(66)


[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

[[[[[[[

[

minus01387533798

minus02031928415

minus09692575354

00

00

00

]]]]]]]

]

[[[[[[[

[

minus08476774112

05304148448

001015375725

00

00

00

]]]]]]]

]

[[[[[[[

[

minus05120454135

minus08230265872

02458388325

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

(67)



[[[

[

2074599999 0000000004356660361 000000003402343994 00 00 00

0000000004356660361 2074600004 000000002232269567 00 00 00

000000003402343994 000000002232269567 2074600000 00 00 00

00 00 00 2074600000 2074600000 00

00 00 00 00 00 00

00 00 00 00 00 2074600000

]]]

]

(68)




[[[[[[[

[

008621257414

01243595697

09884847438

00

00

00

]]]]]]]

]

[[[[[[[

[

08755641188

minus04828494241

minus001561753067

00

00

00

]]]]]]]

]

[[[[[

[

minus04753471023

minus08668282006

01505124700

00

00

00

]]]]]

]

[[[[[

[

00

00

00

10

00

00

]]]]]

]

[[[[[

[

00

00

00

00

10

00

]]]]]

]

[[[[[

[

00

00

00

00

00

10

]]]]]

]

(69)


[[[[[

[

008621257423

01243595698

09884847448

00

00

00

]]]]]

]

[[[[[

[

08755641188

minus04828494241

minus001561753067

00

00

00

]]]]]

]


[[[[[[[

[

minus04753471018

minus08668281997

01505124698

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(70)



[[[

[

1890099999 000000001209663993 minus000000002532733996 00 00 00000000001209663993 1890099991 0000000001701090138 00 00 00minus000000002532733996 0000000001701090138 1890100001 00 00 00

00 00 00 1890099997 1890099997 0000 00 00 00 00 0000 00 00 00 00 1890099997

]]]

]

(71)




[[[[[[[

[

03961520086

06338768023

06642768888

00

00

00

]]]]]]]

]

[[[[[[[

[

09160614623

minus02236797319

minus03328645006

00

00

00

]]]]]]]

]

[[[[[[[

[

006240981414

minus07403833993

06692812840

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(72)


[[[[[[[

[

03961520086

06338768023

06642768888

00

00

00

]]]]]]]

]

[[[[[[[

[

09160614614

minus02236797317

minus03328645003

00

00

00

]]]]]]]

]

[[[[[[[

[

006240981426

minus07403834008

06692812853

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(73)



[[[

[


00 00 00 8772299998 8772299998 0000 00 00 00 00 0000 00 00 00 00 8772299998

]]]

]

(74)




[[[[[[[

[

minus006591789198

minus008975775227

minus09937798435

00

00

00

]]]]]]]

]

[[[[[[[

[

minus08289381135

05593224991

0004466103673

00

00

00

]]]]]]]

]

[[[[[

[

minus05554425577

minus08240763851

01112729817

00

00

00

]]]]]

]

[[[[[

[

00

00

00

10

00

00

]]]]]

]

[[[[[

[

00

00

00

00

10

00

]]]]]

]

[[[[[

[

00

00

00

00

00

10

]]]]]

]


[[[[[[[

[

minus01387533798

minus02031928415

minus09692575354

00

00

00

]]]]]]]

]

[[[[[[[

[

minus08476774112

05304148448

001015375725

00

00

00

]]]]]]]

]


[[[[

[

minus05120454135

minus08230265872

02458388325

00

00

00

]]]]

]

[[[[

[

00

00

00

10

00

00

]]]]

]

[[[[

[

00

00

00

00

10

00

]]]]

]

[[[[

[

00

00

00

00

00

10

]]]]

]

(76)



[[[

[

2228057494 0000000004678921127 000000003654014285 00 00 000000000004678921127 2228057499 000000002397389829 00 00 00000000003654014285 000000002397389829 2228057495 00 00 00

00 00 00 2228057495 2228057495 0000 00 00 00 00 0000 00 00 00 00 2228057495

]]]

]

(77)




[[[[[[[

[

006975010637

01071019008

09917984199

00

00

00

]]]]]]]

]

[[[[[[[

[

09079000793

minus04187725641

minus001862756541

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04133429180

minus09017531389

01264472547

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(78)


[[[[[[[

[

006975010637

01071019008

09917984199

00

00

00

]]]]]]]

]

[[[[[[[

[

09079000793

minus04187725641

minus001862756541

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04133429184

minus09017531398

01264472548

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(79)



[[[

[


00 00 00 2598699998 2598699998 0000 00 00 00 00 0000 00 00 00 00 2598699998

]]]

]

(80)




[[[[[

[

minus00192522847250000

minus00192842313600000

000386151499999998

00

00

00

]]]]]

]

[[[[[

[

000961799224999998

00165029120500000

000436480214750000

00

00

00

]]]]]

]

[[[[[[

[

minus0494444050150000

minus0856906622200000

0142104790200000

00

00

00

]]]]]]

]

[[[[[[

[

00

00

00

10

00

00

]]]]]]

]

[[[[[[

[

00

00

00

00

10

00

]]]]]]

]

[[[[[[

[

00

00

00

00

00

10

]]]]]]

]

(81)



[[[[[[[[[[

[

minus2541745843

minus2545963537

5098090872

00

00

00

]]]]]]]]]]

]

[[[[[[[[[[

[

2505315863

4298714977

1136953303

00

00

00

]]]]]]]]]]

]

[[[[[[[

[

minus04949599718

minus08578007511

01422530677

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(82)



[[[

[

315679169478799 427324383657779 384557503470365 00 00 00427324383657779 618700481877479 889153535276175 00 00 00384557503470365 889153535276175 384768762708609 00 00 00

00 00 00 247795001400000 00 0000 00 00 00 247795001400000 0000 00 00 00 00 247795001400000

]]]

]

(83)




[[[[[[[

[

01135176976

01764907831

09777344911

00

00

00

]]]]]]]

]

[[[[[[[

[

08848520771

minus04654963442

minus001870707734

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04518302069

minus08672739791

02090103088

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(84)


[[[[[[[

[

01135176977

01764907833

09777344921

00

00

00

]]]]]]]

]

[[[[[[[

[

08848520771

minus04654963442

minus001870707734

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04518302069

minus08672739791

02090103088

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(85)



[[[

[

2663700003 000000004768023095 000000003169803038 00 00 00000000004768023095 2663699992 0000000008310743498 00 00 00000000003169803038 0000000008310743498 2663700001 00 00 00

00 00 00 2663699998 2663699998 0000 00 00 00 00 0000 00 00 00 00 2663699998

]]]

]

(86)





6 Conclusions









Disclosure




Acknowledgments



References




















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Table11Th

eeigenvalues

andeigenvectorsfor3

specim

enso

fcancello

usbo

necalculated

usingMAPL

EΛ[119894]sta

ndforthe

eigenvalueso

f119894th

specim

enof

cancellous

boneand

119881[119894]sta

ndfor

thec

orrespon

ding

eigenvectors

Cancellous

bone

EigenvaluesΛ

[119894]

Eigenvectors

119881[119894]

Specim

en1

[ [ [ [ [ [ [ [

1334869880

minus02071167473

8644210768

6837676373

3768806080

4123724855

] ] ] ] ] ] ] ]

[ [ [ [ [ [ [ [

03316115964

08831238953

02356441950

02242668491

005354786914

003787816350

06475092156

minus01099136965

01556396475

minus06940063479

minus01278953723

02154647417

04800906963

minus02832250652

007648827487

02494278793

06410471445

minus04585742261

minus02737532692

minus006354940263

04518342532

minus006214920576

05836952694

06101669758

minus03706582070

03314035780

minus01276606259

minus06337030234

03639432801

minus04499474345

01671829224

01180163925

minus08330343502

002008124471

03109325968

04087706408

] ] ] ] ] ] ] ]

Specim

en2

[ [ [ [ [ [ [ [

1754634171

7773848354

1550688791

2301790452

3445174379

3589156342

] ] ] ] ] ] ] ]

[ [ [ [ [ [ [ [

minus03057290719

minus03005337771

minus09030341401

001584500766

minus002175914011

minus0003741933916

minus04311831740

minus08021570593

04130146809

00001362645283

minus0003854432451

0005396441669

minus08488209390

05154657137

01152135966

minus0004741957786

002229739182

minus0002068264586

00009402467441

minus0005358584867

0003987518000

minus003989381331

003796849613

minus09984595017

minus001058157817

002100470582

002092543658

0006230946613

minus09987520802

minus003826770492

00009823402498

0006977574167

001484146906

09990475909

0008196718814

minus003958286493

] ] ] ] ] ] ] ]

Specim

en3

[ [ [ [ [ [ [ [

1418542670

5838388363

9959058231

3433818493

3024968125

1757492470

] ] ] ] ] ] ] ]

[ [ [ [ [ [ [ [

03244960223

minus0004831643783

minus08451713783

04062092722

01164524794

minus004239310198

06461152835

minus07125334236

02668138654

004131783689

minus002431040234

003665187401

06621315621

07009969173

02633438279

002836289512

minus0001711614840

0005269111592

minus01962401199

0009159859564

03740839141

08850746598

01904577552

004284654369

minus0008597323304

minus0002526763452

001897677365

01738799329

minus06977939665

minus06945566457

001533190599

minus002803230310

006961922985

minus01374446216

06801869735

minus07159517722

] ] ] ] ] ] ] ]


24681012141618

12

34

56

Column Row

12

34

56


12345678

12

34

56

Column Row

12

34

56




120590119894119895= 119862


where 119862119894119895119896119897


120590 = 120590119894119895119890119894otimes 119890

119895 120598 = 120598

119896119897119890119896otimes 119890

119897

C = 119862119894119895119896119897

119890119894otimes 119890

119896otimes 119890

119896otimes 119890

119897

(23)

5

10

15

20

12

34

56

Column Row

12

34

56


5

10

15

20

25

12

34

56

Column Row

12

34

56



E1equiv 119890

1otimes 119890

1

E2equiv 119890

2otimes 119890

2

E3equiv 119890

3otimes 119890

3

(24)


E4equiv

1


2+ 119890

2otimes 119890

1)

E5equiv

1


3+ 119890

3otimes 119890

2)

E6equiv

1


1+ 119890

1otimes 119890

3)

(25)



E7equiv

1


2minus 119890

2otimes 119890

1)

E8equiv

1


3minus 119890

3otimes 119890

2)

E9equiv

1


1minus 119890

1otimes 119890

3)

(26)


120590 = 119878119860E

119860

120598 = 119864119860E

119860

C = 119862119860119861

119890119860otimes 119890

119861

(27)




120590119894119895= 119862

119894119895119896119897120598119896119897

lArrrArr 119878119860= 119862

119860119861E119861

119862119894119895119896119897

120598119896119897

= 120582120598119894119895lArrrArr 119862

119860119861E119861= 120582119864

119860

(28)


((((((((

(

1198781

1198782

1198783

1198784

1198785

1198786

1198787

1198788

1198789

))))))))

)

=

((((((((

(

12059011

12059022

12059033

120590(12)

120590(23)

120590(31)

120590[12]

120590[23]

120590[31]

))))))))

)

((((((((

(

1198641

1198642

1198643

1198644

1198645

1198646

1198647

1198648

1198649

))))))))

)

=

((((((((

(

12059811

12059822

12059833

120598(12)

120598(23)

120598(31)

120598[12]

120598[23]

120598[31]

))))))))

)

(29)


120579(119894119895)

equiv1

radic2(120579119894119895+ 120579

119895119894) 120579

[119894119895]equiv

1

radic2(120579119894119895minus 120579

119895119894) (30)


tensor C are

((((((((

(

11986211

11986212

11986213

11986214

11986215

11986216

11986217

11986218

11986219

11986221

11986222

11986223

11986224

11986225

11986226

11986227

11986228

11986229

11986231

11986232

11986233

11986234

11986235

11986236

11986237

11986238

11986239

11986241

11986242

11986243

11986244

11986245

11986246

11986247

11986248

11986249

11986251

11986252

11986253

11986254

11986255

11986256

11986257

11986258

11986259

11986261

11986262

11986263

11986264

11986265

11986266

11986267

11986268

11986269

11986271

11986272

11986273

11986274

11986275

11986276

11986277

11986278

11986279

11986281

11986282

11986283

11986284

11986285

11986286

11986287

11986288

11986289

11986291

11986292

11986293

11986294

11986295

11986296

11986297

11986298

11986299

))))))))

)

=

1198621111

1198621122

1198621133

1198622211

1198622222

1198623333

1198623311

1198623322

1198623333

11986211(12)

11986211(23)

11986211(31)

11986222(12)

11986222(23)

11986222(31)

11986233(12)

11986233(23)

11986233(31)

11986211[12]

11986211[23]

11986211[31]

11986222[12]

11986222[23]

11986222[31]

11986233[12]

11986233[23]

11986233[31]

119862(12)11

119862(12)22

119862(12)33

119862(23)11

119862(23)22

119862(23)33

119862(31)11

119862(31)22

119862(31)33

119862(1212)

119862(1223)

119862(1231)

119862(2312)

119862(2323)

119862(2331)

119862(3112)

119862(3123)

119862(3131)

119862(12)[12]

119862(12)[23]

119862(12)[31]

119862(23)[12]

119862(23)[23]

119862(23)[31]

119862(31)[12]

119862(31)[23]

119862(31)[31]

119862[12]11

119862[12]22

119862[12]33

119862[23]11

119862[23]22

119862[123]33

119862[21]11

119862[31]22

119862[31]33

119862[12](12)

119862[12](23)

119862[12](21)

119862[23](12)

119862[23](23)

119862[23](21)

119862[31](12)

119862[31](23)

119862[31](21)

119862[1212]

119862[1223]

119862[1231]

119862[2312]

119862[2323]

119862[2331]

119862[3112]

119862[3123]

119862[3131]

(31)

where

119862119894119895(119896119897)

equiv1

radic2(119862

119894119895119896119897+ 119862

119894119895119897119896) 119862

119894119895[119896119897]equiv

1

radic2(119862

119894119895119896119897minus 119862

119894119895119897119896)

119862(119894119895)119896119897

equiv1

radic2(119862

119894119895119896119897+ 119862

119895119894119896119897) 119862

[119894119895]119896119897equiv

1

radic2(119862

119894119895119896119897minus 119862

119895119894119896119897)

119862(119894119895119896119897)

equiv1

2(119862

119894119895119896119897+ 119862

119894119895119897119896+ 119862

119895119894119896119897+ 119862

119895119894119897119896)

119862(119894119895)[119896119897]

equiv1

2(119862

119894119895119896119897minus 119862

119894119895119897119896+ 119862

119895119894119896119897minus 119862

119895119894119897119896)

119862[119894119895](119896119897)

equiv1

2(119862

119894119895119896119897+ 119862

119894119895119897119896minus 119862

119895119894119896119897minus 119862

119895119894119897119896)

119862[119894119895119896119897]

equiv1

2(119862

119894119895119896119897minus 119862

119895119894119896119897minus 119862

119894119895119897119896+ 119862

119895119894119897119896)

(32)


12

34

56

ColumnRow

12

34

56

600005000040000300002000010000


5

10

15

20

25

12

34

56

Column Row

12

34

56






119895119894and 120598

119896119897= 120598

119897119896 the 119862

119894119895119896119897will be

like 119862119894119895119896119897

= 119862119895119894119896119897

= 119862119894119895119897119896


119862119894119895119896119897

= 119862119896119897119894119895












2 sdot 119864energy = 120590119894119895120598119894119895= 120590

119901120598119901 (33)


119901 = 119894120575119894119895+ (1 minus 120575

119894119895) (9 minus 119894 minus 119895)

119902 = 119896120575119896119897+ (1 minus 120575

119896119897) (9 minus 119896 minus 119897)

(34)


119901 119862

119894119895119896119897= 119862

119901119902 and 120598

119902= (2 minus

120575119896119897)120598119896119897



(

(

1205901

1205902

1205903

1205904

1205905

1205906

)

)

= (

(

11986211

11986212

11986213

11986214

11986215

11986216

11986221

11986222

11986223

11986224

11986225

11986226

11986231

11986232

11986233

11986234

11986235

11986236

11986241

11986242

11986243

11986244

11986245

11986246

11986251

11986252

11986253

11986254

11986255

11986256

11986261

11986262

11986263

11986264

11986265

11986266

)

)

(

(

1205981

1205982

1205983

1205984

1205985

1205986

)

)

(35)



(1) the 120590119894119895and 120598


(2) the norms of 120590119894119895 120598119896119897 and 119862




01 02 03 04 05 06 07 08

2

4

6

8

10

12

14

16

(a)

01

02

03

04

05

06

07

08

09

01 02 03 04 05 06 07 08

(b)


01 02 03 04 05 06 07 08

05

1

15

2

25

3

(a)

01 02 03 04 05 06 07 08

05

1

15

2

25

3

(b)




119901 = 119894120575119894119895+ (1 minus 120575

119894119895) (9 minus 119894 minus 119895)

119902 = 119896120575119896119897+ (1 minus 120575

119896119897) (9 minus 119896 minus 119897)

(36)

In this way

120590119901= (120575

119894119895+ radic2 (1 minus 120575

119894119895)) 120598

119894119895

120598119902= (120575

119896119897+ radic2 (1 minus 120575

119896119897)) 120598

119896119897

119862119901119902

= (120575119894119895+ radic2 (1 minus 120575

119894119895)) (120575

119896119897+ radic2 (1 minus 120575

119896119897)) 119862

119894119895119896119897

(37)




120590kelvin

= 120590viogt

120585 120590voigt

=120590kelvin

120585

120598kelvin

= 120598voigt

120585 120598voigt

=120598kelvin

120585

(38)


119862kelvin

=((

(







))

)

119862voigt

(39)


(

(

12059011

12059022

12059033

radic212059023

radic212059013

radic212059012

)

)

= (

1198621111 1198621122 1198621133radic21198621123

radic21198621113radic21198621112

1198622211 1198622222 1198622333radic21198622223

radic21198622213radic21198622212

1198623311 1198623322 1198623333radic21198623323

radic21198623313radic21198623312

radic21198622311radic21198622322

radic21198622333 21198622323 21198622313 21198622312

radic21198621311radic21198621322

radic21198621333 21198621323 21198621313 21198621312

radic21198621211radic21198621222

radic21198621233 21198621223 21198621213 21198621212

)

lowast(

(

12059811

12059822

12059833

radic212059823

radic212059813

radic212059812

)

)

(40)


= C (41)



(

(

12059011

12059022

12059033

radic212059023

radic212059013

radic212059012

)

)

=((

(

11986211

11986212

11986213

radic211986214

radic211986215

radic211986216

11986212

11986222

11986223

radic211986224

radic211986225

radic211986226

11986213

11986223

11986233

radic211986234

radic211986235

radic211986236

radic211986214

radic211986224

radic211986234

211986244

211986245

211986246

radic211986215

radic211986225

radic211986235

211986245

211986255

211986256

radic211986216

radic211986226

radic211986236

211986246

211986256

211986266

))

)

lowast (

(

12059811

12059822

12059833

radic212059823

radic212059813

radic212059812

)

)

(42)


(

(

1

2

3

4

5

6

)

)

=((

(

11986211

11986212

11986213

11986214

11986215

11986216

11986212

11986222

11986223

11986224

11986225

11986226

11986213

11986223

11986233

11986234

11986235

11986236

11986214

11986224

11986234

11986244

11986245

11986246

11986215

11986225

11986235

11986245

11986255

11986256

11986216

11986226

11986236

11986246

11986256

11986266

))

)

lowast (

(

1205981

1205982

1205983

1205984

1205985

1205986

)

)

(43)



































01 02 03 04 05 06 07 08

05

1

15

2

25

3

(a)

01 02 03 04 05 06 07 08

05

1

15

2

25

3

(b)


01 02 03 04 05 06 07 08

2

4

6

8

10

12

14

16





ΛI) N = 0 (44)


119894 119894 = 1 2 6 and

















NNA119896

equiv1

119872

119872

sum

119884=1

N119884119896 (45)



NAVG119896

equiv JNANNA119896

(46)


119896=1NNA119896

otimes NNA119896

| that is

JNA equiv (

6

sum

119896=1

NNA119896

otimes NNA119896

)

minus12

(47)


119896 that is

ΛAVG119902

equiv1

119872

119872

sum

119884=1

6

sum

119896=1

Λ119884

119896(N119884

119896sdot NAVG

119902)2

(48)


gt 0 for all 119896 = 1 2 6

and ΛAVG119896


NA119896


ΛNA119896

equiv1

119872

119872

sum

119884=1

Λ119884

119896 (49)


CAVG=

6

sum

119896=1

ΛAVG119896

NAVG119896

otimes NAVG119896

(50)





[[[[[[[

[

00367834559700000

00453681054800000

0998185540700000

00

00

00

]]]]]]]

]

[[[[[[[

[

0983745905000000

minus0170879918750000

minus00277901141300000

00

00

00

]]]]]]]

]

[[[[[[[

[

minus0169284222800000

minus0982986098000000

00509905132200000

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(51)


[[[[[[[

[

003679134179

004537783172

09983995367

00

00

00

]]]]]]]

]

[[[[[[[

[

09859858256

minus01712690003

minus002785339027

00

00

00

]]]]]]]

]

[[[[[[[

[

minus01697052873

minus09854311019

005111734310

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(52)




[[[

[

334725276837330 0000112116752981933 000198542359441849 00 00 00

0000112116752981933 334773746006714 minus0000991802322788501 00 00 00

000198542359441849 minus0000991802322788501 334013593769726 00 00 00

00 00 00 333950000000000 00 00

00 00 00 00 333950000000000 00

00 00 00 00 00 333950000000000

]]]

]

(53)




[[[[[[[

[

004028666644

006399313350

09971368323

00

00

00

]]]]]]]

]

[[[[[[[

[

09048796003

minus04255671399

minus0009247686096

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04237568827

minus09026613361

007505076253

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(54)


[[[[[[[

[

004028666644

006399313350

09971368323

00

00

00

]]]]]]]

]

[[[[[[[

[

09048795994

minus04255671395

minus0009247686087

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04237568827

minus09026613361

007505076253

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(55)



[[[

[

1458799998 000000001785571215 minus000000001009489597 00 00 00000000001785571215 1458799997 0000000002407020197 00 00 00minus000000001009489597 0000000002407020197 1458799997 00 00 00

00 00 00 1458799998 00 0000 00 00 00 1458799998 0000 00 00 00 00 1458799998

]]]

]

(56)




[[[[[

[

005368516527

007220768570

09959437502

00

00

00

]]]]]

]

[[[[[

[

09266208315

minus03753089731

minus002273783078

00

00

00

]]]]]

]

[[[[[

[

minus03721447810

minus09240829102

008705767064

00

00

00

]]]]]

]

[[[[[

[

00

00

00

00

00

10

]]]]]

]

[[[[[

[

00

00

00

10

00

00

]]]]]

]

[[[[[

[

00

00

00

00

10

00

]]]]]

]

(57)


[[[[[[[[[

[

005368516527

007220768570

09959437502

00

00

00

]]]]]]]]]

]

[[[[[[[[[

[

09266208315

minus03753089731

minus002273783078

00

00

00

]]]]]]]]]

]

[[[[[[[[[

[

minus03721447806

minus09240829093

008705767055

00

00

00

]]]]]]]]]

]

[[[[[[[[[

[

00

00

00

10

00

00

]]]]]]]]]

]

[[[[[[[[[

[

00

00

00

00

10

00

]]]]]]]]]

]

[[[[[[[[[

[

00

00

00

00

00

10

]]]]]]]]]

]

(58)




[[[

[

1615299998 0000000005508173090 minus0000000008722620038 00 00 00

0000000005508173090 1615299996 minus0000000004345156817 00 00 00

minus0000000008722620038 minus0000000004345156817 1615300000 00 00 00

00 00 00 1615299998 00 00

00 00 00 00 1615299998 00

00 00 00 00 00 1615299998

]]]

]

(59)




[[[[[[[

[

minus00598757013800000

minus00768229223150000

minus0995231841750000

00

00

00

]]]]]]]

]

[[[[[[[

[

minus0869358919900000

0493939153600000

00141567750950000

00

00

00

]]]]]]]

]

[[[[[[[

[

minus0490490276500000

minus0866078136900000

00963811082600000

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(60)


[[[[[[[[[[[

[

minus005987730132

minus007682497510

minus09952584354

00

00

00

]]]]]]]]]]]

]

[[[[[[[[[[[

[

minus08693926232

04939583026

001415732393

00

00

00

]]]]]]]]]]]

]

[[[[[[[[[[[

[

00

00

00

10

00

00

]]]]]]]]]]]

]

[[[[[[[[[[[

[

00

00

00

00

10

00

]]]]]]]]]]]

]

[[[[[[[[[[[

[

00

00

00

00

00

10

]]]]]]]]]]]

]

(61)



[[[

[


00 00 00 1819400002 1819400002 0000 00 00 00 00 0000 00 00 00 00 1819400002

]]]

]

(62)




[[[[[[[

[

004967528691

008463937788

09951726186

00

00

00

]]]]]]]

]

[[[[[[[

[

09161715971

minus04006166011

minus001165943224

00

00

00

]]]]]]]

]

[[[[[

[

minus03976958243

minus09123280736

009744493938

00

00

00

]]]]]

]

[[[[[

[

00

00

00

10

00

00

]]]]]

]

[[[[[

[

00

00

00

00

10

00

]]]]]

]

[[[[[

[

00

00

00

00

00

10

]]]]]

]

(63)


[[[[[[[

[

004967528696

008463937796

09951726196

00

00

00

]]]]]]]

]

[[[[[[[

[

09161715980

minus04006166015

minus001165943225

00

00

00

]]]]]]]

]


[[[[[[[

[

minus03976958247

minus09123280745

009744493948

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(64)



[[[

[

1922399998 minus000000001176508811 minus000000001537920006 00 00 00

minus000000001176508811 1922400000 minus0000000007631928036 00 00 00

minus000000001537920006 minus0000000007631928036 1922400000 00 00 00

00 00 00 1922399998 1922399998 00

00 00 00 00 00 00

00 00 00 00 00 1922399998

]]]

]

(65)




[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

[[[[[[[

[

minus01387533797

minus02031928413

minus09692575344

00

00

00

]]]]]]]

]

[[[[[[[

[

minus08476774112

05304148448

001015375725

00

00

00

]]]]]]]

]

[[[[[[[

[

minus05120454135

minus08230265872

02458388325

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

(66)


[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

[[[[[[[

[

minus01387533798

minus02031928415

minus09692575354

00

00

00

]]]]]]]

]

[[[[[[[

[

minus08476774112

05304148448

001015375725

00

00

00

]]]]]]]

]

[[[[[[[

[

minus05120454135

minus08230265872

02458388325

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

(67)



[[[

[

2074599999 0000000004356660361 000000003402343994 00 00 00

0000000004356660361 2074600004 000000002232269567 00 00 00

000000003402343994 000000002232269567 2074600000 00 00 00

00 00 00 2074600000 2074600000 00

00 00 00 00 00 00

00 00 00 00 00 2074600000

]]]

]

(68)




[[[[[[[

[

008621257414

01243595697

09884847438

00

00

00

]]]]]]]

]

[[[[[[[

[

08755641188

minus04828494241

minus001561753067

00

00

00

]]]]]]]

]

[[[[[

[

minus04753471023

minus08668282006

01505124700

00

00

00

]]]]]

]

[[[[[

[

00

00

00

10

00

00

]]]]]

]

[[[[[

[

00

00

00

00

10

00

]]]]]

]

[[[[[

[

00

00

00

00

00

10

]]]]]

]

(69)


[[[[[

[

008621257423

01243595698

09884847448

00

00

00

]]]]]

]

[[[[[

[

08755641188

minus04828494241

minus001561753067

00

00

00

]]]]]

]


[[[[[[[

[

minus04753471018

minus08668281997

01505124698

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(70)



[[[

[

1890099999 000000001209663993 minus000000002532733996 00 00 00000000001209663993 1890099991 0000000001701090138 00 00 00minus000000002532733996 0000000001701090138 1890100001 00 00 00

00 00 00 1890099997 1890099997 0000 00 00 00 00 0000 00 00 00 00 1890099997

]]]

]

(71)




[[[[[[[

[

03961520086

06338768023

06642768888

00

00

00

]]]]]]]

]

[[[[[[[

[

09160614623

minus02236797319

minus03328645006

00

00

00

]]]]]]]

]

[[[[[[[

[

006240981414

minus07403833993

06692812840

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(72)


[[[[[[[

[

03961520086

06338768023

06642768888

00

00

00

]]]]]]]

]

[[[[[[[

[

09160614614

minus02236797317

minus03328645003

00

00

00

]]]]]]]

]

[[[[[[[

[

006240981426

minus07403834008

06692812853

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(73)



[[[

[


00 00 00 8772299998 8772299998 0000 00 00 00 00 0000 00 00 00 00 8772299998

]]]

]

(74)




[[[[[[[

[

minus006591789198

minus008975775227

minus09937798435

00

00

00

]]]]]]]

]

[[[[[[[

[

minus08289381135

05593224991

0004466103673

00

00

00

]]]]]]]

]

[[[[[

[

minus05554425577

minus08240763851

01112729817

00

00

00

]]]]]

]

[[[[[

[

00

00

00

10

00

00

]]]]]

]

[[[[[

[

00

00

00

00

10

00

]]]]]

]

[[[[[

[

00

00

00

00

00

10

]]]]]

]


[[[[[[[

[

minus01387533798

minus02031928415

minus09692575354

00

00

00

]]]]]]]

]

[[[[[[[

[

minus08476774112

05304148448

001015375725

00

00

00

]]]]]]]

]


[[[[

[

minus05120454135

minus08230265872

02458388325

00

00

00

]]]]

]

[[[[

[

00

00

00

10

00

00

]]]]

]

[[[[

[

00

00

00

00

10

00

]]]]

]

[[[[

[

00

00

00

00

00

10

]]]]

]

(76)



[[[

[

2228057494 0000000004678921127 000000003654014285 00 00 000000000004678921127 2228057499 000000002397389829 00 00 00000000003654014285 000000002397389829 2228057495 00 00 00

00 00 00 2228057495 2228057495 0000 00 00 00 00 0000 00 00 00 00 2228057495

]]]

]

(77)




[[[[[[[

[

006975010637

01071019008

09917984199

00

00

00

]]]]]]]

]

[[[[[[[

[

09079000793

minus04187725641

minus001862756541

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04133429180

minus09017531389

01264472547

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(78)


[[[[[[[

[

006975010637

01071019008

09917984199

00

00

00

]]]]]]]

]

[[[[[[[

[

09079000793

minus04187725641

minus001862756541

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04133429184

minus09017531398

01264472548

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(79)



[[[

[


00 00 00 2598699998 2598699998 0000 00 00 00 00 0000 00 00 00 00 2598699998

]]]

]

(80)




[[[[[

[

minus00192522847250000

minus00192842313600000

000386151499999998

00

00

00

]]]]]

]

[[[[[

[

000961799224999998

00165029120500000

000436480214750000

00

00

00

]]]]]

]

[[[[[[

[

minus0494444050150000

minus0856906622200000

0142104790200000

00

00

00

]]]]]]

]

[[[[[[

[

00

00

00

10

00

00

]]]]]]

]

[[[[[[

[

00

00

00

00

10

00

]]]]]]

]

[[[[[[

[

00

00

00

00

00

10

]]]]]]

]

(81)



[[[[[[[[[[

[

minus2541745843

minus2545963537

5098090872

00

00

00

]]]]]]]]]]

]

[[[[[[[[[[

[

2505315863

4298714977

1136953303

00

00

00

]]]]]]]]]]

]

[[[[[[[

[

minus04949599718

minus08578007511

01422530677

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(82)



[[[

[

315679169478799 427324383657779 384557503470365 00 00 00427324383657779 618700481877479 889153535276175 00 00 00384557503470365 889153535276175 384768762708609 00 00 00

00 00 00 247795001400000 00 0000 00 00 00 247795001400000 0000 00 00 00 00 247795001400000

]]]

]

(83)




[[[[[[[

[

01135176976

01764907831

09777344911

00

00

00

]]]]]]]

]

[[[[[[[

[

08848520771

minus04654963442

minus001870707734

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04518302069

minus08672739791

02090103088

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(84)


[[[[[[[

[

01135176977

01764907833

09777344921

00

00

00

]]]]]]]

]

[[[[[[[

[

08848520771

minus04654963442

minus001870707734

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04518302069

minus08672739791

02090103088

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(85)



[[[

[

2663700003 000000004768023095 000000003169803038 00 00 00000000004768023095 2663699992 0000000008310743498 00 00 00000000003169803038 0000000008310743498 2663700001 00 00 00

00 00 00 2663699998 2663699998 0000 00 00 00 00 0000 00 00 00 00 2663699998

]]]

]

(86)





6 Conclusions









Disclosure




Acknowledgments



References




















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










24681012141618

12

34

56

Column Row

12

34

56


12345678

12

34

56

Column Row

12

34

56




120590119894119895= 119862


where 119862119894119895119896119897


120590 = 120590119894119895119890119894otimes 119890

119895 120598 = 120598

119896119897119890119896otimes 119890

119897

C = 119862119894119895119896119897

119890119894otimes 119890

119896otimes 119890

119896otimes 119890

119897

(23)

5

10

15

20

12

34

56

Column Row

12

34

56


5

10

15

20

25

12

34

56

Column Row

12

34

56



E1equiv 119890

1otimes 119890

1

E2equiv 119890

2otimes 119890

2

E3equiv 119890

3otimes 119890

3

(24)


E4equiv

1


2+ 119890

2otimes 119890

1)

E5equiv

1


3+ 119890

3otimes 119890

2)

E6equiv

1


1+ 119890

1otimes 119890

3)

(25)



E7equiv

1


2minus 119890

2otimes 119890

1)

E8equiv

1


3minus 119890

3otimes 119890

2)

E9equiv

1


1minus 119890

1otimes 119890

3)

(26)


120590 = 119878119860E

119860

120598 = 119864119860E

119860

C = 119862119860119861

119890119860otimes 119890

119861

(27)




120590119894119895= 119862

119894119895119896119897120598119896119897

lArrrArr 119878119860= 119862

119860119861E119861

119862119894119895119896119897

120598119896119897

= 120582120598119894119895lArrrArr 119862

119860119861E119861= 120582119864

119860

(28)


((((((((

(

1198781

1198782

1198783

1198784

1198785

1198786

1198787

1198788

1198789

))))))))

)

=

((((((((

(

12059011

12059022

12059033

120590(12)

120590(23)

120590(31)

120590[12]

120590[23]

120590[31]

))))))))

)

((((((((

(

1198641

1198642

1198643

1198644

1198645

1198646

1198647

1198648

1198649

))))))))

)

=

((((((((

(

12059811

12059822

12059833

120598(12)

120598(23)

120598(31)

120598[12]

120598[23]

120598[31]

))))))))

)

(29)


120579(119894119895)

equiv1

radic2(120579119894119895+ 120579

119895119894) 120579

[119894119895]equiv

1

radic2(120579119894119895minus 120579

119895119894) (30)


tensor C are

((((((((

(

11986211

11986212

11986213

11986214

11986215

11986216

11986217

11986218

11986219

11986221

11986222

11986223

11986224

11986225

11986226

11986227

11986228

11986229

11986231

11986232

11986233

11986234

11986235

11986236

11986237

11986238

11986239

11986241

11986242

11986243

11986244

11986245

11986246

11986247

11986248

11986249

11986251

11986252

11986253

11986254

11986255

11986256

11986257

11986258

11986259

11986261

11986262

11986263

11986264

11986265

11986266

11986267

11986268

11986269

11986271

11986272

11986273

11986274

11986275

11986276

11986277

11986278

11986279

11986281

11986282

11986283

11986284

11986285

11986286

11986287

11986288

11986289

11986291

11986292

11986293

11986294

11986295

11986296

11986297

11986298

11986299

))))))))

)

=

1198621111

1198621122

1198621133

1198622211

1198622222

1198623333

1198623311

1198623322

1198623333

11986211(12)

11986211(23)

11986211(31)

11986222(12)

11986222(23)

11986222(31)

11986233(12)

11986233(23)

11986233(31)

11986211[12]

11986211[23]

11986211[31]

11986222[12]

11986222[23]

11986222[31]

11986233[12]

11986233[23]

11986233[31]

119862(12)11

119862(12)22

119862(12)33

119862(23)11

119862(23)22

119862(23)33

119862(31)11

119862(31)22

119862(31)33

119862(1212)

119862(1223)

119862(1231)

119862(2312)

119862(2323)

119862(2331)

119862(3112)

119862(3123)

119862(3131)

119862(12)[12]

119862(12)[23]

119862(12)[31]

119862(23)[12]

119862(23)[23]

119862(23)[31]

119862(31)[12]

119862(31)[23]

119862(31)[31]

119862[12]11

119862[12]22

119862[12]33

119862[23]11

119862[23]22

119862[123]33

119862[21]11

119862[31]22

119862[31]33

119862[12](12)

119862[12](23)

119862[12](21)

119862[23](12)

119862[23](23)

119862[23](21)

119862[31](12)

119862[31](23)

119862[31](21)

119862[1212]

119862[1223]

119862[1231]

119862[2312]

119862[2323]

119862[2331]

119862[3112]

119862[3123]

119862[3131]

(31)

where

119862119894119895(119896119897)

equiv1

radic2(119862

119894119895119896119897+ 119862

119894119895119897119896) 119862

119894119895[119896119897]equiv

1

radic2(119862

119894119895119896119897minus 119862

119894119895119897119896)

119862(119894119895)119896119897

equiv1

radic2(119862

119894119895119896119897+ 119862

119895119894119896119897) 119862

[119894119895]119896119897equiv

1

radic2(119862

119894119895119896119897minus 119862

119895119894119896119897)

119862(119894119895119896119897)

equiv1

2(119862

119894119895119896119897+ 119862

119894119895119897119896+ 119862

119895119894119896119897+ 119862

119895119894119897119896)

119862(119894119895)[119896119897]

equiv1

2(119862

119894119895119896119897minus 119862

119894119895119897119896+ 119862

119895119894119896119897minus 119862

119895119894119897119896)

119862[119894119895](119896119897)

equiv1

2(119862

119894119895119896119897+ 119862

119894119895119897119896minus 119862

119895119894119896119897minus 119862

119895119894119897119896)

119862[119894119895119896119897]

equiv1

2(119862

119894119895119896119897minus 119862

119895119894119896119897minus 119862

119894119895119897119896+ 119862

119895119894119897119896)

(32)


12

34

56

ColumnRow

12

34

56

600005000040000300002000010000


5

10

15

20

25

12

34

56

Column Row

12

34

56






119895119894and 120598

119896119897= 120598

119897119896 the 119862

119894119895119896119897will be

like 119862119894119895119896119897

= 119862119895119894119896119897

= 119862119894119895119897119896


119862119894119895119896119897

= 119862119896119897119894119895












2 sdot 119864energy = 120590119894119895120598119894119895= 120590

119901120598119901 (33)


119901 = 119894120575119894119895+ (1 minus 120575

119894119895) (9 minus 119894 minus 119895)

119902 = 119896120575119896119897+ (1 minus 120575

119896119897) (9 minus 119896 minus 119897)

(34)


119901 119862

119894119895119896119897= 119862

119901119902 and 120598

119902= (2 minus

120575119896119897)120598119896119897



(

(

1205901

1205902

1205903

1205904

1205905

1205906

)

)

= (

(

11986211

11986212

11986213

11986214

11986215

11986216

11986221

11986222

11986223

11986224

11986225

11986226

11986231

11986232

11986233

11986234

11986235

11986236

11986241

11986242

11986243

11986244

11986245

11986246

11986251

11986252

11986253

11986254

11986255

11986256

11986261

11986262

11986263

11986264

11986265

11986266

)

)

(

(

1205981

1205982

1205983

1205984

1205985

1205986

)

)

(35)



(1) the 120590119894119895and 120598


(2) the norms of 120590119894119895 120598119896119897 and 119862




01 02 03 04 05 06 07 08

2

4

6

8

10

12

14

16

(a)

01

02

03

04

05

06

07

08

09

01 02 03 04 05 06 07 08

(b)


01 02 03 04 05 06 07 08

05

1

15

2

25

3

(a)

01 02 03 04 05 06 07 08

05

1

15

2

25

3

(b)




119901 = 119894120575119894119895+ (1 minus 120575

119894119895) (9 minus 119894 minus 119895)

119902 = 119896120575119896119897+ (1 minus 120575

119896119897) (9 minus 119896 minus 119897)

(36)

In this way

120590119901= (120575

119894119895+ radic2 (1 minus 120575

119894119895)) 120598

119894119895

120598119902= (120575

119896119897+ radic2 (1 minus 120575

119896119897)) 120598

119896119897

119862119901119902

= (120575119894119895+ radic2 (1 minus 120575

119894119895)) (120575

119896119897+ radic2 (1 minus 120575

119896119897)) 119862

119894119895119896119897

(37)




120590kelvin

= 120590viogt

120585 120590voigt

=120590kelvin

120585

120598kelvin

= 120598voigt

120585 120598voigt

=120598kelvin

120585

(38)


119862kelvin

=((

(







))

)

119862voigt

(39)


(

(

12059011

12059022

12059033

radic212059023

radic212059013

radic212059012

)

)

= (

1198621111 1198621122 1198621133radic21198621123

radic21198621113radic21198621112

1198622211 1198622222 1198622333radic21198622223

radic21198622213radic21198622212

1198623311 1198623322 1198623333radic21198623323

radic21198623313radic21198623312

radic21198622311radic21198622322

radic21198622333 21198622323 21198622313 21198622312

radic21198621311radic21198621322

radic21198621333 21198621323 21198621313 21198621312

radic21198621211radic21198621222

radic21198621233 21198621223 21198621213 21198621212

)

lowast(

(

12059811

12059822

12059833

radic212059823

radic212059813

radic212059812

)

)

(40)


= C (41)



(

(

12059011

12059022

12059033

radic212059023

radic212059013

radic212059012

)

)

=((

(

11986211

11986212

11986213

radic211986214

radic211986215

radic211986216

11986212

11986222

11986223

radic211986224

radic211986225

radic211986226

11986213

11986223

11986233

radic211986234

radic211986235

radic211986236

radic211986214

radic211986224

radic211986234

211986244

211986245

211986246

radic211986215

radic211986225

radic211986235

211986245

211986255

211986256

radic211986216

radic211986226

radic211986236

211986246

211986256

211986266

))

)

lowast (

(

12059811

12059822

12059833

radic212059823

radic212059813

radic212059812

)

)

(42)


(

(

1

2

3

4

5

6

)

)

=((

(

11986211

11986212

11986213

11986214

11986215

11986216

11986212

11986222

11986223

11986224

11986225

11986226

11986213

11986223

11986233

11986234

11986235

11986236

11986214

11986224

11986234

11986244

11986245

11986246

11986215

11986225

11986235

11986245

11986255

11986256

11986216

11986226

11986236

11986246

11986256

11986266

))

)

lowast (

(

1205981

1205982

1205983

1205984

1205985

1205986

)

)

(43)



































01 02 03 04 05 06 07 08

05

1

15

2

25

3

(a)

01 02 03 04 05 06 07 08

05

1

15

2

25

3

(b)


01 02 03 04 05 06 07 08

2

4

6

8

10

12

14

16





ΛI) N = 0 (44)


119894 119894 = 1 2 6 and

















NNA119896

equiv1

119872

119872

sum

119884=1

N119884119896 (45)



NAVG119896

equiv JNANNA119896

(46)


119896=1NNA119896

otimes NNA119896

| that is

JNA equiv (

6

sum

119896=1

NNA119896

otimes NNA119896

)

minus12

(47)


119896 that is

ΛAVG119902

equiv1

119872

119872

sum

119884=1

6

sum

119896=1

Λ119884

119896(N119884

119896sdot NAVG

119902)2

(48)


gt 0 for all 119896 = 1 2 6

and ΛAVG119896


NA119896


ΛNA119896

equiv1

119872

119872

sum

119884=1

Λ119884

119896 (49)


CAVG=

6

sum

119896=1

ΛAVG119896

NAVG119896

otimes NAVG119896

(50)





[[[[[[[

[

00367834559700000

00453681054800000

0998185540700000

00

00

00

]]]]]]]

]

[[[[[[[

[

0983745905000000

minus0170879918750000

minus00277901141300000

00

00

00

]]]]]]]

]

[[[[[[[

[

minus0169284222800000

minus0982986098000000

00509905132200000

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(51)


[[[[[[[

[

003679134179

004537783172

09983995367

00

00

00

]]]]]]]

]

[[[[[[[

[

09859858256

minus01712690003

minus002785339027

00

00

00

]]]]]]]

]

[[[[[[[

[

minus01697052873

minus09854311019

005111734310

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(52)




[[[

[

334725276837330 0000112116752981933 000198542359441849 00 00 00

0000112116752981933 334773746006714 minus0000991802322788501 00 00 00

000198542359441849 minus0000991802322788501 334013593769726 00 00 00

00 00 00 333950000000000 00 00

00 00 00 00 333950000000000 00

00 00 00 00 00 333950000000000

]]]

]

(53)




[[[[[[[

[

004028666644

006399313350

09971368323

00

00

00

]]]]]]]

]

[[[[[[[

[

09048796003

minus04255671399

minus0009247686096

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04237568827

minus09026613361

007505076253

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(54)


[[[[[[[

[

004028666644

006399313350

09971368323

00

00

00

]]]]]]]

]

[[[[[[[

[

09048795994

minus04255671395

minus0009247686087

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04237568827

minus09026613361

007505076253

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(55)



[[[

[

1458799998 000000001785571215 minus000000001009489597 00 00 00000000001785571215 1458799997 0000000002407020197 00 00 00minus000000001009489597 0000000002407020197 1458799997 00 00 00

00 00 00 1458799998 00 0000 00 00 00 1458799998 0000 00 00 00 00 1458799998

]]]

]

(56)




[[[[[

[

005368516527

007220768570

09959437502

00

00

00

]]]]]

]

[[[[[

[

09266208315

minus03753089731

minus002273783078

00

00

00

]]]]]

]

[[[[[

[

minus03721447810

minus09240829102

008705767064

00

00

00

]]]]]

]

[[[[[

[

00

00

00

00

00

10

]]]]]

]

[[[[[

[

00

00

00

10

00

00

]]]]]

]

[[[[[

[

00

00

00

00

10

00

]]]]]

]

(57)


[[[[[[[[[

[

005368516527

007220768570

09959437502

00

00

00

]]]]]]]]]

]

[[[[[[[[[

[

09266208315

minus03753089731

minus002273783078

00

00

00

]]]]]]]]]

]

[[[[[[[[[

[

minus03721447806

minus09240829093

008705767055

00

00

00

]]]]]]]]]

]

[[[[[[[[[

[

00

00

00

10

00

00

]]]]]]]]]

]

[[[[[[[[[

[

00

00

00

00

10

00

]]]]]]]]]

]

[[[[[[[[[

[

00

00

00

00

00

10

]]]]]]]]]

]

(58)




[[[

[

1615299998 0000000005508173090 minus0000000008722620038 00 00 00

0000000005508173090 1615299996 minus0000000004345156817 00 00 00

minus0000000008722620038 minus0000000004345156817 1615300000 00 00 00

00 00 00 1615299998 00 00

00 00 00 00 1615299998 00

00 00 00 00 00 1615299998

]]]

]

(59)




[[[[[[[

[

minus00598757013800000

minus00768229223150000

minus0995231841750000

00

00

00

]]]]]]]

]

[[[[[[[

[

minus0869358919900000

0493939153600000

00141567750950000

00

00

00

]]]]]]]

]

[[[[[[[

[

minus0490490276500000

minus0866078136900000

00963811082600000

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(60)


[[[[[[[[[[[

[

minus005987730132

minus007682497510

minus09952584354

00

00

00

]]]]]]]]]]]

]

[[[[[[[[[[[

[

minus08693926232

04939583026

001415732393

00

00

00

]]]]]]]]]]]

]

[[[[[[[[[[[

[

00

00

00

10

00

00

]]]]]]]]]]]

]

[[[[[[[[[[[

[

00

00

00

00

10

00

]]]]]]]]]]]

]

[[[[[[[[[[[

[

00

00

00

00

00

10

]]]]]]]]]]]

]

(61)



[[[

[


00 00 00 1819400002 1819400002 0000 00 00 00 00 0000 00 00 00 00 1819400002

]]]

]

(62)




[[[[[[[

[

004967528691

008463937788

09951726186

00

00

00

]]]]]]]

]

[[[[[[[

[

09161715971

minus04006166011

minus001165943224

00

00

00

]]]]]]]

]

[[[[[

[

minus03976958243

minus09123280736

009744493938

00

00

00

]]]]]

]

[[[[[

[

00

00

00

10

00

00

]]]]]

]

[[[[[

[

00

00

00

00

10

00

]]]]]

]

[[[[[

[

00

00

00

00

00

10

]]]]]

]

(63)


[[[[[[[

[

004967528696

008463937796

09951726196

00

00

00

]]]]]]]

]

[[[[[[[

[

09161715980

minus04006166015

minus001165943225

00

00

00

]]]]]]]

]


[[[[[[[

[

minus03976958247

minus09123280745

009744493948

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(64)



[[[

[

1922399998 minus000000001176508811 minus000000001537920006 00 00 00

minus000000001176508811 1922400000 minus0000000007631928036 00 00 00

minus000000001537920006 minus0000000007631928036 1922400000 00 00 00

00 00 00 1922399998 1922399998 00

00 00 00 00 00 00

00 00 00 00 00 1922399998

]]]

]

(65)




[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

[[[[[[[

[

minus01387533797

minus02031928413

minus09692575344

00

00

00

]]]]]]]

]

[[[[[[[

[

minus08476774112

05304148448

001015375725

00

00

00

]]]]]]]

]

[[[[[[[

[

minus05120454135

minus08230265872

02458388325

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

(66)


[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

[[[[[[[

[

minus01387533798

minus02031928415

minus09692575354

00

00

00

]]]]]]]

]

[[[[[[[

[

minus08476774112

05304148448

001015375725

00

00

00

]]]]]]]

]

[[[[[[[

[

minus05120454135

minus08230265872

02458388325

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

(67)



[[[

[

2074599999 0000000004356660361 000000003402343994 00 00 00

0000000004356660361 2074600004 000000002232269567 00 00 00

000000003402343994 000000002232269567 2074600000 00 00 00

00 00 00 2074600000 2074600000 00

00 00 00 00 00 00

00 00 00 00 00 2074600000

]]]

]

(68)




[[[[[[[

[

008621257414

01243595697

09884847438

00

00

00

]]]]]]]

]

[[[[[[[

[

08755641188

minus04828494241

minus001561753067

00

00

00

]]]]]]]

]

[[[[[

[

minus04753471023

minus08668282006

01505124700

00

00

00

]]]]]

]

[[[[[

[

00

00

00

10

00

00

]]]]]

]

[[[[[

[

00

00

00

00

10

00

]]]]]

]

[[[[[

[

00

00

00

00

00

10

]]]]]

]

(69)


[[[[[

[

008621257423

01243595698

09884847448

00

00

00

]]]]]

]

[[[[[

[

08755641188

minus04828494241

minus001561753067

00

00

00

]]]]]

]


[[[[[[[

[

minus04753471018

minus08668281997

01505124698

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(70)



[[[

[

1890099999 000000001209663993 minus000000002532733996 00 00 00000000001209663993 1890099991 0000000001701090138 00 00 00minus000000002532733996 0000000001701090138 1890100001 00 00 00

00 00 00 1890099997 1890099997 0000 00 00 00 00 0000 00 00 00 00 1890099997

]]]

]

(71)




[[[[[[[

[

03961520086

06338768023

06642768888

00

00

00

]]]]]]]

]

[[[[[[[

[

09160614623

minus02236797319

minus03328645006

00

00

00

]]]]]]]

]

[[[[[[[

[

006240981414

minus07403833993

06692812840

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(72)


[[[[[[[

[

03961520086

06338768023

06642768888

00

00

00

]]]]]]]

]

[[[[[[[

[

09160614614

minus02236797317

minus03328645003

00

00

00

]]]]]]]

]

[[[[[[[

[

006240981426

minus07403834008

06692812853

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(73)



[[[

[


00 00 00 8772299998 8772299998 0000 00 00 00 00 0000 00 00 00 00 8772299998

]]]

]

(74)




[[[[[[[

[

minus006591789198

minus008975775227

minus09937798435

00

00

00

]]]]]]]

]

[[[[[[[

[

minus08289381135

05593224991

0004466103673

00

00

00

]]]]]]]

]

[[[[[

[

minus05554425577

minus08240763851

01112729817

00

00

00

]]]]]

]

[[[[[

[

00

00

00

10

00

00

]]]]]

]

[[[[[

[

00

00

00

00

10

00

]]]]]

]

[[[[[

[

00

00

00

00

00

10

]]]]]

]


[[[[[[[

[

minus01387533798

minus02031928415

minus09692575354

00

00

00

]]]]]]]

]

[[[[[[[

[

minus08476774112

05304148448

001015375725

00

00

00

]]]]]]]

]


[[[[

[

minus05120454135

minus08230265872

02458388325

00

00

00

]]]]

]

[[[[

[

00

00

00

10

00

00

]]]]

]

[[[[

[

00

00

00

00

10

00

]]]]

]

[[[[

[

00

00

00

00

00

10

]]]]

]

(76)



[[[

[

2228057494 0000000004678921127 000000003654014285 00 00 000000000004678921127 2228057499 000000002397389829 00 00 00000000003654014285 000000002397389829 2228057495 00 00 00

00 00 00 2228057495 2228057495 0000 00 00 00 00 0000 00 00 00 00 2228057495

]]]

]

(77)




[[[[[[[

[

006975010637

01071019008

09917984199

00

00

00

]]]]]]]

]

[[[[[[[

[

09079000793

minus04187725641

minus001862756541

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04133429180

minus09017531389

01264472547

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(78)


[[[[[[[

[

006975010637

01071019008

09917984199

00

00

00

]]]]]]]

]

[[[[[[[

[

09079000793

minus04187725641

minus001862756541

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04133429184

minus09017531398

01264472548

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(79)



[[[

[


00 00 00 2598699998 2598699998 0000 00 00 00 00 0000 00 00 00 00 2598699998

]]]

]

(80)




[[[[[

[

minus00192522847250000

minus00192842313600000

000386151499999998

00

00

00

]]]]]

]

[[[[[

[

000961799224999998

00165029120500000

000436480214750000

00

00

00

]]]]]

]

[[[[[[

[

minus0494444050150000

minus0856906622200000

0142104790200000

00

00

00

]]]]]]

]

[[[[[[

[

00

00

00

10

00

00

]]]]]]

]

[[[[[[

[

00

00

00

00

10

00

]]]]]]

]

[[[[[[

[

00

00

00

00

00

10

]]]]]]

]

(81)



[[[[[[[[[[

[

minus2541745843

minus2545963537

5098090872

00

00

00

]]]]]]]]]]

]

[[[[[[[[[[

[

2505315863

4298714977

1136953303

00

00

00

]]]]]]]]]]

]

[[[[[[[

[

minus04949599718

minus08578007511

01422530677

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(82)



[[[

[

315679169478799 427324383657779 384557503470365 00 00 00427324383657779 618700481877479 889153535276175 00 00 00384557503470365 889153535276175 384768762708609 00 00 00

00 00 00 247795001400000 00 0000 00 00 00 247795001400000 0000 00 00 00 00 247795001400000

]]]

]

(83)




[[[[[[[

[

01135176976

01764907831

09777344911

00

00

00

]]]]]]]

]

[[[[[[[

[

08848520771

minus04654963442

minus001870707734

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04518302069

minus08672739791

02090103088

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(84)


[[[[[[[

[

01135176977

01764907833

09777344921

00

00

00

]]]]]]]

]

[[[[[[[

[

08848520771

minus04654963442

minus001870707734

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04518302069

minus08672739791

02090103088

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(85)



[[[

[

2663700003 000000004768023095 000000003169803038 00 00 00000000004768023095 2663699992 0000000008310743498 00 00 00000000003169803038 0000000008310743498 2663700001 00 00 00

00 00 00 2663699998 2663699998 0000 00 00 00 00 0000 00 00 00 00 2663699998

]]]

]

(86)





6 Conclusions









Disclosure




Acknowledgments



References




















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











E7equiv

1


2minus 119890

2otimes 119890

1)

E8equiv

1


3minus 119890

3otimes 119890

2)

E9equiv

1


1minus 119890

1otimes 119890

3)

(26)


120590 = 119878119860E

119860

120598 = 119864119860E

119860

C = 119862119860119861

119890119860otimes 119890

119861

(27)




120590119894119895= 119862

119894119895119896119897120598119896119897

lArrrArr 119878119860= 119862

119860119861E119861

119862119894119895119896119897

120598119896119897

= 120582120598119894119895lArrrArr 119862

119860119861E119861= 120582119864

119860

(28)


((((((((

(

1198781

1198782

1198783

1198784

1198785

1198786

1198787

1198788

1198789

))))))))

)

=

((((((((

(

12059011

12059022

12059033

120590(12)

120590(23)

120590(31)

120590[12]

120590[23]

120590[31]

))))))))

)

((((((((

(

1198641

1198642

1198643

1198644

1198645

1198646

1198647

1198648

1198649

))))))))

)

=

((((((((

(

12059811

12059822

12059833

120598(12)

120598(23)

120598(31)

120598[12]

120598[23]

120598[31]

))))))))

)

(29)


120579(119894119895)

equiv1

radic2(120579119894119895+ 120579

119895119894) 120579

[119894119895]equiv

1

radic2(120579119894119895minus 120579

119895119894) (30)


tensor C are

((((((((

(

11986211

11986212

11986213

11986214

11986215

11986216

11986217

11986218

11986219

11986221

11986222

11986223

11986224

11986225

11986226

11986227

11986228

11986229

11986231

11986232

11986233

11986234

11986235

11986236

11986237

11986238

11986239

11986241

11986242

11986243

11986244

11986245

11986246

11986247

11986248

11986249

11986251

11986252

11986253

11986254

11986255

11986256

11986257

11986258

11986259

11986261

11986262

11986263

11986264

11986265

11986266

11986267

11986268

11986269

11986271

11986272

11986273

11986274

11986275

11986276

11986277

11986278

11986279

11986281

11986282

11986283

11986284

11986285

11986286

11986287

11986288

11986289

11986291

11986292

11986293

11986294

11986295

11986296

11986297

11986298

11986299

))))))))

)

=

1198621111

1198621122

1198621133

1198622211

1198622222

1198623333

1198623311

1198623322

1198623333

11986211(12)

11986211(23)

11986211(31)

11986222(12)

11986222(23)

11986222(31)

11986233(12)

11986233(23)

11986233(31)

11986211[12]

11986211[23]

11986211[31]

11986222[12]

11986222[23]

11986222[31]

11986233[12]

11986233[23]

11986233[31]

119862(12)11

119862(12)22

119862(12)33

119862(23)11

119862(23)22

119862(23)33

119862(31)11

119862(31)22

119862(31)33

119862(1212)

119862(1223)

119862(1231)

119862(2312)

119862(2323)

119862(2331)

119862(3112)

119862(3123)

119862(3131)

119862(12)[12]

119862(12)[23]

119862(12)[31]

119862(23)[12]

119862(23)[23]

119862(23)[31]

119862(31)[12]

119862(31)[23]

119862(31)[31]

119862[12]11

119862[12]22

119862[12]33

119862[23]11

119862[23]22

119862[123]33

119862[21]11

119862[31]22

119862[31]33

119862[12](12)

119862[12](23)

119862[12](21)

119862[23](12)

119862[23](23)

119862[23](21)

119862[31](12)

119862[31](23)

119862[31](21)

119862[1212]

119862[1223]

119862[1231]

119862[2312]

119862[2323]

119862[2331]

119862[3112]

119862[3123]

119862[3131]

(31)

where

119862119894119895(119896119897)

equiv1

radic2(119862

119894119895119896119897+ 119862

119894119895119897119896) 119862

119894119895[119896119897]equiv

1

radic2(119862

119894119895119896119897minus 119862

119894119895119897119896)

119862(119894119895)119896119897

equiv1

radic2(119862

119894119895119896119897+ 119862

119895119894119896119897) 119862

[119894119895]119896119897equiv

1

radic2(119862

119894119895119896119897minus 119862

119895119894119896119897)

119862(119894119895119896119897)

equiv1

2(119862

119894119895119896119897+ 119862

119894119895119897119896+ 119862

119895119894119896119897+ 119862

119895119894119897119896)

119862(119894119895)[119896119897]

equiv1

2(119862

119894119895119896119897minus 119862

119894119895119897119896+ 119862

119895119894119896119897minus 119862

119895119894119897119896)

119862[119894119895](119896119897)

equiv1

2(119862

119894119895119896119897+ 119862

119894119895119897119896minus 119862

119895119894119896119897minus 119862

119895119894119897119896)

119862[119894119895119896119897]

equiv1

2(119862

119894119895119896119897minus 119862

119895119894119896119897minus 119862

119894119895119897119896+ 119862

119895119894119897119896)

(32)


12

34

56

ColumnRow

12

34

56

600005000040000300002000010000


5

10

15

20

25

12

34

56

Column Row

12

34

56






119895119894and 120598

119896119897= 120598

119897119896 the 119862

119894119895119896119897will be

like 119862119894119895119896119897

= 119862119895119894119896119897

= 119862119894119895119897119896


119862119894119895119896119897

= 119862119896119897119894119895












2 sdot 119864energy = 120590119894119895120598119894119895= 120590

119901120598119901 (33)


119901 = 119894120575119894119895+ (1 minus 120575

119894119895) (9 minus 119894 minus 119895)

119902 = 119896120575119896119897+ (1 minus 120575

119896119897) (9 minus 119896 minus 119897)

(34)


119901 119862

119894119895119896119897= 119862

119901119902 and 120598

119902= (2 minus

120575119896119897)120598119896119897



(

(

1205901

1205902

1205903

1205904

1205905

1205906

)

)

= (

(

11986211

11986212

11986213

11986214

11986215

11986216

11986221

11986222

11986223

11986224

11986225

11986226

11986231

11986232

11986233

11986234

11986235

11986236

11986241

11986242

11986243

11986244

11986245

11986246

11986251

11986252

11986253

11986254

11986255

11986256

11986261

11986262

11986263

11986264

11986265

11986266

)

)

(

(

1205981

1205982

1205983

1205984

1205985

1205986

)

)

(35)



(1) the 120590119894119895and 120598


(2) the norms of 120590119894119895 120598119896119897 and 119862




01 02 03 04 05 06 07 08

2

4

6

8

10

12

14

16

(a)

01

02

03

04

05

06

07

08

09

01 02 03 04 05 06 07 08

(b)


01 02 03 04 05 06 07 08

05

1

15

2

25

3

(a)

01 02 03 04 05 06 07 08

05

1

15

2

25

3

(b)




119901 = 119894120575119894119895+ (1 minus 120575

119894119895) (9 minus 119894 minus 119895)

119902 = 119896120575119896119897+ (1 minus 120575

119896119897) (9 minus 119896 minus 119897)

(36)

In this way

120590119901= (120575

119894119895+ radic2 (1 minus 120575

119894119895)) 120598

119894119895

120598119902= (120575

119896119897+ radic2 (1 minus 120575

119896119897)) 120598

119896119897

119862119901119902

= (120575119894119895+ radic2 (1 minus 120575

119894119895)) (120575

119896119897+ radic2 (1 minus 120575

119896119897)) 119862

119894119895119896119897

(37)




120590kelvin

= 120590viogt

120585 120590voigt

=120590kelvin

120585

120598kelvin

= 120598voigt

120585 120598voigt

=120598kelvin

120585

(38)


119862kelvin

=((

(







))

)

119862voigt

(39)


(

(

12059011

12059022

12059033

radic212059023

radic212059013

radic212059012

)

)

= (

1198621111 1198621122 1198621133radic21198621123

radic21198621113radic21198621112

1198622211 1198622222 1198622333radic21198622223

radic21198622213radic21198622212

1198623311 1198623322 1198623333radic21198623323

radic21198623313radic21198623312

radic21198622311radic21198622322

radic21198622333 21198622323 21198622313 21198622312

radic21198621311radic21198621322

radic21198621333 21198621323 21198621313 21198621312

radic21198621211radic21198621222

radic21198621233 21198621223 21198621213 21198621212

)

lowast(

(

12059811

12059822

12059833

radic212059823

radic212059813

radic212059812

)

)

(40)


= C (41)



(

(

12059011

12059022

12059033

radic212059023

radic212059013

radic212059012

)

)

=((

(

11986211

11986212

11986213

radic211986214

radic211986215

radic211986216

11986212

11986222

11986223

radic211986224

radic211986225

radic211986226

11986213

11986223

11986233

radic211986234

radic211986235

radic211986236

radic211986214

radic211986224

radic211986234

211986244

211986245

211986246

radic211986215

radic211986225

radic211986235

211986245

211986255

211986256

radic211986216

radic211986226

radic211986236

211986246

211986256

211986266

))

)

lowast (

(

12059811

12059822

12059833

radic212059823

radic212059813

radic212059812

)

)

(42)


(

(

1

2

3

4

5

6

)

)

=((

(

11986211

11986212

11986213

11986214

11986215

11986216

11986212

11986222

11986223

11986224

11986225

11986226

11986213

11986223

11986233

11986234

11986235

11986236

11986214

11986224

11986234

11986244

11986245

11986246

11986215

11986225

11986235

11986245

11986255

11986256

11986216

11986226

11986236

11986246

11986256

11986266

))

)

lowast (

(

1205981

1205982

1205983

1205984

1205985

1205986

)

)

(43)



































01 02 03 04 05 06 07 08

05

1

15

2

25

3

(a)

01 02 03 04 05 06 07 08

05

1

15

2

25

3

(b)


01 02 03 04 05 06 07 08

2

4

6

8

10

12

14

16





ΛI) N = 0 (44)


119894 119894 = 1 2 6 and

















NNA119896

equiv1

119872

119872

sum

119884=1

N119884119896 (45)



NAVG119896

equiv JNANNA119896

(46)


119896=1NNA119896

otimes NNA119896

| that is

JNA equiv (

6

sum

119896=1

NNA119896

otimes NNA119896

)

minus12

(47)


119896 that is

ΛAVG119902

equiv1

119872

119872

sum

119884=1

6

sum

119896=1

Λ119884

119896(N119884

119896sdot NAVG

119902)2

(48)


gt 0 for all 119896 = 1 2 6

and ΛAVG119896


NA119896


ΛNA119896

equiv1

119872

119872

sum

119884=1

Λ119884

119896 (49)


CAVG=

6

sum

119896=1

ΛAVG119896

NAVG119896

otimes NAVG119896

(50)





[[[[[[[

[

00367834559700000

00453681054800000

0998185540700000

00

00

00

]]]]]]]

]

[[[[[[[

[

0983745905000000

minus0170879918750000

minus00277901141300000

00

00

00

]]]]]]]

]

[[[[[[[

[

minus0169284222800000

minus0982986098000000

00509905132200000

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(51)


[[[[[[[

[

003679134179

004537783172

09983995367

00

00

00

]]]]]]]

]

[[[[[[[

[

09859858256

minus01712690003

minus002785339027

00

00

00

]]]]]]]

]

[[[[[[[

[

minus01697052873

minus09854311019

005111734310

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(52)




[[[

[

334725276837330 0000112116752981933 000198542359441849 00 00 00

0000112116752981933 334773746006714 minus0000991802322788501 00 00 00

000198542359441849 minus0000991802322788501 334013593769726 00 00 00

00 00 00 333950000000000 00 00

00 00 00 00 333950000000000 00

00 00 00 00 00 333950000000000

]]]

]

(53)




[[[[[[[

[

004028666644

006399313350

09971368323

00

00

00

]]]]]]]

]

[[[[[[[

[

09048796003

minus04255671399

minus0009247686096

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04237568827

minus09026613361

007505076253

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(54)


[[[[[[[

[

004028666644

006399313350

09971368323

00

00

00

]]]]]]]

]

[[[[[[[

[

09048795994

minus04255671395

minus0009247686087

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04237568827

minus09026613361

007505076253

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(55)



[[[

[

1458799998 000000001785571215 minus000000001009489597 00 00 00000000001785571215 1458799997 0000000002407020197 00 00 00minus000000001009489597 0000000002407020197 1458799997 00 00 00

00 00 00 1458799998 00 0000 00 00 00 1458799998 0000 00 00 00 00 1458799998

]]]

]

(56)




[[[[[

[

005368516527

007220768570

09959437502

00

00

00

]]]]]

]

[[[[[

[

09266208315

minus03753089731

minus002273783078

00

00

00

]]]]]

]

[[[[[

[

minus03721447810

minus09240829102

008705767064

00

00

00

]]]]]

]

[[[[[

[

00

00

00

00

00

10

]]]]]

]

[[[[[

[

00

00

00

10

00

00

]]]]]

]

[[[[[

[

00

00

00

00

10

00

]]]]]

]

(57)


[[[[[[[[[

[

005368516527

007220768570

09959437502

00

00

00

]]]]]]]]]

]

[[[[[[[[[

[

09266208315

minus03753089731

minus002273783078

00

00

00

]]]]]]]]]

]

[[[[[[[[[

[

minus03721447806

minus09240829093

008705767055

00

00

00

]]]]]]]]]

]

[[[[[[[[[

[

00

00

00

10

00

00

]]]]]]]]]

]

[[[[[[[[[

[

00

00

00

00

10

00

]]]]]]]]]

]

[[[[[[[[[

[

00

00

00

00

00

10

]]]]]]]]]

]

(58)




[[[

[

1615299998 0000000005508173090 minus0000000008722620038 00 00 00

0000000005508173090 1615299996 minus0000000004345156817 00 00 00

minus0000000008722620038 minus0000000004345156817 1615300000 00 00 00

00 00 00 1615299998 00 00

00 00 00 00 1615299998 00

00 00 00 00 00 1615299998

]]]

]

(59)




[[[[[[[

[

minus00598757013800000

minus00768229223150000

minus0995231841750000

00

00

00

]]]]]]]

]

[[[[[[[

[

minus0869358919900000

0493939153600000

00141567750950000

00

00

00

]]]]]]]

]

[[[[[[[

[

minus0490490276500000

minus0866078136900000

00963811082600000

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(60)


[[[[[[[[[[[

[

minus005987730132

minus007682497510

minus09952584354

00

00

00

]]]]]]]]]]]

]

[[[[[[[[[[[

[

minus08693926232

04939583026

001415732393

00

00

00

]]]]]]]]]]]

]

[[[[[[[[[[[

[

00

00

00

10

00

00

]]]]]]]]]]]

]

[[[[[[[[[[[

[

00

00

00

00

10

00

]]]]]]]]]]]

]

[[[[[[[[[[[

[

00

00

00

00

00

10

]]]]]]]]]]]

]

(61)



[[[

[


00 00 00 1819400002 1819400002 0000 00 00 00 00 0000 00 00 00 00 1819400002

]]]

]

(62)




[[[[[[[

[

004967528691

008463937788

09951726186

00

00

00

]]]]]]]

]

[[[[[[[

[

09161715971

minus04006166011

minus001165943224

00

00

00

]]]]]]]

]

[[[[[

[

minus03976958243

minus09123280736

009744493938

00

00

00

]]]]]

]

[[[[[

[

00

00

00

10

00

00

]]]]]

]

[[[[[

[

00

00

00

00

10

00

]]]]]

]

[[[[[

[

00

00

00

00

00

10

]]]]]

]

(63)


[[[[[[[

[

004967528696

008463937796

09951726196

00

00

00

]]]]]]]

]

[[[[[[[

[

09161715980

minus04006166015

minus001165943225

00

00

00

]]]]]]]

]


[[[[[[[

[

minus03976958247

minus09123280745

009744493948

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(64)



[[[

[

1922399998 minus000000001176508811 minus000000001537920006 00 00 00

minus000000001176508811 1922400000 minus0000000007631928036 00 00 00

minus000000001537920006 minus0000000007631928036 1922400000 00 00 00

00 00 00 1922399998 1922399998 00

00 00 00 00 00 00

00 00 00 00 00 1922399998

]]]

]

(65)




[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

[[[[[[[

[

minus01387533797

minus02031928413

minus09692575344

00

00

00

]]]]]]]

]

[[[[[[[

[

minus08476774112

05304148448

001015375725

00

00

00

]]]]]]]

]

[[[[[[[

[

minus05120454135

minus08230265872

02458388325

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

(66)


[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

[[[[[[[

[

minus01387533798

minus02031928415

minus09692575354

00

00

00

]]]]]]]

]

[[[[[[[

[

minus08476774112

05304148448

001015375725

00

00

00

]]]]]]]

]

[[[[[[[

[

minus05120454135

minus08230265872

02458388325

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

(67)



[[[

[

2074599999 0000000004356660361 000000003402343994 00 00 00

0000000004356660361 2074600004 000000002232269567 00 00 00

000000003402343994 000000002232269567 2074600000 00 00 00

00 00 00 2074600000 2074600000 00

00 00 00 00 00 00

00 00 00 00 00 2074600000

]]]

]

(68)




[[[[[[[

[

008621257414

01243595697

09884847438

00

00

00

]]]]]]]

]

[[[[[[[

[

08755641188

minus04828494241

minus001561753067

00

00

00

]]]]]]]

]

[[[[[

[

minus04753471023

minus08668282006

01505124700

00

00

00

]]]]]

]

[[[[[

[

00

00

00

10

00

00

]]]]]

]

[[[[[

[

00

00

00

00

10

00

]]]]]

]

[[[[[

[

00

00

00

00

00

10

]]]]]

]

(69)


[[[[[

[

008621257423

01243595698

09884847448

00

00

00

]]]]]

]

[[[[[

[

08755641188

minus04828494241

minus001561753067

00

00

00

]]]]]

]


[[[[[[[

[

minus04753471018

minus08668281997

01505124698

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(70)



[[[

[

1890099999 000000001209663993 minus000000002532733996 00 00 00000000001209663993 1890099991 0000000001701090138 00 00 00minus000000002532733996 0000000001701090138 1890100001 00 00 00

00 00 00 1890099997 1890099997 0000 00 00 00 00 0000 00 00 00 00 1890099997

]]]

]

(71)




[[[[[[[

[

03961520086

06338768023

06642768888

00

00

00

]]]]]]]

]

[[[[[[[

[

09160614623

minus02236797319

minus03328645006

00

00

00

]]]]]]]

]

[[[[[[[

[

006240981414

minus07403833993

06692812840

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(72)


[[[[[[[

[

03961520086

06338768023

06642768888

00

00

00

]]]]]]]

]

[[[[[[[

[

09160614614

minus02236797317

minus03328645003

00

00

00

]]]]]]]

]

[[[[[[[

[

006240981426

minus07403834008

06692812853

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(73)



[[[

[


00 00 00 8772299998 8772299998 0000 00 00 00 00 0000 00 00 00 00 8772299998

]]]

]

(74)




[[[[[[[

[

minus006591789198

minus008975775227

minus09937798435

00

00

00

]]]]]]]

]

[[[[[[[

[

minus08289381135

05593224991

0004466103673

00

00

00

]]]]]]]

]

[[[[[

[

minus05554425577

minus08240763851

01112729817

00

00

00

]]]]]

]

[[[[[

[

00

00

00

10

00

00

]]]]]

]

[[[[[

[

00

00

00

00

10

00

]]]]]

]

[[[[[

[

00

00

00

00

00

10

]]]]]

]


[[[[[[[

[

minus01387533798

minus02031928415

minus09692575354

00

00

00

]]]]]]]

]

[[[[[[[

[

minus08476774112

05304148448

001015375725

00

00

00

]]]]]]]

]


[[[[

[

minus05120454135

minus08230265872

02458388325

00

00

00

]]]]

]

[[[[

[

00

00

00

10

00

00

]]]]

]

[[[[

[

00

00

00

00

10

00

]]]]

]

[[[[

[

00

00

00

00

00

10

]]]]

]

(76)



[[[

[

2228057494 0000000004678921127 000000003654014285 00 00 000000000004678921127 2228057499 000000002397389829 00 00 00000000003654014285 000000002397389829 2228057495 00 00 00

00 00 00 2228057495 2228057495 0000 00 00 00 00 0000 00 00 00 00 2228057495

]]]

]

(77)




[[[[[[[

[

006975010637

01071019008

09917984199

00

00

00

]]]]]]]

]

[[[[[[[

[

09079000793

minus04187725641

minus001862756541

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04133429180

minus09017531389

01264472547

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(78)


[[[[[[[

[

006975010637

01071019008

09917984199

00

00

00

]]]]]]]

]

[[[[[[[

[

09079000793

minus04187725641

minus001862756541

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04133429184

minus09017531398

01264472548

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(79)



[[[

[


00 00 00 2598699998 2598699998 0000 00 00 00 00 0000 00 00 00 00 2598699998

]]]

]

(80)




[[[[[

[

minus00192522847250000

minus00192842313600000

000386151499999998

00

00

00

]]]]]

]

[[[[[

[

000961799224999998

00165029120500000

000436480214750000

00

00

00

]]]]]

]

[[[[[[

[

minus0494444050150000

minus0856906622200000

0142104790200000

00

00

00

]]]]]]

]

[[[[[[

[

00

00

00

10

00

00

]]]]]]

]

[[[[[[

[

00

00

00

00

10

00

]]]]]]

]

[[[[[[

[

00

00

00

00

00

10

]]]]]]

]

(81)



[[[[[[[[[[

[

minus2541745843

minus2545963537

5098090872

00

00

00

]]]]]]]]]]

]

[[[[[[[[[[

[

2505315863

4298714977

1136953303

00

00

00

]]]]]]]]]]

]

[[[[[[[

[

minus04949599718

minus08578007511

01422530677

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(82)



[[[

[

315679169478799 427324383657779 384557503470365 00 00 00427324383657779 618700481877479 889153535276175 00 00 00384557503470365 889153535276175 384768762708609 00 00 00

00 00 00 247795001400000 00 0000 00 00 00 247795001400000 0000 00 00 00 00 247795001400000

]]]

]

(83)




[[[[[[[

[

01135176976

01764907831

09777344911

00

00

00

]]]]]]]

]

[[[[[[[

[

08848520771

minus04654963442

minus001870707734

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04518302069

minus08672739791

02090103088

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(84)


[[[[[[[

[

01135176977

01764907833

09777344921

00

00

00

]]]]]]]

]

[[[[[[[

[

08848520771

minus04654963442

minus001870707734

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04518302069

minus08672739791

02090103088

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(85)



[[[

[

2663700003 000000004768023095 000000003169803038 00 00 00000000004768023095 2663699992 0000000008310743498 00 00 00000000003169803038 0000000008310743498 2663700001 00 00 00

00 00 00 2663699998 2663699998 0000 00 00 00 00 0000 00 00 00 00 2663699998

]]]

]

(86)





6 Conclusions









Disclosure




Acknowledgments



References




















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










12

34

56

ColumnRow

12

34

56

600005000040000300002000010000


5

10

15

20

25

12

34

56

Column Row

12

34

56






119895119894and 120598

119896119897= 120598

119897119896 the 119862

119894119895119896119897will be

like 119862119894119895119896119897

= 119862119895119894119896119897

= 119862119894119895119897119896


119862119894119895119896119897

= 119862119896119897119894119895












2 sdot 119864energy = 120590119894119895120598119894119895= 120590

119901120598119901 (33)


119901 = 119894120575119894119895+ (1 minus 120575

119894119895) (9 minus 119894 minus 119895)

119902 = 119896120575119896119897+ (1 minus 120575

119896119897) (9 minus 119896 minus 119897)

(34)


119901 119862

119894119895119896119897= 119862

119901119902 and 120598

119902= (2 minus

120575119896119897)120598119896119897



(

(

1205901

1205902

1205903

1205904

1205905

1205906

)

)

= (

(

11986211

11986212

11986213

11986214

11986215

11986216

11986221

11986222

11986223

11986224

11986225

11986226

11986231

11986232

11986233

11986234

11986235

11986236

11986241

11986242

11986243

11986244

11986245

11986246

11986251

11986252

11986253

11986254

11986255

11986256

11986261

11986262

11986263

11986264

11986265

11986266

)

)

(

(

1205981

1205982

1205983

1205984

1205985

1205986

)

)

(35)



(1) the 120590119894119895and 120598


(2) the norms of 120590119894119895 120598119896119897 and 119862




01 02 03 04 05 06 07 08

2

4

6

8

10

12

14

16

(a)

01

02

03

04

05

06

07

08

09

01 02 03 04 05 06 07 08

(b)


01 02 03 04 05 06 07 08

05

1

15

2

25

3

(a)

01 02 03 04 05 06 07 08

05

1

15

2

25

3

(b)




119901 = 119894120575119894119895+ (1 minus 120575

119894119895) (9 minus 119894 minus 119895)

119902 = 119896120575119896119897+ (1 minus 120575

119896119897) (9 minus 119896 minus 119897)

(36)

In this way

120590119901= (120575

119894119895+ radic2 (1 minus 120575

119894119895)) 120598

119894119895

120598119902= (120575

119896119897+ radic2 (1 minus 120575

119896119897)) 120598

119896119897

119862119901119902

= (120575119894119895+ radic2 (1 minus 120575

119894119895)) (120575

119896119897+ radic2 (1 minus 120575

119896119897)) 119862

119894119895119896119897

(37)




120590kelvin

= 120590viogt

120585 120590voigt

=120590kelvin

120585

120598kelvin

= 120598voigt

120585 120598voigt

=120598kelvin

120585

(38)


119862kelvin

=((

(







))

)

119862voigt

(39)


(

(

12059011

12059022

12059033

radic212059023

radic212059013

radic212059012

)

)

= (

1198621111 1198621122 1198621133radic21198621123

radic21198621113radic21198621112

1198622211 1198622222 1198622333radic21198622223

radic21198622213radic21198622212

1198623311 1198623322 1198623333radic21198623323

radic21198623313radic21198623312

radic21198622311radic21198622322

radic21198622333 21198622323 21198622313 21198622312

radic21198621311radic21198621322

radic21198621333 21198621323 21198621313 21198621312

radic21198621211radic21198621222

radic21198621233 21198621223 21198621213 21198621212

)

lowast(

(

12059811

12059822

12059833

radic212059823

radic212059813

radic212059812

)

)

(40)


= C (41)



(

(

12059011

12059022

12059033

radic212059023

radic212059013

radic212059012

)

)

=((

(

11986211

11986212

11986213

radic211986214

radic211986215

radic211986216

11986212

11986222

11986223

radic211986224

radic211986225

radic211986226

11986213

11986223

11986233

radic211986234

radic211986235

radic211986236

radic211986214

radic211986224

radic211986234

211986244

211986245

211986246

radic211986215

radic211986225

radic211986235

211986245

211986255

211986256

radic211986216

radic211986226

radic211986236

211986246

211986256

211986266

))

)

lowast (

(

12059811

12059822

12059833

radic212059823

radic212059813

radic212059812

)

)

(42)


(

(

1

2

3

4

5

6

)

)

=((

(

11986211

11986212

11986213

11986214

11986215

11986216

11986212

11986222

11986223

11986224

11986225

11986226

11986213

11986223

11986233

11986234

11986235

11986236

11986214

11986224

11986234

11986244

11986245

11986246

11986215

11986225

11986235

11986245

11986255

11986256

11986216

11986226

11986236

11986246

11986256

11986266

))

)

lowast (

(

1205981

1205982

1205983

1205984

1205985

1205986

)

)

(43)



































01 02 03 04 05 06 07 08

05

1

15

2

25

3

(a)

01 02 03 04 05 06 07 08

05

1

15

2

25

3

(b)


01 02 03 04 05 06 07 08

2

4

6

8

10

12

14

16





ΛI) N = 0 (44)


119894 119894 = 1 2 6 and

















NNA119896

equiv1

119872

119872

sum

119884=1

N119884119896 (45)



NAVG119896

equiv JNANNA119896

(46)


119896=1NNA119896

otimes NNA119896

| that is

JNA equiv (

6

sum

119896=1

NNA119896

otimes NNA119896

)

minus12

(47)


119896 that is

ΛAVG119902

equiv1

119872

119872

sum

119884=1

6

sum

119896=1

Λ119884

119896(N119884

119896sdot NAVG

119902)2

(48)


gt 0 for all 119896 = 1 2 6

and ΛAVG119896


NA119896


ΛNA119896

equiv1

119872

119872

sum

119884=1

Λ119884

119896 (49)


CAVG=

6

sum

119896=1

ΛAVG119896

NAVG119896

otimes NAVG119896

(50)





[[[[[[[

[

00367834559700000

00453681054800000

0998185540700000

00

00

00

]]]]]]]

]

[[[[[[[

[

0983745905000000

minus0170879918750000

minus00277901141300000

00

00

00

]]]]]]]

]

[[[[[[[

[

minus0169284222800000

minus0982986098000000

00509905132200000

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(51)


[[[[[[[

[

003679134179

004537783172

09983995367

00

00

00

]]]]]]]

]

[[[[[[[

[

09859858256

minus01712690003

minus002785339027

00

00

00

]]]]]]]

]

[[[[[[[

[

minus01697052873

minus09854311019

005111734310

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(52)




[[[

[

334725276837330 0000112116752981933 000198542359441849 00 00 00

0000112116752981933 334773746006714 minus0000991802322788501 00 00 00

000198542359441849 minus0000991802322788501 334013593769726 00 00 00

00 00 00 333950000000000 00 00

00 00 00 00 333950000000000 00

00 00 00 00 00 333950000000000

]]]

]

(53)




[[[[[[[

[

004028666644

006399313350

09971368323

00

00

00

]]]]]]]

]

[[[[[[[

[

09048796003

minus04255671399

minus0009247686096

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04237568827

minus09026613361

007505076253

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(54)


[[[[[[[

[

004028666644

006399313350

09971368323

00

00

00

]]]]]]]

]

[[[[[[[

[

09048795994

minus04255671395

minus0009247686087

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04237568827

minus09026613361

007505076253

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(55)



[[[

[

1458799998 000000001785571215 minus000000001009489597 00 00 00000000001785571215 1458799997 0000000002407020197 00 00 00minus000000001009489597 0000000002407020197 1458799997 00 00 00

00 00 00 1458799998 00 0000 00 00 00 1458799998 0000 00 00 00 00 1458799998

]]]

]

(56)




[[[[[

[

005368516527

007220768570

09959437502

00

00

00

]]]]]

]

[[[[[

[

09266208315

minus03753089731

minus002273783078

00

00

00

]]]]]

]

[[[[[

[

minus03721447810

minus09240829102

008705767064

00

00

00

]]]]]

]

[[[[[

[

00

00

00

00

00

10

]]]]]

]

[[[[[

[

00

00

00

10

00

00

]]]]]

]

[[[[[

[

00

00

00

00

10

00

]]]]]

]

(57)


[[[[[[[[[

[

005368516527

007220768570

09959437502

00

00

00

]]]]]]]]]

]

[[[[[[[[[

[

09266208315

minus03753089731

minus002273783078

00

00

00

]]]]]]]]]

]

[[[[[[[[[

[

minus03721447806

minus09240829093

008705767055

00

00

00

]]]]]]]]]

]

[[[[[[[[[

[

00

00

00

10

00

00

]]]]]]]]]

]

[[[[[[[[[

[

00

00

00

00

10

00

]]]]]]]]]

]

[[[[[[[[[

[

00

00

00

00

00

10

]]]]]]]]]

]

(58)




[[[

[

1615299998 0000000005508173090 minus0000000008722620038 00 00 00

0000000005508173090 1615299996 minus0000000004345156817 00 00 00

minus0000000008722620038 minus0000000004345156817 1615300000 00 00 00

00 00 00 1615299998 00 00

00 00 00 00 1615299998 00

00 00 00 00 00 1615299998

]]]

]

(59)




[[[[[[[

[

minus00598757013800000

minus00768229223150000

minus0995231841750000

00

00

00

]]]]]]]

]

[[[[[[[

[

minus0869358919900000

0493939153600000

00141567750950000

00

00

00

]]]]]]]

]

[[[[[[[

[

minus0490490276500000

minus0866078136900000

00963811082600000

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(60)


[[[[[[[[[[[

[

minus005987730132

minus007682497510

minus09952584354

00

00

00

]]]]]]]]]]]

]

[[[[[[[[[[[

[

minus08693926232

04939583026

001415732393

00

00

00

]]]]]]]]]]]

]

[[[[[[[[[[[

[

00

00

00

10

00

00

]]]]]]]]]]]

]

[[[[[[[[[[[

[

00

00

00

00

10

00

]]]]]]]]]]]

]

[[[[[[[[[[[

[

00

00

00

00

00

10

]]]]]]]]]]]

]

(61)



[[[

[


00 00 00 1819400002 1819400002 0000 00 00 00 00 0000 00 00 00 00 1819400002

]]]

]

(62)




[[[[[[[

[

004967528691

008463937788

09951726186

00

00

00

]]]]]]]

]

[[[[[[[

[

09161715971

minus04006166011

minus001165943224

00

00

00

]]]]]]]

]

[[[[[

[

minus03976958243

minus09123280736

009744493938

00

00

00

]]]]]

]

[[[[[

[

00

00

00

10

00

00

]]]]]

]

[[[[[

[

00

00

00

00

10

00

]]]]]

]

[[[[[

[

00

00

00

00

00

10

]]]]]

]

(63)


[[[[[[[

[

004967528696

008463937796

09951726196

00

00

00

]]]]]]]

]

[[[[[[[

[

09161715980

minus04006166015

minus001165943225

00

00

00

]]]]]]]

]


[[[[[[[

[

minus03976958247

minus09123280745

009744493948

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(64)



[[[

[

1922399998 minus000000001176508811 minus000000001537920006 00 00 00

minus000000001176508811 1922400000 minus0000000007631928036 00 00 00

minus000000001537920006 minus0000000007631928036 1922400000 00 00 00

00 00 00 1922399998 1922399998 00

00 00 00 00 00 00

00 00 00 00 00 1922399998

]]]

]

(65)




[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

[[[[[[[

[

minus01387533797

minus02031928413

minus09692575344

00

00

00

]]]]]]]

]

[[[[[[[

[

minus08476774112

05304148448

001015375725

00

00

00

]]]]]]]

]

[[[[[[[

[

minus05120454135

minus08230265872

02458388325

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

(66)


[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

[[[[[[[

[

minus01387533798

minus02031928415

minus09692575354

00

00

00

]]]]]]]

]

[[[[[[[

[

minus08476774112

05304148448

001015375725

00

00

00

]]]]]]]

]

[[[[[[[

[

minus05120454135

minus08230265872

02458388325

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

(67)



[[[

[

2074599999 0000000004356660361 000000003402343994 00 00 00

0000000004356660361 2074600004 000000002232269567 00 00 00

000000003402343994 000000002232269567 2074600000 00 00 00

00 00 00 2074600000 2074600000 00

00 00 00 00 00 00

00 00 00 00 00 2074600000

]]]

]

(68)




[[[[[[[

[

008621257414

01243595697

09884847438

00

00

00

]]]]]]]

]

[[[[[[[

[

08755641188

minus04828494241

minus001561753067

00

00

00

]]]]]]]

]

[[[[[

[

minus04753471023

minus08668282006

01505124700

00

00

00

]]]]]

]

[[[[[

[

00

00

00

10

00

00

]]]]]

]

[[[[[

[

00

00

00

00

10

00

]]]]]

]

[[[[[

[

00

00

00

00

00

10

]]]]]

]

(69)


[[[[[

[

008621257423

01243595698

09884847448

00

00

00

]]]]]

]

[[[[[

[

08755641188

minus04828494241

minus001561753067

00

00

00

]]]]]

]


[[[[[[[

[

minus04753471018

minus08668281997

01505124698

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(70)



[[[

[

1890099999 000000001209663993 minus000000002532733996 00 00 00000000001209663993 1890099991 0000000001701090138 00 00 00minus000000002532733996 0000000001701090138 1890100001 00 00 00

00 00 00 1890099997 1890099997 0000 00 00 00 00 0000 00 00 00 00 1890099997

]]]

]

(71)




[[[[[[[

[

03961520086

06338768023

06642768888

00

00

00

]]]]]]]

]

[[[[[[[

[

09160614623

minus02236797319

minus03328645006

00

00

00

]]]]]]]

]

[[[[[[[

[

006240981414

minus07403833993

06692812840

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(72)


[[[[[[[

[

03961520086

06338768023

06642768888

00

00

00

]]]]]]]

]

[[[[[[[

[

09160614614

minus02236797317

minus03328645003

00

00

00

]]]]]]]

]

[[[[[[[

[

006240981426

minus07403834008

06692812853

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(73)



[[[

[


00 00 00 8772299998 8772299998 0000 00 00 00 00 0000 00 00 00 00 8772299998

]]]

]

(74)




[[[[[[[

[

minus006591789198

minus008975775227

minus09937798435

00

00

00

]]]]]]]

]

[[[[[[[

[

minus08289381135

05593224991

0004466103673

00

00

00

]]]]]]]

]

[[[[[

[

minus05554425577

minus08240763851

01112729817

00

00

00

]]]]]

]

[[[[[

[

00

00

00

10

00

00

]]]]]

]

[[[[[

[

00

00

00

00

10

00

]]]]]

]

[[[[[

[

00

00

00

00

00

10

]]]]]

]


[[[[[[[

[

minus01387533798

minus02031928415

minus09692575354

00

00

00

]]]]]]]

]

[[[[[[[

[

minus08476774112

05304148448

001015375725

00

00

00

]]]]]]]

]


[[[[

[

minus05120454135

minus08230265872

02458388325

00

00

00

]]]]

]

[[[[

[

00

00

00

10

00

00

]]]]

]

[[[[

[

00

00

00

00

10

00

]]]]

]

[[[[

[

00

00

00

00

00

10

]]]]

]

(76)



[[[

[

2228057494 0000000004678921127 000000003654014285 00 00 000000000004678921127 2228057499 000000002397389829 00 00 00000000003654014285 000000002397389829 2228057495 00 00 00

00 00 00 2228057495 2228057495 0000 00 00 00 00 0000 00 00 00 00 2228057495

]]]

]

(77)




[[[[[[[

[

006975010637

01071019008

09917984199

00

00

00

]]]]]]]

]

[[[[[[[

[

09079000793

minus04187725641

minus001862756541

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04133429180

minus09017531389

01264472547

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(78)


[[[[[[[

[

006975010637

01071019008

09917984199

00

00

00

]]]]]]]

]

[[[[[[[

[

09079000793

minus04187725641

minus001862756541

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04133429184

minus09017531398

01264472548

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(79)



[[[

[


00 00 00 2598699998 2598699998 0000 00 00 00 00 0000 00 00 00 00 2598699998

]]]

]

(80)




[[[[[

[

minus00192522847250000

minus00192842313600000

000386151499999998

00

00

00

]]]]]

]

[[[[[

[

000961799224999998

00165029120500000

000436480214750000

00

00

00

]]]]]

]

[[[[[[

[

minus0494444050150000

minus0856906622200000

0142104790200000

00

00

00

]]]]]]

]

[[[[[[

[

00

00

00

10

00

00

]]]]]]

]

[[[[[[

[

00

00

00

00

10

00

]]]]]]

]

[[[[[[

[

00

00

00

00

00

10

]]]]]]

]

(81)



[[[[[[[[[[

[

minus2541745843

minus2545963537

5098090872

00

00

00

]]]]]]]]]]

]

[[[[[[[[[[

[

2505315863

4298714977

1136953303

00

00

00

]]]]]]]]]]

]

[[[[[[[

[

minus04949599718

minus08578007511

01422530677

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(82)



[[[

[

315679169478799 427324383657779 384557503470365 00 00 00427324383657779 618700481877479 889153535276175 00 00 00384557503470365 889153535276175 384768762708609 00 00 00

00 00 00 247795001400000 00 0000 00 00 00 247795001400000 0000 00 00 00 00 247795001400000

]]]

]

(83)




[[[[[[[

[

01135176976

01764907831

09777344911

00

00

00

]]]]]]]

]

[[[[[[[

[

08848520771

minus04654963442

minus001870707734

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04518302069

minus08672739791

02090103088

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(84)


[[[[[[[

[

01135176977

01764907833

09777344921

00

00

00

]]]]]]]

]

[[[[[[[

[

08848520771

minus04654963442

minus001870707734

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04518302069

minus08672739791

02090103088

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(85)



[[[

[

2663700003 000000004768023095 000000003169803038 00 00 00000000004768023095 2663699992 0000000008310743498 00 00 00000000003169803038 0000000008310743498 2663700001 00 00 00

00 00 00 2663699998 2663699998 0000 00 00 00 00 0000 00 00 00 00 2663699998

]]]

]

(86)





6 Conclusions









Disclosure




Acknowledgments



References




















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










01 02 03 04 05 06 07 08

2

4

6

8

10

12

14

16

(a)

01

02

03

04

05

06

07

08

09

01 02 03 04 05 06 07 08

(b)


01 02 03 04 05 06 07 08

05

1

15

2

25

3

(a)

01 02 03 04 05 06 07 08

05

1

15

2

25

3

(b)




119901 = 119894120575119894119895+ (1 minus 120575

119894119895) (9 minus 119894 minus 119895)

119902 = 119896120575119896119897+ (1 minus 120575

119896119897) (9 minus 119896 minus 119897)

(36)

In this way

120590119901= (120575

119894119895+ radic2 (1 minus 120575

119894119895)) 120598

119894119895

120598119902= (120575

119896119897+ radic2 (1 minus 120575

119896119897)) 120598

119896119897

119862119901119902

= (120575119894119895+ radic2 (1 minus 120575

119894119895)) (120575

119896119897+ radic2 (1 minus 120575

119896119897)) 119862

119894119895119896119897

(37)




120590kelvin

= 120590viogt

120585 120590voigt

=120590kelvin

120585

120598kelvin

= 120598voigt

120585 120598voigt

=120598kelvin

120585

(38)


119862kelvin

=((

(







))

)

119862voigt

(39)


(

(

12059011

12059022

12059033

radic212059023

radic212059013

radic212059012

)

)

= (

1198621111 1198621122 1198621133radic21198621123

radic21198621113radic21198621112

1198622211 1198622222 1198622333radic21198622223

radic21198622213radic21198622212

1198623311 1198623322 1198623333radic21198623323

radic21198623313radic21198623312

radic21198622311radic21198622322

radic21198622333 21198622323 21198622313 21198622312

radic21198621311radic21198621322

radic21198621333 21198621323 21198621313 21198621312

radic21198621211radic21198621222

radic21198621233 21198621223 21198621213 21198621212

)

lowast(

(

12059811

12059822

12059833

radic212059823

radic212059813

radic212059812

)

)

(40)


= C (41)



(

(

12059011

12059022

12059033

radic212059023

radic212059013

radic212059012

)

)

=((

(

11986211

11986212

11986213

radic211986214

radic211986215

radic211986216

11986212

11986222

11986223

radic211986224

radic211986225

radic211986226

11986213

11986223

11986233

radic211986234

radic211986235

radic211986236

radic211986214

radic211986224

radic211986234

211986244

211986245

211986246

radic211986215

radic211986225

radic211986235

211986245

211986255

211986256

radic211986216

radic211986226

radic211986236

211986246

211986256

211986266

))

)

lowast (

(

12059811

12059822

12059833

radic212059823

radic212059813

radic212059812

)

)

(42)


(

(

1

2

3

4

5

6

)

)

=((

(

11986211

11986212

11986213

11986214

11986215

11986216

11986212

11986222

11986223

11986224

11986225

11986226

11986213

11986223

11986233

11986234

11986235

11986236

11986214

11986224

11986234

11986244

11986245

11986246

11986215

11986225

11986235

11986245

11986255

11986256

11986216

11986226

11986236

11986246

11986256

11986266

))

)

lowast (

(

1205981

1205982

1205983

1205984

1205985

1205986

)

)

(43)



































01 02 03 04 05 06 07 08

05

1

15

2

25

3

(a)

01 02 03 04 05 06 07 08

05

1

15

2

25

3

(b)


01 02 03 04 05 06 07 08

2

4

6

8

10

12

14

16





ΛI) N = 0 (44)


119894 119894 = 1 2 6 and

















NNA119896

equiv1

119872

119872

sum

119884=1

N119884119896 (45)



NAVG119896

equiv JNANNA119896

(46)


119896=1NNA119896

otimes NNA119896

| that is

JNA equiv (

6

sum

119896=1

NNA119896

otimes NNA119896

)

minus12

(47)


119896 that is

ΛAVG119902

equiv1

119872

119872

sum

119884=1

6

sum

119896=1

Λ119884

119896(N119884

119896sdot NAVG

119902)2

(48)


gt 0 for all 119896 = 1 2 6

and ΛAVG119896


NA119896


ΛNA119896

equiv1

119872

119872

sum

119884=1

Λ119884

119896 (49)


CAVG=

6

sum

119896=1

ΛAVG119896

NAVG119896

otimes NAVG119896

(50)





[[[[[[[

[

00367834559700000

00453681054800000

0998185540700000

00

00

00

]]]]]]]

]

[[[[[[[

[

0983745905000000

minus0170879918750000

minus00277901141300000

00

00

00

]]]]]]]

]

[[[[[[[

[

minus0169284222800000

minus0982986098000000

00509905132200000

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(51)


[[[[[[[

[

003679134179

004537783172

09983995367

00

00

00

]]]]]]]

]

[[[[[[[

[

09859858256

minus01712690003

minus002785339027

00

00

00

]]]]]]]

]

[[[[[[[

[

minus01697052873

minus09854311019

005111734310

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(52)




[[[

[

334725276837330 0000112116752981933 000198542359441849 00 00 00

0000112116752981933 334773746006714 minus0000991802322788501 00 00 00

000198542359441849 minus0000991802322788501 334013593769726 00 00 00

00 00 00 333950000000000 00 00

00 00 00 00 333950000000000 00

00 00 00 00 00 333950000000000

]]]

]

(53)




[[[[[[[

[

004028666644

006399313350

09971368323

00

00

00

]]]]]]]

]

[[[[[[[

[

09048796003

minus04255671399

minus0009247686096

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04237568827

minus09026613361

007505076253

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(54)


[[[[[[[

[

004028666644

006399313350

09971368323

00

00

00

]]]]]]]

]

[[[[[[[

[

09048795994

minus04255671395

minus0009247686087

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04237568827

minus09026613361

007505076253

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(55)



[[[

[

1458799998 000000001785571215 minus000000001009489597 00 00 00000000001785571215 1458799997 0000000002407020197 00 00 00minus000000001009489597 0000000002407020197 1458799997 00 00 00

00 00 00 1458799998 00 0000 00 00 00 1458799998 0000 00 00 00 00 1458799998

]]]

]

(56)




[[[[[

[

005368516527

007220768570

09959437502

00

00

00

]]]]]

]

[[[[[

[

09266208315

minus03753089731

minus002273783078

00

00

00

]]]]]

]

[[[[[

[

minus03721447810

minus09240829102

008705767064

00

00

00

]]]]]

]

[[[[[

[

00

00

00

00

00

10

]]]]]

]

[[[[[

[

00

00

00

10

00

00

]]]]]

]

[[[[[

[

00

00

00

00

10

00

]]]]]

]

(57)


[[[[[[[[[

[

005368516527

007220768570

09959437502

00

00

00

]]]]]]]]]

]

[[[[[[[[[

[

09266208315

minus03753089731

minus002273783078

00

00

00

]]]]]]]]]

]

[[[[[[[[[

[

minus03721447806

minus09240829093

008705767055

00

00

00

]]]]]]]]]

]

[[[[[[[[[

[

00

00

00

10

00

00

]]]]]]]]]

]

[[[[[[[[[

[

00

00

00

00

10

00

]]]]]]]]]

]

[[[[[[[[[

[

00

00

00

00

00

10

]]]]]]]]]

]

(58)




[[[

[

1615299998 0000000005508173090 minus0000000008722620038 00 00 00

0000000005508173090 1615299996 minus0000000004345156817 00 00 00

minus0000000008722620038 minus0000000004345156817 1615300000 00 00 00

00 00 00 1615299998 00 00

00 00 00 00 1615299998 00

00 00 00 00 00 1615299998

]]]

]

(59)




[[[[[[[

[

minus00598757013800000

minus00768229223150000

minus0995231841750000

00

00

00

]]]]]]]

]

[[[[[[[

[

minus0869358919900000

0493939153600000

00141567750950000

00

00

00

]]]]]]]

]

[[[[[[[

[

minus0490490276500000

minus0866078136900000

00963811082600000

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(60)


[[[[[[[[[[[

[

minus005987730132

minus007682497510

minus09952584354

00

00

00

]]]]]]]]]]]

]

[[[[[[[[[[[

[

minus08693926232

04939583026

001415732393

00

00

00

]]]]]]]]]]]

]

[[[[[[[[[[[

[

00

00

00

10

00

00

]]]]]]]]]]]

]

[[[[[[[[[[[

[

00

00

00

00

10

00

]]]]]]]]]]]

]

[[[[[[[[[[[

[

00

00

00

00

00

10

]]]]]]]]]]]

]

(61)



[[[

[


00 00 00 1819400002 1819400002 0000 00 00 00 00 0000 00 00 00 00 1819400002

]]]

]

(62)




[[[[[[[

[

004967528691

008463937788

09951726186

00

00

00

]]]]]]]

]

[[[[[[[

[

09161715971

minus04006166011

minus001165943224

00

00

00

]]]]]]]

]

[[[[[

[

minus03976958243

minus09123280736

009744493938

00

00

00

]]]]]

]

[[[[[

[

00

00

00

10

00

00

]]]]]

]

[[[[[

[

00

00

00

00

10

00

]]]]]

]

[[[[[

[

00

00

00

00

00

10

]]]]]

]

(63)


[[[[[[[

[

004967528696

008463937796

09951726196

00

00

00

]]]]]]]

]

[[[[[[[

[

09161715980

minus04006166015

minus001165943225

00

00

00

]]]]]]]

]


[[[[[[[

[

minus03976958247

minus09123280745

009744493948

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(64)



[[[

[

1922399998 minus000000001176508811 minus000000001537920006 00 00 00

minus000000001176508811 1922400000 minus0000000007631928036 00 00 00

minus000000001537920006 minus0000000007631928036 1922400000 00 00 00

00 00 00 1922399998 1922399998 00

00 00 00 00 00 00

00 00 00 00 00 1922399998

]]]

]

(65)




[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

[[[[[[[

[

minus01387533797

minus02031928413

minus09692575344

00

00

00

]]]]]]]

]

[[[[[[[

[

minus08476774112

05304148448

001015375725

00

00

00

]]]]]]]

]

[[[[[[[

[

minus05120454135

minus08230265872

02458388325

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

(66)


[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

[[[[[[[

[

minus01387533798

minus02031928415

minus09692575354

00

00

00

]]]]]]]

]

[[[[[[[

[

minus08476774112

05304148448

001015375725

00

00

00

]]]]]]]

]

[[[[[[[

[

minus05120454135

minus08230265872

02458388325

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

(67)



[[[

[

2074599999 0000000004356660361 000000003402343994 00 00 00

0000000004356660361 2074600004 000000002232269567 00 00 00

000000003402343994 000000002232269567 2074600000 00 00 00

00 00 00 2074600000 2074600000 00

00 00 00 00 00 00

00 00 00 00 00 2074600000

]]]

]

(68)




[[[[[[[

[

008621257414

01243595697

09884847438

00

00

00

]]]]]]]

]

[[[[[[[

[

08755641188

minus04828494241

minus001561753067

00

00

00

]]]]]]]

]

[[[[[

[

minus04753471023

minus08668282006

01505124700

00

00

00

]]]]]

]

[[[[[

[

00

00

00

10

00

00

]]]]]

]

[[[[[

[

00

00

00

00

10

00

]]]]]

]

[[[[[

[

00

00

00

00

00

10

]]]]]

]

(69)


[[[[[

[

008621257423

01243595698

09884847448

00

00

00

]]]]]

]

[[[[[

[

08755641188

minus04828494241

minus001561753067

00

00

00

]]]]]

]


[[[[[[[

[

minus04753471018

minus08668281997

01505124698

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(70)



[[[

[

1890099999 000000001209663993 minus000000002532733996 00 00 00000000001209663993 1890099991 0000000001701090138 00 00 00minus000000002532733996 0000000001701090138 1890100001 00 00 00

00 00 00 1890099997 1890099997 0000 00 00 00 00 0000 00 00 00 00 1890099997

]]]

]

(71)




[[[[[[[

[

03961520086

06338768023

06642768888

00

00

00

]]]]]]]

]

[[[[[[[

[

09160614623

minus02236797319

minus03328645006

00

00

00

]]]]]]]

]

[[[[[[[

[

006240981414

minus07403833993

06692812840

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(72)


[[[[[[[

[

03961520086

06338768023

06642768888

00

00

00

]]]]]]]

]

[[[[[[[

[

09160614614

minus02236797317

minus03328645003

00

00

00

]]]]]]]

]

[[[[[[[

[

006240981426

minus07403834008

06692812853

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(73)



[[[

[


00 00 00 8772299998 8772299998 0000 00 00 00 00 0000 00 00 00 00 8772299998

]]]

]

(74)




[[[[[[[

[

minus006591789198

minus008975775227

minus09937798435

00

00

00

]]]]]]]

]

[[[[[[[

[

minus08289381135

05593224991

0004466103673

00

00

00

]]]]]]]

]

[[[[[

[

minus05554425577

minus08240763851

01112729817

00

00

00

]]]]]

]

[[[[[

[

00

00

00

10

00

00

]]]]]

]

[[[[[

[

00

00

00

00

10

00

]]]]]

]

[[[[[

[

00

00

00

00

00

10

]]]]]

]


[[[[[[[

[

minus01387533798

minus02031928415

minus09692575354

00

00

00

]]]]]]]

]

[[[[[[[

[

minus08476774112

05304148448

001015375725

00

00

00

]]]]]]]

]


[[[[

[

minus05120454135

minus08230265872

02458388325

00

00

00

]]]]

]

[[[[

[

00

00

00

10

00

00

]]]]

]

[[[[

[

00

00

00

00

10

00

]]]]

]

[[[[

[

00

00

00

00

00

10

]]]]

]

(76)



[[[

[

2228057494 0000000004678921127 000000003654014285 00 00 000000000004678921127 2228057499 000000002397389829 00 00 00000000003654014285 000000002397389829 2228057495 00 00 00

00 00 00 2228057495 2228057495 0000 00 00 00 00 0000 00 00 00 00 2228057495

]]]

]

(77)




[[[[[[[

[

006975010637

01071019008

09917984199

00

00

00

]]]]]]]

]

[[[[[[[

[

09079000793

minus04187725641

minus001862756541

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04133429180

minus09017531389

01264472547

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(78)


[[[[[[[

[

006975010637

01071019008

09917984199

00

00

00

]]]]]]]

]

[[[[[[[

[

09079000793

minus04187725641

minus001862756541

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04133429184

minus09017531398

01264472548

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(79)



[[[

[


00 00 00 2598699998 2598699998 0000 00 00 00 00 0000 00 00 00 00 2598699998

]]]

]

(80)




[[[[[

[

minus00192522847250000

minus00192842313600000

000386151499999998

00

00

00

]]]]]

]

[[[[[

[

000961799224999998

00165029120500000

000436480214750000

00

00

00

]]]]]

]

[[[[[[

[

minus0494444050150000

minus0856906622200000

0142104790200000

00

00

00

]]]]]]

]

[[[[[[

[

00

00

00

10

00

00

]]]]]]

]

[[[[[[

[

00

00

00

00

10

00

]]]]]]

]

[[[[[[

[

00

00

00

00

00

10

]]]]]]

]

(81)



[[[[[[[[[[

[

minus2541745843

minus2545963537

5098090872

00

00

00

]]]]]]]]]]

]

[[[[[[[[[[

[

2505315863

4298714977

1136953303

00

00

00

]]]]]]]]]]

]

[[[[[[[

[

minus04949599718

minus08578007511

01422530677

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(82)



[[[

[

315679169478799 427324383657779 384557503470365 00 00 00427324383657779 618700481877479 889153535276175 00 00 00384557503470365 889153535276175 384768762708609 00 00 00

00 00 00 247795001400000 00 0000 00 00 00 247795001400000 0000 00 00 00 00 247795001400000

]]]

]

(83)




[[[[[[[

[

01135176976

01764907831

09777344911

00

00

00

]]]]]]]

]

[[[[[[[

[

08848520771

minus04654963442

minus001870707734

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04518302069

minus08672739791

02090103088

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(84)


[[[[[[[

[

01135176977

01764907833

09777344921

00

00

00

]]]]]]]

]

[[[[[[[

[

08848520771

minus04654963442

minus001870707734

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04518302069

minus08672739791

02090103088

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(85)



[[[

[

2663700003 000000004768023095 000000003169803038 00 00 00000000004768023095 2663699992 0000000008310743498 00 00 00000000003169803038 0000000008310743498 2663700001 00 00 00

00 00 00 2663699998 2663699998 0000 00 00 00 00 0000 00 00 00 00 2663699998

]]]

]

(86)





6 Conclusions









Disclosure




Acknowledgments



References




















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











120590kelvin

= 120590viogt

120585 120590voigt

=120590kelvin

120585

120598kelvin

= 120598voigt

120585 120598voigt

=120598kelvin

120585

(38)


119862kelvin

=((

(







))

)

119862voigt

(39)


(

(

12059011

12059022

12059033

radic212059023

radic212059013

radic212059012

)

)

= (

1198621111 1198621122 1198621133radic21198621123

radic21198621113radic21198621112

1198622211 1198622222 1198622333radic21198622223

radic21198622213radic21198622212

1198623311 1198623322 1198623333radic21198623323

radic21198623313radic21198623312

radic21198622311radic21198622322

radic21198622333 21198622323 21198622313 21198622312

radic21198621311radic21198621322

radic21198621333 21198621323 21198621313 21198621312

radic21198621211radic21198621222

radic21198621233 21198621223 21198621213 21198621212

)

lowast(

(

12059811

12059822

12059833

radic212059823

radic212059813

radic212059812

)

)

(40)


= C (41)



(

(

12059011

12059022

12059033

radic212059023

radic212059013

radic212059012

)

)

=((

(

11986211

11986212

11986213

radic211986214

radic211986215

radic211986216

11986212

11986222

11986223

radic211986224

radic211986225

radic211986226

11986213

11986223

11986233

radic211986234

radic211986235

radic211986236

radic211986214

radic211986224

radic211986234

211986244

211986245

211986246

radic211986215

radic211986225

radic211986235

211986245

211986255

211986256

radic211986216

radic211986226

radic211986236

211986246

211986256

211986266

))

)

lowast (

(

12059811

12059822

12059833

radic212059823

radic212059813

radic212059812

)

)

(42)


(

(

1

2

3

4

5

6

)

)

=((

(

11986211

11986212

11986213

11986214

11986215

11986216

11986212

11986222

11986223

11986224

11986225

11986226

11986213

11986223

11986233

11986234

11986235

11986236

11986214

11986224

11986234

11986244

11986245

11986246

11986215

11986225

11986235

11986245

11986255

11986256

11986216

11986226

11986236

11986246

11986256

11986266

))

)

lowast (

(

1205981

1205982

1205983

1205984

1205985

1205986

)

)

(43)



































01 02 03 04 05 06 07 08

05

1

15

2

25

3

(a)

01 02 03 04 05 06 07 08

05

1

15

2

25

3

(b)


01 02 03 04 05 06 07 08

2

4

6

8

10

12

14

16





ΛI) N = 0 (44)


119894 119894 = 1 2 6 and

















NNA119896

equiv1

119872

119872

sum

119884=1

N119884119896 (45)



NAVG119896

equiv JNANNA119896

(46)


119896=1NNA119896

otimes NNA119896

| that is

JNA equiv (

6

sum

119896=1

NNA119896

otimes NNA119896

)

minus12

(47)


119896 that is

ΛAVG119902

equiv1

119872

119872

sum

119884=1

6

sum

119896=1

Λ119884

119896(N119884

119896sdot NAVG

119902)2

(48)


gt 0 for all 119896 = 1 2 6

and ΛAVG119896


NA119896


ΛNA119896

equiv1

119872

119872

sum

119884=1

Λ119884

119896 (49)


CAVG=

6

sum

119896=1

ΛAVG119896

NAVG119896

otimes NAVG119896

(50)





[[[[[[[

[

00367834559700000

00453681054800000

0998185540700000

00

00

00

]]]]]]]

]

[[[[[[[

[

0983745905000000

minus0170879918750000

minus00277901141300000

00

00

00

]]]]]]]

]

[[[[[[[

[

minus0169284222800000

minus0982986098000000

00509905132200000

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(51)


[[[[[[[

[

003679134179

004537783172

09983995367

00

00

00

]]]]]]]

]

[[[[[[[

[

09859858256

minus01712690003

minus002785339027

00

00

00

]]]]]]]

]

[[[[[[[

[

minus01697052873

minus09854311019

005111734310

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(52)




[[[

[

334725276837330 0000112116752981933 000198542359441849 00 00 00

0000112116752981933 334773746006714 minus0000991802322788501 00 00 00

000198542359441849 minus0000991802322788501 334013593769726 00 00 00

00 00 00 333950000000000 00 00

00 00 00 00 333950000000000 00

00 00 00 00 00 333950000000000

]]]

]

(53)




[[[[[[[

[

004028666644

006399313350

09971368323

00

00

00

]]]]]]]

]

[[[[[[[

[

09048796003

minus04255671399

minus0009247686096

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04237568827

minus09026613361

007505076253

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(54)


[[[[[[[

[

004028666644

006399313350

09971368323

00

00

00

]]]]]]]

]

[[[[[[[

[

09048795994

minus04255671395

minus0009247686087

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04237568827

minus09026613361

007505076253

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(55)



[[[

[

1458799998 000000001785571215 minus000000001009489597 00 00 00000000001785571215 1458799997 0000000002407020197 00 00 00minus000000001009489597 0000000002407020197 1458799997 00 00 00

00 00 00 1458799998 00 0000 00 00 00 1458799998 0000 00 00 00 00 1458799998

]]]

]

(56)




[[[[[

[

005368516527

007220768570

09959437502

00

00

00

]]]]]

]

[[[[[

[

09266208315

minus03753089731

minus002273783078

00

00

00

]]]]]

]

[[[[[

[

minus03721447810

minus09240829102

008705767064

00

00

00

]]]]]

]

[[[[[

[

00

00

00

00

00

10

]]]]]

]

[[[[[

[

00

00

00

10

00

00

]]]]]

]

[[[[[

[

00

00

00

00

10

00

]]]]]

]

(57)


[[[[[[[[[

[

005368516527

007220768570

09959437502

00

00

00

]]]]]]]]]

]

[[[[[[[[[

[

09266208315

minus03753089731

minus002273783078

00

00

00

]]]]]]]]]

]

[[[[[[[[[

[

minus03721447806

minus09240829093

008705767055

00

00

00

]]]]]]]]]

]

[[[[[[[[[

[

00

00

00

10

00

00

]]]]]]]]]

]

[[[[[[[[[

[

00

00

00

00

10

00

]]]]]]]]]

]

[[[[[[[[[

[

00

00

00

00

00

10

]]]]]]]]]

]

(58)




[[[

[

1615299998 0000000005508173090 minus0000000008722620038 00 00 00

0000000005508173090 1615299996 minus0000000004345156817 00 00 00

minus0000000008722620038 minus0000000004345156817 1615300000 00 00 00

00 00 00 1615299998 00 00

00 00 00 00 1615299998 00

00 00 00 00 00 1615299998

]]]

]

(59)




[[[[[[[

[

minus00598757013800000

minus00768229223150000

minus0995231841750000

00

00

00

]]]]]]]

]

[[[[[[[

[

minus0869358919900000

0493939153600000

00141567750950000

00

00

00

]]]]]]]

]

[[[[[[[

[

minus0490490276500000

minus0866078136900000

00963811082600000

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(60)


[[[[[[[[[[[

[

minus005987730132

minus007682497510

minus09952584354

00

00

00

]]]]]]]]]]]

]

[[[[[[[[[[[

[

minus08693926232

04939583026

001415732393

00

00

00

]]]]]]]]]]]

]

[[[[[[[[[[[

[

00

00

00

10

00

00

]]]]]]]]]]]

]

[[[[[[[[[[[

[

00

00

00

00

10

00

]]]]]]]]]]]

]

[[[[[[[[[[[

[

00

00

00

00

00

10

]]]]]]]]]]]

]

(61)



[[[

[


00 00 00 1819400002 1819400002 0000 00 00 00 00 0000 00 00 00 00 1819400002

]]]

]

(62)




[[[[[[[

[

004967528691

008463937788

09951726186

00

00

00

]]]]]]]

]

[[[[[[[

[

09161715971

minus04006166011

minus001165943224

00

00

00

]]]]]]]

]

[[[[[

[

minus03976958243

minus09123280736

009744493938

00

00

00

]]]]]

]

[[[[[

[

00

00

00

10

00

00

]]]]]

]

[[[[[

[

00

00

00

00

10

00

]]]]]

]

[[[[[

[

00

00

00

00

00

10

]]]]]

]

(63)


[[[[[[[

[

004967528696

008463937796

09951726196

00

00

00

]]]]]]]

]

[[[[[[[

[

09161715980

minus04006166015

minus001165943225

00

00

00

]]]]]]]

]


[[[[[[[

[

minus03976958247

minus09123280745

009744493948

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(64)



[[[

[

1922399998 minus000000001176508811 minus000000001537920006 00 00 00

minus000000001176508811 1922400000 minus0000000007631928036 00 00 00

minus000000001537920006 minus0000000007631928036 1922400000 00 00 00

00 00 00 1922399998 1922399998 00

00 00 00 00 00 00

00 00 00 00 00 1922399998

]]]

]

(65)




[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

[[[[[[[

[

minus01387533797

minus02031928413

minus09692575344

00

00

00

]]]]]]]

]

[[[[[[[

[

minus08476774112

05304148448

001015375725

00

00

00

]]]]]]]

]

[[[[[[[

[

minus05120454135

minus08230265872

02458388325

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

(66)


[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

[[[[[[[

[

minus01387533798

minus02031928415

minus09692575354

00

00

00

]]]]]]]

]

[[[[[[[

[

minus08476774112

05304148448

001015375725

00

00

00

]]]]]]]

]

[[[[[[[

[

minus05120454135

minus08230265872

02458388325

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

(67)



[[[

[

2074599999 0000000004356660361 000000003402343994 00 00 00

0000000004356660361 2074600004 000000002232269567 00 00 00

000000003402343994 000000002232269567 2074600000 00 00 00

00 00 00 2074600000 2074600000 00

00 00 00 00 00 00

00 00 00 00 00 2074600000

]]]

]

(68)




[[[[[[[

[

008621257414

01243595697

09884847438

00

00

00

]]]]]]]

]

[[[[[[[

[

08755641188

minus04828494241

minus001561753067

00

00

00

]]]]]]]

]

[[[[[

[

minus04753471023

minus08668282006

01505124700

00

00

00

]]]]]

]

[[[[[

[

00

00

00

10

00

00

]]]]]

]

[[[[[

[

00

00

00

00

10

00

]]]]]

]

[[[[[

[

00

00

00

00

00

10

]]]]]

]

(69)


[[[[[

[

008621257423

01243595698

09884847448

00

00

00

]]]]]

]

[[[[[

[

08755641188

minus04828494241

minus001561753067

00

00

00

]]]]]

]


[[[[[[[

[

minus04753471018

minus08668281997

01505124698

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(70)



[[[

[

1890099999 000000001209663993 minus000000002532733996 00 00 00000000001209663993 1890099991 0000000001701090138 00 00 00minus000000002532733996 0000000001701090138 1890100001 00 00 00

00 00 00 1890099997 1890099997 0000 00 00 00 00 0000 00 00 00 00 1890099997

]]]

]

(71)




[[[[[[[

[

03961520086

06338768023

06642768888

00

00

00

]]]]]]]

]

[[[[[[[

[

09160614623

minus02236797319

minus03328645006

00

00

00

]]]]]]]

]

[[[[[[[

[

006240981414

minus07403833993

06692812840

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(72)


[[[[[[[

[

03961520086

06338768023

06642768888

00

00

00

]]]]]]]

]

[[[[[[[

[

09160614614

minus02236797317

minus03328645003

00

00

00

]]]]]]]

]

[[[[[[[

[

006240981426

minus07403834008

06692812853

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(73)



[[[

[


00 00 00 8772299998 8772299998 0000 00 00 00 00 0000 00 00 00 00 8772299998

]]]

]

(74)




[[[[[[[

[

minus006591789198

minus008975775227

minus09937798435

00

00

00

]]]]]]]

]

[[[[[[[

[

minus08289381135

05593224991

0004466103673

00

00

00

]]]]]]]

]

[[[[[

[

minus05554425577

minus08240763851

01112729817

00

00

00

]]]]]

]

[[[[[

[

00

00

00

10

00

00

]]]]]

]

[[[[[

[

00

00

00

00

10

00

]]]]]

]

[[[[[

[

00

00

00

00

00

10

]]]]]

]


[[[[[[[

[

minus01387533798

minus02031928415

minus09692575354

00

00

00

]]]]]]]

]

[[[[[[[

[

minus08476774112

05304148448

001015375725

00

00

00

]]]]]]]

]


[[[[

[

minus05120454135

minus08230265872

02458388325

00

00

00

]]]]

]

[[[[

[

00

00

00

10

00

00

]]]]

]

[[[[

[

00

00

00

00

10

00

]]]]

]

[[[[

[

00

00

00

00

00

10

]]]]

]

(76)



[[[

[

2228057494 0000000004678921127 000000003654014285 00 00 000000000004678921127 2228057499 000000002397389829 00 00 00000000003654014285 000000002397389829 2228057495 00 00 00

00 00 00 2228057495 2228057495 0000 00 00 00 00 0000 00 00 00 00 2228057495

]]]

]

(77)




[[[[[[[

[

006975010637

01071019008

09917984199

00

00

00

]]]]]]]

]

[[[[[[[

[

09079000793

minus04187725641

minus001862756541

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04133429180

minus09017531389

01264472547

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(78)


[[[[[[[

[

006975010637

01071019008

09917984199

00

00

00

]]]]]]]

]

[[[[[[[

[

09079000793

minus04187725641

minus001862756541

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04133429184

minus09017531398

01264472548

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(79)



[[[

[


00 00 00 2598699998 2598699998 0000 00 00 00 00 0000 00 00 00 00 2598699998

]]]

]

(80)




[[[[[

[

minus00192522847250000

minus00192842313600000

000386151499999998

00

00

00

]]]]]

]

[[[[[

[

000961799224999998

00165029120500000

000436480214750000

00

00

00

]]]]]

]

[[[[[[

[

minus0494444050150000

minus0856906622200000

0142104790200000

00

00

00

]]]]]]

]

[[[[[[

[

00

00

00

10

00

00

]]]]]]

]

[[[[[[

[

00

00

00

00

10

00

]]]]]]

]

[[[[[[

[

00

00

00

00

00

10

]]]]]]

]

(81)



[[[[[[[[[[

[

minus2541745843

minus2545963537

5098090872

00

00

00

]]]]]]]]]]

]

[[[[[[[[[[

[

2505315863

4298714977

1136953303

00

00

00

]]]]]]]]]]

]

[[[[[[[

[

minus04949599718

minus08578007511

01422530677

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(82)



[[[

[

315679169478799 427324383657779 384557503470365 00 00 00427324383657779 618700481877479 889153535276175 00 00 00384557503470365 889153535276175 384768762708609 00 00 00

00 00 00 247795001400000 00 0000 00 00 00 247795001400000 0000 00 00 00 00 247795001400000

]]]

]

(83)




[[[[[[[

[

01135176976

01764907831

09777344911

00

00

00

]]]]]]]

]

[[[[[[[

[

08848520771

minus04654963442

minus001870707734

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04518302069

minus08672739791

02090103088

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(84)


[[[[[[[

[

01135176977

01764907833

09777344921

00

00

00

]]]]]]]

]

[[[[[[[

[

08848520771

minus04654963442

minus001870707734

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04518302069

minus08672739791

02090103088

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(85)



[[[

[

2663700003 000000004768023095 000000003169803038 00 00 00000000004768023095 2663699992 0000000008310743498 00 00 00000000003169803038 0000000008310743498 2663700001 00 00 00

00 00 00 2663699998 2663699998 0000 00 00 00 00 0000 00 00 00 00 2663699998

]]]

]

(86)





6 Conclusions









Disclosure




Acknowledgments



References




















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation








































01 02 03 04 05 06 07 08

05

1

15

2

25

3

(a)

01 02 03 04 05 06 07 08

05

1

15

2

25

3

(b)


01 02 03 04 05 06 07 08

2

4

6

8

10

12

14

16





ΛI) N = 0 (44)


119894 119894 = 1 2 6 and

















NNA119896

equiv1

119872

119872

sum

119884=1

N119884119896 (45)



NAVG119896

equiv JNANNA119896

(46)


119896=1NNA119896

otimes NNA119896

| that is

JNA equiv (

6

sum

119896=1

NNA119896

otimes NNA119896

)

minus12

(47)


119896 that is

ΛAVG119902

equiv1

119872

119872

sum

119884=1

6

sum

119896=1

Λ119884

119896(N119884

119896sdot NAVG

119902)2

(48)


gt 0 for all 119896 = 1 2 6

and ΛAVG119896


NA119896


ΛNA119896

equiv1

119872

119872

sum

119884=1

Λ119884

119896 (49)


CAVG=

6

sum

119896=1

ΛAVG119896

NAVG119896

otimes NAVG119896

(50)





[[[[[[[

[

00367834559700000

00453681054800000

0998185540700000

00

00

00

]]]]]]]

]

[[[[[[[

[

0983745905000000

minus0170879918750000

minus00277901141300000

00

00

00

]]]]]]]

]

[[[[[[[

[

minus0169284222800000

minus0982986098000000

00509905132200000

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(51)


[[[[[[[

[

003679134179

004537783172

09983995367

00

00

00

]]]]]]]

]

[[[[[[[

[

09859858256

minus01712690003

minus002785339027

00

00

00

]]]]]]]

]

[[[[[[[

[

minus01697052873

minus09854311019

005111734310

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(52)




[[[

[

334725276837330 0000112116752981933 000198542359441849 00 00 00

0000112116752981933 334773746006714 minus0000991802322788501 00 00 00

000198542359441849 minus0000991802322788501 334013593769726 00 00 00

00 00 00 333950000000000 00 00

00 00 00 00 333950000000000 00

00 00 00 00 00 333950000000000

]]]

]

(53)




[[[[[[[

[

004028666644

006399313350

09971368323

00

00

00

]]]]]]]

]

[[[[[[[

[

09048796003

minus04255671399

minus0009247686096

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04237568827

minus09026613361

007505076253

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(54)


[[[[[[[

[

004028666644

006399313350

09971368323

00

00

00

]]]]]]]

]

[[[[[[[

[

09048795994

minus04255671395

minus0009247686087

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04237568827

minus09026613361

007505076253

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(55)



[[[

[

1458799998 000000001785571215 minus000000001009489597 00 00 00000000001785571215 1458799997 0000000002407020197 00 00 00minus000000001009489597 0000000002407020197 1458799997 00 00 00

00 00 00 1458799998 00 0000 00 00 00 1458799998 0000 00 00 00 00 1458799998

]]]

]

(56)




[[[[[

[

005368516527

007220768570

09959437502

00

00

00

]]]]]

]

[[[[[

[

09266208315

minus03753089731

minus002273783078

00

00

00

]]]]]

]

[[[[[

[

minus03721447810

minus09240829102

008705767064

00

00

00

]]]]]

]

[[[[[

[

00

00

00

00

00

10

]]]]]

]

[[[[[

[

00

00

00

10

00

00

]]]]]

]

[[[[[

[

00

00

00

00

10

00

]]]]]

]

(57)


[[[[[[[[[

[

005368516527

007220768570

09959437502

00

00

00

]]]]]]]]]

]

[[[[[[[[[

[

09266208315

minus03753089731

minus002273783078

00

00

00

]]]]]]]]]

]

[[[[[[[[[

[

minus03721447806

minus09240829093

008705767055

00

00

00

]]]]]]]]]

]

[[[[[[[[[

[

00

00

00

10

00

00

]]]]]]]]]

]

[[[[[[[[[

[

00

00

00

00

10

00

]]]]]]]]]

]

[[[[[[[[[

[

00

00

00

00

00

10

]]]]]]]]]

]

(58)




[[[

[

1615299998 0000000005508173090 minus0000000008722620038 00 00 00

0000000005508173090 1615299996 minus0000000004345156817 00 00 00

minus0000000008722620038 minus0000000004345156817 1615300000 00 00 00

00 00 00 1615299998 00 00

00 00 00 00 1615299998 00

00 00 00 00 00 1615299998

]]]

]

(59)




[[[[[[[

[

minus00598757013800000

minus00768229223150000

minus0995231841750000

00

00

00

]]]]]]]

]

[[[[[[[

[

minus0869358919900000

0493939153600000

00141567750950000

00

00

00

]]]]]]]

]

[[[[[[[

[

minus0490490276500000

minus0866078136900000

00963811082600000

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(60)


[[[[[[[[[[[

[

minus005987730132

minus007682497510

minus09952584354

00

00

00

]]]]]]]]]]]

]

[[[[[[[[[[[

[

minus08693926232

04939583026

001415732393

00

00

00

]]]]]]]]]]]

]

[[[[[[[[[[[

[

00

00

00

10

00

00

]]]]]]]]]]]

]

[[[[[[[[[[[

[

00

00

00

00

10

00

]]]]]]]]]]]

]

[[[[[[[[[[[

[

00

00

00

00

00

10

]]]]]]]]]]]

]

(61)



[[[

[


00 00 00 1819400002 1819400002 0000 00 00 00 00 0000 00 00 00 00 1819400002

]]]

]

(62)




[[[[[[[

[

004967528691

008463937788

09951726186

00

00

00

]]]]]]]

]

[[[[[[[

[

09161715971

minus04006166011

minus001165943224

00

00

00

]]]]]]]

]

[[[[[

[

minus03976958243

minus09123280736

009744493938

00

00

00

]]]]]

]

[[[[[

[

00

00

00

10

00

00

]]]]]

]

[[[[[

[

00

00

00

00

10

00

]]]]]

]

[[[[[

[

00

00

00

00

00

10

]]]]]

]

(63)


[[[[[[[

[

004967528696

008463937796

09951726196

00

00

00

]]]]]]]

]

[[[[[[[

[

09161715980

minus04006166015

minus001165943225

00

00

00

]]]]]]]

]


[[[[[[[

[

minus03976958247

minus09123280745

009744493948

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(64)



[[[

[

1922399998 minus000000001176508811 minus000000001537920006 00 00 00

minus000000001176508811 1922400000 minus0000000007631928036 00 00 00

minus000000001537920006 minus0000000007631928036 1922400000 00 00 00

00 00 00 1922399998 1922399998 00

00 00 00 00 00 00

00 00 00 00 00 1922399998

]]]

]

(65)




[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

[[[[[[[

[

minus01387533797

minus02031928413

minus09692575344

00

00

00

]]]]]]]

]

[[[[[[[

[

minus08476774112

05304148448

001015375725

00

00

00

]]]]]]]

]

[[[[[[[

[

minus05120454135

minus08230265872

02458388325

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

(66)


[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

[[[[[[[

[

minus01387533798

minus02031928415

minus09692575354

00

00

00

]]]]]]]

]

[[[[[[[

[

minus08476774112

05304148448

001015375725

00

00

00

]]]]]]]

]

[[[[[[[

[

minus05120454135

minus08230265872

02458388325

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

(67)



[[[

[

2074599999 0000000004356660361 000000003402343994 00 00 00

0000000004356660361 2074600004 000000002232269567 00 00 00

000000003402343994 000000002232269567 2074600000 00 00 00

00 00 00 2074600000 2074600000 00

00 00 00 00 00 00

00 00 00 00 00 2074600000

]]]

]

(68)




[[[[[[[

[

008621257414

01243595697

09884847438

00

00

00

]]]]]]]

]

[[[[[[[

[

08755641188

minus04828494241

minus001561753067

00

00

00

]]]]]]]

]

[[[[[

[

minus04753471023

minus08668282006

01505124700

00

00

00

]]]]]

]

[[[[[

[

00

00

00

10

00

00

]]]]]

]

[[[[[

[

00

00

00

00

10

00

]]]]]

]

[[[[[

[

00

00

00

00

00

10

]]]]]

]

(69)


[[[[[

[

008621257423

01243595698

09884847448

00

00

00

]]]]]

]

[[[[[

[

08755641188

minus04828494241

minus001561753067

00

00

00

]]]]]

]


[[[[[[[

[

minus04753471018

minus08668281997

01505124698

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(70)



[[[

[

1890099999 000000001209663993 minus000000002532733996 00 00 00000000001209663993 1890099991 0000000001701090138 00 00 00minus000000002532733996 0000000001701090138 1890100001 00 00 00

00 00 00 1890099997 1890099997 0000 00 00 00 00 0000 00 00 00 00 1890099997

]]]

]

(71)




[[[[[[[

[

03961520086

06338768023

06642768888

00

00

00

]]]]]]]

]

[[[[[[[

[

09160614623

minus02236797319

minus03328645006

00

00

00

]]]]]]]

]

[[[[[[[

[

006240981414

minus07403833993

06692812840

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(72)


[[[[[[[

[

03961520086

06338768023

06642768888

00

00

00

]]]]]]]

]

[[[[[[[

[

09160614614

minus02236797317

minus03328645003

00

00

00

]]]]]]]

]

[[[[[[[

[

006240981426

minus07403834008

06692812853

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(73)



[[[

[


00 00 00 8772299998 8772299998 0000 00 00 00 00 0000 00 00 00 00 8772299998

]]]

]

(74)




[[[[[[[

[

minus006591789198

minus008975775227

minus09937798435

00

00

00

]]]]]]]

]

[[[[[[[

[

minus08289381135

05593224991

0004466103673

00

00

00

]]]]]]]

]

[[[[[

[

minus05554425577

minus08240763851

01112729817

00

00

00

]]]]]

]

[[[[[

[

00

00

00

10

00

00

]]]]]

]

[[[[[

[

00

00

00

00

10

00

]]]]]

]

[[[[[

[

00

00

00

00

00

10

]]]]]

]


[[[[[[[

[

minus01387533798

minus02031928415

minus09692575354

00

00

00

]]]]]]]

]

[[[[[[[

[

minus08476774112

05304148448

001015375725

00

00

00

]]]]]]]

]


[[[[

[

minus05120454135

minus08230265872

02458388325

00

00

00

]]]]

]

[[[[

[

00

00

00

10

00

00

]]]]

]

[[[[

[

00

00

00

00

10

00

]]]]

]

[[[[

[

00

00

00

00

00

10

]]]]

]

(76)



[[[

[

2228057494 0000000004678921127 000000003654014285 00 00 000000000004678921127 2228057499 000000002397389829 00 00 00000000003654014285 000000002397389829 2228057495 00 00 00

00 00 00 2228057495 2228057495 0000 00 00 00 00 0000 00 00 00 00 2228057495

]]]

]

(77)




[[[[[[[

[

006975010637

01071019008

09917984199

00

00

00

]]]]]]]

]

[[[[[[[

[

09079000793

minus04187725641

minus001862756541

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04133429180

minus09017531389

01264472547

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(78)


[[[[[[[

[

006975010637

01071019008

09917984199

00

00

00

]]]]]]]

]

[[[[[[[

[

09079000793

minus04187725641

minus001862756541

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04133429184

minus09017531398

01264472548

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(79)



[[[

[


00 00 00 2598699998 2598699998 0000 00 00 00 00 0000 00 00 00 00 2598699998

]]]

]

(80)




[[[[[

[

minus00192522847250000

minus00192842313600000

000386151499999998

00

00

00

]]]]]

]

[[[[[

[

000961799224999998

00165029120500000

000436480214750000

00

00

00

]]]]]

]

[[[[[[

[

minus0494444050150000

minus0856906622200000

0142104790200000

00

00

00

]]]]]]

]

[[[[[[

[

00

00

00

10

00

00

]]]]]]

]

[[[[[[

[

00

00

00

00

10

00

]]]]]]

]

[[[[[[

[

00

00

00

00

00

10

]]]]]]

]

(81)



[[[[[[[[[[

[

minus2541745843

minus2545963537

5098090872

00

00

00

]]]]]]]]]]

]

[[[[[[[[[[

[

2505315863

4298714977

1136953303

00

00

00

]]]]]]]]]]

]

[[[[[[[

[

minus04949599718

minus08578007511

01422530677

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(82)



[[[

[

315679169478799 427324383657779 384557503470365 00 00 00427324383657779 618700481877479 889153535276175 00 00 00384557503470365 889153535276175 384768762708609 00 00 00

00 00 00 247795001400000 00 0000 00 00 00 247795001400000 0000 00 00 00 00 247795001400000

]]]

]

(83)




[[[[[[[

[

01135176976

01764907831

09777344911

00

00

00

]]]]]]]

]

[[[[[[[

[

08848520771

minus04654963442

minus001870707734

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04518302069

minus08672739791

02090103088

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(84)


[[[[[[[

[

01135176977

01764907833

09777344921

00

00

00

]]]]]]]

]

[[[[[[[

[

08848520771

minus04654963442

minus001870707734

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04518302069

minus08672739791

02090103088

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(85)



[[[

[

2663700003 000000004768023095 000000003169803038 00 00 00000000004768023095 2663699992 0000000008310743498 00 00 00000000003169803038 0000000008310743498 2663700001 00 00 00

00 00 00 2663699998 2663699998 0000 00 00 00 00 0000 00 00 00 00 2663699998

]]]

]

(86)





6 Conclusions









Disclosure




Acknowledgments



References




















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation

















NNA119896

equiv1

119872

119872

sum

119884=1

N119884119896 (45)



NAVG119896

equiv JNANNA119896

(46)


119896=1NNA119896

otimes NNA119896

| that is

JNA equiv (

6

sum

119896=1

NNA119896

otimes NNA119896

)

minus12

(47)


119896 that is

ΛAVG119902

equiv1

119872

119872

sum

119884=1

6

sum

119896=1

Λ119884

119896(N119884

119896sdot NAVG

119902)2

(48)


gt 0 for all 119896 = 1 2 6

and ΛAVG119896


NA119896


ΛNA119896

equiv1

119872

119872

sum

119884=1

Λ119884

119896 (49)


CAVG=

6

sum

119896=1

ΛAVG119896

NAVG119896

otimes NAVG119896

(50)





[[[[[[[

[

00367834559700000

00453681054800000

0998185540700000

00

00

00

]]]]]]]

]

[[[[[[[

[

0983745905000000

minus0170879918750000

minus00277901141300000

00

00

00

]]]]]]]

]

[[[[[[[

[

minus0169284222800000

minus0982986098000000

00509905132200000

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(51)


[[[[[[[

[

003679134179

004537783172

09983995367

00

00

00

]]]]]]]

]

[[[[[[[

[

09859858256

minus01712690003

minus002785339027

00

00

00

]]]]]]]

]

[[[[[[[

[

minus01697052873

minus09854311019

005111734310

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(52)




[[[

[

334725276837330 0000112116752981933 000198542359441849 00 00 00

0000112116752981933 334773746006714 minus0000991802322788501 00 00 00

000198542359441849 minus0000991802322788501 334013593769726 00 00 00

00 00 00 333950000000000 00 00

00 00 00 00 333950000000000 00

00 00 00 00 00 333950000000000

]]]

]

(53)




[[[[[[[

[

004028666644

006399313350

09971368323

00

00

00

]]]]]]]

]

[[[[[[[

[

09048796003

minus04255671399

minus0009247686096

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04237568827

minus09026613361

007505076253

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(54)


[[[[[[[

[

004028666644

006399313350

09971368323

00

00

00

]]]]]]]

]

[[[[[[[

[

09048795994

minus04255671395

minus0009247686087

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04237568827

minus09026613361

007505076253

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(55)



[[[

[

1458799998 000000001785571215 minus000000001009489597 00 00 00000000001785571215 1458799997 0000000002407020197 00 00 00minus000000001009489597 0000000002407020197 1458799997 00 00 00

00 00 00 1458799998 00 0000 00 00 00 1458799998 0000 00 00 00 00 1458799998

]]]

]

(56)




[[[[[

[

005368516527

007220768570

09959437502

00

00

00

]]]]]

]

[[[[[

[

09266208315

minus03753089731

minus002273783078

00

00

00

]]]]]

]

[[[[[

[

minus03721447810

minus09240829102

008705767064

00

00

00

]]]]]

]

[[[[[

[

00

00

00

00

00

10

]]]]]

]

[[[[[

[

00

00

00

10

00

00

]]]]]

]

[[[[[

[

00

00

00

00

10

00

]]]]]

]

(57)


[[[[[[[[[

[

005368516527

007220768570

09959437502

00

00

00

]]]]]]]]]

]

[[[[[[[[[

[

09266208315

minus03753089731

minus002273783078

00

00

00

]]]]]]]]]

]

[[[[[[[[[

[

minus03721447806

minus09240829093

008705767055

00

00

00

]]]]]]]]]

]

[[[[[[[[[

[

00

00

00

10

00

00

]]]]]]]]]

]

[[[[[[[[[

[

00

00

00

00

10

00

]]]]]]]]]

]

[[[[[[[[[

[

00

00

00

00

00

10

]]]]]]]]]

]

(58)




[[[

[

1615299998 0000000005508173090 minus0000000008722620038 00 00 00

0000000005508173090 1615299996 minus0000000004345156817 00 00 00

minus0000000008722620038 minus0000000004345156817 1615300000 00 00 00

00 00 00 1615299998 00 00

00 00 00 00 1615299998 00

00 00 00 00 00 1615299998

]]]

]

(59)




[[[[[[[

[

minus00598757013800000

minus00768229223150000

minus0995231841750000

00

00

00

]]]]]]]

]

[[[[[[[

[

minus0869358919900000

0493939153600000

00141567750950000

00

00

00

]]]]]]]

]

[[[[[[[

[

minus0490490276500000

minus0866078136900000

00963811082600000

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(60)


[[[[[[[[[[[

[

minus005987730132

minus007682497510

minus09952584354

00

00

00

]]]]]]]]]]]

]

[[[[[[[[[[[

[

minus08693926232

04939583026

001415732393

00

00

00

]]]]]]]]]]]

]

[[[[[[[[[[[

[

00

00

00

10

00

00

]]]]]]]]]]]

]

[[[[[[[[[[[

[

00

00

00

00

10

00

]]]]]]]]]]]

]

[[[[[[[[[[[

[

00

00

00

00

00

10

]]]]]]]]]]]

]

(61)



[[[

[


00 00 00 1819400002 1819400002 0000 00 00 00 00 0000 00 00 00 00 1819400002

]]]

]

(62)




[[[[[[[

[

004967528691

008463937788

09951726186

00

00

00

]]]]]]]

]

[[[[[[[

[

09161715971

minus04006166011

minus001165943224

00

00

00

]]]]]]]

]

[[[[[

[

minus03976958243

minus09123280736

009744493938

00

00

00

]]]]]

]

[[[[[

[

00

00

00

10

00

00

]]]]]

]

[[[[[

[

00

00

00

00

10

00

]]]]]

]

[[[[[

[

00

00

00

00

00

10

]]]]]

]

(63)


[[[[[[[

[

004967528696

008463937796

09951726196

00

00

00

]]]]]]]

]

[[[[[[[

[

09161715980

minus04006166015

minus001165943225

00

00

00

]]]]]]]

]


[[[[[[[

[

minus03976958247

minus09123280745

009744493948

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(64)



[[[

[

1922399998 minus000000001176508811 minus000000001537920006 00 00 00

minus000000001176508811 1922400000 minus0000000007631928036 00 00 00

minus000000001537920006 minus0000000007631928036 1922400000 00 00 00

00 00 00 1922399998 1922399998 00

00 00 00 00 00 00

00 00 00 00 00 1922399998

]]]

]

(65)




[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

[[[[[[[

[

minus01387533797

minus02031928413

minus09692575344

00

00

00

]]]]]]]

]

[[[[[[[

[

minus08476774112

05304148448

001015375725

00

00

00

]]]]]]]

]

[[[[[[[

[

minus05120454135

minus08230265872

02458388325

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

(66)


[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

[[[[[[[

[

minus01387533798

minus02031928415

minus09692575354

00

00

00

]]]]]]]

]

[[[[[[[

[

minus08476774112

05304148448

001015375725

00

00

00

]]]]]]]

]

[[[[[[[

[

minus05120454135

minus08230265872

02458388325

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

(67)



[[[

[

2074599999 0000000004356660361 000000003402343994 00 00 00

0000000004356660361 2074600004 000000002232269567 00 00 00

000000003402343994 000000002232269567 2074600000 00 00 00

00 00 00 2074600000 2074600000 00

00 00 00 00 00 00

00 00 00 00 00 2074600000

]]]

]

(68)




[[[[[[[

[

008621257414

01243595697

09884847438

00

00

00

]]]]]]]

]

[[[[[[[

[

08755641188

minus04828494241

minus001561753067

00

00

00

]]]]]]]

]

[[[[[

[

minus04753471023

minus08668282006

01505124700

00

00

00

]]]]]

]

[[[[[

[

00

00

00

10

00

00

]]]]]

]

[[[[[

[

00

00

00

00

10

00

]]]]]

]

[[[[[

[

00

00

00

00

00

10

]]]]]

]

(69)


[[[[[

[

008621257423

01243595698

09884847448

00

00

00

]]]]]

]

[[[[[

[

08755641188

minus04828494241

minus001561753067

00

00

00

]]]]]

]


[[[[[[[

[

minus04753471018

minus08668281997

01505124698

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(70)



[[[

[

1890099999 000000001209663993 minus000000002532733996 00 00 00000000001209663993 1890099991 0000000001701090138 00 00 00minus000000002532733996 0000000001701090138 1890100001 00 00 00

00 00 00 1890099997 1890099997 0000 00 00 00 00 0000 00 00 00 00 1890099997

]]]

]

(71)




[[[[[[[

[

03961520086

06338768023

06642768888

00

00

00

]]]]]]]

]

[[[[[[[

[

09160614623

minus02236797319

minus03328645006

00

00

00

]]]]]]]

]

[[[[[[[

[

006240981414

minus07403833993

06692812840

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(72)


[[[[[[[

[

03961520086

06338768023

06642768888

00

00

00

]]]]]]]

]

[[[[[[[

[

09160614614

minus02236797317

minus03328645003

00

00

00

]]]]]]]

]

[[[[[[[

[

006240981426

minus07403834008

06692812853

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(73)



[[[

[


00 00 00 8772299998 8772299998 0000 00 00 00 00 0000 00 00 00 00 8772299998

]]]

]

(74)




[[[[[[[

[

minus006591789198

minus008975775227

minus09937798435

00

00

00

]]]]]]]

]

[[[[[[[

[

minus08289381135

05593224991

0004466103673

00

00

00

]]]]]]]

]

[[[[[

[

minus05554425577

minus08240763851

01112729817

00

00

00

]]]]]

]

[[[[[

[

00

00

00

10

00

00

]]]]]

]

[[[[[

[

00

00

00

00

10

00

]]]]]

]

[[[[[

[

00

00

00

00

00

10

]]]]]

]


[[[[[[[

[

minus01387533798

minus02031928415

minus09692575354

00

00

00

]]]]]]]

]

[[[[[[[

[

minus08476774112

05304148448

001015375725

00

00

00

]]]]]]]

]


[[[[

[

minus05120454135

minus08230265872

02458388325

00

00

00

]]]]

]

[[[[

[

00

00

00

10

00

00

]]]]

]

[[[[

[

00

00

00

00

10

00

]]]]

]

[[[[

[

00

00

00

00

00

10

]]]]

]

(76)



[[[

[

2228057494 0000000004678921127 000000003654014285 00 00 000000000004678921127 2228057499 000000002397389829 00 00 00000000003654014285 000000002397389829 2228057495 00 00 00

00 00 00 2228057495 2228057495 0000 00 00 00 00 0000 00 00 00 00 2228057495

]]]

]

(77)




[[[[[[[

[

006975010637

01071019008

09917984199

00

00

00

]]]]]]]

]

[[[[[[[

[

09079000793

minus04187725641

minus001862756541

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04133429180

minus09017531389

01264472547

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(78)


[[[[[[[

[

006975010637

01071019008

09917984199

00

00

00

]]]]]]]

]

[[[[[[[

[

09079000793

minus04187725641

minus001862756541

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04133429184

minus09017531398

01264472548

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(79)



[[[

[


00 00 00 2598699998 2598699998 0000 00 00 00 00 0000 00 00 00 00 2598699998

]]]

]

(80)




[[[[[

[

minus00192522847250000

minus00192842313600000

000386151499999998

00

00

00

]]]]]

]

[[[[[

[

000961799224999998

00165029120500000

000436480214750000

00

00

00

]]]]]

]

[[[[[[

[

minus0494444050150000

minus0856906622200000

0142104790200000

00

00

00

]]]]]]

]

[[[[[[

[

00

00

00

10

00

00

]]]]]]

]

[[[[[[

[

00

00

00

00

10

00

]]]]]]

]

[[[[[[

[

00

00

00

00

00

10

]]]]]]

]

(81)



[[[[[[[[[[

[

minus2541745843

minus2545963537

5098090872

00

00

00

]]]]]]]]]]

]

[[[[[[[[[[

[

2505315863

4298714977

1136953303

00

00

00

]]]]]]]]]]

]

[[[[[[[

[

minus04949599718

minus08578007511

01422530677

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(82)



[[[

[

315679169478799 427324383657779 384557503470365 00 00 00427324383657779 618700481877479 889153535276175 00 00 00384557503470365 889153535276175 384768762708609 00 00 00

00 00 00 247795001400000 00 0000 00 00 00 247795001400000 0000 00 00 00 00 247795001400000

]]]

]

(83)




[[[[[[[

[

01135176976

01764907831

09777344911

00

00

00

]]]]]]]

]

[[[[[[[

[

08848520771

minus04654963442

minus001870707734

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04518302069

minus08672739791

02090103088

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(84)


[[[[[[[

[

01135176977

01764907833

09777344921

00

00

00

]]]]]]]

]

[[[[[[[

[

08848520771

minus04654963442

minus001870707734

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04518302069

minus08672739791

02090103088

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(85)



[[[

[

2663700003 000000004768023095 000000003169803038 00 00 00000000004768023095 2663699992 0000000008310743498 00 00 00000000003169803038 0000000008310743498 2663700001 00 00 00

00 00 00 2663699998 2663699998 0000 00 00 00 00 0000 00 00 00 00 2663699998

]]]

]

(86)





6 Conclusions









Disclosure




Acknowledgments



References




















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










[[[

[

334725276837330 0000112116752981933 000198542359441849 00 00 00

0000112116752981933 334773746006714 minus0000991802322788501 00 00 00

000198542359441849 minus0000991802322788501 334013593769726 00 00 00

00 00 00 333950000000000 00 00

00 00 00 00 333950000000000 00

00 00 00 00 00 333950000000000

]]]

]

(53)




[[[[[[[

[

004028666644

006399313350

09971368323

00

00

00

]]]]]]]

]

[[[[[[[

[

09048796003

minus04255671399

minus0009247686096

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04237568827

minus09026613361

007505076253

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(54)


[[[[[[[

[

004028666644

006399313350

09971368323

00

00

00

]]]]]]]

]

[[[[[[[

[

09048795994

minus04255671395

minus0009247686087

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04237568827

minus09026613361

007505076253

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(55)



[[[

[

1458799998 000000001785571215 minus000000001009489597 00 00 00000000001785571215 1458799997 0000000002407020197 00 00 00minus000000001009489597 0000000002407020197 1458799997 00 00 00

00 00 00 1458799998 00 0000 00 00 00 1458799998 0000 00 00 00 00 1458799998

]]]

]

(56)




[[[[[

[

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007220768570

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00

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]

[[[[[

[

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[[[[[

[

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008705767064

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]

[[[[[

[

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]

[[[[[

[

00

00

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]

[[[[[

[

00

00

00

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10

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]]]]]

]

(57)


[[[[[[[[[

[

005368516527

007220768570

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00

00

00

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]

[[[[[[[[[

[

09266208315

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00

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]

[[[[[[[[[

[

minus03721447806

minus09240829093

008705767055

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00

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]

[[[[[[[[[

[

00

00

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10

00

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]

[[[[[[[[[

[

00

00

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10

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]

[[[[[[[[[

[

00

00

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00

00

10

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]

(58)




[[[

[

1615299998 0000000005508173090 minus0000000008722620038 00 00 00

0000000005508173090 1615299996 minus0000000004345156817 00 00 00

minus0000000008722620038 minus0000000004345156817 1615300000 00 00 00

00 00 00 1615299998 00 00

00 00 00 00 1615299998 00

00 00 00 00 00 1615299998

]]]

]

(59)




[[[[[[[

[

minus00598757013800000

minus00768229223150000

minus0995231841750000

00

00

00

]]]]]]]

]

[[[[[[[

[

minus0869358919900000

0493939153600000

00141567750950000

00

00

00

]]]]]]]

]

[[[[[[[

[

minus0490490276500000

minus0866078136900000

00963811082600000

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

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10

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]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(60)


[[[[[[[[[[[

[

minus005987730132

minus007682497510

minus09952584354

00

00

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]

[[[[[[[[[[[

[

minus08693926232

04939583026

001415732393

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]

[[[[[[[[[[[

[

00

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10

00

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]

[[[[[[[[[[[

[

00

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10

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]

[[[[[[[[[[[

[

00

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10

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]

(61)



[[[

[


00 00 00 1819400002 1819400002 0000 00 00 00 00 0000 00 00 00 00 1819400002

]]]

]

(62)




[[[[[[[

[

004967528691

008463937788

09951726186

00

00

00

]]]]]]]

]

[[[[[[[

[

09161715971

minus04006166011

minus001165943224

00

00

00

]]]]]]]

]

[[[[[

[

minus03976958243

minus09123280736

009744493938

00

00

00

]]]]]

]

[[[[[

[

00

00

00

10

00

00

]]]]]

]

[[[[[

[

00

00

00

00

10

00

]]]]]

]

[[[[[

[

00

00

00

00

00

10

]]]]]

]

(63)


[[[[[[[

[

004967528696

008463937796

09951726196

00

00

00

]]]]]]]

]

[[[[[[[

[

09161715980

minus04006166015

minus001165943225

00

00

00

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]


[[[[[[[

[

minus03976958247

minus09123280745

009744493948

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(64)



[[[

[

1922399998 minus000000001176508811 minus000000001537920006 00 00 00

minus000000001176508811 1922400000 minus0000000007631928036 00 00 00

minus000000001537920006 minus0000000007631928036 1922400000 00 00 00

00 00 00 1922399998 1922399998 00

00 00 00 00 00 00

00 00 00 00 00 1922399998

]]]

]

(65)




[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

[[[[[[[

[

minus01387533797

minus02031928413

minus09692575344

00

00

00

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]

[[[[[[[

[

minus08476774112

05304148448

001015375725

00

00

00

]]]]]]]

]

[[[[[[[

[

minus05120454135

minus08230265872

02458388325

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

(66)


[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

[[[[[[[

[

minus01387533798

minus02031928415

minus09692575354

00

00

00

]]]]]]]

]

[[[[[[[

[

minus08476774112

05304148448

001015375725

00

00

00

]]]]]]]

]

[[[[[[[

[

minus05120454135

minus08230265872

02458388325

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

(67)



[[[

[

2074599999 0000000004356660361 000000003402343994 00 00 00

0000000004356660361 2074600004 000000002232269567 00 00 00

000000003402343994 000000002232269567 2074600000 00 00 00

00 00 00 2074600000 2074600000 00

00 00 00 00 00 00

00 00 00 00 00 2074600000

]]]

]

(68)




[[[[[[[

[

008621257414

01243595697

09884847438

00

00

00

]]]]]]]

]

[[[[[[[

[

08755641188

minus04828494241

minus001561753067

00

00

00

]]]]]]]

]

[[[[[

[

minus04753471023

minus08668282006

01505124700

00

00

00

]]]]]

]

[[[[[

[

00

00

00

10

00

00

]]]]]

]

[[[[[

[

00

00

00

00

10

00

]]]]]

]

[[[[[

[

00

00

00

00

00

10

]]]]]

]

(69)


[[[[[

[

008621257423

01243595698

09884847448

00

00

00

]]]]]

]

[[[[[

[

08755641188

minus04828494241

minus001561753067

00

00

00

]]]]]

]


[[[[[[[

[

minus04753471018

minus08668281997

01505124698

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(70)



[[[

[

1890099999 000000001209663993 minus000000002532733996 00 00 00000000001209663993 1890099991 0000000001701090138 00 00 00minus000000002532733996 0000000001701090138 1890100001 00 00 00

00 00 00 1890099997 1890099997 0000 00 00 00 00 0000 00 00 00 00 1890099997

]]]

]

(71)




[[[[[[[

[

03961520086

06338768023

06642768888

00

00

00

]]]]]]]

]

[[[[[[[

[

09160614623

minus02236797319

minus03328645006

00

00

00

]]]]]]]

]

[[[[[[[

[

006240981414

minus07403833993

06692812840

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(72)


[[[[[[[

[

03961520086

06338768023

06642768888

00

00

00

]]]]]]]

]

[[[[[[[

[

09160614614

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minus03328645003

00

00

00

]]]]]]]

]

[[[[[[[

[

006240981426

minus07403834008

06692812853

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(73)



[[[

[


00 00 00 8772299998 8772299998 0000 00 00 00 00 0000 00 00 00 00 8772299998

]]]

]

(74)




[[[[[[[

[

minus006591789198

minus008975775227

minus09937798435

00

00

00

]]]]]]]

]

[[[[[[[

[

minus08289381135

05593224991

0004466103673

00

00

00

]]]]]]]

]

[[[[[

[

minus05554425577

minus08240763851

01112729817

00

00

00

]]]]]

]

[[[[[

[

00

00

00

10

00

00

]]]]]

]

[[[[[

[

00

00

00

00

10

00

]]]]]

]

[[[[[

[

00

00

00

00

00

10

]]]]]

]


[[[[[[[

[

minus01387533798

minus02031928415

minus09692575354

00

00

00

]]]]]]]

]

[[[[[[[

[

minus08476774112

05304148448

001015375725

00

00

00

]]]]]]]

]


[[[[

[

minus05120454135

minus08230265872

02458388325

00

00

00

]]]]

]

[[[[

[

00

00

00

10

00

00

]]]]

]

[[[[

[

00

00

00

00

10

00

]]]]

]

[[[[

[

00

00

00

00

00

10

]]]]

]

(76)



[[[

[

2228057494 0000000004678921127 000000003654014285 00 00 000000000004678921127 2228057499 000000002397389829 00 00 00000000003654014285 000000002397389829 2228057495 00 00 00

00 00 00 2228057495 2228057495 0000 00 00 00 00 0000 00 00 00 00 2228057495

]]]

]

(77)




[[[[[[[

[

006975010637

01071019008

09917984199

00

00

00

]]]]]]]

]

[[[[[[[

[

09079000793

minus04187725641

minus001862756541

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04133429180

minus09017531389

01264472547

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(78)


[[[[[[[

[

006975010637

01071019008

09917984199

00

00

00

]]]]]]]

]

[[[[[[[

[

09079000793

minus04187725641

minus001862756541

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04133429184

minus09017531398

01264472548

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(79)



[[[

[


00 00 00 2598699998 2598699998 0000 00 00 00 00 0000 00 00 00 00 2598699998

]]]

]

(80)




[[[[[

[

minus00192522847250000

minus00192842313600000

000386151499999998

00

00

00

]]]]]

]

[[[[[

[

000961799224999998

00165029120500000

000436480214750000

00

00

00

]]]]]

]

[[[[[[

[

minus0494444050150000

minus0856906622200000

0142104790200000

00

00

00

]]]]]]

]

[[[[[[

[

00

00

00

10

00

00

]]]]]]

]

[[[[[[

[

00

00

00

00

10

00

]]]]]]

]

[[[[[[

[

00

00

00

00

00

10

]]]]]]

]

(81)



[[[[[[[[[[

[

minus2541745843

minus2545963537

5098090872

00

00

00

]]]]]]]]]]

]

[[[[[[[[[[

[

2505315863

4298714977

1136953303

00

00

00

]]]]]]]]]]

]

[[[[[[[

[

minus04949599718

minus08578007511

01422530677

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(82)



[[[

[

315679169478799 427324383657779 384557503470365 00 00 00427324383657779 618700481877479 889153535276175 00 00 00384557503470365 889153535276175 384768762708609 00 00 00

00 00 00 247795001400000 00 0000 00 00 00 247795001400000 0000 00 00 00 00 247795001400000

]]]

]

(83)




[[[[[[[

[

01135176976

01764907831

09777344911

00

00

00

]]]]]]]

]

[[[[[[[

[

08848520771

minus04654963442

minus001870707734

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04518302069

minus08672739791

02090103088

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(84)


[[[[[[[

[

01135176977

01764907833

09777344921

00

00

00

]]]]]]]

]

[[[[[[[

[

08848520771

minus04654963442

minus001870707734

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04518302069

minus08672739791

02090103088

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(85)



[[[

[

2663700003 000000004768023095 000000003169803038 00 00 00000000004768023095 2663699992 0000000008310743498 00 00 00000000003169803038 0000000008310743498 2663700001 00 00 00

00 00 00 2663699998 2663699998 0000 00 00 00 00 0000 00 00 00 00 2663699998

]]]

]

(86)





6 Conclusions









Disclosure




Acknowledgments



References




















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












[[[

[

1615299998 0000000005508173090 minus0000000008722620038 00 00 00

0000000005508173090 1615299996 minus0000000004345156817 00 00 00

minus0000000008722620038 minus0000000004345156817 1615300000 00 00 00

00 00 00 1615299998 00 00

00 00 00 00 1615299998 00

00 00 00 00 00 1615299998

]]]

]

(59)




[[[[[[[

[

minus00598757013800000

minus00768229223150000

minus0995231841750000

00

00

00

]]]]]]]

]

[[[[[[[

[

minus0869358919900000

0493939153600000

00141567750950000

00

00

00

]]]]]]]

]

[[[[[[[

[

minus0490490276500000

minus0866078136900000

00963811082600000

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(60)


[[[[[[[[[[[

[

minus005987730132

minus007682497510

minus09952584354

00

00

00

]]]]]]]]]]]

]

[[[[[[[[[[[

[

minus08693926232

04939583026

001415732393

00

00

00

]]]]]]]]]]]

]

[[[[[[[[[[[

[

00

00

00

10

00

00

]]]]]]]]]]]

]

[[[[[[[[[[[

[

00

00

00

00

10

00

]]]]]]]]]]]

]

[[[[[[[[[[[

[

00

00

00

00

00

10

]]]]]]]]]]]

]

(61)



[[[

[


00 00 00 1819400002 1819400002 0000 00 00 00 00 0000 00 00 00 00 1819400002

]]]

]

(62)




[[[[[[[

[

004967528691

008463937788

09951726186

00

00

00

]]]]]]]

]

[[[[[[[

[

09161715971

minus04006166011

minus001165943224

00

00

00

]]]]]]]

]

[[[[[

[

minus03976958243

minus09123280736

009744493938

00

00

00

]]]]]

]

[[[[[

[

00

00

00

10

00

00

]]]]]

]

[[[[[

[

00

00

00

00

10

00

]]]]]

]

[[[[[

[

00

00

00

00

00

10

]]]]]

]

(63)


[[[[[[[

[

004967528696

008463937796

09951726196

00

00

00

]]]]]]]

]

[[[[[[[

[

09161715980

minus04006166015

minus001165943225

00

00

00

]]]]]]]

]


[[[[[[[

[

minus03976958247

minus09123280745

009744493948

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(64)



[[[

[

1922399998 minus000000001176508811 minus000000001537920006 00 00 00

minus000000001176508811 1922400000 minus0000000007631928036 00 00 00

minus000000001537920006 minus0000000007631928036 1922400000 00 00 00

00 00 00 1922399998 1922399998 00

00 00 00 00 00 00

00 00 00 00 00 1922399998

]]]

]

(65)




[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

[[[[[[[

[

minus01387533797

minus02031928413

minus09692575344

00

00

00

]]]]]]]

]

[[[[[[[

[

minus08476774112

05304148448

001015375725

00

00

00

]]]]]]]

]

[[[[[[[

[

minus05120454135

minus08230265872

02458388325

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

(66)


[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

[[[[[[[

[

minus01387533798

minus02031928415

minus09692575354

00

00

00

]]]]]]]

]

[[[[[[[

[

minus08476774112

05304148448

001015375725

00

00

00

]]]]]]]

]

[[[[[[[

[

minus05120454135

minus08230265872

02458388325

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

(67)



[[[

[

2074599999 0000000004356660361 000000003402343994 00 00 00

0000000004356660361 2074600004 000000002232269567 00 00 00

000000003402343994 000000002232269567 2074600000 00 00 00

00 00 00 2074600000 2074600000 00

00 00 00 00 00 00

00 00 00 00 00 2074600000

]]]

]

(68)




[[[[[[[

[

008621257414

01243595697

09884847438

00

00

00

]]]]]]]

]

[[[[[[[

[

08755641188

minus04828494241

minus001561753067

00

00

00

]]]]]]]

]

[[[[[

[

minus04753471023

minus08668282006

01505124700

00

00

00

]]]]]

]

[[[[[

[

00

00

00

10

00

00

]]]]]

]

[[[[[

[

00

00

00

00

10

00

]]]]]

]

[[[[[

[

00

00

00

00

00

10

]]]]]

]

(69)


[[[[[

[

008621257423

01243595698

09884847448

00

00

00

]]]]]

]

[[[[[

[

08755641188

minus04828494241

minus001561753067

00

00

00

]]]]]

]


[[[[[[[

[

minus04753471018

minus08668281997

01505124698

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(70)



[[[

[

1890099999 000000001209663993 minus000000002532733996 00 00 00000000001209663993 1890099991 0000000001701090138 00 00 00minus000000002532733996 0000000001701090138 1890100001 00 00 00

00 00 00 1890099997 1890099997 0000 00 00 00 00 0000 00 00 00 00 1890099997

]]]

]

(71)




[[[[[[[

[

03961520086

06338768023

06642768888

00

00

00

]]]]]]]

]

[[[[[[[

[

09160614623

minus02236797319

minus03328645006

00

00

00

]]]]]]]

]

[[[[[[[

[

006240981414

minus07403833993

06692812840

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(72)


[[[[[[[

[

03961520086

06338768023

06642768888

00

00

00

]]]]]]]

]

[[[[[[[

[

09160614614

minus02236797317

minus03328645003

00

00

00

]]]]]]]

]

[[[[[[[

[

006240981426

minus07403834008

06692812853

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(73)



[[[

[


00 00 00 8772299998 8772299998 0000 00 00 00 00 0000 00 00 00 00 8772299998

]]]

]

(74)




[[[[[[[

[

minus006591789198

minus008975775227

minus09937798435

00

00

00

]]]]]]]

]

[[[[[[[

[

minus08289381135

05593224991

0004466103673

00

00

00

]]]]]]]

]

[[[[[

[

minus05554425577

minus08240763851

01112729817

00

00

00

]]]]]

]

[[[[[

[

00

00

00

10

00

00

]]]]]

]

[[[[[

[

00

00

00

00

10

00

]]]]]

]

[[[[[

[

00

00

00

00

00

10

]]]]]

]


[[[[[[[

[

minus01387533798

minus02031928415

minus09692575354

00

00

00

]]]]]]]

]

[[[[[[[

[

minus08476774112

05304148448

001015375725

00

00

00

]]]]]]]

]


[[[[

[

minus05120454135

minus08230265872

02458388325

00

00

00

]]]]

]

[[[[

[

00

00

00

10

00

00

]]]]

]

[[[[

[

00

00

00

00

10

00

]]]]

]

[[[[

[

00

00

00

00

00

10

]]]]

]

(76)



[[[

[

2228057494 0000000004678921127 000000003654014285 00 00 000000000004678921127 2228057499 000000002397389829 00 00 00000000003654014285 000000002397389829 2228057495 00 00 00

00 00 00 2228057495 2228057495 0000 00 00 00 00 0000 00 00 00 00 2228057495

]]]

]

(77)




[[[[[[[

[

006975010637

01071019008

09917984199

00

00

00

]]]]]]]

]

[[[[[[[

[

09079000793

minus04187725641

minus001862756541

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04133429180

minus09017531389

01264472547

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(78)


[[[[[[[

[

006975010637

01071019008

09917984199

00

00

00

]]]]]]]

]

[[[[[[[

[

09079000793

minus04187725641

minus001862756541

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04133429184

minus09017531398

01264472548

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(79)



[[[

[


00 00 00 2598699998 2598699998 0000 00 00 00 00 0000 00 00 00 00 2598699998

]]]

]

(80)




[[[[[

[

minus00192522847250000

minus00192842313600000

000386151499999998

00

00

00

]]]]]

]

[[[[[

[

000961799224999998

00165029120500000

000436480214750000

00

00

00

]]]]]

]

[[[[[[

[

minus0494444050150000

minus0856906622200000

0142104790200000

00

00

00

]]]]]]

]

[[[[[[

[

00

00

00

10

00

00

]]]]]]

]

[[[[[[

[

00

00

00

00

10

00

]]]]]]

]

[[[[[[

[

00

00

00

00

00

10

]]]]]]

]

(81)



[[[[[[[[[[

[

minus2541745843

minus2545963537

5098090872

00

00

00

]]]]]]]]]]

]

[[[[[[[[[[

[

2505315863

4298714977

1136953303

00

00

00

]]]]]]]]]]

]

[[[[[[[

[

minus04949599718

minus08578007511

01422530677

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(82)



[[[

[

315679169478799 427324383657779 384557503470365 00 00 00427324383657779 618700481877479 889153535276175 00 00 00384557503470365 889153535276175 384768762708609 00 00 00

00 00 00 247795001400000 00 0000 00 00 00 247795001400000 0000 00 00 00 00 247795001400000

]]]

]

(83)




[[[[[[[

[

01135176976

01764907831

09777344911

00

00

00

]]]]]]]

]

[[[[[[[

[

08848520771

minus04654963442

minus001870707734

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04518302069

minus08672739791

02090103088

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(84)


[[[[[[[

[

01135176977

01764907833

09777344921

00

00

00

]]]]]]]

]

[[[[[[[

[

08848520771

minus04654963442

minus001870707734

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04518302069

minus08672739791

02090103088

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(85)



[[[

[

2663700003 000000004768023095 000000003169803038 00 00 00000000004768023095 2663699992 0000000008310743498 00 00 00000000003169803038 0000000008310743498 2663700001 00 00 00

00 00 00 2663699998 2663699998 0000 00 00 00 00 0000 00 00 00 00 2663699998

]]]

]

(86)





6 Conclusions









Disclosure




Acknowledgments



References




















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










[[[[[[[

[

minus03976958247

minus09123280745

009744493948

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(64)



[[[

[

1922399998 minus000000001176508811 minus000000001537920006 00 00 00

minus000000001176508811 1922400000 minus0000000007631928036 00 00 00

minus000000001537920006 minus0000000007631928036 1922400000 00 00 00

00 00 00 1922399998 1922399998 00

00 00 00 00 00 00

00 00 00 00 00 1922399998

]]]

]

(65)




[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

[[[[[[[

[

minus01387533797

minus02031928413

minus09692575344

00

00

00

]]]]]]]

]

[[[[[[[

[

minus08476774112

05304148448

001015375725

00

00

00

]]]]]]]

]

[[[[[[[

[

minus05120454135

minus08230265872

02458388325

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

(66)


[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

[[[[[[[

[

minus01387533798

minus02031928415

minus09692575354

00

00

00

]]]]]]]

]

[[[[[[[

[

minus08476774112

05304148448

001015375725

00

00

00

]]]]]]]

]

[[[[[[[

[

minus05120454135

minus08230265872

02458388325

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

(67)



[[[

[

2074599999 0000000004356660361 000000003402343994 00 00 00

0000000004356660361 2074600004 000000002232269567 00 00 00

000000003402343994 000000002232269567 2074600000 00 00 00

00 00 00 2074600000 2074600000 00

00 00 00 00 00 00

00 00 00 00 00 2074600000

]]]

]

(68)




[[[[[[[

[

008621257414

01243595697

09884847438

00

00

00

]]]]]]]

]

[[[[[[[

[

08755641188

minus04828494241

minus001561753067

00

00

00

]]]]]]]

]

[[[[[

[

minus04753471023

minus08668282006

01505124700

00

00

00

]]]]]

]

[[[[[

[

00

00

00

10

00

00

]]]]]

]

[[[[[

[

00

00

00

00

10

00

]]]]]

]

[[[[[

[

00

00

00

00

00

10

]]]]]

]

(69)


[[[[[

[

008621257423

01243595698

09884847448

00

00

00

]]]]]

]

[[[[[

[

08755641188

minus04828494241

minus001561753067

00

00

00

]]]]]

]


[[[[[[[

[

minus04753471018

minus08668281997

01505124698

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(70)



[[[

[

1890099999 000000001209663993 minus000000002532733996 00 00 00000000001209663993 1890099991 0000000001701090138 00 00 00minus000000002532733996 0000000001701090138 1890100001 00 00 00

00 00 00 1890099997 1890099997 0000 00 00 00 00 0000 00 00 00 00 1890099997

]]]

]

(71)




[[[[[[[

[

03961520086

06338768023

06642768888

00

00

00

]]]]]]]

]

[[[[[[[

[

09160614623

minus02236797319

minus03328645006

00

00

00

]]]]]]]

]

[[[[[[[

[

006240981414

minus07403833993

06692812840

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(72)


[[[[[[[

[

03961520086

06338768023

06642768888

00

00

00

]]]]]]]

]

[[[[[[[

[

09160614614

minus02236797317

minus03328645003

00

00

00

]]]]]]]

]

[[[[[[[

[

006240981426

minus07403834008

06692812853

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(73)



[[[

[


00 00 00 8772299998 8772299998 0000 00 00 00 00 0000 00 00 00 00 8772299998

]]]

]

(74)




[[[[[[[

[

minus006591789198

minus008975775227

minus09937798435

00

00

00

]]]]]]]

]

[[[[[[[

[

minus08289381135

05593224991

0004466103673

00

00

00

]]]]]]]

]

[[[[[

[

minus05554425577

minus08240763851

01112729817

00

00

00

]]]]]

]

[[[[[

[

00

00

00

10

00

00

]]]]]

]

[[[[[

[

00

00

00

00

10

00

]]]]]

]

[[[[[

[

00

00

00

00

00

10

]]]]]

]


[[[[[[[

[

minus01387533798

minus02031928415

minus09692575354

00

00

00

]]]]]]]

]

[[[[[[[

[

minus08476774112

05304148448

001015375725

00

00

00

]]]]]]]

]


[[[[

[

minus05120454135

minus08230265872

02458388325

00

00

00

]]]]

]

[[[[

[

00

00

00

10

00

00

]]]]

]

[[[[

[

00

00

00

00

10

00

]]]]

]

[[[[

[

00

00

00

00

00

10

]]]]

]

(76)



[[[

[

2228057494 0000000004678921127 000000003654014285 00 00 000000000004678921127 2228057499 000000002397389829 00 00 00000000003654014285 000000002397389829 2228057495 00 00 00

00 00 00 2228057495 2228057495 0000 00 00 00 00 0000 00 00 00 00 2228057495

]]]

]

(77)




[[[[[[[

[

006975010637

01071019008

09917984199

00

00

00

]]]]]]]

]

[[[[[[[

[

09079000793

minus04187725641

minus001862756541

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04133429180

minus09017531389

01264472547

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(78)


[[[[[[[

[

006975010637

01071019008

09917984199

00

00

00

]]]]]]]

]

[[[[[[[

[

09079000793

minus04187725641

minus001862756541

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04133429184

minus09017531398

01264472548

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(79)



[[[

[


00 00 00 2598699998 2598699998 0000 00 00 00 00 0000 00 00 00 00 2598699998

]]]

]

(80)




[[[[[

[

minus00192522847250000

minus00192842313600000

000386151499999998

00

00

00

]]]]]

]

[[[[[

[

000961799224999998

00165029120500000

000436480214750000

00

00

00

]]]]]

]

[[[[[[

[

minus0494444050150000

minus0856906622200000

0142104790200000

00

00

00

]]]]]]

]

[[[[[[

[

00

00

00

10

00

00

]]]]]]

]

[[[[[[

[

00

00

00

00

10

00

]]]]]]

]

[[[[[[

[

00

00

00

00

00

10

]]]]]]

]

(81)



[[[[[[[[[[

[

minus2541745843

minus2545963537

5098090872

00

00

00

]]]]]]]]]]

]

[[[[[[[[[[

[

2505315863

4298714977

1136953303

00

00

00

]]]]]]]]]]

]

[[[[[[[

[

minus04949599718

minus08578007511

01422530677

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(82)



[[[

[

315679169478799 427324383657779 384557503470365 00 00 00427324383657779 618700481877479 889153535276175 00 00 00384557503470365 889153535276175 384768762708609 00 00 00

00 00 00 247795001400000 00 0000 00 00 00 247795001400000 0000 00 00 00 00 247795001400000

]]]

]

(83)




[[[[[[[

[

01135176976

01764907831

09777344911

00

00

00

]]]]]]]

]

[[[[[[[

[

08848520771

minus04654963442

minus001870707734

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04518302069

minus08672739791

02090103088

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(84)


[[[[[[[

[

01135176977

01764907833

09777344921

00

00

00

]]]]]]]

]

[[[[[[[

[

08848520771

minus04654963442

minus001870707734

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04518302069

minus08672739791

02090103088

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(85)



[[[

[

2663700003 000000004768023095 000000003169803038 00 00 00000000004768023095 2663699992 0000000008310743498 00 00 00000000003169803038 0000000008310743498 2663700001 00 00 00

00 00 00 2663699998 2663699998 0000 00 00 00 00 0000 00 00 00 00 2663699998

]]]

]

(86)





6 Conclusions









Disclosure




Acknowledgments



References




















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










[[[[[[[

[

minus04753471018

minus08668281997

01505124698

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00

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]

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[

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[

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[[[[[[[

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]

(70)



[[[

[

1890099999 000000001209663993 minus000000002532733996 00 00 00000000001209663993 1890099991 0000000001701090138 00 00 00minus000000002532733996 0000000001701090138 1890100001 00 00 00

00 00 00 1890099997 1890099997 0000 00 00 00 00 0000 00 00 00 00 1890099997

]]]

]

(71)




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]

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]

(72)


[[[[[[[

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]

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[[[[[[[

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06692812853

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]

[[[[[[[

[

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]

[[[[[[[

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]

(73)



[[[

[


00 00 00 8772299998 8772299998 0000 00 00 00 00 0000 00 00 00 00 8772299998

]]]

]

(74)




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05593224991

0004466103673

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00

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]

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minus08240763851

01112729817

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]

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[

00

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00

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]

[[[[[

[

00

00

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]

[[[[[

[

00

00

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10

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]


[[[[[[[

[

minus01387533798

minus02031928415

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00

00

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]

[[[[[[[

[

minus08476774112

05304148448

001015375725

00

00

00

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]


[[[[

[

minus05120454135

minus08230265872

02458388325

00

00

00

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]

[[[[

[

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]

[[[[

[

00

00

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10

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[[[[

[

00

00

00

00

00

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]

(76)



[[[

[

2228057494 0000000004678921127 000000003654014285 00 00 000000000004678921127 2228057499 000000002397389829 00 00 00000000003654014285 000000002397389829 2228057495 00 00 00

00 00 00 2228057495 2228057495 0000 00 00 00 00 0000 00 00 00 00 2228057495

]]]

]

(77)




[[[[[[[

[

006975010637

01071019008

09917984199

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00

00

]]]]]]]

]

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[

09079000793

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00

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]

[[[[[[[

[

minus04133429180

minus09017531389

01264472547

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00

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]

[[[[[[[

[

00

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]

[[[[[[[

[

00

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]

[[[[[[[

[

00

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]

(78)


[[[[[[[

[

006975010637

01071019008

09917984199

00

00

00

]]]]]]]

]

[[[[[[[

[

09079000793

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]

[[[[[[[

[

minus04133429184

minus09017531398

01264472548

00

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]

[[[[[[[

[

00

00

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10

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]

[[[[[[[

[

00

00

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10

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]

[[[[[[[

[

00

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]

(79)



[[[

[


00 00 00 2598699998 2598699998 0000 00 00 00 00 0000 00 00 00 00 2598699998

]]]

]

(80)




[[[[[

[

minus00192522847250000

minus00192842313600000

000386151499999998

00

00

00

]]]]]

]

[[[[[

[

000961799224999998

00165029120500000

000436480214750000

00

00

00

]]]]]

]

[[[[[[

[

minus0494444050150000

minus0856906622200000

0142104790200000

00

00

00

]]]]]]

]

[[[[[[

[

00

00

00

10

00

00

]]]]]]

]

[[[[[[

[

00

00

00

00

10

00

]]]]]]

]

[[[[[[

[

00

00

00

00

00

10

]]]]]]

]

(81)



[[[[[[[[[[

[

minus2541745843

minus2545963537

5098090872

00

00

00

]]]]]]]]]]

]

[[[[[[[[[[

[

2505315863

4298714977

1136953303

00

00

00

]]]]]]]]]]

]

[[[[[[[

[

minus04949599718

minus08578007511

01422530677

00

00

00

]]]]]]]

]

[[[[[[[

[

00

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00

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]

[[[[[[[

[

00

00

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]

[[[[[[[

[

00

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00

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]

(82)



[[[

[

315679169478799 427324383657779 384557503470365 00 00 00427324383657779 618700481877479 889153535276175 00 00 00384557503470365 889153535276175 384768762708609 00 00 00

00 00 00 247795001400000 00 0000 00 00 00 247795001400000 0000 00 00 00 00 247795001400000

]]]

]

(83)




[[[[[[[

[

01135176976

01764907831

09777344911

00

00

00

]]]]]]]

]

[[[[[[[

[

08848520771

minus04654963442

minus001870707734

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04518302069

minus08672739791

02090103088

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

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10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

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10

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]

[[[[[[[

[

00

00

00

00

00

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]

(84)


[[[[[[[

[

01135176977

01764907833

09777344921

00

00

00

]]]]]]]

]

[[[[[[[

[

08848520771

minus04654963442

minus001870707734

00

00

00

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]

[[[[[[[

[

minus04518302069

minus08672739791

02090103088

00

00

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]

[[[[[[[

[

00

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00

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]

[[[[[[[

[

00

00

00

00

10

00

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]

[[[[[[[

[

00

00

00

00

00

10

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]

(85)



[[[

[

2663700003 000000004768023095 000000003169803038 00 00 00000000004768023095 2663699992 0000000008310743498 00 00 00000000003169803038 0000000008310743498 2663700001 00 00 00

00 00 00 2663699998 2663699998 0000 00 00 00 00 0000 00 00 00 00 2663699998

]]]

]

(86)





6 Conclusions









Disclosure




Acknowledgments



References




















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










[[[[

[

minus05120454135

minus08230265872

02458388325

00

00

00

]]]]

]

[[[[

[

00

00

00

10

00

00

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]

[[[[

[

00

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[[[[

[

00

00

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00

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(76)



[[[

[

2228057494 0000000004678921127 000000003654014285 00 00 000000000004678921127 2228057499 000000002397389829 00 00 00000000003654014285 000000002397389829 2228057495 00 00 00

00 00 00 2228057495 2228057495 0000 00 00 00 00 0000 00 00 00 00 2228057495

]]]

]

(77)




[[[[[[[

[

006975010637

01071019008

09917984199

00

00

00

]]]]]]]

]

[[[[[[[

[

09079000793

minus04187725641

minus001862756541

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04133429180

minus09017531389

01264472547

00

00

00

]]]]]]]

]

[[[[[[[

[

00

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10

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]

[[[[[[[

[

00

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]

[[[[[[[

[

00

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]

(78)


[[[[[[[

[

006975010637

01071019008

09917984199

00

00

00

]]]]]]]

]

[[[[[[[

[

09079000793

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minus001862756541

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]

[[[[[[[

[

minus04133429184

minus09017531398

01264472548

00

00

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]]]]]]]

]

[[[[[[[

[

00

00

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]

[[[[[[[

[

00

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10

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]

[[[[[[[

[

00

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]

(79)



[[[

[


00 00 00 2598699998 2598699998 0000 00 00 00 00 0000 00 00 00 00 2598699998

]]]

]

(80)




[[[[[

[

minus00192522847250000

minus00192842313600000

000386151499999998

00

00

00

]]]]]

]

[[[[[

[

000961799224999998

00165029120500000

000436480214750000

00

00

00

]]]]]

]

[[[[[[

[

minus0494444050150000

minus0856906622200000

0142104790200000

00

00

00

]]]]]]

]

[[[[[[

[

00

00

00

10

00

00

]]]]]]

]

[[[[[[

[

00

00

00

00

10

00

]]]]]]

]

[[[[[[

[

00

00

00

00

00

10

]]]]]]

]

(81)



[[[[[[[[[[

[

minus2541745843

minus2545963537

5098090872

00

00

00

]]]]]]]]]]

]

[[[[[[[[[[

[

2505315863

4298714977

1136953303

00

00

00

]]]]]]]]]]

]

[[[[[[[

[

minus04949599718

minus08578007511

01422530677

00

00

00

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]

[[[[[[[

[

00

00

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10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

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]

[[[[[[[

[

00

00

00

00

00

10

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]

(82)



[[[

[

315679169478799 427324383657779 384557503470365 00 00 00427324383657779 618700481877479 889153535276175 00 00 00384557503470365 889153535276175 384768762708609 00 00 00

00 00 00 247795001400000 00 0000 00 00 00 247795001400000 0000 00 00 00 00 247795001400000

]]]

]

(83)




[[[[[[[

[

01135176976

01764907831

09777344911

00

00

00

]]]]]]]

]

[[[[[[[

[

08848520771

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minus001870707734

00

00

00

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]

[[[[[[[

[

minus04518302069

minus08672739791

02090103088

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

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10

00

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]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

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]

(84)


[[[[[[[

[

01135176977

01764907833

09777344921

00

00

00

]]]]]]]

]

[[[[[[[

[

08848520771

minus04654963442

minus001870707734

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04518302069

minus08672739791

02090103088

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

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10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

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]

(85)



[[[

[

2663700003 000000004768023095 000000003169803038 00 00 00000000004768023095 2663699992 0000000008310743498 00 00 00000000003169803038 0000000008310743498 2663700001 00 00 00

00 00 00 2663699998 2663699998 0000 00 00 00 00 0000 00 00 00 00 2663699998

]]]

]

(86)





6 Conclusions









Disclosure




Acknowledgments



References




















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











[[[[[[[[[[

[

minus2541745843

minus2545963537

5098090872

00

00

00

]]]]]]]]]]

]

[[[[[[[[[[

[

2505315863

4298714977

1136953303

00

00

00

]]]]]]]]]]

]

[[[[[[[

[

minus04949599718

minus08578007511

01422530677

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(82)



[[[

[

315679169478799 427324383657779 384557503470365 00 00 00427324383657779 618700481877479 889153535276175 00 00 00384557503470365 889153535276175 384768762708609 00 00 00

00 00 00 247795001400000 00 0000 00 00 00 247795001400000 0000 00 00 00 00 247795001400000

]]]

]

(83)




[[[[[[[

[

01135176976

01764907831

09777344911

00

00

00

]]]]]]]

]

[[[[[[[

[

08848520771

minus04654963442

minus001870707734

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04518302069

minus08672739791

02090103088

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(84)


[[[[[[[

[

01135176977

01764907833

09777344921

00

00

00

]]]]]]]

]

[[[[[[[

[

08848520771

minus04654963442

minus001870707734

00

00

00

]]]]]]]

]

[[[[[[[

[

minus04518302069

minus08672739791

02090103088

00

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

10

00

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

10

00

]]]]]]]

]

[[[[[[[

[

00

00

00

00

00

10

]]]]]]]

]

(85)



[[[

[

2663700003 000000004768023095 000000003169803038 00 00 00000000004768023095 2663699992 0000000008310743498 00 00 00000000003169803038 0000000008310743498 2663700001 00 00 00

00 00 00 2663699998 2663699998 0000 00 00 00 00 0000 00 00 00 00 2663699998

]]]

]

(86)





6 Conclusions









Disclosure




Acknowledgments



References




















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










6 Conclusions









Disclosure




Acknowledgments



References




















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









research article a cas approach to handle the anisotropic

Documents