research article vibration analysis of conical shells by...

Research ArticleVibration Analysis of Conical Shells by the Improved FourierExpansion-Based Differential Quadrature Method

Wanyou Li Gang Wang and Jingtao Du

College of Power and Energy Engineering Harbin Engineering University Harbin 150001 China

Correspondence should be addressed to Wanyou Li hrbeu ripet lwy163com

Received 10 August 2015 Accepted 13 September 2015

Academic Editor Laurent Mevel

Copyright copy 2016 Wanyou Li et al 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

An improved Fourier expansion-based differential quadrature (DQ) algorithm is proposed to study the free vibration behavior oftruncated conical shells with different boundary conditionsTheoriginal function is expressed as the Fourier cosine series combinedwith close-form auxiliary functions Those auxiliary functions are introduced to ensure and accelerate the convergence of seriesexpansion The grid points are uniformly distributed along the space The weighting coefficients in the DQ method are easilyobtained by the inverse of the coefficient matrixThe derivatives in both the governing equations and the boundaries are discretizedby the DQ method Natural frequencies and modal shapes can be easily obtained by solving the numerical eigenvalue equationsThe accuracy and stability of this proposed method are validated against the results in the literature and a very good agreement isobserved The centrosymmetric properties of these newly proposed weighting coefficients are also validated Studies on the effectsof semivertex angle and the ratio of length to radius are reported

1 Introduction

Conical shells are widely used in various engineering fieldssuch as aerospace and ship industries The development ofaccurate shell theories has been the subject of significantresearch interest for many years and a large number of shelltheories based on different approximations and assumptionshave been proposed However more work is focused onthe vibration of cylindrical shells compared with the conicalshell Since the conical coordinate system is function of themeridional direction the equations of motion for conicalshells consist of a set of partial differential equations withvariable coefficients Current methods for the free vibrationanalysis of thin conical shells can be classified as to analyticalmethods and numerical methods Saunders et al [1] Garnetand Kempner [2] Siu [3] and Lim and Liew [4] have studiedthe free vibration of uniform conical shells by Rayleigh-RitzmethodUeda [5] analyzed the same problems using the finiteelement method Irie et al [6] studied a conical shell withvariable thickness by the transfer matrix method The DQmethod was employed to analyze the free vibration of a uni-form conical shell [7] Jin et al [8] studied the free and forcedvibration of conical shell using the improved Fourier series

method by considering the general boundary conditionsThekernel particle (k-p) functions were employed in hybridizedform with harmonic functions to study the vibration of theconical shell based on Ritz method [9]

Besides the studies of the isotropic conical shells lam-inated and functional graded conical shells have also beenfully studied by various methods [10ndash16] The differentialquadrature (DQ)methodwas adopted to solve the differentialgoverning equations of the conical shell in those researches[10ndash13] For the vibration of the rotating conical shell inwhich the centrifuge force should be taken into consider-ation the DQ method was also extensively used to studythose problems [10 17] The reason that the DQ methodis widely adopted to study the vibration behavior of theconical shell is the convenience of transforming the partialdifferential governing equations approximately into a set oflinear algebraic governing equations Imposing the givenboundary conditions the numerical eigenvalue equations forthe free vibration of the (rotating or composite) conical shellare derived and solved

The differential quadrature method is a numerical tech-nique for solving the differential equations It was firstdeveloped by Bellman et al [18 19] and their associates in

Hindawi Publishing CorporationShock and VibrationVolume 2016 Article ID 9617957 10 pageshttpdxdoiorg10115520169617957

2 Shock and Vibration

the early 1970s The DQ method akin to the conventionalintegral quadrature method approximates the derivative ofa function at any location by a linear summation of all thefunctional values along a mesh line The key procedure inthe DQ method applications lies in the determination ofthe weighting coefficients Shu [7 12 20 21] proposed twotypes of weighting coefficients obtained by the polynomialsand the truncated Fourier series among which the Lagrangeinterpolation functions are widely used for their simplicityexplicitness [7 21] Although it is well known that Lagrangeinterpolation functions are limited by the number of inter-polation points and severe oscillation may take place if theorder is large the use of the Gauss-Chebyshev points [7 21]can accelerate the convergence rate of the DQmethod Someworks focusing on improving the accuracy and stability ofthe DQ method are presented by proposing different waysto generate the weighting coefficients and to determine thedistribution of grid points [22ndash24]

The Fourier series with auxiliary functions was firstproposed by Li [25] to study the vibration problems of thebeam structure This method is extensively used to study the2D and 3D structural vibration and vibroacoustic problems[26ndash30] The auxiliary functions are introduced to acceleratethe convergence and deal with all the possible discontinuitiesat the end points or edges associatedwith the original Fouriercosine series This improved Fourier series method becomesa promising method to study the structural vibrationproblems

In this paper the improved Fourier expansion-baseddifferential quadrature method is proposed to solve thefree vibration behavior of the truncated conical shell Theweighting coefficients are obtained based on this improvedFourier series in a much easier way The centrosymmetricproperties of these newly proposed weighting coefficientsare also validated The following sections will illustrate thedevelopment of this hybrid method and numerical resultsare then presented to validate the effectiveness accuracyand stability of this current method on predicting the modalcharacteristics of the conical shell

2 Theoretical Formulation

For a continuous function 119891(119909) defined on [0 119871] with anabsolutely integrable derivative it can be expanded in Fouriercosine series

119891 (119909) =infin

sum119898=0

119886119898cos 119898120587

119871119909 0 lt 119909 lt 119871 (1)

The first-order derivative of 119891(119909) can be done term-by-term

1198911015840 (119909) = minusinfin

sum119898=0

119886119898

119898120587

119871sin 119898120587

119871119909 (2)

The second-order derivative of 119891(119909) cannot be obtainedterm-by-term which is shown as

11989110158401015840 (119909) = minus1198911015840 (119871) minus 1198911015840 (0)

119871

minusinfin

sum119898=1

(2

119871[(minus1)119898 119891 (119871) minus 119891 (0)] +

119898120587

119871119886119898)

sdot119898120587

119871cos 119898120587

119871119909

(3)

These formulations basically tell that while a cosine seriescan always be differentiated term-by-term this can be doneto a sine series only if 1198911015840(0) = 1198911015840(119871) = 0 To implementthe differential quadrature algorithm the auxiliary functionsare added to traditional Fourier cosine series to cover thediscontinuity of the function at the end points and to get thederivatives term-by-termA function can be expanded as [27]

119891 (119909) =infin

sum119898=0

119886119898cos 119898120587

119871119909 +4

sum119905=1

119862119905120585119905(119909) (4)

where

1205851(119909) =

9119871

4120587sin(120587119909

2119871) minus

119871

12120587sin(3120587119909

2119871) (5a)

1205852(119909) = minus

9119871

4120587cos(120587119909

2119871) minus

119871

12120587cos(3120587119909

2119871) (5b)

1205853(119909) =

1198713

1205873sin(120587119909

2119871) minus

1198713

31205873sin(3120587119909

2119871) (5c)

1205854(119909) = minus

1198713

1205873cos(120587119909

2119871) minus

1198713

31205873cos(3120587119909

2119871) (5d)

where 119886119898

and 119862119905represent the unknown Fourier expan-

sion coefficients The supplementary functions 120585119905(119909) can be

represented as arbitrary continuous functions regardless ofthe boundaries It is easy to verify that 1205851015840

1(0) = 120585101584010158401015840

3(0) =

120585101584010158402(0) = 120585101584010158401015840

4(0) = 1 and all the other 1st-order and 3rd-order

derivatives are identically equal to zero at both ends Themain purpose of introducing these supplementary functionsto standard Fourier series is to get the first four derivativesof the Fourier cosine series term-by-term As an immediatenumerical benefit the Fourier series in (4) will convergeuniformly at an accelerated rate

To implement the differential quadrature method 119873points are equally distributed on [0 119871]

119909119894=(119894 minus 1) 119871

119873 minus 1 119894 = 1 2 3 119873 (6)

The functional values at those grid points can be deter-mined as

119891 (119909119894) =infin

sum119898=0

119886119898cos 119898120587

119871119909119894+4

sum119905=1

119862119905120585119905(119909119894) (7)

The Fourier series is truncated to119898 = 119872 Rewrite (7) intothe matrix form

Shock and Vibration 3

119891(1199091)

119891 (1199092)

119891 (119909119873)

=

[[[[[[[[[[

[

cos 01205871198711199091

cos11205871198711199091sdot sdot sdot cos119872120587

11987111990911205851(1199091) 1205852(1199091) 1205853(1199091) 1205854(1199091)

cos 01205871198711199092


11987111990921205851(1199092) 1205852(1199092) 1205853(1199092) 1205854(1199092)

d

cos 0120587119871119909119873


1198711199091198731205851(119909119873) 1205852(119909119873) 1205853(119909119873) 1205854(119909119873)

]]]]]]]]]]

]

1198860

1198861

119886119872

1198621

1198622

1198623

1198624

(8)

The Fourier series coefficients can be obtained by theinverse of the matrix

1198860

1198861

119886119872

1198621

1198622

1198623

1198624

= Rminus1

119891(1199091)

119891 (1199092)

119891 (119909119873)

(9)

in which

R =

[[[[[[[[[[

[

cos01205871198711199091


11987111990911205851(1199091) 1205852(1199091) 1205853(1199091) 1205854(1199091)

cos01205871198711199092


11987111990921205851(1199092) 1205852(1199092) 1205853(1199092) 1205854(1199092)

d

cos0120587119871119909119873


1198711199091198731205851(119909119873) 1205852(119909119873) 1205853(119909119873) 1205854(119909119873)

]]]]]]]]]]

]

(10)

In this proposed method the number of truncatedFourier series and the number of grid points follow therelation that 119873 = 119872 + 5 to ensure R is a square matrix tolet the inverse be more accurate Once the constant matrixR is determined the approximated Fourier series coefficientsare obtained When the DQ method was first developedpolynomials were adopted to follow this procedure to gen-erate weighting coefficients which would lead to highly illcondition when119873 is largeThe Fourier series however showmuchmore stability to derive the coefficients by the inversionof 119877 which will be validated in the results section

The first-order derivatives at those grid points are

1198911015840 (119909119894) =119872

sum119898=0

119886119898(minus119898120587

119871) sin 119898120587

119871119909119894+4

sum119905=1

1198621199051205851015840119905(119909119894)

119894 = 1 2 3 119873

(11)

Rewrite (11) into the matrix form

1198911015840 (1199091)

1198911015840 (1199092)

1198911015840 (119909119873)

= R(1) lowast

1198860

1198861

119886119872

1198621

1198622

1198623

1198624


= R(1) lowast Rminus1 lowast

119891(1199091)

119891 (1199092)

119891 (119909119873)

= c(1) lowast

119891(1199091)

119891 (1199092)

119891 (119909119873)

(12)

in which R(1) is the first-order derivative of R and c(1) is thefirst-order weighting coefficient matrix of the DQ methodc(1) = R(1) lowast Rminus1 It is obvious from the above equation thatthe weighting coefficients of the second- and higher-orderderivatives can be completely determined through the sameway which are expressed as

119891(119894) (1199091)

119891(119894) (1199092)

119891(119894) (119909119873)


119891(1199091)

119891 (1199092)

119891 (119909119873)

= c(119894) lowast

119891(1199091)

119891 (1199092)

119891 (119909119873)

(13)

in which R(119894) (119894 = 1 2 3 4) is the 119894th-order derivative of Rand c(119894) is the 119894th-orderweighting coefficientmatrix of theDQmethod c(119894) = R(119894) lowast Rminus1

In this paper only four supplementary functions areadded to the Fourier cosine series which will ensure the first-four-order derivatives to converge at a high rate and to keepstability of the Fourier series Consequently first-four-orderweighting coefficient matrixes can be obtained which aresufficiently enough to study the vibration of a conical shellAdding more supplementary functions to the Fourier cosineseries will give the capability to study the correspondinghigher-order partial differential equations

3 Free Vibration Behavior of a Conical Shell

The free vibration behavior of conical shells has been studiedby Shu [7] by the DQmethodThis model is adopted again tovalidate the efficiency accuracy and stability of this proposedmethod

r

120579

R(x)

R(x)

R0

w

u

a

L

Figure 1 Geometry of a conical shell and the coordinate system

Consider a conical shell structure with semivertex angle119886 and the radius of the large edge is 119877

0 as shown in Figure 1

The displacement fields of the conical shell in 119909 120579 and 119903directions are denoted by 119906 V and 119908 respectively If thecouplings between these three displacement components areignored the field functions can be expressed as

119906 = 119880 (119909) sdot cos (119899120579) cos (120596119905)

V = 119881 (119909) sdot sin (119899120579) cos (120596119905)

119908 = 119882 (119909) sdot cos (119899120579) cos (120596119905)

(14)

in which 119899 and 120596 are the circumferential wave number andthe frequency in radsec respectively

The differential governing equations of the conical shellbased upon Flugge theory are written as

11987111119906 + 11987112V + 11987113119908 = minus120588ℎ1205962119880

11987121119906 + 11987122V + 11987123119908 = minus120588ℎ1205962119881

11987131119906 + 11987132V + 11987133119908 = minus120588ℎ1205962119882

(15)

in which 120588 and ℎ denote the density of the shell and the shellthickness respectively The differential operators 119871

119894119895(119894 119895 =

1 2 3) can be referred to [7]


Substituting (14) into (15) and applying (13) then

119878110119880119894+119873

sum119896=1

(119878111119888(1)119894119896+ 119878112119888(2)119894119896)119880119896+ 119878120119881119894

+119873

sum119896=1

119878121119888(1)119894119896119881119896+ 119878130119882119894+119873

sum119896=1

119878131119888(1)119894119896119882119896

= minus120588ℎ1205962119880119894

(16a)

119878210119880119894+119873

sum119896=1

119878211119888(1)119894119896119880119896+ 119878220119881119894

+119873

sum119896=1

(119878221119888(1)119894119896+ 119878222119888(2)119894119896)119881119896+ 119878230119882119894

+119873

sum119896=1

(119878231119888(1)119894119896+ 119878232119888(2)119894119896)119882119896= minus120588ℎ1205962119881

119894

(16b)

119878310119880119894+119873

sum119896=1

119878311119888(1)119894119896119880119896+ 119878320119881119894

+119873

sum119896=1

(119878321119888(1)119894119896+ 119878322119888(2)119894119896)119881119896+ 119878330119882119894

+119873

sum119896=1

(119878331119888(1)119894119896+ 119878232119888(2)119894119896+ 119878333119888(3)119894119896+ 119878334119888(4)119894119896)119882119896

= minus120588ℎ1205962119882119894

(16c)

where 119888(119899)119894119895

represents the weighting coefficients 119894 =

3 4 119873 minus 2The boundary conditions for the conical shell are as

follows

simple-supported boundary condition (S)

119881 = 0

119882 = 0

119873119909= 0

119872119909= 0

(17)

clamped boundary condition (C)

119881 = 0

119882 = 0

119880 = 0

119882(1) = 0

(18)

The boundary conditions can also be expressed in thedifferential forms by substituting (13) into them Simple

supported condition at the small edge is chosen as anexample

1198811= 0 (19a)

1198821= 0 (19b)

(V sin 1198861198771

+ 119888(1)11)1198801+ 119888(1)1119873119880119873= minus119873minus1

sum119896=2

119888(1)1119896119880119896

(19c)

(V sin 1198861198771

119888(1)12+ 119888(2)12)1198822

+ (V sin 1198861198771

119888(1)1(119873minus1)

+ 119888(2)1(119873minus1)

)119882119873minus1

= minus119873minus2

sum119896=3

(119888(2)1119896+V sin 1198861198772

119888(1)1119896)119882119896

(19d)

Other boundary conditions can also be formulated insimilar formsThose four boundary formulations are appliedat the grid points of 119894 = 1 2119873 minus 2119873 Rewrite (16a) (16b)(16c) (19a) (19b) (19c) (19d) and boundary conditions intothe matrix form

PX = ΩX (20)

whereΩ = 120588ℎ1205962P is amatrixwith the dimension of 3119873times3119873and

X = 1198801 1198802 119880

119873 1198811 1198812 119881

11987311988211198822

119882119873119879

(21)

It is clear from (20) that the natural frequencies andmodal shapes for the conical shell can now be directlyobtained by solving the standard matrix eigenvalue problemThe natural frequency parameter is defined as

Ω119894= 1198770120596119894

radic120588 (1 minus V2)119864

(22)

4 Numerical Results and Discussion

41 Convergence Study To study the convergence of this pro-posed method different numbers of grid points or truncatedFourier series (119873 = 119872 + 5) are selected The geometric andmaterial parameters of the conical shell are Youngrsquos modulus119864 = 70GPa Poissonrsquos ratio V = 03 120588 = 2700 kgm3ℎ1198770= 001 119886 = 45∘ and 119871 sin 119886119877

0= 05 The effect of the

number of grid points that affects the natural frequencies isstudied The results derived by finite element method (FEM)are adopted to compare with those obtained by this methodFigure 2 shows the natural frequency parameters under S-Sboundary conditions when the circumferential wave numberis 119899 = 0 and axial number is 119898 = 1 2 3 Figure 3 shows thenatural frequency parameter under S-C boundary conditionwhen the circumferential wave number is 119899 = 3 and axialnumber is 119898 = 1 2 3 By comparing the results derived


Table 1 Comparison of the natural frequency parameters under variable circumferential wave numbers (119886 = 45∘ 119871 sin 1198861198772= 05 and

ℎ1198770= 001)

119899C-C S-S F-C F-S

Present Shu [7] Irie et al [6] Present Shu [7] Irie et al [6] Present Irie et al [6] Present Irie et al [6]0 08731 08732 08731 02234 02233 02233 08696 08696 01435 014411 08120 08120 08120 05460 05463 05462 07634 07634 01660 016672 06696 06696 06696 06307 06310 06310 05289 05292 01152 011583 05430 05428 05430 05063 05062 05065 0363 03637 01007 010174 04568 04566 04570 03944 03942 03947 02818 02829 01467 014745 04092 04089 04095 03341 03340 03348 02767 02779 02089 020936 03965 03963 03970 03239 03239 03248 03184 03196 02729 027437 04144 04143 04151 03512 03514 03524 03762 03775 03345 033618 04569 04568 04577 04019 04023 04033 04398 04411 03970 039859 05177 05177 05186 04670 04676 04684 05103 05116 04655 04670

FEM

Freq

uenc

y pa

ram

eter

s

Number of grid points

m = 3

m = 2

m = 1

12

1

08

06

04

02

0

100806040200

Figure 2 Natural frequency parameters under S-S boundaryconditions (119899 = 0119898 = 1 2 3)

by this method and FEM the fast convergence behaviorand high stability of this method are observed The naturalfrequency parameters keep stable evenwith a large number ofgrid points In the following calculation the number of gridpoints will be chosen as119873 = 30

42 Validation of This Proposed Method To validate theaccuracy of the present method an example reported byIrie et al [6] and Shu [7] is adopted again Shu [7] stud-ied the free vibration behavior of this problem by theDQ method in which Lagrange interpolation functionsand Gauss-Chebyshev points are employed The geometricparameters are 119886 = 45∘ 119871 sin 119886119877

0= 05 and ℎ119877

0=

001 Table 1 shows the comparison between current naturalfrequency parameters and results studied by Irie et al [6] andShu [7] for the conical shells with C-C S-S F-C and F-Sboundary conditions The small discrepancies show a good

Freq

uenc

y pa

ram

eter

s


14

12

1

08

06

04

02

100806040200

FEMm = 3

m = 2

m = 1

Figure 3 Natural frequency parameters under S-C boundaryconditions (119899 = 3119898 = 1 2 3)

agreement Figures 4 and 5 show some selected modal shapesof the conical shell with different parameters

43 Effects of the Auxiliary Functions To study the advan-tage of introducing the auxiliary functions the weightingcoefficients obtained without auxiliary functions are adoptedto study the eigenvalue problems The constant matrix Rwithout the auxiliary functions can be rewritten as

Rcos =

[[[[[[[[[[

[

cos01205871198711199091


1198711199091

cos01205871198711199092


1198711199092

d

cos0120587119871119909119873


119871119909119873

]]]]]]]]]]

]

(23)


(a) 119899 = 1119898 = 2 (b) 119899 = 2119898 = 2 (c) 119899 = 4119898 = 3

Figure 4 Modal shapes of the conical shell under S-S boundary conditions

(a) 119899 = 3119898 = 1 (b) 119899 = 3119898 = 2 (c) 119899 = 3119898 = 4

Figure 5 Modal shapes of the conical shell under F-C boundary conditions

Table 2 Natural frequency parameters under different boundaryconditions obtained by two types of weighting coefficients (119899 = 0119898 = 0)

F-S S-S F-CWith auxiliary functions 01435 02234 08696Without auxiliary functions 08442 08772 08403Irie et al [6] 01441 02233 08696

The weighting coefficient matrix is then derived in thesame way as (13)

c(119894)cos = R(119894)cos lowast Rminus1cos (24)

in which c(119894)cos is the 119894th-order weighting coefficient matrix andR(119894)cos is the 119894th-order derivative of Rcos By using this typeof weighting coefficients natural frequency parameters arederived again to compare with the results obtained beforeTable 2 shows the frequency comparison between the resultsderived by these two types of weighting coefficients Thenumber of grid points is chosen as119873 = 35 It is clear that theintroduction of auxiliary functions will improve the accuracyof this method

44 Relation between the Numbers of Truncated FourierSeries and Grid Points For the study above the numbers oftruncated Fourier series and grid points follow the relationthat 119873 = 119872 + 5 to ensure that R is a square matrix It iswell known that pseudoinverse of R could also be adopted to

derive weighting coefficients even when119873 = 119872+5 Figure 6shows the natural frequency parameters when 119872 is set to119872 = 30 and 119873 = 10ndash100 It is clear that only if 119873 = 35that is119873 = 119872+5 the accurate results could be obtained Toimplement this method the relation between the numbers oftruncated Fourier series and grid points should be strictly setto119873 = 119872 + 5

45 The Centrosymmetric Properties of the Weighting Coef-ficients When DQ method was developed the centrosym-metric and skew centrosymmetric properties were observedshown as [21]

c(119894) = c(119894minus1) lowast c(1) (25)Equation (25) shows that the DQ weighting coefficient

matrix is skew centrosymmetric for odd derivatives (119894 is odd)and centrosymmetric for even order derivatives (119894 is even)when the grid distribution is symmetric with respect to thecenter point This conclusion is true for both uniform andnonuniform grids

In this proposedmethod the centrosymmetric propertiesare also validated Two types of weighting coefficients calcu-lated in different ways are employed to study the eigenvalueproblems those are

c(119894) = R(119894) lowast Rminus1 (26)

c(119894) = c(119894minus1) lowast c(1) (27)

To implement (27) c(1) is first derived based on (26)Table 3 shows the natural frequency parameters obtained by


Freq

uenc

y pa

ram

eter

s


Y 08731X 35

12

1

08

06

04

02

0

100806040200

(a) C-C

Freq

uenc

y pa

ram

eter

s


Y 01435X 35

12

1

08

06

04

02

0

100806040200

(b) F-S

Figure 6 Natural frequency parameters derived by different numbers of grid points when119872 = 30 ((a) C-C (b) F-S)

n = 2

n = 1

n = 0

Freq

uenc

y pa

ram

eter

s

Length parameters

1

09

08

07

06

05

04121110908

(a)

Freq

uenc

y pa

ram

eter

s

Length parameters

1

09

08

07

06

05

04

121110908

n = 2

n = 1

n = 0

(b)

Figure 7 Natural frequency parameters for the conical shell with variation of length to radius (119886 = 45∘ ℎ1198772= 001 (a) C-C (b) F-C)

these two types of weighting coefficients The small discrep-ancies between these two results show a good agreement Itis concluded that the weighting coefficient matrix derivedby this proposed method also obeys the centrosymmetricproperty

46 Effects of Geometric Parameters The geometric parame-ters play an important role in affecting the natural frequenciesof a conical shell In this part two parameters are studied tostudy their effects on the free vibration behavior of the conicalshells Figure 7 shows the natural frequency parameterschanging with variable ratio of length to radius of the conicalshell underC-C and F-C boundary conditionsThe geometricparameters are chosen as 119886 = 45∘ ℎ119877

2= 001 and

Table 3 Natural frequency parameters obtained by employing (26)and (27) (119899 = 0119898 = 0)

F-C C-C F-SBy employing (27) 08689 08730 01440By employing (26) 08689 08731 01435Irie et al [6] 08689 08731 01441

variable 119871 sin 1198861198770= 05ndash09 The frequency parameters

nearly keep constant when 119899 = 0 as increasing the ratio oflength to radius The frequency parameters decrease when119899 = 1 2 except for the case of 119899 = 2 for F-C boundaryconditions Next the effect of semivertex angle is studiedwith


Freq

uenc

y pa

ram

eter

s08

07

06

05

04

03

02

01

0

90807060504030

n = 2

n = 1

n = 0

Semivertex angle (∘)

(a)

12

1

08

06

04

02

0

90807060504030

Freq

uenc

y pa

ram

eter

s

n = 2

n = 1

n = 0


(b)

Figure 8 Natural frequency parameters of the conical shell with the variation of semivertex angle (ℎ1198772= 001 119871 sin 119886119877

2= 05 (a) S-S (b)

S-C)

the geometric parameters ℎ1198770= 001 119871 sin 119886119877

0= 05

and variable semivertex angle 119886 = 30∘ndash90∘ Figure 8 showsthe natural frequency parameters of different circumferentialwave numbers As semivertex angle increasing to 90∘ thefrequencies converge to one value This phenomenon can beexplained by the fact that the conical shell degenerates to acircular plate when semivertex angle is 90∘

5 Conclusions

In this paper a new method is proposed to generate theweighting coefficients of the DQ method The functionsin the DQ method are expressed as the Fourier cosineseries combined with close-form auxiliary functions Theweighting coefficients are directly derived by the inverse ofthe constant matrix which presents a much easier way Theboundary conditions and differential governing equations arediscretized to form the numerical eigenvalue equations Theresults obtained by this method are compared with thoseavailable in the literature and a good agreement is observedThe centrosymmetric properties of these newly proposedweighting coefficients are also validated By increasing thenumber of grid points the efficiency and high stability arepresented in this method The effect of those parameterswhich may affect the dynamic characteristics of the shell isalso studied

This method gives a much easier way to generate weight-ing coefficients in DQ algorithm It can also be extendedto study higher-order partial differential equations just byadding more corresponding supplementary functions to theFourier cosine series

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The research work is supported by the National NaturalScience Foundation of China (Grant no 51375104) and Hei-longjiang Province Funds for Distinguished Young Scientists(Grant no JC 201405)

References

[1] H Saunders E J Wisniewski and P R Paslay ldquoVibrations ofconical shellsrdquoThe Journal of the Acoustical Society of Americavol 32 pp 765ndash772 1960

[2] H Garnet and J Kempner ldquoAxisymmetric free vibrations ofconical shellsrdquo Journal of Applied Mechanics vol 31 no 3 pp458ndash466 1966

[3] C C Siu ldquoBert Free vibration analysis of sandwich conicalshells with free edgesrdquo The Journal of the Acoustical Society ofAmerica vol 47 no 3 pp 943ndash945 1970

[4] CW Lim andKM Liew ldquoVibratory behaviour of shallow con-ical shells by a global Ritz formulationrdquo Engineering Structuresvol 17 no 1 pp 63ndash70 1995

[5] T Ueda ldquoNon-linear free vibrations of conical shellsrdquo Journal ofSound and Vibration vol 64 no 1 pp 85ndash95 1979

[6] T Irie G Yamada and K Tanaka ldquoNatural frequencies oftruncated conical shellsrdquo Journal of Sound and Vibration vol92 no 3 pp 447ndash453 1984

[7] C Shu ldquoAn efficient approach for free vibration analysis ofconical shellsrdquo International Journal of Mechanical Sciences vol38 no 8-9 pp 935ndash949 1996

[8] G Jin XMa S Shi T Ye and Z Liu ldquoAmodified Fourier seriessolution for vibration analysis of truncated conical shells withgeneral boundary conditionsrdquoApplied Acoustics vol 85 pp 82ndash96 2014

[9] K M Liew T Y Ng and X Zhao ldquoFree vibration analysis ofconical shells via the element-free kp-Ritz methodrdquo Journal ofSound and Vibration vol 281 no 3-5 pp 627ndash645 2005


[10] T Y Ng H Li and K Y Lam ldquoGeneralized differentialquadrature for free vibration of rotating composite laminatedconical shell with various boundary conditionsrdquo InternationalJournal of Mechanical Sciences vol 45 no 3 pp 567ndash587 2003

[11] F Tornabene ldquoFree vibration analysis of functionally gradedconical cylindrical shell and annular plate structures with afour-parameter power-law distributionrdquo Computer Methods inAppliedMechanics and Engineering vol 198 no 37-40 pp 2911ndash2935 2009

[12] C Shu ldquoFree vibration analysis of composite laminated conicalshells by generalized differential quadraturerdquo Journal of Soundand Vibration vol 194 no 4 pp 587ndash604 1996

[13] C-PWu andC-Y Lee ldquoDifferential quadrature solution for thefree vibration analysis of laminated conical shells with variablestiffnessrdquo International Journal of Mechanical Sciences vol 43no 8 pp 1853ndash1869 2001

[14] X Zhao and KM Liew ldquoFree vibration analysis of functionallygraded conical shell panels by a meshless methodrdquo CompositeStructures vol 93 no 2 pp 649ndash664 2011

[15] A Korjakin R Rikards A Chate and H Altenbach ldquoAnalysisof free damped vibrations of laminated composite conicalshellsrdquo Composite Structures vol 41 no 1 pp 39ndash47 1998

[16] A A Lakis A Selmane and A Toledano ldquoNon-linear freevibration analysis of laminated orthotropic cylindrical shellsrdquoInternational Journal of Mechanical Sciences vol 40 no 1 pp27ndash49 1998

[17] O Civalek ldquoAn efficient method for free vibration analysisof rotating truncated conical shellsrdquo International Journal ofPressure Vessels and Piping vol 83 no 1 pp 1ndash12 2006

[18] R Bellman and J Casti ldquoDifferential quadrature and long-termintegrationrdquo Journal of Mathematical Analysis and Applicationsvol 34 pp 235ndash238 1971

[19] R Bellman BG Kashef and J Casti ldquoDifferential quadrature atechnique for the rapid solution of nonlinear partial differentialequationsrdquo Journal of Computational Physics vol 10 pp 40ndash521972

[20] C ShuGeneralized differential-integral quadrature and applica-tion to the simulation of incompressible viscous flows includingparallel computation [PhD thesis] University of Glasgow 1991

[21] C Shu Differential Quadrature and Its Application in Engineer-ing Springer Science amp Business Media London UK 2000

[22] Z Zong ldquoA variable order approach to improve differentialquadrature accuracy in dynamic analysisrdquo Journal of Sound andVibration vol 266 no 2 pp 307ndash323 2003

[23] A G Striz X Wang and C W Bert ldquoHarmonic differentialquadrature method and applications to analysis of structuralcomponentsrdquo Acta Mechanica vol 111 no 1-2 pp 85ndash94 1995

[24] H Zhong ldquoSpline-based differential quadrature for fourthorder differential equations and its application to Kirchhoffplatesrdquo Applied Mathematical Modelling vol 28 no 4 pp 353ndash366 2004

[25] W L Li ldquoFree vibrations of beams with general boundaryconditionsrdquo Journal of Sound and Vibration vol 237 no 4 pp709ndash725 2000

[26] W L Li ldquoVibration analysis of rectangular plates with generalelastic boundary supportsrdquo Journal of Sound and Vibration vol273 no 3 pp 619ndash635 2004

[27] W L Li X Zhang J Du and Z Liu ldquoAn exact series solutionfor the transverse vibration of rectangular plates with generalelastic boundary supportsrdquo Journal of Sound and Vibration vol321 no 1-2 pp 254ndash269 2009

[28] L Dai T Yang J Du W L Li and M J Brennan ldquoAn exactseries solution for the vibration analysis of cylindrical shellswith arbitrary boundary conditionsrdquo Applied Acoustics vol 74no 3 pp 440ndash449 2013

[29] J T Du W L Li H A Xu and Z G Liu ldquoVibro-acousticanalysis of a rectangular cavity bounded by a flexible panel withelastically restrained edgesrdquoThe Journal of the Acoustical Societyof America vol 131 no 4 pp 2799ndash2810 2012

[30] G Jin T Ye Y Chen Z Su andY Yan ldquoAn exact solution for thefree vibration analysis of laminated composite cylindrical shellswith general elastic boundary conditionsrdquoComposite Structuresvol 106 pp 114ndash127 2013

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



the early 1970s The DQ method akin to the conventionalintegral quadrature method approximates the derivative ofa function at any location by a linear summation of all thefunctional values along a mesh line The key procedure inthe DQ method applications lies in the determination ofthe weighting coefficients Shu [7 12 20 21] proposed twotypes of weighting coefficients obtained by the polynomialsand the truncated Fourier series among which the Lagrangeinterpolation functions are widely used for their simplicityexplicitness [7 21] Although it is well known that Lagrangeinterpolation functions are limited by the number of inter-polation points and severe oscillation may take place if theorder is large the use of the Gauss-Chebyshev points [7 21]can accelerate the convergence rate of the DQmethod Someworks focusing on improving the accuracy and stability ofthe DQ method are presented by proposing different waysto generate the weighting coefficients and to determine thedistribution of grid points [22ndash24]

The Fourier series with auxiliary functions was firstproposed by Li [25] to study the vibration problems of thebeam structure This method is extensively used to study the2D and 3D structural vibration and vibroacoustic problems[26ndash30] The auxiliary functions are introduced to acceleratethe convergence and deal with all the possible discontinuitiesat the end points or edges associatedwith the original Fouriercosine series This improved Fourier series method becomesa promising method to study the structural vibrationproblems

In this paper the improved Fourier expansion-baseddifferential quadrature method is proposed to solve thefree vibration behavior of the truncated conical shell Theweighting coefficients are obtained based on this improvedFourier series in a much easier way The centrosymmetricproperties of these newly proposed weighting coefficientsare also validated The following sections will illustrate thedevelopment of this hybrid method and numerical resultsare then presented to validate the effectiveness accuracyand stability of this current method on predicting the modalcharacteristics of the conical shell

2 Theoretical Formulation

For a continuous function 119891(119909) defined on [0 119871] with anabsolutely integrable derivative it can be expanded in Fouriercosine series

119891 (119909) =infin

sum119898=0

119886119898cos 119898120587

119871119909 0 lt 119909 lt 119871 (1)

The first-order derivative of 119891(119909) can be done term-by-term

1198911015840 (119909) = minusinfin

sum119898=0

119886119898

119898120587

119871sin 119898120587

119871119909 (2)

The second-order derivative of 119891(119909) cannot be obtainedterm-by-term which is shown as

11989110158401015840 (119909) = minus1198911015840 (119871) minus 1198911015840 (0)

119871

minusinfin

sum119898=1

(2

119871[(minus1)119898 119891 (119871) minus 119891 (0)] +

119898120587

119871119886119898)

sdot119898120587

119871cos 119898120587

119871119909

(3)

These formulations basically tell that while a cosine seriescan always be differentiated term-by-term this can be doneto a sine series only if 1198911015840(0) = 1198911015840(119871) = 0 To implementthe differential quadrature algorithm the auxiliary functionsare added to traditional Fourier cosine series to cover thediscontinuity of the function at the end points and to get thederivatives term-by-termA function can be expanded as [27]

119891 (119909) =infin

sum119898=0

119886119898cos 119898120587

119871119909 +4

sum119905=1

119862119905120585119905(119909) (4)

where

1205851(119909) =

9119871

4120587sin(120587119909

2119871) minus

119871

12120587sin(3120587119909

2119871) (5a)

1205852(119909) = minus

9119871

4120587cos(120587119909

2119871) minus

119871

12120587cos(3120587119909

2119871) (5b)

1205853(119909) =

1198713

1205873sin(120587119909

2119871) minus

1198713

31205873sin(3120587119909

2119871) (5c)

1205854(119909) = minus

1198713

1205873cos(120587119909

2119871) minus

1198713

31205873cos(3120587119909

2119871) (5d)

where 119886119898

and 119862119905represent the unknown Fourier expan-

sion coefficients The supplementary functions 120585119905(119909) can be

represented as arbitrary continuous functions regardless ofthe boundaries It is easy to verify that 1205851015840

1(0) = 120585101584010158401015840

3(0) =

120585101584010158402(0) = 120585101584010158401015840

4(0) = 1 and all the other 1st-order and 3rd-order

derivatives are identically equal to zero at both ends Themain purpose of introducing these supplementary functionsto standard Fourier series is to get the first four derivativesof the Fourier cosine series term-by-term As an immediatenumerical benefit the Fourier series in (4) will convergeuniformly at an accelerated rate

To implement the differential quadrature method 119873points are equally distributed on [0 119871]

119909119894=(119894 minus 1) 119871

119873 minus 1 119894 = 1 2 3 119873 (6)

The functional values at those grid points can be deter-mined as

119891 (119909119894) =infin

sum119898=0

119886119898cos 119898120587

119871119909119894+4

sum119905=1

119862119905120585119905(119909119894) (7)

The Fourier series is truncated to119898 = 119872 Rewrite (7) intothe matrix form


119891(1199091)

119891 (1199092)

119891 (119909119873)

=

[[[[[[[[[[

[

cos 01205871198711199091


11987111990911205851(1199091) 1205852(1199091) 1205853(1199091) 1205854(1199091)

cos 01205871198711199092


11987111990921205851(1199092) 1205852(1199092) 1205853(1199092) 1205854(1199092)

d

cos 0120587119871119909119873


1198711199091198731205851(119909119873) 1205852(119909119873) 1205853(119909119873) 1205854(119909119873)

]]]]]]]]]]

]

1198860

1198861

119886119872

1198621

1198622

1198623

1198624

(8)


1198860

1198861

119886119872

1198621

1198622

1198623

1198624

= Rminus1

119891(1199091)

119891 (1199092)

119891 (119909119873)

(9)

in which

R =

[[[[[[[[[[

[

cos01205871198711199091


11987111990911205851(1199091) 1205852(1199091) 1205853(1199091) 1205854(1199091)

cos01205871198711199092


11987111990921205851(1199092) 1205852(1199092) 1205853(1199092) 1205854(1199092)

d

cos0120587119871119909119873


1198711199091198731205851(119909119873) 1205852(119909119873) 1205853(119909119873) 1205854(119909119873)

]]]]]]]]]]

]

(10)



1198911015840 (119909119894) =119872

sum119898=0

119886119898(minus119898120587

119871) sin 119898120587

119871119909119894+4

sum119905=1

1198621199051205851015840119905(119909119894)

119894 = 1 2 3 119873

(11)


1198911015840 (1199091)

1198911015840 (1199092)

1198911015840 (119909119873)

= R(1) lowast

1198860

1198861

119886119872

1198621

1198622

1198623

1198624



119891(1199091)

119891 (1199092)

119891 (119909119873)

= c(1) lowast

119891(1199091)

119891 (1199092)

119891 (119909119873)

(12)


119891(119894) (1199091)

119891(119894) (1199092)

119891(119894) (119909119873)


119891(1199091)

119891 (1199092)

119891 (119909119873)

= c(119894) lowast

119891(1199091)

119891 (1199092)

119891 (119909119873)

(13)





r

120579

R(x)

R(x)

R0

w

u

a

L





119906 = 119880 (119909) sdot cos (119899120579) cos (120596119905)

V = 119881 (119909) sdot sin (119899120579) cos (120596119905)

119908 = 119882 (119909) sdot cos (119899120579) cos (120596119905)

(14)



11987111119906 + 11987112V + 11987113119908 = minus120588ℎ1205962119880

11987121119906 + 11987122V + 11987123119908 = minus120588ℎ1205962119881

11987131119906 + 11987132V + 11987133119908 = minus120588ℎ1205962119882

(15)


119894119895(119894 119895 =




119878110119880119894+119873

sum119896=1

(119878111119888(1)119894119896+ 119878112119888(2)119894119896)119880119896+ 119878120119881119894

+119873

sum119896=1

119878121119888(1)119894119896119881119896+ 119878130119882119894+119873

sum119896=1

119878131119888(1)119894119896119882119896

= minus120588ℎ1205962119880119894

(16a)

119878210119880119894+119873

sum119896=1

119878211119888(1)119894119896119880119896+ 119878220119881119894

+119873

sum119896=1

(119878221119888(1)119894119896+ 119878222119888(2)119894119896)119881119896+ 119878230119882119894

+119873

sum119896=1

(119878231119888(1)119894119896+ 119878232119888(2)119894119896)119882119896= minus120588ℎ1205962119881

119894

(16b)

119878310119880119894+119873

sum119896=1

119878311119888(1)119894119896119880119896+ 119878320119881119894

+119873

sum119896=1

(119878321119888(1)119894119896+ 119878322119888(2)119894119896)119881119896+ 119878330119882119894

+119873

sum119896=1

(119878331119888(1)119894119896+ 119878232119888(2)119894119896+ 119878333119888(3)119894119896+ 119878334119888(4)119894119896)119882119896

= minus120588ℎ1205962119882119894

(16c)

where 119888(119899)119894119895



follows


119881 = 0

119882 = 0

119873119909= 0

119872119909= 0

(17)


119881 = 0

119882 = 0

119880 = 0

119882(1) = 0

(18)



1198811= 0 (19a)

1198821= 0 (19b)

(V sin 1198861198771

+ 119888(1)11)1198801+ 119888(1)1119873119880119873= minus119873minus1

sum119896=2

119888(1)1119896119880119896

(19c)

(V sin 1198861198771

119888(1)12+ 119888(2)12)1198822

+ (V sin 1198861198771

119888(1)1(119873minus1)

+ 119888(2)1(119873minus1)

)119882119873minus1

= minus119873minus2

sum119896=3

(119888(2)1119896+V sin 1198861198772

119888(1)1119896)119882119896

(19d)


PX = ΩX (20)


X = 1198801 1198802 119880

119873 1198811 1198812 119881

11987311988211198822

119882119873119879

(21)


Ω119894= 1198770120596119894

radic120588 (1 minus V2)119864

(22)







ℎ1198770= 001)

119899C-C S-S F-C F-S


FEM

Freq

uenc

y pa

ram

eter

s


m = 3

m = 2

m = 1

12

1

08

06

04

02

0

100806040200




0= 05 and ℎ119877

0=


Freq

uenc

y pa

ram

eter

s


14

12

1

08

06

04

02

100806040200

FEMm = 3

m = 2

m = 1




Rcos =

[[[[[[[[[[

[

cos01205871198711199091


1198711199091

cos01205871198711199092


1198711199092

d

cos0120587119871119909119873


119871119909119873

]]]]]]]]]]

]

(23)


(a) 119899 = 1119898 = 2 (b) 119899 = 2119898 = 2 (c) 119899 = 4119898 = 3


(a) 119899 = 3119898 = 1 (b) 119899 = 3119898 = 2 (c) 119899 = 3119898 = 4













c(119894) = R(119894) lowast Rminus1 (26)

c(119894) = c(119894minus1) lowast c(1) (27)



Freq

uenc

y pa

ram

eter

s


Y 08731X 35

12

1

08

06

04

02

0

100806040200

(a) C-C

Freq

uenc

y pa

ram

eter

s


Y 01435X 35

12

1

08

06

04

02

0

100806040200

(b) F-S


n = 2

n = 1

n = 0

Freq

uenc

y pa

ram

eter

s

Length parameters

1

09

08

07

06

05

04121110908

(a)

Freq

uenc

y pa

ram

eter

s

Length parameters

1

09

08

07

06

05

04

121110908

n = 2

n = 1

n = 0

(b)




2= 001 and






Freq

uenc

y pa

ram

eter

s08

07

06

05

04

03

02

01

0

90807060504030

n = 2

n = 1

n = 0


(a)

12

1

08

06

04

02

0

90807060504030

Freq

uenc

y pa

ram

eter

s

n = 2

n = 1

n = 0


(b)


2= 05 (a) S-S (b)

S-C)


0= 05


5 Conclusions





Acknowledgments


References


































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










119891(1199091)

119891 (1199092)

119891 (119909119873)

=

[[[[[[[[[[

[

cos 01205871198711199091


11987111990911205851(1199091) 1205852(1199091) 1205853(1199091) 1205854(1199091)

cos 01205871198711199092


11987111990921205851(1199092) 1205852(1199092) 1205853(1199092) 1205854(1199092)

d

cos 0120587119871119909119873


1198711199091198731205851(119909119873) 1205852(119909119873) 1205853(119909119873) 1205854(119909119873)

]]]]]]]]]]

]

1198860

1198861

119886119872

1198621

1198622

1198623

1198624

(8)


1198860

1198861

119886119872

1198621

1198622

1198623

1198624

= Rminus1

119891(1199091)

119891 (1199092)

119891 (119909119873)

(9)

in which

R =

[[[[[[[[[[

[

cos01205871198711199091


11987111990911205851(1199091) 1205852(1199091) 1205853(1199091) 1205854(1199091)

cos01205871198711199092


11987111990921205851(1199092) 1205852(1199092) 1205853(1199092) 1205854(1199092)

d

cos0120587119871119909119873


1198711199091198731205851(119909119873) 1205852(119909119873) 1205853(119909119873) 1205854(119909119873)

]]]]]]]]]]

]

(10)



1198911015840 (119909119894) =119872

sum119898=0

119886119898(minus119898120587

119871) sin 119898120587

119871119909119894+4

sum119905=1

1198621199051205851015840119905(119909119894)

119894 = 1 2 3 119873

(11)


1198911015840 (1199091)

1198911015840 (1199092)

1198911015840 (119909119873)

= R(1) lowast

1198860

1198861

119886119872

1198621

1198622

1198623

1198624



119891(1199091)

119891 (1199092)

119891 (119909119873)

= c(1) lowast

119891(1199091)

119891 (1199092)

119891 (119909119873)

(12)


119891(119894) (1199091)

119891(119894) (1199092)

119891(119894) (119909119873)


119891(1199091)

119891 (1199092)

119891 (119909119873)

= c(119894) lowast

119891(1199091)

119891 (1199092)

119891 (119909119873)

(13)





r

120579

R(x)

R(x)

R0

w

u

a

L





119906 = 119880 (119909) sdot cos (119899120579) cos (120596119905)

V = 119881 (119909) sdot sin (119899120579) cos (120596119905)

119908 = 119882 (119909) sdot cos (119899120579) cos (120596119905)

(14)



11987111119906 + 11987112V + 11987113119908 = minus120588ℎ1205962119880

11987121119906 + 11987122V + 11987123119908 = minus120588ℎ1205962119881

11987131119906 + 11987132V + 11987133119908 = minus120588ℎ1205962119882

(15)


119894119895(119894 119895 =




119878110119880119894+119873

sum119896=1

(119878111119888(1)119894119896+ 119878112119888(2)119894119896)119880119896+ 119878120119881119894

+119873

sum119896=1

119878121119888(1)119894119896119881119896+ 119878130119882119894+119873

sum119896=1

119878131119888(1)119894119896119882119896

= minus120588ℎ1205962119880119894

(16a)

119878210119880119894+119873

sum119896=1

119878211119888(1)119894119896119880119896+ 119878220119881119894

+119873

sum119896=1

(119878221119888(1)119894119896+ 119878222119888(2)119894119896)119881119896+ 119878230119882119894

+119873

sum119896=1

(119878231119888(1)119894119896+ 119878232119888(2)119894119896)119882119896= minus120588ℎ1205962119881

119894

(16b)

119878310119880119894+119873

sum119896=1

119878311119888(1)119894119896119880119896+ 119878320119881119894

+119873

sum119896=1

(119878321119888(1)119894119896+ 119878322119888(2)119894119896)119881119896+ 119878330119882119894

+119873

sum119896=1

(119878331119888(1)119894119896+ 119878232119888(2)119894119896+ 119878333119888(3)119894119896+ 119878334119888(4)119894119896)119882119896

= minus120588ℎ1205962119882119894

(16c)

where 119888(119899)119894119895



follows


119881 = 0

119882 = 0

119873119909= 0

119872119909= 0

(17)


119881 = 0

119882 = 0

119880 = 0

119882(1) = 0

(18)



1198811= 0 (19a)

1198821= 0 (19b)

(V sin 1198861198771

+ 119888(1)11)1198801+ 119888(1)1119873119880119873= minus119873minus1

sum119896=2

119888(1)1119896119880119896

(19c)

(V sin 1198861198771

119888(1)12+ 119888(2)12)1198822

+ (V sin 1198861198771

119888(1)1(119873minus1)

+ 119888(2)1(119873minus1)

)119882119873minus1

= minus119873minus2

sum119896=3

(119888(2)1119896+V sin 1198861198772

119888(1)1119896)119882119896

(19d)


PX = ΩX (20)


X = 1198801 1198802 119880

119873 1198811 1198812 119881

11987311988211198822

119882119873119879

(21)


Ω119894= 1198770120596119894

radic120588 (1 minus V2)119864

(22)







ℎ1198770= 001)

119899C-C S-S F-C F-S


FEM

Freq

uenc

y pa

ram

eter

s


m = 3

m = 2

m = 1

12

1

08

06

04

02

0

100806040200




0= 05 and ℎ119877

0=


Freq

uenc

y pa

ram

eter

s


14

12

1

08

06

04

02

100806040200

FEMm = 3

m = 2

m = 1




Rcos =

[[[[[[[[[[

[

cos01205871198711199091


1198711199091

cos01205871198711199092


1198711199092

d

cos0120587119871119909119873


119871119909119873

]]]]]]]]]]

]

(23)


(a) 119899 = 1119898 = 2 (b) 119899 = 2119898 = 2 (c) 119899 = 4119898 = 3


(a) 119899 = 3119898 = 1 (b) 119899 = 3119898 = 2 (c) 119899 = 3119898 = 4













c(119894) = R(119894) lowast Rminus1 (26)

c(119894) = c(119894minus1) lowast c(1) (27)



Freq

uenc

y pa

ram

eter

s


Y 08731X 35

12

1

08

06

04

02

0

100806040200

(a) C-C

Freq

uenc

y pa

ram

eter

s


Y 01435X 35

12

1

08

06

04

02

0

100806040200

(b) F-S


n = 2

n = 1

n = 0

Freq

uenc

y pa

ram

eter

s

Length parameters

1

09

08

07

06

05

04121110908

(a)

Freq

uenc

y pa

ram

eter

s

Length parameters

1

09

08

07

06

05

04

121110908

n = 2

n = 1

n = 0

(b)




2= 001 and






Freq

uenc

y pa

ram

eter

s08

07

06

05

04

03

02

01

0

90807060504030

n = 2

n = 1

n = 0


(a)

12

1

08

06

04

02

0

90807060504030

Freq

uenc

y pa

ram

eter

s

n = 2

n = 1

n = 0


(b)


2= 05 (a) S-S (b)

S-C)


0= 05


5 Conclusions





Acknowledgments


References


































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119891(1199091)

119891 (1199092)

119891 (119909119873)

= c(1) lowast

119891(1199091)

119891 (1199092)

119891 (119909119873)

(12)


119891(119894) (1199091)

119891(119894) (1199092)

119891(119894) (119909119873)


119891(1199091)

119891 (1199092)

119891 (119909119873)

= c(119894) lowast

119891(1199091)

119891 (1199092)

119891 (119909119873)

(13)





r

120579

R(x)

R(x)

R0

w

u

a

L





119906 = 119880 (119909) sdot cos (119899120579) cos (120596119905)

V = 119881 (119909) sdot sin (119899120579) cos (120596119905)

119908 = 119882 (119909) sdot cos (119899120579) cos (120596119905)

(14)



11987111119906 + 11987112V + 11987113119908 = minus120588ℎ1205962119880

11987121119906 + 11987122V + 11987123119908 = minus120588ℎ1205962119881

11987131119906 + 11987132V + 11987133119908 = minus120588ℎ1205962119882

(15)


119894119895(119894 119895 =




119878110119880119894+119873

sum119896=1

(119878111119888(1)119894119896+ 119878112119888(2)119894119896)119880119896+ 119878120119881119894

+119873

sum119896=1

119878121119888(1)119894119896119881119896+ 119878130119882119894+119873

sum119896=1

119878131119888(1)119894119896119882119896

= minus120588ℎ1205962119880119894

(16a)

119878210119880119894+119873

sum119896=1

119878211119888(1)119894119896119880119896+ 119878220119881119894

+119873

sum119896=1

(119878221119888(1)119894119896+ 119878222119888(2)119894119896)119881119896+ 119878230119882119894

+119873

sum119896=1

(119878231119888(1)119894119896+ 119878232119888(2)119894119896)119882119896= minus120588ℎ1205962119881

119894

(16b)

119878310119880119894+119873

sum119896=1

119878311119888(1)119894119896119880119896+ 119878320119881119894

+119873

sum119896=1

(119878321119888(1)119894119896+ 119878322119888(2)119894119896)119881119896+ 119878330119882119894

+119873

sum119896=1

(119878331119888(1)119894119896+ 119878232119888(2)119894119896+ 119878333119888(3)119894119896+ 119878334119888(4)119894119896)119882119896

= minus120588ℎ1205962119882119894

(16c)

where 119888(119899)119894119895



follows


119881 = 0

119882 = 0

119873119909= 0

119872119909= 0

(17)


119881 = 0

119882 = 0

119880 = 0

119882(1) = 0

(18)



1198811= 0 (19a)

1198821= 0 (19b)

(V sin 1198861198771

+ 119888(1)11)1198801+ 119888(1)1119873119880119873= minus119873minus1

sum119896=2

119888(1)1119896119880119896

(19c)

(V sin 1198861198771

119888(1)12+ 119888(2)12)1198822

+ (V sin 1198861198771

119888(1)1(119873minus1)

+ 119888(2)1(119873minus1)

)119882119873minus1

= minus119873minus2

sum119896=3

(119888(2)1119896+V sin 1198861198772

119888(1)1119896)119882119896

(19d)


PX = ΩX (20)


X = 1198801 1198802 119880

119873 1198811 1198812 119881

11987311988211198822

119882119873119879

(21)


Ω119894= 1198770120596119894

radic120588 (1 minus V2)119864

(22)







ℎ1198770= 001)

119899C-C S-S F-C F-S


FEM

Freq

uenc

y pa

ram

eter

s


m = 3

m = 2

m = 1

12

1

08

06

04

02

0

100806040200




0= 05 and ℎ119877

0=


Freq

uenc

y pa

ram

eter

s


14

12

1

08

06

04

02

100806040200

FEMm = 3

m = 2

m = 1




Rcos =

[[[[[[[[[[

[

cos01205871198711199091


1198711199091

cos01205871198711199092


1198711199092

d

cos0120587119871119909119873


119871119909119873

]]]]]]]]]]

]

(23)


(a) 119899 = 1119898 = 2 (b) 119899 = 2119898 = 2 (c) 119899 = 4119898 = 3


(a) 119899 = 3119898 = 1 (b) 119899 = 3119898 = 2 (c) 119899 = 3119898 = 4













c(119894) = R(119894) lowast Rminus1 (26)

c(119894) = c(119894minus1) lowast c(1) (27)



Freq

uenc

y pa

ram

eter

s


Y 08731X 35

12

1

08

06

04

02

0

100806040200

(a) C-C

Freq

uenc

y pa

ram

eter

s


Y 01435X 35

12

1

08

06

04

02

0

100806040200

(b) F-S


n = 2

n = 1

n = 0

Freq

uenc

y pa

ram

eter

s

Length parameters

1

09

08

07

06

05

04121110908

(a)

Freq

uenc

y pa

ram

eter

s

Length parameters

1

09

08

07

06

05

04

121110908

n = 2

n = 1

n = 0

(b)




2= 001 and






Freq

uenc

y pa

ram

eter

s08

07

06

05

04

03

02

01

0

90807060504030

n = 2

n = 1

n = 0


(a)

12

1

08

06

04

02

0

90807060504030

Freq

uenc

y pa

ram

eter

s

n = 2

n = 1

n = 0


(b)


2= 05 (a) S-S (b)

S-C)


0= 05


5 Conclusions





Acknowledgments


References


































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119878110119880119894+119873

sum119896=1

(119878111119888(1)119894119896+ 119878112119888(2)119894119896)119880119896+ 119878120119881119894

+119873

sum119896=1

119878121119888(1)119894119896119881119896+ 119878130119882119894+119873

sum119896=1

119878131119888(1)119894119896119882119896

= minus120588ℎ1205962119880119894

(16a)

119878210119880119894+119873

sum119896=1

119878211119888(1)119894119896119880119896+ 119878220119881119894

+119873

sum119896=1

(119878221119888(1)119894119896+ 119878222119888(2)119894119896)119881119896+ 119878230119882119894

+119873

sum119896=1

(119878231119888(1)119894119896+ 119878232119888(2)119894119896)119882119896= minus120588ℎ1205962119881

119894

(16b)

119878310119880119894+119873

sum119896=1

119878311119888(1)119894119896119880119896+ 119878320119881119894

+119873

sum119896=1

(119878321119888(1)119894119896+ 119878322119888(2)119894119896)119881119896+ 119878330119882119894

+119873

sum119896=1

(119878331119888(1)119894119896+ 119878232119888(2)119894119896+ 119878333119888(3)119894119896+ 119878334119888(4)119894119896)119882119896

= minus120588ℎ1205962119882119894

(16c)

where 119888(119899)119894119895



follows


119881 = 0

119882 = 0

119873119909= 0

119872119909= 0

(17)


119881 = 0

119882 = 0

119880 = 0

119882(1) = 0

(18)



1198811= 0 (19a)

1198821= 0 (19b)

(V sin 1198861198771

+ 119888(1)11)1198801+ 119888(1)1119873119880119873= minus119873minus1

sum119896=2

119888(1)1119896119880119896

(19c)

(V sin 1198861198771

119888(1)12+ 119888(2)12)1198822

+ (V sin 1198861198771

119888(1)1(119873minus1)

+ 119888(2)1(119873minus1)

)119882119873minus1

= minus119873minus2

sum119896=3

(119888(2)1119896+V sin 1198861198772

119888(1)1119896)119882119896

(19d)


PX = ΩX (20)


X = 1198801 1198802 119880

119873 1198811 1198812 119881

11987311988211198822

119882119873119879

(21)


Ω119894= 1198770120596119894

radic120588 (1 minus V2)119864

(22)







ℎ1198770= 001)

119899C-C S-S F-C F-S


FEM

Freq

uenc

y pa

ram

eter

s


m = 3

m = 2

m = 1

12

1

08

06

04

02

0

100806040200




0= 05 and ℎ119877

0=


Freq

uenc

y pa

ram

eter

s


14

12

1

08

06

04

02

100806040200

FEMm = 3

m = 2

m = 1




Rcos =

[[[[[[[[[[

[

cos01205871198711199091


1198711199091

cos01205871198711199092


1198711199092

d

cos0120587119871119909119873


119871119909119873

]]]]]]]]]]

]

(23)


(a) 119899 = 1119898 = 2 (b) 119899 = 2119898 = 2 (c) 119899 = 4119898 = 3


(a) 119899 = 3119898 = 1 (b) 119899 = 3119898 = 2 (c) 119899 = 3119898 = 4













c(119894) = R(119894) lowast Rminus1 (26)

c(119894) = c(119894minus1) lowast c(1) (27)



Freq

uenc

y pa

ram

eter

s


Y 08731X 35

12

1

08

06

04

02

0

100806040200

(a) C-C

Freq

uenc

y pa

ram

eter

s


Y 01435X 35

12

1

08

06

04

02

0

100806040200

(b) F-S


n = 2

n = 1

n = 0

Freq

uenc

y pa

ram

eter

s

Length parameters

1

09

08

07

06

05

04121110908

(a)

Freq

uenc

y pa

ram

eter

s

Length parameters

1

09

08

07

06

05

04

121110908

n = 2

n = 1

n = 0

(b)




2= 001 and






Freq

uenc

y pa

ram

eter

s08

07

06

05

04

03

02

01

0

90807060504030

n = 2

n = 1

n = 0


(a)

12

1

08

06

04

02

0

90807060504030

Freq

uenc

y pa

ram

eter

s

n = 2

n = 1

n = 0


(b)


2= 05 (a) S-S (b)

S-C)


0= 05


5 Conclusions





Acknowledgments


References


































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











ℎ1198770= 001)

119899C-C S-S F-C F-S


FEM

Freq

uenc

y pa

ram

eter

s


m = 3

m = 2

m = 1

12

1

08

06

04

02

0

100806040200




0= 05 and ℎ119877

0=


Freq

uenc

y pa

ram

eter

s


14

12

1

08

06

04

02

100806040200

FEMm = 3

m = 2

m = 1




Rcos =

[[[[[[[[[[

[

cos01205871198711199091


1198711199091

cos01205871198711199092


1198711199092

d

cos0120587119871119909119873


119871119909119873

]]]]]]]]]]

]

(23)


(a) 119899 = 1119898 = 2 (b) 119899 = 2119898 = 2 (c) 119899 = 4119898 = 3


(a) 119899 = 3119898 = 1 (b) 119899 = 3119898 = 2 (c) 119899 = 3119898 = 4













c(119894) = R(119894) lowast Rminus1 (26)

c(119894) = c(119894minus1) lowast c(1) (27)



Freq

uenc

y pa

ram

eter

s


Y 08731X 35

12

1

08

06

04

02

0

100806040200

(a) C-C

Freq

uenc

y pa

ram

eter

s


Y 01435X 35

12

1

08

06

04

02

0

100806040200

(b) F-S


n = 2

n = 1

n = 0

Freq

uenc

y pa

ram

eter

s

Length parameters

1

09

08

07

06

05

04121110908

(a)

Freq

uenc

y pa

ram

eter

s

Length parameters

1

09

08

07

06

05

04

121110908

n = 2

n = 1

n = 0

(b)




2= 001 and






Freq

uenc

y pa

ram

eter

s08

07

06

05

04

03

02

01

0

90807060504030

n = 2

n = 1

n = 0


(a)

12

1

08

06

04

02

0

90807060504030

Freq

uenc

y pa

ram

eter

s

n = 2

n = 1

n = 0


(b)


2= 05 (a) S-S (b)

S-C)


0= 05


5 Conclusions





Acknowledgments


References


































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










(a) 119899 = 1119898 = 2 (b) 119899 = 2119898 = 2 (c) 119899 = 4119898 = 3


(a) 119899 = 3119898 = 1 (b) 119899 = 3119898 = 2 (c) 119899 = 3119898 = 4













c(119894) = R(119894) lowast Rminus1 (26)

c(119894) = c(119894minus1) lowast c(1) (27)



Freq

uenc

y pa

ram

eter

s


Y 08731X 35

12

1

08

06

04

02

0

100806040200

(a) C-C

Freq

uenc

y pa

ram

eter

s


Y 01435X 35

12

1

08

06

04

02

0

100806040200

(b) F-S


n = 2

n = 1

n = 0

Freq

uenc

y pa

ram

eter

s

Length parameters

1

09

08

07

06

05

04121110908

(a)

Freq

uenc

y pa

ram

eter

s

Length parameters

1

09

08

07

06

05

04

121110908

n = 2

n = 1

n = 0

(b)




2= 001 and






Freq

uenc

y pa

ram

eter

s08

07

06

05

04

03

02

01

0

90807060504030

n = 2

n = 1

n = 0


(a)

12

1

08

06

04

02

0

90807060504030

Freq

uenc

y pa

ram

eter

s

n = 2

n = 1

n = 0


(b)


2= 05 (a) S-S (b)

S-C)


0= 05


5 Conclusions





Acknowledgments


References


































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Freq

uenc

y pa

ram

eter

s


Y 08731X 35

12

1

08

06

04

02

0

100806040200

(a) C-C

Freq

uenc

y pa

ram

eter

s


Y 01435X 35

12

1

08

06

04

02

0

100806040200

(b) F-S


n = 2

n = 1

n = 0

Freq

uenc

y pa

ram

eter

s

Length parameters

1

09

08

07

06

05

04121110908

(a)

Freq

uenc

y pa

ram

eter

s

Length parameters

1

09

08

07

06

05

04

121110908

n = 2

n = 1

n = 0

(b)




2= 001 and






Freq

uenc

y pa

ram

eter

s08

07

06

05

04

03

02

01

0

90807060504030

n = 2

n = 1

n = 0


(a)

12

1

08

06

04

02

0

90807060504030

Freq

uenc

y pa

ram

eter

s

n = 2

n = 1

n = 0


(b)


2= 05 (a) S-S (b)

S-C)


0= 05


5 Conclusions





Acknowledgments


References


































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Freq

uenc

y pa

ram

eter

s08

07

06

05

04

03

02

01

0

90807060504030

n = 2

n = 1

n = 0


(a)

12

1

08

06

04

02

0

90807060504030

Freq

uenc

y pa

ram

eter

s

n = 2

n = 1

n = 0


(b)


2= 05 (a) S-S (b)

S-C)


0= 05


5 Conclusions





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









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