research article sensitivity analysis of deviation source

Research ArticleSensitivity Analysis of Deviation Source for Fast AssemblyPrecision Optimization

Jianjun Tang Xitian Tian and Junhao Geng

Institute of CAPP ampManufacturing Engineering Software Northwestern Polytechnical University Xirsquoan 710072 China

Correspondence should be addressed to Junhao Geng gengjunhaonwpueducn

Received 26 December 2013 Accepted 28 February 2014 Published 17 April 2014

Academic Editor Manyu Xiao

Copyright copy 2014 Jianjun Tang 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

Assembly precision optimization of complex product has a huge benefit in improving the quality of our products Due to theimpact of a variety of deviation source coupling phenomena the goal of assembly precision optimization is difficult to be confirmedaccurately In order to achieve optimization of assembly precision accurately and rapidly sensitivity analysis of deviation source isproposed First deviation source sensitivity is defined as the ratio of assembly dimension variation and deviation source dimensionvariation Second according to assembly constraint relations assembly sequences and locating deviation transmission paths areestablished by locating the joints between the adjacent parts and establishing each partrsquos datum reference frame Third assemblymultidimensional vector loops are created using deviation transmission paths and the corresponding scalar equations of eachdimension are established Then assembly deviation source sensitivity is calculated by using a first-order Taylor expansion andmatrix transformation method Finally taking assembly precision optimization of wing flap rocker as an example the effectivenessand efficiency of the deviation source sensitivity analysis method are verified

1 Introduction

In the aerospace industry products are more and more com-plex and productsrsquo precision requirements are also increas-ing Precision performance of complex products is mainlyguaranteed by the assembly process The final assemblyprecision is affected by multiple assembly deviation sourcesThe degree of influence is called sensitivity And the differentsensitivities lead to the result that the difficulty of precisionoptimization [1] is different Therefore assembly precisioncan be optimized accurately and rapidly by analyzing eachdeviation sourcersquos sensitivity and reducing tolerance of thedeviation source which has large sensitivity

A typical assembly precision model requires input dataincluding component geometry and tolerance specificationsand assembly information (such as assembly sequenceslocating and clamping) to produce the desired dimensionaloutput Many commercial software packages exist for thispurpose such as vis VSA [2] and CETOL6120590 [3] usingMonte Carlo simulations These tools are built on com-monly accepted GDampT standards [4] and adopt more recent

research results as reported in [5ndash7]The basic assumption isthat the product is comprised of rigid bodies

Tolerance design in computer aided process planningneeds to obtain an appropriate set of manufacturing tol-erances for the various manufacturing operations involvedconsidering process capability of the machines andmanufac-turing allowance for each operation in succession Aimingat drawbacks of tolerance design [8ndash11] a few authorsproposed a series of optimizationmethods [12ndash15] Chen andChung [16] introduced a model to determine the inspectionprecision and the optimal number of repeated measurementsin order to maximize the net expected profit per itemKannan and Jayabalan [17] used a Genetic Algorithm tosolve the problem by generating six partitions for eachcomponent (subassembly) One kind of method used smalldisplacements torsor (SDT) to model the process planning[18]

Assembly precision optimization [19ndash22] is an importantmeans to ensure product quality Assembly deviation sourcesensitivity analysis [23ndash26] provides critical informationfor assembly precision optimization ANSELMETTI [27]

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 148360 7 pageshttpdxdoiorg1011552014148360

2 Mathematical Problems in Engineering

proposed the concept of deviation sources Deviation trans-mission mechanism was analyzed but the effect of devi-ation source sensitivity to cumulative deviation was notconsidered in the multidimensional environment Mansuyet al [28] introduced segmented geometric elements alongthe three-dimensional vector direction and then calculatedthe sensitivity and predicted the precision Laperriere andElMaraghy [29] proposed a method of tolerance analysis bydescribing the tolerance vector with six scalar equations inthree-dimensional space But it did not consider the role ofassembly datum in the process of deviation transmission

Because the physical world is predominantly nonlin-ear Monte Carlo simulation is naturally the most accu-rate and sometimes the only method for tolerance analysisfor a generic assembly product However computationalinefficiency from Monte Carlo simulations hinders manyadvanced features such as assembly precision and sequenceoptimization which all require a significant number ofanalysis iterations

For most engineering applications including the wingassembly the nonlinear kinematic relations can be approx-imated by linear models through a first-order Taylor seriesexpansionTherefore developing an efficient linear deviationsource sensitivity analysis method is both imperative andfeasible In multidimensional space for a comprehensivesensitivity analysis of mutual coupled deviation source anassembly deviation source sensitivity calculation method isproposed based on multidimensional vector loops Combin-ing the characteristic of complex product assembly multi-dimensional vector loops are built Based on the loops thesensitivity of each deviation source is calculated by reduced-order operation

This paper is organized as follows Section 2 gives theprinciple of assembly precision optimization Section 3 pro-poses a method of deviation source sensitivity analysisSection 4 presents a computational example to demonstratethe method Section 5 comes up with a summary for thispaper

2 Principle of AssemblyPrecision Optimization

Tolerance analysis methods have the worst case and RSSmethod shown as follows [21]

119905wc =

119899

sum

119894=1

1003816100381610038161003816119886119894119905119894

1003816100381610038161003816 (1)

119905rss = radic

119899

sum

119894=1

(119886119894119905119894)2

(2)

Assembly precision optimization is a process of solvingthe minimum value of assembly deviation value Anotherway to determine value of sensitivity 119886

119894 is developed for fast

assembly precision optimization

min 119905wc = min119899

sum

119894=1

1003816100381610038161003816119886119894119905119894

1003816100381610038161003816

min 119905rss = radicmin119899

sum

119894=1

(119886119894119905119894)2

(3)

According to (3) 119886119894determines the influence of 119905

119894to

119905wc and 119905rss In process of assembly precision optimizationby determining the 119886

119894 the deviation source which needs to

be optimized can be quickly determined then min 119905wc andmin 119905rss can be obtained So the most important thing ofassembly precision optimization is calculating each of thedeviation source sensitivities

In one-dimensional space 119886119894is defined as plusmn1 (increases

ring 119886119894= +1 decreases ring 119886

119894= minus1) In multidimensional

space because of the uncertainty of deviation vector direc-tion 119886

119894can be dramatically magnified So every deviation

source sensitivity 119886119894should be calculated Determining which

119886119894is bigger the corresponding deviation source is the one

which needs to be reduced By decreasing value of deviationsource assembly precision optimization is realized effectively

The result of assembly deviation accumulation expressesthe variable between actual assembly dimension and designdimension The sensitivities are solved by taking partialderivatives with respect to each variable

Assume that 1199101 1199102 119910

119899 are the corresponding

partrsquos dimensions of deviation sources and a functionΘ(1199101 1199102 119910

119899) expresses value of assembly dimension The

sensitivity of Θ with respect to 119910119894is

119886119894=

120597Θ

120597119910119894

10038161003816100381610038161003816100381610038161003816NominalValues

(4)

The partial derivative at the nominal values of eachvariable is evaluated And the nominal value for each variableis the center of a tolerance range or the value of the dimensionwhen the tolerances are equal bilaterally

3 Deviation Source SensitivityAnalysis Method

In the process of complicated product assembly a varietyof joint types are required to describe the mating partsrsquocontact points Different joint types form different kinds ofdeviation source couplings And different couplings lead todifferent results of deviation transmission Now deviationtransmission path vector loop equations and sensitivitycalculation method are discussed respectively

31 Deviation Transmission Path Based on Datum ReferenceFrames Assembly parts are located through the locatingdatum in the process of product assembly In the process offorming deviation transmission path partrsquos locating datumis called datum reference frame (DRF) DRF is mainly usedfor locating every needed feature of a part Datum path (DP)is a path from the joint to the DRF which is connected withnominal dimension vectors A deviation transmission pathof two constrained parts is along a DP and through a joint

Mathematical Problems in Engineering 3

It must obey certain modeling rules as it passes through apart It must

(1) enter a part through a joint(2) follow the DP to the DRF in the part(3) follow a second DP leading to another joint(4) exit to the next adjacent part from the joint in the

assemblyThe deviation transmission path is illustrated in Figure 1

119869 (Part1 Part2) is the joint of Part1 and Part2 There are fourDPs in Part1 and Part2 (DP11 DP12 DP21 and DP22)They arecreated by dimension vectors (a b c d e f g and h) All thedimension vectors are important for sensitivity analysis

32 Dimension Vector Loops and Vector Equations Accord-ing to productrsquos assembly sequence and locating a dimen-sion vector loop is formed by connecting all the deviationtransmission paths Dimension vector loop is divided intoclosed loop and opened loop Closed loop describes relationsbetween nominal dimensions of parts and assembly dimen-sions Opened loop describes the influence of partsrsquo nominaldimensions to the key characteristics in the assembly Mod-eling rules for dimension vector loops include the following

(1) Loops must pass through every part and every jointin the assembly

(2) A single vector loop passes through the same joint nomore than once but it may start and end in the samepart

(3) If a vector loop includes the exact same dimensiontwice in opposite directions the dimension must beomitted

(4) For an assembly the number of closed loop 119871 can beexpressed as

119871 = 119869 minus 119875 + 1 (5)

where 119869 is the number of joints and 119875 is the number of partsAssuming that an assembly contains part dimensions

1199091 1199092 119909

119899 and assembly dimensions 119880

1 1198802 119880

119898

multidimensional vector closed loops can be expressed asℎ119863

= 119863 (1199091 1199092 119909

119899 1198801 1198802 119880

119898) = 0 (6)

where 119863 is the vector direction of dimensions for examplein three-dimensional space119863 = (119906 V 119908 120572 120573 120574) and in two-dimensional space 119863 = (119909 119910 120579) The parameters (119906 V 119908 119909and 119910) are location parameters of a vector The parameters(120572 120573 120574 and 120579) are direction parameters of a vector

Multidimensional vector opened loops can be expressedas

Gap = Γ (1199091 1199092 119909

119899 1198801 1198802 119880

119898) (7)

where Γ is the vector direction along the opened loopdistance

By (4) the explicit expression of 119909 and 119880 should becalculated But (6) and (7) are nonlinear and implicitThey contain products and trigonometric functions of thevariables So certain mathematical methods are needed forsensitivity analysis

33 Order Reduction of Vector Loop Equations The assemblydeviation describes a small variable of dimension For explicitexpression of 119909 and 119880 first-order Taylorrsquos series expansion[30] of (6) and (7) is used The following equations show theexplicit expression for closed loop and opened loop

120575ℎ119863

=

120597ℎ119863

1205971199091

1205751199091+ sdot sdot sdot +

120597ℎ119863

120597119909119899

120575119909119899

+

120597ℎ119863

1205971198801

1205751198801+ sdot sdot sdot +

120597ℎ119863

120597119880119899

120575119880119899= 0

(8)

120575Gap =

120597Gap1205971199091

1205751199091+ sdot sdot sdot +

120597Gap120597119909119899

120575119909119899

+

120597Gap1205971198801

1205751198801+ sdot sdot sdot +

120597Gap120597119880119899

120575119880119899

(9)

Equations (8) and (9) may be written in matrix form andsolved for the deviation source sensitivities bymatrix algebraEquation (8) can be expressed in matrix form as follows

[119860] [120575119909] + [119861] [120575119880] = 0 (10)

Themultidimensional vector opened loop scalar equation (9)can be expressed in matrix form as follows

[120575Gap] = [119862] [120575119909] + [119864] [120575119880] (11)

where [119860] is the partial derivative matrix of a closed loopscalar equation to the part dimension variables [119861] is thepartial derivative matrix of a closed loop scalar equationto the assembly dimension variables [120575119909] is the vector ofsmall variations in the part dimensions [120575119880] is the vectorof small variations in the assembly dimensions [119862] is thepartial derivative matrix of an opened loop scalar equationto the part dimension variables [119864] is the partial derivativematrix of an opened loop scalar equation to the assemblydimension variables and [120575Gap] is the vector of variationsin the assembly key characteristics

By matrix transformation (10) and (11) are expressed inthe following form

[120575119880] = minus [119861minus1

119860] [120575119909]

[120575Gap] = [119862 minus 119864119861minus1

119860] [120575119909]

(12)

The matrix minus[119861minus1

119860] is the sensitivity matrix of assemblydimension variables with respect to deviation source dimen-sion variables The matrix [119862 minus 119864119861

minus1

119860] is the sensitivitymatrix of assembly key characteristics dimension variableswith respect to deviation source dimension variables

4 Computational Experiment

Taking assembly deviation source sensitivity analysis of wingflap rocker as a research example the deviation transmissionpaths and dimension vector loops are established The maininfluence factors of assembly precision optimization areanalyzed based on calculating deviation source sensitivity


Planar Cylindrical slider Edge slider Parallel cylinders

120601R

U U U

120601 R2R1

120601

DRFi

DRFj

DRFiDRFiDRFi

DRFjDRFj

DRFj

DRF1

ab

cd

DP 11

DP12

DRF2

e

f gDP 21

DP22

h

J (part1 part2)

Part1 Part2

Figure 1 Deviation transmission path formed by four kinds of joint types

x2

02x1

x5

x6

x3 x8

x9

x10

x11 x12 U1

x13U2

x4 x7

AA

ProckerPpin

Pwing body

Pfront connector

Pback connector

Figure 2 A profile of wing flap rocker assembly

Wing flap rocker mainly contains five components119875wing body 119875front connector 119875back connector 119875rocker and 119875pin shownin Figure 2 The deviation sources are 119909

1 1199092 119909

13and

perp| 02 | A All the deviation sources dimensionsrsquo nominalvalues and tolerances are shown in Table 1

The process of wing flap rocker assembly deviationsource sensitivity analysis based on dimension vector loopsis illustrated in Figure 3

First of all DPs are established According to assemblyconstraints the connection types and joints (119869(119875rocker119875pin) 119869(119875pin 119875front connector) 119869(119875front connector 119875wing body)119869(119875wing body 119875back connector) and 119869(119875back connector 119875rocker))in the assembly are located DRFs (DRFrockerDRFpinDRFfront connector DRFwing body and DRFback connector) aredefined based on factors such as component designreferences and assembly locating datum DPs from jointsto DRFs are created along the nominal dimension vectordirections as shown in Figure 4

Then the deviation transmission paths and multidimen-sional vector loops are created based on the DPs as shown inFigure 5 Figure 5(a) shows assembly deviation transmissionvector closed loop Because of the perp| 02 | A 119886

14becomes

a deviation source Figure 5(b) shows assembly deviation

transmission vector opened loop Gap means the distancefrom DRFwing body to 119909

1

Finally (13) is generated according to the dimensionality119863 = 119909 119910 120579 The parameters 120575119909 120575119880 119860 119861 119862 and 119864 aresolved by first-order Taylorrsquos series expansion The param-eters are shown in (14) Sensitivities are solved by matrixoperation as shown in Table 2

ℎ119909= 1199091cos (0) + 119909

3cos (119909

14) + 1199094cos (0) + 119909

8cos (minus119880

3)

+ 1199099cos (90) + 119909

10cos (90) + 119880

1cos (minus119909

12)

+ 11990913cos (minus119880

2minus 90) + 119909

7cos (90)

+ 1199095cos (minus180 + 119909

6) + 1199092cos (minus180) = 0

ℎ119910= 1199091sin (0) + 119909

3sin (11988614) + 1199094sin (0) + 119909

8sin (minus119880

3)

+ 1199099sin (90) + 119909

10sin (90) + 119880

1sin (minus119909

12)

+ 11990913sin (minus119880

2minus 90) + 119909

7sin (90)

+ 1199095sin (minus180 + 119886

6) + 1199092sin (minus180) = 0

ℎ120579= 0 + 119909

14minus 90 minus 119880

3+ 180 minus (90 + 119909

12)

minus 90 minus (180 minus 1198802) + 90 minus 119909

6+ 180 = 0


Table 1 The deviation sources dimensionsrsquo nominal values and tolerances

Dimension parameter 1199091mm 119909

2mm 119909

3mm 119909

4mm 119909

5mm 119909

6deg 119909

7mm 119909

8mm 119909

9mm

Nominal value 260 320 42 20 221 9 28 10 30Tolerance plusmn03 plusmn03 plusmn03 plusmn03 plusmn03 plusmn05 plusmn03 plusmn03 plusmn03Dimension parameter 119909

10mm 119909

11mm 119909

12deg 119909

13mm 119909

14deg 119880

1mm 119880

2deg 119880

3deg

Nominal value 10 150 6 80 90 251 15 90Tolerance plusmn03 plusmn03 plusmn05 plusmn03 plusmn02 plusmn05 plusmn1 plusmn1

Set up the wing flap rocker

assembly graph

Locate joints by assembly constraint relations J(Pi Pj)

Define datum referenceframes for every part based

Generate everydimensionalityrsquosscalar equation

on locating DRFs

Create datum pathsfrom J(Pi Pj) to

DRFs DPs

Create deviationtransmission pathand vector loops

Calculate derivativesand form matrix

equations

Solve fordeviation source

sensitivities

Figure 3 The process of assembly deviation source sensitivity analysis

Gap = (11990911

+ 1198801) sin (minus119909

12) + 11990913sin (minus119880

2minus 90)

+ 1199097sin (90) + 119909

5sin (minus180 + 119909

6)

(13)

[119860] =

[[[[[[[[[[

[

120597ℎ119909

1205971199091

120597ℎ119909

1205971199092

sdot sdot sdot

120597ℎ119909

12059711990914

120597ℎ119910

1205971199091

120597ℎ119910

1205971199092

sdot sdot sdot

120597ℎ119910

12059711990914

120597ℎ120579

1205971199091

120597ℎ120579

1205971199092

sdot sdot sdot

120597ℎ120579

12059711990914

]]]]]]]]]]

]

[119861] =

[[[[[[[[[[

[

120597ℎ119909

1205971198801

120597ℎ119909

1205971198802

120597ℎ119909

1205971198803

120597ℎ119910

1205971198801

120597ℎ119910

1205971198802

120597ℎ119910

1205971198803

120597ℎ120579

1205971198801

120597ℎ120579

1205971198802

120597ℎ120579

1205971198803

]]]]]]]]]]

]

[119862] = [

120597Gap1205971199091

120597Gap1205971199092

sdot sdot sdot

120597Gap12059711990914

]

[119864] = [

120597Gap1205971198801

120597Gap1205971198802

120597Gap1205971198803

]

[120575119909] = [1205751199091

1205751199092

sdot sdot sdot 12057511990914]Τ

[120575119880] = [1205751198801

1205751198802

1205751198803]Τ

(14)

As shown in Table 2 the greatest impact on the assemblyprecision is the deviation sources 119909

6 11990912 and 119909

14 The angle

dimensions 1199096 11990912 and 119909

14correspond to length dimensions

1199095 1198801 and 119909

3 It can be seen that the larger the length

dimension the greater the sensitivity of the correspondingangle

According to (1) assembly precision 1205751198802= 05394

The first assembly precision optimization method is thatthe tolerances of deviation sources 119909

1 1199092 1199093 1199094 1199095 1199097 1199098

1199099 11990910 11990911 11990913 and 119909

14are reduced by 50 An optimized

assembly precision is obtained 1205751198802= 03940

The second assembly precision optimization method isthat the tolerances of deviation sources119909

6and11990912are reduced

by 50 An optimized assembly precision is obtained 1205751198802=

02533 So the goal of assembly precision optimization is thetolerances of deviation sources 119909

6and 119909

12

The results indicate that it would be more conducive tooptimize the assembly precision by reducing the deviationswhich have large sensitivity

5 Conclusions

This paper presents an approach for fast assembly preci-sion optimization of complex products based on deviationsources sensitivities analysis The joints between the adjacentparts and each partrsquos datum reference frame are defined forcreating deviation transmission paths and multidimensionaldimension vector loops Sensitivity calculations of assemblydeviation source are established by linearizing all themultidi-mensional vector loop scalar equations which can be gottenusing first-order Taylorrsquos series expansion andmatrix algebra

In practice we find that the sensitivity of deviation sourceis not always +1 and minus1 In the multidimensional spacesensitivity of deviation source is enlarged dramatically undercertain conditions If a list of deviation sources has the samevector directions the sensitivities of an assembly dimensionto the deviation sources are the same If the vector direction


x2

x5x6

x10

x11x12

x13U2

DRFrockerx7

DRF

J(P Pwing body)

J(P Procker)

DRF

DRFwing body

back connector

front connectorback connector

front connector

Figure 4 DRFs and DPs

x2

x1 x5

x6

x8x3

x9

x10x11

x12

U3

x13U2

x4

x7

x11

x14

U1

(a) The closed loop

x2

x5

x6

x11x12

U1

x13U2

a7

Gap

(b) The opened loop

Figure 5 Deviation transmission vector loops

Table 2 The value of deviation source sensitivities

1199091

1199092

1199093

1199094

1199095

1199096

1199097

1199098

1199099

11990910

11990911

11990912

11990913

11990914

Sensitivity of 1198801

minus18 18 minus79 minus18 31 1644 minus79 79 minus79 minus79 0 2005 81 993Sensitivity of 119880

20 0 minus01 0 0 188 minus01 01 minus01 minus01 0 222 01 05


0 0 minus01 0 0 178 minus01 01 minus01 minus01 0 212 01 15Sensitivity of Gap 0 0 minus1 0 0 0 0 1 minus1 minus1 minus01045 minus149 0 0

of an assembly dimension and a deviation source is the samethe sensitivity of the assembly dimension to the deviationsource is +1 If the vector direction of an assembly dimensionand a deviation source is opposite the sensitivity of theassembly dimension to the deviation source isminus1 If the vectordirection of an assembly dimension and a deviation source isperpendicular the sensitivity of the assembly dimension tothe deviation source is 0

Deviation source sensitivity is an important indicator ofassembly precision optimization in the aerospace industryTo further improve the flexibility of our approach fourkinds of joint types and multidimensional vector loops areused Considering the deviation source sensitivity analysisof complex product assembly our future work also includesimproving the algorithm by providing a more traceable androbust method

Conflict of Interests

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

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China under Grant no 51105313 and the Doc-torate Foundation of Northwestern Polytechnical Universityunder Grant no CX201313

References

[1] S Shin P Kongsuwon and B R Cho ldquoDevelopment ofthe parametric tolerance modeling and optimization schemesand cost-effective solutionsrdquo European Journal of OperationalResearch vol 207 no 3 pp 1728ndash1741 2010

[2] Z Shen ldquoTolerance analysis with EDSVisVSArdquo Journal ofComputing and Information Science in Engineering vol 3 no1 pp 95ndash99 2003

[3] CETOL6120590 Sigmetrix LLC httpwwwsigmetrixcom[4] American Society of Mechanical Engineers ANSIASME

Y145M-1994 Dimensioning and tolerancing 1994[5] Y Wu J J Shah and J K Davidson ldquoComputer modeling

of geometric variations in mechanical parts and assembliesrdquoJournal of Computing and Information Science in Engineeringvol 3 no 1 pp 54ndash63 2003


[6] P K Singh S C Jain and P K Jain ldquoAdvanced optimaltolerance design of mechanical assemblies with interrelateddimension chains and process precision limitsrdquo Computers inIndustry vol 56 no 2 pp 179ndash194 2005

[7] A J Qureshi J-Y Dantan V Sabri P Beaucaire and NGayton ldquoA statistical tolerance analysis approach for over-constrained mechanism based on optimization and MonteCarlo simulationrdquo CAD Computer Aided Design vol 44 no 2pp 132ndash142 2012

[8] Y Zhang Z Li J Gao J Hong F Villecco and Y Li ldquoAmethod for designing assembly tolerance networks of mechan-ical assembliesrdquo Mathematical Problems in Engineering vol2012 Article ID 513958 26 pages 2012

[9] G Zhang ldquoSimultaneous tolerancing for design and manufac-turingrdquo International Journal of Production Research vol 34 no12 pp 3361ndash3382 1996

[10] E A Lehtihet S Ranade and P Dewan ldquoComparative evalua-tion of tolerance control chart modelrdquo International Journal ofProduction Research vol 38 no 7 pp 1539ndash1556 2000

[11] Y S Hong and T-C Chang ldquoA comprehensive review of toler-ancing researchrdquo International Journal of Production Researchvol 40 no 11 pp 2425ndash2459 2002

[12] H P Peng X Q Jiang and X J Liu ldquoConcurrent optimalallocation of design and process tolerances for mechanicalassemblies with interrelated dimension chainsrdquo InternationalJournal of Production Research vol 46 no 24 pp 6963ndash69792008

[13] F W Ciarallo and C C Yang ldquoOptimization of propagation ininterval constraint networks for tolerance designrdquo in Proceed-ings of the IEEE International Conference on Systems Man andCybernetics pp 1924ndash1929 October 1997

[14] B-W Cheng and SMaghsoodloo ldquoOptimization ofmechanicalassembly tolerances by incorporating Taguchirsquos quality lossfunctionrdquo Journal of Manufacturing Systems vol 14 no 4 pp264ndash276 1995

[15] S Jung D-H Choi B-L Choi and J H Kim ldquoToleranceoptimization of a mobile phone camera lens systemrdquo AppliedOptics vol 50 no 23 pp 4688ndash4700 2011

[16] S-L Chen and K-J Chung ldquoSelection of the optimal precisionlevel and target value for a production process the lower-specification-limit caserdquo IIE Transactions vol 28 no 12 pp979ndash985 1996

[17] S M Kannan and V Jayabalan ldquoA new grouping method forminimizing the surplus parts in selective assemblyrdquo QualityEngineering vol 14 no 1 pp 65ndash75 2001

[18] F Villeneuve O Legoff and Y Landon ldquoTolerancing for manu-facturing a three-dimensional modelrdquo International Journal ofProduction Research vol 39 no 8 pp 1625ndash1648 2001

[19] S Xu and J Keyser ldquoGeometric computation and optimizationon tolerance dimensioningrdquo Computer-Aided Design vol 46pp 129ndash137 2014

[20] R Musa J-P Arnaout and F Frank Chen ldquoOptimization-simulation-optimization based approach for proactive variationreduction in assemblyrdquo Robotics and Computer-IntegratedMan-ufacturing vol 28 no 5 pp 613ndash620 2012

[21] S H Huang Q Liu and R Musa ldquoTolerance-based processplan evaluation using Monte Carlo simulationrdquo InternationalJournal of Production Research vol 42 no 23 pp 4871ndash48912004

[22] M V Raj S S Sankar and S G Ponnambalam ldquoOptimizationof assembly tolerance variation and manufacturing system effi-ciency by using genetic algorithm in batch selective assemblyrdquo

International Journal of Advanced Manufacturing Technologyvol 55 no 9-12 pp 1193ndash1208 2011

[23] C C Yang and V N A Naikan ldquoOptimum tolerance designfor complex assemblies using hierarchical interval constraintnetworksrdquo Computers and Industrial Engineering vol 45 no 3pp 511ndash543 2003

[24] P K Singh S C Jain and P K Jain ldquoConcurrent optimaladjustment of nominal dimensions and selection of toler-ances considering alternative machinesrdquo CAD Computer AidedDesign vol 38 no 10 pp 1074ndash1087 2006

[25] W Cai ldquoA new tolerance modeling and analysis methodologythrough a two-step linearization with applications in automo-tive body assemblyrdquo Journal of Manufacturing Systems vol 27no 1 pp 26ndash35 2008

[26] C C Yang and V N Achutha Naikan ldquoOptimum design ofcomponent tolerances of assemblies using constraint networksrdquoInternational Journal of Production Economics vol 84 no 2 pp149ndash163 2003

[27] B Anselmetti ldquoGeneration of functional tolerancing based onpositioning featuresrdquo CAD Computer Aided Design vol 38 no8 pp 902ndash919 2006

[28] MMansuyM Giordano and P Hernandez ldquoA new calculationmethod for the worst case tolerance analysis and synthesis instack-type assembliesrdquo CAD Computer Aided Design vol 43no 9 pp 1118ndash1125 2011

[29] L Laperriere and H ElMaraghy ldquoTolerance analysis and syn-thesis using jacobian-transformsrdquo CIRP-Annals vol 49 no 1pp 359ndash362 2000

[30] X Huang and Y Zhang ldquoProbabilistic approach to systemreliability of mechanism with correlated failure modelsrdquoMath-ematical Problems in Engineering vol 2012 Article ID 46585311 pages 2012

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


proposed the concept of deviation sources Deviation trans-mission mechanism was analyzed but the effect of devi-ation source sensitivity to cumulative deviation was notconsidered in the multidimensional environment Mansuyet al [28] introduced segmented geometric elements alongthe three-dimensional vector direction and then calculatedthe sensitivity and predicted the precision Laperriere andElMaraghy [29] proposed a method of tolerance analysis bydescribing the tolerance vector with six scalar equations inthree-dimensional space But it did not consider the role ofassembly datum in the process of deviation transmission

Because the physical world is predominantly nonlin-ear Monte Carlo simulation is naturally the most accu-rate and sometimes the only method for tolerance analysisfor a generic assembly product However computationalinefficiency from Monte Carlo simulations hinders manyadvanced features such as assembly precision and sequenceoptimization which all require a significant number ofanalysis iterations

For most engineering applications including the wingassembly the nonlinear kinematic relations can be approx-imated by linear models through a first-order Taylor seriesexpansionTherefore developing an efficient linear deviationsource sensitivity analysis method is both imperative andfeasible In multidimensional space for a comprehensivesensitivity analysis of mutual coupled deviation source anassembly deviation source sensitivity calculation method isproposed based on multidimensional vector loops Combin-ing the characteristic of complex product assembly multi-dimensional vector loops are built Based on the loops thesensitivity of each deviation source is calculated by reduced-order operation

This paper is organized as follows Section 2 gives theprinciple of assembly precision optimization Section 3 pro-poses a method of deviation source sensitivity analysisSection 4 presents a computational example to demonstratethe method Section 5 comes up with a summary for thispaper

2 Principle of AssemblyPrecision Optimization

Tolerance analysis methods have the worst case and RSSmethod shown as follows [21]

119905wc =

119899

sum

119894=1

1003816100381610038161003816119886119894119905119894

1003816100381610038161003816 (1)

119905rss = radic

119899

sum

119894=1

(119886119894119905119894)2

(2)

Assembly precision optimization is a process of solvingthe minimum value of assembly deviation value Anotherway to determine value of sensitivity 119886

119894 is developed for fast

assembly precision optimization

min 119905wc = min119899

sum

119894=1

1003816100381610038161003816119886119894119905119894

1003816100381610038161003816

min 119905rss = radicmin119899

sum

119894=1

(119886119894119905119894)2

(3)

According to (3) 119886119894determines the influence of 119905

119894to

119905wc and 119905rss In process of assembly precision optimizationby determining the 119886

119894 the deviation source which needs to

be optimized can be quickly determined then min 119905wc andmin 119905rss can be obtained So the most important thing ofassembly precision optimization is calculating each of thedeviation source sensitivities

In one-dimensional space 119886119894is defined as plusmn1 (increases

ring 119886119894= +1 decreases ring 119886

119894= minus1) In multidimensional

space because of the uncertainty of deviation vector direc-tion 119886

119894can be dramatically magnified So every deviation

source sensitivity 119886119894should be calculated Determining which

119886119894is bigger the corresponding deviation source is the one

which needs to be reduced By decreasing value of deviationsource assembly precision optimization is realized effectively

The result of assembly deviation accumulation expressesthe variable between actual assembly dimension and designdimension The sensitivities are solved by taking partialderivatives with respect to each variable

Assume that 1199101 1199102 119910

119899 are the corresponding

partrsquos dimensions of deviation sources and a functionΘ(1199101 1199102 119910

119899) expresses value of assembly dimension The

sensitivity of Θ with respect to 119910119894is

119886119894=

120597Θ

120597119910119894

10038161003816100381610038161003816100381610038161003816NominalValues

(4)

The partial derivative at the nominal values of eachvariable is evaluated And the nominal value for each variableis the center of a tolerance range or the value of the dimensionwhen the tolerances are equal bilaterally

3 Deviation Source SensitivityAnalysis Method

In the process of complicated product assembly a varietyof joint types are required to describe the mating partsrsquocontact points Different joint types form different kinds ofdeviation source couplings And different couplings lead todifferent results of deviation transmission Now deviationtransmission path vector loop equations and sensitivitycalculation method are discussed respectively

31 Deviation Transmission Path Based on Datum ReferenceFrames Assembly parts are located through the locatingdatum in the process of product assembly In the process offorming deviation transmission path partrsquos locating datumis called datum reference frame (DRF) DRF is mainly usedfor locating every needed feature of a part Datum path (DP)is a path from the joint to the DRF which is connected withnominal dimension vectors A deviation transmission pathof two constrained parts is along a DP and through a joint











119871 = 119869 minus 119875 + 1 (5)


1199091 1199092 119909


1 1198802 119880

119898


= 119863 (1199091 1199092 119909

119899 1198801 1198802 119880

119898) = 0 (6)



Gap = Γ (1199091 1199092 119909

119899 1198801 1198802 119880

119898) (7)




120575ℎ119863

=

120597ℎ119863

1205971199091

1205751199091+ sdot sdot sdot +

120597ℎ119863

120597119909119899

120575119909119899

+

120597ℎ119863

1205971198801

1205751198801+ sdot sdot sdot +

120597ℎ119863

120597119880119899

120575119880119899= 0

(8)

120575Gap =

120597Gap1205971199091

1205751199091+ sdot sdot sdot +

120597Gap120597119909119899

120575119909119899

+

120597Gap1205971198801

1205751198801+ sdot sdot sdot +

120597Gap120597119880119899

120575119880119899

(9)


[119860] [120575119909] + [119861] [120575119880] = 0 (10)


[120575Gap] = [119862] [120575119909] + [119864] [120575119880] (11)



[120575119880] = minus [119861minus1

119860] [120575119909]

[120575Gap] = [119862 minus 119864119861minus1

119860] [120575119909]

(12)



minus1






120601R

U U U

120601 R2R1

120601

DRFi

DRFj

DRFiDRFiDRFi

DRFjDRFj

DRFj

DRF1

ab

cd

DP 11

DP12

DRF2

e

f gDP 21

DP22

h

J (part1 part2)

Part1 Part2


x2

02x1

x5

x6

x3 x8

x9

x10

x11 x12 U1

x13U2

x4 x7

AA

ProckerPpin

Pwing body

Pfront connector

Pback connector



1 1199092 119909

13and





14becomes



1


ℎ119909= 1199091cos (0) + 119909

3cos (119909

14) + 1199094cos (0) + 119909

8cos (minus119880

3)

+ 1199099cos (90) + 119909

10cos (90) + 119880

1cos (minus119909

12)

+ 11990913cos (minus119880

2minus 90) + 119909

7cos (90)

+ 1199095cos (minus180 + 119909

6) + 1199092cos (minus180) = 0

ℎ119910= 1199091sin (0) + 119909

3sin (11988614) + 1199094sin (0) + 119909

8sin (minus119880

3)

+ 1199099sin (90) + 119909

10sin (90) + 119880

1sin (minus119909

12)

+ 11990913sin (minus119880

2minus 90) + 119909

7sin (90)

+ 1199095sin (minus180 + 119886

6) + 1199092sin (minus180) = 0

ℎ120579= 0 + 119909


3+ 180 minus (90 + 119909

12)


6+ 180 = 0




2mm 119909

3mm 119909

4mm 119909

5mm 119909

6deg 119909

7mm 119909

8mm 119909

9mm


10mm 119909

11mm 119909

12deg 119909

13mm 119909

14deg 119880

1mm 119880

2deg 119880

3deg



assembly graph




on locating DRFs


DRFs DPs



equations


sensitivities


Gap = (11990911

+ 1198801) sin (minus119909

12) + 11990913sin (minus119880

2minus 90)

+ 1199097sin (90) + 119909

5sin (minus180 + 119909

6)

(13)

[119860] =

[[[[[[[[[[

[

120597ℎ119909

1205971199091

120597ℎ119909

1205971199092

sdot sdot sdot

120597ℎ119909

12059711990914

120597ℎ119910

1205971199091

120597ℎ119910

1205971199092

sdot sdot sdot

120597ℎ119910

12059711990914

120597ℎ120579

1205971199091

120597ℎ120579

1205971199092

sdot sdot sdot

120597ℎ120579

12059711990914

]]]]]]]]]]

]

[119861] =

[[[[[[[[[[

[

120597ℎ119909

1205971198801

120597ℎ119909

1205971198802

120597ℎ119909

1205971198803

120597ℎ119910

1205971198801

120597ℎ119910

1205971198802

120597ℎ119910

1205971198803

120597ℎ120579

1205971198801

120597ℎ120579

1205971198802

120597ℎ120579

1205971198803

]]]]]]]]]]

]

[119862] = [

120597Gap1205971199091

120597Gap1205971199092

sdot sdot sdot

120597Gap12059711990914

]

[119864] = [

120597Gap1205971198801

120597Gap1205971198802

120597Gap1205971198803

]

[120575119909] = [1205751199091

1205751199092


[120575119880] = [1205751198801

1205751198802

1205751198803]Τ

(14)


6 11990912 and 119909

14 The angle

dimensions 1199096 11990912 and 119909


1199095 1198801 and 119909





1 1199092 1199093 1199094 1199095 1199097 1199098

1199099 11990910 11990911 11990913 and 119909







6and 119909

12


5 Conclusions




x2

x5x6

x10

x11x12

x13U2

DRFrockerx7

DRF

J(P Pwing body)

J(P Procker)

DRF

DRFwing body

back connector


front connector


x2

x1 x5

x6

x8x3

x9

x10x11

x12

U3

x13U2

x4

x7

x11

x14

U1

(a) The closed loop

x2

x5

x6

x11x12

U1

x13U2

a7

Gap

(b) The opened loop



1199091

1199092

1199093

1199094

1199095

1199096

1199097

1199098

1199099

11990910

11990911

11990912

11990913

11990914










Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra



















119871 = 119869 minus 119875 + 1 (5)


1199091 1199092 119909


1 1198802 119880

119898


= 119863 (1199091 1199092 119909

119899 1198801 1198802 119880

119898) = 0 (6)



Gap = Γ (1199091 1199092 119909

119899 1198801 1198802 119880

119898) (7)




120575ℎ119863

=

120597ℎ119863

1205971199091

1205751199091+ sdot sdot sdot +

120597ℎ119863

120597119909119899

120575119909119899

+

120597ℎ119863

1205971198801

1205751198801+ sdot sdot sdot +

120597ℎ119863

120597119880119899

120575119880119899= 0

(8)

120575Gap =

120597Gap1205971199091

1205751199091+ sdot sdot sdot +

120597Gap120597119909119899

120575119909119899

+

120597Gap1205971198801

1205751198801+ sdot sdot sdot +

120597Gap120597119880119899

120575119880119899

(9)


[119860] [120575119909] + [119861] [120575119880] = 0 (10)


[120575Gap] = [119862] [120575119909] + [119864] [120575119880] (11)



[120575119880] = minus [119861minus1

119860] [120575119909]

[120575Gap] = [119862 minus 119864119861minus1

119860] [120575119909]

(12)



minus1






120601R

U U U

120601 R2R1

120601

DRFi

DRFj

DRFiDRFiDRFi

DRFjDRFj

DRFj

DRF1

ab

cd

DP 11

DP12

DRF2

e

f gDP 21

DP22

h

J (part1 part2)

Part1 Part2


x2

02x1

x5

x6

x3 x8

x9

x10

x11 x12 U1

x13U2

x4 x7

AA

ProckerPpin

Pwing body

Pfront connector

Pback connector



1 1199092 119909

13and





14becomes



1


ℎ119909= 1199091cos (0) + 119909

3cos (119909

14) + 1199094cos (0) + 119909

8cos (minus119880

3)

+ 1199099cos (90) + 119909

10cos (90) + 119880

1cos (minus119909

12)

+ 11990913cos (minus119880

2minus 90) + 119909

7cos (90)

+ 1199095cos (minus180 + 119909

6) + 1199092cos (minus180) = 0

ℎ119910= 1199091sin (0) + 119909

3sin (11988614) + 1199094sin (0) + 119909

8sin (minus119880

3)

+ 1199099sin (90) + 119909

10sin (90) + 119880

1sin (minus119909

12)

+ 11990913sin (minus119880

2minus 90) + 119909

7sin (90)

+ 1199095sin (minus180 + 119886

6) + 1199092sin (minus180) = 0

ℎ120579= 0 + 119909


3+ 180 minus (90 + 119909

12)


6+ 180 = 0




2mm 119909

3mm 119909

4mm 119909

5mm 119909

6deg 119909

7mm 119909

8mm 119909

9mm


10mm 119909

11mm 119909

12deg 119909

13mm 119909

14deg 119880

1mm 119880

2deg 119880

3deg



assembly graph




on locating DRFs


DRFs DPs



equations


sensitivities


Gap = (11990911

+ 1198801) sin (minus119909

12) + 11990913sin (minus119880

2minus 90)

+ 1199097sin (90) + 119909

5sin (minus180 + 119909

6)

(13)

[119860] =

[[[[[[[[[[

[

120597ℎ119909

1205971199091

120597ℎ119909

1205971199092

sdot sdot sdot

120597ℎ119909

12059711990914

120597ℎ119910

1205971199091

120597ℎ119910

1205971199092

sdot sdot sdot

120597ℎ119910

12059711990914

120597ℎ120579

1205971199091

120597ℎ120579

1205971199092

sdot sdot sdot

120597ℎ120579

12059711990914

]]]]]]]]]]

]

[119861] =

[[[[[[[[[[

[

120597ℎ119909

1205971198801

120597ℎ119909

1205971198802

120597ℎ119909

1205971198803

120597ℎ119910

1205971198801

120597ℎ119910

1205971198802

120597ℎ119910

1205971198803

120597ℎ120579

1205971198801

120597ℎ120579

1205971198802

120597ℎ120579

1205971198803

]]]]]]]]]]

]

[119862] = [

120597Gap1205971199091

120597Gap1205971199092

sdot sdot sdot

120597Gap12059711990914

]

[119864] = [

120597Gap1205971198801

120597Gap1205971198802

120597Gap1205971198803

]

[120575119909] = [1205751199091

1205751199092


[120575119880] = [1205751198801

1205751198802

1205751198803]Τ

(14)


6 11990912 and 119909

14 The angle

dimensions 1199096 11990912 and 119909


1199095 1198801 and 119909





1 1199092 1199093 1199094 1199095 1199097 1199098

1199099 11990910 11990911 11990913 and 119909







6and 119909

12


5 Conclusions




x2

x5x6

x10

x11x12

x13U2

DRFrockerx7

DRF

J(P Pwing body)

J(P Procker)

DRF

DRFwing body

back connector


front connector


x2

x1 x5

x6

x8x3

x9

x10x11

x12

U3

x13U2

x4

x7

x11

x14

U1

(a) The closed loop

x2

x5

x6

x11x12

U1

x13U2

a7

Gap

(b) The opened loop



1199091

1199092

1199093

1199094

1199095

1199096

1199097

1199098

1199099

11990910

11990911

11990912

11990913

11990914










Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120601R

U U U

120601 R2R1

120601

DRFi

DRFj

DRFiDRFiDRFi

DRFjDRFj

DRFj

DRF1

ab

cd

DP 11

DP12

DRF2

e

f gDP 21

DP22

h

J (part1 part2)

Part1 Part2


x2

02x1

x5

x6

x3 x8

x9

x10

x11 x12 U1

x13U2

x4 x7

AA

ProckerPpin

Pwing body

Pfront connector

Pback connector



1 1199092 119909

13and





14becomes



1


ℎ119909= 1199091cos (0) + 119909

3cos (119909

14) + 1199094cos (0) + 119909

8cos (minus119880

3)

+ 1199099cos (90) + 119909

10cos (90) + 119880

1cos (minus119909

12)

+ 11990913cos (minus119880

2minus 90) + 119909

7cos (90)

+ 1199095cos (minus180 + 119909

6) + 1199092cos (minus180) = 0

ℎ119910= 1199091sin (0) + 119909

3sin (11988614) + 1199094sin (0) + 119909

8sin (minus119880

3)

+ 1199099sin (90) + 119909

10sin (90) + 119880

1sin (minus119909

12)

+ 11990913sin (minus119880

2minus 90) + 119909

7sin (90)

+ 1199095sin (minus180 + 119886

6) + 1199092sin (minus180) = 0

ℎ120579= 0 + 119909


3+ 180 minus (90 + 119909

12)


6+ 180 = 0




2mm 119909

3mm 119909

4mm 119909

5mm 119909

6deg 119909

7mm 119909

8mm 119909

9mm


10mm 119909

11mm 119909

12deg 119909

13mm 119909

14deg 119880

1mm 119880

2deg 119880

3deg



assembly graph




on locating DRFs


DRFs DPs



equations


sensitivities


Gap = (11990911

+ 1198801) sin (minus119909

12) + 11990913sin (minus119880

2minus 90)

+ 1199097sin (90) + 119909

5sin (minus180 + 119909

6)

(13)

[119860] =

[[[[[[[[[[

[

120597ℎ119909

1205971199091

120597ℎ119909

1205971199092

sdot sdot sdot

120597ℎ119909

12059711990914

120597ℎ119910

1205971199091

120597ℎ119910

1205971199092

sdot sdot sdot

120597ℎ119910

12059711990914

120597ℎ120579

1205971199091

120597ℎ120579

1205971199092

sdot sdot sdot

120597ℎ120579

12059711990914

]]]]]]]]]]

]

[119861] =

[[[[[[[[[[

[

120597ℎ119909

1205971198801

120597ℎ119909

1205971198802

120597ℎ119909

1205971198803

120597ℎ119910

1205971198801

120597ℎ119910

1205971198802

120597ℎ119910

1205971198803

120597ℎ120579

1205971198801

120597ℎ120579

1205971198802

120597ℎ120579

1205971198803

]]]]]]]]]]

]

[119862] = [

120597Gap1205971199091

120597Gap1205971199092

sdot sdot sdot

120597Gap12059711990914

]

[119864] = [

120597Gap1205971198801

120597Gap1205971198802

120597Gap1205971198803

]

[120575119909] = [1205751199091

1205751199092


[120575119880] = [1205751198801

1205751198802

1205751198803]Τ

(14)


6 11990912 and 119909

14 The angle

dimensions 1199096 11990912 and 119909


1199095 1198801 and 119909





1 1199092 1199093 1199094 1199095 1199097 1199098

1199099 11990910 11990911 11990913 and 119909







6and 119909

12


5 Conclusions




x2

x5x6

x10

x11x12

x13U2

DRFrockerx7

DRF

J(P Pwing body)

J(P Procker)

DRF

DRFwing body

back connector


front connector


x2

x1 x5

x6

x8x3

x9

x10x11

x12

U3

x13U2

x4

x7

x11

x14

U1

(a) The closed loop

x2

x5

x6

x11x12

U1

x13U2

a7

Gap

(b) The opened loop



1199091

1199092

1199093

1199094

1199095

1199096

1199097

1199098

1199099

11990910

11990911

11990912

11990913

11990914










Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












2mm 119909

3mm 119909

4mm 119909

5mm 119909

6deg 119909

7mm 119909

8mm 119909

9mm


10mm 119909

11mm 119909

12deg 119909

13mm 119909

14deg 119880

1mm 119880

2deg 119880

3deg



assembly graph




on locating DRFs


DRFs DPs



equations


sensitivities


Gap = (11990911

+ 1198801) sin (minus119909

12) + 11990913sin (minus119880

2minus 90)

+ 1199097sin (90) + 119909

5sin (minus180 + 119909

6)

(13)

[119860] =

[[[[[[[[[[

[

120597ℎ119909

1205971199091

120597ℎ119909

1205971199092

sdot sdot sdot

120597ℎ119909

12059711990914

120597ℎ119910

1205971199091

120597ℎ119910

1205971199092

sdot sdot sdot

120597ℎ119910

12059711990914

120597ℎ120579

1205971199091

120597ℎ120579

1205971199092

sdot sdot sdot

120597ℎ120579

12059711990914

]]]]]]]]]]

]

[119861] =

[[[[[[[[[[

[

120597ℎ119909

1205971198801

120597ℎ119909

1205971198802

120597ℎ119909

1205971198803

120597ℎ119910

1205971198801

120597ℎ119910

1205971198802

120597ℎ119910

1205971198803

120597ℎ120579

1205971198801

120597ℎ120579

1205971198802

120597ℎ120579

1205971198803

]]]]]]]]]]

]

[119862] = [

120597Gap1205971199091

120597Gap1205971199092

sdot sdot sdot

120597Gap12059711990914

]

[119864] = [

120597Gap1205971198801

120597Gap1205971198802

120597Gap1205971198803

]

[120575119909] = [1205751199091

1205751199092


[120575119880] = [1205751198801

1205751198802

1205751198803]Τ

(14)


6 11990912 and 119909

14 The angle

dimensions 1199096 11990912 and 119909


1199095 1198801 and 119909





1 1199092 1199093 1199094 1199095 1199097 1199098

1199099 11990910 11990911 11990913 and 119909







6and 119909

12


5 Conclusions




x2

x5x6

x10

x11x12

x13U2

DRFrockerx7

DRF

J(P Pwing body)

J(P Procker)

DRF

DRFwing body

back connector


front connector


x2

x1 x5

x6

x8x3

x9

x10x11

x12

U3

x13U2

x4

x7

x11

x14

U1

(a) The closed loop

x2

x5

x6

x11x12

U1

x13U2

a7

Gap

(b) The opened loop



1199091

1199092

1199093

1199094

1199095

1199096

1199097

1199098

1199099

11990910

11990911

11990912

11990913

11990914










Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










x2

x5x6

x10

x11x12

x13U2

DRFrockerx7

DRF

J(P Pwing body)

J(P Procker)

DRF

DRFwing body

back connector


front connector


x2

x1 x5

x6

x8x3

x9

x10x11

x12

U3

x13U2

x4

x7

x11

x14

U1

(a) The closed loop

x2

x5

x6

x11x12

U1

x13U2

a7

Gap

(b) The opened loop



1199091

1199092

1199093

1199094

1199095

1199096

1199097

1199098

1199099

11990910

11990911

11990912

11990913

11990914










Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article sensitivity analysis of deviation source

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