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Research ArticleThree-Dimensional Vibration Analysis of RectangularThick Plates on Pasternak Foundation with ArbitraryBoundary Conditions

Huimin Liu12 Fanming Liu1 Xin Jing1 ZhenpengWang1 and Linlin Xia3

1College of Automation Harbin Engineering University Harbin China2College of Mechanical and Electrical Engineering Qingdao Agricultural University Qingdao China3School of Automation Engineering Northeast Dianli University Jilin China

Correspondence should be addressed to Fanming Liu hrblfm407126com

Received 9 December 2016 Revised 10 March 2017 Accepted 3 April 2017 Published 22 May 2017

Academic Editor Anindya Ghoshal

Copyright copy 2017 Huimin Liu 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

This paper presents the first known vibration characteristic of rectangular thick plates on Pasternak foundation with arbitraryboundary conditions on the basis of the three-dimensional elasticity theory The arbitrary boundary conditions are obtained bylaying out three types of linear springs on all edges The modified Fourier series are chosen as the basis functions of the admissiblefunction of the thick plates to eliminate all the relevant discontinuities of the displacements and their derivatives at the edges Theexact solution is obtained based on the RayleighndashRitz procedure by the energy functions of the thick plate The excellent accuracyand reliability of current solutions are demonstrated by numerical examples and comparisons with the results available in theliterature In addition the influence of the foundation coefficients as well as the boundary restraint parameters is also analyzedwhich can serve as the benchmark data for the future research technique

1 Introduction

Rectangular plates as the common structural componentshave been extensively used in various engineering fieldssuch as aerospace military and marine industries Thusthe knowledge of vibration characteristics of the rectangularplates is of particular importance for the predesign of theengineering structures In general the analysis for the vibra-tion problem of the rectangular plates is based on the clas-sical thin plate theory [1] and two-dimensional approximatetheories [2ndash5] that is first-order shear deformation theoryand higher-order deformation theory However the presentresults indicate that the two-dimensional plate theories haveshortcomings [6ndash8] For the classical thin plate theory itneglects the effect of the transverse shear deformation by thesimplified assumption that the normal to the undeformedmid-plane remains normal after deformation For case ofthe two-dimensional approximate theories the shear factorstrongly relies on the boundary condition and they oftenneed more hardware resources to obtain proper accuracy

solution Based on the above analysis the three-dimensionalelastic theory is presented to overcome the weakness of thetwo-dimensional plate theories which does not rely on anyhypotheses or deserve any numerical precision and can beused to solve the vibration problem of thick rectangularplates

Over the past several decades the three-dimensionalvibration analyses of thick rectangular plates have been inves-tigated by many researchers Srinivas et al [9] analyzed thefree vibration of thick rectangular plates by the direct methodbased on the three-dimensional linear small deformationtheory In their study the boundary condition is limitedto the all simply supported case on four edges Cheungand Chakrabarti [10] used the finite layer method which isan extension of the well-known finite element method tostudy the free vibration of rectangular plates with variousclassical boundary conditions on the basis of the three-dimensional linear small deformation theory Hutchinsonand Zillmer [11] used the series solution method to analyzethe free vibration of a completely free parallelepiped Fromme

HindawiShock and VibrationVolume 2017 Article ID 3425298 10 pageshttpsdoiorg10115520173425298

2 Shock and Vibration

and Leissa [12] extended the Fourier method to investigatethe free vibration of the rectangular parallelepiped withsimple classical boundary conditions based on the three-dimensional elasticity theory On the basis of the three-dimensional elasticity theory and differential quadraturemethod (DQM) Malik and Bert [13] Liew and Teo [14] andLiew et al [15] investigated the free vibration characteristicsof rectangular plates with some selected classical boundaryconditions Leissa and Zhang [16] employed the Ritz methodto study the three-dimensional problem of determining thefree vibration frequencies and mode shapes for a rectangularparallelepiped which is completely fixed on one face andfree on other five faces In this research the displacementsof the rectangular parallelepiped are assumed in the formof algebraic polynomials Also Liew et al [17ndash19] appliedthe Ritz method and the three-dimensional elastic theory toestablish the continuum three-dimensional Ritz formulationto study the vibration characteristic of the thick rectangularplates with classical boundary conditions The orthogonallygenerated polynomial functions are used as admissible func-tions to make analysis in such plates Itakura [20] adoptedthe three-dimensional theory to study the free vibration ofrectangular thick plate with classical boundary conditionsLim et al [21 22] presented a concise formulation andan efficient method of solution to study the free vibrationof thick shear deformable plates with classical boundaryconditions Zhou et al [23] used the Chebyshev polynomialsas admissible functions and applied Ritz method to studythe three-dimensional vibration of rectangular plates withclassical boundary conditions

By means of the review of the above references most ofthe studies for the vibration problem of the thick rectangularplates do not consider the elastic foundations Howevera lot of engineering problems boil down to a rectangularplate on elastic foundations such as footing of buildingspavement of roads and base of heavy machines In thepractical application the Pasternakmodel (also referred to asthe two-parameter model) [24ndash26] is widely used to describethemechanical behavior of the foundation in which the well-known Winkler model [27] is a special case In additionthe boundary condition may not always be classical casein reality and a variety of possible boundary conditionsincluding classical boundary conditions elastic boundaryrestraints and the combinations of two or more conditionsmay be encountered [6ndash8 28ndash47] Based on the open pub-lished paper we can know that merely Zhou et al [4849] extended the Chebyshev-Ritz method to study the freevibration characteristics of rectangular thick plates and thickcircular plates resting on elastic foundations respectivelyHowever it should be noted that the boundary conditionsare limited to the FFFF SSSS and CCCC classical boundaryconditions There is no work reported on the free vibrationbehavior of rectangular thick plates on Pasternak foundationwith general boundary conditions Thus to present a newthree-dimensional exact solution for the free vibrations ofrectangular thick plates with arbitrary boundary conditionsis of crucial importance

In this paper the first known three-dimensional vibra-tion characteristic of rectangular thick plates on Pasternak

foundation with arbitrary boundary conditions is stud-ied The theoretical formulations are based on the three-dimensional linear small-strain elasticity theory The mod-ified Fourier series are chosen as the basis functions of theadmissible function of the thick plates in the Ritz methodThe arbitrary boundary conditions are obtained by layingout three types of liner springs on all edges The excellentaccuracy and reliability of current solutions are demonstratedby numerical examples and comparisons with the resultsavailable in the literature In addition the influence of thefoundation coefficients as well as the boundary restraintparameters are also analyzed which can serve as the bench-mark data for the future research technique

2 Theoretical Formulations

Consider a rectangular thick plate on Pasternak foundationwith arbitrary boundary conditions as shown in Figure 1 ACartesian coordinate (119909 119910 119911) is also shown in the Figure 1which will be used in the analysis 119880 119881 and 119882 denotethe displacement components in 119909 119910 and 119911 directionsLength width and thickness of the rectangular thick platesare assumed as 119886 119887 and ℎ respectively As shown in Figure 1three groups of linear restraint springs are arranged at allsides of the rectangular thick plates to separately simulate thearbitrary boundary conditions The undersurface of the plateis continuously rested on an elastic foundation representedby the WinklerPasternak model in which the Winkler andshear stiffness are denoted by 119870119908 and 119870119904 respectively seenfrom Figure 1

According to the three-dimensional elasticity theorythe strain components of rectangular thick plates can beexpressed as

120576119909 = 120597119906120597119909 120576119910 = 120597V120597119910 120576119911 = 120597119908120597119911

(1a)

120574119909119910 = 120597V120597119909 + 120597119906120597119910 120574119909119911 = 120597119908120597119909 + 120597119906120597119911 120574119910119911 = 120597V120597119911 + 120597119908120597119910

(1b)

Based on Hookersquos law the corresponding stress-strainrelations of the rectangular thick plates can be written as120590119909 = (Υ + 2Θ) 120576119909 + Υ120576119910 + Υ120576119911120590119910 = Υ120576119909 + (Υ + 2Θ) 120576119910 + Υ120576119911120590119911 = Υ120576119909 + Υ120576119910 + (Υ + 2Θ) 120576119911120591119909119910 = Θ120574119909119910

Shock and Vibration 3

z

x

y

a

b

ℎ

W

U V

Ground

Middle surface

KS shearing layerKW Winkler layer

Figure 1 The rectangular thick plates in contact with elastic foundation with general elastic boundary conditions

120591119909119911 = Θ120574119909119911120591119910119911 = Θ120574119910119911(2)

where Υ and Θ are as follows

Υ = 119864120583(1 + 120583) (1 minus 2120583) Θ = 1198642 (1 + 120583) (3)

in which 119864 and 120583 are Youngrsquos modulus and Poissonrsquos ratioof the rectangular thick plates The strain energy 119880 forrectangular thick plates is given in integral form by

119880 = 12 intintint119881 120576119909120590119909 + 120576119910120590119910 + 120576119911120590119911 + 120574119909119910120591119909119910 + 120574119909119911120591119909119911+ 120574119910119911120591119910119911 119889119909 119889119910 119889119911= 12 int1198860 int1198870 intℎ0 (Υ + 2Θ) [(120597119906120597119909)2 + (120597V120597119910)2 + (120597119908120597119911 )2]+ 2Υ[120597119906120597119909 120597V120597119910 + 120597V120597119910 120597119908120597119911 + 120597119908120597119911 120597119906120597119909]+ 2Θ[120597119906120597119910 120597V120597119909 + 120597V120597119911 120597119908120597119910 + 120597119908120597119909 120597119906120597119911]+ Θ[(120597119906120597119910)2 + ( 120597V120597119909)2 + (120597119908120597119909 )2+(120597119906120597119911)2 + (120597V120597119911)2 + (120597119908120597119910 )2]119889119909119889119910119889119911

(4)

The kinetic energy 119879 of the rectangular thick plates isdepicted as

119879 = 12sdot 120588int1198860int1198870intℎ0[(120597119906120597119905 )2 + (120597V120597119905 )2 + (120597119908120597119905 )2]119889119909119889119910119889119911 (5)

As mentioned before in this paper the boundary con-ditions of the rectangular thick plates are the arbitrary

boundary condition and obtained by laying out three typesof linear springs on all the edges Thus the correspondingboundary conditions can be described as120590119909 + 1198961199061199090119906 = 0120591119909119910 + 119896V1199090V = 0120591119909119911 + 1198961199081199090119908 = 0 119909 = 0120590119909 minus 1198961199061199091119906 = 0120591119909119910 minus 119896V1199091V = 0120591119909119911 minus 1198961199081199091119908 = 0 119909 = 119886120591119909119910 + 1198961199061199100119906 = 0120590119910 + 119896V1199100V = 0120591119910119911 + 1198961199081205790119908 = 0 119910 = 0120591119909119910 minus 119896119906119910119887119906 = 0120590119910 minus 119896V119910119887V = 0120591119910119911 minus 119896119908119910119887119908 = 0 119910 = 119887

(6)

and the potential energy P stored in the boundary springs canbe expressed as

119875 = 12 int1198870 intℎ0 [11989611990611990901199062 + 119896V1199090V2 + 11989611990811990901199082]119909=0+ [1198961199061199091198861199062 + 119896V119909119886V2 + 1198961199081199091198861199082]119909=119886 119889119910 119889119911+ 12 int1198860 intℎ0 [11989611990611991001199062 + 119896V1199100V2 + 11989611990811991001199082]119910=0+ [11989611990611991011199062 + 119896V1199101V2 + 11989611990811991011199082]119910=119887 119889119909 119889119911

(7)


The potential energy of the elastic foundation (119880119901119891) isgiven as

119880119901119891 = 12sdot int1198860int1198870(1198701199081199082 + 119870119904 (120597119908120597119909 )2 + 119870119904 (120597119908120597119910 )2)119889119909119889119910 (8)

where Kw and Ks are the stiffness of the Winkler layer andshear layer respectively

The total energy functional for the rectangular thickplates can be expressed as

prod = 119879 minus 119880 minus 119875 minus 119880119901119891 (9)

For the free vibration of the plate the displacementcomponents can be expressed in terms of the displacementamplitude functions

119906 (119909 119910 119911 119905) = 119880 (119909 119910 119911) 119890119894120596119905 (10a)

V (119909 119910 119911 119905) = 119881 (119909 119910 119911) 119890119894120596119905 (10b)119908 (119909 119910 119911 119905) = 119882 (119909 119910 119911) 119890119894120596119905 (10c)

where120596 denotes the eigenfrequency of the plate and 119894 = radicminus1Each of the displacement amplitude functions 119880(119909 119910 119911)119881(119909 119910 119911) and 119882(119909 119910 119911) is expressed respectively in the

form of a three-dimensional (3D) Fourier cosine seriessupplemented with closed-form auxiliary functions intro-duced to eliminate all the relevant discontinuities of thedisplacement function and its derivatives at the edges thatis 119880 (119909 119910 119911)

= infinsum119898=0

infinsum119899=0

infinsum119902=0

119860119898119899119902 cos 120582119898119909119909 cos 120582119899119910119910 cos 120582119902119911119911+ infinsum119898=0

infinsum119899=0

(11988611198981198991205771119911 (119911) + 11988621198981198991205772119911 (119911)) cos 120582119898119909119909 cos 120582119899119910119910+ infinsum119898=0

infinsum119902=0

(11988631198981199021205771119910 (119910) + 11988641198981199021205772119910 (119910)) cos 120582119898119909119909sdot cos 120582119902119911119911 + infinsum

119899=0

infinsum119902=0

(11988651198991199021205771119909 (119909) + 11988661198991199021205772119909 (119909)) cos 120582119899119910119910sdot cos 120582119902119911119911119881 (119909 119910 119911)= infinsum119898=0

infinsum119899=0

infinsum119902=0


infinsum119899=0

(11988711198981198991205771119911 (119911) + 11988721198981198991205772119911 (119911)) cos 120582119898119909119909 cos 120582119899119910119910

+ infinsum119898=0

infinsum119902=0

(11988731198981199021205771119910 (119910) + 11988741198981199021205772119910 (119910)) cos 120582119898119909119909 cos 120582119902119911119911+ infinsum119899=0

infinsum119902=0

(11988751198991199021205771119909 (119909) + 11988761198991199021205772119909 (119909)) cos 120582119899119910119910 cos 120582119902119911119911119882 (119909 119910 119911)

= infinsum119898=0

infinsum119899=0

infinsum119902=0


infinsum119899=0

(11988811198981198991205771119911 (119911) + 11988821198981198991205772119911 (119911)) cos 120582119898119909119909 cos 120582119899119910119910+ infinsum119898=0

infinsum119902=0

(11988831198981199021205771119910 (119910) + 11988841198981199021205772119910 (119910)) cos 120582119898119909119909 cos 120582119902119911119911+ infinsum119899=0

infinsum119902=0

(11988851198991199021205771119909 (119909) + 11988861198991199021205772119909 (119909)) cos 120582119899119910119910 cos 120582119902119911119911(11)

where 120582119898119909 = 119898120587119886 120582119899119910 = 119899120587119887 120582119902119911 = 119902120587ℎ and119860119898119899119902 119861119898119899119902 and 119862119898119899119902 are the Fourier coefficients of three-dimensional Fourier series expansions for the displacementsfunctions respectively a1mn a

2mn a

3mq a

4mq a

5nq a

6nq b

1mn b

2mn

b3mq b4mq b

5nq b

6nq c

1mn c

2mn c

3mq c

4mq c

5nq and c6nq are the

supplemented coefficients of the auxiliary functions Theclosed-form auxiliary functions are given as follows

1205771119909 (119909) = 1198862120587 sin(1205871199092119886 ) + 1198862120587 sin(31205871199092119886 ) 1205772119909 (119909) = minus 1198862120587 cos(1205871199092119886 ) + 1198862120587 cos(31205871199092119886 ) (12a)

1205771119910 (119910) = 1198872120587 sin(1205871199102119887 ) + 1198872120587 sin(31205871199102119887 ) 1205772119910 (119910) = minus 1198872120587 cos(1205871199102119887 ) + 1198872120587 cos(31205871199102119887 ) (12b)

1205771119911 (119911) = ℎ2120587 sin(1205871199112ℎ ) + 1206012120587 sin(31205871199112ℎ ) 1205772119911 (119911) = minus ℎ2120587 cos(1205871199112ℎ ) + ℎ2120587 cos(31205871199112ℎ ) (12c)

It is easy to verify that

1205771119909 (0) = 1205771119909 (119886) = 12057711199091015840 (119886) = 0 12057711199091015840 (0) = 1 (13a)

1205772119909 (0) = 1205772119909 (119886) = 12057721199091015840 (0) = 0 12057721199091015840 (119886) = 1 (13b)

1205771119910 (0) = 1205771119910 (119887) = 12057711199101015840 (119887) = 0 12057711199101015840 (0) = 1 (13c)

1205772119910 (0) = 1205772119910 (119887) = 12057721199101015840 (0) = 0 12057721199101015840 (119887) = 1 (13d)

1205771119911 (0) = 1205771119911 (ℎ) = 12057711199111015840 (ℎ) = 0 12057711199111015840 (0) = 1 (13e)


1st mode2nd mode

3rd mode4th mode

5

6

Ω 7

8

9

3 6 9 120

M N Q

(a)

1st mode2nd mode

3rd mode4th mode

3

4

Ω

5

6

3 6 9 120

M N Q

(b)

1st mode2nd mode

3rd mode4th mode

2

3

Ω

4

3 6 9 120

M N Q

(c)

Figure 2 Convergence of frequency parametersΩ for a thick rectangular plate with different boundary conditions (a) CCCC (b) SSSS (c)FFFF

1205772119911 (0) = 1205772119911 (ℎ) = 12057721199111015840 (0) = 0 12057721199111015840 (ℎ) = 1 (13f)

Substituting (11) into (9) and then minimizing the func-tional prod with respect to the coefficients of the admissiblefunctions that is120597prod120597120580 = 0

120580 = 119860119906119898119899 119886119897119898 119887119897119899 119861V119898119899 119888119897119898 119889119897119899 119864119910119898119899 119892119897119898 ℎ119897119899 (14)

the equations of motion for the rectangular plates can beyielded and are given in the matrix form

([[[[[Κ119906119906] [Κ119906V] [Κ119906119908][Κ119906V]119879 [ΚVV] [ΚV119908][Κ119906119908]119879 [ΚV119908]119879 [Κ119908119908]

]]]]minus 1205962 [[[[

[M119906119906] 0 00 [MVV] 00 0 [M119908119908]

]]]])ABC

= 0(15)

where [K119894119895] and [M119894119895] (119894 119895 = 119906 V 119908) respectively representthe stiffness submatrix and mass submatrix The columnvectors of the unknown coefficients A B and C areshown as follows

A = 119860000 119860119898119899119902 119860119872119873119876 119886100 1198861119872119873 119886200 1198862119872119873 119886300 1198863119872119876 119886400 1198864119872119876 119886500 1198865119873119876 119886600 1198866119873119876 (16a)

B = 119861000 119861119898119899119902 119861119872119873119876 119887100 1198871119872119873 119887200 1198872119872119873 119887300 1198873119872119876 119887400 1198874119872119876 119887500 1198875119873119876 119887600 1198876119873119876 (16b)

C = 119862000 119862119898119899119902 119862119872119873119876 119888100 1198881119872119873 119888200 1198882119872119873 119888300 1198883119872119876 119888400 1198884119872119876 119888500 1198885119873119876 119888600 1198886119873119876 (16c)The exact natural frequencies and mode shapes of the

rectangular plates can be easily obtained by solving (15)

3 Numerical Results and Discussion

Since the natural frequencies and mode shapes are obtainedfrom the Ritz method and the number of terms of theadmissible functions being chosen as infinity is unrealistic inthe actual calculations thus it is very important to make theconvergence study of the present method to understand theconvergence rate and the accuracy of the method For gen-eral purposes in the future the nondimensional foundationparameters and frequency parameters are used for numericalresults as follows119870119882 = 119870119908times1198864119863119870119878 = 119870119904times1198862119863 andΩ =1205961198872(120588ℎ119863)12 where 119863 = 119864ℎ312(1 minus 1205832) Figure 2 showsthe convergence studies of the first four frequency parametersfor a rectangular plate with CFCF boundary condition anddifferent truncated numbers 119872 119873 and 119876 The geometricaldimensions of the plates are used 119886 = 1m 119887 = 2mand ℎ = 05m The foundation coefficients are 119870119882 = 10and 119870119878 = 10 It is observed that the proposed method hasfast convergence and good stability In view of the excellent


Table 1 Comparison of the frequency parameters Ω for square thick plate with different boundary conditions

BC ℎ119886 Method Mode number1 2 3 4 5 6 7

SSSS02

Reference [20] 17528 32192 32192 38502 38502 45526 55843Reference [17] 17526 32192 32192 38483 38483 45526 55787

Present 17525 32184 32184 38480 38480 45526 55781

05 Reference [17] 12426 12877 12877 18210 23007 23007 25753Present 12424 12874 12874 18210 23005 23005 25747

CCCC02


Present 26966 47192 47192 61949 61949 63453 72404


FFFF02 Reference [18] 11710 17433 21252 27647 27647 40191 42776

Present 11719 17436 21254 27659 27659 40192 42776


Table 2 Comparison of the frequency parameters Ω for thick rectangular plate with different foundation coefficients

BC ℎ119886 119870119882 119870119878 Method Mode number1 2 3 4

SSSS

001

100 10Reference [48] 26551 55717 55717 85406Reference [50] 26551 55718 55718 85405

Present 26551 55717 55717 85404


Present 33400 59287 59287 87774

01


Present 27837 53013 53013 77215


Present 39802 60052 60052 82148

CCCC 0015 13902 16683Reference [48] 81675 128230 128230 168330Reference [50] 81375 128980 128980 169320

Present 81397 128995 128995 139349

numerical behavior of the current solution the truncationnumbers will be simply set as 119872 = 119873 = 119876 = 12 inFigure 2

Next the comparison study will be carried out bythe present method and other methods presented in theexisting publications Table 1 shows the comparison of thefirst seven frequency parameters Ω for a square thick platewith SSSS CCCC and FFFF boundary conditions In thisexample the thick plate is without considering the foun-dation parameters In order to check the present methodthe results are compared with other published solutions byusing the 3D Ritz method with general orthogonal poly-nomials [17] the 3D Ritz method with general orthogonalpolynomials using the GramndashSchmidt process [18] and the3D Ritz method with simple algebraic polynomials [20]From the comparison we can see a consistent agreement of

the results taken from the current method and referentialdata

Furthermore the rectangular thick plate with variousboundary conditions and elastic foundations is examinedComparison is presented in Table 2 In this table first fourfrequency parameters Ω are obtained for various boundaryconditions and foundation coefficients according to thepresent formulation and compared with those given byZhou et al [48] and Omurtag et al [50] based on differentnumericalmethods and elastic theories It is observed that thefrequencies are in excellent agreement with those obtainedfrom FEM which verifies the accuracy and efficiency of theproposed model

From Figure 2 and Tables 1 and 2 we can draw theconclusion that the present approach has good convergenceand excellent accuracy and reliability to study the vibration


04

02

0

minus02

0

05

1

15

2

1

050

ki = 104

04

06

08

02

0

05

1

15

2

1

050

ki = 106

04

02

0

minus02

0

05

1

15

2

1

050

ki = 108

04

02

0

minus02

0

05

1

15

2

1

05

0

ki = 1012 ki = 1014

0

05

1

15

2

1

05

0

04

06

02

04

06

02

0

05

1

15

2

1

050

ki = 1010

04

02

0

minus02

0

05

1

15

2

1

050

ki = 1016

04

02

0

minus02

0

05

1

15

2

1

05

0

ki = 1020

Figure 3 The corresponding fundamental mode shapes of Table 3

analysis of rectangular thick plates on Pasternak foundationwith arbitrary boundary conditionsThus based on the com-parison studies the first known results of rectangular thickplates rested on Pasternak foundation with various elasticboundary conditions as well as foundation coefficients arepresented inTable 3The geometrical dimensions of the platesare the same as those in Figure 1 The boundary condition isclamped at 119910 = constants and 119909 = 0 and the elastic restraintson the 119909 = 119886 are uniformly (119896119906 = 119896V = 119896119908) changed from104 to 1020 From the table first we can see that the frequencyparameters gradually increase as the elastic restraints increaseand then they almost keep unchanged while the stiffness islarger than 1018 In order to enhance the understanding of the

phenomenon the corresponding fundamental mode shapesof Table 3 are depicted in Figure 3 From Figure 3 it is directlyseen that the boundary elastic restraints have a significanteffect on the vibration behavior of the rectangular thickplate

Lastly the influence of the foundation coefficients ofthe Pasternak foundation on the rectangular thick platessubjected to various boundary conditions is reported forthe first time as shown in Figure 4 The elastic boundarycondition is defined as 119896119908 = 119896V = 119896119908 = 1010 Thegeometric parameters are the same as Figure 2 It can be easilyobtained that regardless of the boundary conditions of therectangular thick plates there exists a certain range of the


Table 3 The first frequency parameters Ω for rectangular thick plate with different elastic boundary condition and foundation coefficient

119896119906 = 119896V = 119896119908 (119870119882 119870119878) = (10 10) (119870119882 119870119878) = (100 10)1 2 3 4 1 2 3 4

104 30361 48053 49183 51693 33626 49027 49199 51746106 30361 48053 49183 51693 33627 49027 49200 51746108 30381 48064 49191 51701 33645 49038 49208 517541010 32188 49129 50023 52473 35330 50040 50084 525301012 49669 62598 68032 78515 51577 63305 68046 785181014 54078 65994 71711 83654 55706 66627 71724 839071016 54959 66322 71760 83794 56004 66785 71770 839921018 55041 66356 71761 83808 56037 66801 71771 840001020 55041 66356 71761 83808 56037 66801 71771 84000

58565452

Ω

105

105

100

100

10minus5

10minus5

KW

K S

Ω

105

100

100

10minus5

4644424

38

105

10minus5

KW

K SΩ

105

100

100

10minus5

105104103102101

105

10minus5

KW

K S

CCCC SSSS

EEEE

Figure 4 Variation of the frequency parameters Ω versus the elastic foundation coefficients for rectangular thick plates with differentboundary conditions

elastic foundation coefficients during which the frequencyparameter Ω increases and out of which the influence onfrequency parameterΩ is negligible

4 Conclusions

In this paper a unified method is developed for the vibra-tion analysis of the rectangular thick plates on Pasternakfoundation with arbitrary boundary conditions based on

the linear small-strain 3D elasticity theory The methodcombines the advantages of the Ritzmethod and themodifiedFourier expansion Under the current framework regardlessof boundary conditions each displacement function of theplate is expanded as the superposition of a standard three-dimensional cosine Fourier series and several auxiliary func-tions are introduced to remove any potential discontinu-ousness of the original rectangular plates unknown and itsderivatives at the edges The good convergence and excellent


accuracy and reliability are checked and validated by thecomparison with the results presented by other contributorsNew results for free vibration of rectangular thick plates withelastic boundary conditions are presentedwhichmay be usedfor benchmarking of researchers in the field In addition bymeans of the influences of elastic restrain parameters andfoundation coefficients on free vibration characteristic of therectangular thick plates we can know that the frequencyparameters increase rapidly as the restraint parameters andfoundation coefficients increase in the certain range

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The authors gratefully acknowledge the financial supportfrom the National Natural Science Foundation of China (no61633008 and no 61503073) and the Natural Research Fundof Jilin Province (no 20170101125JC)

References

[1] S Timoshenko and S Woinowsky-KriegerTheory of Plates andShells 1959

[2] R D Mindlin ldquoInfluence of rotary inertia and shear onflexural motions of isotropic elastic platesrdquo Journal of AppliedMechanics vol 18 pp 1031ndash1036 1951

[3] R D Mindlin A Schacknow and H Deresiewicz ldquoFlexuralvibrations of rectangular platesrdquo vol 23 no 3 pp 430ndash4361956

[4] N FHanna andAW Leissa ldquoA higher order shear deformationtheory for the vibration of thick platesrdquo Journal of Sound andVibration vol 170 no 4 pp 545ndash555 1994

[5] J N Reddy ldquoA simple higher order theory for laminatedcomposite platesrdquo Journal of Applied Mechanics vol 51 no 4pp 745ndash752 1984

[6] Q Wang D Shi Q Liang and X Shi ldquoA unified solution forvibration analysis of functionally graded circular annular andsector plates with general boundary conditionsrdquo CompositesPart B Engineering vol 88 pp 264ndash294 2016

[7] Q Wang D Shi and X Shi ldquoA modified solution for the freevibration analysis of moderately thick orthotropic rectangularplates with general boundary conditions internal line supportsand resting on elastic foundationrdquoMeccanica vol 51 no 8 pp1985ndash2017 2016

[8] Q Wang D Shi F Pang and F e Ahad ldquoBenchmark solutionfor free vibration of thick open cylindrical shells on Pasternakfoundation with general boundary conditionsrdquoMeccanica vol52 no 1 pp 457ndash482 2017

[9] S Srinivas C V Joga Rao and A K Rao ldquoAn exact analysisfor vibration of simply-supported homogeneous and laminatedthick rectangular platesrdquo Journal of Sound andVibration vol 12no 2 pp 187ndash199 1970

[10] Y K Cheung and S Chakrabarti ldquoFree vibration of thicklayered rectangular plates by a finite layer methodrdquo Journal ofSound and Vibration vol 21 no 3 pp 277ndash284 1972

[11] J RHutchinson and SD Zillmer ldquoVibration of a free rectangu-lar parallelepipedrdquo Journal of Applied Mechanics TransactionsASME vol 50 no 1 pp 123ndash130 1983

[12] J A Fromme and A W Leissa ldquoFree vibration of the rectangu-lar parallelepipedrdquo Journal of the Acoustical Society of Americavol 48 no 1 pp 290ndash298 1970

[13] M Malik and C W Bert ldquoThree-dimensional elasticity solu-tions for free vibrations of rectangular plates by the differentialquadrature methodrdquo International Journal of Solids and Struc-tures vol 35 no 3-4 pp 299ndash318 1998

[14] K M Liew and T M Teo ldquoThree-dimensional vibrationanalysis of rectangular plates based on differential quadraturemethodrdquo Journal of Sound and Vibration vol 220 no 4 pp577ndash599 1999

[15] K M Liew T M Teo and J-B Han ldquoComparative accuracy ofDQ and HDQ methods for three-dimensional vibration anal-ysis of rectangular platesrdquo International Journal for NumericalMethods in Engineering vol 45 no 12 pp 1831ndash1848 1999

[16] A Leissa and Z-D Zhang ldquoOn the three-dimensional vibra-tions of the cantilevered rectangular parallelepipedrdquo Journal ofthe Acoustical Society of America vol 73 no 6 pp 2013ndash20211983

[17] K M Liew K C Hung and M K Lim ldquoA continuum three-dimensional vibration analysis of thick rectangular platesrdquoInternational Journal of Solids and Structures vol 30 no 24 pp3357ndash3379 1993

[18] K M Liew K C Hung and M K Lim ldquoFree vibrationstudies on stress-free three-dimensional elastic solidsrdquo Journalof Applied Mechanics vol 62 no 1 pp 159ndash165 1995

[19] K M Liew K C Hung and M K Lim ldquoThree-dimensionalvibration of rectangular plates effects of thickness and edgeconstraintsrdquo Journal of Sound and Vibration vol 182 no 5 pp709ndash727 1995

[20] K Itakura ldquoFree vibration analysis of thick skewed plateshaving arbitrary boundary conditionsrdquo Journal of Structural andConstruction Engineering (Transactions of AIJ) vol 62 no 492pp 37ndash45 1997

[21] C W Lim S Kitipornchai and K M Liew ldquoNumericalaspects for free vibration of thick plates Part II numericalefficiency and vibration frequenciesrdquo Computer Methods inApplied Mechanics and Engineering vol 156 no 1-4 pp 31ndash441998

[22] C W Lim K M Liew and S Kitipornchai ldquoNumericalaspects for free vibration of thick plates Part I formulationand verificationrdquo Computer Methods in Applied Mechanics andEngineering vol 156 no 1-4 pp 15ndash29 1998

[23] D Zhou Y K Cheung F T K Au and S H Lo ldquoThree-dimensional vibration analysis of thick rectangular plates usingChebyshev polynomial and Ritz methodrdquo International Journalof Solids and Structures vol 39 no 26 pp 6339ndash6353 2002

[24] P L Pasternak ldquoOn a newmethod of analysis of an elastic foun-dation bymeans of two foundation constantsrdquoGosudarstvennoeIzdatelstvo Literaturi Po Stroitelstvu I Arkhitekture pp 1ndash561954

[25] H S Shen J Yang and L Zhang ldquoFree and forced vibrationof Reissner-Mindlin plates with free edges resting on elasticfoundationsrdquo Journal of Sound and Vibration vol 244 no 2 pp299ndash320 2001

[26] A Kutlu B Ugurlu M H Omurtag and A Ergin ldquoDynamicresponse of Mindlin plates resting on arbitrarily orthotropicPasternak foundation and partially in contact with fluidrdquoOceanEngineering vol 42 pp 112ndash125 2012


[27] E Winkler Die Lehre von der Elastizitat und Festigkeit 1867[28] K Zhang D Shi W Wang and Q Wang ldquoMechanical charac-

terization of hybrid lattice-to-steel joint with pyramidal CFRPtruss for marine applicationrdquo Composite Structures vol 160 pp1198ndash1204 2017

[29] H Zhang D Shi and Q Wang ldquoAn improved Fourier seriessolution for free vibration analysis of the moderately thick lam-inated composite rectangular platewith non-uniformboundaryconditionsrdquo International Journal of Mechanical Sciences vol121 pp 1ndash20 2017

[30] Q Wang D Shi Q Liang and F Pang ldquoFree vibration offour-parameter functionally graded moderately thick doubly-curved panels and shells of revolution with general boundaryconditionsrdquo Applied Mathematical Modelling vol 42 pp 705ndash734 2017

[31] Q Wang D Shi Q Liang F Pang et al ldquoFree vibrationsof composite laminated doubly-curved shells and panels ofrevolutionwith general elastic restraintsrdquoAppliedMathematicalModelling vol 46 pp 227ndash262 2017

[32] D Shao S Hu Q Wang and F Pang ldquoAn enhancedreverberation-raymatrix approach for transient response analy-sis of composite laminated shallow shells with general boundaryconditionsrdquo Composite Structures vol 162 pp 133ndash155 2017

[33] D Shao S Hu Q Wang and F Pang ldquoFree vibration ofrefined higher-order shear deformation composite laminatedbeams with general boundary conditionsrdquo Composites Part BEngineering vol 108 pp 75ndash90 2017

[34] Q Wang D Shi Q Liang and F e Ahad ldquoA unified solutionfor free in-plane vibration of orthotropic circular annularand sector plates with general boundary conditionsrdquo AppliedMathematical Modelling vol 40 no 21-22 pp 9228ndash9253 2016

[35] Q Wang D Shi Q Liang and F Ahad ldquoAn improved Fourierseries solution for the dynamic analysis of laminated compositeannular circular and sector plate with general boundary condi-tionsrdquo Journal of Composite Materials vol 50 no 30 pp 4199ndash4233 2016

[36] QWang D Shi andQ Liang ldquoFree vibration analysis of axiallyloaded laminated composite beams with general boundaryconditions by using a modified Fourier-Ritz approachrdquo Journalof Composite Materials vol 50 no 15 pp 2111ndash2135 2016

[37] D Shao S Hu QWang and F Pang ldquoA unified analysis for thetransient response of composite laminated curved beam witharbitrary lamination schemes and general boundary restraintsrdquoComposite Structures vol 154 pp 507ndash526 2016

[38] D Shao F Hu Q Wang F Pang and S Hu ldquoTransientresponse analysis of cross-ply composite laminated rectangularplates with general boundary restraints by the method ofreverberation ray matrixrdquo Composite Structures vol 152 pp168ndash182 2016

[39] D Shi Q Wang X Shi and F Pang ldquoA series solution forthe in-plane vibration analysis of orthotropic rectangular plateswith non-uniform elastic boundary constraints and internalline supportsrdquo Archive of Applied Mechanics vol 85 no 1 pp51ndash73 2015

[40] Q Wang D Shi Q Liang and F Pang ldquoA unified solutionfor vibration analysis of moderately thick functionally gradedrectangular plates with general boundary restraints and internalline supportsrdquoMechanics of Advanced Materials and Structuresvol 24 no 11 pp 961ndash973 2017

[41] T Ye G Jin Z Su and Y Chen ldquoA modified Fourier solutionfor vibration analysis of moderately thick laminated plates

with general boundary restraints and internal line supportsrdquoInternational Journal of Mechanical Sciences vol 80 pp 29ndash462014

[42] T Ye G Jin and Z Su ldquoThree-dimensional vibration analysisof laminated functionally graded spherical shells with generalboundary conditionsrdquo Composite Structures vol 116 no 1 pp571ndash588 2014

[43] T Ye G Jin S Shi and X Ma ldquoThree-dimensional freevibration analysis of thick cylindrical shells with general endconditions and resting on elastic foundationsrdquo InternationalJournal of Mechanical Sciences vol 84 pp 120ndash137 2014

[44] Z Su G Jin and T Ye ldquoFree vibration analysis of moderatelythick functionally graded open shells with general boundaryconditionsrdquo Composite Structures vol 117 no 1 pp 169ndash1862014

[45] D Shi Q Wang X Shi and F Pang ldquoAn accurate solu-tion method for the vibration analysis of Timoshenko beamswith general elastic supportsrdquo Proceedings of the Institution ofMechanical Engineers Part C Journal ofMechanical EngineeringScience vol 229 no 13 pp 2327ndash2340 2014

[46] D Shi Y Zhao Q Wang X Teng and F Pang ldquoA unifiedspectro-geometric-ritz method for vibration analysis of openand closed shells with arbitrary boundary conditionsrdquo Shockand Vibration vol 2016 Article ID 4097123 30 pages 2016

[47] QWangD Shi F Pang andQ Liang ldquoVibrations of compositelaminated circular panels and shells of revolution with generalelastic boundary conditions via Fourier-Ritz methodrdquo Curvedand Layered Structures vol 3 no 1 pp 105ndash136 2016

[48] D Zhou Y K Cheung S H Lo and F T K Au ldquoThree-dimensional vibration analysis of rectangular thick plates onPasternak foundationrdquo International Journal for NumericalMethods in Engineering vol 59 no 10 pp 1313ndash1334 2004

[49] D Zhou S H Lo F T K Au and Y K Cheung ldquoThree-dimensional free vibration of thick circular plates on Pasternakfoundationrdquo Journal of Sound and Vibration vol 292 no 3ndash5pp 726ndash741 2006

[50] MH Omurtag A Ozutok A Y Akoz and Y Ozcelikors ldquoFreevibration analysis of Kirchhoff plates resting on elastic foun-dation by mixed finite element formulation based on Gateauxdifferentialrdquo International Journal for Numerical Methods inEngineering vol 40 no 2 pp 295ndash317 1997

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of


International Journal of

RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal of

Volume 201

Submit your manuscripts athttpswwwhindawicom

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



and Leissa [12] extended the Fourier method to investigatethe free vibration of the rectangular parallelepiped withsimple classical boundary conditions based on the three-dimensional elasticity theory On the basis of the three-dimensional elasticity theory and differential quadraturemethod (DQM) Malik and Bert [13] Liew and Teo [14] andLiew et al [15] investigated the free vibration characteristicsof rectangular plates with some selected classical boundaryconditions Leissa and Zhang [16] employed the Ritz methodto study the three-dimensional problem of determining thefree vibration frequencies and mode shapes for a rectangularparallelepiped which is completely fixed on one face andfree on other five faces In this research the displacementsof the rectangular parallelepiped are assumed in the formof algebraic polynomials Also Liew et al [17ndash19] appliedthe Ritz method and the three-dimensional elastic theory toestablish the continuum three-dimensional Ritz formulationto study the vibration characteristic of the thick rectangularplates with classical boundary conditions The orthogonallygenerated polynomial functions are used as admissible func-tions to make analysis in such plates Itakura [20] adoptedthe three-dimensional theory to study the free vibration ofrectangular thick plate with classical boundary conditionsLim et al [21 22] presented a concise formulation andan efficient method of solution to study the free vibrationof thick shear deformable plates with classical boundaryconditions Zhou et al [23] used the Chebyshev polynomialsas admissible functions and applied Ritz method to studythe three-dimensional vibration of rectangular plates withclassical boundary conditions

By means of the review of the above references most ofthe studies for the vibration problem of the thick rectangularplates do not consider the elastic foundations Howevera lot of engineering problems boil down to a rectangularplate on elastic foundations such as footing of buildingspavement of roads and base of heavy machines In thepractical application the Pasternakmodel (also referred to asthe two-parameter model) [24ndash26] is widely used to describethemechanical behavior of the foundation in which the well-known Winkler model [27] is a special case In additionthe boundary condition may not always be classical casein reality and a variety of possible boundary conditionsincluding classical boundary conditions elastic boundaryrestraints and the combinations of two or more conditionsmay be encountered [6ndash8 28ndash47] Based on the open pub-lished paper we can know that merely Zhou et al [4849] extended the Chebyshev-Ritz method to study the freevibration characteristics of rectangular thick plates and thickcircular plates resting on elastic foundations respectivelyHowever it should be noted that the boundary conditionsare limited to the FFFF SSSS and CCCC classical boundaryconditions There is no work reported on the free vibrationbehavior of rectangular thick plates on Pasternak foundationwith general boundary conditions Thus to present a newthree-dimensional exact solution for the free vibrations ofrectangular thick plates with arbitrary boundary conditionsis of crucial importance

In this paper the first known three-dimensional vibra-tion characteristic of rectangular thick plates on Pasternak

foundation with arbitrary boundary conditions is stud-ied The theoretical formulations are based on the three-dimensional linear small-strain elasticity theory The mod-ified Fourier series are chosen as the basis functions of theadmissible function of the thick plates in the Ritz methodThe arbitrary boundary conditions are obtained by layingout three types of liner springs on all edges The excellentaccuracy and reliability of current solutions are demonstratedby numerical examples and comparisons with the resultsavailable in the literature In addition the influence of thefoundation coefficients as well as the boundary restraintparameters are also analyzed which can serve as the bench-mark data for the future research technique

2 Theoretical Formulations

Consider a rectangular thick plate on Pasternak foundationwith arbitrary boundary conditions as shown in Figure 1 ACartesian coordinate (119909 119910 119911) is also shown in the Figure 1which will be used in the analysis 119880 119881 and 119882 denotethe displacement components in 119909 119910 and 119911 directionsLength width and thickness of the rectangular thick platesare assumed as 119886 119887 and ℎ respectively As shown in Figure 1three groups of linear restraint springs are arranged at allsides of the rectangular thick plates to separately simulate thearbitrary boundary conditions The undersurface of the plateis continuously rested on an elastic foundation representedby the WinklerPasternak model in which the Winkler andshear stiffness are denoted by 119870119908 and 119870119904 respectively seenfrom Figure 1

According to the three-dimensional elasticity theorythe strain components of rectangular thick plates can beexpressed as

120576119909 = 120597119906120597119909 120576119910 = 120597V120597119910 120576119911 = 120597119908120597119911

(1a)

120574119909119910 = 120597V120597119909 + 120597119906120597119910 120574119909119911 = 120597119908120597119909 + 120597119906120597119911 120574119910119911 = 120597V120597119911 + 120597119908120597119910

(1b)

Based on Hookersquos law the corresponding stress-strainrelations of the rectangular thick plates can be written as120590119909 = (Υ + 2Θ) 120576119909 + Υ120576119910 + Υ120576119911120590119910 = Υ120576119909 + (Υ + 2Θ) 120576119910 + Υ120576119911120590119911 = Υ120576119909 + Υ120576119910 + (Υ + 2Θ) 120576119911120591119909119910 = Θ120574119909119910


z

x

y

a

b

ℎ

W

U V

Ground

Middle surface



120591119909119911 = Θ120574119909119911120591119910119911 = Θ120574119910119911(2)


Υ = 119864120583(1 + 120583) (1 minus 2120583) Θ = 1198642 (1 + 120583) (3)



(4)





(6)



(7)



119880119901119891 = 12sdot int1198860int1198870(1198701199081199082 + 119870119904 (120597119908120597119909 )2 + 119870119904 (120597119908120597119910 )2)119889119909119889119910 (8)





119906 (119909 119910 119911 119905) = 119880 (119909 119910 119911) 119890119894120596119905 (10a)

V (119909 119910 119911 119905) = 119881 (119909 119910 119911) 119890119894120596119905 (10b)119908 (119909 119910 119911 119905) = 119882 (119909 119910 119911) 119890119894120596119905 (10c)



= infinsum119898=0

infinsum119899=0

infinsum119902=0


infinsum119899=0

(11988611198981198991205771119911 (119911) + 11988621198981198991205772119911 (119911)) cos 120582119898119909119909 cos 120582119899119910119910+ infinsum119898=0

infinsum119902=0


119899=0

infinsum119902=0


infinsum119899=0

infinsum119902=0


infinsum119899=0

(11988711198981198991205771119911 (119911) + 11988721198981198991205772119911 (119911)) cos 120582119898119909119909 cos 120582119899119910119910

+ infinsum119898=0

infinsum119902=0

(11988731198981199021205771119910 (119910) + 11988741198981199021205772119910 (119910)) cos 120582119898119909119909 cos 120582119902119911119911+ infinsum119899=0

infinsum119902=0

(11988751198991199021205771119909 (119909) + 11988761198991199021205772119909 (119909)) cos 120582119899119910119910 cos 120582119902119911119911119882 (119909 119910 119911)

= infinsum119898=0

infinsum119899=0

infinsum119902=0


infinsum119899=0

(11988811198981198991205771119911 (119911) + 11988821198981198991205772119911 (119911)) cos 120582119898119909119909 cos 120582119899119910119910+ infinsum119898=0

infinsum119902=0

(11988831198981199021205771119910 (119910) + 11988841198981199021205772119910 (119910)) cos 120582119898119909119909 cos 120582119902119911119911+ infinsum119899=0

infinsum119902=0

(11988851198991199021205771119909 (119909) + 11988861198991199021205772119909 (119909)) cos 120582119899119910119910 cos 120582119902119911119911(11)


2mn a

3mq a

4mq a

5nq a

6nq b

1mn b

2mn

b3mq b4mq b

5nq b

6nq c

1mn c

2mn c

3mq c

4mq c







1205771119909 (0) = 1205771119909 (119886) = 12057711199091015840 (119886) = 0 12057711199091015840 (0) = 1 (13a)

1205772119909 (0) = 1205772119909 (119886) = 12057721199091015840 (0) = 0 12057721199091015840 (119886) = 1 (13b)

1205771119910 (0) = 1205771119910 (119887) = 12057711199101015840 (119887) = 0 12057711199101015840 (0) = 1 (13c)

1205772119910 (0) = 1205772119910 (119887) = 12057721199101015840 (0) = 0 12057721199101015840 (119887) = 1 (13d)

1205771119911 (0) = 1205771119911 (ℎ) = 12057711199111015840 (ℎ) = 0 12057711199111015840 (0) = 1 (13e)


1st mode2nd mode

3rd mode4th mode

5

6

Ω 7

8

9

3 6 9 120

M N Q

(a)

1st mode2nd mode

3rd mode4th mode

3

4

Ω

5

6

3 6 9 120

M N Q

(b)

1st mode2nd mode

3rd mode4th mode

2

3

Ω

4

3 6 9 120

M N Q

(c)


1205772119911 (0) = 1205772119911 (ℎ) = 12057721199111015840 (0) = 0 12057721199111015840 (ℎ) = 1 (13f)


120580 = 119860119906119898119899 119886119897119898 119887119897119899 119861V119898119899 119888119897119898 119889119897119899 119864119910119898119899 119892119897119898 ℎ119897119899 (14)



]]]]minus 1205962 [[[[

[M119906119906] 0 00 [MVV] 00 0 [M119908119908]

]]]])ABC

= 0(15)


A = 119860000 119860119898119899119902 119860119872119873119876 119886100 1198861119872119873 119886200 1198862119872119873 119886300 1198863119872119876 119886400 1198864119872119876 119886500 1198865119873119876 119886600 1198866119873119876 (16a)

B = 119861000 119861119898119899119902 119861119872119873119876 119887100 1198871119872119873 119887200 1198872119872119873 119887300 1198873119872119876 119887400 1198874119872119876 119887500 1198875119873119876 119887600 1198876119873119876 (16b)








SSSS02


Present 17525 32184 32184 38480 38480 45526 55781


CCCC02


Present 26966 47192 47192 61949 61949 63453 72404


FFFF02 Reference [18] 11710 17433 21252 27647 27647 40191 42776

Present 11719 17436 21254 27659 27659 40192 42776




SSSS

001


Present 26551 55717 55717 85404


Present 33400 59287 59287 87774

01


Present 27837 53013 53013 77215


Present 39802 60052 60052 82148


Present 81397 128995 128995 139349







04

02

0

minus02

0

05

1

15

2

1

050

ki = 104

04

06

08

02

0

05

1

15

2

1

050

ki = 106

04

02

0

minus02

0

05

1

15

2

1

050

ki = 108

04

02

0

minus02

0

05

1

15

2

1

05

0

ki = 1012 ki = 1014

0

05

1

15

2

1

05

0

04

06

02

04

06

02

0

05

1

15

2

1

050

ki = 1010

04

02

0

minus02

0

05

1

15

2

1

050

ki = 1016

04

02

0

minus02

0

05

1

15

2

1

05

0

ki = 1020







119896119906 = 119896V = 119896119908 (119870119882 119870119878) = (10 10) (119870119882 119870119878) = (100 10)1 2 3 4 1 2 3 4

104 30361 48053 49183 51693 33626 49027 49199 51746106 30361 48053 49183 51693 33627 49027 49200 51746108 30381 48064 49191 51701 33645 49038 49208 517541010 32188 49129 50023 52473 35330 50040 50084 525301012 49669 62598 68032 78515 51577 63305 68046 785181014 54078 65994 71711 83654 55706 66627 71724 839071016 54959 66322 71760 83794 56004 66785 71770 839921018 55041 66356 71761 83808 56037 66801 71771 840001020 55041 66356 71761 83808 56037 66801 71771 84000

58565452

Ω

105

105

100

100

10minus5

10minus5

KW

K S

Ω

105

100

100

10minus5

4644424

38

105

10minus5

KW

K SΩ

105

100

100

10minus5

105104103102101

105

10minus5

KW

K S

CCCC SSSS

EEEE



4 Conclusions







Acknowledgments


References





















































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










z

x

y

a

b

ℎ

W

U V

Ground

Middle surface



120591119909119911 = Θ120574119909119911120591119910119911 = Θ120574119910119911(2)


Υ = 119864120583(1 + 120583) (1 minus 2120583) Θ = 1198642 (1 + 120583) (3)



(4)





(6)



(7)



119880119901119891 = 12sdot int1198860int1198870(1198701199081199082 + 119870119904 (120597119908120597119909 )2 + 119870119904 (120597119908120597119910 )2)119889119909119889119910 (8)





119906 (119909 119910 119911 119905) = 119880 (119909 119910 119911) 119890119894120596119905 (10a)

V (119909 119910 119911 119905) = 119881 (119909 119910 119911) 119890119894120596119905 (10b)119908 (119909 119910 119911 119905) = 119882 (119909 119910 119911) 119890119894120596119905 (10c)



= infinsum119898=0

infinsum119899=0

infinsum119902=0


infinsum119899=0

(11988611198981198991205771119911 (119911) + 11988621198981198991205772119911 (119911)) cos 120582119898119909119909 cos 120582119899119910119910+ infinsum119898=0

infinsum119902=0


119899=0

infinsum119902=0


infinsum119899=0

infinsum119902=0


infinsum119899=0

(11988711198981198991205771119911 (119911) + 11988721198981198991205772119911 (119911)) cos 120582119898119909119909 cos 120582119899119910119910

+ infinsum119898=0

infinsum119902=0

(11988731198981199021205771119910 (119910) + 11988741198981199021205772119910 (119910)) cos 120582119898119909119909 cos 120582119902119911119911+ infinsum119899=0

infinsum119902=0

(11988751198991199021205771119909 (119909) + 11988761198991199021205772119909 (119909)) cos 120582119899119910119910 cos 120582119902119911119911119882 (119909 119910 119911)

= infinsum119898=0

infinsum119899=0

infinsum119902=0


infinsum119899=0

(11988811198981198991205771119911 (119911) + 11988821198981198991205772119911 (119911)) cos 120582119898119909119909 cos 120582119899119910119910+ infinsum119898=0

infinsum119902=0

(11988831198981199021205771119910 (119910) + 11988841198981199021205772119910 (119910)) cos 120582119898119909119909 cos 120582119902119911119911+ infinsum119899=0

infinsum119902=0

(11988851198991199021205771119909 (119909) + 11988861198991199021205772119909 (119909)) cos 120582119899119910119910 cos 120582119902119911119911(11)


2mn a

3mq a

4mq a

5nq a

6nq b

1mn b

2mn

b3mq b4mq b

5nq b

6nq c

1mn c

2mn c

3mq c

4mq c







1205771119909 (0) = 1205771119909 (119886) = 12057711199091015840 (119886) = 0 12057711199091015840 (0) = 1 (13a)

1205772119909 (0) = 1205772119909 (119886) = 12057721199091015840 (0) = 0 12057721199091015840 (119886) = 1 (13b)

1205771119910 (0) = 1205771119910 (119887) = 12057711199101015840 (119887) = 0 12057711199101015840 (0) = 1 (13c)

1205772119910 (0) = 1205772119910 (119887) = 12057721199101015840 (0) = 0 12057721199101015840 (119887) = 1 (13d)

1205771119911 (0) = 1205771119911 (ℎ) = 12057711199111015840 (ℎ) = 0 12057711199111015840 (0) = 1 (13e)


1st mode2nd mode

3rd mode4th mode

5

6

Ω 7

8

9

3 6 9 120

M N Q

(a)

1st mode2nd mode

3rd mode4th mode

3

4

Ω

5

6

3 6 9 120

M N Q

(b)

1st mode2nd mode

3rd mode4th mode

2

3

Ω

4

3 6 9 120

M N Q

(c)


1205772119911 (0) = 1205772119911 (ℎ) = 12057721199111015840 (0) = 0 12057721199111015840 (ℎ) = 1 (13f)


120580 = 119860119906119898119899 119886119897119898 119887119897119899 119861V119898119899 119888119897119898 119889119897119899 119864119910119898119899 119892119897119898 ℎ119897119899 (14)



]]]]minus 1205962 [[[[

[M119906119906] 0 00 [MVV] 00 0 [M119908119908]

]]]])ABC

= 0(15)


A = 119860000 119860119898119899119902 119860119872119873119876 119886100 1198861119872119873 119886200 1198862119872119873 119886300 1198863119872119876 119886400 1198864119872119876 119886500 1198865119873119876 119886600 1198866119873119876 (16a)

B = 119861000 119861119898119899119902 119861119872119873119876 119887100 1198871119872119873 119887200 1198872119872119873 119887300 1198873119872119876 119887400 1198874119872119876 119887500 1198875119873119876 119887600 1198876119873119876 (16b)








SSSS02


Present 17525 32184 32184 38480 38480 45526 55781


CCCC02


Present 26966 47192 47192 61949 61949 63453 72404


FFFF02 Reference [18] 11710 17433 21252 27647 27647 40191 42776

Present 11719 17436 21254 27659 27659 40192 42776




SSSS

001


Present 26551 55717 55717 85404


Present 33400 59287 59287 87774

01


Present 27837 53013 53013 77215


Present 39802 60052 60052 82148


Present 81397 128995 128995 139349







04

02

0

minus02

0

05

1

15

2

1

050

ki = 104

04

06

08

02

0

05

1

15

2

1

050

ki = 106

04

02

0

minus02

0

05

1

15

2

1

050

ki = 108

04

02

0

minus02

0

05

1

15

2

1

05

0

ki = 1012 ki = 1014

0

05

1

15

2

1

05

0

04

06

02

04

06

02

0

05

1

15

2

1

050

ki = 1010

04

02

0

minus02

0

05

1

15

2

1

050

ki = 1016

04

02

0

minus02

0

05

1

15

2

1

05

0

ki = 1020







119896119906 = 119896V = 119896119908 (119870119882 119870119878) = (10 10) (119870119882 119870119878) = (100 10)1 2 3 4 1 2 3 4

104 30361 48053 49183 51693 33626 49027 49199 51746106 30361 48053 49183 51693 33627 49027 49200 51746108 30381 48064 49191 51701 33645 49038 49208 517541010 32188 49129 50023 52473 35330 50040 50084 525301012 49669 62598 68032 78515 51577 63305 68046 785181014 54078 65994 71711 83654 55706 66627 71724 839071016 54959 66322 71760 83794 56004 66785 71770 839921018 55041 66356 71761 83808 56037 66801 71771 840001020 55041 66356 71761 83808 56037 66801 71771 84000

58565452

Ω

105

105

100

100

10minus5

10minus5

KW

K S

Ω

105

100

100

10minus5

4644424

38

105

10minus5

KW

K SΩ

105

100

100

10minus5

105104103102101

105

10minus5

KW

K S

CCCC SSSS

EEEE



4 Conclusions







Acknowledgments


References





















































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119880119901119891 = 12sdot int1198860int1198870(1198701199081199082 + 119870119904 (120597119908120597119909 )2 + 119870119904 (120597119908120597119910 )2)119889119909119889119910 (8)





119906 (119909 119910 119911 119905) = 119880 (119909 119910 119911) 119890119894120596119905 (10a)

V (119909 119910 119911 119905) = 119881 (119909 119910 119911) 119890119894120596119905 (10b)119908 (119909 119910 119911 119905) = 119882 (119909 119910 119911) 119890119894120596119905 (10c)



= infinsum119898=0

infinsum119899=0

infinsum119902=0


infinsum119899=0

(11988611198981198991205771119911 (119911) + 11988621198981198991205772119911 (119911)) cos 120582119898119909119909 cos 120582119899119910119910+ infinsum119898=0

infinsum119902=0


119899=0

infinsum119902=0


infinsum119899=0

infinsum119902=0


infinsum119899=0

(11988711198981198991205771119911 (119911) + 11988721198981198991205772119911 (119911)) cos 120582119898119909119909 cos 120582119899119910119910

+ infinsum119898=0

infinsum119902=0

(11988731198981199021205771119910 (119910) + 11988741198981199021205772119910 (119910)) cos 120582119898119909119909 cos 120582119902119911119911+ infinsum119899=0

infinsum119902=0

(11988751198991199021205771119909 (119909) + 11988761198991199021205772119909 (119909)) cos 120582119899119910119910 cos 120582119902119911119911119882 (119909 119910 119911)

= infinsum119898=0

infinsum119899=0

infinsum119902=0


infinsum119899=0

(11988811198981198991205771119911 (119911) + 11988821198981198991205772119911 (119911)) cos 120582119898119909119909 cos 120582119899119910119910+ infinsum119898=0

infinsum119902=0

(11988831198981199021205771119910 (119910) + 11988841198981199021205772119910 (119910)) cos 120582119898119909119909 cos 120582119902119911119911+ infinsum119899=0

infinsum119902=0

(11988851198991199021205771119909 (119909) + 11988861198991199021205772119909 (119909)) cos 120582119899119910119910 cos 120582119902119911119911(11)


2mn a

3mq a

4mq a

5nq a

6nq b

1mn b

2mn

b3mq b4mq b

5nq b

6nq c

1mn c

2mn c

3mq c

4mq c







1205771119909 (0) = 1205771119909 (119886) = 12057711199091015840 (119886) = 0 12057711199091015840 (0) = 1 (13a)

1205772119909 (0) = 1205772119909 (119886) = 12057721199091015840 (0) = 0 12057721199091015840 (119886) = 1 (13b)

1205771119910 (0) = 1205771119910 (119887) = 12057711199101015840 (119887) = 0 12057711199101015840 (0) = 1 (13c)

1205772119910 (0) = 1205772119910 (119887) = 12057721199101015840 (0) = 0 12057721199101015840 (119887) = 1 (13d)

1205771119911 (0) = 1205771119911 (ℎ) = 12057711199111015840 (ℎ) = 0 12057711199111015840 (0) = 1 (13e)


1st mode2nd mode

3rd mode4th mode

5

6

Ω 7

8

9

3 6 9 120

M N Q

(a)

1st mode2nd mode

3rd mode4th mode

3

4

Ω

5

6

3 6 9 120

M N Q

(b)

1st mode2nd mode

3rd mode4th mode

2

3

Ω

4

3 6 9 120

M N Q

(c)


1205772119911 (0) = 1205772119911 (ℎ) = 12057721199111015840 (0) = 0 12057721199111015840 (ℎ) = 1 (13f)


120580 = 119860119906119898119899 119886119897119898 119887119897119899 119861V119898119899 119888119897119898 119889119897119899 119864119910119898119899 119892119897119898 ℎ119897119899 (14)



]]]]minus 1205962 [[[[

[M119906119906] 0 00 [MVV] 00 0 [M119908119908]

]]]])ABC

= 0(15)


A = 119860000 119860119898119899119902 119860119872119873119876 119886100 1198861119872119873 119886200 1198862119872119873 119886300 1198863119872119876 119886400 1198864119872119876 119886500 1198865119873119876 119886600 1198866119873119876 (16a)

B = 119861000 119861119898119899119902 119861119872119873119876 119887100 1198871119872119873 119887200 1198872119872119873 119887300 1198873119872119876 119887400 1198874119872119876 119887500 1198875119873119876 119887600 1198876119873119876 (16b)








SSSS02


Present 17525 32184 32184 38480 38480 45526 55781


CCCC02


Present 26966 47192 47192 61949 61949 63453 72404


FFFF02 Reference [18] 11710 17433 21252 27647 27647 40191 42776

Present 11719 17436 21254 27659 27659 40192 42776




SSSS

001


Present 26551 55717 55717 85404


Present 33400 59287 59287 87774

01


Present 27837 53013 53013 77215


Present 39802 60052 60052 82148


Present 81397 128995 128995 139349







04

02

0

minus02

0

05

1

15

2

1

050

ki = 104

04

06

08

02

0

05

1

15

2

1

050

ki = 106

04

02

0

minus02

0

05

1

15

2

1

050

ki = 108

04

02

0

minus02

0

05

1

15

2

1

05

0

ki = 1012 ki = 1014

0

05

1

15

2

1

05

0

04

06

02

04

06

02

0

05

1

15

2

1

050

ki = 1010

04

02

0

minus02

0

05

1

15

2

1

050

ki = 1016

04

02

0

minus02

0

05

1

15

2

1

05

0

ki = 1020







119896119906 = 119896V = 119896119908 (119870119882 119870119878) = (10 10) (119870119882 119870119878) = (100 10)1 2 3 4 1 2 3 4

104 30361 48053 49183 51693 33626 49027 49199 51746106 30361 48053 49183 51693 33627 49027 49200 51746108 30381 48064 49191 51701 33645 49038 49208 517541010 32188 49129 50023 52473 35330 50040 50084 525301012 49669 62598 68032 78515 51577 63305 68046 785181014 54078 65994 71711 83654 55706 66627 71724 839071016 54959 66322 71760 83794 56004 66785 71770 839921018 55041 66356 71761 83808 56037 66801 71771 840001020 55041 66356 71761 83808 56037 66801 71771 84000

58565452

Ω

105

105

100

100

10minus5

10minus5

KW

K S

Ω

105

100

100

10minus5

4644424

38

105

10minus5

KW

K SΩ

105

100

100

10minus5

105104103102101

105

10minus5

KW

K S

CCCC SSSS

EEEE



4 Conclusions







Acknowledgments


References





















































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










1st mode2nd mode

3rd mode4th mode

5

6

Ω 7

8

9

3 6 9 120

M N Q

(a)

1st mode2nd mode

3rd mode4th mode

3

4

Ω

5

6

3 6 9 120

M N Q

(b)

1st mode2nd mode

3rd mode4th mode

2

3

Ω

4

3 6 9 120

M N Q

(c)


1205772119911 (0) = 1205772119911 (ℎ) = 12057721199111015840 (0) = 0 12057721199111015840 (ℎ) = 1 (13f)


120580 = 119860119906119898119899 119886119897119898 119887119897119899 119861V119898119899 119888119897119898 119889119897119899 119864119910119898119899 119892119897119898 ℎ119897119899 (14)



]]]]minus 1205962 [[[[

[M119906119906] 0 00 [MVV] 00 0 [M119908119908]

]]]])ABC

= 0(15)


A = 119860000 119860119898119899119902 119860119872119873119876 119886100 1198861119872119873 119886200 1198862119872119873 119886300 1198863119872119876 119886400 1198864119872119876 119886500 1198865119873119876 119886600 1198866119873119876 (16a)

B = 119861000 119861119898119899119902 119861119872119873119876 119887100 1198871119872119873 119887200 1198872119872119873 119887300 1198873119872119876 119887400 1198874119872119876 119887500 1198875119873119876 119887600 1198876119873119876 (16b)








SSSS02


Present 17525 32184 32184 38480 38480 45526 55781


CCCC02


Present 26966 47192 47192 61949 61949 63453 72404


FFFF02 Reference [18] 11710 17433 21252 27647 27647 40191 42776

Present 11719 17436 21254 27659 27659 40192 42776




SSSS

001


Present 26551 55717 55717 85404


Present 33400 59287 59287 87774

01


Present 27837 53013 53013 77215


Present 39802 60052 60052 82148


Present 81397 128995 128995 139349







04

02

0

minus02

0

05

1

15

2

1

050

ki = 104

04

06

08

02

0

05

1

15

2

1

050

ki = 106

04

02

0

minus02

0

05

1

15

2

1

050

ki = 108

04

02

0

minus02

0

05

1

15

2

1

05

0

ki = 1012 ki = 1014

0

05

1

15

2

1

05

0

04

06

02

04

06

02

0

05

1

15

2

1

050

ki = 1010

04

02

0

minus02

0

05

1

15

2

1

050

ki = 1016

04

02

0

minus02

0

05

1

15

2

1

05

0

ki = 1020







119896119906 = 119896V = 119896119908 (119870119882 119870119878) = (10 10) (119870119882 119870119878) = (100 10)1 2 3 4 1 2 3 4

104 30361 48053 49183 51693 33626 49027 49199 51746106 30361 48053 49183 51693 33627 49027 49200 51746108 30381 48064 49191 51701 33645 49038 49208 517541010 32188 49129 50023 52473 35330 50040 50084 525301012 49669 62598 68032 78515 51577 63305 68046 785181014 54078 65994 71711 83654 55706 66627 71724 839071016 54959 66322 71760 83794 56004 66785 71770 839921018 55041 66356 71761 83808 56037 66801 71771 840001020 55041 66356 71761 83808 56037 66801 71771 84000

58565452

Ω

105

105

100

100

10minus5

10minus5

KW

K S

Ω

105

100

100

10minus5

4644424

38

105

10minus5

KW

K SΩ

105

100

100

10minus5

105104103102101

105

10minus5

KW

K S

CCCC SSSS

EEEE



4 Conclusions







Acknowledgments


References





















































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












SSSS02


Present 17525 32184 32184 38480 38480 45526 55781


CCCC02


Present 26966 47192 47192 61949 61949 63453 72404


FFFF02 Reference [18] 11710 17433 21252 27647 27647 40191 42776

Present 11719 17436 21254 27659 27659 40192 42776




SSSS

001


Present 26551 55717 55717 85404


Present 33400 59287 59287 87774

01


Present 27837 53013 53013 77215


Present 39802 60052 60052 82148


Present 81397 128995 128995 139349







04

02

0

minus02

0

05

1

15

2

1

050

ki = 104

04

06

08

02

0

05

1

15

2

1

050

ki = 106

04

02

0

minus02

0

05

1

15

2

1

050

ki = 108

04

02

0

minus02

0

05

1

15

2

1

05

0

ki = 1012 ki = 1014

0

05

1

15

2

1

05

0

04

06

02

04

06

02

0

05

1

15

2

1

050

ki = 1010

04

02

0

minus02

0

05

1

15

2

1

050

ki = 1016

04

02

0

minus02

0

05

1

15

2

1

05

0

ki = 1020







119896119906 = 119896V = 119896119908 (119870119882 119870119878) = (10 10) (119870119882 119870119878) = (100 10)1 2 3 4 1 2 3 4

104 30361 48053 49183 51693 33626 49027 49199 51746106 30361 48053 49183 51693 33627 49027 49200 51746108 30381 48064 49191 51701 33645 49038 49208 517541010 32188 49129 50023 52473 35330 50040 50084 525301012 49669 62598 68032 78515 51577 63305 68046 785181014 54078 65994 71711 83654 55706 66627 71724 839071016 54959 66322 71760 83794 56004 66785 71770 839921018 55041 66356 71761 83808 56037 66801 71771 840001020 55041 66356 71761 83808 56037 66801 71771 84000

58565452

Ω

105

105

100

100

10minus5

10minus5

KW

K S

Ω

105

100

100

10minus5

4644424

38

105

10minus5

KW

K SΩ

105

100

100

10minus5

105104103102101

105

10minus5

KW

K S

CCCC SSSS

EEEE



4 Conclusions







Acknowledgments


References





















































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










04

02

0

minus02

0

05

1

15

2

1

050

ki = 104

04

06

08

02

0

05

1

15

2

1

050

ki = 106

04

02

0

minus02

0

05

1

15

2

1

050

ki = 108

04

02

0

minus02

0

05

1

15

2

1

05

0

ki = 1012 ki = 1014

0

05

1

15

2

1

05

0

04

06

02

04

06

02

0

05

1

15

2

1

050

ki = 1010

04

02

0

minus02

0

05

1

15

2

1

050

ki = 1016

04

02

0

minus02

0

05

1

15

2

1

05

0

ki = 1020







119896119906 = 119896V = 119896119908 (119870119882 119870119878) = (10 10) (119870119882 119870119878) = (100 10)1 2 3 4 1 2 3 4

104 30361 48053 49183 51693 33626 49027 49199 51746106 30361 48053 49183 51693 33627 49027 49200 51746108 30381 48064 49191 51701 33645 49038 49208 517541010 32188 49129 50023 52473 35330 50040 50084 525301012 49669 62598 68032 78515 51577 63305 68046 785181014 54078 65994 71711 83654 55706 66627 71724 839071016 54959 66322 71760 83794 56004 66785 71770 839921018 55041 66356 71761 83808 56037 66801 71771 840001020 55041 66356 71761 83808 56037 66801 71771 84000

58565452

Ω

105

105

100

100

10minus5

10minus5

KW

K S

Ω

105

100

100

10minus5

4644424

38

105

10minus5

KW

K SΩ

105

100

100

10minus5

105104103102101

105

10minus5

KW

K S

CCCC SSSS

EEEE



4 Conclusions







Acknowledgments


References





















































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119896119906 = 119896V = 119896119908 (119870119882 119870119878) = (10 10) (119870119882 119870119878) = (100 10)1 2 3 4 1 2 3 4

104 30361 48053 49183 51693 33626 49027 49199 51746106 30361 48053 49183 51693 33627 49027 49200 51746108 30381 48064 49191 51701 33645 49038 49208 517541010 32188 49129 50023 52473 35330 50040 50084 525301012 49669 62598 68032 78515 51577 63305 68046 785181014 54078 65994 71711 83654 55706 66627 71724 839071016 54959 66322 71760 83794 56004 66785 71770 839921018 55041 66356 71761 83808 56037 66801 71771 840001020 55041 66356 71761 83808 56037 66801 71771 84000

58565452

Ω

105

105

100

100

10minus5

10minus5

KW

K S

Ω

105

100

100

10minus5

4644424

38

105

10minus5

KW

K SΩ

105

100

100

10minus5

105104103102101

105

10minus5

KW

K S

CCCC SSSS

EEEE



4 Conclusions







Acknowledgments


References





















































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













Acknowledgments


References





















































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation



































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Documents

Three-Dimensional Vibration Analysis of Rectangular …downloads.hindawi.com/journals/sv/2017/3425298.pdf · Three-Dimensional Vibration Analysis of Rectangular ... a lot of engineering