research article three-dimensional exact free vibration analysis...

Research ArticleThree-Dimensional Exact Free Vibration Analysis of SphericalCylindrical and Flat One-Layered Panels

Salvatore Brischetto

Department of Mechanical and Aerospace Engineering Politecnico di Torino Corso Duca degli Abruzzi 24 10129 Torino Italy

Correspondence should be addressed to Salvatore Brischetto salvatorebrischettopolitoit

Received 29 July 2013 Revised 16 January 2014 Accepted 18 January 2014 Published 9 March 2014

Academic Editor Ahmet S Yigit

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

The paper proposes a three-dimensional elastic analysis of the free vibration problem of one-layered spherical cylindrical and flatpanels The exact solution is developed for the differential equations of equilibrium written in orthogonal curvilinear coordinatesfor the free vibrations of simply supported structuresThese equations consider an exact geometry for shells without simplificationsThe main novelty is the possibility of a general formulation for different geometries The equations written in general orthogonalcurvilinear coordinates allow the analysis of spherical shell panels and they automatically degenerate into cylindrical shell panelcylindrical closed shell and plate cases Results are proposed for isotropic and orthotropic structures An exhaustive overview isgiven of the vibrationmodes for a number of thickness ratios imposed wave numbers geometries embeddedmaterials and anglesof orthotropyThese results can also be used as reference solutions to validate two-dimensional models for plates and shells in bothanalytical and numerical form (eg closed solutions finite element method differential quadrature method and global collocationmethod)

1 Introduction

Exact solutions for the three-dimensional analysis of platesand shells have been developed by several researchers Inparticular three-dimensional static and dynamic analysisof cylinders and cylindricalspherical shells has receivedconsiderable attention in the past few yearsThe developmentof plate and shell elements highlights the importance ofresearch on the design of these geometries because they arefundamental for the analysis of general structures in theengineering field For this reason plate and shell elementsneed accurate validation Three-dimensional exact solutionsallow such validations and checks to be made and theyalso give further details about three-dimensional effectsand their importance In the literature works about exactthree-dimensional solutions analyze separately the variousgeometries and they do not give a general overview of plateand shell elements The present paper aims to fill this gap byproposing a general formulation for the equations of motionin orthogonal curvilinear coordinates that is valid for squareand rectangular plates cylindrical shell panels spherical shell

panels and cylinders A general overview is given for thosereaders interested in both plate and shell analysis This paperexactly solves the equations of motion in general curvilinearorthogonal coordinates including an exact geometry for shellstructures without simplifications To the best of the presentauthorrsquos knowledge this is the first time that this solution isproposed by means of the exponential matrix method for thethree-dimensional elastic free vibration analysis of plates andshells

The next paragraph discusses the most relevant worksabout three-dimensional shell analysis Chen et al [1] exactlystudied the coupled free vibrations of a transversely isotropiccylindrical shell embedded in an elastic mediumThe behav-ior of the cylindrical shell was analyzed based on three-dimensional elasticity and three displacement functionswere chosen to represent three displacement componentsto decouple the three-dimensional equations of motion ofa transversely isotropic body Fan and Zhang [2] presentedthe exact solutions for statics dynamics and buckling ofthick open laminated cylindrical shells by means of theCayley-Hamilton theoremThe state equation for orthotropy

Hindawi Publishing CorporationShock and VibrationVolume 2014 Article ID 479738 29 pageshttpdxdoiorg1011552014479738

2 Shock and Vibration

was established in a cylindrical coordinate system withoutany assumption about displacement models and stress dis-tribution Gasemzadeh et al [3] studied free vibrations ofcylindrical shells on the basis of three-dimensional exacttheory Simply supported boundary conditions were stud-ied and extensive frequency parameters were obtained bysolving frequency equations The free vibrations of simplysupported cross-ply cylindrical and doubly curved laminateswere investigated in [4] The three-dimensional equations ofmotion expressed in terms of displacements were reducedto a system of coupled ordinary differential equations thatwere solved using the power series method Sharma et al [5]studied the three-dimensional free vibrations of a homoge-nous isotropic viscothermoelastic hollow sphere whose sur-faces were subjected to stress-free thermally insulated orisothermal boundary conditions The governing partial dif-ferential equations were solved into a coupled system ofordinary differential equations Frobenious matrix methodwas employed to obtain the solution The same method wasalso used for the exact three-dimensional vibration analysisof a transradially isotropic thermoelastic solid sphere [6]Soldatos and Ye [7] proposed exact three-dimensional freevibration analysis of angle-ply laminated cylinders Resultswere presented and discussed for relatively thick cylindershaving a regular symmetric or a regular antisymmetric angle-ply lay-up Armenakas et al [8] proposed a self-containedtreatment of the problem of plane harmonic waves propaga-tion along a hollow circular cylinder in the framework of thethree-dimensional theory of elasticity The results presentedfrequencies of free vibrations of such cylinders for a widerange of axial and circumferential wave-lengths and of shelldimensions An interesting comparison between frequenciesobtained from a refined two-dimensional analysis a sheardeformation theory the Flugge theory and an exact elasticityanalysis were proposed in [9] Further details about theFlugge classical thin shell theory concerning the free vibra-tions of cylindrical shells with elastic boundary conditionscan be found in [10] Other comparisons between two-dimensional closed form solutions and available exact 3Delastic and analytical solutions for the free vibration analysisof simply supported and clamped homogenous isotropiccircular cylindrical shells were also proposed in [11] Exactelasticity solutions were extended to functionally gradedcylindrical shells in [12] The three-dimensional linear elas-todynamics equations simplified to the case of generalizedplane strain deformation in the axial direction were solvedusing suitable displacement functions that identically satisfythe boundary conditions Loy and Lam [13] proposed alayerwise approach to study the vibration of thick circularcylindrical shells on the basis of three-dimensional theoryof elasticity The governing equation was obtained using anenergy minimization principle Another interesting exten-sion was proposed by Wang et al [14] who investigatedthe free vibrations ofmagnetoelectroelastic cylindrical panelsbased on the three-dimensional theory Further results aboutthree-dimensional analysis of shells that are not given inclosed form were shown by Efraim and Eisenberger [15]for the dynamic stiffness matrix method and by Kang and

Leissa [16] and Liew et al [17] for the three-dimensional Ritzmethod for vibration of spherical shells

Three-dimensional analysis of plates is usually performedby using simpler equations that do not allow the analysis ofshell geometriesDemasi [18] proposed aNavier-typemethodfor finding the exact three-dimensional solution for isotropicthick and thin rectangular platesThemethod used themixedform of Hooke Law to write the boundary conditions onthe top and bottom surfaces of the plate directly in terms oftransverse stresses Aimmanee and Batra [19] proposed ananalytical solution for free vibrations of a simply supportedrectangular plate made of an incompressible homogeneouslinear elastic isotropic material Some frequencies missing inprevious analytical solutions for plates made of compressiblematerials were identified Some of the in-plane modes of freevibration of a simply supported rectangular plate that weremissed in previous exact solutions were also proposed in[20] A three-dimensional linear elastic small deformationtheory obtained by the direct method was developed for thefree vibration of simply supported homogeneous isotropicthick rectangular plates in [21] The same method was usedin [22] for the flexure of simply supported homogeneousisotropic thick rectangular plates under arbitrary loadingThis solution in terms of infinite series was formally exactand yielded accurate numerical results without undue effortUseful comparisons between two-dimensional models andan exact three-dimensional solution for the free vibrationsof a simply supported rectangular orthotropic thick plate areshown in [23] On the basis of three-dimensional elasticity Ye[24] presented a free vibration analysis of cross-ply laminatedrectangular plates with clamped boundariesThe analysis wasbased on a recursive solution suitable for three-dimensionalvibration analysis of simply supported plates Comparisonsbetween 2D-displacement-based models and exact results ofthe linear three-dimensional elasticity were also proposed in[25] for natural frequencies displacement and stress quanti-ties in multilayered plates Exact three-dimensional elasticitytheory was investigated in [26] for isosceles triangular platesbymeans of a global three-dimensional Ritz formulationThethree-dimensional elasticity free vibration analysis of a circu-lar plate was analyzed in [27] by means of the Ritz methodwith a set of orthogonal polynomial series to approximate thespatial displacements Theoretical analysis of high frequencyvibrations is indispensable in a variety of engineering designsZhao et al [28] introduced a novel computational approachthe discrete singular convolution (DSC) algorithm for highfrequency vibration analysis of plate structures A completelyindependent approach the Levy method was employed toprovide exact solutions to validate the proposed methodThe importance of high frequency analysis of multilayeredcomposite plates was also confirmed in [29] Further worksabout the analysis of high frequency vibrations of structuresare given in [30] where the discrete singular convolution(DSC) algorithm was used and then compared with a com-pletely independent approach (the Levy method) In [31]the first nine frequency parameters of circular and annularplates with variable thickness and combined boundary con-ditions were computed for different thickness to radius ratiosThree-dimensional elasticity theory was used in [31] and the

Shock and Vibration 3

Ritz method was applied to derive the eigenvalue equationXing and Liu [32] proposed the separation of variables tosolve the Hamiltonian dual form of eigenvalue problem fortransverse free vibrations of thin plates and formulation ofthe natural modes in closed form was performed The closedform natural mode exactly satisfied the governing equationof the eigenvalue problem of thin plate and was applicable forany types of boundary conditions The extension of 3D exactanalysis to functionally graded plates was performed in [33ndash35] Exact closed form solutions (based on three-dimensionalelasticity theory) were carried out in [33] for both in-plane and out-of-plane free vibration of thick homogeneoussimply supported rectangular plates coated by a functionallygraded (FG) layer Vel and Batra [34] presented a three-dimensional exact solution for free and forced vibrationsof simply supported functionally graded rectangular platesand displacement functions that identically satisfy boundaryconditions Xu and Zhou [35] used the three-dimensionalelasticity theory to derive the general expressions for thedisplacements and stresses of the plate under static loadswhich exactly satisfy the governing differential equations andthe simply supported boundary conditions at four edges ofthe plate The extension of three-dimensional solutions toplates embedding piezoelectric layers was given in [36ndash41]The three-dimensional exact solution by Haojiang et al [36]allowed the free vibration analysis of a circular plate made ofpiezoelectric material Baillargeon and Vel [37] obtained anexact three-dimensional solution for the cylindrical bendingvibration of simply supported laminated composite plateswith an embedded piezoelectric shear actuator The bendingand free vibration of an imperfectly bonded orthotropicpiezoelectric rectangular laminate were investigated in [38]using a three-dimensional state-space approach Cheng et al[39] presented an exact three-dimensional method for thedistribution of mechanical and electric quantities in interiorpoints of a symmetrically inhomogeneous and laminatedpiezoelectric plate in the framework of linear theory ofpiezoelectricity An exact three-dimensional (3D) piezother-moelasticity solution was presented in [40] for static freevibration and steady state harmonic response of simply sup-ported cross-ply piezoelectric (hybrid) laminated rectangularplates with interlaminar bonding imperfections Zhong andShang [41] presented an exact three-dimensional analysis fora functionally graded piezoelectric rectangular plate that wassimply supported and grounded along its four edgesThe stateequations of the functionally graded piezoelectric plate weredeveloped based on the state space approach

The revision of the literature about three-dimensionalanalysis of shells and plates demonstrates that there area variety of interesting works concerning plate or shellgeometry and they include static and dynamic analysisfunctionally graded composite and piezoelectric materialsdisplacement and mixed models The present work givesa general formulation for all the geometries (square andrectangular plates cylindrical and spherical shell panels andcylindrical closed shells) The equations of motion for thedynamic case are written in general orthogonal curvilinearcoordinates by using an exact geometry for shellsThe systemof second order differential equations is reduced to a system

Z

h2

120572120573

Z

in

i120572 i120573

minush2

Ω0 Ω0

dz

Figure 1 Notation and reference system for shells

of first order differential equations and afterwards it is exactlysolved by using the exponential matrix method Similarapproaches have been used in [25] for the three-dimensionalanalysis of plates in rectilinear orthogonal coordinates and in[7] for an exact three-dimensional free vibration analysis ofangle-ply laminated cylinders in cylindrical coordinates Theequations of motion written in orthogonal curvilinear coor-dinates allow general exact solutions for simply supportedplate and shell geometries with constant radii of curvatureThese equations are exactly solved for the first time in thispaper by means of the exponential matrix method

2 Geometrical and Constitutive Equations

A shell is a three-dimensional body bounded by two closelyspaced curved surfaces where one dimension (the distancebetween the two surfaces) is small in comparison with theother two dimensions in the plane directions The middlesurface of the shell is the locus of points which lie midwaybetween these surfaces The distance between the surfacesmeasured along the normal to the middle surface is thethickness of the shell at that point [42 43] Geometry and thereference system are indicated in Figure 1 where curvilinearorthogonal coordinates (120572 120573 119911) are shown The three-dimensional space (the space in which the three independentvariables 120572 120573 and 119911 vary) is an Euclidean space 119877

120572and 119877

120573

are the principal radii of curvature along the coordinates 120572and 120573 respectively 119911 is the thickness coordinate that variesfrom minusℎ2 to +ℎ2 (where ℎ is the thickness of the structureas shown in Figure 1)

In this work we will focus on shells with constant radiiof curvature (eg cylindrical and spherical geometries) Thegeometrical relations written for shells with constant radiiof curvature are a particular case of the strain-displacementequations of three-dimensional theory of elasticity in orthog-onal curvilinear coordinates (see also [44ndash46])

120598120572120572=

1

119867120572

120597119906

120597120572

+

119908

119867120572119877120572

120598120573120573=

1

119867120573

120597V120597120573

+

119908

119867120573119877120573


120598119911119911=

120597119908

120597119911

120574120572120573=

1

119867120572

120597V120597120572

+

1

119867120573

120597119906

120597120573

120574120572119911=

1

119867120572

120597119908

120597120572

+

120597119906

120597119911

minus

119906

119867120572119877120572

120574120573119911=

1

119867120573

120597119908

120597120573

+

120597V120597119911

minus

V119867120573119877120573

(1)

where 119906 V and 119908 are the displacement components in120572 120573 and 119911 direction respectively Symbol 120597 indicates thepartial derivatives the parametric coefficients for shellswith constant radii of curvature that use the orthogonalcurvilinear coordinate system are

119867120572= (1 +

119911

119877120572

) 119867120573= (1 +

119911

119877120573

) 119867119911= 1 (2)

General geometrical relations for spherical shells degenerateinto geometrical relations for cylindrical shells when119877

120572or119877120573

is infinite (with 119867120572or 119867120573equals one) and they degenerate

into geometrical relations for plates when both 119877120572and 119877

120573are

infinite (with119867120572=119867120573= 1) [46]

Geometrical relations (see (2)) are substituted in three-dimensional linear elastic constitutive equations in orthogo-nal curvilinear coordinates (120572 120573 119911) for orthotropic materialin the structural reference system (see [47 48]) Partialderivatives 120597120597120572 120597120597120573 and 120597120597119911 are indicatedwith subscripts120572120573 and

119911 A closed form solution of differential equations

of equilibrium for shells is obtained when coefficients 11986216

1198622611986236 and119862

45are set to zero (this means orthotropic angle

120579 equals 0∘ or 90∘)

120590120572120572=

11986211

119867120572

119906120572+

11986211

119867120572119877120572

119908 +

11986212

119867120573

V120573+

11986212

119867120573119877120573

119908 + 11986213119908119911

120590120573120573=

11986212

119867120572

119906120572+

11986212

119867120572119877120572

119908 +

11986222

119867120573

V120573+

11986222

119867120573119877120573

119908 + 11986223119908119911

120590119911119911=

11986213

119867120572

119906120572+

11986213

119867120572119877120572

119908 +

11986223

119867120573

V120573+

11986223

119867120573119877120573

119908 + 11986233119908119911

120590120573119911=

11986244

119867120573

119908120573+ 11986244V119911minus

11986244

119867120573119877120573

V

120590120572119911=

11986255

119867120572

119908120572+ 11986255119906119911minus

11986255

119867120572119877120572

119906

120590120572120573=

11986266

119867120572

V120572+

11986266

119867120573

119906120573

(3)

3 Equilibrium Equations

The three differential equations of equilibriumwritten for thecase of free vibration analysis of spherical shells with constant

radii of curvature 119877120572and 119877

120573are here given (the most general

form for variable radii of curvature can be found in [45 49])

119867120573

120597120590120572120572

120597120572

+ 119867120572

120597120590120572120573

120597120573

+ 119867120572119867120573

120597120590120572119911

120597119911

+ (

2119867120573

119877120572

+

119867120572

119877120573

)120590120572119911= 120588119867120572119867120573

119867120573

120597120590120572120573

120597120572

+ 119867120572

120597120590120573120573

120597120573

+ 119867120572119867120573

120597120590120573119911

120597119911

+ (

2119867120572

119877120573

+

119867120573

119877120572

)120590120573119911= 120588119867120572119867120573V

119867120573

120597120590120572119911

120597120572

+ 119867120572

120597120590120573119911

120597120573

+ 119867120572119867120573

120597120590119911119911

120597119911

minus

119867120573

119877120572

120590120572120572minus

119867120572

119877120573

120590120573120573

+ (

119867120573

119877120572

+

119867120572

119877120573

)120590119911119911= 120588119867120572119867120573

(4)

where120588 is themass density and V and indicate the secondtemporal derivative of the three displacement components

Equation (3) is introduced in (4) and the closed formis obtained for simply supported shells and plates made ofisotropic material or orthotropic material with 0

∘ or 90∘orthotropic angle (in both cases 119862

16= 11986226= 11986236= 11986245=

0) The three displacement components have the followingharmonic form

119906 (120572 120573 119911 119905) = 119880 (119911) 119890119894120596119905 cos (120572120572) sin (120573120573)

V (120572 120573 119911 119905) = 119881 (119911) 119890119894120596119905 sin (120572120572) cos (120573120573)

119908 (120572 120573 119911 119905) = 119882 (119911) 119890119894120596119905 sin (120572120572) sin (120573120573)

(5)

where 119880 119881 and 119882 are the displacement amplitudes in 120572120573 and 119911 directions respectively 119894 is the coefficient of theimaginary unit 120596 = 2120587119891 is the circular frequency where 119891is the frequency value and 119905 is the time In coefficients 120572 =

119898120587119886 and 120573 = 119899120587119887119898 and 119899 are the half-wave numbers and119886 and 119887 are the shell dimensions in 120572 and 120573 directions respec-tively Equilibrium equations are for spherical shell panelsthey automatically degenerate into equilibrium equations forcylindrical closedopen shell panels when 119877

120572or 119877120573is infinite

(with 119867120572or 119867120573equals one) and into equilibrium equations

for plates when 119877120572and 119877

120573are infinite (with119867

120572and119867

120573equal

one) In this way a unique and general closed formulation ispossible for any geometry

(minus

11986255119867120573

1198671205721198772

120572

minus

11986255

119877120572119877120573

minus 120572211986211119867120573

119867120572

minus120573

211986266119867120572

119867120573

+ 1205881198671205721198671205731205962

)119880 + (minus12057212057311986212minus 120572120573119862

66)119881


Table 1 Assessment 1 Simply supported aluminum square plate with Poisson ratio ] = 03 First six exact natural circular frequencies in no-dimensional form 120596 = 120596(119886

2

ℎ)radic120588119864 for imposed half-wave numbers 119898 = 119899 = 1 Comparison between present three-dimensional analysisand three-dimensional analyses by Vel and Batra [34] and Srinivas et al [22] 119872 is the number of fictitious layers and 119873 is the order ofexpansion for the exponential matrix

ModeI II III IV V VI

119886ℎ = 10

Present 3D (119872 = 1119873 = 3) 53370 27554 46493 15439 16809 26836Present 3D (119872 = 1119873 = 7) 57769 27554 46502 19291 19798 32935Present 3D (119872 = 1119873 = 10) 57769 27554 46502 19621 20187 30834Present 3D (119872 = 1119873 = 15) 57769 27554 46502 19677 20134 35823Present 3D (119872 = 1119873 = 18) 57769 27554 46502 19677 20134 35742Present 3D (119872 = 10119873 = 3) 57769 27554 46502 19671 20127 35540Present 3D (119872 = 50119873 = 3) 57769 27554 46502 19677 20134 35190Present 3D (119872 = 100119873 = 3) 57769 27554 46502 19677 20134 357423D [34] 57769 27554 46503 19677 20134 357423D [22] 57769 27554 46502 19677 20134 35742

119886ℎ = radic10

Present 3D (119872 = 1119873 = 3) 87132 14182 17512 Present 3D (119872 = 1119873 = 7) 46598 87132 14463 20987 23999 29925Present 3D (119872 = 1119873 = 10) 46582 87132 14463 21383 25009 31399Present 3D (119872 = 1119873 = 15) 46581 87132 14463 21343 24830 34049Present 3D (119872 = 1119873 = 18) 46581 87132 14463 21343 24830 33982Present 3D (119872 = 10119873 = 3) 46580 87132 14463 21337 24821 33939Present 3D (119872 = 50119873 = 3) 46581 87132 14463 21343 24830 33923Present 3D (119872 = 100119873 = 3) 46581 87132 14463 21343 24830 339823D [34] 46582 87132 14463 21343 24830 339823D [22] 46582 87134 14462 21343 24830 33982

+ (120572

11986211119867120573

119867120572119877120572

+ 120572

11986212

119877120573

+ 120572

11986255119867120573

119867120572119877120572

+ 120572

11986255

119877120573

)119882

+ (

11986255119867120573

119877120572

+

11986255119867120572

119877120573

)119880119911+ (120572119862

13119867120573+ 12057211986255119867120573)119882119911

+ (11986255119867120572119867120573)119880119911119911= 0

(minus12057212057311986266minus 120572120573119862

12)119880

+ (minus

11986244119867120572

1198671205731198772

120573

minus

11986244

119877120572119877120573

minus 120572211986266119867120573

119867120572

minus 120573

211986222119867120572

119867120573

+ 1205881198671205721198671205731205962

)119881

+ (120573

11986244119867120572

119867120573119877120573

+ 120573

11986244

119877120572

+ 120573

11986222119867120572

119867120573119877120573

+ 120573

11986212

119877120572

)119882

+ (

11986244119867120572

119877120573

+

11986244119867120573

119877120572

)119881119911+ (120573119862

44119867120572+ 12057311986223119867120572)119882119911

+ (11986244119867120572119867120573)119881119911119911= 0

(120572

11986255119867120573

119867120572119877120572

minus 120572

11986213

119877120573

+ 120572

11986211119867120573

119867120572119877120572

+ 120572

11986212

119877120573

)119880

+ (120573

11986244119867120572

119867120573119877120573

minus 120573

11986223

119877120572

+ 120573

11986222119867120572

119867120573119877120573

+ 120573

11986212

119877120572

)119881

+ (

11986213

119877120572119877120573

+

11986223

119877120572119877120573

minus

11986211119867120573

1198671205721198772

120572

minus

211986212

119877120572119877120573

minus

11986222119867120572

1198671205731198772

120573

minus120572211986255119867120573

119867120572

minus 120573

211986244119867120572

119867120573

+ 1205881198671205721198671205731205962

)119882

+ (minus12057211986255119867120573minus 12057211986213119867120573)119880119911+ (minus120573119862

44119867120572minus 12057311986223119867120572)119881119911

+ (

11986233119867120573

119877120572

+

11986233119867120572

119877120573

)119882119911+ (11986233119867120572119867120573)119882119911119911= 0

(6)


Table 2 Assessment 2 (part I) Simply supported aluminum cylinder with Poisson ratio ] = 03 and thickness ratio ℎ119877120572= 012 Lowest

exact natural circular frequencies in no-dimensional form 120596 = 120596(ℎ120587)radic120588119866 for imposed axial half-wave number 119899 = 1 and several half-wave numbers 119898 in circumferential direction Comparison between present three-dimensional analysis and three-dimensional analysis byArmenakas et al [8] as adapted in Bhimaraddi [9]119872 is the number of fictitious layers and119873 is the order of expansion for the exponentialmatrix

ℎ119877120572= 012

119898 = 2 119898 = 4 119898 = 6 119898 = 8

119887119877120572= 2

Present 3D (119872 = 1119873 = 3) 003724 002323 002175 002846Present 3D (119872 = 10119873 = 3) 003730 002360 002464 003687Present 3D (119872 = 50119873 = 3) 003730 002359 002463 003686Present 3D (119872 = 100119873 = 3) 003730 002359 002462 0036863D [8 9] 003730 002359 002462 003686

119887119877120572= 1


119887119877120572= 05


119887119877120572= 025


The system of equation (6) can be written in a compact formby introducing coefficients 119860

119903for each block ()

1198601119880 + 119860

2119881 + 119860

3119882+ 119860

4119880119911+ 1198605119882119911+ 1198606119880119911119911= 0

1198607119880 + 119860

8119881 + 119860

9119882+119860

10119881119911+ 11986011119882119911+ 11986012119881119911119911= 0

11986013119880 + 119860

14119881 + 119860

15119882+119860

16119880119911

+ 11986017119881119911+ 11986018119882119911+ 11986019119882119911119911= 0

(7)

Equation (7) is a system of three second order differentialequations They are written for spherical shell panels withconstant radii of curvature but they automatically degenerateinto equations for cylindrical shells and plates (see also [725])

31 Solution for the System of Differential Equations Thesystem of second order differential equations can be reducedto a system of first order differential equations by using the

method described in [50 51] This methodology allows thefollowing system to be obtained

[

[

[

[

[

[

[

[

1198606

0 0 0 0 0

0 11986012

0 0 0 0

0 0 11986019

0 0 0

0 0 0 1198606

0 0

0 0 0 0 11986012

0

0 0 0 0 0 11986019

]

]

]

]

]

]

]

]

[

[

[

[

[

[

[

[

119880

119881

119882

1198801015840

1198811015840

1198821015840

]

]

]

]

]

]

]

]

1015840

=

[

[

[

[

[

[

[

[

0 0 0 1198606

0 0

0 0 0 0 11986012

0

0 0 0 0 0 11986019

minus1198601

minus1198602

minus1198603

minus1198604

0 minus1198605

minus1198607

minus1198608

minus1198609

0 minus11986010

minus11986011

minus11986013

minus11986014

minus11986015

minus11986016

minus11986017

minus11986018

]

]

]

]

]

]

]

]

[

[

[

[

[

[

[

[

119880

119881

119882

1198801015840

1198811015840

1198821015840

]

]

]

]

]

]

]

]

(8)

Equation (8) can be written in a compact form as

D120597U120597

= AU (9)


Table 3 Assessment 2 (part II) Simply supported aluminum cylinder with Poisson ratio ] = 03 and thickness ratio ℎ119877120572= 018 Lowest


ℎ119877120572= 018

119898 = 2 119898 = 4 119898 = 6 119898 = 8

119887119877120572= 2


119887119877120572= 1


119887119877120572= 05


119887119877120572= 025


where 120597U120597 = U1015840 and U = [119880 119881 119882 1198801015840

1198811015840

1198821015840

] Equation(9) can be written as

DU1015840 = AU (10)

U1015840 = Dminus1AU (11)

U1015840 = AlowastU (12)

with Alowast = Dminus1AIn the case of plate geometry coefficients119860

31198604119860911986010

11986013 11986014 and 119860

18are zero because the radii of curvature 119877

120572

and119877120573are infiniteThe other coefficients119860

11198602119860511986061198607

1198608 11986011 11986012 11986015 11986016 11986017 and 119860

19are constant because

parametric coefficients119867120572= 119867120573= 1 and they do not depend

on the thickness coordinate therefore matrices D A andAlowast are constant in the plate case The solution of (12) can bewritten as [51 52]

U () = exp (Alowast)U (0) with isin [0 ℎ] (13)where is the thickness coordinate referred to the bottom ofthe structure that has values from 0 at the bottom to ℎ at the

top (see Figure 2) The exponential matrix is calculated with = ℎ as

Alowastlowast = exp (Alowastℎ) = I + Alowastℎ + Alowast2

2

ℎ2

+

Alowast3

3

ℎ3

+ sdot sdot sdot +

Alowast119873

119873

ℎ119873

(14)

where I is the 6 times 6 identity matrix This expansion has a fastconvergence as indicated in [53] and it is not time consumingfrom the computational point of view In the case of one-layered plate there is not a necessity to impose the continuityconditions at the interfaces for displacements and transverseshearnormal stresses for this reason the transfer matrix T isequal to the identity matrix I

T = I (15)

Equation (15) allows the calculation of the matrixH as

H = AlowastlowastI = Alowastlowast (16)


Table 4 Assessment 3 Simply supported orthotropic cylinder with thickness ratio ℎ119877120572= 03 and orthotropic angle 120579 equal to 0∘ or 90∘

First three exact natural circular frequencies in no-dimensional form 120596 = 120596(119877120587)radic12058811986612for imposed half-wave numbers119898 = 0 and 119899 = 1

Comparison between present three-dimensional analysis and three-dimensional analysis by Soldatos and Ye [7]119872 is the number of fictitiouslayers and119873 is the order of expansion for the exponential matrix

ModeI II III I II III

(119898 = 0 119899 = 1) 120579 = 0∘ (119898 = 0 119899 = 1) 120579 = 90∘

Present 3D (119872 = 50119873 = 3) 102407 255671 466395 020406 049393 091564Present 3D (119872 = 60119873 = 3) 032274 041827 329830 030493 081346 751166Present 3D (119872 = 80119873 = 3) 099999 129840 185418 095308 100000 304558Present 3D (119872 = 90119873 = 3) 100000 129714 248260 095308 100000 325249Present 3D (119872 = 100119873 = 3) 100000 129714 248260 095308 100000 3252493D [7] 100000 129730 248260 095307 100000 325250

Table 5 Benchmark 1 Simply supported isotropic square plate Firstthree vibrationmodes in term of no-dimensional circular frequency120596 = 120596(119886

2

ℎ)radic120588119864 for several combinations of half-wave numbers119898 and 119899119872 = 1 fictitious layers and119873 = 18 order of expansion forthe exponential matrix

119886ℎ 100 50 10 5(119898 = 0 119899 = 1)

I mode 29861 29846 29360 28012II mode 19483 97417 19483 97417III mode 32933 16466 32908 16415

(119898 = 1 119899 = 0)I mode 29861 29846 29360 28012II mode 19483 97417 19483 97417III mode 32933 16466 32908 16415





this last matrix permits linking vectorU calculated at the topof the plate ( = ℎ) with vector U calculated at the bottom ofthe plate ( = 0)

U (ℎ) = HU (0) (17)

In the case of shell geometry matrices D A and Alowast arenot constant because of parametric coefficients 119867

120572= (1 +

119911119877120572) = (1 + ( minus ℎ2)119877

120572) and 119867

120573= (1 + 119911119877

120573) =

(1 + ( minus ℎ2)119877120573) that depend on 119911 or coordinate (see

Figure 2) A first method could be the use of hypothesis119911119877120572= 119911119877

120573= 0 (it is valid only for very thin shells)

that means 119867120572= 119867120573= 1 In this case the solution is the

same already seen for the plate because matrices D A andAlowast are constant This method is not used in the benchmarksproposed in this paper because it is an approximation that isvalid only for very thin shells and it does not consider theexact geometry of the structure However some comparisonsbetween the method using exact geometry and the methodusing 119867

120572= 119867120573= 1 will be given at the end of the section

for results The second method (used in this paper) is theintroduction of several 119895 fictitious layers where 119867

120572and 119867

120573

can exactly be calculated with the thickness coordinate ofthe middle of the fictitious layers in this way (12) (13) and(14) are valid for each 119895 layer because Alowast

119895is here constant

The exponential matrix Alowastlowast119895

is calculated in each fictitious 119895layer with the thickness ℎ

119895of the layer The results proposed

in this paper for the benchmarks are given with a numberof fictitious layers 119895 = 119872 = 100 this value guaranteesthe exact convergence to the correct geometry of shells foreach case investigated as demonstrated in the section for thepreliminary results In the case of119872 layers for shell geometryit is necessary to calculate119872 minus 1 transfer matrices T

119895minus1119895by

using for each interface the following conditions

119906119887

119895= 119906119905

119895minus1 V119887

119895= V119905119895minus1 119908

119887

119895= 119908119905

119895minus1

120590119887

119911119911119895= 120590119905

119911119911119895minus1 120590

119887

120572119911119895= 120590119905

120572119911119895minus1 120590

119887

120573119911119895= 120590119905

120573119911119895minus1

(18)

that means each displacement and transverse stress compo-nent at the top (119905) of the 119895minus1 layer is equal to displacement andtransverse stress component at the bottom (119887) of the 119895 layerThe calculated T

119895minus1119895matrices allow linking U at the bottom

of the 119895 layer with U at the top of the 119895 minus 1 layer

U119895(0) = T

119895minus1119895U119895minus1

(ℎ119895minus1) (19)

U at the top of the 119895 layer is linked with U at the bottom ofthe same layer 119895 by means of the exponential matrix Alowastlowast

119895

U119895(ℎ119895) = Alowastlowast

119895U119895(0) (20)


Table 6 Benchmark 2 Simply supported orthotropic square plate First three vibration modes in term of no-dimensional circular frequency120596 = 120596(119886

2

ℎ)radic1205881198642for several combinations of half-wave numbers 119898 and 119899119872 = 1 fictitious layers and119873 = 18 order of expansion for the

exponential matrix

119886ℎ 100 50 10 5 100 50 10 5(119898 = 0 119899 = 1) 120579 = 0∘ (119898 = 0 119899 = 1) 120579 = 90∘

I mode 28553 28537 28058 26731 10007 99712 90075 72059II mode 22777 11388 22777 11388 22777 11388 22777 11388III mode 31489 15744 31457 15679 11047 55235 11041 46861

(119898 = 1 119899 = 0) 120579 = 0∘ (119898 = 1 119899 = 0) 120579 = 90∘


(119898 = 1 119899 = 1) 120579 = 0∘

(119898 = 1 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 2) 120579 = 0∘

(119898 = 1 119899 = 2) 120579 = 90∘


(119898 = 2 119899 = 1) 120579 = 0∘

(119898 = 2 119899 = 1) 120579 = 90∘


(119898 = 2 119899 = 2) 120579 = 0∘

(119898 = 2 119899 = 2) 120579 = 90∘


Ω0

+h2

120572 120573

minush2

zz

h

0

0

(a)

Ω0

00

+h2

120572 120573

minush2 j = 1

j = 2

j

j = M

zMhM

zz

h

0 0

z1 h1

(b)

Figure 2 Thickness coordinates for plates and shells

Equation (19) can recursively be introduced in (20) for the119872minus 1 interfaces to obtain

U119872(ℎ119872)

= Alowastlowast119872T119872minus1119872

Alowastlowast119872minus1

T119872minus2119872minus1

sdot sdot sdotAlowastlowast2

T12

Alowastlowast1U1(0)

(21)by defining thematrixH

119898for themulti-fictitious-layer shells

it is possible to write (21) in analogy with (17)U119872(ℎ119872) = H

119898U1(0) (22)

It links U calculated at the top of the last 119872 fictitious layerwithU calculated at the bottomof the first layerMatricesAlowastlowast

119895

are constant because they are evaluated with 119877120572 119877120573 120572 and 120573

calculated in the midsurfaceΩ0of the shell and with119867

120572and

119867120573calculated in the middle of each 119895 layer Matrices T

119895minus1119895

are constant because they are evaluated with 119877120572 119877120573 120572 and

120573 calculated in the midsurface Ω0of the shell and with 119867

120572

and119867120573calculated at each fictitious interface Equation (17) is

valid for one-layered plates and (22) is valid for one-layered


Table 7 Benchmark 3 Simply supported isotropic rectangularplate First three vibration modes in terms of no-dimensionalcircular frequency 120596 = 120596(119886

2

ℎ)radic120588119864 for several combinations ofhalf-wave numbers 119898 and 119899 119872 = 1 fictitious layers and 119873 = 18

order of expansion for the exponential matrix

119886ℎ 100 50 10 5(119898 = 0 119899 = 1)


(119898 = 1 119899 = 0)


(119898 = 1 119899 = 1)



(119898 = 2 119899 = 1)



shells divided in 119895 fictitious layers to correctly consider thegeometry

The structures are simply supported and free stresses atthe top and at the bottom this feature means

120590119911119911= 120590120572119911= 120590120573119911= 0 for 119911 = minusℎ

2

+ℎ

2

or = 0 ℎ (23)

119908 = V = 0 120590120572120572= 0 for 120572 = 0 119886 (24)

119908 = 119906 = 0 120590120573120573= 0 for 120573 = 0 119887 (25)

Equation (23) imposed at the top of the structure has 119877120572 119877120573

120572 and 120573 calculated in the midsurfaceΩ0of the shell and119867119905

120572

and119867119905120573are calculated at the top of the shell for = ℎ

120590119905

119911119911= minus120572

11986213

119867119905

120572

119880119905

+

11986213

119867119905

120572119877120572

119882119905

minus 120573

11986223

119867119905

120573

119881119905

+

11986223

119867119905

120573119877120573

119882119905

+ 11986233119882119905

119911= 0

120590119905

120573119911= 120573

11986244

119867119905

120573

119882119905

+ 11986244119881119905

119911minus

11986244

119867119905

120573119877120573

119881119905

= 0

120590119905

120572119911= 120572

11986255

119867119905

120572

119882119905

+ 11986255119880119905

119911minus

11986255

119867119905

120572119877120572

119880119905

= 0

(26)

Equation (23) imposed at the bottom of the structure has 119877120572

119877120573 120572 and 120573 calculated in the midsurfaceΩ

0of the shell and

119867119887

120572and119867119887

120573are calculated at the bottom of the shell for = 0

120590119887

119911119911= minus120572

11986213

119867119887

120572

119880119887

+

11986213

119867119887

120572119877120572

119882119887

minus 120573

11986223

119867119887

120573

119881119887

+

11986223

119867119887

120573119877120573

119882119887

+ 11986233119882119887

119911= 0

120590119887

120573119911= 120573

11986244

119867119887

120573

119882119887

+ 11986244119881119887

119911minus

11986244

119867119887

120573119877120573

119881119887

= 0

120590119887

120572119911= 120572

11986255

119867119887

120572

119882119887

+ 11986255119880119887

119911minus

11986255

119867119887

120572119877120572

119880119887

= 0

(27)

Equations (26) in matrix form are (U119872(ℎ119872) means U calcu-

lated at the top of the shell)

[

[

[

[

[

[

[

[

[

[

[

[

[

minus120572

11986213

119867119905

120572

minus120573

11986223

119867119905

120573

(

11986213

119867119905

120572119877120572

+

11986223

119867119905

120573119877120573

) 0 0 11986233

0 minus

11986244

119867119905

120573119877120573

120573

11986244

119867119905

120573

0 11986244

0

minus

11986255

119867119905

120572119877120572

0 120572

11986255

119867119905

120572

11986255

0 0

]

]

]

]

]

]

]

]

]

]

]

]

]

times

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

119880119872(ℎ119872)

119881119872(ℎ119872)

119882119872(ℎ119872)

1198801015840

119872(ℎ119872)

1198811015840

119872(ℎ119872)

1198821015840

119872(ℎ119872)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

=[

[

0

0

0

]

]

(28)


Table 8 Benchmark 4 Simply supported orthotropic rectangular plate First three vibration modes in terms of no-dimensional circularfrequency 120596 = 120596(1198862ℎ)radic120588119864

2for several combinations of half-wave numbers119898 and 119899119872 = 1 fictitious layers and119873 = 18 order of expansion

for the exponential matrix

119886ℎ 100 50 10 5 100 50 10 5(119898 = 0 119899 = 1) 120579 = 0∘ (119898 = 0 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 0) 120579 = 0∘ (119898 = 1 119899 = 0) 120579 = 90∘


(119898 = 1 119899 = 1) 120579 = 0∘ (119898 = 1 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 2) 120579 = 0∘ (119898 = 1 119899 = 2) 120579 = 90∘


(119898 = 2 119899 = 1) 120579 = 0∘ (119898 = 2 119899 = 1) 120579 = 90∘


(119898 = 2 119899 = 2) 120579 = 0∘ (119898 = 2 119899 = 2) 120579 = 90∘


Equation (27) in matrix form is (U1(0)meansU calculated at

the bottom of the shell)

[

[

[

[

[

[

[

[

[

[

[

minus120572

11986213

119867119887

120572

minus120573

11986223

119867119887

120573

(

11986213

119867119887

120572119877120572

+

11986223

119867119887

120573119877120573

) 0 0 11986233

0 minus

11986244

119867119887

120573119877120573

120573

11986244

119867119887

120573

0 11986244

0

minus

11986255

119867119887

120572119877120572

0 120572

11986255

119867119887

120572

11986255

0 0

]

]

]

]

]

]

]

]

]

]

]

times

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

1198801(0)

1198811(0)

1198821(0)

1198801015840

1(0)

1198811015840

1(0)

1198821015840

1(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

=[

[

0

0

0

]

]

(29)

Equations (28)-(29) in compact form to express the free stressstate at the top and bottom of the shell are

B119872(ℎ119872)U119872(ℎ119872) = 0 (30)

B1(0)U1(0) = 0 (31)

in the case of plate there are no 119895 fictitious layers

B (ℎ)U (ℎ) = 0

B (0)U (0) = 0(32)

Equation (22) can be substituted in (30)

B119872(ℎ119872)H119898U1(0) = 0

B1(0)U1(0) = 0

(33)

in this way (33) are grouped in the following system

[

B119872(ℎ119872)H119898

B1(0)

] [U1(0)] = [0] (34)

and introducing the (6 times 6) Ematrix (34) becomes

[E] [U1(0)] = [0] (35)


Table 9 Benchmark 5 Simply supported isotropic cylindrical shellpanel First three vibration modes in terms of no-dimensionalcircular frequency 120596 = 120596(119886

2

ℎ)radic120588119864 for several combinations ofhalf-wave numbers 119898 and 119899119872 = 100 fictitious layers and 119873 = 3


119877120572ℎ 1000 100 10 5

(119898 = 0 119899 = 1)


(119898 = 1 119899 = 0)


(119898 = 1 119899 = 1)


(119898 = 1 119899 = 2)


(119898 = 2 119899 = 1)


(119898 = 2 119899 = 2)


Equation (35) is also valid for plate case where the fictitiouslayers are not introduced andU(0) = U

1(0)H = H

119898 andB =

B119872(ℎ119872) = B

1(0) The solution is implemented in a Matlab

code where only the spherical shell method is considered itautomatically degenerates into cylindrical openclosed shelland plate methods

The free vibration problem means finding the nontrivialsolution of U

1(0) in (35) this means imposing the determi-

nant of matrix E equal to zero

det [E] = 0 (36)

Equation (36) needs to find the roots of a higher orderpolynomial in 120582 = 120596

2 For each pair of half-wave numbers(119898 119899) a certain number of circular frequencies are obtaineddepending on the order 119873 chosen for the exponentialmatrices Alowastlowast

119895and the119872 number of fictitious layers

32 Vibration Modes A certain number of circular frequen-cies 120596

119897are found when half-wave numbers 119898 and 119899 are

imposed in the structures For each frequency120596119897 it is possible

to find the vibration mode through the thickness in termsof the three displacement components If the frequency 120596

119897

is substituted in (6 times 6) matrix E this last matrix has six

eigenvalues We are interested to the null space of matrixE that means finding the (6 times 1) eigenvector related to theminimum of the six eigenvalues proposed This null space isfor the chosen frequency 120596

119897 the vector U calculated at the

bottom

U1120596119897(0) = [119880

1(0) 119881

1(0) 119882

1(0) 119880

1015840

1(0) 119881

1015840

1(0) 119882

1015840

1(0)]

119879

120596119897

(37)

119879 means the transpose of the vector and the subscript 120596119897

means that the null space is calculated for the circularfrequency 120596

119897

It is possible to findU119895120596119897(119895) (with the three displacement

components 119880119895120596119897(119895) 119881119895120596119897(119895) and 119882

119895120596119897(119895) through the

thickness) for each 119895 layer by using (19)ndash(22) the thicknesscoordinate can assume all the values from the bottom to thetop of the structure For the plate case the procedure is simplerbecause there are no 119895 fictitious layers and the 119872 numberequals 1

4 Results

The three-dimensional exact solution proposed has beenvalidated by means of a comparison with three assessmentsgiven in the literature for a one-layered isotropic plate [2234] one-layered isotropic cylinder [8 9] and one-layeredorthotropic cylinder [7] After this preliminary validationand the appropriate convergence study the method hasbeen used to investigate the free vibration analysis of one-layered isotropic and orthotropic square and rectangularplates cylindrical shell panels cylinders and spherical shellpanels (see Figure 3)The effects of the curvature terms in119867

120572

and119867120573parametric coefficients have also been investigated

41 Preliminary Assessments The assessment proposed inVel and Batra [34] compares their 3D solution with the 3Dsolution by Srinivas et al [22] The square plate is simplysupported embedding one aluminium alloy layer with Youngmodulus 119864 = 70GPa Poisson ratio ] = 03 and mass density120588 = 2702 kgm3 The thickness value is ℎ = 1 Two differentthickness ratios are considered that is 119886ℎ = 10 and 119886ℎ =radic10 The first six circular frequencies in no-dimensionalform 120596 = 120596(119886

2

ℎ)radic120588119864 are given in Table 1 for imposedhalf-wave numbers 119898 = 119899 = 1 The 3D exact solutionproposed in this paper coincides with those proposed by Veland Batra [34] and by Srinivas et al [22] for both thicknessratios and for each vibration mode (from the first to the sixthone) A convergence study has been proposed where the119872number of fictitious layers and the119873 order of expansion forthe exponential matrix have conveniently been chosen Forthe plate case one fictitious layer is used because there is noeffect of parametric coefficients 119867

120572and 119867

120573 For 119872 = 1

an order of expansion 119873 = 18 for the exponential matrixalways gives the correct solution for each vibrationmode andthickness ratio 119886ℎThe calculation of the exponential matrixis not time consuming An alternative method could be theuse of 119872 fictitious layers and a lower order of expansion119873 = 3 for the exponential matrix In this last case 119872 =

100 fictitious layers always give the correct solution for each


Table 10 Benchmark 6 Simply supported orthotropic cylindrical shell panel First three vibration modes in term of no-dimensional circularfrequency120596 = 120596(1198862ℎ)radic120588119864



119877120572ℎ 1000 100 10 5 1000 100 10 5

(119898 = 0 119899 = 1) 120579 = 0∘

(119898 = 0 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 0) 120579 = 0∘

(119898 = 1 119899 = 0) 120579 = 90∘


(119898 = 1 119899 = 1) 120579 = 0∘

(119898 = 1 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 2) 120579 = 0∘

(119898 = 1 119899 = 2) 120579 = 90∘


(119898 = 2 119899 = 1) 120579 = 0∘

(119898 = 2 119899 = 1) 120579 = 90∘


(119898 = 2 119899 = 2) 120579 = 0∘

(119898 = 2 119899 = 2) 120579 = 90∘


a

z

y

x

hb

(a)

a

zy

x

hb

(b)

a

z

b

R120572

120573

120572

(c)

z

b

R120572

120573

120572

a = 2120587R120572

(d)

a

z

b

R120572

R120573

120573

120572

(e)

Figure 3 Geometries considered for the assessments and benchmarks (a) square plate (b) rectangular plate (c) cylindrical shell panel (d)cylinder and (e) spherical shell panel


Table 11 Benchmark 7 Simply supported isotropic cylinder Firstthree vibrationmodes in term of no-dimensional circular frequency120596 = 120596(119886

2


119877120572ℎ 1000 100 10 5





vibrationmode and thickness ratioTheuse of119872 = 100 layersis not time consuming because the Ematrix has always 6 times 6dimension for any value of 119872 layers even if the proposedmethod uses a layerwise approach Solutions with119872 = 1 and119873 = 18 or with 119872 = 100 and 119873 = 3 are coincident forthe plate case and their time solution needs a few seconds Inconclusion the three-dimensional exact solution presentedhere is correct for the free vibration analysis of one-layeredplates even when the structures are thick and higher ordervibration modes are considered

The assessment proposed by Khalili et al [11] comparedtheir 2D closed form solutions for isotropic circular cylindri-cal shells with the three-dimensional analysis by Armenakaset al [8] adapted from Bhimaraddi [9] The cylinder isisotropic and one-layered with Poisson ratio ] = 03 Thestructure is simply supported and it has radius of curvature119877120572= 1 and infinite radius of curvature 119877

120573in 120573 direction

The circular dimension is 119886 = 2120587119877120572 the thickness ℎ

is 012 and 018 that means thickness ratios are ℎ119877120572=

012 and ℎ119877120572= 018 respectively Several ratios cylinder

lengthradius of curvature (119887119877120572) are investigated in the case

of 119887 = 2 1 05 025 The fundamental circular frequency inno-dimensional form 120596 = 120596(ℎ120587)radic120588119866 is given in Tables2 and 3 (thickness ratio ℎ119877

120572equals 012 ad 018 resp) for

axial half-wave number 119899 = 1 and several circular half-wave numbers 119898 = 2 4 6 8 The three-dimensional exactsolution proposed here is coincident with the 3D analysis byArmenakas et al [8] for each half-wave number thicknessratio and cylinder length proposed For the case of shellthe use of fictitious layers is necessary to approximate theparametric coefficients119867

120572and119867

120573 The convergence study is

shown for a value119873 = 3 for the expansion of the exponentialmatrix 119872 = 100 fictitious layers always give the correct

solution for each thickness ratio ℎ119877120572(see Tables 2 and

3) length 119887 of the cylinder and half-wave numbers 119898 incircumferential direction The present method is not timeconsuming119872 = 100 layers solution gives the correct resultsin a few seconds In conclusion the solution proposed can beconsidered as validated for one-layered isotropic shells

The assessment proposed by Soldatos and Ye [7] givesthree-dimensional results for multilayered orthotropiccylinders(minus120579 + 120579)

119895 A comparison with the present method

is made for the cases of orthotropic angle 120579 equal to 0∘ or90∘ (they allow the closed form solution and it also means

one-layered structure) The cylinder is simply supportedwith a great degree of orthotropy (119864

11198642= 11986411198643= 40

119866121198642= 119866131198642= 06 119866

231198642= 05 ]

12= ]13= ]23= 025

and homogeneous mass density 120588) The length 119887 is 10 andthe radius of curvature 119877

120572equals 10 The circular dimension

is 119886 = 2120587119877120572 The cylinder is very thick (ℎ = 3) that means

thickness ratio is ℎ119877120572= 03 In Table 4 the first three circular

frequencies in no-dimensional form 120596 = 120596(119877120587)radic12058811986612

are given for half-wave numbers 119898 = 0 and 119899 = 1 andorthotropic angle 120579 equals 0∘ or 90∘ The 3D solution bySoldatos and Ye [7] is correctly obtained for each mode andangle 120579 by using119872 = 100 fictitious layers and 119873 = 3 orderof expansion (119872 = 100 119873 = 3) solution is not heavy fromthe computational point of view because the E matrix hasalways 6 times 6 dimension even if the method uses a layerwiseapproach In conclusion the solution is also validated fororthotropic single-layered thick shells

All the new benchmarks proposed in Section 42 will use(119872 = 1119873 = 18) solution for plate cases and (119872 = 100119873 =

3) solution for shell cases Both solutions are very convenientfrom the computational point of view (they need only a fewseconds) and they allow the correct analysis for each thickand thin isotropicorthotropic plate and shell

42 Benchmarks Ten different benchmarks are proposed toshow a complete overview of the free vibration analysis ofone-layered plates and shells Five different geometries aretaken into account in Figure 3 All these structures are simplysupported The square plate has dimensions 119886 = 119887 = 1 andthickness ratios 119886ℎ = 100 50 10 5 The rectangular platehas dimensions 119886 = 1 and 119887 = 3119886 and the thickness ratios are119886ℎ = 100 50 10 5The cylindrical shell panel has a radius ofcurvature 119877

120572= 10 and an infinite radius of curvature 119877

120573in

120573 direction The dimensions are 119886 = (1205873)119877120572and 119887 = 20

and the thickness ratios are 119877120572ℎ = 1000 100 10 5 The

cylinder has the same radii of curvature of the cylindricalshell panel but it is closed in circumferential direction thatmeans 119886 = 2120587119877

120572 The other dimension is 119887 = 100 and

the thickness ratios are 119877120572ℎ = 1000 100 10 5 The last

geometry is the spherical shell panel with radii of curvature119877120572= 119877120573= 10 dimensions 119886 = 119887 = (1205873)119877

120572 and thickness

ratios 119877120572ℎ = 1000 100 10 5 Each geometry includes two

materials one isotropic layer in aluminium alloy Al2024 withYoung modulus 119864 = 73GPa Poisson ratio ] = 03 andmass density 120588 = 2800 kgm3 and one orthotropic layer inGrEp with Young modulus components 119864

1= 13238GPa

and 1198642= 1198643= 10756GPa shear modulus components


Table 12 Benchmark 8 Simply supported orthotropic cylinder First three vibration modes in terms of no-dimensional circular frequency120596 = 120596(119886

2

ℎ)radic1205881198642for several combinations of half-wave numbers119898 and 119899119872 = 100 fictitious layers and119873 = 3 order of expansion for the

exponential matrix

119877120572ℎ 1000 100 10 5 1000 100 10 5

(119898 = 0 119899 = 1) 120579 = 0∘ (119898 = 0 119899 = 1) 120579 = 90∘


(119898 = 2 119899 = 1) 120579 = 0∘ (119898 = 2 119899 = 1) 120579 = 90∘


(119898 = 2 119899 = 2) 120579 = 0∘ (119898 = 2 119899 = 2) 120579 = 90∘


(119898 = 2 119899 = 3) 120579 = 0∘ (119898 = 2 119899 = 3) 120579 = 90∘


11986612= 11986613= 56537GPa and 119866

23= 3603GPa Poisson ratio

components ]12

= ]13

= 024 and ]23

= 049 and massdensity 120588 = 1600 kgm3 The benchmarks are one-layeredisotropic square plate (see Table 5) one-layered orthotropicsquare plate (see Table 6) one-layered isotropic rectangularplate (see Table 7) one-layered orthotropic rectangular plate(see Table 8) one-layered isotropic cylindrical shell panel(see Table 9) one-layered orthotropic cylindrical shell panel(see Table 10) one-layered isotropic cylinder (see Table 11)one-layered orthotropic cylinder (see Table 12) one-layeredisotropic spherical shell panel (see Table 13) and one-layeredorthotropic spherical shell panel (see Table 14) The firstthree circular frequencies in no-dimensional form 120596 =

120596(1198862

ℎ)radic1205881198642are calculated in Tables 5ndash14 for various

pairs of half-wave numbers and several thickness ratios Thevibration modes plotted in Figures 4ndash12 are given in terms ofno-dimensional values such as 119906lowast = 119906|119906max| V

lowast

= V|Vmax|119908lowast

= 119908|119908max| and 119911lowast

= ℎTable 5 presents a square isotropic plate (benchmark 1)

It shows the first three vibration modes for each possiblecombination of half-wave numbers (119898 119899) from (0 1) untilto (2 2) Both thick and thin plates are analyzed The plate issquare and isotropic therefore results for half-wave numbers(0 1) are equal to those for (1 0) and the same considerationsare made for the half-wave numbers (1 2) and (2 1) The firstthree vibration modes through the thickness of a thick plate(119886ℎ = 10) in terms of the three displacement components119906lowast Vlowast and 119908lowast are given in Figure 4 for half-wave numbers(1 0) (1 1) (2 1) and (2 2) The second vibration mode isan in-planemode for each pair of half-wave numbers becausetransverse displacement 119908lowast through the thickness is zeroThe first mode has a constant transverse displacement 119908lowastthrough the thickness and linear in-plane displacements 119906lowastand Vlowast The transverse displacement is not perfectly constant

for higher values of half-wave number (effect of higher ordermodes)The in-plane displacement Vlowast is zero for the first casebecause the half-wave number 119899 = 0 The third vibrationmode always has linear transverse displacement 119908lowast throughthe thickness for each pair of half-wave numbers In-planedisplacements 119906lowast and Vlowast are constant (they are not perfectlyconstant for higher values of half-wave numbers it is a typicaleffect of higher order modes) The in-plane displacement Vlowastis zero for the first case because of zero half-wave number 119899

Table 6 shows a square orthotropic plate (benchmark 2)and lists the first three vibration modes for each possiblecombination of half-wave numbers (119898 119899) from (0 1) until to(2 2) and for two different fibre orientation angles 120579 (0∘ and90∘) For the angle 120579 = 0∘ results for half-wave numbers (0 1)

are not equal to those for (1 0) and the same considerationsare made for the half-wave numbers (1 2) and (2 1) Thisis due to the in-plane orthotropy The same considerationsare valid for the angle 120579 = 90

∘ It is important to noticethat frequencies for (0 1) and 120579 = 0

∘ are equal to those for(1 0) and 120579 = 90

∘ and frequencies for (1 0) and 120579 = 0∘ are

equal to those for (0 1) and 120579 = 90∘ The same considerationsremain valid for (1 2) and (2 1) These properties are validfor square plates and they are not true for rectangular platesFigures 5 and 6 show the first three vibration modes throughthe thickness in terms of the three displacement components119906lowast Vlowast and119908lowast for half-wave numbers (1 0) (1 1) (2 1) and(2 2) and for a thick plate (119886ℎ = 10) with fiber orientation120579 = 0

∘ or 120579 = 90∘ The considerations already made for the

first benchmark in Figure 4 are also valid in this case Theorthotropy also generates a coupling between in-plane andout-of-plane behavior that gives displacements that are notconstant or linear though the thickness even when low half-wave numbers are consideredMoreover thicknessmodes aremore complicated for higher half-wave numbers For these


Table 13 Benchmark 9 Simply supported isotropic spherical shellpanel First three vibration modes in terms of no-dimensionalcircular frequency 120596 = 120596(119886

2



119877120572ℎ 1000 100 10 5







reasons a 2D model will have more difficulties to obtainthe 3D behavior of the benchmark 2 with respect to the3D behavior of the isotropic benchmark 1 The change oforthotropic angle 120579 has an important effect on the frequencyvalues and a negligible effect on the vibration modes throughthe thickness

Tables 7 and 8 show the same cases already seen inTables 5 and 6 but a rectangular plate is analyzed in placeof a square plate This geometry gives a difference in Table 7(benchmark 3) between modes with (0 1) and those with(1 0) and between modes with (1 2) and those with (2 1)in fact the two in-plane dimensions 119886 and 119887 are not the same(119887 = 3119886) This difference due to the in-plane dimensions isalso important in Table 8 (benchmark 4) where frequenciesfor (0 1) and 120579 = 0∘ are not equal to the frequencies for (1 0)and 120579 = 90

∘ and and frequencies for (1 0) and 120579 = 0∘ are

not equal to the frequencies for (0 1) and 120579 = 90∘ The sameconsiderations can be made for higher half-wave numbers(1 2) and (2 1)

Table 9 gives the first three vibration modes of theisotropic cylindrical shell panel (benchmark 5) for severalcombinations of half-wave numbers (119898 119899) The shell isisotropic but the frequencies are always different for eachcouple of (119898 119899) because of the geometry with different 119886

and 119887 dimensions The same considerations are also validfor benchmark 6 in Table 10 (the orthotropic cylindrical shellpanel) The geometry with different 119886 and 119887 dimensions doesnot give any similarity when angle 120579 changes in combinationwith the half-wave numbers The orthotropic effects give acoupling between in-plane and out-of-plane behavior thatwill require higher order 2Dmodels to obtain the 3D analysisin terms of frequencies and vibration modes

Tables 11 and 12 show results for isotropic and orthotropicone-layered cylinders respectively Results do not add furtherconsiderations with respect to those already given for theopen cylindrical shell The main difference is in terms ofvibration modes through the thickness as shown in Figure 7for isotropic cylinder (benchmark 7) and in Figures 8 and 9for orthotropic cylinder (benchmark 8with orthotropic angle120579 equals 0∘ or 90∘) In Figure 7 the first mode for each pair of(119898 119899) always has constant displacements 119906lowast and 119908lowast and aquasi-constant Vlowast displacement but in the case 119898 = 0 and119899 = 1 the only component different from zero is 119906lowast Thesecond mode always has constant displacements Vlowast and 119908lowastand a linear 119906lowast displacement but in the case 119898 = 0 and119899 = 1119906

lowast is zero and 119908lowast and Vlowast are constant The third modehas quasi-constant 119906lowast Vlowast and 119908lowast displacements but in thecase 119898 = 0 and 119899 = 1 the component 119906lowast is zero In Figure 8the first three modes are given for the orthotropic case with120579 = 0

∘ The considerations about the first vibration modefor each pair of (119898 119899) are the same as those already givenfor the isotropic benchmark The second and third modesare different with respect to the isotropic case because of thecoupling due to the in-plane orthotropy (see in particular119908lowast displacement for the second mode and Vlowast displacement

for the third mode) The change of angle of orthotropy 120579gives large differences in terms of frequency values and smalldifferences in terms of vibration modes (see the comparisonbetween Figure 8 for 120579 = 0∘ and Figure 9 for 120579 = 90∘)

Tables 13 and 14 investigate isotropic spherical shellpanel (benchmark 9) and orthotropic spherical shell panel(benchmark 10) respectively The shell geometry consideredhere has the same radii of curvature in directions 120572 and120573 and the same dimensions 119886 and 119887 For this reason theconsiderations made for the square plate (Tables 5 and 6)are also valid in these two last cases results for half-wavenumbers (0 1) are equal to those for (1 0) and the samesimilarity is valid for the half-wave numbers (1 2) and (2 1)(isotropic shell) For the angle 120579 = 0

∘ results for half-wavenumbers (0 1) are not equal to those for (1 0) and the sameconsiderations are made for the half-wave numbers (1 2)and (2 1) This is due to the in-plane orthotropy Similarconsiderations are also valid for the angle 120579 = 90

∘ It isimportant to notice that frequencies for (0 1) and 120579 = 0∘ areequal to the frequencies for (1 0) and 120579 = 90∘ and frequenciesfor (1 0) and 120579 = 0∘ are equal to those for (0 1) and 120579 = 90∘The same feature is confirmed for (1 2) and (2 1) Figure 10shows the vibration modes for isotropic case while Figures11 and 12 show the vibration modes for 120579 = 0

∘ and 120579 =

90∘ orthotropic cases respectively The curvature effect is

very important and gives rather complicated modes throughthe thickness (in particular for the orthotropic sphericalshells in Figures 11 and 12) These demonstrate that the


Table 14 Benchmark 10 Simply supported orthotropic spherical shell panel First three vibration modes in terms of no-dimensional circularfrequency120596 = 120596(1198862ℎ)radic120588119864



119877120572ℎ 1000 100 10 5 1000 100 10 5

(119898 = 0 119899 = 1) 120579 = 0∘ (119898 = 0 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 0) 120579 = 0∘ (119898 = 1 119899 = 0) 120579 = 90∘


(119898 = 1 119899 = 1) 120579 = 0∘ (119898 = 1 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 2) 120579 = 0∘ (119898 = 1 119899 = 2) 120579 = 90∘


(119898 = 2 119899 = 1) 120579 = 0∘ (119898 = 2 119899 = 1) 120579 = 90∘


(119898 = 2 119899 = 2) 120579 = 0∘ (119898 = 2 119899 = 2) 120579 = 90∘


use of 3D analysis or refined 2D models is fundamental tocalculate these frequencies and modes The change of angleof orthotropy has a large influence on the frequency valuesbut this influence is smaller for vibration modes in Figures 11and 12

The parametric coefficient effects in particular the 119867120572

effect in cylindrical shell panels and both 119867120572and 119867

120573effects

in spherical shell panels are investigated in Tables 15 and16 Table 15 shows the comparison between the present 3Dsolution using the exact geometry and the same solutionusing 119867

120572= 1 (which means 119911119877

120572≃ 0) for the benchmark 5

concerning the isotropic cylindrical shell panel The approxi-mation119867

120572= 1 is valid for very thin shells (119877

120572ℎ equals 1000

or 100) even if it could have some problems for particularvibration modes and higher values of half-wave numbers(119898 119899) For thick shells the use of the exact geometry withoutthe approximation 119867

120572= 1 is mandatory Table 16 shows the

comparison between the present 3D solution using the exactgeometry and the same solution using the approximation119867120572

= 119867120573

= 1 (which means 119911119877120572

= 119911119877120573

≃ 0)for the benchmark 10 concerning the orthotropic sphericalshell panel The approximation 119867

120572= 119867120573= 1 is valid

for thin shells (119877120572ℎ equals 1000 or 100) for low values of

half-wave numbers and for the lower vibration modes Theapproximation gives significative errors for thick shells wherethe hypothesis 119911119877

120572= 119911119877

120573≃ 0 is not valid

5 Conclusions

Thedifferential equations of equilibrium in orthogonal curvi-linear coordinates for the free vibrations of simply supportedstructures have been exactly solved in three-dimensionalform by using the exponential matrix method The proposedgeneral 3D formulation uses an exact geometry for shells andit allows results for spherical open cylindrical closed cylin-drical and flat panels to be obtained The first three vibrationmodes have been investigated for several geometries bothisotropic and orthotropic layers various thickness ratios andimposed half-wave numbers The modes plotted through thethickness make it possible to recognize the most complicatedcases and these results will be useful benchmarks to validatefuture refined 2DmodelsThemethod is simple and intuitiveand it will be extended to multilayered structures (alsoembedding functionally graded layers) This extension willgive a global three-dimensional overview of the free vibrationproblem of one-layered and multilayered plates and shells


Table 15 Effects of119867120572= (1 + 119911119877

120572) terms in benchmark 5 for simply supported isotropic cylindrical shell panel First three vibration modes

in terms of no-dimensional circular frequency 120596 = 120596(1198862ℎ)radic120588119864 for several combinations of half-wave numbers119898 and 119899119872 = 100 fictitiouslayers and119873 = 3 order of expansion for the exponential matrix

119877120572ℎ 1000 100 10 5

(119898 = 0 119899 = 1)I mode 10671 10671 10683 53415II mode 10683 10683 10690 53727III mode 18557 18557 18557 92797I mode (119867

120572= 1) 10671 10671 10686 53486

II mode (119867120572= 1) 10683 10683 10688 53684

III mode (119867120572= 1) 18557 18556 18550 92651


120572= 1) 27747 27473 27029 25828

II mode (119867120572= 1) 20403 20403 20403 10201

III mode (119867120572= 1) 36353 36352 36303 18076


120572= 1) 22273 22535 40440 33738

II mode (119867120572= 1) 23645 23645 23668 11868

III mode (119867120572= 1) 40174 40173 40115 19965

Table 16 Effects of 119867120572= (1 + 119911119877

120572) and 119867

120573= (1 + 119911119877

120573) terms in benchmark 10 for simply supported orthotropic spherical shell panel

First three vibration modes in term of no-dimensional circular frequency 120596 = 120596(1198862ℎ)radic1205881198642for several combinations of half-wave numbers

119898 and 119899119872 = 100 fictitious layers and119873 = 3 order of expansion for the exponential matrix

119877120572ℎ 1000 100 10 5 1000 100 10 5

(119898 = 0 119899 = 1) 120579 = 0∘ (119898 = 0 119899 = 1) 120579 = 90∘

I mode 23852 23851 23802 11825 10379 10408 12585 77554II mode 29020 29020 29026 14513 23852 23851 23802 11824III mode 43716 43715 43559 21518 12223 12223 12165 51888I mode (119867

120572= 119867120573= 1) 23852 23852 23858 11939 10379 10409 12602 77410

II mode (119867120572= 119867120573= 1) 29020 29020 29030 14529 23852 23852 23856 11934

III mode (119867120572= 119867120573= 1) 43716 43715 43623 21667 12223 12223 12204 51743

(119898 = 1 119899 = 0) 120579 = 0∘ (119898 = 1 119899 = 0) 120579 = 90∘


120572= 119867120573= 1) 10379 10409 12602 77410 23852 23852 23858 11939

II mode (119867120572= 119867120573= 1) 23852 23852 23856 11934 29020 29020 29030 14529

III mode (119867120572= 119867120573= 1) 12223 12223 12204 51743 43716 43715 43623 21667

(119898 = 1 119899 = 1) 120579 = 0∘ (119898 = 1 119899 = 1) 120579 = 90∘


120572= 119867120573= 1) 12097 12133 14810 92169 12097 12133 14810 92169

II mode (119867120572= 119867120573= 1) 40165 40164 40137 20022 40165 40164 40137 20022

III mode (119867120572= 119867120573= 1) 12456 12456 12429 54312 12456 12456 12429 54312

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ulowast lowast wlowastulowast lowast wlowast


Figure 4 Benchmark 1 simply supported isotropic square plate First three vibration modes in terms of displacement components throughthe thickness for thickness ratio 119886ℎ = 10 and several combinations of half-wave numbers

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ulowast lowast wlowast ulowast lowast wlowast ulowast lowast wlowast

Figure 5 Benchmark 2 simply supported orthotropic square plate (120579 = 0∘) First three vibrationmodes in terms of displacement componentsthrough the thickness for thickness ratio 119886ℎ = 10 and several combinations of half-wave numbers














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Figure 6 Benchmark 2 simply supported orthotropic square plate (120579 = 90∘) First three vibration modes in terms of displacement

components through the thickness for thickness ratio 119886ℎ = 10 and several combinations of half-wave numbers


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Figure 7 Benchmark 7 simply supported isotropic cylinder First three vibration modes in terms of displacement components through thethickness for thickness ratio 119877

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01



zlowast

1

09

08

07

06

05

04

03

02

01



zlowast

1

09

08

07

06

05

04

03

02

01



zlowast

1

09

08

07

06

05

04

03

02

01



zlowast

1

09

08

07

06

05

04

03

02

01



zlowast

1

09

08

07

06

05

04

03

02

01



zlowast

1

09

08

07

06

05

04

03

02

01



zlowast

1

09

08

07

06

05

04

03

02

01
















Figure 8 Benchmark 8 simply supported orthotropic cylinder (120579 = 0∘) First three vibration modes in terms of displacement componentsthrough the thickness for thickness ratio 119877



1

09

08

07

06

05

04

03

02

01



zlowast

1

09

08

07

06

05

04

03

02

01



zlowast

1

09

08

07

06

05

04

03

02

01



zlowast

1

09

08

07

06

05

04

03

02

01



zlowast

1

09

08

07

06

05

04

03

02

01



zlowast

1

09

08

07

06

05

04

03

02

01



zlowast

1

09

08

07

06

05

04

03

02

01



zlowast

1

09

08

07

06

05

04

03

02

01



zlowast

1

09

08

07

06

05

04

03

02

01



zlowast

1

09

08

07

06

05

04

03

02

01



zlowast

1

09

08

07

06

05

04

03

02

01



zlowast

1

09

08

07

06

05

04

03

02

01



















1

09

08

07

06

05

04

03

02

01



zlowast

1

09

08

07

06

05

04

03

02

01



zlowast

1

09

08

07

06

05

04

03

02

01



zlowast

1

09

08

07

06

05

04

03

02

01



zlowast

1

09

08

07

06

05

04

03

02

01



zlowast

1

09

08

07

06

05

04

03

02

01



zlowast

1

09

08

07

06

05

04

03

02

01



zlowast

1

09

08

07

06

05

04

03

02

01



zlowast

1

09

08

07

06

05

04

03

02

01



zlowast

1

09

08

07

06

05

04

03

02

01



zlowast

1

09

08

07

06

05

04

03

02

01



zlowast

1

09

08

07

06

05

04

03

02

01












First mode for m = 2 and n = 2 Second mode for m = 2 and n = 2 Third mode for m = 2 and n = 2


Figure 10 Benchmark 9 simply supported isotropic spherical shell panel First three vibration modes in terms of displacement componentsthrough the thickness for thickness ratio 119877



1

09

08

07

06

05

04

03

02

01



zlowast

1

09

08

07

06

05

04

03

02

01



zlowast

1

09

08

07

06

05

04

03

02

01



zlowast

1

09

08

07

06

05

04

03

02

01



zlowast

1

09

08

07

06

05

04

03

02

01



zlowast

1

09

08

07

06

05

04

03

02

01



zlowast

1

09

08

07

06

05

04

03

02

01



zlowast

1

09

08

07

06

05

04

03

02

01



zlowast

1

09

08

07

06

05

04

03

02

01



zlowast

1

09

08

07

06

05

04

03

02

01



zlowast

1

09

08

07

06

05

04

03

02

01



zlowast

1

09

08

07

06

05

04

03

02

01














Figure 11 Benchmark 10 simply supported orthotropic spherical shell panel (120579 = 0∘) First three vibration modes in terms of displacementcomponents through the thickness for thickness ratio 119877















zlowast

1

09

08

07

06

05

04

03

02

01



zlowast

1

09

08

07

06

05

04

03

02

01



zlowast

1

09

08

07

06

05

04

03

02

01



zlowast

1

09

08

07

06

05

04

03

02

01



zlowast

1

09

08

07

06

05

04

03

02

01



zlowast

1

09

08

07

06

05

04

03

02

01



zlowast

1

09

08

07

06

05

04

03

02

01



zlowast

1

09

08

07

06

05

04

03

02

01



zlowast

1

09

08

07

06

05

04

03

02

01



zlowast

1

09

08

07

06

05

04

03

02

01



zlowast

1

09

08

07

06

05

04

03

02

01



zlowast

1

09

08

07

06

05

04

03

02

01







The present method has a fast convergence and it is not heavyfrom the computational point of view for each geometry(plates and shells) This feature is an important advantagewith respect to other methods proposed in the literature thatare valid only for a chosen geometry

Conflict of Interests

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

References

[1] W-Q Chen H-J Ding and R-Q Xu ldquoOn exact analysis offree vibrations of embedded transversely isotropic cylindricalshellsrdquo International Journal of Pressure Vessels and Piping vol75 no 13 pp 961ndash966 1998

[2] J-R Fan and J-Y Zhang ldquoExact solutions for thick laminatedshellsrdquo Science in China vol 35 no 11 pp 1343ndash1355 1992

[3] B Gasemzadeh R Azarafza Y Sahebi and A MotallebildquoAnalysis of free vibration of cylindrical shells on the basisof three dimensional exact elasticity theoryrdquo Indian Journal ofScience amp Technology vol 5 no 9 pp 3260ndash3262 2012

[4] N N Huang ldquoExact analysis for three-dimensional free vibra-tions of cross-ply cylindrical and doubly-curved laminatesrdquoActa Mechanica vol 108 no 1ndash4 pp 23ndash34 1995

[5] J N Sharma D K Sharma and S S Dhaliwal ldquoThree-dimensional free vibration analysis of a viscothermoelastichollow sphererdquoOpen Journal of Acoustics vol 2 pp 12ndash24 2012

[6] J N Sharma and N Sharma ldquoThree-dimensional free vibrationanalysis of a homogeneous transradially isotropic thermoelasticsphererdquo Journal of AppliedMechanics vol 77 no 2 pp 1ndash9 2010

[7] K P Soldatos and J Ye ldquoAxisymmetric static and dynamicanalysis of laminated hollow cylinders composed ofmonoclinicelastic layersrdquo Journal of Sound and Vibration vol 184 no 2 pp245ndash259 1995

[8] A E ArmenakasDCGazis andGHerrmannFreeVibrationsof Circular Cylindrical Shells PergamonPressOxfordUK 1969

[9] A Bhimaraddi ldquoA higher order theory for free vibrationanalysis of circular cylindrical shellsrdquo International Journal ofSolids and Structures vol 20 no 7 pp 623ndash630 1984

[10] H Zhou W Li B Lv and W L Li ldquoFree vibrations ofcylindrical shells with elastic-support boundary conditionsrdquoApplied Acoustics vol 73 no 8 pp 751ndash756 2012

[11] S M R Khalili A Davar and K M Fard ldquoFree vibrationanalysis of homogeneous isotropic circular cylindrical shellsbased on a new three-dimensional refined higher-order theoryrdquoInternational Journal of Mechanical Sciences vol 56 no 1 pp 1ndash25 2012

[12] S S Vel ldquoExact elasticity solution for the vibration of function-ally graded anisotropic cylindrical shellsrdquoComposite Structuresvol 92 no 11 pp 2712ndash2727 2010

[13] C T Loy and K Y Lam ldquoVibration of thick cylindrical shellson the basis of three-dimensional theory of elasticityrdquo Journalof Sound and Vibration vol 226 no 4 pp 719ndash737 1999

[14] Y Wang R Xu H Ding and J Chen ldquoThree-dimensionalexact solutions for free vibrations of simply supportedmagneto-electro-elastic cylindrical panelsrdquo International Journal of Engi-neering Science vol 48 no 12 pp 1778ndash1796 2010

[15] E Efraim and M Eisenberger ldquoExact vibration frequencies ofsegmented axisymmetric shellsrdquoThin-Walled Structures vol 44no 3 pp 281ndash289 2006

[16] J-H Kang and A W Leissa ldquoThree-dimensional vibrationsof thick spherical shell segments with variable thicknessrdquoInternational Journal of Solids and Structures vol 37 no 35 pp4811ndash4823 2000

[17] K M Liew L X Peng and T Y Ng ldquoThree-dimensionalvibration analysis of spherical shell panels subjected to differ-ent boundary conditionsrdquo International Journal of MechanicalSciences vol 44 no 10 pp 2103ndash2117 2002

[18] L Demasi ldquoThree-dimensional closed form solutions and exactthin plate theories for isotropic platesrdquo Composite Structuresvol 80 no 2 pp 183ndash195 2007

[19] S Aimmanee and R C Batra ldquoAnalytical solution for vibrationof an incompressible isotropic linear elastic rectangular plateand frequencies missed in previous solutionsrdquo Journal of Soundand Vibration vol 302 no 3 pp 613ndash620 2007

[20] R C Batra and S Aimmanee ldquoLetter to the editor missingfrequencies in previous exact solutions of free vibrations ofsimply supported rectangular platesrdquo Journal of Sound andVibration vol 265 no 4 pp 887ndash896 2003

[21] 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

[22] S Srinivas AK Rao andCV JogaRao ldquoFlexure of simply sup-ported thick homogeneous and laminated rectangular platesrdquoZeitschrift fur Angewandte Mathematik und Mechanik vol 49no 8 pp 449ndash458 1969

[23] R C Batra S Vidoli and F Vestroni ldquoPlane wave solutions andmodal analysis in higher order shear and normal deformableplate theoriesrdquo Journal of Sound and Vibration vol 257 no 1pp 63ndash88 2002

[24] J Q Ye ldquoA three-dimensional free vibration analysis of cross-ply laminated rectangular plates with clamped edgesrdquoComputerMethods in AppliedMechanics and Engineering vol 140 no 3-4pp 383ndash392 1997

[25] A Messina ldquoThree dimensional free vibration analysis ofcross-ply laminated plates through 2D and exact modelsrdquo inProceedings of the 3rd International Conference on IntegrityReliability and Failure Porto Portugal July 2009

[26] Y K Cheung and D Zhou ldquoThree-dimensional vibrationanalysis of cantilevered and completely free isosceles triangularplatesrdquo International Journal of Solids and Structures vol 39 no3 pp 673ndash687 2002

[27] KM Liew and B Yang ldquoThree-dimensional elasticity solutionsfor free vibrations of circular plates a polynomials-Ritz analy-sisrdquo Computer Methods in Applied Mechanics and Engineeringvol 175 no 1-2 pp 189ndash201 1999

[28] Y B Zhao G W Wei and Y Xiang ldquoDiscrete singularconvolution for the prediction of high frequency vibration ofplatesrdquo International Journal of Solids and Structures vol 39 no1 pp 65ndash88 2002

[29] S Brischetto and E Carrera ldquoImportance of higher ordermodes and refined theories in free vibration analysis of com-posite platesrdquo Journal of Applied Mechanics vol 77 no 1 pp1ndash14 2009

[30] G W Wei Y B Zhao and Y Xiang ldquoA novel approach forthe analysis of high-frequency vibrationsrdquo Journal of Sound andVibration vol 257 no 2 pp 207ndash246 2002


[31] H R D Taher M Omidi A A Zadpoor and A A NikooyanldquoFree vibration of circular and annular plates with variablethickness and different combinations of boundary conditionsrdquoJournal of Sound and Vibration vol 296 no 4-5 pp 1084ndash10922006

[32] Y Xing and B Liu ldquoNew exact solutions for free vibrationsof rectangular thin plates by symplectic dual methodrdquo ActaMechanica Sinica vol 25 no 2 pp 265ndash270 2009

[33] Sh Hosseini-Hashemi H Salehipour and S R AtashipourldquoExact three-dimensional free vibration analysis of thick homo-geneous plates coated by a functionally graded layerrdquo ActaMechanica vol 223 no 10 pp 2153ndash2166 2012

[34] S S Vel and R C Batra ldquoThree-dimensional exact solution forthe vibration of functionally graded rectangular platesrdquo Journalof Sound and Vibration vol 272 no 3ndash5 pp 703ndash730 2004

[35] Y Xu and D Zhou ldquoThree-dimensional elasticity solution offunctionally graded rectangular plates with variable thicknessrdquoComposite Structures vol 91 no 1 pp 56ndash65 2009

[36] D Haojiang X Rongqiao and C Weiqiu ldquoExact solutions forfree vibration of transversely isotropic piezoelectric circularplatesrdquo Acta Mechanica Sinica vol 16 no 2 pp 146ndash147 2000

[37] B P Baillargeon and S S Vel ldquoExact solution for the vibrationand active damping of composite plates with piezoelectric shearactuatorsrdquo Journal of Sound and Vibration vol 282 no 3ndash5 pp781ndash804 2005

[38] W Q Chen J B Cai G R Ye and Y F Wang ldquoExact three-dimensional solutions of laminated orthotropic piezoelectricrectangular plates featuring interlaminar bonding imperfec-tions modeled by a general spring layerrdquo International Journalof Solids and Structures vol 41 no 18-19 pp 5247ndash5263 2004

[39] Z-Q Cheng C W Lim and S Kitipornchai ldquoThree-dimen-sional exact solution for inhomogeneous and laminated piezo-electric platesrdquo International Journal of Engineering Science vol37 no 11 pp 1425ndash1439 1999

[40] S Kapuria and P G Nair ldquoExact three-dimensional piezother-moelasticity solution for dynamics of rectangular cross-plyhybrid plates featuring interlaminar bonding imperfectionsrdquoComposites Science and Technology vol 70 no 5 pp 752ndash7622010

[41] Z Zhong and E T Shang ldquoThree-dimensional exact analysisof a simply supported functionally gradient piezoelectric platerdquoInternational Journal of Solids and Structures vol 40 no 20 pp5335ndash5352 2003

[42] A W Leissa ldquoVibration of platesrdquo Tech Rep SP-160 NASAWashington DC USA 1969

[43] A W Leissa ldquoVibration of shellsrdquo Tech Rep SP-288 NASAWashington DC USA 1973

[44] W Soedel Vibration of Shells and Plates Marcel Dekker NewYork NY USA 2004

[45] F B Hildebrand E Reissner and G B Thomas Notes on theFoundations of theTheory of Small Displacements of OrthotropicShells NACATechnicalNote no 1833NACAWashingtonDCUSA 1949

[46] E Carrera S Brischetto and P Nali Plates and Shells for SmartStructures Classical and Advanced Theories for Modeling andAnalysis John Wiley amp Sons New Delhi India 2011

[47] E Carrera and S Brischetto ldquoAnalysis of thickness lockingin classical refined and mixed multilayered plate theoriesrdquoComposite Structures vol 82 no 4 pp 549ndash562 2008

[48] E Carrera and S Brischetto ldquoAnalysis of thickness lockingin classical refined and mixed theories for layered shellsrdquoComposite Structures vol 85 no 1 pp 83ndash90 2008

[49] F Tornabene Meccanica delle Strutture a Guscio in MaterialeComposito Societa Editrice Esculapio Bologna Italy 2012

[50] Open Document ldquoSystems of Differential Equationsrdquo 2013httpwwwmathutahedusimgustafso2250systems-depdf

[51] W E Boyce and R C DiPrima Elementary Differential Equa-tions and Boundary Value Problems John Wiley amp Sons NewYork NY USA 2001

[52] D Zwillinger Handbook of Differential Equations AcademicPress New York NY USA 1997

[53] C Moler and C van Loan ldquoNineteen dubious ways to computethe exponential of a matrix twenty-five years laterrdquo SIAMReview vol 45 no 1 pp 3ndash49 2003

International Journal of

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Navigation and Observation



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was established in a cylindrical coordinate system withoutany assumption about displacement models and stress dis-tribution Gasemzadeh et al [3] studied free vibrations ofcylindrical shells on the basis of three-dimensional exacttheory Simply supported boundary conditions were stud-ied and extensive frequency parameters were obtained bysolving frequency equations The free vibrations of simplysupported cross-ply cylindrical and doubly curved laminateswere investigated in [4] The three-dimensional equations ofmotion expressed in terms of displacements were reducedto a system of coupled ordinary differential equations thatwere solved using the power series method Sharma et al [5]studied the three-dimensional free vibrations of a homoge-nous isotropic viscothermoelastic hollow sphere whose sur-faces were subjected to stress-free thermally insulated orisothermal boundary conditions The governing partial dif-ferential equations were solved into a coupled system ofordinary differential equations Frobenious matrix methodwas employed to obtain the solution The same method wasalso used for the exact three-dimensional vibration analysisof a transradially isotropic thermoelastic solid sphere [6]Soldatos and Ye [7] proposed exact three-dimensional freevibration analysis of angle-ply laminated cylinders Resultswere presented and discussed for relatively thick cylindershaving a regular symmetric or a regular antisymmetric angle-ply lay-up Armenakas et al [8] proposed a self-containedtreatment of the problem of plane harmonic waves propaga-tion along a hollow circular cylinder in the framework of thethree-dimensional theory of elasticity The results presentedfrequencies of free vibrations of such cylinders for a widerange of axial and circumferential wave-lengths and of shelldimensions An interesting comparison between frequenciesobtained from a refined two-dimensional analysis a sheardeformation theory the Flugge theory and an exact elasticityanalysis were proposed in [9] Further details about theFlugge classical thin shell theory concerning the free vibra-tions of cylindrical shells with elastic boundary conditionscan be found in [10] Other comparisons between two-dimensional closed form solutions and available exact 3Delastic and analytical solutions for the free vibration analysisof simply supported and clamped homogenous isotropiccircular cylindrical shells were also proposed in [11] Exactelasticity solutions were extended to functionally gradedcylindrical shells in [12] The three-dimensional linear elas-todynamics equations simplified to the case of generalizedplane strain deformation in the axial direction were solvedusing suitable displacement functions that identically satisfythe boundary conditions Loy and Lam [13] proposed alayerwise approach to study the vibration of thick circularcylindrical shells on the basis of three-dimensional theoryof elasticity The governing equation was obtained using anenergy minimization principle Another interesting exten-sion was proposed by Wang et al [14] who investigatedthe free vibrations ofmagnetoelectroelastic cylindrical panelsbased on the three-dimensional theory Further results aboutthree-dimensional analysis of shells that are not given inclosed form were shown by Efraim and Eisenberger [15]for the dynamic stiffness matrix method and by Kang and

Leissa [16] and Liew et al [17] for the three-dimensional Ritzmethod for vibration of spherical shells

Three-dimensional analysis of plates is usually performedby using simpler equations that do not allow the analysis ofshell geometriesDemasi [18] proposed aNavier-typemethodfor finding the exact three-dimensional solution for isotropicthick and thin rectangular platesThemethod used themixedform of Hooke Law to write the boundary conditions onthe top and bottom surfaces of the plate directly in terms oftransverse stresses Aimmanee and Batra [19] proposed ananalytical solution for free vibrations of a simply supportedrectangular plate made of an incompressible homogeneouslinear elastic isotropic material Some frequencies missing inprevious analytical solutions for plates made of compressiblematerials were identified Some of the in-plane modes of freevibration of a simply supported rectangular plate that weremissed in previous exact solutions were also proposed in[20] A three-dimensional linear elastic small deformationtheory obtained by the direct method was developed for thefree vibration of simply supported homogeneous isotropicthick rectangular plates in [21] The same method was usedin [22] for the flexure of simply supported homogeneousisotropic thick rectangular plates under arbitrary loadingThis solution in terms of infinite series was formally exactand yielded accurate numerical results without undue effortUseful comparisons between two-dimensional models andan exact three-dimensional solution for the free vibrationsof a simply supported rectangular orthotropic thick plate areshown in [23] On the basis of three-dimensional elasticity Ye[24] presented a free vibration analysis of cross-ply laminatedrectangular plates with clamped boundariesThe analysis wasbased on a recursive solution suitable for three-dimensionalvibration analysis of simply supported plates Comparisonsbetween 2D-displacement-based models and exact results ofthe linear three-dimensional elasticity were also proposed in[25] for natural frequencies displacement and stress quanti-ties in multilayered plates Exact three-dimensional elasticitytheory was investigated in [26] for isosceles triangular platesbymeans of a global three-dimensional Ritz formulationThethree-dimensional elasticity free vibration analysis of a circu-lar plate was analyzed in [27] by means of the Ritz methodwith a set of orthogonal polynomial series to approximate thespatial displacements Theoretical analysis of high frequencyvibrations is indispensable in a variety of engineering designsZhao et al [28] introduced a novel computational approachthe discrete singular convolution (DSC) algorithm for highfrequency vibration analysis of plate structures A completelyindependent approach the Levy method was employed toprovide exact solutions to validate the proposed methodThe importance of high frequency analysis of multilayeredcomposite plates was also confirmed in [29] Further worksabout the analysis of high frequency vibrations of structuresare given in [30] where the discrete singular convolution(DSC) algorithm was used and then compared with a com-pletely independent approach (the Levy method) In [31]the first nine frequency parameters of circular and annularplates with variable thickness and combined boundary con-ditions were computed for different thickness to radius ratiosThree-dimensional elasticity theory was used in [31] and the




Z

h2

120572120573

Z

in

i120572 i120573

minush2

Ω0 Ω0

dz





120572and 119877

120573



120598120572120572=

1

119867120572

120597119906

120597120572

+

119908

119867120572119877120572

120598120573120573=

1

119867120573

120597V120597120573

+

119908

119867120573119877120573


120598119911119911=

120597119908

120597119911

120574120572120573=

1

119867120572

120597V120597120572

+

1

119867120573

120597119906

120597120573

120574120572119911=

1

119867120572

120597119908

120597120572

+

120597119906

120597119911

minus

119906

119867120572119877120572

120574120573119911=

1

119867120573

120597119908

120597120573

+

120597V120597119911

minus

V119867120573119877120573

(1)


119867120572= (1 +

119911

119877120572

) 119867120573= (1 +

119911

119877120573

) 119867119911= 1 (2)


120572or119877120573



120573are

infinite (with119867120572=119867120573= 1) [46]




1198622611986236 and119862


120579 equals 0∘ or 90∘)

120590120572120572=

11986211

119867120572

119906120572+

11986211

119867120572119877120572

119908 +

11986212

119867120573

V120573+

11986212

119867120573119877120573

119908 + 11986213119908119911

120590120573120573=

11986212

119867120572

119906120572+

11986212

119867120572119877120572

119908 +

11986222

119867120573

V120573+

11986222

119867120573119877120573

119908 + 11986223119908119911

120590119911119911=

11986213

119867120572

119906120572+

11986213

119867120572119877120572

119908 +

11986223

119867120573

V120573+

11986223

119867120573119877120573

119908 + 11986233119908119911

120590120573119911=

11986244

119867120573

119908120573+ 11986244V119911minus

11986244

119867120573119877120573

V

120590120572119911=

11986255

119867120572

119908120572+ 11986255119906119911minus

11986255

119867120572119877120572

119906

120590120572120573=

11986266

119867120572

V120572+

11986266

119867120573

119906120573

(3)






119867120573

120597120590120572120572

120597120572

+ 119867120572

120597120590120572120573

120597120573

+ 119867120572119867120573

120597120590120572119911

120597119911

+ (

2119867120573

119877120572

+

119867120572

119877120573

)120590120572119911= 120588119867120572119867120573

119867120573

120597120590120572120573

120597120572

+ 119867120572

120597120590120573120573

120597120573

+ 119867120572119867120573

120597120590120573119911

120597119911

+ (

2119867120572

119877120573

+

119867120573

119877120572

)120590120573119911= 120588119867120572119867120573V

119867120573

120597120590120572119911

120597120572

+ 119867120572

120597120590120573119911

120597120573

+ 119867120572119867120573

120597120590119911119911

120597119911

minus

119867120573

119877120572

120590120572120572minus

119867120572

119877120573

120590120573120573

+ (

119867120573

119877120572

+

119867120572

119877120573

)120590119911119911= 120588119867120572119867120573

(4)




16= 11986226= 11986236= 11986245=


119906 (120572 120573 119911 119905) = 119880 (119911) 119890119894120596119905 cos (120572120572) sin (120573120573)

V (120572 120573 119911 119905) = 119881 (119911) 119890119894120596119905 sin (120572120572) cos (120573120573)

119908 (120572 120573 119911 119905) = 119882 (119911) 119890119894120596119905 sin (120572120572) sin (120573120573)

(5)



120572or 119877120573is infinite




120572and119867

120573equal


(minus

11986255119867120573

1198671205721198772

120572

minus

11986255

119877120572119877120573

minus 120572211986211119867120573

119867120572

minus120573

211986266119867120572

119867120573

+ 1205881198671205721198671205731205962

)119880 + (minus12057212057311986212minus 120572120573119862

66)119881



2



119886ℎ = 10


119886ℎ = radic10


+ (120572

11986211119867120573

119867120572119877120572

+ 120572

11986212

119877120573

+ 120572

11986255119867120573

119867120572119877120572

+ 120572

11986255

119877120573

)119882

+ (

11986255119867120573

119877120572

+

11986255119867120572

119877120573

)119880119911+ (120572119862

13119867120573+ 12057211986255119867120573)119882119911

+ (11986255119867120572119867120573)119880119911119911= 0

(minus12057212057311986266minus 120572120573119862

12)119880

+ (minus

11986244119867120572

1198671205731198772

120573

minus

11986244

119877120572119877120573

minus 120572211986266119867120573

119867120572

minus 120573

211986222119867120572

119867120573

+ 1205881198671205721198671205731205962

)119881

+ (120573

11986244119867120572

119867120573119877120573

+ 120573

11986244

119877120572

+ 120573

11986222119867120572

119867120573119877120573

+ 120573

11986212

119877120572

)119882

+ (

11986244119867120572

119877120573

+

11986244119867120573

119877120572

)119881119911+ (120573119862

44119867120572+ 12057311986223119867120572)119882119911

+ (11986244119867120572119867120573)119881119911119911= 0

(120572

11986255119867120573

119867120572119877120572

minus 120572

11986213

119877120573

+ 120572

11986211119867120573

119867120572119877120572

+ 120572

11986212

119877120573

)119880

+ (120573

11986244119867120572

119867120573119877120573

minus 120573

11986223

119877120572

+ 120573

11986222119867120572

119867120573119877120573

+ 120573

11986212

119877120572

)119881

+ (

11986213

119877120572119877120573

+

11986223

119877120572119877120573

minus

11986211119867120573

1198671205721198772

120572

minus

211986212

119877120572119877120573

minus

11986222119867120572

1198671205731198772

120573

minus120572211986255119867120573

119867120572

minus 120573

211986244119867120572

119867120573

+ 1205881198671205721198671205731205962

)119882


44119867120572minus 12057311986223119867120572)119881119911

+ (

11986233119867120573

119877120572

+

11986233119867120572

119877120573

)119882119911+ (11986233119867120572119867120573)119882119911119911= 0

(6)




ℎ119877120572= 012

119898 = 2 119898 = 4 119898 = 6 119898 = 8

119887119877120572= 2


119887119877120572= 1


119887119877120572= 05


119887119877120572= 025




1198601119880 + 119860

2119881 + 119860

3119882+ 119860

4119880119911+ 1198605119882119911+ 1198606119880119911119911= 0

1198607119880 + 119860

8119881 + 119860

9119882+119860

10119881119911+ 11986011119882119911+ 11986012119881119911119911= 0

11986013119880 + 119860

14119881 + 119860

15119882+119860

16119880119911

+ 11986017119881119911+ 11986018119882119911+ 11986019119882119911119911= 0

(7)




[

[

[

[

[

[

[

[

1198606

0 0 0 0 0

0 11986012

0 0 0 0

0 0 11986019

0 0 0

0 0 0 1198606

0 0

0 0 0 0 11986012

0

0 0 0 0 0 11986019

]

]

]

]

]

]

]

]

[

[

[

[

[

[

[

[

119880

119881

119882

1198801015840

1198811015840

1198821015840

]

]

]

]

]

]

]

]

1015840

=

[

[

[

[

[

[

[

[

0 0 0 1198606

0 0

0 0 0 0 11986012

0

0 0 0 0 0 11986019

minus1198601

minus1198602

minus1198603

minus1198604

0 minus1198605

minus1198607

minus1198608

minus1198609

0 minus11986010

minus11986011

minus11986013

minus11986014

minus11986015

minus11986016

minus11986017

minus11986018

]

]

]

]

]

]

]

]

[

[

[

[

[

[

[

[

119880

119881

119882

1198801015840

1198811015840

1198821015840

]

]

]

]

]

]

]

]

(8)


D120597U120597

= AU (9)




ℎ119877120572= 018

119898 = 2 119898 = 4 119898 = 6 119898 = 8

119887119877120572= 2


119887119877120572= 1


119887119877120572= 05


119887119877120572= 025


where 120597U120597 = U1015840 and U = [119880 119881 119882 1198801015840

1198811015840

1198821015840


DU1015840 = AU (10)

U1015840 = Dminus1AU (11)

U1015840 = AlowastU (12)


31198604119860911986010

11986013 11986014 and 119860


120572


11198602119860511986061198607

1198608 11986011 11986012 11986015 11986016 11986017 and 119860







2

ℎ2

+

Alowast3

3

ℎ3

+ sdot sdot sdot +

Alowast119873

119873

ℎ119873

(14)


T = I (15)








(119898 = 0 119899 = 1) 120579 = 0∘ (119898 = 0 119899 = 1) 120579 = 90∘



2


119886ℎ 100 50 10 5(119898 = 0 119899 = 1)








U (ℎ) = HU (0) (17)


120572= (1 +

119911119877120572) = (1 + ( minus ℎ2)119877

120572) and 119867

120573= (1 + 119911119877

120573) =








120572and 119867

120573







119895minus1119895by


119906119887

119895= 119906119905

119895minus1 V119887

119895= V119905119895minus1 119908

119887

119895= 119908119905

119895minus1

120590119887

119911119911119895= 120590119905

119911119911119895minus1 120590

119887

120572119911119895= 120590119905

120572119911119895minus1 120590

119887

120573119911119895= 120590119905

120573119911119895minus1

(18)




U119895(0) = T

119895minus1119895U119895minus1

(ℎ119895minus1) (19)


119895


119895U119895(0) (20)



2


exponential matrix

119886ℎ 100 50 10 5 100 50 10 5(119898 = 0 119899 = 1) 120579 = 0∘ (119898 = 0 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 0) 120579 = 0∘ (119898 = 1 119899 = 0) 120579 = 90∘


(119898 = 1 119899 = 1) 120579 = 0∘

(119898 = 1 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 2) 120579 = 0∘

(119898 = 1 119899 = 2) 120579 = 90∘


(119898 = 2 119899 = 1) 120579 = 0∘

(119898 = 2 119899 = 1) 120579 = 90∘


(119898 = 2 119899 = 2) 120579 = 0∘

(119898 = 2 119899 = 2) 120579 = 90∘


Ω0

+h2

120572 120573

minush2

zz

h

0

0

(a)

Ω0

00

+h2

120572 120573

minush2 j = 1

j = 2

j

j = M

zMhM

zz

h

0 0

z1 h1

(b)



U119872(ℎ119872)





T12

Alowastlowast1U1(0)




119898U1(0) (22)


119895



120572and


119895minus1119895



120572





2



119886ℎ 100 50 10 5(119898 = 0 119899 = 1)


(119898 = 1 119899 = 0)


(119898 = 1 119899 = 1)



(119898 = 2 119899 = 1)





120590119911119911= 120590120572119911= 120590120573119911= 0 for 119911 = minusℎ

2

+ℎ

2

or = 0 ℎ (23)

119908 = V = 0 120590120572120572= 0 for 120572 = 0 119886 (24)

119908 = 119906 = 0 120590120573120573= 0 for 120573 = 0 119887 (25)



120572


120590119905

119911119911= minus120572

11986213

119867119905

120572

119880119905

+

11986213

119867119905

120572119877120572

119882119905

minus 120573

11986223

119867119905

120573

119881119905

+

11986223

119867119905

120573119877120573

119882119905

+ 11986233119882119905

119911= 0

120590119905

120573119911= 120573

11986244

119867119905

120573

119882119905

+ 11986244119881119905

119911minus

11986244

119867119905

120573119877120573

119881119905

= 0

120590119905

120572119911= 120572

11986255

119867119905

120572

119882119905

+ 11986255119880119905

119911minus

11986255

119867119905

120572119877120572

119880119905

= 0

(26)



0of the shell and

119867119887

120572and119867119887


120590119887

119911119911= minus120572

11986213

119867119887

120572

119880119887

+

11986213

119867119887

120572119877120572

119882119887

minus 120573

11986223

119867119887

120573

119881119887

+

11986223

119867119887

120573119877120573

119882119887

+ 11986233119882119887

119911= 0

120590119887

120573119911= 120573

11986244

119867119887

120573

119882119887

+ 11986244119881119887

119911minus

11986244

119867119887

120573119877120573

119881119887

= 0

120590119887

120572119911= 120572

11986255

119867119887

120572

119882119887

+ 11986255119880119887

119911minus

11986255

119867119887

120572119877120572

119880119887

= 0

(27)



[

[

[

[

[

[

[

[

[

[

[

[

[

minus120572

11986213

119867119905

120572

minus120573

11986223

119867119905

120573

(

11986213

119867119905

120572119877120572

+

11986223

119867119905

120573119877120573

) 0 0 11986233

0 minus

11986244

119867119905

120573119877120573

120573

11986244

119867119905

120573

0 11986244

0

minus

11986255

119867119905

120572119877120572

0 120572

11986255

119867119905

120572

11986255

0 0

]

]

]

]

]

]

]

]

]

]

]

]

]

times

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

119880119872(ℎ119872)

119881119872(ℎ119872)

119882119872(ℎ119872)

1198801015840

119872(ℎ119872)

1198811015840

119872(ℎ119872)

1198821015840

119872(ℎ119872)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

=[

[

0

0

0

]

]

(28)





119886ℎ 100 50 10 5 100 50 10 5(119898 = 0 119899 = 1) 120579 = 0∘ (119898 = 0 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 0) 120579 = 0∘ (119898 = 1 119899 = 0) 120579 = 90∘


(119898 = 1 119899 = 1) 120579 = 0∘ (119898 = 1 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 2) 120579 = 0∘ (119898 = 1 119899 = 2) 120579 = 90∘


(119898 = 2 119899 = 1) 120579 = 0∘ (119898 = 2 119899 = 1) 120579 = 90∘


(119898 = 2 119899 = 2) 120579 = 0∘ (119898 = 2 119899 = 2) 120579 = 90∘




[

[

[

[

[

[

[

[

[

[

[

minus120572

11986213

119867119887

120572

minus120573

11986223

119867119887

120573

(

11986213

119867119887

120572119877120572

+

11986223

119867119887

120573119877120573

) 0 0 11986233

0 minus

11986244

119867119887

120573119877120573

120573

11986244

119867119887

120573

0 11986244

0

minus

11986255

119867119887

120572119877120572

0 120572

11986255

119867119887

120572

11986255

0 0

]

]

]

]

]

]

]

]

]

]

]

times

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

1198801(0)

1198811(0)

1198821(0)

1198801015840

1(0)

1198811015840

1(0)

1198821015840

1(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

=[

[

0

0

0

]

]

(29)


B119872(ℎ119872)U119872(ℎ119872) = 0 (30)

B1(0)U1(0) = 0 (31)


B (ℎ)U (ℎ) = 0

B (0)U (0) = 0(32)


B119872(ℎ119872)H119898U1(0) = 0

B1(0)U1(0) = 0

(33)


[

B119872(ℎ119872)H119898

B1(0)

] [U1(0)] = [0] (34)


[E] [U1(0)] = [0] (35)



2



119877120572ℎ 1000 100 10 5

(119898 = 0 119899 = 1)


(119898 = 1 119899 = 0)


(119898 = 1 119899 = 1)


(119898 = 1 119899 = 2)


(119898 = 2 119899 = 1)


(119898 = 2 119899 = 2)



1(0)H = H

119898 andB =

B119872(ℎ119872) = B






det [E] = 0 (36)








119897




bottom

U1120596119897(0) = [119880

1(0) 119881

1(0) 119882

1(0) 119880

1015840

1(0) 119881

1015840

1(0) 119882

1015840

1(0)]

119879

120596119897

(37)



119897


components 119880119895120596119897(119895) 119881119895120596119897(119895) and 119882

119895120596119897(119895) through the


4 Results


120572



2


120572and 119867

120573 For 119872 = 1







119877120572ℎ 1000 100 10 5 1000 100 10 5

(119898 = 0 119899 = 1) 120579 = 0∘

(119898 = 0 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 0) 120579 = 0∘

(119898 = 1 119899 = 0) 120579 = 90∘


(119898 = 1 119899 = 1) 120579 = 0∘

(119898 = 1 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 2) 120579 = 0∘

(119898 = 1 119899 = 2) 120579 = 90∘


(119898 = 2 119899 = 1) 120579 = 0∘

(119898 = 2 119899 = 1) 120579 = 90∘


(119898 = 2 119899 = 2) 120579 = 0∘

(119898 = 2 119899 = 2) 120579 = 90∘


a

z

y

x

hb

(a)

a

zy

x

hb

(b)

a

z

b

R120572

120573

120572

(c)

z

b

R120572

120573

120572

a = 2120587R120572

(d)

a

z

b

R120572

R120573

120573

120572

(e)




2


119877120572ℎ 1000 100 10 5















120572and119867









11198642= 11986411198643= 40

119866121198642= 119866131198642= 06 119866

231198642= 05 ]

12= ]13= ]23= 025











120573in










1= 13238GPa




2


exponential matrix

119877120572ℎ 1000 100 10 5 1000 100 10 5

(119898 = 0 119899 = 1) 120579 = 0∘ (119898 = 0 119899 = 1) 120579 = 90∘


(119898 = 2 119899 = 1) 120579 = 0∘ (119898 = 2 119899 = 1) 120579 = 90∘


(119898 = 2 119899 = 2) 120579 = 0∘ (119898 = 2 119899 = 2) 120579 = 90∘


(119898 = 2 119899 = 3) 120579 = 0∘ (119898 = 2 119899 = 3) 120579 = 90∘


11986612= 11986613= 56537GPa and 119866


components ]12

= ]13

= 024 and ]23


120596(1198862



lowast


= 119908|119908max| and 119911lowast














2



119877120572ℎ 1000 100 10 5




















∘ and 120579 =







119877120572ℎ 1000 100 10 5 1000 100 10 5

(119898 = 0 119899 = 1) 120579 = 0∘ (119898 = 0 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 0) 120579 = 0∘ (119898 = 1 119899 = 0) 120579 = 90∘


(119898 = 1 119899 = 1) 120579 = 0∘ (119898 = 1 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 2) 120579 = 0∘ (119898 = 1 119899 = 2) 120579 = 90∘


(119898 = 2 119899 = 1) 120579 = 0∘ (119898 = 2 119899 = 1) 120579 = 90∘


(119898 = 2 119899 = 2) 120579 = 0∘ (119898 = 2 119899 = 2) 120579 = 90∘





120573effects


120572= 1 (which means 119911119877




120572ℎ equals 1000




= 119867120573

= 1 (which means 119911119877120572

= 119911119877120573


120572= 119867120573= 1 is valid



120572= 119911119877


5 Conclusions



Table 15 Effects of119867120572= (1 + 119911119877



119877120572ℎ 1000 100 10 5


120572= 1) 10671 10671 10686 53486

II mode (119867120572= 1) 10683 10683 10688 53684

III mode (119867120572= 1) 18557 18556 18550 92651


120572= 1) 27747 27473 27029 25828

II mode (119867120572= 1) 20403 20403 20403 10201

III mode (119867120572= 1) 36353 36352 36303 18076


120572= 1) 22273 22535 40440 33738

II mode (119867120572= 1) 23645 23645 23668 11868

III mode (119867120572= 1) 40174 40173 40115 19965

Table 16 Effects of 119867120572= (1 + 119911119877

120572) and 119867

120573= (1 + 119911119877




119877120572ℎ 1000 100 10 5 1000 100 10 5

(119898 = 0 119899 = 1) 120579 = 0∘ (119898 = 0 119899 = 1) 120579 = 90∘


120572= 119867120573= 1) 23852 23852 23858 11939 10379 10409 12602 77410

II mode (119867120572= 119867120573= 1) 29020 29020 29030 14529 23852 23852 23856 11934

III mode (119867120572= 119867120573= 1) 43716 43715 43623 21667 12223 12223 12204 51743

(119898 = 1 119899 = 0) 120579 = 0∘ (119898 = 1 119899 = 0) 120579 = 90∘


120572= 119867120573= 1) 10379 10409 12602 77410 23852 23852 23858 11939

II mode (119867120572= 119867120573= 1) 23852 23852 23856 11934 29020 29020 29030 14529

III mode (119867120572= 119867120573= 1) 12223 12223 12204 51743 43716 43715 43623 21667

(119898 = 1 119899 = 1) 120579 = 0∘ (119898 = 1 119899 = 1) 120579 = 90∘


120572= 119867120573= 1) 12097 12133 14810 92169 12097 12133 14810 92169

II mode (119867120572= 119867120573= 1) 40165 40164 40137 20022 40165 40164 40137 20022

III mode (119867120572= 119867120573= 1) 12456 12456 12429 54312 12456 12456 12429 54312


1

09

08

07

06

05

04

03

02

01



1

09

08

07

06

05

04

03

02

01



zlowast

1

09

08

07

06

05

04

03

02

01



zlowast

1

09

08

07

06

05

04

03

02

01



zlowast

1

09

08

07

06

05

04

03

02

01



zlowast

1

09

08

07

06

05

04

03

02

01



zlowast

1

09

08

07

06

05

04

03

02

01



zlowast

1

09

08

07

06

05

04

03

02

01



zlowast

1

09

08

07

06

05

04

03

02

01



zlowast

1

09

08

07

06

05

04

03

02

01



zlowast

1

09

08

07

06

05

04

03

02

01



zlowast

1

09

08

07

06

05

04

03

02

01



















1

09

08

07

06

05

04

03

02

01



1

09

08

07

06

05

04

03

02

01



zlowast

1

09

08

07

06

05

04

03

02

01



zlowast

1

09

08

07

06

05

04

03

02

01



zlowast

1

09

08

07

06

05

04

03

02

01



zlowast

1

09

08

07

06

05

04

03

02

01



zlowast

1

09

08

07

06

05

04

03

02

01



zlowast

1

09

08

07

06

05

04

03

02

01



zlowast

1

09

08

07

06

05

04

03

02

01



zlowast

1

09

08

07

06

05

04

03

02

01



zlowast

1

09

08

07

06

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References

























































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












Z

h2

120572120573

Z

in

i120572 i120573

minush2

Ω0 Ω0

dz





120572and 119877

120573



120598120572120572=

1

119867120572

120597119906

120597120572

+

119908

119867120572119877120572

120598120573120573=

1

119867120573

120597V120597120573

+

119908

119867120573119877120573


120598119911119911=

120597119908

120597119911

120574120572120573=

1

119867120572

120597V120597120572

+

1

119867120573

120597119906

120597120573

120574120572119911=

1

119867120572

120597119908

120597120572

+

120597119906

120597119911

minus

119906

119867120572119877120572

120574120573119911=

1

119867120573

120597119908

120597120573

+

120597V120597119911

minus

V119867120573119877120573

(1)


119867120572= (1 +

119911

119877120572

) 119867120573= (1 +

119911

119877120573

) 119867119911= 1 (2)


120572or119877120573



120573are

infinite (with119867120572=119867120573= 1) [46]




1198622611986236 and119862


120579 equals 0∘ or 90∘)

120590120572120572=

11986211

119867120572

119906120572+

11986211

119867120572119877120572

119908 +

11986212

119867120573

V120573+

11986212

119867120573119877120573

119908 + 11986213119908119911

120590120573120573=

11986212

119867120572

119906120572+

11986212

119867120572119877120572

119908 +

11986222

119867120573

V120573+

11986222

119867120573119877120573

119908 + 11986223119908119911

120590119911119911=

11986213

119867120572

119906120572+

11986213

119867120572119877120572

119908 +

11986223

119867120573

V120573+

11986223

119867120573119877120573

119908 + 11986233119908119911

120590120573119911=

11986244

119867120573

119908120573+ 11986244V119911minus

11986244

119867120573119877120573

V

120590120572119911=

11986255

119867120572

119908120572+ 11986255119906119911minus

11986255

119867120572119877120572

119906

120590120572120573=

11986266

119867120572

V120572+

11986266

119867120573

119906120573

(3)






119867120573

120597120590120572120572

120597120572

+ 119867120572

120597120590120572120573

120597120573

+ 119867120572119867120573

120597120590120572119911

120597119911

+ (

2119867120573

119877120572

+

119867120572

119877120573

)120590120572119911= 120588119867120572119867120573

119867120573

120597120590120572120573

120597120572

+ 119867120572

120597120590120573120573

120597120573

+ 119867120572119867120573

120597120590120573119911

120597119911

+ (

2119867120572

119877120573

+

119867120573

119877120572

)120590120573119911= 120588119867120572119867120573V

119867120573

120597120590120572119911

120597120572

+ 119867120572

120597120590120573119911

120597120573

+ 119867120572119867120573

120597120590119911119911

120597119911

minus

119867120573

119877120572

120590120572120572minus

119867120572

119877120573

120590120573120573

+ (

119867120573

119877120572

+

119867120572

119877120573

)120590119911119911= 120588119867120572119867120573

(4)




16= 11986226= 11986236= 11986245=


119906 (120572 120573 119911 119905) = 119880 (119911) 119890119894120596119905 cos (120572120572) sin (120573120573)

V (120572 120573 119911 119905) = 119881 (119911) 119890119894120596119905 sin (120572120572) cos (120573120573)

119908 (120572 120573 119911 119905) = 119882 (119911) 119890119894120596119905 sin (120572120572) sin (120573120573)

(5)



120572or 119877120573is infinite




120572and119867

120573equal


(minus

11986255119867120573

1198671205721198772

120572

minus

11986255

119877120572119877120573

minus 120572211986211119867120573

119867120572

minus120573

211986266119867120572

119867120573

+ 1205881198671205721198671205731205962

)119880 + (minus12057212057311986212minus 120572120573119862

66)119881



2



119886ℎ = 10


119886ℎ = radic10


+ (120572

11986211119867120573

119867120572119877120572

+ 120572

11986212

119877120573

+ 120572

11986255119867120573

119867120572119877120572

+ 120572

11986255

119877120573

)119882

+ (

11986255119867120573

119877120572

+

11986255119867120572

119877120573

)119880119911+ (120572119862

13119867120573+ 12057211986255119867120573)119882119911

+ (11986255119867120572119867120573)119880119911119911= 0

(minus12057212057311986266minus 120572120573119862

12)119880

+ (minus

11986244119867120572

1198671205731198772

120573

minus

11986244

119877120572119877120573

minus 120572211986266119867120573

119867120572

minus 120573

211986222119867120572

119867120573

+ 1205881198671205721198671205731205962

)119881

+ (120573

11986244119867120572

119867120573119877120573

+ 120573

11986244

119877120572

+ 120573

11986222119867120572

119867120573119877120573

+ 120573

11986212

119877120572

)119882

+ (

11986244119867120572

119877120573

+

11986244119867120573

119877120572

)119881119911+ (120573119862

44119867120572+ 12057311986223119867120572)119882119911

+ (11986244119867120572119867120573)119881119911119911= 0

(120572

11986255119867120573

119867120572119877120572

minus 120572

11986213

119877120573

+ 120572

11986211119867120573

119867120572119877120572

+ 120572

11986212

119877120573

)119880

+ (120573

11986244119867120572

119867120573119877120573

minus 120573

11986223

119877120572

+ 120573

11986222119867120572

119867120573119877120573

+ 120573

11986212

119877120572

)119881

+ (

11986213

119877120572119877120573

+

11986223

119877120572119877120573

minus

11986211119867120573

1198671205721198772

120572

minus

211986212

119877120572119877120573

minus

11986222119867120572

1198671205731198772

120573

minus120572211986255119867120573

119867120572

minus 120573

211986244119867120572

119867120573

+ 1205881198671205721198671205731205962

)119882


44119867120572minus 12057311986223119867120572)119881119911

+ (

11986233119867120573

119877120572

+

11986233119867120572

119877120573

)119882119911+ (11986233119867120572119867120573)119882119911119911= 0

(6)




ℎ119877120572= 012

119898 = 2 119898 = 4 119898 = 6 119898 = 8

119887119877120572= 2


119887119877120572= 1


119887119877120572= 05


119887119877120572= 025




1198601119880 + 119860

2119881 + 119860

3119882+ 119860

4119880119911+ 1198605119882119911+ 1198606119880119911119911= 0

1198607119880 + 119860

8119881 + 119860

9119882+119860

10119881119911+ 11986011119882119911+ 11986012119881119911119911= 0

11986013119880 + 119860

14119881 + 119860

15119882+119860

16119880119911

+ 11986017119881119911+ 11986018119882119911+ 11986019119882119911119911= 0

(7)




[

[

[

[

[

[

[

[

1198606

0 0 0 0 0

0 11986012

0 0 0 0

0 0 11986019

0 0 0

0 0 0 1198606

0 0

0 0 0 0 11986012

0

0 0 0 0 0 11986019

]

]

]

]

]

]

]

]

[

[

[

[

[

[

[

[

119880

119881

119882

1198801015840

1198811015840

1198821015840

]

]

]

]

]

]

]

]

1015840

=

[

[

[

[

[

[

[

[

0 0 0 1198606

0 0

0 0 0 0 11986012

0

0 0 0 0 0 11986019

minus1198601

minus1198602

minus1198603

minus1198604

0 minus1198605

minus1198607

minus1198608

minus1198609

0 minus11986010

minus11986011

minus11986013

minus11986014

minus11986015

minus11986016

minus11986017

minus11986018

]

]

]

]

]

]

]

]

[

[

[

[

[

[

[

[

119880

119881

119882

1198801015840

1198811015840

1198821015840

]

]

]

]

]

]

]

]

(8)


D120597U120597

= AU (9)




ℎ119877120572= 018

119898 = 2 119898 = 4 119898 = 6 119898 = 8

119887119877120572= 2


119887119877120572= 1


119887119877120572= 05


119887119877120572= 025


where 120597U120597 = U1015840 and U = [119880 119881 119882 1198801015840

1198811015840

1198821015840


DU1015840 = AU (10)

U1015840 = Dminus1AU (11)

U1015840 = AlowastU (12)


31198604119860911986010

11986013 11986014 and 119860


120572


11198602119860511986061198607

1198608 11986011 11986012 11986015 11986016 11986017 and 119860







2

ℎ2

+

Alowast3

3

ℎ3

+ sdot sdot sdot +

Alowast119873

119873

ℎ119873

(14)


T = I (15)








(119898 = 0 119899 = 1) 120579 = 0∘ (119898 = 0 119899 = 1) 120579 = 90∘



2


119886ℎ 100 50 10 5(119898 = 0 119899 = 1)








U (ℎ) = HU (0) (17)


120572= (1 +

119911119877120572) = (1 + ( minus ℎ2)119877

120572) and 119867

120573= (1 + 119911119877

120573) =








120572and 119867

120573







119895minus1119895by


119906119887

119895= 119906119905

119895minus1 V119887

119895= V119905119895minus1 119908

119887

119895= 119908119905

119895minus1

120590119887

119911119911119895= 120590119905

119911119911119895minus1 120590

119887

120572119911119895= 120590119905

120572119911119895minus1 120590

119887

120573119911119895= 120590119905

120573119911119895minus1

(18)




U119895(0) = T

119895minus1119895U119895minus1

(ℎ119895minus1) (19)


119895


119895U119895(0) (20)



2


exponential matrix

119886ℎ 100 50 10 5 100 50 10 5(119898 = 0 119899 = 1) 120579 = 0∘ (119898 = 0 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 0) 120579 = 0∘ (119898 = 1 119899 = 0) 120579 = 90∘


(119898 = 1 119899 = 1) 120579 = 0∘

(119898 = 1 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 2) 120579 = 0∘

(119898 = 1 119899 = 2) 120579 = 90∘


(119898 = 2 119899 = 1) 120579 = 0∘

(119898 = 2 119899 = 1) 120579 = 90∘


(119898 = 2 119899 = 2) 120579 = 0∘

(119898 = 2 119899 = 2) 120579 = 90∘


Ω0

+h2

120572 120573

minush2

zz

h

0

0

(a)

Ω0

00

+h2

120572 120573

minush2 j = 1

j = 2

j

j = M

zMhM

zz

h

0 0

z1 h1

(b)



U119872(ℎ119872)





T12

Alowastlowast1U1(0)




119898U1(0) (22)


119895



120572and


119895minus1119895



120572





2



119886ℎ 100 50 10 5(119898 = 0 119899 = 1)


(119898 = 1 119899 = 0)


(119898 = 1 119899 = 1)



(119898 = 2 119899 = 1)





120590119911119911= 120590120572119911= 120590120573119911= 0 for 119911 = minusℎ

2

+ℎ

2

or = 0 ℎ (23)

119908 = V = 0 120590120572120572= 0 for 120572 = 0 119886 (24)

119908 = 119906 = 0 120590120573120573= 0 for 120573 = 0 119887 (25)



120572


120590119905

119911119911= minus120572

11986213

119867119905

120572

119880119905

+

11986213

119867119905

120572119877120572

119882119905

minus 120573

11986223

119867119905

120573

119881119905

+

11986223

119867119905

120573119877120573

119882119905

+ 11986233119882119905

119911= 0

120590119905

120573119911= 120573

11986244

119867119905

120573

119882119905

+ 11986244119881119905

119911minus

11986244

119867119905

120573119877120573

119881119905

= 0

120590119905

120572119911= 120572

11986255

119867119905

120572

119882119905

+ 11986255119880119905

119911minus

11986255

119867119905

120572119877120572

119880119905

= 0

(26)



0of the shell and

119867119887

120572and119867119887


120590119887

119911119911= minus120572

11986213

119867119887

120572

119880119887

+

11986213

119867119887

120572119877120572

119882119887

minus 120573

11986223

119867119887

120573

119881119887

+

11986223

119867119887

120573119877120573

119882119887

+ 11986233119882119887

119911= 0

120590119887

120573119911= 120573

11986244

119867119887

120573

119882119887

+ 11986244119881119887

119911minus

11986244

119867119887

120573119877120573

119881119887

= 0

120590119887

120572119911= 120572

11986255

119867119887

120572

119882119887

+ 11986255119880119887

119911minus

11986255

119867119887

120572119877120572

119880119887

= 0

(27)



[

[

[

[

[

[

[

[

[

[

[

[

[

minus120572

11986213

119867119905

120572

minus120573

11986223

119867119905

120573

(

11986213

119867119905

120572119877120572

+

11986223

119867119905

120573119877120573

) 0 0 11986233

0 minus

11986244

119867119905

120573119877120573

120573

11986244

119867119905

120573

0 11986244

0

minus

11986255

119867119905

120572119877120572

0 120572

11986255

119867119905

120572

11986255

0 0

]

]

]

]

]

]

]

]

]

]

]

]

]

times

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

119880119872(ℎ119872)

119881119872(ℎ119872)

119882119872(ℎ119872)

1198801015840

119872(ℎ119872)

1198811015840

119872(ℎ119872)

1198821015840

119872(ℎ119872)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

=[

[

0

0

0

]

]

(28)





119886ℎ 100 50 10 5 100 50 10 5(119898 = 0 119899 = 1) 120579 = 0∘ (119898 = 0 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 0) 120579 = 0∘ (119898 = 1 119899 = 0) 120579 = 90∘


(119898 = 1 119899 = 1) 120579 = 0∘ (119898 = 1 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 2) 120579 = 0∘ (119898 = 1 119899 = 2) 120579 = 90∘


(119898 = 2 119899 = 1) 120579 = 0∘ (119898 = 2 119899 = 1) 120579 = 90∘


(119898 = 2 119899 = 2) 120579 = 0∘ (119898 = 2 119899 = 2) 120579 = 90∘




[

[

[

[

[

[

[

[

[

[

[

minus120572

11986213

119867119887

120572

minus120573

11986223

119867119887

120573

(

11986213

119867119887

120572119877120572

+

11986223

119867119887

120573119877120573

) 0 0 11986233

0 minus

11986244

119867119887

120573119877120573

120573

11986244

119867119887

120573

0 11986244

0

minus

11986255

119867119887

120572119877120572

0 120572

11986255

119867119887

120572

11986255

0 0

]

]

]

]

]

]

]

]

]

]

]

times

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

1198801(0)

1198811(0)

1198821(0)

1198801015840

1(0)

1198811015840

1(0)

1198821015840

1(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

=[

[

0

0

0

]

]

(29)


B119872(ℎ119872)U119872(ℎ119872) = 0 (30)

B1(0)U1(0) = 0 (31)


B (ℎ)U (ℎ) = 0

B (0)U (0) = 0(32)


B119872(ℎ119872)H119898U1(0) = 0

B1(0)U1(0) = 0

(33)


[

B119872(ℎ119872)H119898

B1(0)

] [U1(0)] = [0] (34)


[E] [U1(0)] = [0] (35)



2



119877120572ℎ 1000 100 10 5

(119898 = 0 119899 = 1)


(119898 = 1 119899 = 0)


(119898 = 1 119899 = 1)


(119898 = 1 119899 = 2)


(119898 = 2 119899 = 1)


(119898 = 2 119899 = 2)



1(0)H = H

119898 andB =

B119872(ℎ119872) = B






det [E] = 0 (36)








119897




bottom

U1120596119897(0) = [119880

1(0) 119881

1(0) 119882

1(0) 119880

1015840

1(0) 119881

1015840

1(0) 119882

1015840

1(0)]

119879

120596119897

(37)



119897


components 119880119895120596119897(119895) 119881119895120596119897(119895) and 119882

119895120596119897(119895) through the


4 Results


120572



2


120572and 119867

120573 For 119872 = 1







119877120572ℎ 1000 100 10 5 1000 100 10 5

(119898 = 0 119899 = 1) 120579 = 0∘

(119898 = 0 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 0) 120579 = 0∘

(119898 = 1 119899 = 0) 120579 = 90∘


(119898 = 1 119899 = 1) 120579 = 0∘

(119898 = 1 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 2) 120579 = 0∘

(119898 = 1 119899 = 2) 120579 = 90∘


(119898 = 2 119899 = 1) 120579 = 0∘

(119898 = 2 119899 = 1) 120579 = 90∘


(119898 = 2 119899 = 2) 120579 = 0∘

(119898 = 2 119899 = 2) 120579 = 90∘


a

z

y

x

hb

(a)

a

zy

x

hb

(b)

a

z

b

R120572

120573

120572

(c)

z

b

R120572

120573

120572

a = 2120587R120572

(d)

a

z

b

R120572

R120573

120573

120572

(e)




2


119877120572ℎ 1000 100 10 5















120572and119867









11198642= 11986411198643= 40

119866121198642= 119866131198642= 06 119866

231198642= 05 ]

12= ]13= ]23= 025











120573in










1= 13238GPa




2


exponential matrix

119877120572ℎ 1000 100 10 5 1000 100 10 5

(119898 = 0 119899 = 1) 120579 = 0∘ (119898 = 0 119899 = 1) 120579 = 90∘


(119898 = 2 119899 = 1) 120579 = 0∘ (119898 = 2 119899 = 1) 120579 = 90∘


(119898 = 2 119899 = 2) 120579 = 0∘ (119898 = 2 119899 = 2) 120579 = 90∘


(119898 = 2 119899 = 3) 120579 = 0∘ (119898 = 2 119899 = 3) 120579 = 90∘


11986612= 11986613= 56537GPa and 119866


components ]12

= ]13

= 024 and ]23


120596(1198862



lowast


= 119908|119908max| and 119911lowast














2



119877120572ℎ 1000 100 10 5




















∘ and 120579 =







119877120572ℎ 1000 100 10 5 1000 100 10 5

(119898 = 0 119899 = 1) 120579 = 0∘ (119898 = 0 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 0) 120579 = 0∘ (119898 = 1 119899 = 0) 120579 = 90∘


(119898 = 1 119899 = 1) 120579 = 0∘ (119898 = 1 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 2) 120579 = 0∘ (119898 = 1 119899 = 2) 120579 = 90∘


(119898 = 2 119899 = 1) 120579 = 0∘ (119898 = 2 119899 = 1) 120579 = 90∘


(119898 = 2 119899 = 2) 120579 = 0∘ (119898 = 2 119899 = 2) 120579 = 90∘





120573effects


120572= 1 (which means 119911119877




120572ℎ equals 1000




= 119867120573

= 1 (which means 119911119877120572

= 119911119877120573


120572= 119867120573= 1 is valid



120572= 119911119877


5 Conclusions



Table 15 Effects of119867120572= (1 + 119911119877



119877120572ℎ 1000 100 10 5


120572= 1) 10671 10671 10686 53486

II mode (119867120572= 1) 10683 10683 10688 53684

III mode (119867120572= 1) 18557 18556 18550 92651


120572= 1) 27747 27473 27029 25828

II mode (119867120572= 1) 20403 20403 20403 10201

III mode (119867120572= 1) 36353 36352 36303 18076


120572= 1) 22273 22535 40440 33738

II mode (119867120572= 1) 23645 23645 23668 11868

III mode (119867120572= 1) 40174 40173 40115 19965

Table 16 Effects of 119867120572= (1 + 119911119877

120572) and 119867

120573= (1 + 119911119877




119877120572ℎ 1000 100 10 5 1000 100 10 5

(119898 = 0 119899 = 1) 120579 = 0∘ (119898 = 0 119899 = 1) 120579 = 90∘


120572= 119867120573= 1) 23852 23852 23858 11939 10379 10409 12602 77410

II mode (119867120572= 119867120573= 1) 29020 29020 29030 14529 23852 23852 23856 11934

III mode (119867120572= 119867120573= 1) 43716 43715 43623 21667 12223 12223 12204 51743

(119898 = 1 119899 = 0) 120579 = 0∘ (119898 = 1 119899 = 0) 120579 = 90∘


120572= 119867120573= 1) 10379 10409 12602 77410 23852 23852 23858 11939

II mode (119867120572= 119867120573= 1) 23852 23852 23856 11934 29020 29020 29030 14529

III mode (119867120572= 119867120573= 1) 12223 12223 12204 51743 43716 43715 43623 21667

(119898 = 1 119899 = 1) 120579 = 0∘ (119898 = 1 119899 = 1) 120579 = 90∘


120572= 119867120573= 1) 12097 12133 14810 92169 12097 12133 14810 92169

II mode (119867120572= 119867120573= 1) 40165 40164 40137 20022 40165 40164 40137 20022

III mode (119867120572= 119867120573= 1) 12456 12456 12429 54312 12456 12456 12429 54312


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References

























































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










120598119911119911=

120597119908

120597119911

120574120572120573=

1

119867120572

120597V120597120572

+

1

119867120573

120597119906

120597120573

120574120572119911=

1

119867120572

120597119908

120597120572

+

120597119906

120597119911

minus

119906

119867120572119877120572

120574120573119911=

1

119867120573

120597119908

120597120573

+

120597V120597119911

minus

V119867120573119877120573

(1)


119867120572= (1 +

119911

119877120572

) 119867120573= (1 +

119911

119877120573

) 119867119911= 1 (2)


120572or119877120573



120573are

infinite (with119867120572=119867120573= 1) [46]




1198622611986236 and119862


120579 equals 0∘ or 90∘)

120590120572120572=

11986211

119867120572

119906120572+

11986211

119867120572119877120572

119908 +

11986212

119867120573

V120573+

11986212

119867120573119877120573

119908 + 11986213119908119911

120590120573120573=

11986212

119867120572

119906120572+

11986212

119867120572119877120572

119908 +

11986222

119867120573

V120573+

11986222

119867120573119877120573

119908 + 11986223119908119911

120590119911119911=

11986213

119867120572

119906120572+

11986213

119867120572119877120572

119908 +

11986223

119867120573

V120573+

11986223

119867120573119877120573

119908 + 11986233119908119911

120590120573119911=

11986244

119867120573

119908120573+ 11986244V119911minus

11986244

119867120573119877120573

V

120590120572119911=

11986255

119867120572

119908120572+ 11986255119906119911minus

11986255

119867120572119877120572

119906

120590120572120573=

11986266

119867120572

V120572+

11986266

119867120573

119906120573

(3)






119867120573

120597120590120572120572

120597120572

+ 119867120572

120597120590120572120573

120597120573

+ 119867120572119867120573

120597120590120572119911

120597119911

+ (

2119867120573

119877120572

+

119867120572

119877120573

)120590120572119911= 120588119867120572119867120573

119867120573

120597120590120572120573

120597120572

+ 119867120572

120597120590120573120573

120597120573

+ 119867120572119867120573

120597120590120573119911

120597119911

+ (

2119867120572

119877120573

+

119867120573

119877120572

)120590120573119911= 120588119867120572119867120573V

119867120573

120597120590120572119911

120597120572

+ 119867120572

120597120590120573119911

120597120573

+ 119867120572119867120573

120597120590119911119911

120597119911

minus

119867120573

119877120572

120590120572120572minus

119867120572

119877120573

120590120573120573

+ (

119867120573

119877120572

+

119867120572

119877120573

)120590119911119911= 120588119867120572119867120573

(4)




16= 11986226= 11986236= 11986245=


119906 (120572 120573 119911 119905) = 119880 (119911) 119890119894120596119905 cos (120572120572) sin (120573120573)

V (120572 120573 119911 119905) = 119881 (119911) 119890119894120596119905 sin (120572120572) cos (120573120573)

119908 (120572 120573 119911 119905) = 119882 (119911) 119890119894120596119905 sin (120572120572) sin (120573120573)

(5)



120572or 119877120573is infinite




120572and119867

120573equal


(minus

11986255119867120573

1198671205721198772

120572

minus

11986255

119877120572119877120573

minus 120572211986211119867120573

119867120572

minus120573

211986266119867120572

119867120573

+ 1205881198671205721198671205731205962

)119880 + (minus12057212057311986212minus 120572120573119862

66)119881



2



119886ℎ = 10


119886ℎ = radic10


+ (120572

11986211119867120573

119867120572119877120572

+ 120572

11986212

119877120573

+ 120572

11986255119867120573

119867120572119877120572

+ 120572

11986255

119877120573

)119882

+ (

11986255119867120573

119877120572

+

11986255119867120572

119877120573

)119880119911+ (120572119862

13119867120573+ 12057211986255119867120573)119882119911

+ (11986255119867120572119867120573)119880119911119911= 0

(minus12057212057311986266minus 120572120573119862

12)119880

+ (minus

11986244119867120572

1198671205731198772

120573

minus

11986244

119877120572119877120573

minus 120572211986266119867120573

119867120572

minus 120573

211986222119867120572

119867120573

+ 1205881198671205721198671205731205962

)119881

+ (120573

11986244119867120572

119867120573119877120573

+ 120573

11986244

119877120572

+ 120573

11986222119867120572

119867120573119877120573

+ 120573

11986212

119877120572

)119882

+ (

11986244119867120572

119877120573

+

11986244119867120573

119877120572

)119881119911+ (120573119862

44119867120572+ 12057311986223119867120572)119882119911

+ (11986244119867120572119867120573)119881119911119911= 0

(120572

11986255119867120573

119867120572119877120572

minus 120572

11986213

119877120573

+ 120572

11986211119867120573

119867120572119877120572

+ 120572

11986212

119877120573

)119880

+ (120573

11986244119867120572

119867120573119877120573

minus 120573

11986223

119877120572

+ 120573

11986222119867120572

119867120573119877120573

+ 120573

11986212

119877120572

)119881

+ (

11986213

119877120572119877120573

+

11986223

119877120572119877120573

minus

11986211119867120573

1198671205721198772

120572

minus

211986212

119877120572119877120573

minus

11986222119867120572

1198671205731198772

120573

minus120572211986255119867120573

119867120572

minus 120573

211986244119867120572

119867120573

+ 1205881198671205721198671205731205962

)119882


44119867120572minus 12057311986223119867120572)119881119911

+ (

11986233119867120573

119877120572

+

11986233119867120572

119877120573

)119882119911+ (11986233119867120572119867120573)119882119911119911= 0

(6)




ℎ119877120572= 012

119898 = 2 119898 = 4 119898 = 6 119898 = 8

119887119877120572= 2


119887119877120572= 1


119887119877120572= 05


119887119877120572= 025




1198601119880 + 119860

2119881 + 119860

3119882+ 119860

4119880119911+ 1198605119882119911+ 1198606119880119911119911= 0

1198607119880 + 119860

8119881 + 119860

9119882+119860

10119881119911+ 11986011119882119911+ 11986012119881119911119911= 0

11986013119880 + 119860

14119881 + 119860

15119882+119860

16119880119911

+ 11986017119881119911+ 11986018119882119911+ 11986019119882119911119911= 0

(7)




[

[

[

[

[

[

[

[

1198606

0 0 0 0 0

0 11986012

0 0 0 0

0 0 11986019

0 0 0

0 0 0 1198606

0 0

0 0 0 0 11986012

0

0 0 0 0 0 11986019

]

]

]

]

]

]

]

]

[

[

[

[

[

[

[

[

119880

119881

119882

1198801015840

1198811015840

1198821015840

]

]

]

]

]

]

]

]

1015840

=

[

[

[

[

[

[

[

[

0 0 0 1198606

0 0

0 0 0 0 11986012

0

0 0 0 0 0 11986019

minus1198601

minus1198602

minus1198603

minus1198604

0 minus1198605

minus1198607

minus1198608

minus1198609

0 minus11986010

minus11986011

minus11986013

minus11986014

minus11986015

minus11986016

minus11986017

minus11986018

]

]

]

]

]

]

]

]

[

[

[

[

[

[

[

[

119880

119881

119882

1198801015840

1198811015840

1198821015840

]

]

]

]

]

]

]

]

(8)


D120597U120597

= AU (9)




ℎ119877120572= 018

119898 = 2 119898 = 4 119898 = 6 119898 = 8

119887119877120572= 2


119887119877120572= 1


119887119877120572= 05


119887119877120572= 025


where 120597U120597 = U1015840 and U = [119880 119881 119882 1198801015840

1198811015840

1198821015840


DU1015840 = AU (10)

U1015840 = Dminus1AU (11)

U1015840 = AlowastU (12)


31198604119860911986010

11986013 11986014 and 119860


120572


11198602119860511986061198607

1198608 11986011 11986012 11986015 11986016 11986017 and 119860







2

ℎ2

+

Alowast3

3

ℎ3

+ sdot sdot sdot +

Alowast119873

119873

ℎ119873

(14)


T = I (15)








(119898 = 0 119899 = 1) 120579 = 0∘ (119898 = 0 119899 = 1) 120579 = 90∘



2


119886ℎ 100 50 10 5(119898 = 0 119899 = 1)








U (ℎ) = HU (0) (17)


120572= (1 +

119911119877120572) = (1 + ( minus ℎ2)119877

120572) and 119867

120573= (1 + 119911119877

120573) =








120572and 119867

120573







119895minus1119895by


119906119887

119895= 119906119905

119895minus1 V119887

119895= V119905119895minus1 119908

119887

119895= 119908119905

119895minus1

120590119887

119911119911119895= 120590119905

119911119911119895minus1 120590

119887

120572119911119895= 120590119905

120572119911119895minus1 120590

119887

120573119911119895= 120590119905

120573119911119895minus1

(18)




U119895(0) = T

119895minus1119895U119895minus1

(ℎ119895minus1) (19)


119895


119895U119895(0) (20)



2


exponential matrix

119886ℎ 100 50 10 5 100 50 10 5(119898 = 0 119899 = 1) 120579 = 0∘ (119898 = 0 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 0) 120579 = 0∘ (119898 = 1 119899 = 0) 120579 = 90∘


(119898 = 1 119899 = 1) 120579 = 0∘

(119898 = 1 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 2) 120579 = 0∘

(119898 = 1 119899 = 2) 120579 = 90∘


(119898 = 2 119899 = 1) 120579 = 0∘

(119898 = 2 119899 = 1) 120579 = 90∘


(119898 = 2 119899 = 2) 120579 = 0∘

(119898 = 2 119899 = 2) 120579 = 90∘


Ω0

+h2

120572 120573

minush2

zz

h

0

0

(a)

Ω0

00

+h2

120572 120573

minush2 j = 1

j = 2

j

j = M

zMhM

zz

h

0 0

z1 h1

(b)



U119872(ℎ119872)





T12

Alowastlowast1U1(0)




119898U1(0) (22)


119895



120572and


119895minus1119895



120572





2



119886ℎ 100 50 10 5(119898 = 0 119899 = 1)


(119898 = 1 119899 = 0)


(119898 = 1 119899 = 1)



(119898 = 2 119899 = 1)





120590119911119911= 120590120572119911= 120590120573119911= 0 for 119911 = minusℎ

2

+ℎ

2

or = 0 ℎ (23)

119908 = V = 0 120590120572120572= 0 for 120572 = 0 119886 (24)

119908 = 119906 = 0 120590120573120573= 0 for 120573 = 0 119887 (25)



120572


120590119905

119911119911= minus120572

11986213

119867119905

120572

119880119905

+

11986213

119867119905

120572119877120572

119882119905

minus 120573

11986223

119867119905

120573

119881119905

+

11986223

119867119905

120573119877120573

119882119905

+ 11986233119882119905

119911= 0

120590119905

120573119911= 120573

11986244

119867119905

120573

119882119905

+ 11986244119881119905

119911minus

11986244

119867119905

120573119877120573

119881119905

= 0

120590119905

120572119911= 120572

11986255

119867119905

120572

119882119905

+ 11986255119880119905

119911minus

11986255

119867119905

120572119877120572

119880119905

= 0

(26)



0of the shell and

119867119887

120572and119867119887


120590119887

119911119911= minus120572

11986213

119867119887

120572

119880119887

+

11986213

119867119887

120572119877120572

119882119887

minus 120573

11986223

119867119887

120573

119881119887

+

11986223

119867119887

120573119877120573

119882119887

+ 11986233119882119887

119911= 0

120590119887

120573119911= 120573

11986244

119867119887

120573

119882119887

+ 11986244119881119887

119911minus

11986244

119867119887

120573119877120573

119881119887

= 0

120590119887

120572119911= 120572

11986255

119867119887

120572

119882119887

+ 11986255119880119887

119911minus

11986255

119867119887

120572119877120572

119880119887

= 0

(27)



[

[

[

[

[

[

[

[

[

[

[

[

[

minus120572

11986213

119867119905

120572

minus120573

11986223

119867119905

120573

(

11986213

119867119905

120572119877120572

+

11986223

119867119905

120573119877120573

) 0 0 11986233

0 minus

11986244

119867119905

120573119877120573

120573

11986244

119867119905

120573

0 11986244

0

minus

11986255

119867119905

120572119877120572

0 120572

11986255

119867119905

120572

11986255

0 0

]

]

]

]

]

]

]

]

]

]

]

]

]

times

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

119880119872(ℎ119872)

119881119872(ℎ119872)

119882119872(ℎ119872)

1198801015840

119872(ℎ119872)

1198811015840

119872(ℎ119872)

1198821015840

119872(ℎ119872)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

=[

[

0

0

0

]

]

(28)





119886ℎ 100 50 10 5 100 50 10 5(119898 = 0 119899 = 1) 120579 = 0∘ (119898 = 0 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 0) 120579 = 0∘ (119898 = 1 119899 = 0) 120579 = 90∘


(119898 = 1 119899 = 1) 120579 = 0∘ (119898 = 1 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 2) 120579 = 0∘ (119898 = 1 119899 = 2) 120579 = 90∘


(119898 = 2 119899 = 1) 120579 = 0∘ (119898 = 2 119899 = 1) 120579 = 90∘


(119898 = 2 119899 = 2) 120579 = 0∘ (119898 = 2 119899 = 2) 120579 = 90∘




[

[

[

[

[

[

[

[

[

[

[

minus120572

11986213

119867119887

120572

minus120573

11986223

119867119887

120573

(

11986213

119867119887

120572119877120572

+

11986223

119867119887

120573119877120573

) 0 0 11986233

0 minus

11986244

119867119887

120573119877120573

120573

11986244

119867119887

120573

0 11986244

0

minus

11986255

119867119887

120572119877120572

0 120572

11986255

119867119887

120572

11986255

0 0

]

]

]

]

]

]

]

]

]

]

]

times

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

1198801(0)

1198811(0)

1198821(0)

1198801015840

1(0)

1198811015840

1(0)

1198821015840

1(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

=[

[

0

0

0

]

]

(29)


B119872(ℎ119872)U119872(ℎ119872) = 0 (30)

B1(0)U1(0) = 0 (31)


B (ℎ)U (ℎ) = 0

B (0)U (0) = 0(32)


B119872(ℎ119872)H119898U1(0) = 0

B1(0)U1(0) = 0

(33)


[

B119872(ℎ119872)H119898

B1(0)

] [U1(0)] = [0] (34)


[E] [U1(0)] = [0] (35)



2



119877120572ℎ 1000 100 10 5

(119898 = 0 119899 = 1)


(119898 = 1 119899 = 0)


(119898 = 1 119899 = 1)


(119898 = 1 119899 = 2)


(119898 = 2 119899 = 1)


(119898 = 2 119899 = 2)



1(0)H = H

119898 andB =

B119872(ℎ119872) = B






det [E] = 0 (36)








119897




bottom

U1120596119897(0) = [119880

1(0) 119881

1(0) 119882

1(0) 119880

1015840

1(0) 119881

1015840

1(0) 119882

1015840

1(0)]

119879

120596119897

(37)



119897


components 119880119895120596119897(119895) 119881119895120596119897(119895) and 119882

119895120596119897(119895) through the


4 Results


120572



2


120572and 119867

120573 For 119872 = 1







119877120572ℎ 1000 100 10 5 1000 100 10 5

(119898 = 0 119899 = 1) 120579 = 0∘

(119898 = 0 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 0) 120579 = 0∘

(119898 = 1 119899 = 0) 120579 = 90∘


(119898 = 1 119899 = 1) 120579 = 0∘

(119898 = 1 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 2) 120579 = 0∘

(119898 = 1 119899 = 2) 120579 = 90∘


(119898 = 2 119899 = 1) 120579 = 0∘

(119898 = 2 119899 = 1) 120579 = 90∘


(119898 = 2 119899 = 2) 120579 = 0∘

(119898 = 2 119899 = 2) 120579 = 90∘


a

z

y

x

hb

(a)

a

zy

x

hb

(b)

a

z

b

R120572

120573

120572

(c)

z

b

R120572

120573

120572

a = 2120587R120572

(d)

a

z

b

R120572

R120573

120573

120572

(e)




2


119877120572ℎ 1000 100 10 5















120572and119867









11198642= 11986411198643= 40

119866121198642= 119866131198642= 06 119866

231198642= 05 ]

12= ]13= ]23= 025











120573in










1= 13238GPa




2


exponential matrix

119877120572ℎ 1000 100 10 5 1000 100 10 5

(119898 = 0 119899 = 1) 120579 = 0∘ (119898 = 0 119899 = 1) 120579 = 90∘


(119898 = 2 119899 = 1) 120579 = 0∘ (119898 = 2 119899 = 1) 120579 = 90∘


(119898 = 2 119899 = 2) 120579 = 0∘ (119898 = 2 119899 = 2) 120579 = 90∘


(119898 = 2 119899 = 3) 120579 = 0∘ (119898 = 2 119899 = 3) 120579 = 90∘


11986612= 11986613= 56537GPa and 119866


components ]12

= ]13

= 024 and ]23


120596(1198862



lowast


= 119908|119908max| and 119911lowast














2



119877120572ℎ 1000 100 10 5




















∘ and 120579 =







119877120572ℎ 1000 100 10 5 1000 100 10 5

(119898 = 0 119899 = 1) 120579 = 0∘ (119898 = 0 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 0) 120579 = 0∘ (119898 = 1 119899 = 0) 120579 = 90∘


(119898 = 1 119899 = 1) 120579 = 0∘ (119898 = 1 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 2) 120579 = 0∘ (119898 = 1 119899 = 2) 120579 = 90∘


(119898 = 2 119899 = 1) 120579 = 0∘ (119898 = 2 119899 = 1) 120579 = 90∘


(119898 = 2 119899 = 2) 120579 = 0∘ (119898 = 2 119899 = 2) 120579 = 90∘





120573effects


120572= 1 (which means 119911119877




120572ℎ equals 1000




= 119867120573

= 1 (which means 119911119877120572

= 119911119877120573


120572= 119867120573= 1 is valid



120572= 119911119877


5 Conclusions



Table 15 Effects of119867120572= (1 + 119911119877



119877120572ℎ 1000 100 10 5


120572= 1) 10671 10671 10686 53486

II mode (119867120572= 1) 10683 10683 10688 53684

III mode (119867120572= 1) 18557 18556 18550 92651


120572= 1) 27747 27473 27029 25828

II mode (119867120572= 1) 20403 20403 20403 10201

III mode (119867120572= 1) 36353 36352 36303 18076


120572= 1) 22273 22535 40440 33738

II mode (119867120572= 1) 23645 23645 23668 11868

III mode (119867120572= 1) 40174 40173 40115 19965

Table 16 Effects of 119867120572= (1 + 119911119877

120572) and 119867

120573= (1 + 119911119877




119877120572ℎ 1000 100 10 5 1000 100 10 5

(119898 = 0 119899 = 1) 120579 = 0∘ (119898 = 0 119899 = 1) 120579 = 90∘


120572= 119867120573= 1) 23852 23852 23858 11939 10379 10409 12602 77410

II mode (119867120572= 119867120573= 1) 29020 29020 29030 14529 23852 23852 23856 11934

III mode (119867120572= 119867120573= 1) 43716 43715 43623 21667 12223 12223 12204 51743

(119898 = 1 119899 = 0) 120579 = 0∘ (119898 = 1 119899 = 0) 120579 = 90∘


120572= 119867120573= 1) 10379 10409 12602 77410 23852 23852 23858 11939

II mode (119867120572= 119867120573= 1) 23852 23852 23856 11934 29020 29020 29030 14529

III mode (119867120572= 119867120573= 1) 12223 12223 12204 51743 43716 43715 43623 21667

(119898 = 1 119899 = 1) 120579 = 0∘ (119898 = 1 119899 = 1) 120579 = 90∘


120572= 119867120573= 1) 12097 12133 14810 92169 12097 12133 14810 92169

II mode (119867120572= 119867120573= 1) 40165 40164 40137 20022 40165 40164 40137 20022

III mode (119867120572= 119867120573= 1) 12456 12456 12429 54312 12456 12456 12429 54312


1

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References

























































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











2



119886ℎ = 10


119886ℎ = radic10


+ (120572

11986211119867120573

119867120572119877120572

+ 120572

11986212

119877120573

+ 120572

11986255119867120573

119867120572119877120572

+ 120572

11986255

119877120573

)119882

+ (

11986255119867120573

119877120572

+

11986255119867120572

119877120573

)119880119911+ (120572119862

13119867120573+ 12057211986255119867120573)119882119911

+ (11986255119867120572119867120573)119880119911119911= 0

(minus12057212057311986266minus 120572120573119862

12)119880

+ (minus

11986244119867120572

1198671205731198772

120573

minus

11986244

119877120572119877120573

minus 120572211986266119867120573

119867120572

minus 120573

211986222119867120572

119867120573

+ 1205881198671205721198671205731205962

)119881

+ (120573

11986244119867120572

119867120573119877120573

+ 120573

11986244

119877120572

+ 120573

11986222119867120572

119867120573119877120573

+ 120573

11986212

119877120572

)119882

+ (

11986244119867120572

119877120573

+

11986244119867120573

119877120572

)119881119911+ (120573119862

44119867120572+ 12057311986223119867120572)119882119911

+ (11986244119867120572119867120573)119881119911119911= 0

(120572

11986255119867120573

119867120572119877120572

minus 120572

11986213

119877120573

+ 120572

11986211119867120573

119867120572119877120572

+ 120572

11986212

119877120573

)119880

+ (120573

11986244119867120572

119867120573119877120573

minus 120573

11986223

119877120572

+ 120573

11986222119867120572

119867120573119877120573

+ 120573

11986212

119877120572

)119881

+ (

11986213

119877120572119877120573

+

11986223

119877120572119877120573

minus

11986211119867120573

1198671205721198772

120572

minus

211986212

119877120572119877120573

minus

11986222119867120572

1198671205731198772

120573

minus120572211986255119867120573

119867120572

minus 120573

211986244119867120572

119867120573

+ 1205881198671205721198671205731205962

)119882


44119867120572minus 12057311986223119867120572)119881119911

+ (

11986233119867120573

119877120572

+

11986233119867120572

119877120573

)119882119911+ (11986233119867120572119867120573)119882119911119911= 0

(6)




ℎ119877120572= 012

119898 = 2 119898 = 4 119898 = 6 119898 = 8

119887119877120572= 2


119887119877120572= 1


119887119877120572= 05


119887119877120572= 025




1198601119880 + 119860

2119881 + 119860

3119882+ 119860

4119880119911+ 1198605119882119911+ 1198606119880119911119911= 0

1198607119880 + 119860

8119881 + 119860

9119882+119860

10119881119911+ 11986011119882119911+ 11986012119881119911119911= 0

11986013119880 + 119860

14119881 + 119860

15119882+119860

16119880119911

+ 11986017119881119911+ 11986018119882119911+ 11986019119882119911119911= 0

(7)




[

[

[

[

[

[

[

[

1198606

0 0 0 0 0

0 11986012

0 0 0 0

0 0 11986019

0 0 0

0 0 0 1198606

0 0

0 0 0 0 11986012

0

0 0 0 0 0 11986019

]

]

]

]

]

]

]

]

[

[

[

[

[

[

[

[

119880

119881

119882

1198801015840

1198811015840

1198821015840

]

]

]

]

]

]

]

]

1015840

=

[

[

[

[

[

[

[

[

0 0 0 1198606

0 0

0 0 0 0 11986012

0

0 0 0 0 0 11986019

minus1198601

minus1198602

minus1198603

minus1198604

0 minus1198605

minus1198607

minus1198608

minus1198609

0 minus11986010

minus11986011

minus11986013

minus11986014

minus11986015

minus11986016

minus11986017

minus11986018

]

]

]

]

]

]

]

]

[

[

[

[

[

[

[

[

119880

119881

119882

1198801015840

1198811015840

1198821015840

]

]

]

]

]

]

]

]

(8)


D120597U120597

= AU (9)




ℎ119877120572= 018

119898 = 2 119898 = 4 119898 = 6 119898 = 8

119887119877120572= 2


119887119877120572= 1


119887119877120572= 05


119887119877120572= 025


where 120597U120597 = U1015840 and U = [119880 119881 119882 1198801015840

1198811015840

1198821015840


DU1015840 = AU (10)

U1015840 = Dminus1AU (11)

U1015840 = AlowastU (12)


31198604119860911986010

11986013 11986014 and 119860


120572


11198602119860511986061198607

1198608 11986011 11986012 11986015 11986016 11986017 and 119860







2

ℎ2

+

Alowast3

3

ℎ3

+ sdot sdot sdot +

Alowast119873

119873

ℎ119873

(14)


T = I (15)








(119898 = 0 119899 = 1) 120579 = 0∘ (119898 = 0 119899 = 1) 120579 = 90∘



2


119886ℎ 100 50 10 5(119898 = 0 119899 = 1)








U (ℎ) = HU (0) (17)


120572= (1 +

119911119877120572) = (1 + ( minus ℎ2)119877

120572) and 119867

120573= (1 + 119911119877

120573) =








120572and 119867

120573







119895minus1119895by


119906119887

119895= 119906119905

119895minus1 V119887

119895= V119905119895minus1 119908

119887

119895= 119908119905

119895minus1

120590119887

119911119911119895= 120590119905

119911119911119895minus1 120590

119887

120572119911119895= 120590119905

120572119911119895minus1 120590

119887

120573119911119895= 120590119905

120573119911119895minus1

(18)




U119895(0) = T

119895minus1119895U119895minus1

(ℎ119895minus1) (19)


119895


119895U119895(0) (20)



2


exponential matrix

119886ℎ 100 50 10 5 100 50 10 5(119898 = 0 119899 = 1) 120579 = 0∘ (119898 = 0 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 0) 120579 = 0∘ (119898 = 1 119899 = 0) 120579 = 90∘


(119898 = 1 119899 = 1) 120579 = 0∘

(119898 = 1 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 2) 120579 = 0∘

(119898 = 1 119899 = 2) 120579 = 90∘


(119898 = 2 119899 = 1) 120579 = 0∘

(119898 = 2 119899 = 1) 120579 = 90∘


(119898 = 2 119899 = 2) 120579 = 0∘

(119898 = 2 119899 = 2) 120579 = 90∘


Ω0

+h2

120572 120573

minush2

zz

h

0

0

(a)

Ω0

00

+h2

120572 120573

minush2 j = 1

j = 2

j

j = M

zMhM

zz

h

0 0

z1 h1

(b)



U119872(ℎ119872)





T12

Alowastlowast1U1(0)




119898U1(0) (22)


119895



120572and


119895minus1119895



120572





2



119886ℎ 100 50 10 5(119898 = 0 119899 = 1)


(119898 = 1 119899 = 0)


(119898 = 1 119899 = 1)



(119898 = 2 119899 = 1)





120590119911119911= 120590120572119911= 120590120573119911= 0 for 119911 = minusℎ

2

+ℎ

2

or = 0 ℎ (23)

119908 = V = 0 120590120572120572= 0 for 120572 = 0 119886 (24)

119908 = 119906 = 0 120590120573120573= 0 for 120573 = 0 119887 (25)



120572


120590119905

119911119911= minus120572

11986213

119867119905

120572

119880119905

+

11986213

119867119905

120572119877120572

119882119905

minus 120573

11986223

119867119905

120573

119881119905

+

11986223

119867119905

120573119877120573

119882119905

+ 11986233119882119905

119911= 0

120590119905

120573119911= 120573

11986244

119867119905

120573

119882119905

+ 11986244119881119905

119911minus

11986244

119867119905

120573119877120573

119881119905

= 0

120590119905

120572119911= 120572

11986255

119867119905

120572

119882119905

+ 11986255119880119905

119911minus

11986255

119867119905

120572119877120572

119880119905

= 0

(26)



0of the shell and

119867119887

120572and119867119887


120590119887

119911119911= minus120572

11986213

119867119887

120572

119880119887

+

11986213

119867119887

120572119877120572

119882119887

minus 120573

11986223

119867119887

120573

119881119887

+

11986223

119867119887

120573119877120573

119882119887

+ 11986233119882119887

119911= 0

120590119887

120573119911= 120573

11986244

119867119887

120573

119882119887

+ 11986244119881119887

119911minus

11986244

119867119887

120573119877120573

119881119887

= 0

120590119887

120572119911= 120572

11986255

119867119887

120572

119882119887

+ 11986255119880119887

119911minus

11986255

119867119887

120572119877120572

119880119887

= 0

(27)



[

[

[

[

[

[

[

[

[

[

[

[

[

minus120572

11986213

119867119905

120572

minus120573

11986223

119867119905

120573

(

11986213

119867119905

120572119877120572

+

11986223

119867119905

120573119877120573

) 0 0 11986233

0 minus

11986244

119867119905

120573119877120573

120573

11986244

119867119905

120573

0 11986244

0

minus

11986255

119867119905

120572119877120572

0 120572

11986255

119867119905

120572

11986255

0 0

]

]

]

]

]

]

]

]

]

]

]

]

]

times

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

119880119872(ℎ119872)

119881119872(ℎ119872)

119882119872(ℎ119872)

1198801015840

119872(ℎ119872)

1198811015840

119872(ℎ119872)

1198821015840

119872(ℎ119872)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

=[

[

0

0

0

]

]

(28)





119886ℎ 100 50 10 5 100 50 10 5(119898 = 0 119899 = 1) 120579 = 0∘ (119898 = 0 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 0) 120579 = 0∘ (119898 = 1 119899 = 0) 120579 = 90∘


(119898 = 1 119899 = 1) 120579 = 0∘ (119898 = 1 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 2) 120579 = 0∘ (119898 = 1 119899 = 2) 120579 = 90∘


(119898 = 2 119899 = 1) 120579 = 0∘ (119898 = 2 119899 = 1) 120579 = 90∘


(119898 = 2 119899 = 2) 120579 = 0∘ (119898 = 2 119899 = 2) 120579 = 90∘




[

[

[

[

[

[

[

[

[

[

[

minus120572

11986213

119867119887

120572

minus120573

11986223

119867119887

120573

(

11986213

119867119887

120572119877120572

+

11986223

119867119887

120573119877120573

) 0 0 11986233

0 minus

11986244

119867119887

120573119877120573

120573

11986244

119867119887

120573

0 11986244

0

minus

11986255

119867119887

120572119877120572

0 120572

11986255

119867119887

120572

11986255

0 0

]

]

]

]

]

]

]

]

]

]

]

times

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

1198801(0)

1198811(0)

1198821(0)

1198801015840

1(0)

1198811015840

1(0)

1198821015840

1(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

=[

[

0

0

0

]

]

(29)


B119872(ℎ119872)U119872(ℎ119872) = 0 (30)

B1(0)U1(0) = 0 (31)


B (ℎ)U (ℎ) = 0

B (0)U (0) = 0(32)


B119872(ℎ119872)H119898U1(0) = 0

B1(0)U1(0) = 0

(33)


[

B119872(ℎ119872)H119898

B1(0)

] [U1(0)] = [0] (34)


[E] [U1(0)] = [0] (35)



2



119877120572ℎ 1000 100 10 5

(119898 = 0 119899 = 1)


(119898 = 1 119899 = 0)


(119898 = 1 119899 = 1)


(119898 = 1 119899 = 2)


(119898 = 2 119899 = 1)


(119898 = 2 119899 = 2)



1(0)H = H

119898 andB =

B119872(ℎ119872) = B






det [E] = 0 (36)








119897




bottom

U1120596119897(0) = [119880

1(0) 119881

1(0) 119882

1(0) 119880

1015840

1(0) 119881

1015840

1(0) 119882

1015840

1(0)]

119879

120596119897

(37)



119897


components 119880119895120596119897(119895) 119881119895120596119897(119895) and 119882

119895120596119897(119895) through the


4 Results


120572



2


120572and 119867

120573 For 119872 = 1







119877120572ℎ 1000 100 10 5 1000 100 10 5

(119898 = 0 119899 = 1) 120579 = 0∘

(119898 = 0 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 0) 120579 = 0∘

(119898 = 1 119899 = 0) 120579 = 90∘


(119898 = 1 119899 = 1) 120579 = 0∘

(119898 = 1 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 2) 120579 = 0∘

(119898 = 1 119899 = 2) 120579 = 90∘


(119898 = 2 119899 = 1) 120579 = 0∘

(119898 = 2 119899 = 1) 120579 = 90∘


(119898 = 2 119899 = 2) 120579 = 0∘

(119898 = 2 119899 = 2) 120579 = 90∘


a

z

y

x

hb

(a)

a

zy

x

hb

(b)

a

z

b

R120572

120573

120572

(c)

z

b

R120572

120573

120572

a = 2120587R120572

(d)

a

z

b

R120572

R120573

120573

120572

(e)




2


119877120572ℎ 1000 100 10 5















120572and119867









11198642= 11986411198643= 40

119866121198642= 119866131198642= 06 119866

231198642= 05 ]

12= ]13= ]23= 025











120573in










1= 13238GPa




2


exponential matrix

119877120572ℎ 1000 100 10 5 1000 100 10 5

(119898 = 0 119899 = 1) 120579 = 0∘ (119898 = 0 119899 = 1) 120579 = 90∘


(119898 = 2 119899 = 1) 120579 = 0∘ (119898 = 2 119899 = 1) 120579 = 90∘


(119898 = 2 119899 = 2) 120579 = 0∘ (119898 = 2 119899 = 2) 120579 = 90∘


(119898 = 2 119899 = 3) 120579 = 0∘ (119898 = 2 119899 = 3) 120579 = 90∘


11986612= 11986613= 56537GPa and 119866


components ]12

= ]13

= 024 and ]23


120596(1198862



lowast


= 119908|119908max| and 119911lowast














2



119877120572ℎ 1000 100 10 5




















∘ and 120579 =







119877120572ℎ 1000 100 10 5 1000 100 10 5

(119898 = 0 119899 = 1) 120579 = 0∘ (119898 = 0 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 0) 120579 = 0∘ (119898 = 1 119899 = 0) 120579 = 90∘


(119898 = 1 119899 = 1) 120579 = 0∘ (119898 = 1 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 2) 120579 = 0∘ (119898 = 1 119899 = 2) 120579 = 90∘


(119898 = 2 119899 = 1) 120579 = 0∘ (119898 = 2 119899 = 1) 120579 = 90∘


(119898 = 2 119899 = 2) 120579 = 0∘ (119898 = 2 119899 = 2) 120579 = 90∘





120573effects


120572= 1 (which means 119911119877




120572ℎ equals 1000




= 119867120573

= 1 (which means 119911119877120572

= 119911119877120573


120572= 119867120573= 1 is valid



120572= 119911119877


5 Conclusions



Table 15 Effects of119867120572= (1 + 119911119877



119877120572ℎ 1000 100 10 5


120572= 1) 10671 10671 10686 53486

II mode (119867120572= 1) 10683 10683 10688 53684

III mode (119867120572= 1) 18557 18556 18550 92651


120572= 1) 27747 27473 27029 25828

II mode (119867120572= 1) 20403 20403 20403 10201

III mode (119867120572= 1) 36353 36352 36303 18076


120572= 1) 22273 22535 40440 33738

II mode (119867120572= 1) 23645 23645 23668 11868

III mode (119867120572= 1) 40174 40173 40115 19965

Table 16 Effects of 119867120572= (1 + 119911119877

120572) and 119867

120573= (1 + 119911119877




119877120572ℎ 1000 100 10 5 1000 100 10 5

(119898 = 0 119899 = 1) 120579 = 0∘ (119898 = 0 119899 = 1) 120579 = 90∘


120572= 119867120573= 1) 23852 23852 23858 11939 10379 10409 12602 77410

II mode (119867120572= 119867120573= 1) 29020 29020 29030 14529 23852 23852 23856 11934

III mode (119867120572= 119867120573= 1) 43716 43715 43623 21667 12223 12223 12204 51743

(119898 = 1 119899 = 0) 120579 = 0∘ (119898 = 1 119899 = 0) 120579 = 90∘


120572= 119867120573= 1) 10379 10409 12602 77410 23852 23852 23858 11939

II mode (119867120572= 119867120573= 1) 23852 23852 23856 11934 29020 29020 29030 14529

III mode (119867120572= 119867120573= 1) 12223 12223 12204 51743 43716 43715 43623 21667

(119898 = 1 119899 = 1) 120579 = 0∘ (119898 = 1 119899 = 1) 120579 = 90∘


120572= 119867120573= 1) 12097 12133 14810 92169 12097 12133 14810 92169

II mode (119867120572= 119867120573= 1) 40165 40164 40137 20022 40165 40164 40137 20022

III mode (119867120572= 119867120573= 1) 12456 12456 12429 54312 12456 12456 12429 54312


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References

























































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












ℎ119877120572= 012

119898 = 2 119898 = 4 119898 = 6 119898 = 8

119887119877120572= 2


119887119877120572= 1


119887119877120572= 05


119887119877120572= 025




1198601119880 + 119860

2119881 + 119860

3119882+ 119860

4119880119911+ 1198605119882119911+ 1198606119880119911119911= 0

1198607119880 + 119860

8119881 + 119860

9119882+119860

10119881119911+ 11986011119882119911+ 11986012119881119911119911= 0

11986013119880 + 119860

14119881 + 119860

15119882+119860

16119880119911

+ 11986017119881119911+ 11986018119882119911+ 11986019119882119911119911= 0

(7)




[

[

[

[

[

[

[

[

1198606

0 0 0 0 0

0 11986012

0 0 0 0

0 0 11986019

0 0 0

0 0 0 1198606

0 0

0 0 0 0 11986012

0

0 0 0 0 0 11986019

]

]

]

]

]

]

]

]

[

[

[

[

[

[

[

[

119880

119881

119882

1198801015840

1198811015840

1198821015840

]

]

]

]

]

]

]

]

1015840

=

[

[

[

[

[

[

[

[

0 0 0 1198606

0 0

0 0 0 0 11986012

0

0 0 0 0 0 11986019

minus1198601

minus1198602

minus1198603

minus1198604

0 minus1198605

minus1198607

minus1198608

minus1198609

0 minus11986010

minus11986011

minus11986013

minus11986014

minus11986015

minus11986016

minus11986017

minus11986018

]

]

]

]

]

]

]

]

[

[

[

[

[

[

[

[

119880

119881

119882

1198801015840

1198811015840

1198821015840

]

]

]

]

]

]

]

]

(8)


D120597U120597

= AU (9)




ℎ119877120572= 018

119898 = 2 119898 = 4 119898 = 6 119898 = 8

119887119877120572= 2


119887119877120572= 1


119887119877120572= 05


119887119877120572= 025


where 120597U120597 = U1015840 and U = [119880 119881 119882 1198801015840

1198811015840

1198821015840


DU1015840 = AU (10)

U1015840 = Dminus1AU (11)

U1015840 = AlowastU (12)


31198604119860911986010

11986013 11986014 and 119860


120572


11198602119860511986061198607

1198608 11986011 11986012 11986015 11986016 11986017 and 119860







2

ℎ2

+

Alowast3

3

ℎ3

+ sdot sdot sdot +

Alowast119873

119873

ℎ119873

(14)


T = I (15)








(119898 = 0 119899 = 1) 120579 = 0∘ (119898 = 0 119899 = 1) 120579 = 90∘



2


119886ℎ 100 50 10 5(119898 = 0 119899 = 1)








U (ℎ) = HU (0) (17)


120572= (1 +

119911119877120572) = (1 + ( minus ℎ2)119877

120572) and 119867

120573= (1 + 119911119877

120573) =








120572and 119867

120573







119895minus1119895by


119906119887

119895= 119906119905

119895minus1 V119887

119895= V119905119895minus1 119908

119887

119895= 119908119905

119895minus1

120590119887

119911119911119895= 120590119905

119911119911119895minus1 120590

119887

120572119911119895= 120590119905

120572119911119895minus1 120590

119887

120573119911119895= 120590119905

120573119911119895minus1

(18)




U119895(0) = T

119895minus1119895U119895minus1

(ℎ119895minus1) (19)


119895


119895U119895(0) (20)



2


exponential matrix

119886ℎ 100 50 10 5 100 50 10 5(119898 = 0 119899 = 1) 120579 = 0∘ (119898 = 0 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 0) 120579 = 0∘ (119898 = 1 119899 = 0) 120579 = 90∘


(119898 = 1 119899 = 1) 120579 = 0∘

(119898 = 1 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 2) 120579 = 0∘

(119898 = 1 119899 = 2) 120579 = 90∘


(119898 = 2 119899 = 1) 120579 = 0∘

(119898 = 2 119899 = 1) 120579 = 90∘


(119898 = 2 119899 = 2) 120579 = 0∘

(119898 = 2 119899 = 2) 120579 = 90∘


Ω0

+h2

120572 120573

minush2

zz

h

0

0

(a)

Ω0

00

+h2

120572 120573

minush2 j = 1

j = 2

j

j = M

zMhM

zz

h

0 0

z1 h1

(b)



U119872(ℎ119872)





T12

Alowastlowast1U1(0)




119898U1(0) (22)


119895



120572and


119895minus1119895



120572





2



119886ℎ 100 50 10 5(119898 = 0 119899 = 1)


(119898 = 1 119899 = 0)


(119898 = 1 119899 = 1)



(119898 = 2 119899 = 1)





120590119911119911= 120590120572119911= 120590120573119911= 0 for 119911 = minusℎ

2

+ℎ

2

or = 0 ℎ (23)

119908 = V = 0 120590120572120572= 0 for 120572 = 0 119886 (24)

119908 = 119906 = 0 120590120573120573= 0 for 120573 = 0 119887 (25)



120572


120590119905

119911119911= minus120572

11986213

119867119905

120572

119880119905

+

11986213

119867119905

120572119877120572

119882119905

minus 120573

11986223

119867119905

120573

119881119905

+

11986223

119867119905

120573119877120573

119882119905

+ 11986233119882119905

119911= 0

120590119905

120573119911= 120573

11986244

119867119905

120573

119882119905

+ 11986244119881119905

119911minus

11986244

119867119905

120573119877120573

119881119905

= 0

120590119905

120572119911= 120572

11986255

119867119905

120572

119882119905

+ 11986255119880119905

119911minus

11986255

119867119905

120572119877120572

119880119905

= 0

(26)



0of the shell and

119867119887

120572and119867119887


120590119887

119911119911= minus120572

11986213

119867119887

120572

119880119887

+

11986213

119867119887

120572119877120572

119882119887

minus 120573

11986223

119867119887

120573

119881119887

+

11986223

119867119887

120573119877120573

119882119887

+ 11986233119882119887

119911= 0

120590119887

120573119911= 120573

11986244

119867119887

120573

119882119887

+ 11986244119881119887

119911minus

11986244

119867119887

120573119877120573

119881119887

= 0

120590119887

120572119911= 120572

11986255

119867119887

120572

119882119887

+ 11986255119880119887

119911minus

11986255

119867119887

120572119877120572

119880119887

= 0

(27)



[

[

[

[

[

[

[

[

[

[

[

[

[

minus120572

11986213

119867119905

120572

minus120573

11986223

119867119905

120573

(

11986213

119867119905

120572119877120572

+

11986223

119867119905

120573119877120573

) 0 0 11986233

0 minus

11986244

119867119905

120573119877120573

120573

11986244

119867119905

120573

0 11986244

0

minus

11986255

119867119905

120572119877120572

0 120572

11986255

119867119905

120572

11986255

0 0

]

]

]

]

]

]

]

]

]

]

]

]

]

times

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

119880119872(ℎ119872)

119881119872(ℎ119872)

119882119872(ℎ119872)

1198801015840

119872(ℎ119872)

1198811015840

119872(ℎ119872)

1198821015840

119872(ℎ119872)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

=[

[

0

0

0

]

]

(28)





119886ℎ 100 50 10 5 100 50 10 5(119898 = 0 119899 = 1) 120579 = 0∘ (119898 = 0 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 0) 120579 = 0∘ (119898 = 1 119899 = 0) 120579 = 90∘


(119898 = 1 119899 = 1) 120579 = 0∘ (119898 = 1 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 2) 120579 = 0∘ (119898 = 1 119899 = 2) 120579 = 90∘


(119898 = 2 119899 = 1) 120579 = 0∘ (119898 = 2 119899 = 1) 120579 = 90∘


(119898 = 2 119899 = 2) 120579 = 0∘ (119898 = 2 119899 = 2) 120579 = 90∘




[

[

[

[

[

[

[

[

[

[

[

minus120572

11986213

119867119887

120572

minus120573

11986223

119867119887

120573

(

11986213

119867119887

120572119877120572

+

11986223

119867119887

120573119877120573

) 0 0 11986233

0 minus

11986244

119867119887

120573119877120573

120573

11986244

119867119887

120573

0 11986244

0

minus

11986255

119867119887

120572119877120572

0 120572

11986255

119867119887

120572

11986255

0 0

]

]

]

]

]

]

]

]

]

]

]

times

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

1198801(0)

1198811(0)

1198821(0)

1198801015840

1(0)

1198811015840

1(0)

1198821015840

1(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

=[

[

0

0

0

]

]

(29)


B119872(ℎ119872)U119872(ℎ119872) = 0 (30)

B1(0)U1(0) = 0 (31)


B (ℎ)U (ℎ) = 0

B (0)U (0) = 0(32)


B119872(ℎ119872)H119898U1(0) = 0

B1(0)U1(0) = 0

(33)


[

B119872(ℎ119872)H119898

B1(0)

] [U1(0)] = [0] (34)


[E] [U1(0)] = [0] (35)



2



119877120572ℎ 1000 100 10 5

(119898 = 0 119899 = 1)


(119898 = 1 119899 = 0)


(119898 = 1 119899 = 1)


(119898 = 1 119899 = 2)


(119898 = 2 119899 = 1)


(119898 = 2 119899 = 2)



1(0)H = H

119898 andB =

B119872(ℎ119872) = B






det [E] = 0 (36)








119897




bottom

U1120596119897(0) = [119880

1(0) 119881

1(0) 119882

1(0) 119880

1015840

1(0) 119881

1015840

1(0) 119882

1015840

1(0)]

119879

120596119897

(37)



119897


components 119880119895120596119897(119895) 119881119895120596119897(119895) and 119882

119895120596119897(119895) through the


4 Results


120572



2


120572and 119867

120573 For 119872 = 1







119877120572ℎ 1000 100 10 5 1000 100 10 5

(119898 = 0 119899 = 1) 120579 = 0∘

(119898 = 0 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 0) 120579 = 0∘

(119898 = 1 119899 = 0) 120579 = 90∘


(119898 = 1 119899 = 1) 120579 = 0∘

(119898 = 1 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 2) 120579 = 0∘

(119898 = 1 119899 = 2) 120579 = 90∘


(119898 = 2 119899 = 1) 120579 = 0∘

(119898 = 2 119899 = 1) 120579 = 90∘


(119898 = 2 119899 = 2) 120579 = 0∘

(119898 = 2 119899 = 2) 120579 = 90∘


a

z

y

x

hb

(a)

a

zy

x

hb

(b)

a

z

b

R120572

120573

120572

(c)

z

b

R120572

120573

120572

a = 2120587R120572

(d)

a

z

b

R120572

R120573

120573

120572

(e)




2


119877120572ℎ 1000 100 10 5















120572and119867









11198642= 11986411198643= 40

119866121198642= 119866131198642= 06 119866

231198642= 05 ]

12= ]13= ]23= 025











120573in










1= 13238GPa




2


exponential matrix

119877120572ℎ 1000 100 10 5 1000 100 10 5

(119898 = 0 119899 = 1) 120579 = 0∘ (119898 = 0 119899 = 1) 120579 = 90∘


(119898 = 2 119899 = 1) 120579 = 0∘ (119898 = 2 119899 = 1) 120579 = 90∘


(119898 = 2 119899 = 2) 120579 = 0∘ (119898 = 2 119899 = 2) 120579 = 90∘


(119898 = 2 119899 = 3) 120579 = 0∘ (119898 = 2 119899 = 3) 120579 = 90∘


11986612= 11986613= 56537GPa and 119866


components ]12

= ]13

= 024 and ]23


120596(1198862



lowast


= 119908|119908max| and 119911lowast














2



119877120572ℎ 1000 100 10 5




















∘ and 120579 =







119877120572ℎ 1000 100 10 5 1000 100 10 5

(119898 = 0 119899 = 1) 120579 = 0∘ (119898 = 0 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 0) 120579 = 0∘ (119898 = 1 119899 = 0) 120579 = 90∘


(119898 = 1 119899 = 1) 120579 = 0∘ (119898 = 1 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 2) 120579 = 0∘ (119898 = 1 119899 = 2) 120579 = 90∘


(119898 = 2 119899 = 1) 120579 = 0∘ (119898 = 2 119899 = 1) 120579 = 90∘


(119898 = 2 119899 = 2) 120579 = 0∘ (119898 = 2 119899 = 2) 120579 = 90∘





120573effects


120572= 1 (which means 119911119877




120572ℎ equals 1000




= 119867120573

= 1 (which means 119911119877120572

= 119911119877120573


120572= 119867120573= 1 is valid



120572= 119911119877


5 Conclusions



Table 15 Effects of119867120572= (1 + 119911119877



119877120572ℎ 1000 100 10 5


120572= 1) 10671 10671 10686 53486

II mode (119867120572= 1) 10683 10683 10688 53684

III mode (119867120572= 1) 18557 18556 18550 92651


120572= 1) 27747 27473 27029 25828

II mode (119867120572= 1) 20403 20403 20403 10201

III mode (119867120572= 1) 36353 36352 36303 18076


120572= 1) 22273 22535 40440 33738

II mode (119867120572= 1) 23645 23645 23668 11868

III mode (119867120572= 1) 40174 40173 40115 19965

Table 16 Effects of 119867120572= (1 + 119911119877

120572) and 119867

120573= (1 + 119911119877




119877120572ℎ 1000 100 10 5 1000 100 10 5

(119898 = 0 119899 = 1) 120579 = 0∘ (119898 = 0 119899 = 1) 120579 = 90∘


120572= 119867120573= 1) 23852 23852 23858 11939 10379 10409 12602 77410

II mode (119867120572= 119867120573= 1) 29020 29020 29030 14529 23852 23852 23856 11934

III mode (119867120572= 119867120573= 1) 43716 43715 43623 21667 12223 12223 12204 51743

(119898 = 1 119899 = 0) 120579 = 0∘ (119898 = 1 119899 = 0) 120579 = 90∘


120572= 119867120573= 1) 10379 10409 12602 77410 23852 23852 23858 11939

II mode (119867120572= 119867120573= 1) 23852 23852 23856 11934 29020 29020 29030 14529

III mode (119867120572= 119867120573= 1) 12223 12223 12204 51743 43716 43715 43623 21667

(119898 = 1 119899 = 1) 120579 = 0∘ (119898 = 1 119899 = 1) 120579 = 90∘


120572= 119867120573= 1) 12097 12133 14810 92169 12097 12133 14810 92169

II mode (119867120572= 119867120573= 1) 40165 40164 40137 20022 40165 40164 40137 20022

III mode (119867120572= 119867120573= 1) 12456 12456 12429 54312 12456 12456 12429 54312


1

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References

























































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












ℎ119877120572= 018

119898 = 2 119898 = 4 119898 = 6 119898 = 8

119887119877120572= 2


119887119877120572= 1


119887119877120572= 05


119887119877120572= 025


where 120597U120597 = U1015840 and U = [119880 119881 119882 1198801015840

1198811015840

1198821015840


DU1015840 = AU (10)

U1015840 = Dminus1AU (11)

U1015840 = AlowastU (12)


31198604119860911986010

11986013 11986014 and 119860


120572


11198602119860511986061198607

1198608 11986011 11986012 11986015 11986016 11986017 and 119860







2

ℎ2

+

Alowast3

3

ℎ3

+ sdot sdot sdot +

Alowast119873

119873

ℎ119873

(14)


T = I (15)








(119898 = 0 119899 = 1) 120579 = 0∘ (119898 = 0 119899 = 1) 120579 = 90∘



2


119886ℎ 100 50 10 5(119898 = 0 119899 = 1)








U (ℎ) = HU (0) (17)


120572= (1 +

119911119877120572) = (1 + ( minus ℎ2)119877

120572) and 119867

120573= (1 + 119911119877

120573) =








120572and 119867

120573







119895minus1119895by


119906119887

119895= 119906119905

119895minus1 V119887

119895= V119905119895minus1 119908

119887

119895= 119908119905

119895minus1

120590119887

119911119911119895= 120590119905

119911119911119895minus1 120590

119887

120572119911119895= 120590119905

120572119911119895minus1 120590

119887

120573119911119895= 120590119905

120573119911119895minus1

(18)




U119895(0) = T

119895minus1119895U119895minus1

(ℎ119895minus1) (19)


119895


119895U119895(0) (20)



2


exponential matrix

119886ℎ 100 50 10 5 100 50 10 5(119898 = 0 119899 = 1) 120579 = 0∘ (119898 = 0 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 0) 120579 = 0∘ (119898 = 1 119899 = 0) 120579 = 90∘


(119898 = 1 119899 = 1) 120579 = 0∘

(119898 = 1 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 2) 120579 = 0∘

(119898 = 1 119899 = 2) 120579 = 90∘


(119898 = 2 119899 = 1) 120579 = 0∘

(119898 = 2 119899 = 1) 120579 = 90∘


(119898 = 2 119899 = 2) 120579 = 0∘

(119898 = 2 119899 = 2) 120579 = 90∘


Ω0

+h2

120572 120573

minush2

zz

h

0

0

(a)

Ω0

00

+h2

120572 120573

minush2 j = 1

j = 2

j

j = M

zMhM

zz

h

0 0

z1 h1

(b)



U119872(ℎ119872)





T12

Alowastlowast1U1(0)




119898U1(0) (22)


119895



120572and


119895minus1119895



120572





2



119886ℎ 100 50 10 5(119898 = 0 119899 = 1)


(119898 = 1 119899 = 0)


(119898 = 1 119899 = 1)



(119898 = 2 119899 = 1)





120590119911119911= 120590120572119911= 120590120573119911= 0 for 119911 = minusℎ

2

+ℎ

2

or = 0 ℎ (23)

119908 = V = 0 120590120572120572= 0 for 120572 = 0 119886 (24)

119908 = 119906 = 0 120590120573120573= 0 for 120573 = 0 119887 (25)



120572


120590119905

119911119911= minus120572

11986213

119867119905

120572

119880119905

+

11986213

119867119905

120572119877120572

119882119905

minus 120573

11986223

119867119905

120573

119881119905

+

11986223

119867119905

120573119877120573

119882119905

+ 11986233119882119905

119911= 0

120590119905

120573119911= 120573

11986244

119867119905

120573

119882119905

+ 11986244119881119905

119911minus

11986244

119867119905

120573119877120573

119881119905

= 0

120590119905

120572119911= 120572

11986255

119867119905

120572

119882119905

+ 11986255119880119905

119911minus

11986255

119867119905

120572119877120572

119880119905

= 0

(26)



0of the shell and

119867119887

120572and119867119887


120590119887

119911119911= minus120572

11986213

119867119887

120572

119880119887

+

11986213

119867119887

120572119877120572

119882119887

minus 120573

11986223

119867119887

120573

119881119887

+

11986223

119867119887

120573119877120573

119882119887

+ 11986233119882119887

119911= 0

120590119887

120573119911= 120573

11986244

119867119887

120573

119882119887

+ 11986244119881119887

119911minus

11986244

119867119887

120573119877120573

119881119887

= 0

120590119887

120572119911= 120572

11986255

119867119887

120572

119882119887

+ 11986255119880119887

119911minus

11986255

119867119887

120572119877120572

119880119887

= 0

(27)



[

[

[

[

[

[

[

[

[

[

[

[

[

minus120572

11986213

119867119905

120572

minus120573

11986223

119867119905

120573

(

11986213

119867119905

120572119877120572

+

11986223

119867119905

120573119877120573

) 0 0 11986233

0 minus

11986244

119867119905

120573119877120573

120573

11986244

119867119905

120573

0 11986244

0

minus

11986255

119867119905

120572119877120572

0 120572

11986255

119867119905

120572

11986255

0 0

]

]

]

]

]

]

]

]

]

]

]

]

]

times

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

119880119872(ℎ119872)

119881119872(ℎ119872)

119882119872(ℎ119872)

1198801015840

119872(ℎ119872)

1198811015840

119872(ℎ119872)

1198821015840

119872(ℎ119872)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

=[

[

0

0

0

]

]

(28)





119886ℎ 100 50 10 5 100 50 10 5(119898 = 0 119899 = 1) 120579 = 0∘ (119898 = 0 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 0) 120579 = 0∘ (119898 = 1 119899 = 0) 120579 = 90∘


(119898 = 1 119899 = 1) 120579 = 0∘ (119898 = 1 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 2) 120579 = 0∘ (119898 = 1 119899 = 2) 120579 = 90∘


(119898 = 2 119899 = 1) 120579 = 0∘ (119898 = 2 119899 = 1) 120579 = 90∘


(119898 = 2 119899 = 2) 120579 = 0∘ (119898 = 2 119899 = 2) 120579 = 90∘




[

[

[

[

[

[

[

[

[

[

[

minus120572

11986213

119867119887

120572

minus120573

11986223

119867119887

120573

(

11986213

119867119887

120572119877120572

+

11986223

119867119887

120573119877120573

) 0 0 11986233

0 minus

11986244

119867119887

120573119877120573

120573

11986244

119867119887

120573

0 11986244

0

minus

11986255

119867119887

120572119877120572

0 120572

11986255

119867119887

120572

11986255

0 0

]

]

]

]

]

]

]

]

]

]

]

times

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

1198801(0)

1198811(0)

1198821(0)

1198801015840

1(0)

1198811015840

1(0)

1198821015840

1(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

=[

[

0

0

0

]

]

(29)


B119872(ℎ119872)U119872(ℎ119872) = 0 (30)

B1(0)U1(0) = 0 (31)


B (ℎ)U (ℎ) = 0

B (0)U (0) = 0(32)


B119872(ℎ119872)H119898U1(0) = 0

B1(0)U1(0) = 0

(33)


[

B119872(ℎ119872)H119898

B1(0)

] [U1(0)] = [0] (34)


[E] [U1(0)] = [0] (35)



2



119877120572ℎ 1000 100 10 5

(119898 = 0 119899 = 1)


(119898 = 1 119899 = 0)


(119898 = 1 119899 = 1)


(119898 = 1 119899 = 2)


(119898 = 2 119899 = 1)


(119898 = 2 119899 = 2)



1(0)H = H

119898 andB =

B119872(ℎ119872) = B






det [E] = 0 (36)








119897




bottom

U1120596119897(0) = [119880

1(0) 119881

1(0) 119882

1(0) 119880

1015840

1(0) 119881

1015840

1(0) 119882

1015840

1(0)]

119879

120596119897

(37)



119897


components 119880119895120596119897(119895) 119881119895120596119897(119895) and 119882

119895120596119897(119895) through the


4 Results


120572



2


120572and 119867

120573 For 119872 = 1







119877120572ℎ 1000 100 10 5 1000 100 10 5

(119898 = 0 119899 = 1) 120579 = 0∘

(119898 = 0 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 0) 120579 = 0∘

(119898 = 1 119899 = 0) 120579 = 90∘


(119898 = 1 119899 = 1) 120579 = 0∘

(119898 = 1 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 2) 120579 = 0∘

(119898 = 1 119899 = 2) 120579 = 90∘


(119898 = 2 119899 = 1) 120579 = 0∘

(119898 = 2 119899 = 1) 120579 = 90∘


(119898 = 2 119899 = 2) 120579 = 0∘

(119898 = 2 119899 = 2) 120579 = 90∘


a

z

y

x

hb

(a)

a

zy

x

hb

(b)

a

z

b

R120572

120573

120572

(c)

z

b

R120572

120573

120572

a = 2120587R120572

(d)

a

z

b

R120572

R120573

120573

120572

(e)




2


119877120572ℎ 1000 100 10 5















120572and119867









11198642= 11986411198643= 40

119866121198642= 119866131198642= 06 119866

231198642= 05 ]

12= ]13= ]23= 025











120573in










1= 13238GPa




2


exponential matrix

119877120572ℎ 1000 100 10 5 1000 100 10 5

(119898 = 0 119899 = 1) 120579 = 0∘ (119898 = 0 119899 = 1) 120579 = 90∘


(119898 = 2 119899 = 1) 120579 = 0∘ (119898 = 2 119899 = 1) 120579 = 90∘


(119898 = 2 119899 = 2) 120579 = 0∘ (119898 = 2 119899 = 2) 120579 = 90∘


(119898 = 2 119899 = 3) 120579 = 0∘ (119898 = 2 119899 = 3) 120579 = 90∘


11986612= 11986613= 56537GPa and 119866


components ]12

= ]13

= 024 and ]23


120596(1198862



lowast


= 119908|119908max| and 119911lowast














2



119877120572ℎ 1000 100 10 5




















∘ and 120579 =







119877120572ℎ 1000 100 10 5 1000 100 10 5

(119898 = 0 119899 = 1) 120579 = 0∘ (119898 = 0 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 0) 120579 = 0∘ (119898 = 1 119899 = 0) 120579 = 90∘


(119898 = 1 119899 = 1) 120579 = 0∘ (119898 = 1 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 2) 120579 = 0∘ (119898 = 1 119899 = 2) 120579 = 90∘


(119898 = 2 119899 = 1) 120579 = 0∘ (119898 = 2 119899 = 1) 120579 = 90∘


(119898 = 2 119899 = 2) 120579 = 0∘ (119898 = 2 119899 = 2) 120579 = 90∘





120573effects


120572= 1 (which means 119911119877




120572ℎ equals 1000




= 119867120573

= 1 (which means 119911119877120572

= 119911119877120573


120572= 119867120573= 1 is valid



120572= 119911119877


5 Conclusions



Table 15 Effects of119867120572= (1 + 119911119877



119877120572ℎ 1000 100 10 5


120572= 1) 10671 10671 10686 53486

II mode (119867120572= 1) 10683 10683 10688 53684

III mode (119867120572= 1) 18557 18556 18550 92651


120572= 1) 27747 27473 27029 25828

II mode (119867120572= 1) 20403 20403 20403 10201

III mode (119867120572= 1) 36353 36352 36303 18076


120572= 1) 22273 22535 40440 33738

II mode (119867120572= 1) 23645 23645 23668 11868

III mode (119867120572= 1) 40174 40173 40115 19965

Table 16 Effects of 119867120572= (1 + 119911119877

120572) and 119867

120573= (1 + 119911119877




119877120572ℎ 1000 100 10 5 1000 100 10 5

(119898 = 0 119899 = 1) 120579 = 0∘ (119898 = 0 119899 = 1) 120579 = 90∘


120572= 119867120573= 1) 23852 23852 23858 11939 10379 10409 12602 77410

II mode (119867120572= 119867120573= 1) 29020 29020 29030 14529 23852 23852 23856 11934

III mode (119867120572= 119867120573= 1) 43716 43715 43623 21667 12223 12223 12204 51743

(119898 = 1 119899 = 0) 120579 = 0∘ (119898 = 1 119899 = 0) 120579 = 90∘


120572= 119867120573= 1) 10379 10409 12602 77410 23852 23852 23858 11939

II mode (119867120572= 119867120573= 1) 23852 23852 23856 11934 29020 29020 29030 14529

III mode (119867120572= 119867120573= 1) 12223 12223 12204 51743 43716 43715 43623 21667

(119898 = 1 119899 = 1) 120579 = 0∘ (119898 = 1 119899 = 1) 120579 = 90∘


120572= 119867120573= 1) 12097 12133 14810 92169 12097 12133 14810 92169

II mode (119867120572= 119867120573= 1) 40165 40164 40137 20022 40165 40164 40137 20022

III mode (119867120572= 119867120573= 1) 12456 12456 12429 54312 12456 12456 12429 54312


1

09

08

07

06

05

04

03

02

01



1

09

08

07

06

05

04

03

02

01



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1

09

08

07

06

05

04

03

02

01



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09

08

07

06

05

04

03

02

01



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09

08

07

06

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01



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09

08

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06

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09

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05

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zlowast

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06

05

04

03

02

01



zlowast

1

09

08

07

06

05

04

03

02

01



zlowast

1

09

08

07

06

05

04

03

02

01







1

09

08

07

06

05

04

03

02

01



zlowast

1

09

08

07

06

05

04

03

02

01



zlowast

1

09

08

07

06

05

04

03

02

01



zlowast

1

09

08

07

06

05

04

03

02

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zlowast

1

09

08

07

06

05

04

03

02

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zlowast

1

09

08

07

06

05

04

03

02

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zlowast

1

09

08

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06

05

04

03

02

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zlowast

1

09

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zlowast

1

09

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1

09

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1

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References

























































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














(119898 = 0 119899 = 1) 120579 = 0∘ (119898 = 0 119899 = 1) 120579 = 90∘



2


119886ℎ 100 50 10 5(119898 = 0 119899 = 1)








U (ℎ) = HU (0) (17)


120572= (1 +

119911119877120572) = (1 + ( minus ℎ2)119877

120572) and 119867

120573= (1 + 119911119877

120573) =








120572and 119867

120573







119895minus1119895by


119906119887

119895= 119906119905

119895minus1 V119887

119895= V119905119895minus1 119908

119887

119895= 119908119905

119895minus1

120590119887

119911119911119895= 120590119905

119911119911119895minus1 120590

119887

120572119911119895= 120590119905

120572119911119895minus1 120590

119887

120573119911119895= 120590119905

120573119911119895minus1

(18)




U119895(0) = T

119895minus1119895U119895minus1

(ℎ119895minus1) (19)


119895


119895U119895(0) (20)



2


exponential matrix

119886ℎ 100 50 10 5 100 50 10 5(119898 = 0 119899 = 1) 120579 = 0∘ (119898 = 0 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 0) 120579 = 0∘ (119898 = 1 119899 = 0) 120579 = 90∘


(119898 = 1 119899 = 1) 120579 = 0∘

(119898 = 1 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 2) 120579 = 0∘

(119898 = 1 119899 = 2) 120579 = 90∘


(119898 = 2 119899 = 1) 120579 = 0∘

(119898 = 2 119899 = 1) 120579 = 90∘


(119898 = 2 119899 = 2) 120579 = 0∘

(119898 = 2 119899 = 2) 120579 = 90∘


Ω0

+h2

120572 120573

minush2

zz

h

0

0

(a)

Ω0

00

+h2

120572 120573

minush2 j = 1

j = 2

j

j = M

zMhM

zz

h

0 0

z1 h1

(b)



U119872(ℎ119872)





T12

Alowastlowast1U1(0)




119898U1(0) (22)


119895



120572and


119895minus1119895



120572





2



119886ℎ 100 50 10 5(119898 = 0 119899 = 1)


(119898 = 1 119899 = 0)


(119898 = 1 119899 = 1)



(119898 = 2 119899 = 1)





120590119911119911= 120590120572119911= 120590120573119911= 0 for 119911 = minusℎ

2

+ℎ

2

or = 0 ℎ (23)

119908 = V = 0 120590120572120572= 0 for 120572 = 0 119886 (24)

119908 = 119906 = 0 120590120573120573= 0 for 120573 = 0 119887 (25)



120572


120590119905

119911119911= minus120572

11986213

119867119905

120572

119880119905

+

11986213

119867119905

120572119877120572

119882119905

minus 120573

11986223

119867119905

120573

119881119905

+

11986223

119867119905

120573119877120573

119882119905

+ 11986233119882119905

119911= 0

120590119905

120573119911= 120573

11986244

119867119905

120573

119882119905

+ 11986244119881119905

119911minus

11986244

119867119905

120573119877120573

119881119905

= 0

120590119905

120572119911= 120572

11986255

119867119905

120572

119882119905

+ 11986255119880119905

119911minus

11986255

119867119905

120572119877120572

119880119905

= 0

(26)



0of the shell and

119867119887

120572and119867119887


120590119887

119911119911= minus120572

11986213

119867119887

120572

119880119887

+

11986213

119867119887

120572119877120572

119882119887

minus 120573

11986223

119867119887

120573

119881119887

+

11986223

119867119887

120573119877120573

119882119887

+ 11986233119882119887

119911= 0

120590119887

120573119911= 120573

11986244

119867119887

120573

119882119887

+ 11986244119881119887

119911minus

11986244

119867119887

120573119877120573

119881119887

= 0

120590119887

120572119911= 120572

11986255

119867119887

120572

119882119887

+ 11986255119880119887

119911minus

11986255

119867119887

120572119877120572

119880119887

= 0

(27)



[

[

[

[

[

[

[

[

[

[

[

[

[

minus120572

11986213

119867119905

120572

minus120573

11986223

119867119905

120573

(

11986213

119867119905

120572119877120572

+

11986223

119867119905

120573119877120573

) 0 0 11986233

0 minus

11986244

119867119905

120573119877120573

120573

11986244

119867119905

120573

0 11986244

0

minus

11986255

119867119905

120572119877120572

0 120572

11986255

119867119905

120572

11986255

0 0

]

]

]

]

]

]

]

]

]

]

]

]

]

times

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

119880119872(ℎ119872)

119881119872(ℎ119872)

119882119872(ℎ119872)

1198801015840

119872(ℎ119872)

1198811015840

119872(ℎ119872)

1198821015840

119872(ℎ119872)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

=[

[

0

0

0

]

]

(28)





119886ℎ 100 50 10 5 100 50 10 5(119898 = 0 119899 = 1) 120579 = 0∘ (119898 = 0 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 0) 120579 = 0∘ (119898 = 1 119899 = 0) 120579 = 90∘


(119898 = 1 119899 = 1) 120579 = 0∘ (119898 = 1 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 2) 120579 = 0∘ (119898 = 1 119899 = 2) 120579 = 90∘


(119898 = 2 119899 = 1) 120579 = 0∘ (119898 = 2 119899 = 1) 120579 = 90∘


(119898 = 2 119899 = 2) 120579 = 0∘ (119898 = 2 119899 = 2) 120579 = 90∘




[

[

[

[

[

[

[

[

[

[

[

minus120572

11986213

119867119887

120572

minus120573

11986223

119867119887

120573

(

11986213

119867119887

120572119877120572

+

11986223

119867119887

120573119877120573

) 0 0 11986233

0 minus

11986244

119867119887

120573119877120573

120573

11986244

119867119887

120573

0 11986244

0

minus

11986255

119867119887

120572119877120572

0 120572

11986255

119867119887

120572

11986255

0 0

]

]

]

]

]

]

]

]

]

]

]

times

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

1198801(0)

1198811(0)

1198821(0)

1198801015840

1(0)

1198811015840

1(0)

1198821015840

1(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

=[

[

0

0

0

]

]

(29)


B119872(ℎ119872)U119872(ℎ119872) = 0 (30)

B1(0)U1(0) = 0 (31)


B (ℎ)U (ℎ) = 0

B (0)U (0) = 0(32)


B119872(ℎ119872)H119898U1(0) = 0

B1(0)U1(0) = 0

(33)


[

B119872(ℎ119872)H119898

B1(0)

] [U1(0)] = [0] (34)


[E] [U1(0)] = [0] (35)



2



119877120572ℎ 1000 100 10 5

(119898 = 0 119899 = 1)


(119898 = 1 119899 = 0)


(119898 = 1 119899 = 1)


(119898 = 1 119899 = 2)


(119898 = 2 119899 = 1)


(119898 = 2 119899 = 2)



1(0)H = H

119898 andB =

B119872(ℎ119872) = B






det [E] = 0 (36)








119897




bottom

U1120596119897(0) = [119880

1(0) 119881

1(0) 119882

1(0) 119880

1015840

1(0) 119881

1015840

1(0) 119882

1015840

1(0)]

119879

120596119897

(37)



119897


components 119880119895120596119897(119895) 119881119895120596119897(119895) and 119882

119895120596119897(119895) through the


4 Results


120572



2


120572and 119867

120573 For 119872 = 1







119877120572ℎ 1000 100 10 5 1000 100 10 5

(119898 = 0 119899 = 1) 120579 = 0∘

(119898 = 0 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 0) 120579 = 0∘

(119898 = 1 119899 = 0) 120579 = 90∘


(119898 = 1 119899 = 1) 120579 = 0∘

(119898 = 1 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 2) 120579 = 0∘

(119898 = 1 119899 = 2) 120579 = 90∘


(119898 = 2 119899 = 1) 120579 = 0∘

(119898 = 2 119899 = 1) 120579 = 90∘


(119898 = 2 119899 = 2) 120579 = 0∘

(119898 = 2 119899 = 2) 120579 = 90∘


a

z

y

x

hb

(a)

a

zy

x

hb

(b)

a

z

b

R120572

120573

120572

(c)

z

b

R120572

120573

120572

a = 2120587R120572

(d)

a

z

b

R120572

R120573

120573

120572

(e)




2


119877120572ℎ 1000 100 10 5















120572and119867









11198642= 11986411198643= 40

119866121198642= 119866131198642= 06 119866

231198642= 05 ]

12= ]13= ]23= 025











120573in










1= 13238GPa




2


exponential matrix

119877120572ℎ 1000 100 10 5 1000 100 10 5

(119898 = 0 119899 = 1) 120579 = 0∘ (119898 = 0 119899 = 1) 120579 = 90∘


(119898 = 2 119899 = 1) 120579 = 0∘ (119898 = 2 119899 = 1) 120579 = 90∘


(119898 = 2 119899 = 2) 120579 = 0∘ (119898 = 2 119899 = 2) 120579 = 90∘


(119898 = 2 119899 = 3) 120579 = 0∘ (119898 = 2 119899 = 3) 120579 = 90∘


11986612= 11986613= 56537GPa and 119866


components ]12

= ]13

= 024 and ]23


120596(1198862



lowast


= 119908|119908max| and 119911lowast














2



119877120572ℎ 1000 100 10 5




















∘ and 120579 =







119877120572ℎ 1000 100 10 5 1000 100 10 5

(119898 = 0 119899 = 1) 120579 = 0∘ (119898 = 0 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 0) 120579 = 0∘ (119898 = 1 119899 = 0) 120579 = 90∘


(119898 = 1 119899 = 1) 120579 = 0∘ (119898 = 1 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 2) 120579 = 0∘ (119898 = 1 119899 = 2) 120579 = 90∘


(119898 = 2 119899 = 1) 120579 = 0∘ (119898 = 2 119899 = 1) 120579 = 90∘


(119898 = 2 119899 = 2) 120579 = 0∘ (119898 = 2 119899 = 2) 120579 = 90∘





120573effects


120572= 1 (which means 119911119877




120572ℎ equals 1000




= 119867120573

= 1 (which means 119911119877120572

= 119911119877120573


120572= 119867120573= 1 is valid



120572= 119911119877


5 Conclusions



Table 15 Effects of119867120572= (1 + 119911119877



119877120572ℎ 1000 100 10 5


120572= 1) 10671 10671 10686 53486

II mode (119867120572= 1) 10683 10683 10688 53684

III mode (119867120572= 1) 18557 18556 18550 92651


120572= 1) 27747 27473 27029 25828

II mode (119867120572= 1) 20403 20403 20403 10201

III mode (119867120572= 1) 36353 36352 36303 18076


120572= 1) 22273 22535 40440 33738

II mode (119867120572= 1) 23645 23645 23668 11868

III mode (119867120572= 1) 40174 40173 40115 19965

Table 16 Effects of 119867120572= (1 + 119911119877

120572) and 119867

120573= (1 + 119911119877




119877120572ℎ 1000 100 10 5 1000 100 10 5

(119898 = 0 119899 = 1) 120579 = 0∘ (119898 = 0 119899 = 1) 120579 = 90∘


120572= 119867120573= 1) 23852 23852 23858 11939 10379 10409 12602 77410

II mode (119867120572= 119867120573= 1) 29020 29020 29030 14529 23852 23852 23856 11934

III mode (119867120572= 119867120573= 1) 43716 43715 43623 21667 12223 12223 12204 51743

(119898 = 1 119899 = 0) 120579 = 0∘ (119898 = 1 119899 = 0) 120579 = 90∘


120572= 119867120573= 1) 10379 10409 12602 77410 23852 23852 23858 11939

II mode (119867120572= 119867120573= 1) 23852 23852 23856 11934 29020 29020 29030 14529

III mode (119867120572= 119867120573= 1) 12223 12223 12204 51743 43716 43715 43623 21667

(119898 = 1 119899 = 1) 120579 = 0∘ (119898 = 1 119899 = 1) 120579 = 90∘


120572= 119867120573= 1) 12097 12133 14810 92169 12097 12133 14810 92169

II mode (119867120572= 119867120573= 1) 40165 40164 40137 20022 40165 40164 40137 20022

III mode (119867120572= 119867120573= 1) 12456 12456 12429 54312 12456 12456 12429 54312


1

09

08

07

06

05

04

03

02

01



1

09

08

07

06

05

04

03

02

01



zlowast

1

09

08

07

06

05

04

03

02

01



zlowast

1

09

08

07

06

05

04

03

02

01



zlowast

1

09

08

07

06

05

04

03

02

01



zlowast

1

09

08

07

06

05

04

03

02

01



zlowast

1

09

08

07

06

05

04

03

02

01



zlowast

1

09

08

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1

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1

09

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1

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References

























































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











2


exponential matrix

119886ℎ 100 50 10 5 100 50 10 5(119898 = 0 119899 = 1) 120579 = 0∘ (119898 = 0 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 0) 120579 = 0∘ (119898 = 1 119899 = 0) 120579 = 90∘


(119898 = 1 119899 = 1) 120579 = 0∘

(119898 = 1 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 2) 120579 = 0∘

(119898 = 1 119899 = 2) 120579 = 90∘


(119898 = 2 119899 = 1) 120579 = 0∘

(119898 = 2 119899 = 1) 120579 = 90∘


(119898 = 2 119899 = 2) 120579 = 0∘

(119898 = 2 119899 = 2) 120579 = 90∘


Ω0

+h2

120572 120573

minush2

zz

h

0

0

(a)

Ω0

00

+h2

120572 120573

minush2 j = 1

j = 2

j

j = M

zMhM

zz

h

0 0

z1 h1

(b)



U119872(ℎ119872)





T12

Alowastlowast1U1(0)




119898U1(0) (22)


119895



120572and


119895minus1119895



120572





2



119886ℎ 100 50 10 5(119898 = 0 119899 = 1)


(119898 = 1 119899 = 0)


(119898 = 1 119899 = 1)



(119898 = 2 119899 = 1)





120590119911119911= 120590120572119911= 120590120573119911= 0 for 119911 = minusℎ

2

+ℎ

2

or = 0 ℎ (23)

119908 = V = 0 120590120572120572= 0 for 120572 = 0 119886 (24)

119908 = 119906 = 0 120590120573120573= 0 for 120573 = 0 119887 (25)



120572


120590119905

119911119911= minus120572

11986213

119867119905

120572

119880119905

+

11986213

119867119905

120572119877120572

119882119905

minus 120573

11986223

119867119905

120573

119881119905

+

11986223

119867119905

120573119877120573

119882119905

+ 11986233119882119905

119911= 0

120590119905

120573119911= 120573

11986244

119867119905

120573

119882119905

+ 11986244119881119905

119911minus

11986244

119867119905

120573119877120573

119881119905

= 0

120590119905

120572119911= 120572

11986255

119867119905

120572

119882119905

+ 11986255119880119905

119911minus

11986255

119867119905

120572119877120572

119880119905

= 0

(26)



0of the shell and

119867119887

120572and119867119887


120590119887

119911119911= minus120572

11986213

119867119887

120572

119880119887

+

11986213

119867119887

120572119877120572

119882119887

minus 120573

11986223

119867119887

120573

119881119887

+

11986223

119867119887

120573119877120573

119882119887

+ 11986233119882119887

119911= 0

120590119887

120573119911= 120573

11986244

119867119887

120573

119882119887

+ 11986244119881119887

119911minus

11986244

119867119887

120573119877120573

119881119887

= 0

120590119887

120572119911= 120572

11986255

119867119887

120572

119882119887

+ 11986255119880119887

119911minus

11986255

119867119887

120572119877120572

119880119887

= 0

(27)



[

[

[

[

[

[

[

[

[

[

[

[

[

minus120572

11986213

119867119905

120572

minus120573

11986223

119867119905

120573

(

11986213

119867119905

120572119877120572

+

11986223

119867119905

120573119877120573

) 0 0 11986233

0 minus

11986244

119867119905

120573119877120573

120573

11986244

119867119905

120573

0 11986244

0

minus

11986255

119867119905

120572119877120572

0 120572

11986255

119867119905

120572

11986255

0 0

]

]

]

]

]

]

]

]

]

]

]

]

]

times

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

119880119872(ℎ119872)

119881119872(ℎ119872)

119882119872(ℎ119872)

1198801015840

119872(ℎ119872)

1198811015840

119872(ℎ119872)

1198821015840

119872(ℎ119872)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

=[

[

0

0

0

]

]

(28)





119886ℎ 100 50 10 5 100 50 10 5(119898 = 0 119899 = 1) 120579 = 0∘ (119898 = 0 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 0) 120579 = 0∘ (119898 = 1 119899 = 0) 120579 = 90∘


(119898 = 1 119899 = 1) 120579 = 0∘ (119898 = 1 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 2) 120579 = 0∘ (119898 = 1 119899 = 2) 120579 = 90∘


(119898 = 2 119899 = 1) 120579 = 0∘ (119898 = 2 119899 = 1) 120579 = 90∘


(119898 = 2 119899 = 2) 120579 = 0∘ (119898 = 2 119899 = 2) 120579 = 90∘




[

[

[

[

[

[

[

[

[

[

[

minus120572

11986213

119867119887

120572

minus120573

11986223

119867119887

120573

(

11986213

119867119887

120572119877120572

+

11986223

119867119887

120573119877120573

) 0 0 11986233

0 minus

11986244

119867119887

120573119877120573

120573

11986244

119867119887

120573

0 11986244

0

minus

11986255

119867119887

120572119877120572

0 120572

11986255

119867119887

120572

11986255

0 0

]

]

]

]

]

]

]

]

]

]

]

times

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

1198801(0)

1198811(0)

1198821(0)

1198801015840

1(0)

1198811015840

1(0)

1198821015840

1(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

=[

[

0

0

0

]

]

(29)


B119872(ℎ119872)U119872(ℎ119872) = 0 (30)

B1(0)U1(0) = 0 (31)


B (ℎ)U (ℎ) = 0

B (0)U (0) = 0(32)


B119872(ℎ119872)H119898U1(0) = 0

B1(0)U1(0) = 0

(33)


[

B119872(ℎ119872)H119898

B1(0)

] [U1(0)] = [0] (34)


[E] [U1(0)] = [0] (35)



2



119877120572ℎ 1000 100 10 5

(119898 = 0 119899 = 1)


(119898 = 1 119899 = 0)


(119898 = 1 119899 = 1)


(119898 = 1 119899 = 2)


(119898 = 2 119899 = 1)


(119898 = 2 119899 = 2)



1(0)H = H

119898 andB =

B119872(ℎ119872) = B






det [E] = 0 (36)








119897




bottom

U1120596119897(0) = [119880

1(0) 119881

1(0) 119882

1(0) 119880

1015840

1(0) 119881

1015840

1(0) 119882

1015840

1(0)]

119879

120596119897

(37)



119897


components 119880119895120596119897(119895) 119881119895120596119897(119895) and 119882

119895120596119897(119895) through the


4 Results


120572



2


120572and 119867

120573 For 119872 = 1







119877120572ℎ 1000 100 10 5 1000 100 10 5

(119898 = 0 119899 = 1) 120579 = 0∘

(119898 = 0 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 0) 120579 = 0∘

(119898 = 1 119899 = 0) 120579 = 90∘


(119898 = 1 119899 = 1) 120579 = 0∘

(119898 = 1 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 2) 120579 = 0∘

(119898 = 1 119899 = 2) 120579 = 90∘


(119898 = 2 119899 = 1) 120579 = 0∘

(119898 = 2 119899 = 1) 120579 = 90∘


(119898 = 2 119899 = 2) 120579 = 0∘

(119898 = 2 119899 = 2) 120579 = 90∘


a

z

y

x

hb

(a)

a

zy

x

hb

(b)

a

z

b

R120572

120573

120572

(c)

z

b

R120572

120573

120572

a = 2120587R120572

(d)

a

z

b

R120572

R120573

120573

120572

(e)




2


119877120572ℎ 1000 100 10 5















120572and119867









11198642= 11986411198643= 40

119866121198642= 119866131198642= 06 119866

231198642= 05 ]

12= ]13= ]23= 025











120573in










1= 13238GPa




2


exponential matrix

119877120572ℎ 1000 100 10 5 1000 100 10 5

(119898 = 0 119899 = 1) 120579 = 0∘ (119898 = 0 119899 = 1) 120579 = 90∘


(119898 = 2 119899 = 1) 120579 = 0∘ (119898 = 2 119899 = 1) 120579 = 90∘


(119898 = 2 119899 = 2) 120579 = 0∘ (119898 = 2 119899 = 2) 120579 = 90∘


(119898 = 2 119899 = 3) 120579 = 0∘ (119898 = 2 119899 = 3) 120579 = 90∘


11986612= 11986613= 56537GPa and 119866


components ]12

= ]13

= 024 and ]23


120596(1198862



lowast


= 119908|119908max| and 119911lowast














2



119877120572ℎ 1000 100 10 5




















∘ and 120579 =







119877120572ℎ 1000 100 10 5 1000 100 10 5

(119898 = 0 119899 = 1) 120579 = 0∘ (119898 = 0 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 0) 120579 = 0∘ (119898 = 1 119899 = 0) 120579 = 90∘


(119898 = 1 119899 = 1) 120579 = 0∘ (119898 = 1 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 2) 120579 = 0∘ (119898 = 1 119899 = 2) 120579 = 90∘


(119898 = 2 119899 = 1) 120579 = 0∘ (119898 = 2 119899 = 1) 120579 = 90∘


(119898 = 2 119899 = 2) 120579 = 0∘ (119898 = 2 119899 = 2) 120579 = 90∘





120573effects


120572= 1 (which means 119911119877




120572ℎ equals 1000




= 119867120573

= 1 (which means 119911119877120572

= 119911119877120573


120572= 119867120573= 1 is valid



120572= 119911119877


5 Conclusions



Table 15 Effects of119867120572= (1 + 119911119877



119877120572ℎ 1000 100 10 5


120572= 1) 10671 10671 10686 53486

II mode (119867120572= 1) 10683 10683 10688 53684

III mode (119867120572= 1) 18557 18556 18550 92651


120572= 1) 27747 27473 27029 25828

II mode (119867120572= 1) 20403 20403 20403 10201

III mode (119867120572= 1) 36353 36352 36303 18076


120572= 1) 22273 22535 40440 33738

II mode (119867120572= 1) 23645 23645 23668 11868

III mode (119867120572= 1) 40174 40173 40115 19965

Table 16 Effects of 119867120572= (1 + 119911119877

120572) and 119867

120573= (1 + 119911119877




119877120572ℎ 1000 100 10 5 1000 100 10 5

(119898 = 0 119899 = 1) 120579 = 0∘ (119898 = 0 119899 = 1) 120579 = 90∘


120572= 119867120573= 1) 23852 23852 23858 11939 10379 10409 12602 77410

II mode (119867120572= 119867120573= 1) 29020 29020 29030 14529 23852 23852 23856 11934

III mode (119867120572= 119867120573= 1) 43716 43715 43623 21667 12223 12223 12204 51743

(119898 = 1 119899 = 0) 120579 = 0∘ (119898 = 1 119899 = 0) 120579 = 90∘


120572= 119867120573= 1) 10379 10409 12602 77410 23852 23852 23858 11939

II mode (119867120572= 119867120573= 1) 23852 23852 23856 11934 29020 29020 29030 14529

III mode (119867120572= 119867120573= 1) 12223 12223 12204 51743 43716 43715 43623 21667

(119898 = 1 119899 = 1) 120579 = 0∘ (119898 = 1 119899 = 1) 120579 = 90∘


120572= 119867120573= 1) 12097 12133 14810 92169 12097 12133 14810 92169

II mode (119867120572= 119867120573= 1) 40165 40164 40137 20022 40165 40164 40137 20022

III mode (119867120572= 119867120573= 1) 12456 12456 12429 54312 12456 12456 12429 54312


1

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References

























































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











2



119886ℎ 100 50 10 5(119898 = 0 119899 = 1)


(119898 = 1 119899 = 0)


(119898 = 1 119899 = 1)



(119898 = 2 119899 = 1)





120590119911119911= 120590120572119911= 120590120573119911= 0 for 119911 = minusℎ

2

+ℎ

2

or = 0 ℎ (23)

119908 = V = 0 120590120572120572= 0 for 120572 = 0 119886 (24)

119908 = 119906 = 0 120590120573120573= 0 for 120573 = 0 119887 (25)



120572


120590119905

119911119911= minus120572

11986213

119867119905

120572

119880119905

+

11986213

119867119905

120572119877120572

119882119905

minus 120573

11986223

119867119905

120573

119881119905

+

11986223

119867119905

120573119877120573

119882119905

+ 11986233119882119905

119911= 0

120590119905

120573119911= 120573

11986244

119867119905

120573

119882119905

+ 11986244119881119905

119911minus

11986244

119867119905

120573119877120573

119881119905

= 0

120590119905

120572119911= 120572

11986255

119867119905

120572

119882119905

+ 11986255119880119905

119911minus

11986255

119867119905

120572119877120572

119880119905

= 0

(26)



0of the shell and

119867119887

120572and119867119887


120590119887

119911119911= minus120572

11986213

119867119887

120572

119880119887

+

11986213

119867119887

120572119877120572

119882119887

minus 120573

11986223

119867119887

120573

119881119887

+

11986223

119867119887

120573119877120573

119882119887

+ 11986233119882119887

119911= 0

120590119887

120573119911= 120573

11986244

119867119887

120573

119882119887

+ 11986244119881119887

119911minus

11986244

119867119887

120573119877120573

119881119887

= 0

120590119887

120572119911= 120572

11986255

119867119887

120572

119882119887

+ 11986255119880119887

119911minus

11986255

119867119887

120572119877120572

119880119887

= 0

(27)



[

[

[

[

[

[

[

[

[

[

[

[

[

minus120572

11986213

119867119905

120572

minus120573

11986223

119867119905

120573

(

11986213

119867119905

120572119877120572

+

11986223

119867119905

120573119877120573

) 0 0 11986233

0 minus

11986244

119867119905

120573119877120573

120573

11986244

119867119905

120573

0 11986244

0

minus

11986255

119867119905

120572119877120572

0 120572

11986255

119867119905

120572

11986255

0 0

]

]

]

]

]

]

]

]

]

]

]

]

]

times

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

119880119872(ℎ119872)

119881119872(ℎ119872)

119882119872(ℎ119872)

1198801015840

119872(ℎ119872)

1198811015840

119872(ℎ119872)

1198821015840

119872(ℎ119872)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

=[

[

0

0

0

]

]

(28)





119886ℎ 100 50 10 5 100 50 10 5(119898 = 0 119899 = 1) 120579 = 0∘ (119898 = 0 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 0) 120579 = 0∘ (119898 = 1 119899 = 0) 120579 = 90∘


(119898 = 1 119899 = 1) 120579 = 0∘ (119898 = 1 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 2) 120579 = 0∘ (119898 = 1 119899 = 2) 120579 = 90∘


(119898 = 2 119899 = 1) 120579 = 0∘ (119898 = 2 119899 = 1) 120579 = 90∘


(119898 = 2 119899 = 2) 120579 = 0∘ (119898 = 2 119899 = 2) 120579 = 90∘




[

[

[

[

[

[

[

[

[

[

[

minus120572

11986213

119867119887

120572

minus120573

11986223

119867119887

120573

(

11986213

119867119887

120572119877120572

+

11986223

119867119887

120573119877120573

) 0 0 11986233

0 minus

11986244

119867119887

120573119877120573

120573

11986244

119867119887

120573

0 11986244

0

minus

11986255

119867119887

120572119877120572

0 120572

11986255

119867119887

120572

11986255

0 0

]

]

]

]

]

]

]

]

]

]

]

times

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

1198801(0)

1198811(0)

1198821(0)

1198801015840

1(0)

1198811015840

1(0)

1198821015840

1(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

=[

[

0

0

0

]

]

(29)


B119872(ℎ119872)U119872(ℎ119872) = 0 (30)

B1(0)U1(0) = 0 (31)


B (ℎ)U (ℎ) = 0

B (0)U (0) = 0(32)


B119872(ℎ119872)H119898U1(0) = 0

B1(0)U1(0) = 0

(33)


[

B119872(ℎ119872)H119898

B1(0)

] [U1(0)] = [0] (34)


[E] [U1(0)] = [0] (35)



2



119877120572ℎ 1000 100 10 5

(119898 = 0 119899 = 1)


(119898 = 1 119899 = 0)


(119898 = 1 119899 = 1)


(119898 = 1 119899 = 2)


(119898 = 2 119899 = 1)


(119898 = 2 119899 = 2)



1(0)H = H

119898 andB =

B119872(ℎ119872) = B






det [E] = 0 (36)








119897




bottom

U1120596119897(0) = [119880

1(0) 119881

1(0) 119882

1(0) 119880

1015840

1(0) 119881

1015840

1(0) 119882

1015840

1(0)]

119879

120596119897

(37)



119897


components 119880119895120596119897(119895) 119881119895120596119897(119895) and 119882

119895120596119897(119895) through the


4 Results


120572



2


120572and 119867

120573 For 119872 = 1







119877120572ℎ 1000 100 10 5 1000 100 10 5

(119898 = 0 119899 = 1) 120579 = 0∘

(119898 = 0 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 0) 120579 = 0∘

(119898 = 1 119899 = 0) 120579 = 90∘


(119898 = 1 119899 = 1) 120579 = 0∘

(119898 = 1 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 2) 120579 = 0∘

(119898 = 1 119899 = 2) 120579 = 90∘


(119898 = 2 119899 = 1) 120579 = 0∘

(119898 = 2 119899 = 1) 120579 = 90∘


(119898 = 2 119899 = 2) 120579 = 0∘

(119898 = 2 119899 = 2) 120579 = 90∘


a

z

y

x

hb

(a)

a

zy

x

hb

(b)

a

z

b

R120572

120573

120572

(c)

z

b

R120572

120573

120572

a = 2120587R120572

(d)

a

z

b

R120572

R120573

120573

120572

(e)




2


119877120572ℎ 1000 100 10 5















120572and119867









11198642= 11986411198643= 40

119866121198642= 119866131198642= 06 119866

231198642= 05 ]

12= ]13= ]23= 025











120573in










1= 13238GPa




2


exponential matrix

119877120572ℎ 1000 100 10 5 1000 100 10 5

(119898 = 0 119899 = 1) 120579 = 0∘ (119898 = 0 119899 = 1) 120579 = 90∘


(119898 = 2 119899 = 1) 120579 = 0∘ (119898 = 2 119899 = 1) 120579 = 90∘


(119898 = 2 119899 = 2) 120579 = 0∘ (119898 = 2 119899 = 2) 120579 = 90∘


(119898 = 2 119899 = 3) 120579 = 0∘ (119898 = 2 119899 = 3) 120579 = 90∘


11986612= 11986613= 56537GPa and 119866


components ]12

= ]13

= 024 and ]23


120596(1198862



lowast


= 119908|119908max| and 119911lowast














2



119877120572ℎ 1000 100 10 5




















∘ and 120579 =







119877120572ℎ 1000 100 10 5 1000 100 10 5

(119898 = 0 119899 = 1) 120579 = 0∘ (119898 = 0 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 0) 120579 = 0∘ (119898 = 1 119899 = 0) 120579 = 90∘


(119898 = 1 119899 = 1) 120579 = 0∘ (119898 = 1 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 2) 120579 = 0∘ (119898 = 1 119899 = 2) 120579 = 90∘


(119898 = 2 119899 = 1) 120579 = 0∘ (119898 = 2 119899 = 1) 120579 = 90∘


(119898 = 2 119899 = 2) 120579 = 0∘ (119898 = 2 119899 = 2) 120579 = 90∘





120573effects


120572= 1 (which means 119911119877




120572ℎ equals 1000




= 119867120573

= 1 (which means 119911119877120572

= 119911119877120573


120572= 119867120573= 1 is valid



120572= 119911119877


5 Conclusions



Table 15 Effects of119867120572= (1 + 119911119877



119877120572ℎ 1000 100 10 5


120572= 1) 10671 10671 10686 53486

II mode (119867120572= 1) 10683 10683 10688 53684

III mode (119867120572= 1) 18557 18556 18550 92651


120572= 1) 27747 27473 27029 25828

II mode (119867120572= 1) 20403 20403 20403 10201

III mode (119867120572= 1) 36353 36352 36303 18076


120572= 1) 22273 22535 40440 33738

II mode (119867120572= 1) 23645 23645 23668 11868

III mode (119867120572= 1) 40174 40173 40115 19965

Table 16 Effects of 119867120572= (1 + 119911119877

120572) and 119867

120573= (1 + 119911119877




119877120572ℎ 1000 100 10 5 1000 100 10 5

(119898 = 0 119899 = 1) 120579 = 0∘ (119898 = 0 119899 = 1) 120579 = 90∘


120572= 119867120573= 1) 23852 23852 23858 11939 10379 10409 12602 77410

II mode (119867120572= 119867120573= 1) 29020 29020 29030 14529 23852 23852 23856 11934

III mode (119867120572= 119867120573= 1) 43716 43715 43623 21667 12223 12223 12204 51743

(119898 = 1 119899 = 0) 120579 = 0∘ (119898 = 1 119899 = 0) 120579 = 90∘


120572= 119867120573= 1) 10379 10409 12602 77410 23852 23852 23858 11939

II mode (119867120572= 119867120573= 1) 23852 23852 23856 11934 29020 29020 29030 14529

III mode (119867120572= 119867120573= 1) 12223 12223 12204 51743 43716 43715 43623 21667

(119898 = 1 119899 = 1) 120579 = 0∘ (119898 = 1 119899 = 1) 120579 = 90∘


120572= 119867120573= 1) 12097 12133 14810 92169 12097 12133 14810 92169

II mode (119867120572= 119867120573= 1) 40165 40164 40137 20022 40165 40164 40137 20022

III mode (119867120572= 119867120573= 1) 12456 12456 12429 54312 12456 12456 12429 54312


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References

























































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













119886ℎ 100 50 10 5 100 50 10 5(119898 = 0 119899 = 1) 120579 = 0∘ (119898 = 0 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 0) 120579 = 0∘ (119898 = 1 119899 = 0) 120579 = 90∘


(119898 = 1 119899 = 1) 120579 = 0∘ (119898 = 1 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 2) 120579 = 0∘ (119898 = 1 119899 = 2) 120579 = 90∘


(119898 = 2 119899 = 1) 120579 = 0∘ (119898 = 2 119899 = 1) 120579 = 90∘


(119898 = 2 119899 = 2) 120579 = 0∘ (119898 = 2 119899 = 2) 120579 = 90∘




[

[

[

[

[

[

[

[

[

[

[

minus120572

11986213

119867119887

120572

minus120573

11986223

119867119887

120573

(

11986213

119867119887

120572119877120572

+

11986223

119867119887

120573119877120573

) 0 0 11986233

0 minus

11986244

119867119887

120573119877120573

120573

11986244

119867119887

120573

0 11986244

0

minus

11986255

119867119887

120572119877120572

0 120572

11986255

119867119887

120572

11986255

0 0

]

]

]

]

]

]

]

]

]

]

]

times

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

1198801(0)

1198811(0)

1198821(0)

1198801015840

1(0)

1198811015840

1(0)

1198821015840

1(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

=[

[

0

0

0

]

]

(29)


B119872(ℎ119872)U119872(ℎ119872) = 0 (30)

B1(0)U1(0) = 0 (31)


B (ℎ)U (ℎ) = 0

B (0)U (0) = 0(32)


B119872(ℎ119872)H119898U1(0) = 0

B1(0)U1(0) = 0

(33)


[

B119872(ℎ119872)H119898

B1(0)

] [U1(0)] = [0] (34)


[E] [U1(0)] = [0] (35)



2



119877120572ℎ 1000 100 10 5

(119898 = 0 119899 = 1)


(119898 = 1 119899 = 0)


(119898 = 1 119899 = 1)


(119898 = 1 119899 = 2)


(119898 = 2 119899 = 1)


(119898 = 2 119899 = 2)



1(0)H = H

119898 andB =

B119872(ℎ119872) = B






det [E] = 0 (36)








119897




bottom

U1120596119897(0) = [119880

1(0) 119881

1(0) 119882

1(0) 119880

1015840

1(0) 119881

1015840

1(0) 119882

1015840

1(0)]

119879

120596119897

(37)



119897


components 119880119895120596119897(119895) 119881119895120596119897(119895) and 119882

119895120596119897(119895) through the


4 Results


120572



2


120572and 119867

120573 For 119872 = 1







119877120572ℎ 1000 100 10 5 1000 100 10 5

(119898 = 0 119899 = 1) 120579 = 0∘

(119898 = 0 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 0) 120579 = 0∘

(119898 = 1 119899 = 0) 120579 = 90∘


(119898 = 1 119899 = 1) 120579 = 0∘

(119898 = 1 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 2) 120579 = 0∘

(119898 = 1 119899 = 2) 120579 = 90∘


(119898 = 2 119899 = 1) 120579 = 0∘

(119898 = 2 119899 = 1) 120579 = 90∘


(119898 = 2 119899 = 2) 120579 = 0∘

(119898 = 2 119899 = 2) 120579 = 90∘


a

z

y

x

hb

(a)

a

zy

x

hb

(b)

a

z

b

R120572

120573

120572

(c)

z

b

R120572

120573

120572

a = 2120587R120572

(d)

a

z

b

R120572

R120573

120573

120572

(e)




2


119877120572ℎ 1000 100 10 5















120572and119867









11198642= 11986411198643= 40

119866121198642= 119866131198642= 06 119866

231198642= 05 ]

12= ]13= ]23= 025











120573in










1= 13238GPa




2


exponential matrix

119877120572ℎ 1000 100 10 5 1000 100 10 5

(119898 = 0 119899 = 1) 120579 = 0∘ (119898 = 0 119899 = 1) 120579 = 90∘


(119898 = 2 119899 = 1) 120579 = 0∘ (119898 = 2 119899 = 1) 120579 = 90∘


(119898 = 2 119899 = 2) 120579 = 0∘ (119898 = 2 119899 = 2) 120579 = 90∘


(119898 = 2 119899 = 3) 120579 = 0∘ (119898 = 2 119899 = 3) 120579 = 90∘


11986612= 11986613= 56537GPa and 119866


components ]12

= ]13

= 024 and ]23


120596(1198862



lowast


= 119908|119908max| and 119911lowast














2



119877120572ℎ 1000 100 10 5




















∘ and 120579 =







119877120572ℎ 1000 100 10 5 1000 100 10 5

(119898 = 0 119899 = 1) 120579 = 0∘ (119898 = 0 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 0) 120579 = 0∘ (119898 = 1 119899 = 0) 120579 = 90∘


(119898 = 1 119899 = 1) 120579 = 0∘ (119898 = 1 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 2) 120579 = 0∘ (119898 = 1 119899 = 2) 120579 = 90∘


(119898 = 2 119899 = 1) 120579 = 0∘ (119898 = 2 119899 = 1) 120579 = 90∘


(119898 = 2 119899 = 2) 120579 = 0∘ (119898 = 2 119899 = 2) 120579 = 90∘





120573effects


120572= 1 (which means 119911119877




120572ℎ equals 1000




= 119867120573

= 1 (which means 119911119877120572

= 119911119877120573


120572= 119867120573= 1 is valid



120572= 119911119877


5 Conclusions



Table 15 Effects of119867120572= (1 + 119911119877



119877120572ℎ 1000 100 10 5


120572= 1) 10671 10671 10686 53486

II mode (119867120572= 1) 10683 10683 10688 53684

III mode (119867120572= 1) 18557 18556 18550 92651


120572= 1) 27747 27473 27029 25828

II mode (119867120572= 1) 20403 20403 20403 10201

III mode (119867120572= 1) 36353 36352 36303 18076


120572= 1) 22273 22535 40440 33738

II mode (119867120572= 1) 23645 23645 23668 11868

III mode (119867120572= 1) 40174 40173 40115 19965

Table 16 Effects of 119867120572= (1 + 119911119877

120572) and 119867

120573= (1 + 119911119877




119877120572ℎ 1000 100 10 5 1000 100 10 5

(119898 = 0 119899 = 1) 120579 = 0∘ (119898 = 0 119899 = 1) 120579 = 90∘


120572= 119867120573= 1) 23852 23852 23858 11939 10379 10409 12602 77410

II mode (119867120572= 119867120573= 1) 29020 29020 29030 14529 23852 23852 23856 11934

III mode (119867120572= 119867120573= 1) 43716 43715 43623 21667 12223 12223 12204 51743

(119898 = 1 119899 = 0) 120579 = 0∘ (119898 = 1 119899 = 0) 120579 = 90∘


120572= 119867120573= 1) 10379 10409 12602 77410 23852 23852 23858 11939

II mode (119867120572= 119867120573= 1) 23852 23852 23856 11934 29020 29020 29030 14529

III mode (119867120572= 119867120573= 1) 12223 12223 12204 51743 43716 43715 43623 21667

(119898 = 1 119899 = 1) 120579 = 0∘ (119898 = 1 119899 = 1) 120579 = 90∘


120572= 119867120573= 1) 12097 12133 14810 92169 12097 12133 14810 92169

II mode (119867120572= 119867120573= 1) 40165 40164 40137 20022 40165 40164 40137 20022

III mode (119867120572= 119867120573= 1) 12456 12456 12429 54312 12456 12456 12429 54312


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References

























































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











2



119877120572ℎ 1000 100 10 5

(119898 = 0 119899 = 1)


(119898 = 1 119899 = 0)


(119898 = 1 119899 = 1)


(119898 = 1 119899 = 2)


(119898 = 2 119899 = 1)


(119898 = 2 119899 = 2)



1(0)H = H

119898 andB =

B119872(ℎ119872) = B






det [E] = 0 (36)








119897




bottom

U1120596119897(0) = [119880

1(0) 119881

1(0) 119882

1(0) 119880

1015840

1(0) 119881

1015840

1(0) 119882

1015840

1(0)]

119879

120596119897

(37)



119897


components 119880119895120596119897(119895) 119881119895120596119897(119895) and 119882

119895120596119897(119895) through the


4 Results


120572



2


120572and 119867

120573 For 119872 = 1







119877120572ℎ 1000 100 10 5 1000 100 10 5

(119898 = 0 119899 = 1) 120579 = 0∘

(119898 = 0 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 0) 120579 = 0∘

(119898 = 1 119899 = 0) 120579 = 90∘


(119898 = 1 119899 = 1) 120579 = 0∘

(119898 = 1 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 2) 120579 = 0∘

(119898 = 1 119899 = 2) 120579 = 90∘


(119898 = 2 119899 = 1) 120579 = 0∘

(119898 = 2 119899 = 1) 120579 = 90∘


(119898 = 2 119899 = 2) 120579 = 0∘

(119898 = 2 119899 = 2) 120579 = 90∘


a

z

y

x

hb

(a)

a

zy

x

hb

(b)

a

z

b

R120572

120573

120572

(c)

z

b

R120572

120573

120572

a = 2120587R120572

(d)

a

z

b

R120572

R120573

120573

120572

(e)




2


119877120572ℎ 1000 100 10 5















120572and119867









11198642= 11986411198643= 40

119866121198642= 119866131198642= 06 119866

231198642= 05 ]

12= ]13= ]23= 025











120573in










1= 13238GPa




2


exponential matrix

119877120572ℎ 1000 100 10 5 1000 100 10 5

(119898 = 0 119899 = 1) 120579 = 0∘ (119898 = 0 119899 = 1) 120579 = 90∘


(119898 = 2 119899 = 1) 120579 = 0∘ (119898 = 2 119899 = 1) 120579 = 90∘


(119898 = 2 119899 = 2) 120579 = 0∘ (119898 = 2 119899 = 2) 120579 = 90∘


(119898 = 2 119899 = 3) 120579 = 0∘ (119898 = 2 119899 = 3) 120579 = 90∘


11986612= 11986613= 56537GPa and 119866


components ]12

= ]13

= 024 and ]23


120596(1198862



lowast


= 119908|119908max| and 119911lowast














2



119877120572ℎ 1000 100 10 5




















∘ and 120579 =







119877120572ℎ 1000 100 10 5 1000 100 10 5

(119898 = 0 119899 = 1) 120579 = 0∘ (119898 = 0 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 0) 120579 = 0∘ (119898 = 1 119899 = 0) 120579 = 90∘


(119898 = 1 119899 = 1) 120579 = 0∘ (119898 = 1 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 2) 120579 = 0∘ (119898 = 1 119899 = 2) 120579 = 90∘


(119898 = 2 119899 = 1) 120579 = 0∘ (119898 = 2 119899 = 1) 120579 = 90∘


(119898 = 2 119899 = 2) 120579 = 0∘ (119898 = 2 119899 = 2) 120579 = 90∘





120573effects


120572= 1 (which means 119911119877




120572ℎ equals 1000




= 119867120573

= 1 (which means 119911119877120572

= 119911119877120573


120572= 119867120573= 1 is valid



120572= 119911119877


5 Conclusions



Table 15 Effects of119867120572= (1 + 119911119877



119877120572ℎ 1000 100 10 5


120572= 1) 10671 10671 10686 53486

II mode (119867120572= 1) 10683 10683 10688 53684

III mode (119867120572= 1) 18557 18556 18550 92651


120572= 1) 27747 27473 27029 25828

II mode (119867120572= 1) 20403 20403 20403 10201

III mode (119867120572= 1) 36353 36352 36303 18076


120572= 1) 22273 22535 40440 33738

II mode (119867120572= 1) 23645 23645 23668 11868

III mode (119867120572= 1) 40174 40173 40115 19965

Table 16 Effects of 119867120572= (1 + 119911119877

120572) and 119867

120573= (1 + 119911119877




119877120572ℎ 1000 100 10 5 1000 100 10 5

(119898 = 0 119899 = 1) 120579 = 0∘ (119898 = 0 119899 = 1) 120579 = 90∘


120572= 119867120573= 1) 23852 23852 23858 11939 10379 10409 12602 77410

II mode (119867120572= 119867120573= 1) 29020 29020 29030 14529 23852 23852 23856 11934

III mode (119867120572= 119867120573= 1) 43716 43715 43623 21667 12223 12223 12204 51743

(119898 = 1 119899 = 0) 120579 = 0∘ (119898 = 1 119899 = 0) 120579 = 90∘


120572= 119867120573= 1) 10379 10409 12602 77410 23852 23852 23858 11939

II mode (119867120572= 119867120573= 1) 23852 23852 23856 11934 29020 29020 29030 14529

III mode (119867120572= 119867120573= 1) 12223 12223 12204 51743 43716 43715 43623 21667

(119898 = 1 119899 = 1) 120579 = 0∘ (119898 = 1 119899 = 1) 120579 = 90∘


120572= 119867120573= 1) 12097 12133 14810 92169 12097 12133 14810 92169

II mode (119867120572= 119867120573= 1) 40165 40164 40137 20022 40165 40164 40137 20022

III mode (119867120572= 119867120573= 1) 12456 12456 12429 54312 12456 12456 12429 54312


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References

























































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













119877120572ℎ 1000 100 10 5 1000 100 10 5

(119898 = 0 119899 = 1) 120579 = 0∘

(119898 = 0 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 0) 120579 = 0∘

(119898 = 1 119899 = 0) 120579 = 90∘


(119898 = 1 119899 = 1) 120579 = 0∘

(119898 = 1 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 2) 120579 = 0∘

(119898 = 1 119899 = 2) 120579 = 90∘


(119898 = 2 119899 = 1) 120579 = 0∘

(119898 = 2 119899 = 1) 120579 = 90∘


(119898 = 2 119899 = 2) 120579 = 0∘

(119898 = 2 119899 = 2) 120579 = 90∘


a

z

y

x

hb

(a)

a

zy

x

hb

(b)

a

z

b

R120572

120573

120572

(c)

z

b

R120572

120573

120572

a = 2120587R120572

(d)

a

z

b

R120572

R120573

120573

120572

(e)




2


119877120572ℎ 1000 100 10 5















120572and119867









11198642= 11986411198643= 40

119866121198642= 119866131198642= 06 119866

231198642= 05 ]

12= ]13= ]23= 025











120573in










1= 13238GPa




2


exponential matrix

119877120572ℎ 1000 100 10 5 1000 100 10 5

(119898 = 0 119899 = 1) 120579 = 0∘ (119898 = 0 119899 = 1) 120579 = 90∘


(119898 = 2 119899 = 1) 120579 = 0∘ (119898 = 2 119899 = 1) 120579 = 90∘


(119898 = 2 119899 = 2) 120579 = 0∘ (119898 = 2 119899 = 2) 120579 = 90∘


(119898 = 2 119899 = 3) 120579 = 0∘ (119898 = 2 119899 = 3) 120579 = 90∘


11986612= 11986613= 56537GPa and 119866


components ]12

= ]13

= 024 and ]23


120596(1198862



lowast


= 119908|119908max| and 119911lowast














2



119877120572ℎ 1000 100 10 5




















∘ and 120579 =







119877120572ℎ 1000 100 10 5 1000 100 10 5

(119898 = 0 119899 = 1) 120579 = 0∘ (119898 = 0 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 0) 120579 = 0∘ (119898 = 1 119899 = 0) 120579 = 90∘


(119898 = 1 119899 = 1) 120579 = 0∘ (119898 = 1 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 2) 120579 = 0∘ (119898 = 1 119899 = 2) 120579 = 90∘


(119898 = 2 119899 = 1) 120579 = 0∘ (119898 = 2 119899 = 1) 120579 = 90∘


(119898 = 2 119899 = 2) 120579 = 0∘ (119898 = 2 119899 = 2) 120579 = 90∘





120573effects


120572= 1 (which means 119911119877




120572ℎ equals 1000




= 119867120573

= 1 (which means 119911119877120572

= 119911119877120573


120572= 119867120573= 1 is valid



120572= 119911119877


5 Conclusions



Table 15 Effects of119867120572= (1 + 119911119877



119877120572ℎ 1000 100 10 5


120572= 1) 10671 10671 10686 53486

II mode (119867120572= 1) 10683 10683 10688 53684

III mode (119867120572= 1) 18557 18556 18550 92651


120572= 1) 27747 27473 27029 25828

II mode (119867120572= 1) 20403 20403 20403 10201

III mode (119867120572= 1) 36353 36352 36303 18076


120572= 1) 22273 22535 40440 33738

II mode (119867120572= 1) 23645 23645 23668 11868

III mode (119867120572= 1) 40174 40173 40115 19965

Table 16 Effects of 119867120572= (1 + 119911119877

120572) and 119867

120573= (1 + 119911119877




119877120572ℎ 1000 100 10 5 1000 100 10 5

(119898 = 0 119899 = 1) 120579 = 0∘ (119898 = 0 119899 = 1) 120579 = 90∘


120572= 119867120573= 1) 23852 23852 23858 11939 10379 10409 12602 77410

II mode (119867120572= 119867120573= 1) 29020 29020 29030 14529 23852 23852 23856 11934

III mode (119867120572= 119867120573= 1) 43716 43715 43623 21667 12223 12223 12204 51743

(119898 = 1 119899 = 0) 120579 = 0∘ (119898 = 1 119899 = 0) 120579 = 90∘


120572= 119867120573= 1) 10379 10409 12602 77410 23852 23852 23858 11939

II mode (119867120572= 119867120573= 1) 23852 23852 23856 11934 29020 29020 29030 14529

III mode (119867120572= 119867120573= 1) 12223 12223 12204 51743 43716 43715 43623 21667

(119898 = 1 119899 = 1) 120579 = 0∘ (119898 = 1 119899 = 1) 120579 = 90∘


120572= 119867120573= 1) 12097 12133 14810 92169 12097 12133 14810 92169

II mode (119867120572= 119867120573= 1) 40165 40164 40137 20022 40165 40164 40137 20022

III mode (119867120572= 119867120573= 1) 12456 12456 12429 54312 12456 12456 12429 54312


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References

























































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











2


119877120572ℎ 1000 100 10 5















120572and119867









11198642= 11986411198643= 40

119866121198642= 119866131198642= 06 119866

231198642= 05 ]

12= ]13= ]23= 025











120573in










1= 13238GPa




2


exponential matrix

119877120572ℎ 1000 100 10 5 1000 100 10 5

(119898 = 0 119899 = 1) 120579 = 0∘ (119898 = 0 119899 = 1) 120579 = 90∘


(119898 = 2 119899 = 1) 120579 = 0∘ (119898 = 2 119899 = 1) 120579 = 90∘


(119898 = 2 119899 = 2) 120579 = 0∘ (119898 = 2 119899 = 2) 120579 = 90∘


(119898 = 2 119899 = 3) 120579 = 0∘ (119898 = 2 119899 = 3) 120579 = 90∘


11986612= 11986613= 56537GPa and 119866


components ]12

= ]13

= 024 and ]23


120596(1198862



lowast


= 119908|119908max| and 119911lowast














2



119877120572ℎ 1000 100 10 5




















∘ and 120579 =







119877120572ℎ 1000 100 10 5 1000 100 10 5

(119898 = 0 119899 = 1) 120579 = 0∘ (119898 = 0 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 0) 120579 = 0∘ (119898 = 1 119899 = 0) 120579 = 90∘


(119898 = 1 119899 = 1) 120579 = 0∘ (119898 = 1 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 2) 120579 = 0∘ (119898 = 1 119899 = 2) 120579 = 90∘


(119898 = 2 119899 = 1) 120579 = 0∘ (119898 = 2 119899 = 1) 120579 = 90∘


(119898 = 2 119899 = 2) 120579 = 0∘ (119898 = 2 119899 = 2) 120579 = 90∘





120573effects


120572= 1 (which means 119911119877




120572ℎ equals 1000




= 119867120573

= 1 (which means 119911119877120572

= 119911119877120573


120572= 119867120573= 1 is valid



120572= 119911119877


5 Conclusions



Table 15 Effects of119867120572= (1 + 119911119877



119877120572ℎ 1000 100 10 5


120572= 1) 10671 10671 10686 53486

II mode (119867120572= 1) 10683 10683 10688 53684

III mode (119867120572= 1) 18557 18556 18550 92651


120572= 1) 27747 27473 27029 25828

II mode (119867120572= 1) 20403 20403 20403 10201

III mode (119867120572= 1) 36353 36352 36303 18076


120572= 1) 22273 22535 40440 33738

II mode (119867120572= 1) 23645 23645 23668 11868

III mode (119867120572= 1) 40174 40173 40115 19965

Table 16 Effects of 119867120572= (1 + 119911119877

120572) and 119867

120573= (1 + 119911119877




119877120572ℎ 1000 100 10 5 1000 100 10 5

(119898 = 0 119899 = 1) 120579 = 0∘ (119898 = 0 119899 = 1) 120579 = 90∘


120572= 119867120573= 1) 23852 23852 23858 11939 10379 10409 12602 77410

II mode (119867120572= 119867120573= 1) 29020 29020 29030 14529 23852 23852 23856 11934

III mode (119867120572= 119867120573= 1) 43716 43715 43623 21667 12223 12223 12204 51743

(119898 = 1 119899 = 0) 120579 = 0∘ (119898 = 1 119899 = 0) 120579 = 90∘


120572= 119867120573= 1) 10379 10409 12602 77410 23852 23852 23858 11939

II mode (119867120572= 119867120573= 1) 23852 23852 23856 11934 29020 29020 29030 14529

III mode (119867120572= 119867120573= 1) 12223 12223 12204 51743 43716 43715 43623 21667

(119898 = 1 119899 = 1) 120579 = 0∘ (119898 = 1 119899 = 1) 120579 = 90∘


120572= 119867120573= 1) 12097 12133 14810 92169 12097 12133 14810 92169

II mode (119867120572= 119867120573= 1) 40165 40164 40137 20022 40165 40164 40137 20022

III mode (119867120572= 119867120573= 1) 12456 12456 12429 54312 12456 12456 12429 54312


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References

























































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











2


exponential matrix

119877120572ℎ 1000 100 10 5 1000 100 10 5

(119898 = 0 119899 = 1) 120579 = 0∘ (119898 = 0 119899 = 1) 120579 = 90∘


(119898 = 2 119899 = 1) 120579 = 0∘ (119898 = 2 119899 = 1) 120579 = 90∘


(119898 = 2 119899 = 2) 120579 = 0∘ (119898 = 2 119899 = 2) 120579 = 90∘


(119898 = 2 119899 = 3) 120579 = 0∘ (119898 = 2 119899 = 3) 120579 = 90∘


11986612= 11986613= 56537GPa and 119866


components ]12

= ]13

= 024 and ]23


120596(1198862



lowast


= 119908|119908max| and 119911lowast














2



119877120572ℎ 1000 100 10 5




















∘ and 120579 =







119877120572ℎ 1000 100 10 5 1000 100 10 5

(119898 = 0 119899 = 1) 120579 = 0∘ (119898 = 0 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 0) 120579 = 0∘ (119898 = 1 119899 = 0) 120579 = 90∘


(119898 = 1 119899 = 1) 120579 = 0∘ (119898 = 1 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 2) 120579 = 0∘ (119898 = 1 119899 = 2) 120579 = 90∘


(119898 = 2 119899 = 1) 120579 = 0∘ (119898 = 2 119899 = 1) 120579 = 90∘


(119898 = 2 119899 = 2) 120579 = 0∘ (119898 = 2 119899 = 2) 120579 = 90∘





120573effects


120572= 1 (which means 119911119877




120572ℎ equals 1000




= 119867120573

= 1 (which means 119911119877120572

= 119911119877120573


120572= 119867120573= 1 is valid



120572= 119911119877


5 Conclusions



Table 15 Effects of119867120572= (1 + 119911119877



119877120572ℎ 1000 100 10 5


120572= 1) 10671 10671 10686 53486

II mode (119867120572= 1) 10683 10683 10688 53684

III mode (119867120572= 1) 18557 18556 18550 92651


120572= 1) 27747 27473 27029 25828

II mode (119867120572= 1) 20403 20403 20403 10201

III mode (119867120572= 1) 36353 36352 36303 18076


120572= 1) 22273 22535 40440 33738

II mode (119867120572= 1) 23645 23645 23668 11868

III mode (119867120572= 1) 40174 40173 40115 19965

Table 16 Effects of 119867120572= (1 + 119911119877

120572) and 119867

120573= (1 + 119911119877




119877120572ℎ 1000 100 10 5 1000 100 10 5

(119898 = 0 119899 = 1) 120579 = 0∘ (119898 = 0 119899 = 1) 120579 = 90∘


120572= 119867120573= 1) 23852 23852 23858 11939 10379 10409 12602 77410

II mode (119867120572= 119867120573= 1) 29020 29020 29030 14529 23852 23852 23856 11934

III mode (119867120572= 119867120573= 1) 43716 43715 43623 21667 12223 12223 12204 51743

(119898 = 1 119899 = 0) 120579 = 0∘ (119898 = 1 119899 = 0) 120579 = 90∘


120572= 119867120573= 1) 10379 10409 12602 77410 23852 23852 23858 11939

II mode (119867120572= 119867120573= 1) 23852 23852 23856 11934 29020 29020 29030 14529

III mode (119867120572= 119867120573= 1) 12223 12223 12204 51743 43716 43715 43623 21667

(119898 = 1 119899 = 1) 120579 = 0∘ (119898 = 1 119899 = 1) 120579 = 90∘


120572= 119867120573= 1) 12097 12133 14810 92169 12097 12133 14810 92169

II mode (119867120572= 119867120573= 1) 40165 40164 40137 20022 40165 40164 40137 20022

III mode (119867120572= 119867120573= 1) 12456 12456 12429 54312 12456 12456 12429 54312


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References

























































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











2



119877120572ℎ 1000 100 10 5




















∘ and 120579 =







119877120572ℎ 1000 100 10 5 1000 100 10 5

(119898 = 0 119899 = 1) 120579 = 0∘ (119898 = 0 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 0) 120579 = 0∘ (119898 = 1 119899 = 0) 120579 = 90∘


(119898 = 1 119899 = 1) 120579 = 0∘ (119898 = 1 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 2) 120579 = 0∘ (119898 = 1 119899 = 2) 120579 = 90∘


(119898 = 2 119899 = 1) 120579 = 0∘ (119898 = 2 119899 = 1) 120579 = 90∘


(119898 = 2 119899 = 2) 120579 = 0∘ (119898 = 2 119899 = 2) 120579 = 90∘





120573effects


120572= 1 (which means 119911119877




120572ℎ equals 1000




= 119867120573

= 1 (which means 119911119877120572

= 119911119877120573


120572= 119867120573= 1 is valid



120572= 119911119877


5 Conclusions



Table 15 Effects of119867120572= (1 + 119911119877



119877120572ℎ 1000 100 10 5


120572= 1) 10671 10671 10686 53486

II mode (119867120572= 1) 10683 10683 10688 53684

III mode (119867120572= 1) 18557 18556 18550 92651


120572= 1) 27747 27473 27029 25828

II mode (119867120572= 1) 20403 20403 20403 10201

III mode (119867120572= 1) 36353 36352 36303 18076


120572= 1) 22273 22535 40440 33738

II mode (119867120572= 1) 23645 23645 23668 11868

III mode (119867120572= 1) 40174 40173 40115 19965

Table 16 Effects of 119867120572= (1 + 119911119877

120572) and 119867

120573= (1 + 119911119877




119877120572ℎ 1000 100 10 5 1000 100 10 5

(119898 = 0 119899 = 1) 120579 = 0∘ (119898 = 0 119899 = 1) 120579 = 90∘


120572= 119867120573= 1) 23852 23852 23858 11939 10379 10409 12602 77410

II mode (119867120572= 119867120573= 1) 29020 29020 29030 14529 23852 23852 23856 11934

III mode (119867120572= 119867120573= 1) 43716 43715 43623 21667 12223 12223 12204 51743

(119898 = 1 119899 = 0) 120579 = 0∘ (119898 = 1 119899 = 0) 120579 = 90∘


120572= 119867120573= 1) 10379 10409 12602 77410 23852 23852 23858 11939

II mode (119867120572= 119867120573= 1) 23852 23852 23856 11934 29020 29020 29030 14529

III mode (119867120572= 119867120573= 1) 12223 12223 12204 51743 43716 43715 43623 21667

(119898 = 1 119899 = 1) 120579 = 0∘ (119898 = 1 119899 = 1) 120579 = 90∘


120572= 119867120573= 1) 12097 12133 14810 92169 12097 12133 14810 92169

II mode (119867120572= 119867120573= 1) 40165 40164 40137 20022 40165 40164 40137 20022

III mode (119867120572= 119867120573= 1) 12456 12456 12429 54312 12456 12456 12429 54312


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References

























































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













119877120572ℎ 1000 100 10 5 1000 100 10 5

(119898 = 0 119899 = 1) 120579 = 0∘ (119898 = 0 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 0) 120579 = 0∘ (119898 = 1 119899 = 0) 120579 = 90∘


(119898 = 1 119899 = 1) 120579 = 0∘ (119898 = 1 119899 = 1) 120579 = 90∘


(119898 = 1 119899 = 2) 120579 = 0∘ (119898 = 1 119899 = 2) 120579 = 90∘


(119898 = 2 119899 = 1) 120579 = 0∘ (119898 = 2 119899 = 1) 120579 = 90∘


(119898 = 2 119899 = 2) 120579 = 0∘ (119898 = 2 119899 = 2) 120579 = 90∘





120573effects


120572= 1 (which means 119911119877




120572ℎ equals 1000




= 119867120573

= 1 (which means 119911119877120572

= 119911119877120573


120572= 119867120573= 1 is valid



120572= 119911119877


5 Conclusions



Table 15 Effects of119867120572= (1 + 119911119877



119877120572ℎ 1000 100 10 5


120572= 1) 10671 10671 10686 53486

II mode (119867120572= 1) 10683 10683 10688 53684

III mode (119867120572= 1) 18557 18556 18550 92651


120572= 1) 27747 27473 27029 25828

II mode (119867120572= 1) 20403 20403 20403 10201

III mode (119867120572= 1) 36353 36352 36303 18076


120572= 1) 22273 22535 40440 33738

II mode (119867120572= 1) 23645 23645 23668 11868

III mode (119867120572= 1) 40174 40173 40115 19965

Table 16 Effects of 119867120572= (1 + 119911119877

120572) and 119867

120573= (1 + 119911119877




119877120572ℎ 1000 100 10 5 1000 100 10 5

(119898 = 0 119899 = 1) 120579 = 0∘ (119898 = 0 119899 = 1) 120579 = 90∘


120572= 119867120573= 1) 23852 23852 23858 11939 10379 10409 12602 77410

II mode (119867120572= 119867120573= 1) 29020 29020 29030 14529 23852 23852 23856 11934

III mode (119867120572= 119867120573= 1) 43716 43715 43623 21667 12223 12223 12204 51743

(119898 = 1 119899 = 0) 120579 = 0∘ (119898 = 1 119899 = 0) 120579 = 90∘


120572= 119867120573= 1) 10379 10409 12602 77410 23852 23852 23858 11939

II mode (119867120572= 119867120573= 1) 23852 23852 23856 11934 29020 29020 29030 14529

III mode (119867120572= 119867120573= 1) 12223 12223 12204 51743 43716 43715 43623 21667

(119898 = 1 119899 = 1) 120579 = 0∘ (119898 = 1 119899 = 1) 120579 = 90∘


120572= 119867120573= 1) 12097 12133 14810 92169 12097 12133 14810 92169

II mode (119867120572= 119867120573= 1) 40165 40164 40137 20022 40165 40164 40137 20022

III mode (119867120572= 119867120573= 1) 12456 12456 12429 54312 12456 12456 12429 54312


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References

























































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Table 15 Effects of119867120572= (1 + 119911119877



119877120572ℎ 1000 100 10 5


120572= 1) 10671 10671 10686 53486

II mode (119867120572= 1) 10683 10683 10688 53684

III mode (119867120572= 1) 18557 18556 18550 92651


120572= 1) 27747 27473 27029 25828

II mode (119867120572= 1) 20403 20403 20403 10201

III mode (119867120572= 1) 36353 36352 36303 18076


120572= 1) 22273 22535 40440 33738

II mode (119867120572= 1) 23645 23645 23668 11868

III mode (119867120572= 1) 40174 40173 40115 19965

Table 16 Effects of 119867120572= (1 + 119911119877

120572) and 119867

120573= (1 + 119911119877




119877120572ℎ 1000 100 10 5 1000 100 10 5

(119898 = 0 119899 = 1) 120579 = 0∘ (119898 = 0 119899 = 1) 120579 = 90∘


120572= 119867120573= 1) 23852 23852 23858 11939 10379 10409 12602 77410

II mode (119867120572= 119867120573= 1) 29020 29020 29030 14529 23852 23852 23856 11934

III mode (119867120572= 119867120573= 1) 43716 43715 43623 21667 12223 12223 12204 51743

(119898 = 1 119899 = 0) 120579 = 0∘ (119898 = 1 119899 = 0) 120579 = 90∘


120572= 119867120573= 1) 10379 10409 12602 77410 23852 23852 23858 11939

II mode (119867120572= 119867120573= 1) 23852 23852 23856 11934 29020 29020 29030 14529

III mode (119867120572= 119867120573= 1) 12223 12223 12204 51743 43716 43715 43623 21667

(119898 = 1 119899 = 1) 120579 = 0∘ (119898 = 1 119899 = 1) 120579 = 90∘


120572= 119867120573= 1) 12097 12133 14810 92169 12097 12133 14810 92169

II mode (119867120572= 119867120573= 1) 40165 40164 40137 20022 40165 40164 40137 20022

III mode (119867120572= 119867120573= 1) 12456 12456 12429 54312 12456 12456 12429 54312


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References

























































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










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RoboticsJournal of





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RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










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References

























































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










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References

























































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










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RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



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Volume 2014


SensorsJournal of





Propagation










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RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










1

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References

























































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation






















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References

























































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













References

























































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation



































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









research article three-dimensional exact free vibration analysis...

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