researcharticle free vibration analysis of rings via wave...

Research ArticleFree Vibration Analysis of Rings via Wave Approach

Wang Zhipeng LiuWei Yuan Yunbo Shuai Zhijun Guo Yibin andWang Donghua

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

Correspondence should be addressed to Shuai Zhijun shuaizhijunhrbeueducn

Received 1 February 2018 Revised 28 March 2018 Accepted 2 April 2018 Published 14 May 2018

Academic Editor Tai Thai

Copyright copy 2018 Wang Zhipeng et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Free vibration of rings is presented via wave approach theoretically Firstly based on the solutions of out-of-plane vibrationpropagation reflection and coordinationmatrices are derived for the case of a fixed boundary at inner surface and a free boundaryat outer surface Then assembling these matrices characteristic equation of natural frequency is obtained Wave approach isemployed to study the free vibration of these ring structures Natural frequencies calculated by wave approach are compared withthose obtained by classical method and Finite Element Method (FEM) Afterwards natural frequencies of four type boundaries arecalculated Transverse vibration transmissibility of rings propagating from outer to inner and from inner to outer is investigatedFinally the effects of structural and material parameters on free vibration are discussed in detail

1 Introduction

Noise and vibration problem of rings has been a hotspotthat makes it attract lots of attention because of the wideapplications such as gear transmission system and aircraftstructures Since most of components within these structurescan be regarded as a simple model of ring structure dynamicproperties of these structures in recent years have been thesubject of present studies Based on the StodolandashVianellomethod Gutierrez et al [1] considered the free vibrationof annular membranes with continuously variable densityby using Rayleigh method differential quadrature methodand finite elements simulation By adopting the dynamicstiffness method Jabareen and Eisenberger [2] investigatedthe natural characteristics of the nonhomogeneous annularstructure Wang [3] discussed the natural frequency withthe boundaries of fixed free and simply supported Andthe researches indicated clearly that complex modes wouldswitch correspondingly with the radius increasing graduallyBased on the first-order shear deformation theory Roshanand Rashmi [4] analyzed the free vibration of axisymmetricsandwich circular plate with relatively stiff thickness Oveisiand Shakeri [5] constructed a sandwich composite circularplate containing piezoelectric material finding that the feed-back control gain had an effective control for suppressing the

transverse vibration Hosseini-Hashemi et al [6] presentedthe free vibration of functionally graded circular plate withstepped thickness via Mindlin plate theory By employ-ing harmonic differential quadrature method for obtainingnumerical solution of circular plate Civalek and Uelker [7]made a further analysis for the behavior of the boundaryconditions of fixed and simply supported Moreover theresults obtained by finite difference method were comparedwith those calculated by harmonic differential quadraturemethod whose feasibility is verified by Bakhshi Khaniki andHosseini-Hashemi [8] Liu et al [9] construct the compositethin annular plate by wave approach However compositestructures with multilayer are not studied Different bound-aries and parameter effects are also not analyzed

Additionally in terms of the vibration of structures thereis an alternative method called ldquowave approachrdquo which isvery efficient and widely used for calculating the naturalfrequency of structures such as beams plates rings andperiodic structure by describing waves in the matrix formFor example as early as 1984 Mace [10] applied waveapproach to analyze the wave behaviors in Euler beam Bydividing the waves into propagation and attenuation matri-ces he derived the reflection matrices under three boundaryconditions which established a theoretical foundation forwave approach Adopting wave approach Mei [11] analyzed

HindawiShock and VibrationVolume 2018 Article ID 6181204 14 pageshttpsdoiorg10115520186181204

2 Shock and Vibration

the flexural vibrationwith addedmass for Timoshenko beamand the influence of lumped mass on natural frequency isalso discussed in detail Kang et al [12] divided the realand imaginary parts of wave solutions of curved beam intofour cases and they calculated the natural frequencies bycombining propagation transmission and reflection matri-ces Lee et al [13 14] considered the power flow when wavepropagated in curved beam Furthermore they applied theFlugge theory to analyze the free vibration of a single curvedbeam and their result was compared with Kang et al whichverified the correctness of the numerical results Huang et al[15] investigated the free vibration of planar rotating ringsThe effect of cross section on natural frequency was alsodiscussed Bahrami and Teimourian [16] studied the freevibration of composite plates consisting of two layers andthey also made a comparison between classical results andwave propagation results Tan and Kang [17] concentratedon the free vibration of rotating Timoshenko shaft with axialforce and discussed the effect of continuous condition andcross section on natural frequencies From the wave pointof view Bahrami and Teimourian [18] analyzed the freevibration of nanobeams for the first time Ilkhani et al [19]studied the free vibration of thin rectangular plate It shouldbe noted that the above scholars have done lots of studies forfree vibration of structures by usingwave approach while fewreports for the analysis of natural frequency for transversevibration of rings can be found In fact it is well knownthat the nature of vibration is the propagation of wavesAnalyzing free vibration in terms of wave propagation andattenuating can have a better understanding for usMoreoverone advantage of using wave approach to analyze the freevibration is its conciseness of matrices that makes the naturalfrequencies be calculated easily Wave approach is a strongtool for studying the behavior of wave transmission andreflection in waveguides providing a practical engineeringapplication such as filters

The emphasis of this paper is focused on the free vibrationof rings This paper is organized into five parts Section 1is introduction In Section 2 propagation coordinationand reflection matrices are deduced in forms of matrix Inaddition the characteristic equation of natural frequencyis obtained using classical method and wave approach InSection 3 natural frequencies of rings are calculated by com-bining these matrices Meanwhile vibration transmissibilityof rings propagating from outer to inner and from inner toouter is obtained In Section 4 the influence of structuraland material parameters on natural frequencies is discussedSection 5 is the conclusion

2 Theoretical Analysis

21 Classical Method for Free Vibration

211 Solution of Transverse Vibration Consider sandwichrings consisting of two different materials depicted in Fig-ure 1 Adhesive can be employed for connecting the ringsSame material is selected for the first and third layers Theother material is selected for the second layer Radius of thefirst and third layers is 1199030 and 119903119888 Radius of the intermediate

I

II

I

0

bminus

b+

ra r0

rb

a1

rc

a2a1

a+

aminus aminus

a+a+

aminus bminusbminus

b+b+

Figure 1 Composite rings

layers is 119903119886 and 119903119887 Radial span of the first and third layersis 1198861 Radial span of the second layer is 1198862 119908 is bendingdeflection ℎ is thickness At the boundaries of 119903 = 119903119886 and119903 = 119903119887 positivendashgoing and negativendashgoing wave vectorsare b+1 b

minus1 a+2 aminus2 b+2 bminus2 a+3 aminus3 Also considering another

two boundaries at 119903 = 1199030 and 119903 = 119903119888 positivendashgoingand negativendashgoing wave vectors are a+1 aminus1 b+3 bminus3 Incylindrical coordinates the radius is assumed to be largeenough compared to thickness which means that it satisfiesthe small deformation theory Transverse solution is given byWang [3]

119882 = 11986011198690 (119896119903) + 11986121198840 (119896119903) + 11986231198680 (119896119903) + 11986341198700 (119896119903) (1)

where 1198601 1198612 1198623 1198634 are constants which are determinedby boundaries 1198690(119896119903) and 1198840(119896119903) are Bessel functions offirst and second kinds respectively 1198680(119896119903) and 1198700(119896119903) aremodified Bessel functions of first and second kinds 119896 =(412058721198912120588ℎ119863)025 is wave number and 119863 = 119864ℎ312(1 minus 1205902)is stiffness

212 Solution of Classical BesselMethod With regard to ringssubjected to bending excitation expression of transversedisplacement rotational angle shear force and bendingmoment within the first and third layers can be written as

1198821 (119903) = 119860111198690 (1198961119903) + 119861111198840 (1198961119903) + 119862111198680 (1198961119903)+ 119863111198700 (1198961119903)

120597119882120597119903 1 (119903) = minus1198961119860111198691 (1198961119903) minus 1198961119861111198841 (1198961119903)

+ 1198961119862111198681 (1198961119903) minus 1198961119863111198701 (1198961119903)

Shock and Vibration 3

1198721 (119903) = 11986311986011 [1198961119903 1198691 (1198961119903) minus 12059011198961119903 1198691 (1198961119903)minus 119896121198690 (1198961119903)] + 11986111 [1198961119903 1198841 (1198961119903) minus 12059011198961119903 1198841 (1198961119903)minus 119896121198840 (1198961119903)] + 11986211 [119896121198680 (1198961119903) minus 1198961119903 1198681 (1198961119903)+ 12059011198961119903 1198681 (1198961119903)] + 11986311 [119896121198700 (1198961119903) + 1198961119903 1198701 (1198961119903)minus 12059011198961119903 1198701 (1198961119903)]

1198761 (119903) = 119863 11986011119896131198691 (1198961119903) + 11986111119896131198841 (1198961119903)+ 11986211119896131198681 (1198961119903) minus 11986311119896131198701 (1198961119903)

(2)

Applying fixed boundary condition at 119903 = 1199030 obtains[1198690 (11989611199030) 1198840 (11989611199030) 1198680 (11989611199030) 1198700 (11989611199030)]Ψ11 = 0 (3)[minus11989611198691 (11989611199030) minus11989611198841 (11989611199030) 11989611198681 (11989611199030) minus11989611198701 (11989611199030)]Ψ11= 0 (4)

whereΨ11 = [11986011 11986111 11986211 11986311]119879Free boundary condition is selected at 119903 = 119903119888 then

119863

[1198961119903119888 1198691 (1198961119903119888) minus 12059011198961119903119888 1198691 (1198961119903119888) minus 119896121198690 (1198961119903119888)] [1198961119903119888 1198841 (1198961119903119888) minus 12059011198961119903119888 1198841 (1198961119903119888) minus 119896121198840 (1198961119903119888)][119896121198680 (1198961119903119888) minus 1198961119903119888 1198681 (1198961119903119888) + 12059011198961119903119888 1198681 (1198961119903119888)] [119896121198700 (1198961119903119888) + 1198961119903119888 1198701 (1198961119903119888) minus 12059011198961119903119888 1198701 (1198961119903119888)]

Ψ13 = 0

119863 [119896131198691 (1198961119903119888) 119896131198841 (1198961119903119888) 119896131198681 (1198961119903119888) minus119896131198701 (1198961119903119888)]Ψ13 = 0(5)

whereΨ13 = [11986013 11986113 11986213 11986313]119879 In order to obtain the natural frequencies substituting(A9) into (5) and combining (3)-(4) it reduces to

[[[[[[

1198690 (11989611199030) 1198840 (11989611199030) 1198680 (11989611199030) 1198700 (11989611199030)minus11989611198691 (11989611199030) minus11989611198841 (11989611199030) 11989611198681 (11989611199030) minus11989611198701 (11989611199030)

J1 Y1 I1 K1

119896131198691 (1198961119903119888) times T13 119896131198841 (1198961119903119888) times T13 119896131198681 (1198961119903119888) times T13 minus119896131198701 (1198961119903119888) times T13

]]]]]]Ψ11 = 0 (6)

where the specific theoretical derivation of T13 in (6) ispresented in the Appendix And each element is defined as

J1 = [1198961119903119888 1198691 (1198961119903119888) minus 12059011198961119903119888 1198691 (1198961119903119888) minus 119896121198690 (1198961119903119888)]times T13

(7a)

Y1 = [1198961119903119888 1198841 (1198961119903119888) minus 12059011198961119903119888 1198841 (1198961119903119888) minus 119896121198840 (1198961119903119888)]times T13

(7b)

I1 = [119896121198680 (1198961119903119888) minus 1198961119903119888 1198681 (1198961119903119888) + 12059011198961119903119888 1198681 (1198961119903119888)]times T13

(7c)

K1 = [119896121198700 (1198961119903119888) + 1198961119903119888 1198701 (1198961119903119888) minus 12059011198961119903119888 1198701 (1198961119903119888)]times T13

(7d)

Therefore (6) can be written as a 4 times 4 determinant

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816


J1 Y1 I1 K1119896131198691 (1198961119903119888) times T13 119896131198841 (1198961119903119888) times T13 119896131198681 (1198961119903119888) times T13 minus119896131198701 (1198961119903119888) times T13

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816= 0 (8)

where (8) is the characteristic equation of natural frequencyBy searching the root natural frequency of rings can be

calculated with a fixed boundary at inner surface and a freeboundary at outer surface


213 Solution of Classical Hankel Method The solution isobtained in (1) However it also can be expressed in a Hankelform

119882 = 119860+1119867(2)0 (1198961119903) + 119860minus1119867(1)0 (1198961119903) + 119861+11198700 (1198961119903)+ 119861minus1 1198680 (1198961119903)

(9)

where 119867(1)0 (1198961119903) and 119867(2)0 (1198961119903) are the Hankel functions ofsecond and first kinds respectively They can be defined as

119867(1)0 (1198961119903) = 1198690 (1198961119903) + 1198941198840 (1198961119903)119867(2)0 (1198961119903) = 1198690 (1198961119903) minus 1198941198840 (1198961119903)

(10)

Similarly expression of parameters within the first andthird layers can be written as

119882(119903) = 119860+1 [1198690 (1198961119903) minus 1198941198840 (1198961119903)] + 119860minus1 [1198690 (1198961119903)+ 1198941198840 (1198961119903)] + 119861+11198700 (1198961119903) + 119861minus1 1198680 (1198961119903) (11)

120597119882120597119903 1 (119903) = 119860+1 [minus11989611198691 (1198961119903) + 11989411989611198841 (1198961119903)]

+ 119860minus1 [minus11989611198691 (1198961119903) minus 11989411989611198841 (1198961119903)] minus 119861+111989611198701 (1198961119903)+ 119861minus111989611198681 (1198961119903)

(12)

1198721 (119903) = 119863119860+1 [1198961119903 1198691 (1198961119903) minus 119896121198690 (1198961119903)+ 119894119896211198840 (1198961119903) minus 1198941198961119903 1198841 (1198961119903)+ 120590

119903 [minus11989611198691 (1198961119903) + 11989411989611198841 (1198961119903)]]+ 119860minus1 [1198961119903 1198691 (1198961119903) minus 119896121198690 (1198961119903) minus 119894119896211198840 (1198961119903)+ 1198941198961119903 1198841 (1198961119903) + 120590

119903 [minus11989611198691 (1198961119903) minus 11989411989611198841 (1198961119903)]]+ 119861+1 [119896121198700 (1198961119903) + 1198961119903 1198701 (1198961119903) minus 120590119903 11989611198701 (1198961119903)]+ 119861minus1 [119896121198680 (1198961119903) minus 1198961119903 1198681 (1198961119903) + 12059011198961119903 1198681 (1198961119903)]

(13)

1198761 (119903) = 119863 119860+1 [minus119894119896311198841 (1198961119903) + 119896311198691 (1198961119903)]minus 119861+1119896311198701 (1198961119903) + 119860minus1 [119894119896311198841 (1198961119903) + 119896311198691 (1198961119903)]+ 119861minus1119896131198681 (1198961119903)

(14)

Natural frequencies of transverse vibration can be calcu-lated using classical Hankel method Characteristic equationof natural frequency can be deuced like the process of (3)ndash(8)In order to avoid repeating herein it is ignored

22 Wave Approach for Free Vibration In this section thesolution is presented in terms of cylindrical waves for

this ring Meanwhile positivendashgoing propagation negativendashgoing propagation coordination and reflection matrices arealso deduced By combining these matrices natural frequen-cies are calculated using wave approach

221 Propagation Matrices Wave propagates along thepositivendashgoing and negativendashgoing directions when propa-gating within structures as is shown in Figure 1 Waves willnot propagate at the boundaries but only can be reflectedMoreover parameters are continuous for the connection Inrecent years many researchers describe the waves in thematrix forms [8ndash17]

By considering (11) positivendashgoing waves can bedescribed as

a+1 = [119860+1 1198690 (11989611199030) minus 1198941198840 (11989611199030)119861+11198700 (11989611199030) ] (15a)

b+1 = [119860+1 1198690 (1198961119903119886) minus 1198941198840 (1198961119903119886)119861+11198700 (1198961119903119886) ] (15b)

a+2 = [119860+1 1198690 (1198962119903119886) minus 1198941198840 (1198962119903119886)119861+11198700 (1198962119903119886) ] (16a)

b+2 = [119860+1 1198690 (1198962119903119887) minus 1198941198840 (1198962119903119887)119861+11198700 (1198962119903119887) ] (16b)

a+3 = [119860+1 1198690 (1198961119903119888) minus 1198941198840 (1198961119903119888)119861+11198700 (1198961119903119888) ] (17a)

b+3 = [119860+1 1198690 (1198961119903119888) minus 1198941198840 (1198961119903119888)119861+11198700 (1198961119903119888) ] (17b)

These wave vectors are related byb+1 = f+1 (ra minus r0) a+1 (18a)

b+2 = f+2 (rb minus ra) a+2 (18b)

b+3 = f+3 (rc minus rb) a+3 (18c)Substituting matrices (15a) (15b) (16a) (16b) (17a) (17b)

into (18a)ndash(18c) positivendashgoing propagation matrices areobtained as

f+1 = [[[[[

1198690 (1198961119903119886) minus 1198941198840 (1198961119903119886)1198690 (11989611199030) minus 1198941198840 (11989611199030) 00 1198700 (1198961119903119886)1198700 (11989611199030)

]]]]]

(19a)

f+2 = [[[[[

1198690 (1198961119903119887) minus 1198941198840 (1198961119903119887)1198690 (1198961119903119886) minus 1198941198840 (1198961119903119886) 00 1198700 (1198961119903119887)1198700 (1198961119903119886)

]]]]]

(19b)

f+3 = [[[[[

1198690 (1198961119903119888) minus 1198941198840 (1198961119903119888)1198690 (1198961119903119887) minus 1198941198840 (1198961119903119887) 00 1198700 (1198961119903119888)1198700 (1198961119903119887)

]]]]]

(19c)


Similarly negativendashgoing waves can be rewritten as

aminus1 = [119860minus1 1198690 (11989611199030) + 1198941198840 (11989611199030)119861minus1 1198680 (11989611199030) ] (20a)

bminus1 = [119860minus1 1198690 (1198961119903119886) + 1198941198840 (1198961119903119886)119861minus1 1198680 (1198961119903119886) ] (20b)





These wave vectors are related by

aminus1 = fminus1 (r0 minus ra) bminus1 (23a)

aminus2 = fminus2 (ra minus rb) bminus2 (23b)

aminus3 = fminus3 (rb minus rc) bminus3 (23c)

Substituting matrices (20a) (20b) (21a) (21b) (22a)(22b) into (23a)ndash(23c) negativendashgoing propagation matricesare obtained

fminus1 = [[[[[

1198690 (11989611199030) + 1198941198840 (11989611199030)1198690 (1198961119903119886) + 1198941198840 (1198961119903119886) 00 1198680 (11989611199030)1198680 (1198961119903119886)

]]]]]

(24a)

fminus2 = [[[[[

1198690 (1198961119903119886) + 1198941198840 (1198961119903119886)1198690 (1198961119903119887) + 1198941198840 (1198961119903119887) 00 1198680 (1198961119903119886)1198680 (1198961119903119887)

]]]]]

(24b)

fminus3 = [[[[[

1198690 (1198961119903119887) + 1198941198840 (1198961119903119887)1198690 (1198961119903119888) + 1198941198840 (1198961119903119888) 00 1198680 (1198961119903119887)1198680 (1198961119903119888)

]]]]]

(24c)

222 Reflection Matrices Keeping the boundary conditionof 119903 = 1199030 fixed thus displacements and rotational angle aretaken as

119860+1 [1198690 (11989611199030) minus 1198941198840 (11989611199030)]+ 119860minus1 [1198690 (11989611199030) + 1198941198840 (11989611199030)] + 119861+11198700 (11989611199030)+ 119861minus1 1198680 (11989611199030) = 0

119860+1 [minus11989611198691 (11989611199030) + 11989411989611198841 (11989611199030)]+ 119860minus1 [minus11989611198691 (11989611199030) minus 11989411989611198841 (11989611199030)]minus 119861+111989611198701 (11989611199030) + 119861minus111989611198681 (11989611199030) = 0

(25)

The relationship of incident wave a+1 and reflected waveaminus1 is related by

a+1 = R0aminus

1 (26)

Substituting (25) into (26) the reflection matrices can beobtained as followsR0

= minus[[[[[

1198690 (11989611199030) minus 1198941198840 (11989611199030)1198690 (11989611199030) minus 1198941198840 (11989611199030)1198700 (11989611199030)1198700 (11989611199030)minus11989611198691 (11989611199030) + 11989411989611198841 (11989611199030)1198690 (11989611199030) minus 1198941198840 (11989611199030)

minus11989611198701 (11989611199030)1198700 (11989611199030)]]]]]

minus1

sdot [[[[[

1198690 (11989611199030) + 1198941198840 (11989611199030)1198690 (11989611199030) + 1198941198840 (11989611199030)1198680 (11989611199030)1198680 (11989611199030)minus11989611198691 (11989611199030) minus 11989411989611198841 (11989611199030)1198690 (11989611199030) + 1198941198840 (11989611199030)

11989611198681 (11989611199030)1198680 (11989611199030)]]]]]

(27)

Keeping the boundary condition of 119903 = 119903119888 free gives119863119860+1 [1198961119903119888 1198691 (1198961119903119888) minus 119896121198690 (1198961119903119888) + 119894119896211198840 (1198961119903119888)

minus 1198941198961119903119888 1198841 (1198961119903119888) + 120590119903119888 [minus11989611198691 (1198961119903119888) + 11989411989611198841 (1198961119903119888)]]+ 119860minus1 [1198961119903119888 1198691 (1198961119903119888) minus 119896121198690 (1198961119903119888) minus 119894119896211198840 (1198961119903119888)+ 1198941198961119903119888 1198841 (1198961119903119888) + 120590119903119888 [minus11989611198691 (1198961119903119888) minus 11989411989611198841 (1198961119903119888)]]+ 119861+1 [119896121198700 (1198961119903119888) + 1198961119903119888 1198701 (1198961119903119888)minus 120590119903 11989611198701 (1198961119903119888)] + 119861minus1 [119896121198680 (1198961119903119888) minus 1198961119903119888 1198681 (1198961119903119888)+ 12059011198961119903119888 1198681 (1198961119903119888)] = 0

119863 119860+1 [minus119894119896311198841 (1198961119903119888) + 119896311198691 (1198961119903119888)] minus 119861+1119896311198701 (1198961119903119888)+ 119860minus1 [119894119896311198841 (1198961119903119888) + 119896311198691 (1198961119903119888)]+ 119861minus1119896131198681 (1198961119903119888) = 0

(28)

The relationship of incident wave b+3 and reflected wavebminus3 is

bminus3 = R3b+

3 (29)


Substituting (28) into (29) reflection matrices are calcu-lated as

R3

= minus[[[[[

(1198961119903119888) 1198691 (1198961119903119888) minus 119896121198690 (1198961119903119888) minus 119894119896211198840 (1198961119903119888) + (1198941198961119903119888) 1198841 (1198961119903119888) + (120590119903119888) [minus11989611198691 (1198961119903119888) minus 11989411989611198841 (1198961119903119888)]1198690 (1198961119903119888) + 1198941198840 (1198961119903119888)119896121198680 (1198961119903119888) minus (1198961119903119888) 1198681 (1198961119903119888) + (12059011198961119903119888) 1198681 (1198961119903119888)1198680 (1198961119903119888)119894119896311198841 (1198961119903119888) + 119896311198691 (1198961119903119888)1198690 (1198961119903119888) + 1198941198840 (1198961119903119888)

119896131198681 (1198961119903119888)1198680 (1198961119903119888)]]]]]

minus1

times [[[[[

(1198961119903119888) 1198691 (1198961119903119888) minus 119896121198690 (1198961119903119888) + 119894119896211198840 (1198961119903119888) minus (1198941198961119903119888) 1198841 (1198961119903119888) + (120590119903119888) [minus11989611198691 (1198961119903119888) + 11989411989611198841 (1198961119903119888)]1198690 (1198961119903119888) minus 1198941198840 (1198961119903119888)119896121198700 (1198961119903119888) + (1198961119903119888)1198701 (1198961119903119888) minus (120590119903) 11989611198701 (1198961119903119888)1198700 (1198961119903119888)minus119894119896311198841 (1198961119903119888) + 119896311198691 (1198961119903119888)1198690 (1198961119903119888) minus 1198941198840 (1198961119903119888)

minus119896311198701 (1198961119903119888)1198700 (1198961119903119888)]]]]]

(30)

223 Coordination Matrices By imposing the geometriccontinuity at 119903 = 119903119886 yields

[ 1198690 (1198961119903119886) minus 1198941198840 (1198961119903119886) 1198700 (1198961119903119886)minus11989611198691 (1198961119903119886) + 11989411989611198841 (1198961119903119886) minus11989611198701 (1198961119903119886)] b+1 + [ 1198690 (1198961119903119886) + 1198941198840 (1198961119903119886) 1198680 (1198961119903119886)minus11989611198691 (1198961119903119886) minus 11989411989611198841 (1198961119903119886) 11989611198681 (1198961119903119886)] bminus1 = [ 1198690 (1198962119903119886) minus 1198941198840 (1198962119903119886) 1198700 (1198962119903119886)minus11989621198691 (1198962119903119886) + 11989411989621198841 (1198962119903119886) minus11989621198701 (1198962119903119886)] a+2

+ [ 1198690 (1198962119903119886) + 1198941198840 (1198962119903119886) 1198680 (1198962119903119886)minus11989621198691 (1198962119903119886) minus 11989411989621198841 (1198962119903119886) 11989621198681 (1198962119903119886)] aminus2

[[

1198961119903119886 1198691 (1198961119903119886) minus 119896121198690 (1198961119903119886) + 119894119896211198840 (1198961119903119886) minus 1198941198961119903119886 1198841 (1198961119903119886) + 1205901119903119886 [minus11989611198691 (1198961119903119886) + 11989411989611198841 (1198961119903119886)] 119896121198700 (1198961119903119886) + 11989611199031198861198701 (1198961119903119886) minus 1205901119903119886 11989611198701 (1198961119903119886)minus119894119896311198841 (1198961119903119886) + 119896311198691 (1198961119903119886) minus119896311198701 (1198961119903119886)]]b+1

+ [[

1198961119903119886 1198691 (1198961119903119886) minus 119896121198690 (1198961119903119886) minus 119894119896211198840 (1198961119903119886) + 1198941198961119903119886 1198841 (1198961119903119886) + 1205901119903119886 [minus11989611198691 (1198961119903119886) minus 11989411989611198841 (1198961119903119886)] 119896121198680 (1198961119903119886) minus 1198961119903119886 1198681 (1198961119903119886) + 12059011198961119903119886 1198681 (1198961119903119886)119894119896311198841 (1198961119903119886) + 119896311198691 (1198961119903119886) 119896131198681 (1198961119903119886)

]]bminus1

= [[

1198962119903119886 1198691 (1198962119903119886) minus 119896221198690 (1198962119903119886) + 119894119896221198840 (1198962119903119886) minus 1198941198962119903119886 1198841 (1198962119903119886) + 1205902119903119886 [minus11989621198691 (1198962119903119886) + 11989411989621198841 (1198962119903119886)] 119896221198700 (1198962119903119886) + 11989621199031198861198701 (1198962119903119886) minus 1205902119903119886 11989621198701 (1198962119903119886)minus119894119896321198841 (1198962119903119886) + 119896321198691 (1198962119903119886) minus119896321198701 (1198962119903119886)]]a+2

+ [[


]]aminus2

(31)

Equations (31) can be rewritten as

R+a1b+

1 + Rminusa1bminus

1 = T+a2a+

2 + Tminusa2aminus

2

R+a3b+

1 + Rminusa3bminus

1 = T+a4a+

2 + Tminusa4aminus

2 (32)

According to the continuity at 119903 = 119903119887 shear force andbending moment are required that

[ 1198690 (1198962119903119887) minus 1198941198840 (1198962119903119887) 1198700 (1198962119903119887)minus11989621198691 (1198962119903119887) + 11989411989621198841 (1198962119903119887) minus11989621198701 (1198962119903119887)] b+2 + [ 1198690 (1198962119903119887) + 1198941198840 (1198962119903119887) 1198680 (1198962119903119887)

minus11989621198691 (1198962119903119887) minus 11989411989621198841 (1198962119903119887) 11989621198681 (1198962119903119887)] bminus2 = [ 1198690 (11989611199031) minus 1198941198840 (1198961119903119887) 1198700 (1198961119903119887)minus11989611198691 (1198961119903119887) + 11989411989611198841 (1198961119903119887) minus11989611198701 (1198961119903119887)] a+3


[[

1198962119903119887 1198691 (1198962119903119887) minus 119896221198690 (1198962119903119887) + 119894119896221198840 (1198962119903119887) minus 11989411989621199032 1198841 (1198962119903119887) + 1205902119903119887 [minus11989621198691 (1198962119903119887) + 11989411989621198841 (1198962119903119887)] 119896221198700 (1198962119903119887) + 1198962119903119887 1198701 (1198962119903119887) minus 1205902119903119887 11989621198701 (1198962119903119887)minus119894119896321198841 (1198962119903119887) + 119896321198691 (1198962119903119887) minus119896321198701 (1198962119903119887)]]b+2

+ [[


]]bminus2

= [[

1198961119903119887 1198691 (1198962119903119887) minus 119896121198690 (1198961119903119887) + 119894119896211198840 (1198961119903119887) minus 1198941198961119903119887 1198841 (1198961119903119887) + 1205901119903119887 [minus11989611198691 (1198961119903119887) + 11989411989611198841 (1198961119903119887)] 119896121198700 (1198961119903119887) + 1198961119903119887 1198701 (1198961119903119887) minus 1205901119903119887 11989611198701 (1198961119903119887)minus119894119896311198841 (1198961119903119886) + 119896311198691 (1198961119903119887) minus119896311198701 (1198961119903119887)]]a+3

+ [[


]]aminus3

(33)


Equations (33) can be written as

R+b1b+

1 + Rminusb1bminus

1 = T+b2a+

2 + Tminusb2aminus

2

R+b3b+

1 + Rminusb3bminus

1 = T+b4a+

2 + Tminusb4aminus

2 (34)

224 Characteristic Equation of Natural Frequency Combin-ing propagation matrices reflection matrices and coordina-tion matrices derived in Section 22 natural frequencies ofcomposite rings can be calculated smoothly Figure 1 presentsa clear description of incident and reflected waves Thus thewave matrices described by (18a)ndash(18c) (23a)ndash(23c) (26)(29) (32) and (34) are assembled as

b+1 = f+1 (ra minus r0) a+1aminus1 = fminus1 (r0 minus ra) bminus1b+3 = f+3 (rc minus rb) a+3a+1 = R0a

minus

1

b+2 = f+2 (rb minus ra) a+2aminus2 = fminus2 (ra minus rb) bminus2aminus3 = fminus3 (rb minus rc) bminus3bminus3 = R3b

+

3

R+a1b+

1 + Rminusa1bminus

1 = T+a2a+

2 + Tminusa2aminus

2

R+b1b+

2 + Rminusb1bminus

2 = T+b2a+

3 + Tminusb2aminus

3

R+a3b+

1 + Rminusa3bminus

1 = T+a4a+

2 + Tminusa4aminus

2

R+b3b+

2 + Rminusb3bminus

2 = T+b4a+

3 + Tminusb4aminus

3 (35)

In order to obtain the natural frequency (35) can berewritten in a matrix form

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

minusI2times2 R0 0 0 0 0 0 0 0 0 0 00 minusI2times2 0 fminus1 0 0 0 0 0 0 0 00 0 0 0 0 minusI2times2 0 fminus2 0 0 0 00 0 0 0 0 0 0 0 0 minusI2times2 0 fminus3f+1 0 minusI2times2 0 0 0 0 0 0 0 0 00 0 0 0 f+2 0 minusI2times2 0 0 0 0 00 0 0 0 0 0 0 0 f+3 0 minusI2times2 00 0 R+a1 Rminusa1 T+a2 Tminusa2 0 0 0 0 0 00 0 R+a3 Rminusa3 T+a4 Tminusa4 0 0 0 0 0 00 0 0 0 0 0 R+b1 Rminusb1 T+b2 Tminusb2 0 00 0 0 0 0 0 R+b3 Rminusb3 T+b4 Tminusb4 0 00 0 0 0 0 0 0 0 0 0 R3 minusI2times2

]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

a+1aminus1b+1bminus1a+2aminus2b+2bminus2a+3aminus3b+3bminus3

]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

= 119865 (119891)

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[


]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

= 0 (36)

119865(119891) is a matrix of 12times12 If (36) has solution it requiresthat

1003816100381610038161003816119865 (119891)1003816100381610038161003816 = 0 (37)

By solving the roots of characteristic equation (37) onecan calculate the real and imaginary parts It is important hereto note that the natural frequencies can be found by searchingthe intersections in 119909-axis3 Numerical Results and Discussion

In this section free vibration of rings is calculated by usingwave approach and the results are also compared with thoseobtained by classical method Material RESIN is selected forthe first and third layers Material STEEL is selected for the

middle layers Material and structural parameters are givenin Table 1

Based on Bessel and Hankel solutions calculated byclassical method theoretically natural frequency curves arepresented by solving characteristic equation (8) depictedin Figure 2 Furthermore (37) is calculated using waveapproach It can be seen that the real and imaginary partsintersect at multiple points simultaneously in 119909-axis It isimportant here to note that the roots of the characteristiccurves are natural frequencieswhen the values of longitudinalcoordinates are zero

In Figure 2 two different natural frequencies can beclearly presented in the range of 450ndash1500Hz that is124422Hz and 144331Hz However the values are very smallin the range of 0ndash450Hz In order to find whether the


Table 1 Material and structural parameters

Material parameters Density 120588 (kgm3) Young modulus 119864 (Pa) Poissonrsquos ratioI (RESIN) 1180 0435 times 1010 03679II (STEEL) 7780 2106 times 1010 03Structural parameters 119903119886 = 91199030 119903119888 = 119903119887 + 40 h(mm) 45 125 1

Table 2 Results calculated by classical method wave approach and FEM

Method 1st mode 2nd mode 3rd mode 4th mode 5th modeClassical Bessel 3765Hz 16754Hz 41427Hz 124422Hz 144331HzClassical Hankel 3765Hz 16754Hz 41427Hz 124422Hz 144331HzWave approach 3765Hz 16754Hz 41427Hz 124422Hz 144331HzFEM 3776Hz 16830Hz 41519Hz 124790Hz 144811 Hz

Table 3 Comparison of free vibration by FEM for four type boundaries

Different boundaries 1st mode 2nd mode 3rd mode 4th mode 5th modeInner free outer free 14340Hz 33410Hz 56955Hz 133650Hz 184516HzInner fixed outer free 3776Hz 16830Hz 41519Hz 124790Hz 144810HzInner free outer fixed 7044Hz 32196Hz 57294Hz 138886Hz 184867HzInner fixed outer fixed 10107Hz 41328Hz 127237Hz 147974Hz 292738Hz

0 200 400 600 800 1000 1200 1400 1600Frequency (Hz)

minus4

minus3

minus2

minus1

0

1

2

3

4

Classical Bessel solutionClassical Hankel solution

Wave solution (imag)Wave solution (real)

F(f

)

Figure 2 Natural frequency obtained by classical method and waveapproach

values in this range also intersect at one point three zoomedfigures are drawn for the purpose of better illustration aboutthe natural frequencies of characteristic curves which aredescribed in Figure 3

Natural frequencies calculated by these two methodsare compared Modal analysis is carried out by FEM Thenatural frequencies are presented in Table 2 from which itcan be observed that the first five-order modes calculated

by these three methods are in good agreement Obviouslyit also can be found that natural frequencies obtained byANSYS software are larger than the results calculated byclassic method and wave approach which is mainly causedby the mesh and simplified solid model in FEM Howeverthese errors are within an acceptable range which verifiesthe correctness of theoretical calculations To assess thedeformation of rings Figure 4 is employed to describe themode shape It can be found that themaximum deformationsof the first three mode shapes occur in the outermost surfaceThe fourth and fifth mode shapes appear in the innermostsurface

Adopting FEM method the first five natural frequenciesare calculated for four type boundaries as is shown in Table 3It shows that the first natural frequency is 3776Hz (Min) atthe case of inner boundary fixed and outer boundary freeThefirst natural frequency is 14340Hz (Max) at the case of innerand outer boundaries both free

Harmonic Response Analysis of rings is carried out byusing ANSYS 145 software RESIN is chosen for the first andthird layer The second layer is selected as STEEL Elementcan be selected as Solid 45 which is shown in red and bluein Figure 5(a)Through loading transverse displacement ontothe innermost layer and picking the transverse displacementonto the outermost layer vibration transmissibility of ringspropagating from inner to outer is obtained by using formula119889119861 = 20 log (119889outer119889inner) Similarly through loading trans-verse displacement onto the outermost layer and picking thetransverse displacement onto the innermost layer vibrationtransmissibility propagating from outer to inner is obtainedby using formula 119889119861 = 20 log (119889inner119889outer)


0 10 15 20 25 30 35 40Frequency (Hz)

minus5

0

5


Classical Hankel solutionClassical Bessel solution

5

F(f

)

times10minus8

(a) 0ndash40Hz

40 60 80 100 120 140 160 180minus10

minus8

minus6

minus4

minus2

0

2

4

6



F(f

)

times10minus6

Frequency (Hz)

(b) 40ndash180Hz

200 250 300 350 400 450minus6

minus4

minus2

0

2

4

F(f

)

times10minus3

Frequency (Hz)



(c) 180ndash450Hz

Figure 3 Characteristic curves in the range of 0ndash450Hz

Figure 5(b) indicates that there is no vibration attenuationin the range of 0ndash1500Hz when transverse vibration propa-gates from outer to inner Also four resonance frequenciesappear namely 7044Hz 32196Hz 57294Hz 138886Hzwhich coincide with the first four-order natural frequenciesin Table 3 at the case of innermost layer free and outermostlayer fixed Compared with the case of vibration propagationfrom outer to inner there is vibration attenuation whenvibration propagates from inner to outer In addition five res-onance frequencies also appear namely 3776Hz 16830Hz41519Hz 12479Hz and 14481 Hz which coincide with theresults obtained by wave approach classical Hankel andclassical Bessel methods shown in Table 2

4 Effects of Structural andMaterial Parameters

41 Structural Parameters The effects of structural param-eters such as thickness inner radius and radial span areinvestigated in Figure 6 Adopting single variable principle

herein only change one parameter Figure 6(a) shows clearlythat with thickness increasing the first modes change from3776Hz to 18815Hz and the remaining threemodes increaseobviously which indicates that thickness has great effecton the first four natural frequencies In fact characteristicequation of natural frequency is determined by thicknessdensity and elastic modulus which is shown by the expres-sion of wave number 119896 = (412058721198912120588ℎ119863)025 and stiffness119863 = 119864ℎ312(1 minus 1205902) Therefore thickness is used to adjustthe natural frequency directly through varying wave number119896 = (412058721198912120588ℎ119863)025 in (36)

From the wave number 119896 = (412058721198912120588ℎ119863)025 it canbe found that inner radius is not related to the naturalfrequency Thus inner radius almost has no effect on thenatural frequency shown in Figure 6(b)

In Figure 6(c) there are five different types analyzed forthe radial span ratios of RESIN and STEEL that is 1198861 1198862 =1 times 00422 12 times 00422 1198861 1198862 = 1 times 00421 11times00421 1198861 1198862 = 004 004 1198861 1198862 = 11 times 00421 1 times00421 1198861 1198862 = 12 times 00422 1 times 00422 respectively


1

NODAL SOLUTIONFREQ = 3776USUM (AVG)RSYS = 0DMX = 330732SMX = 330732

0

367

48

734

96

110

244

146

992

183

74

220

488

257

236

293

984

330

732

(a)

1

NODAL SOLUTIONFREQ = 1683USUM (AVG)RSYS = 0DMX = 853383

SMX = 853383

0

948

203

189

641

284

461

379

281

47410

2

568

922

663742

7585

62

853383

(b)

0

683

9

136

78

205

17

273

56

341

95

410

34

478

73

547

12

615

51

1

NODAL SOLUTIONFREQ = 41519 USUM (AVG)RSYS = 0DMX = 61551

SMX = 61551

(c)

0

143

273

286

545

429

818

573

09

716

363

859

635

10029

1

114

618

12894

5

1


SMX = 128945

(d)

0

140

312

280

623

420

935

561

247

701

558

841

87

982

181

112

249

12628

1


SMX = 12628

(e)

Figure 4 Mode shapes of natural frequencies (a) First mode (b) Second mode (c) Third mode (d) Fourth mode (e) Fifth mode


(a) The meshing modeminus40

minus30

minus20

minus10

0

10

20

30

40

50

60

410

019

629

238

848

458

067

677

286

896

410

6011

5612

5213

4814

44

Outer to inner

Inner to outer

(Hz)

Tran

smiss

ibili

ty (d

B)

14481 Hz12479 Hz

41519 Hz3776Hz

138886 Hz57294Hz32196Hz

7044Hz

1683 Hz

(b) Vibration response

Figure 5

0

1000

2000

3000

4000

5000

6000

7000

0001 0002 0003 0004 0005

(Hz)

1st mode

2nd mode

3rd mode

4th mode

(a) Thickness

(Hz)

0

200

400

600

800

1000

1200

1400

0001 0006 0014 0019

1st mode

2nd mode

3rd mode

4th mode

(b) Inner radius

(Hz)

0

5000

10000

15000

20000

25000

30000

35000

40000

112 111 1 111 121

2nd mode1st mode

3rd mode

4th mode

(c) Radial span

Figure 6 Effect of structural parameters


Table 4 Material parameters

Method Density 120588 (kgm3) Young modulus 119864 (Pa) Poissonrsquos ratioPMMA 1062 032 times 1010 03333Al 2799 721 times 1010 03451Pb 11600 408 times 1010 03691Ti 4540 117 times 1010 032

0200400600800

100012001400160018002000

1062 2799 4540 7780 11600

PMMA

Al Ti STEEL Pb

Fre (

Hz)

1st mode

2nd mode

3rd mode

4th mode

Density (kgＧ3)

(a) Density

Fre (

Hz)

0

200

400

600

800

1000

1200

1400

032 408 721 117 2106

PMMA

PbAl

Ti STEEL

1st mode2nd mode

3rd mode

4th mode

Elastic modulus (Pa)

(b) Elastic modulus

Figure 7 Effect of material parameters

When radial span is equal to 1 this means that the size ofRESIN and STEEL is 1 1 namely 1198861 1198862 = 004 004For this case the total size of composite ring is max so themode is min Additionally symmetrical five types cause theapproximate symmetry of Figure 6(c) It also can be foundthat as radial span increases natural frequencies appear as asimilar trend namely decrease afterwards increase

42 Material Parameters Adopting single variable principledensity of middle material STEEL is replaced by the densityof PMMA Al Pb Ti Similar with the study on effectsof structural parameters the effects of density and elasticmodulus are studied for the case of keeping the material andstructural parameters unchanged Also material parametersof PMMA Al Pb and Ti are presented in Table 4

Figure 7(a) indicates that as density increases the firstmode decreases but not very obviously However the secondthird and fourth modes reduce significantly Figure 7(b)shows that when elastic modulus increases gradually the firstmode increases but not significantly The second third andfourth modes increase rapidly

5 Conclusion

This paper focuses on calculating natural frequency forrings via classical method and wave approach Based onthe solutions of transverse vibration expression of rota-tional angle shear force and bending moment are obtainedWave propagation matrices within structure coordinationmatrices between the two materials and reflection matricesat the boundary conditions are also deduced Additionallycharacteristic equation of natural frequencies is obtained by

assembling these wavematricesThe real and imaginary partscalculated by wave approach intersect at the same point withthe results obtained by classical method which verifies thecorrectness of theoretical calculations

A further analysis for the influence of different bound-aries on natural frequencies is discussed It can be found thatthe first natural frequency is Min 3776Hz at the case of innerboundary fixed and outer boundary free In addition it alsoshows that there exists vibration attenuation when vibrationpropagates from inner to outerHowever there is no vibrationattenuation when vibration propagates from outer to innerStructural andmaterial parameters have strong sensitivity forthe free vibration

Finally the behavior of wave propagation is studied indetail which is of great significance to the design of naturalfrequency for the vibration analysis of rotating rings and shaftsystems

Appendix

Derivation of the Transfer Matrix

Due to the continuity at 119903 = 119903119886 the following is obtained1198821 (119903119886) = 1198822 (119903119886)

120597119882120597119903 1 (119903119886) = 120597119882

120597119903 2 (119903119886)1198721 (119903119886) = 1198722 (119903119886)1198761 (119903119886) = 1198762 (119903119886)

(A1)


Equation (A1) can be organized as

[[[[[[[[[


J2 Y2 I2 K2

119896311198691 (1198961119903119886) 119896311198841 (1198961119903119886) 119896311198681 (1198961119903119886) minus119896311198701 (1198961119903119886)

]]]]]]]]]

Ψ11

=[[[[[[[[


J3 Y3 I3 K3

119896321198691 (1198962119903119886) 119896321198841 (1198962119903119886) 119896321198681 (1198962119903119886) minus119896321198701 (1198962119903119886)

]]]]]]]]Ψ12

(A2)

where Ψ12 = [11986012 11986112 11986212 11986312]119879 and each element isdefined as

J2 = 1198961119903119886 1198691 (1198961119903119886) minus 12059011198691 (1198961119903119886) minus 119896211198690 (1198961119903119886)

Y2 = 1198961119903119886 1198841 (1198961119903119886) minus 12059011198841 (1198961119903119886) minus 119896211198840 (1198961119903119886)

I2 = 1198961119903119886 12059011198681 (1198961119903119886) minus 1198681 (1198961119903119886) + 119896211198680 (1198961119903119886)

K2 = 1198961119903119886 1198701 (1198961119903119886) minus 12059011198701 (1198961119903119886) + 119896211198700 (1198961119903119886)

J3 = 1198962119903119886 1198691 (1198962119903119886) minus 12059021198691 (1198962119903119886) minus 119896221198690 (1198962119903119886)

Y3 = 1198962119903119886 1198841 (1198962119903119886) minus 12059021198841 (1198962119903119886) minus 119896221198840 (1198962119903119886)

I3 = 1198962119903119886 12059021198681 (1198962119903119886) minus 1198681 (1198962119903119886) + 119896221198680 (1198962119903119886) K3 = 1198962119903119886 1198701 (1198962119903119886) minus 12059021198701 (1198962119903119886) + 119896221198700 (1198962119903119886)

(A3)

Hence (A2) can be written as

H1Ψ11 = K1Ψ12 (A4)

Similarly by imposing the geometric continuity at 119903 = 119903119887the following is obtained

1198822 (119903119887) = 1198821 (119903119887)120597119882120597119903 2 (119903119887) = 120597119882120597119903 1 (119903119887)1198722 (119903119887) = 1198721 (119903119887)1198762 (119903119887) = 1198761 (119903119887)

(A5)

Arranging (A5) yields

[[[[[[[[


J4 Y4 I4 K4

119896321198691 (1198962119903119887) 119896321198841 (1198962119903119887) 119896321198681 (1198962119903119887) minus119896321198701 (1198962119903119887)

]]]]]]]]Ψ12

=[[[[[[[[[


J5 Y5 I5 K5

119896311198691 (1198961119903119887) 119896311198841 (1198961119903119887) 119896311198681 (1198961119903119887) minus119896311198701 (1198961119903119887)

]]]]]]]]]

Ψ13

(A6)


and each element is defined as

J4 = 1198962119903119887 1198691 (1198962119903119887) minus 12059021198691 (1198962119903119887) minus 119896221198690 (1198962119903119887) Y4 = 1198962119903119887 1198841 (1198962119903119887) minus 12059021198841 (1198962119903119887) minus 119896221198840 (1198962119903119887) I4 = 1198962119903119887 12059021198681 (1198962119903119887) minus 1198681 (1198962119903119887) + 119896221198680 (1198962119903119887) K4 = 1198962119903119887 1198701 (1198962119903119887) minus 12059021198701 (1198962119903119887) + 119896221198700 (1198962119903119887) J5 = 1198961119903119887 1198691 (1198961119903119887) minus 12059011198691 (1198961119903119887) minus 119896211198690 (1198961119903119887) Y5 = 1198961119903119887 1198841 (1198961119903119887) minus 12059011198841 (1198961119903119887) minus 119896211198840 (1198961119903119887) I5 = 1198961119903119887 12059011198681 (1198961119903119887) minus 1198681 (1198961119903119887) + 119896211198680 (1198961119903119887) K5 = 1198961119903119887 1198701 (1198961119903119887) minus 12059011198701 (1198961119903119887) + 119896211198700 (1198961119903119887)

(A7)

Equation (A6) can be simplified as

K2Ψ12 = H2Ψ13 (A8)

Combining (A4) and (A8) gives

Ψ13 = T13Ψ11 = Hminus12 K2Kminus11 H1Ψ11 (A9)

where T13 is the transfer matrix of flexural wave from innerto outer

Conflicts of Interest

There are no conflicts of interest regarding the publication ofthis paper

Acknowledgments

The research was funded by Heilongjiang Province Funds forDistinguished Young Scientists (Grant no JC 201405) ChinaPostdoctoral Science Foundation (Grant no 2015M581433)and Postdoctoral Science Foundation of HeilongjiangProvince (Grant no LBH-Z15038)

References

[1] R H Gutierrez P A A Laura D V Bambill V A Jederlinicand D H Hodges ldquoAxisymmetric vibrations of solid circularand annular membranes with continuously varying densityrdquoJournal of Sound and Vibration vol 212 no 4 pp 611ndash622 1998

[2] M Jabareen and M Eisenberger ldquoFree vibrations of non-homogeneous circular and annular membranesrdquo Journal ofSound and Vibration vol 240 no 3 pp 409ndash429 2001

[3] C Y Wang ldquoThe vibration modes of concentrically supportedfree circular platesrdquo Journal of Sound and Vibration vol 333 no3 pp 835ndash847 2014

[4] L Roshan and R Rashmi ldquoOn radially symmetric vibrationsof circular sandwich plates of non-uniform thicknessrdquo Interna-tional Journal ofMechanical Sciences vol 99 article no 2981 pp29ndash39 2015

[5] A Oveisi and R Shakeri ldquoRobust reliable control in vibrationsuppression of sandwich circular platesrdquo Engineering Structuresvol 116 pp 1ndash11 2016

[6] S Hosseini-Hashemi M Derakhshani and M Fadaee ldquoAnaccurate mathematical study on the free vibration of steppedthickness circularannular Mindlin functionally graded platesrdquoApplied Mathematical Modelling Simulation and Computationfor Engineering and Environmental Systems vol 37 no 6 pp4147ndash4164 2013

[7] O Civalek and M Uelker ldquoHarmonic differential quadrature(HDQ) for axisymmetric bending analysis of thin isotropiccircular platesrdquo Structural Engineering and Mechanics vol 17no 1 pp 1ndash14 2004

[8] H Bakhshi Khaniki and S Hosseini-Hashemi ldquoDynamic trans-verse vibration characteristics of nonuniform nonlocal straingradient beams using the generalized differential quadraturemethodrdquo The European Physical Journal Plus vol 132 no 11article no 500 2017

[9] W Liu D Wang and T Li ldquoTransverse vibration analysis ofcomposite thin annular plate by wave approachrdquo Journal ofVibration and Control p 107754631773220 2017

[10] B R Mace ldquoWave reflection and transmission in beamsrdquoJournal of Sound and Vibration vol 97 no 2 pp 237ndash246 1984

[11] C Mei ldquoStudying the effects of lumped end mass on vibrationsof a Timoshenko beam using a wave-based approachrdquo Journalof Vibration and Control vol 18 no 5 pp 733ndash742 2012

[12] B Kang C H Riedel and C A Tan ldquoFree vibration analysisof planar curved beams by wave propagationrdquo Journal of Soundand Vibration vol 260 no 1 pp 19ndash44 2003

[13] S-K Lee B R Mace and M J Brennan ldquoWave propagationreflection and transmission in curved beamsrdquo Journal of Soundand Vibration vol 306 no 3-5 pp 636ndash656 2007

[14] S K Lee Wave Reflection Transmission and Propagation inStructural Waveguides [PhD thesis] Southampton University2006

[15] D Huang L Tang and R Cao ldquoFree vibration analysis ofplanar rotating rings by wave propagationrdquo Journal of Soundand Vibration vol 332 no 20 pp 4979ndash4997 2013

[16] A Bahrami and A Teimourian ldquoFree vibration analysis ofcomposite circular annular membranes using wave propaga-tion approachrdquo Applied Mathematical Modelling Simulationand Computation for Engineering and Environmental Systemsvol 39 no 16 pp 4781ndash4796 2015

[17] C A Tan andB Kang ldquoFree vibration of axially loaded rotatingTimoshenko shaft systems by the wave-train closure principlerdquoInternational Journal of Solids and Structures vol 36 no 26 pp4031ndash4049 1999

[18] A Bahrami and A Teimourian ldquoNonlocal scale effects onbuckling vibration and wave reflection in nanobeams via wavepropagation approachrdquo Composite Structures vol 134 pp 1061ndash1075 2015

[19] M R Ilkhani A Bahrami and S H Hosseini-Hashemi ldquoFreevibrations of thin rectangular nano-plates using wave propa-gation approachrdquo Applied Mathematical Modelling Simulationand Computation for Engineering and Environmental Systemsvol 40 no 2 pp 1287ndash1299 2016

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wwwhindawicom Volume 2018

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Submit your manuscripts atwwwhindawicom


the flexural vibrationwith addedmass for Timoshenko beamand the influence of lumped mass on natural frequency isalso discussed in detail Kang et al [12] divided the realand imaginary parts of wave solutions of curved beam intofour cases and they calculated the natural frequencies bycombining propagation transmission and reflection matri-ces Lee et al [13 14] considered the power flow when wavepropagated in curved beam Furthermore they applied theFlugge theory to analyze the free vibration of a single curvedbeam and their result was compared with Kang et al whichverified the correctness of the numerical results Huang et al[15] investigated the free vibration of planar rotating ringsThe effect of cross section on natural frequency was alsodiscussed Bahrami and Teimourian [16] studied the freevibration of composite plates consisting of two layers andthey also made a comparison between classical results andwave propagation results Tan and Kang [17] concentratedon the free vibration of rotating Timoshenko shaft with axialforce and discussed the effect of continuous condition andcross section on natural frequencies From the wave pointof view Bahrami and Teimourian [18] analyzed the freevibration of nanobeams for the first time Ilkhani et al [19]studied the free vibration of thin rectangular plate It shouldbe noted that the above scholars have done lots of studies forfree vibration of structures by usingwave approach while fewreports for the analysis of natural frequency for transversevibration of rings can be found In fact it is well knownthat the nature of vibration is the propagation of wavesAnalyzing free vibration in terms of wave propagation andattenuating can have a better understanding for usMoreoverone advantage of using wave approach to analyze the freevibration is its conciseness of matrices that makes the naturalfrequencies be calculated easily Wave approach is a strongtool for studying the behavior of wave transmission andreflection in waveguides providing a practical engineeringapplication such as filters

The emphasis of this paper is focused on the free vibrationof rings This paper is organized into five parts Section 1is introduction In Section 2 propagation coordinationand reflection matrices are deduced in forms of matrix Inaddition the characteristic equation of natural frequencyis obtained using classical method and wave approach InSection 3 natural frequencies of rings are calculated by com-bining these matrices Meanwhile vibration transmissibilityof rings propagating from outer to inner and from inner toouter is obtained In Section 4 the influence of structuraland material parameters on natural frequencies is discussedSection 5 is the conclusion

2 Theoretical Analysis

21 Classical Method for Free Vibration

211 Solution of Transverse Vibration Consider sandwichrings consisting of two different materials depicted in Fig-ure 1 Adhesive can be employed for connecting the ringsSame material is selected for the first and third layers Theother material is selected for the second layer Radius of thefirst and third layers is 1199030 and 119903119888 Radius of the intermediate

I

II

I

0

bminus

b+

ra r0

rb

a1

rc

a2a1

a+

aminus aminus

a+a+

aminus bminusbminus

b+b+

Figure 1 Composite rings

layers is 119903119886 and 119903119887 Radial span of the first and third layersis 1198861 Radial span of the second layer is 1198862 119908 is bendingdeflection ℎ is thickness At the boundaries of 119903 = 119903119886 and119903 = 119903119887 positivendashgoing and negativendashgoing wave vectorsare b+1 b

minus1 a+2 aminus2 b+2 bminus2 a+3 aminus3 Also considering another

two boundaries at 119903 = 1199030 and 119903 = 119903119888 positivendashgoingand negativendashgoing wave vectors are a+1 aminus1 b+3 bminus3 Incylindrical coordinates the radius is assumed to be largeenough compared to thickness which means that it satisfiesthe small deformation theory Transverse solution is given byWang [3]

119882 = 11986011198690 (119896119903) + 11986121198840 (119896119903) + 11986231198680 (119896119903) + 11986341198700 (119896119903) (1)

where 1198601 1198612 1198623 1198634 are constants which are determinedby boundaries 1198690(119896119903) and 1198840(119896119903) are Bessel functions offirst and second kinds respectively 1198680(119896119903) and 1198700(119896119903) aremodified Bessel functions of first and second kinds 119896 =(412058721198912120588ℎ119863)025 is wave number and 119863 = 119864ℎ312(1 minus 1205902)is stiffness

212 Solution of Classical BesselMethod With regard to ringssubjected to bending excitation expression of transversedisplacement rotational angle shear force and bendingmoment within the first and third layers can be written as

1198821 (119903) = 119860111198690 (1198961119903) + 119861111198840 (1198961119903) + 119862111198680 (1198961119903)+ 119863111198700 (1198961119903)

120597119882120597119903 1 (119903) = minus1198961119860111198691 (1198961119903) minus 1198961119861111198841 (1198961119903)

+ 1198961119862111198681 (1198961119903) minus 1198961119863111198701 (1198961119903)



1198761 (119903) = 119863 11986011119896131198691 (1198961119903) + 11986111119896131198841 (1198961119903)+ 11986211119896131198681 (1198961119903) minus 11986311119896131198701 (1198961119903)

(2)



119863


Ψ13 = 0

119863 [119896131198691 (1198961119903119888) 119896131198841 (1198961119903119888) 119896131198681 (1198961119903119888) minus119896131198701 (1198961119903119888)]Ψ13 = 0(5)


[[[[[[


J1 Y1 I1 K1


]]]]]]Ψ11 = 0 (6)



(7a)


(7b)

I1 = [119896121198680 (1198961119903119888) minus 1198961119903119888 1198681 (1198961119903119888) + 12059011198961119903119888 1198681 (1198961119903119888)]times T13

(7c)

K1 = [119896121198700 (1198961119903119888) + 1198961119903119888 1198701 (1198961119903119888) minus 12059011198961119903119888 1198701 (1198961119903119888)]times T13

(7d)


1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816



1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816= 0 (8)





119882 = 119860+1119867(2)0 (1198961119903) + 119860minus1119867(1)0 (1198961119903) + 119861+11198700 (1198961119903)+ 119861minus1 1198680 (1198961119903)

(9)


119867(1)0 (1198961119903) = 1198690 (1198961119903) + 1198941198840 (1198961119903)119867(2)0 (1198961119903) = 1198690 (1198961119903) minus 1198941198840 (1198961119903)

(10)



120597119882120597119903 1 (119903) = 119860+1 [minus11989611198691 (1198961119903) + 11989411989611198841 (1198961119903)]


(12)

1198721 (119903) = 119863119860+1 [1198961119903 1198691 (1198961119903) minus 119896121198690 (1198961119903)+ 119894119896211198840 (1198961119903) minus 1198941198961119903 1198841 (1198961119903)+ 120590



(13)


(14)






a+1 = [119860+1 1198690 (11989611199030) minus 1198941198840 (11989611199030)119861+11198700 (11989611199030) ] (15a)

b+1 = [119860+1 1198690 (1198961119903119886) minus 1198941198840 (1198961119903119886)119861+11198700 (1198961119903119886) ] (15b)

a+2 = [119860+1 1198690 (1198962119903119886) minus 1198941198840 (1198962119903119886)119861+11198700 (1198962119903119886) ] (16a)

b+2 = [119860+1 1198690 (1198962119903119887) minus 1198941198840 (1198962119903119887)119861+11198700 (1198962119903119887) ] (16b)

a+3 = [119860+1 1198690 (1198961119903119888) minus 1198941198840 (1198961119903119888)119861+11198700 (1198961119903119888) ] (17a)

b+3 = [119860+1 1198690 (1198961119903119888) minus 1198941198840 (1198961119903119888)119861+11198700 (1198961119903119888) ] (17b)


b+2 = f+2 (rb minus ra) a+2 (18b)



f+1 = [[[[[

1198690 (1198961119903119886) minus 1198941198840 (1198961119903119886)1198690 (11989611199030) minus 1198941198840 (11989611199030) 00 1198700 (1198961119903119886)1198700 (11989611199030)

]]]]]

(19a)

f+2 = [[[[[

1198690 (1198961119903119887) minus 1198941198840 (1198961119903119887)1198690 (1198961119903119886) minus 1198941198840 (1198961119903119886) 00 1198700 (1198961119903119887)1198700 (1198961119903119886)

]]]]]

(19b)

f+3 = [[[[[

1198690 (1198961119903119888) minus 1198941198840 (1198961119903119888)1198690 (1198961119903119887) minus 1198941198840 (1198961119903119887) 00 1198700 (1198961119903119888)1198700 (1198961119903119887)

]]]]]

(19c)














fminus1 = [[[[[

1198690 (11989611199030) + 1198941198840 (11989611199030)1198690 (1198961119903119886) + 1198941198840 (1198961119903119886) 00 1198680 (11989611199030)1198680 (1198961119903119886)

]]]]]

(24a)

fminus2 = [[[[[

1198690 (1198961119903119886) + 1198941198840 (1198961119903119886)1198690 (1198961119903119887) + 1198941198840 (1198961119903119887) 00 1198680 (1198961119903119886)1198680 (1198961119903119887)

]]]]]

(24b)

fminus3 = [[[[[

1198690 (1198961119903119887) + 1198941198840 (1198961119903119887)1198690 (1198961119903119888) + 1198941198840 (1198961119903119888) 00 1198680 (1198961119903119887)1198680 (1198961119903119888)

]]]]]

(24c)




(25)


a+1 = R0aminus

1 (26)


= minus[[[[[


minus11989611198701 (11989611199030)1198700 (11989611199030)]]]]]

minus1

sdot [[[[[

1198690 (11989611199030) + 1198941198840 (11989611199030)1198690 (11989611199030) + 1198941198840 (11989611199030)1198680 (11989611199030)1198680 (11989611199030)minus11989611198691 (11989611199030) minus 11989411989611198841 (11989611199030)1198690 (11989611199030) + 1198941198840 (11989611199030)

11989611198681 (11989611199030)1198680 (11989611199030)]]]]]

(27)




(28)


bminus3 = R3b+

3 (29)



R3

= minus[[[[[


119896131198681 (1198961119903119888)1198680 (1198961119903119888)]]]]]

minus1

times [[[[[


minus119896311198701 (1198961119903119888)1198700 (1198961119903119888)]]]]]

(30)




[[


+ [[


]]bminus1

= [[


+ [[


]]aminus2

(31)


R+a1b+

1 + Rminusa1bminus

1 = T+a2a+

2 + Tminusa2aminus

2

R+a3b+

1 + Rminusa3bminus

1 = T+a4a+

2 + Tminusa4aminus

2 (32)





[[


+ [[


]]bminus2

= [[


+ [[


]]aminus3

(33)



R+b1b+

1 + Rminusb1bminus

1 = T+b2a+

2 + Tminusb2aminus

2

R+b3b+

1 + Rminusb3bminus

1 = T+b4a+

2 + Tminusb4aminus

2 (34)



minus

1


+

3

R+a1b+

1 + Rminusa1bminus

1 = T+a2a+

2 + Tminusa2aminus

2

R+b1b+

2 + Rminusb1bminus

2 = T+b2a+

3 + Tminusb2aminus

3

R+a3b+

1 + Rminusa3bminus

1 = T+a4a+

2 + Tminusa4aminus

2

R+b3b+

2 + Rminusb3bminus

2 = T+b4a+

3 + Tminusb4aminus

3 (35)


[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[


]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[


]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

= 119865 (119891)

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[


]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

= 0 (36)


1003816100381610038161003816119865 (119891)1003816100381610038161003816 = 0 (37)













0 200 400 600 800 1000 1200 1400 1600Frequency (Hz)

minus4

minus3

minus2

minus1

0

1

2

3

4



F(f

)








0 10 15 20 25 30 35 40Frequency (Hz)

minus5

0

5



5

F(f

)

times10minus8

(a) 0ndash40Hz

40 60 80 100 120 140 160 180minus10

minus8

minus6

minus4

minus2

0

2

4

6



F(f

)

times10minus6

Frequency (Hz)

(b) 40ndash180Hz

200 250 300 350 400 450minus6

minus4

minus2

0

2

4

F(f

)

times10minus3

Frequency (Hz)



(c) 180ndash450Hz









1


0

367

48

734

96

110

244

146

992

183

74

220

488

257

236

293

984

330

732

(a)

1


SMX = 853383

0

948

203

189

641

284

461

379

281

47410

2

568

922

663742

7585

62

853383

(b)

0

683

9

136

78

205

17

273

56

341

95

410

34

478

73

547

12

615

51

1


SMX = 61551

(c)

0

143

273

286

545

429

818

573

09

716

363

859

635

10029

1

114

618

12894

5

1


SMX = 128945

(d)

0

140

312

280

623

420

935

561

247

701

558

841

87

982

181

112

249

12628

1


SMX = 12628

(e)




minus30

minus20

minus10

0

10

20

30

40

50

60

410

019

629

238

848

458

067

677

286

896

410

6011

5612

5213

4814

44

Outer to inner

Inner to outer

(Hz)

Tran

smiss

ibili

ty (d

B)

14481 Hz12479 Hz

41519 Hz3776Hz

138886 Hz57294Hz32196Hz

7044Hz

1683 Hz


Figure 5

0

1000

2000

3000

4000

5000

6000

7000

0001 0002 0003 0004 0005

(Hz)

1st mode

2nd mode

3rd mode

4th mode

(a) Thickness

(Hz)

0

200

400

600

800

1000

1200

1400

0001 0006 0014 0019

1st mode

2nd mode

3rd mode

4th mode

(b) Inner radius

(Hz)

0

5000

10000

15000

20000

25000

30000

35000

40000

112 111 1 111 121

2nd mode1st mode

3rd mode

4th mode

(c) Radial span





0200400600800

100012001400160018002000

1062 2799 4540 7780 11600

PMMA

Al Ti STEEL Pb

Fre (

Hz)

1st mode

2nd mode

3rd mode

4th mode

Density (kgＧ3)

(a) Density

Fre (

Hz)

0

200

400

600

800

1000

1200

1400

032 408 721 117 2106

PMMA

PbAl

Ti STEEL

1st mode2nd mode

3rd mode

4th mode


(b) Elastic modulus





5 Conclusion





Appendix



120597119882120597119903 1 (119903119886) = 120597119882

120597119903 2 (119903119886)1198721 (119903119886) = 1198722 (119903119886)1198761 (119903119886) = 1198762 (119903119886)

(A1)



[[[[[[[[[


J2 Y2 I2 K2

119896311198691 (1198961119903119886) 119896311198841 (1198961119903119886) 119896311198681 (1198961119903119886) minus119896311198701 (1198961119903119886)

]]]]]]]]]

Ψ11

=[[[[[[[[


J3 Y3 I3 K3

119896321198691 (1198962119903119886) 119896321198841 (1198962119903119886) 119896321198681 (1198962119903119886) minus119896321198701 (1198962119903119886)

]]]]]]]]Ψ12

(A2)


J2 = 1198961119903119886 1198691 (1198961119903119886) minus 12059011198691 (1198961119903119886) minus 119896211198690 (1198961119903119886)

Y2 = 1198961119903119886 1198841 (1198961119903119886) minus 12059011198841 (1198961119903119886) minus 119896211198840 (1198961119903119886)

I2 = 1198961119903119886 12059011198681 (1198961119903119886) minus 1198681 (1198961119903119886) + 119896211198680 (1198961119903119886)

K2 = 1198961119903119886 1198701 (1198961119903119886) minus 12059011198701 (1198961119903119886) + 119896211198700 (1198961119903119886)

J3 = 1198962119903119886 1198691 (1198962119903119886) minus 12059021198691 (1198962119903119886) minus 119896221198690 (1198962119903119886)

Y3 = 1198962119903119886 1198841 (1198962119903119886) minus 12059021198841 (1198962119903119886) minus 119896221198840 (1198962119903119886)

I3 = 1198962119903119886 12059021198681 (1198962119903119886) minus 1198681 (1198962119903119886) + 119896221198680 (1198962119903119886) K3 = 1198962119903119886 1198701 (1198962119903119886) minus 12059021198701 (1198962119903119886) + 119896221198700 (1198962119903119886)

(A3)


H1Ψ11 = K1Ψ12 (A4)


1198822 (119903119887) = 1198821 (119903119887)120597119882120597119903 2 (119903119887) = 120597119882120597119903 1 (119903119887)1198722 (119903119887) = 1198721 (119903119887)1198762 (119903119887) = 1198761 (119903119887)

(A5)


[[[[[[[[


J4 Y4 I4 K4

119896321198691 (1198962119903119887) 119896321198841 (1198962119903119887) 119896321198681 (1198962119903119887) minus119896321198701 (1198962119903119887)

]]]]]]]]Ψ12

=[[[[[[[[[


J5 Y5 I5 K5

119896311198691 (1198961119903119887) 119896311198841 (1198961119903119887) 119896311198681 (1198961119903119887) minus119896311198701 (1198961119903119887)

]]]]]]]]]

Ψ13

(A6)




(A7)


K2Ψ12 = H2Ψ13 (A8)






Acknowledgments


References






















RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




1198761 (119903) = 119863 11986011119896131198691 (1198961119903) + 11986111119896131198841 (1198961119903)+ 11986211119896131198681 (1198961119903) minus 11986311119896131198701 (1198961119903)

(2)



119863


Ψ13 = 0

119863 [119896131198691 (1198961119903119888) 119896131198841 (1198961119903119888) 119896131198681 (1198961119903119888) minus119896131198701 (1198961119903119888)]Ψ13 = 0(5)


[[[[[[


J1 Y1 I1 K1


]]]]]]Ψ11 = 0 (6)



(7a)


(7b)

I1 = [119896121198680 (1198961119903119888) minus 1198961119903119888 1198681 (1198961119903119888) + 12059011198961119903119888 1198681 (1198961119903119888)]times T13

(7c)

K1 = [119896121198700 (1198961119903119888) + 1198961119903119888 1198701 (1198961119903119888) minus 12059011198961119903119888 1198701 (1198961119903119888)]times T13

(7d)


1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816



1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816= 0 (8)





119882 = 119860+1119867(2)0 (1198961119903) + 119860minus1119867(1)0 (1198961119903) + 119861+11198700 (1198961119903)+ 119861minus1 1198680 (1198961119903)

(9)


119867(1)0 (1198961119903) = 1198690 (1198961119903) + 1198941198840 (1198961119903)119867(2)0 (1198961119903) = 1198690 (1198961119903) minus 1198941198840 (1198961119903)

(10)



120597119882120597119903 1 (119903) = 119860+1 [minus11989611198691 (1198961119903) + 11989411989611198841 (1198961119903)]


(12)

1198721 (119903) = 119863119860+1 [1198961119903 1198691 (1198961119903) minus 119896121198690 (1198961119903)+ 119894119896211198840 (1198961119903) minus 1198941198961119903 1198841 (1198961119903)+ 120590



(13)


(14)






a+1 = [119860+1 1198690 (11989611199030) minus 1198941198840 (11989611199030)119861+11198700 (11989611199030) ] (15a)

b+1 = [119860+1 1198690 (1198961119903119886) minus 1198941198840 (1198961119903119886)119861+11198700 (1198961119903119886) ] (15b)

a+2 = [119860+1 1198690 (1198962119903119886) minus 1198941198840 (1198962119903119886)119861+11198700 (1198962119903119886) ] (16a)

b+2 = [119860+1 1198690 (1198962119903119887) minus 1198941198840 (1198962119903119887)119861+11198700 (1198962119903119887) ] (16b)

a+3 = [119860+1 1198690 (1198961119903119888) minus 1198941198840 (1198961119903119888)119861+11198700 (1198961119903119888) ] (17a)

b+3 = [119860+1 1198690 (1198961119903119888) minus 1198941198840 (1198961119903119888)119861+11198700 (1198961119903119888) ] (17b)


b+2 = f+2 (rb minus ra) a+2 (18b)



f+1 = [[[[[

1198690 (1198961119903119886) minus 1198941198840 (1198961119903119886)1198690 (11989611199030) minus 1198941198840 (11989611199030) 00 1198700 (1198961119903119886)1198700 (11989611199030)

]]]]]

(19a)

f+2 = [[[[[

1198690 (1198961119903119887) minus 1198941198840 (1198961119903119887)1198690 (1198961119903119886) minus 1198941198840 (1198961119903119886) 00 1198700 (1198961119903119887)1198700 (1198961119903119886)

]]]]]

(19b)

f+3 = [[[[[

1198690 (1198961119903119888) minus 1198941198840 (1198961119903119888)1198690 (1198961119903119887) minus 1198941198840 (1198961119903119887) 00 1198700 (1198961119903119888)1198700 (1198961119903119887)

]]]]]

(19c)














fminus1 = [[[[[

1198690 (11989611199030) + 1198941198840 (11989611199030)1198690 (1198961119903119886) + 1198941198840 (1198961119903119886) 00 1198680 (11989611199030)1198680 (1198961119903119886)

]]]]]

(24a)

fminus2 = [[[[[

1198690 (1198961119903119886) + 1198941198840 (1198961119903119886)1198690 (1198961119903119887) + 1198941198840 (1198961119903119887) 00 1198680 (1198961119903119886)1198680 (1198961119903119887)

]]]]]

(24b)

fminus3 = [[[[[

1198690 (1198961119903119887) + 1198941198840 (1198961119903119887)1198690 (1198961119903119888) + 1198941198840 (1198961119903119888) 00 1198680 (1198961119903119887)1198680 (1198961119903119888)

]]]]]

(24c)




(25)


a+1 = R0aminus

1 (26)


= minus[[[[[


minus11989611198701 (11989611199030)1198700 (11989611199030)]]]]]

minus1

sdot [[[[[

1198690 (11989611199030) + 1198941198840 (11989611199030)1198690 (11989611199030) + 1198941198840 (11989611199030)1198680 (11989611199030)1198680 (11989611199030)minus11989611198691 (11989611199030) minus 11989411989611198841 (11989611199030)1198690 (11989611199030) + 1198941198840 (11989611199030)

11989611198681 (11989611199030)1198680 (11989611199030)]]]]]

(27)




(28)


bminus3 = R3b+

3 (29)



R3

= minus[[[[[


119896131198681 (1198961119903119888)1198680 (1198961119903119888)]]]]]

minus1

times [[[[[


minus119896311198701 (1198961119903119888)1198700 (1198961119903119888)]]]]]

(30)




[[


+ [[


]]bminus1

= [[


+ [[


]]aminus2

(31)


R+a1b+

1 + Rminusa1bminus

1 = T+a2a+

2 + Tminusa2aminus

2

R+a3b+

1 + Rminusa3bminus

1 = T+a4a+

2 + Tminusa4aminus

2 (32)





[[


+ [[


]]bminus2

= [[


+ [[


]]aminus3

(33)



R+b1b+

1 + Rminusb1bminus

1 = T+b2a+

2 + Tminusb2aminus

2

R+b3b+

1 + Rminusb3bminus

1 = T+b4a+

2 + Tminusb4aminus

2 (34)



minus

1


+

3

R+a1b+

1 + Rminusa1bminus

1 = T+a2a+

2 + Tminusa2aminus

2

R+b1b+

2 + Rminusb1bminus

2 = T+b2a+

3 + Tminusb2aminus

3

R+a3b+

1 + Rminusa3bminus

1 = T+a4a+

2 + Tminusa4aminus

2

R+b3b+

2 + Rminusb3bminus

2 = T+b4a+

3 + Tminusb4aminus

3 (35)


[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[


]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[


]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

= 119865 (119891)

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[


]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

= 0 (36)


1003816100381610038161003816119865 (119891)1003816100381610038161003816 = 0 (37)













0 200 400 600 800 1000 1200 1400 1600Frequency (Hz)

minus4

minus3

minus2

minus1

0

1

2

3

4



F(f

)








0 10 15 20 25 30 35 40Frequency (Hz)

minus5

0

5



5

F(f

)

times10minus8

(a) 0ndash40Hz

40 60 80 100 120 140 160 180minus10

minus8

minus6

minus4

minus2

0

2

4

6



F(f

)

times10minus6

Frequency (Hz)

(b) 40ndash180Hz

200 250 300 350 400 450minus6

minus4

minus2

0

2

4

F(f

)

times10minus3

Frequency (Hz)



(c) 180ndash450Hz









1


0

367

48

734

96

110

244

146

992

183

74

220

488

257

236

293

984

330

732

(a)

1


SMX = 853383

0

948

203

189

641

284

461

379

281

47410

2

568

922

663742

7585

62

853383

(b)

0

683

9

136

78

205

17

273

56

341

95

410

34

478

73

547

12

615

51

1


SMX = 61551

(c)

0

143

273

286

545

429

818

573

09

716

363

859

635

10029

1

114

618

12894

5

1


SMX = 128945

(d)

0

140

312

280

623

420

935

561

247

701

558

841

87

982

181

112

249

12628

1


SMX = 12628

(e)




minus30

minus20

minus10

0

10

20

30

40

50

60

410

019

629

238

848

458

067

677

286

896

410

6011

5612

5213

4814

44

Outer to inner

Inner to outer

(Hz)

Tran

smiss

ibili

ty (d

B)

14481 Hz12479 Hz

41519 Hz3776Hz

138886 Hz57294Hz32196Hz

7044Hz

1683 Hz


Figure 5

0

1000

2000

3000

4000

5000

6000

7000

0001 0002 0003 0004 0005

(Hz)

1st mode

2nd mode

3rd mode

4th mode

(a) Thickness

(Hz)

0

200

400

600

800

1000

1200

1400

0001 0006 0014 0019

1st mode

2nd mode

3rd mode

4th mode

(b) Inner radius

(Hz)

0

5000

10000

15000

20000

25000

30000

35000

40000

112 111 1 111 121

2nd mode1st mode

3rd mode

4th mode

(c) Radial span





0200400600800

100012001400160018002000

1062 2799 4540 7780 11600

PMMA

Al Ti STEEL Pb

Fre (

Hz)

1st mode

2nd mode

3rd mode

4th mode

Density (kgＧ3)

(a) Density

Fre (

Hz)

0

200

400

600

800

1000

1200

1400

032 408 721 117 2106

PMMA

PbAl

Ti STEEL

1st mode2nd mode

3rd mode

4th mode


(b) Elastic modulus





5 Conclusion





Appendix



120597119882120597119903 1 (119903119886) = 120597119882

120597119903 2 (119903119886)1198721 (119903119886) = 1198722 (119903119886)1198761 (119903119886) = 1198762 (119903119886)

(A1)



[[[[[[[[[


J2 Y2 I2 K2

119896311198691 (1198961119903119886) 119896311198841 (1198961119903119886) 119896311198681 (1198961119903119886) minus119896311198701 (1198961119903119886)

]]]]]]]]]

Ψ11

=[[[[[[[[


J3 Y3 I3 K3

119896321198691 (1198962119903119886) 119896321198841 (1198962119903119886) 119896321198681 (1198962119903119886) minus119896321198701 (1198962119903119886)

]]]]]]]]Ψ12

(A2)


J2 = 1198961119903119886 1198691 (1198961119903119886) minus 12059011198691 (1198961119903119886) minus 119896211198690 (1198961119903119886)

Y2 = 1198961119903119886 1198841 (1198961119903119886) minus 12059011198841 (1198961119903119886) minus 119896211198840 (1198961119903119886)

I2 = 1198961119903119886 12059011198681 (1198961119903119886) minus 1198681 (1198961119903119886) + 119896211198680 (1198961119903119886)

K2 = 1198961119903119886 1198701 (1198961119903119886) minus 12059011198701 (1198961119903119886) + 119896211198700 (1198961119903119886)

J3 = 1198962119903119886 1198691 (1198962119903119886) minus 12059021198691 (1198962119903119886) minus 119896221198690 (1198962119903119886)

Y3 = 1198962119903119886 1198841 (1198962119903119886) minus 12059021198841 (1198962119903119886) minus 119896221198840 (1198962119903119886)

I3 = 1198962119903119886 12059021198681 (1198962119903119886) minus 1198681 (1198962119903119886) + 119896221198680 (1198962119903119886) K3 = 1198962119903119886 1198701 (1198962119903119886) minus 12059021198701 (1198962119903119886) + 119896221198700 (1198962119903119886)

(A3)


H1Ψ11 = K1Ψ12 (A4)


1198822 (119903119887) = 1198821 (119903119887)120597119882120597119903 2 (119903119887) = 120597119882120597119903 1 (119903119887)1198722 (119903119887) = 1198721 (119903119887)1198762 (119903119887) = 1198761 (119903119887)

(A5)


[[[[[[[[


J4 Y4 I4 K4

119896321198691 (1198962119903119887) 119896321198841 (1198962119903119887) 119896321198681 (1198962119903119887) minus119896321198701 (1198962119903119887)

]]]]]]]]Ψ12

=[[[[[[[[[


J5 Y5 I5 K5

119896311198691 (1198961119903119887) 119896311198841 (1198961119903119887) 119896311198681 (1198961119903119887) minus119896311198701 (1198961119903119887)

]]]]]]]]]

Ψ13

(A6)




(A7)


K2Ψ12 = H2Ψ13 (A8)






Acknowledgments


References






















RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




119882 = 119860+1119867(2)0 (1198961119903) + 119860minus1119867(1)0 (1198961119903) + 119861+11198700 (1198961119903)+ 119861minus1 1198680 (1198961119903)

(9)


119867(1)0 (1198961119903) = 1198690 (1198961119903) + 1198941198840 (1198961119903)119867(2)0 (1198961119903) = 1198690 (1198961119903) minus 1198941198840 (1198961119903)

(10)



120597119882120597119903 1 (119903) = 119860+1 [minus11989611198691 (1198961119903) + 11989411989611198841 (1198961119903)]


(12)

1198721 (119903) = 119863119860+1 [1198961119903 1198691 (1198961119903) minus 119896121198690 (1198961119903)+ 119894119896211198840 (1198961119903) minus 1198941198961119903 1198841 (1198961119903)+ 120590



(13)


(14)






a+1 = [119860+1 1198690 (11989611199030) minus 1198941198840 (11989611199030)119861+11198700 (11989611199030) ] (15a)

b+1 = [119860+1 1198690 (1198961119903119886) minus 1198941198840 (1198961119903119886)119861+11198700 (1198961119903119886) ] (15b)

a+2 = [119860+1 1198690 (1198962119903119886) minus 1198941198840 (1198962119903119886)119861+11198700 (1198962119903119886) ] (16a)

b+2 = [119860+1 1198690 (1198962119903119887) minus 1198941198840 (1198962119903119887)119861+11198700 (1198962119903119887) ] (16b)

a+3 = [119860+1 1198690 (1198961119903119888) minus 1198941198840 (1198961119903119888)119861+11198700 (1198961119903119888) ] (17a)

b+3 = [119860+1 1198690 (1198961119903119888) minus 1198941198840 (1198961119903119888)119861+11198700 (1198961119903119888) ] (17b)


b+2 = f+2 (rb minus ra) a+2 (18b)



f+1 = [[[[[

1198690 (1198961119903119886) minus 1198941198840 (1198961119903119886)1198690 (11989611199030) minus 1198941198840 (11989611199030) 00 1198700 (1198961119903119886)1198700 (11989611199030)

]]]]]

(19a)

f+2 = [[[[[

1198690 (1198961119903119887) minus 1198941198840 (1198961119903119887)1198690 (1198961119903119886) minus 1198941198840 (1198961119903119886) 00 1198700 (1198961119903119887)1198700 (1198961119903119886)

]]]]]

(19b)

f+3 = [[[[[

1198690 (1198961119903119888) minus 1198941198840 (1198961119903119888)1198690 (1198961119903119887) minus 1198941198840 (1198961119903119887) 00 1198700 (1198961119903119888)1198700 (1198961119903119887)

]]]]]

(19c)














fminus1 = [[[[[

1198690 (11989611199030) + 1198941198840 (11989611199030)1198690 (1198961119903119886) + 1198941198840 (1198961119903119886) 00 1198680 (11989611199030)1198680 (1198961119903119886)

]]]]]

(24a)

fminus2 = [[[[[

1198690 (1198961119903119886) + 1198941198840 (1198961119903119886)1198690 (1198961119903119887) + 1198941198840 (1198961119903119887) 00 1198680 (1198961119903119886)1198680 (1198961119903119887)

]]]]]

(24b)

fminus3 = [[[[[

1198690 (1198961119903119887) + 1198941198840 (1198961119903119887)1198690 (1198961119903119888) + 1198941198840 (1198961119903119888) 00 1198680 (1198961119903119887)1198680 (1198961119903119888)

]]]]]

(24c)




(25)


a+1 = R0aminus

1 (26)


= minus[[[[[


minus11989611198701 (11989611199030)1198700 (11989611199030)]]]]]

minus1

sdot [[[[[

1198690 (11989611199030) + 1198941198840 (11989611199030)1198690 (11989611199030) + 1198941198840 (11989611199030)1198680 (11989611199030)1198680 (11989611199030)minus11989611198691 (11989611199030) minus 11989411989611198841 (11989611199030)1198690 (11989611199030) + 1198941198840 (11989611199030)

11989611198681 (11989611199030)1198680 (11989611199030)]]]]]

(27)




(28)


bminus3 = R3b+

3 (29)



R3

= minus[[[[[


119896131198681 (1198961119903119888)1198680 (1198961119903119888)]]]]]

minus1

times [[[[[


minus119896311198701 (1198961119903119888)1198700 (1198961119903119888)]]]]]

(30)




[[


+ [[


]]bminus1

= [[


+ [[


]]aminus2

(31)


R+a1b+

1 + Rminusa1bminus

1 = T+a2a+

2 + Tminusa2aminus

2

R+a3b+

1 + Rminusa3bminus

1 = T+a4a+

2 + Tminusa4aminus

2 (32)





[[


+ [[


]]bminus2

= [[


+ [[


]]aminus3

(33)



R+b1b+

1 + Rminusb1bminus

1 = T+b2a+

2 + Tminusb2aminus

2

R+b3b+

1 + Rminusb3bminus

1 = T+b4a+

2 + Tminusb4aminus

2 (34)



minus

1


+

3

R+a1b+

1 + Rminusa1bminus

1 = T+a2a+

2 + Tminusa2aminus

2

R+b1b+

2 + Rminusb1bminus

2 = T+b2a+

3 + Tminusb2aminus

3

R+a3b+

1 + Rminusa3bminus

1 = T+a4a+

2 + Tminusa4aminus

2

R+b3b+

2 + Rminusb3bminus

2 = T+b4a+

3 + Tminusb4aminus

3 (35)


[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[


]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[


]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

= 119865 (119891)

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[


]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

= 0 (36)


1003816100381610038161003816119865 (119891)1003816100381610038161003816 = 0 (37)













0 200 400 600 800 1000 1200 1400 1600Frequency (Hz)

minus4

minus3

minus2

minus1

0

1

2

3

4



F(f

)








0 10 15 20 25 30 35 40Frequency (Hz)

minus5

0

5



5

F(f

)

times10minus8

(a) 0ndash40Hz

40 60 80 100 120 140 160 180minus10

minus8

minus6

minus4

minus2

0

2

4

6



F(f

)

times10minus6

Frequency (Hz)

(b) 40ndash180Hz

200 250 300 350 400 450minus6

minus4

minus2

0

2

4

F(f

)

times10minus3

Frequency (Hz)



(c) 180ndash450Hz









1


0

367

48

734

96

110

244

146

992

183

74

220

488

257

236

293

984

330

732

(a)

1


SMX = 853383

0

948

203

189

641

284

461

379

281

47410

2

568

922

663742

7585

62

853383

(b)

0

683

9

136

78

205

17

273

56

341

95

410

34

478

73

547

12

615

51

1


SMX = 61551

(c)

0

143

273

286

545

429

818

573

09

716

363

859

635

10029

1

114

618

12894

5

1


SMX = 128945

(d)

0

140

312

280

623

420

935

561

247

701

558

841

87

982

181

112

249

12628

1


SMX = 12628

(e)




minus30

minus20

minus10

0

10

20

30

40

50

60

410

019

629

238

848

458

067

677

286

896

410

6011

5612

5213

4814

44

Outer to inner

Inner to outer

(Hz)

Tran

smiss

ibili

ty (d

B)

14481 Hz12479 Hz

41519 Hz3776Hz

138886 Hz57294Hz32196Hz

7044Hz

1683 Hz


Figure 5

0

1000

2000

3000

4000

5000

6000

7000

0001 0002 0003 0004 0005

(Hz)

1st mode

2nd mode

3rd mode

4th mode

(a) Thickness

(Hz)

0

200

400

600

800

1000

1200

1400

0001 0006 0014 0019

1st mode

2nd mode

3rd mode

4th mode

(b) Inner radius

(Hz)

0

5000

10000

15000

20000

25000

30000

35000

40000

112 111 1 111 121

2nd mode1st mode

3rd mode

4th mode

(c) Radial span





0200400600800

100012001400160018002000

1062 2799 4540 7780 11600

PMMA

Al Ti STEEL Pb

Fre (

Hz)

1st mode

2nd mode

3rd mode

4th mode

Density (kgＧ3)

(a) Density

Fre (

Hz)

0

200

400

600

800

1000

1200

1400

032 408 721 117 2106

PMMA

PbAl

Ti STEEL

1st mode2nd mode

3rd mode

4th mode


(b) Elastic modulus





5 Conclusion





Appendix



120597119882120597119903 1 (119903119886) = 120597119882

120597119903 2 (119903119886)1198721 (119903119886) = 1198722 (119903119886)1198761 (119903119886) = 1198762 (119903119886)

(A1)



[[[[[[[[[


J2 Y2 I2 K2

119896311198691 (1198961119903119886) 119896311198841 (1198961119903119886) 119896311198681 (1198961119903119886) minus119896311198701 (1198961119903119886)

]]]]]]]]]

Ψ11

=[[[[[[[[


J3 Y3 I3 K3

119896321198691 (1198962119903119886) 119896321198841 (1198962119903119886) 119896321198681 (1198962119903119886) minus119896321198701 (1198962119903119886)

]]]]]]]]Ψ12

(A2)


J2 = 1198961119903119886 1198691 (1198961119903119886) minus 12059011198691 (1198961119903119886) minus 119896211198690 (1198961119903119886)

Y2 = 1198961119903119886 1198841 (1198961119903119886) minus 12059011198841 (1198961119903119886) minus 119896211198840 (1198961119903119886)

I2 = 1198961119903119886 12059011198681 (1198961119903119886) minus 1198681 (1198961119903119886) + 119896211198680 (1198961119903119886)

K2 = 1198961119903119886 1198701 (1198961119903119886) minus 12059011198701 (1198961119903119886) + 119896211198700 (1198961119903119886)

J3 = 1198962119903119886 1198691 (1198962119903119886) minus 12059021198691 (1198962119903119886) minus 119896221198690 (1198962119903119886)

Y3 = 1198962119903119886 1198841 (1198962119903119886) minus 12059021198841 (1198962119903119886) minus 119896221198840 (1198962119903119886)

I3 = 1198962119903119886 12059021198681 (1198962119903119886) minus 1198681 (1198962119903119886) + 119896221198680 (1198962119903119886) K3 = 1198962119903119886 1198701 (1198962119903119886) minus 12059021198701 (1198962119903119886) + 119896221198700 (1198962119903119886)

(A3)


H1Ψ11 = K1Ψ12 (A4)


1198822 (119903119887) = 1198821 (119903119887)120597119882120597119903 2 (119903119887) = 120597119882120597119903 1 (119903119887)1198722 (119903119887) = 1198721 (119903119887)1198762 (119903119887) = 1198761 (119903119887)

(A5)


[[[[[[[[


J4 Y4 I4 K4

119896321198691 (1198962119903119887) 119896321198841 (1198962119903119887) 119896321198681 (1198962119903119887) minus119896321198701 (1198962119903119887)

]]]]]]]]Ψ12

=[[[[[[[[[


J5 Y5 I5 K5

119896311198691 (1198961119903119887) 119896311198841 (1198961119903119887) 119896311198681 (1198961119903119887) minus119896311198701 (1198961119903119887)

]]]]]]]]]

Ψ13

(A6)




(A7)


K2Ψ12 = H2Ψ13 (A8)






Acknowledgments


References






















RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia















fminus1 = [[[[[

1198690 (11989611199030) + 1198941198840 (11989611199030)1198690 (1198961119903119886) + 1198941198840 (1198961119903119886) 00 1198680 (11989611199030)1198680 (1198961119903119886)

]]]]]

(24a)

fminus2 = [[[[[

1198690 (1198961119903119886) + 1198941198840 (1198961119903119886)1198690 (1198961119903119887) + 1198941198840 (1198961119903119887) 00 1198680 (1198961119903119886)1198680 (1198961119903119887)

]]]]]

(24b)

fminus3 = [[[[[

1198690 (1198961119903119887) + 1198941198840 (1198961119903119887)1198690 (1198961119903119888) + 1198941198840 (1198961119903119888) 00 1198680 (1198961119903119887)1198680 (1198961119903119888)

]]]]]

(24c)




(25)


a+1 = R0aminus

1 (26)


= minus[[[[[


minus11989611198701 (11989611199030)1198700 (11989611199030)]]]]]

minus1

sdot [[[[[

1198690 (11989611199030) + 1198941198840 (11989611199030)1198690 (11989611199030) + 1198941198840 (11989611199030)1198680 (11989611199030)1198680 (11989611199030)minus11989611198691 (11989611199030) minus 11989411989611198841 (11989611199030)1198690 (11989611199030) + 1198941198840 (11989611199030)

11989611198681 (11989611199030)1198680 (11989611199030)]]]]]

(27)




(28)


bminus3 = R3b+

3 (29)



R3

= minus[[[[[


119896131198681 (1198961119903119888)1198680 (1198961119903119888)]]]]]

minus1

times [[[[[


minus119896311198701 (1198961119903119888)1198700 (1198961119903119888)]]]]]

(30)




[[


+ [[


]]bminus1

= [[


+ [[


]]aminus2

(31)


R+a1b+

1 + Rminusa1bminus

1 = T+a2a+

2 + Tminusa2aminus

2

R+a3b+

1 + Rminusa3bminus

1 = T+a4a+

2 + Tminusa4aminus

2 (32)





[[


+ [[


]]bminus2

= [[


+ [[


]]aminus3

(33)



R+b1b+

1 + Rminusb1bminus

1 = T+b2a+

2 + Tminusb2aminus

2

R+b3b+

1 + Rminusb3bminus

1 = T+b4a+

2 + Tminusb4aminus

2 (34)



minus

1


+

3

R+a1b+

1 + Rminusa1bminus

1 = T+a2a+

2 + Tminusa2aminus

2

R+b1b+

2 + Rminusb1bminus

2 = T+b2a+

3 + Tminusb2aminus

3

R+a3b+

1 + Rminusa3bminus

1 = T+a4a+

2 + Tminusa4aminus

2

R+b3b+

2 + Rminusb3bminus

2 = T+b4a+

3 + Tminusb4aminus

3 (35)


[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[


]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[


]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

= 119865 (119891)

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[


]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

= 0 (36)


1003816100381610038161003816119865 (119891)1003816100381610038161003816 = 0 (37)













0 200 400 600 800 1000 1200 1400 1600Frequency (Hz)

minus4

minus3

minus2

minus1

0

1

2

3

4



F(f

)








0 10 15 20 25 30 35 40Frequency (Hz)

minus5

0

5



5

F(f

)

times10minus8

(a) 0ndash40Hz

40 60 80 100 120 140 160 180minus10

minus8

minus6

minus4

minus2

0

2

4

6



F(f

)

times10minus6

Frequency (Hz)

(b) 40ndash180Hz

200 250 300 350 400 450minus6

minus4

minus2

0

2

4

F(f

)

times10minus3

Frequency (Hz)



(c) 180ndash450Hz









1


0

367

48

734

96

110

244

146

992

183

74

220

488

257

236

293

984

330

732

(a)

1


SMX = 853383

0

948

203

189

641

284

461

379

281

47410

2

568

922

663742

7585

62

853383

(b)

0

683

9

136

78

205

17

273

56

341

95

410

34

478

73

547

12

615

51

1


SMX = 61551

(c)

0

143

273

286

545

429

818

573

09

716

363

859

635

10029

1

114

618

12894

5

1


SMX = 128945

(d)

0

140

312

280

623

420

935

561

247

701

558

841

87

982

181

112

249

12628

1


SMX = 12628

(e)




minus30

minus20

minus10

0

10

20

30

40

50

60

410

019

629

238

848

458

067

677

286

896

410

6011

5612

5213

4814

44

Outer to inner

Inner to outer

(Hz)

Tran

smiss

ibili

ty (d

B)

14481 Hz12479 Hz

41519 Hz3776Hz

138886 Hz57294Hz32196Hz

7044Hz

1683 Hz


Figure 5

0

1000

2000

3000

4000

5000

6000

7000

0001 0002 0003 0004 0005

(Hz)

1st mode

2nd mode

3rd mode

4th mode

(a) Thickness

(Hz)

0

200

400

600

800

1000

1200

1400

0001 0006 0014 0019

1st mode

2nd mode

3rd mode

4th mode

(b) Inner radius

(Hz)

0

5000

10000

15000

20000

25000

30000

35000

40000

112 111 1 111 121

2nd mode1st mode

3rd mode

4th mode

(c) Radial span





0200400600800

100012001400160018002000

1062 2799 4540 7780 11600

PMMA

Al Ti STEEL Pb

Fre (

Hz)

1st mode

2nd mode

3rd mode

4th mode

Density (kgＧ3)

(a) Density

Fre (

Hz)

0

200

400

600

800

1000

1200

1400

032 408 721 117 2106

PMMA

PbAl

Ti STEEL

1st mode2nd mode

3rd mode

4th mode


(b) Elastic modulus





5 Conclusion





Appendix



120597119882120597119903 1 (119903119886) = 120597119882

120597119903 2 (119903119886)1198721 (119903119886) = 1198722 (119903119886)1198761 (119903119886) = 1198762 (119903119886)

(A1)



[[[[[[[[[


J2 Y2 I2 K2

119896311198691 (1198961119903119886) 119896311198841 (1198961119903119886) 119896311198681 (1198961119903119886) minus119896311198701 (1198961119903119886)

]]]]]]]]]

Ψ11

=[[[[[[[[


J3 Y3 I3 K3

119896321198691 (1198962119903119886) 119896321198841 (1198962119903119886) 119896321198681 (1198962119903119886) minus119896321198701 (1198962119903119886)

]]]]]]]]Ψ12

(A2)


J2 = 1198961119903119886 1198691 (1198961119903119886) minus 12059011198691 (1198961119903119886) minus 119896211198690 (1198961119903119886)

Y2 = 1198961119903119886 1198841 (1198961119903119886) minus 12059011198841 (1198961119903119886) minus 119896211198840 (1198961119903119886)

I2 = 1198961119903119886 12059011198681 (1198961119903119886) minus 1198681 (1198961119903119886) + 119896211198680 (1198961119903119886)

K2 = 1198961119903119886 1198701 (1198961119903119886) minus 12059011198701 (1198961119903119886) + 119896211198700 (1198961119903119886)

J3 = 1198962119903119886 1198691 (1198962119903119886) minus 12059021198691 (1198962119903119886) minus 119896221198690 (1198962119903119886)

Y3 = 1198962119903119886 1198841 (1198962119903119886) minus 12059021198841 (1198962119903119886) minus 119896221198840 (1198962119903119886)

I3 = 1198962119903119886 12059021198681 (1198962119903119886) minus 1198681 (1198962119903119886) + 119896221198680 (1198962119903119886) K3 = 1198962119903119886 1198701 (1198962119903119886) minus 12059021198701 (1198962119903119886) + 119896221198700 (1198962119903119886)

(A3)


H1Ψ11 = K1Ψ12 (A4)


1198822 (119903119887) = 1198821 (119903119887)120597119882120597119903 2 (119903119887) = 120597119882120597119903 1 (119903119887)1198722 (119903119887) = 1198721 (119903119887)1198762 (119903119887) = 1198761 (119903119887)

(A5)


[[[[[[[[


J4 Y4 I4 K4

119896321198691 (1198962119903119887) 119896321198841 (1198962119903119887) 119896321198681 (1198962119903119887) minus119896321198701 (1198962119903119887)

]]]]]]]]Ψ12

=[[[[[[[[[


J5 Y5 I5 K5

119896311198691 (1198961119903119887) 119896311198841 (1198961119903119887) 119896311198681 (1198961119903119887) minus119896311198701 (1198961119903119887)

]]]]]]]]]

Ψ13

(A6)




(A7)


K2Ψ12 = H2Ψ13 (A8)






Acknowledgments


References






















RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




R3

= minus[[[[[


119896131198681 (1198961119903119888)1198680 (1198961119903119888)]]]]]

minus1

times [[[[[


minus119896311198701 (1198961119903119888)1198700 (1198961119903119888)]]]]]

(30)




[[


+ [[


]]bminus1

= [[


+ [[


]]aminus2

(31)


R+a1b+

1 + Rminusa1bminus

1 = T+a2a+

2 + Tminusa2aminus

2

R+a3b+

1 + Rminusa3bminus

1 = T+a4a+

2 + Tminusa4aminus

2 (32)





[[


+ [[


]]bminus2

= [[


+ [[


]]aminus3

(33)



R+b1b+

1 + Rminusb1bminus

1 = T+b2a+

2 + Tminusb2aminus

2

R+b3b+

1 + Rminusb3bminus

1 = T+b4a+

2 + Tminusb4aminus

2 (34)



minus

1


+

3

R+a1b+

1 + Rminusa1bminus

1 = T+a2a+

2 + Tminusa2aminus

2

R+b1b+

2 + Rminusb1bminus

2 = T+b2a+

3 + Tminusb2aminus

3

R+a3b+

1 + Rminusa3bminus

1 = T+a4a+

2 + Tminusa4aminus

2

R+b3b+

2 + Rminusb3bminus

2 = T+b4a+

3 + Tminusb4aminus

3 (35)


[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[


]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[


]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

= 119865 (119891)

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[


]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

= 0 (36)


1003816100381610038161003816119865 (119891)1003816100381610038161003816 = 0 (37)













0 200 400 600 800 1000 1200 1400 1600Frequency (Hz)

minus4

minus3

minus2

minus1

0

1

2

3

4



F(f

)








0 10 15 20 25 30 35 40Frequency (Hz)

minus5

0

5



5

F(f

)

times10minus8

(a) 0ndash40Hz

40 60 80 100 120 140 160 180minus10

minus8

minus6

minus4

minus2

0

2

4

6



F(f

)

times10minus6

Frequency (Hz)

(b) 40ndash180Hz

200 250 300 350 400 450minus6

minus4

minus2

0

2

4

F(f

)

times10minus3

Frequency (Hz)



(c) 180ndash450Hz









1


0

367

48

734

96

110

244

146

992

183

74

220

488

257

236

293

984

330

732

(a)

1


SMX = 853383

0

948

203

189

641

284

461

379

281

47410

2

568

922

663742

7585

62

853383

(b)

0

683

9

136

78

205

17

273

56

341

95

410

34

478

73

547

12

615

51

1


SMX = 61551

(c)

0

143

273

286

545

429

818

573

09

716

363

859

635

10029

1

114

618

12894

5

1


SMX = 128945

(d)

0

140

312

280

623

420

935

561

247

701

558

841

87

982

181

112

249

12628

1


SMX = 12628

(e)




minus30

minus20

minus10

0

10

20

30

40

50

60

410

019

629

238

848

458

067

677

286

896

410

6011

5612

5213

4814

44

Outer to inner

Inner to outer

(Hz)

Tran

smiss

ibili

ty (d

B)

14481 Hz12479 Hz

41519 Hz3776Hz

138886 Hz57294Hz32196Hz

7044Hz

1683 Hz


Figure 5

0

1000

2000

3000

4000

5000

6000

7000

0001 0002 0003 0004 0005

(Hz)

1st mode

2nd mode

3rd mode

4th mode

(a) Thickness

(Hz)

0

200

400

600

800

1000

1200

1400

0001 0006 0014 0019

1st mode

2nd mode

3rd mode

4th mode

(b) Inner radius

(Hz)

0

5000

10000

15000

20000

25000

30000

35000

40000

112 111 1 111 121

2nd mode1st mode

3rd mode

4th mode

(c) Radial span





0200400600800

100012001400160018002000

1062 2799 4540 7780 11600

PMMA

Al Ti STEEL Pb

Fre (

Hz)

1st mode

2nd mode

3rd mode

4th mode

Density (kgＧ3)

(a) Density

Fre (

Hz)

0

200

400

600

800

1000

1200

1400

032 408 721 117 2106

PMMA

PbAl

Ti STEEL

1st mode2nd mode

3rd mode

4th mode


(b) Elastic modulus





5 Conclusion





Appendix



120597119882120597119903 1 (119903119886) = 120597119882

120597119903 2 (119903119886)1198721 (119903119886) = 1198722 (119903119886)1198761 (119903119886) = 1198762 (119903119886)

(A1)



[[[[[[[[[


J2 Y2 I2 K2

119896311198691 (1198961119903119886) 119896311198841 (1198961119903119886) 119896311198681 (1198961119903119886) minus119896311198701 (1198961119903119886)

]]]]]]]]]

Ψ11

=[[[[[[[[


J3 Y3 I3 K3

119896321198691 (1198962119903119886) 119896321198841 (1198962119903119886) 119896321198681 (1198962119903119886) minus119896321198701 (1198962119903119886)

]]]]]]]]Ψ12

(A2)


J2 = 1198961119903119886 1198691 (1198961119903119886) minus 12059011198691 (1198961119903119886) minus 119896211198690 (1198961119903119886)

Y2 = 1198961119903119886 1198841 (1198961119903119886) minus 12059011198841 (1198961119903119886) minus 119896211198840 (1198961119903119886)

I2 = 1198961119903119886 12059011198681 (1198961119903119886) minus 1198681 (1198961119903119886) + 119896211198680 (1198961119903119886)

K2 = 1198961119903119886 1198701 (1198961119903119886) minus 12059011198701 (1198961119903119886) + 119896211198700 (1198961119903119886)

J3 = 1198962119903119886 1198691 (1198962119903119886) minus 12059021198691 (1198962119903119886) minus 119896221198690 (1198962119903119886)

Y3 = 1198962119903119886 1198841 (1198962119903119886) minus 12059021198841 (1198962119903119886) minus 119896221198840 (1198962119903119886)

I3 = 1198962119903119886 12059021198681 (1198962119903119886) minus 1198681 (1198962119903119886) + 119896221198680 (1198962119903119886) K3 = 1198962119903119886 1198701 (1198962119903119886) minus 12059021198701 (1198962119903119886) + 119896221198700 (1198962119903119886)

(A3)


H1Ψ11 = K1Ψ12 (A4)


1198822 (119903119887) = 1198821 (119903119887)120597119882120597119903 2 (119903119887) = 120597119882120597119903 1 (119903119887)1198722 (119903119887) = 1198721 (119903119887)1198762 (119903119887) = 1198761 (119903119887)

(A5)


[[[[[[[[


J4 Y4 I4 K4

119896321198691 (1198962119903119887) 119896321198841 (1198962119903119887) 119896321198681 (1198962119903119887) minus119896321198701 (1198962119903119887)

]]]]]]]]Ψ12

=[[[[[[[[[


J5 Y5 I5 K5

119896311198691 (1198961119903119887) 119896311198841 (1198961119903119887) 119896311198681 (1198961119903119887) minus119896311198701 (1198961119903119887)

]]]]]]]]]

Ψ13

(A6)




(A7)


K2Ψ12 = H2Ψ13 (A8)






Acknowledgments


References






















RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




R+b1b+

1 + Rminusb1bminus

1 = T+b2a+

2 + Tminusb2aminus

2

R+b3b+

1 + Rminusb3bminus

1 = T+b4a+

2 + Tminusb4aminus

2 (34)



minus

1


+

3

R+a1b+

1 + Rminusa1bminus

1 = T+a2a+

2 + Tminusa2aminus

2

R+b1b+

2 + Rminusb1bminus

2 = T+b2a+

3 + Tminusb2aminus

3

R+a3b+

1 + Rminusa3bminus

1 = T+a4a+

2 + Tminusa4aminus

2

R+b3b+

2 + Rminusb3bminus

2 = T+b4a+

3 + Tminusb4aminus

3 (35)


[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[


]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[


]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

= 119865 (119891)

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[


]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

= 0 (36)


1003816100381610038161003816119865 (119891)1003816100381610038161003816 = 0 (37)













0 200 400 600 800 1000 1200 1400 1600Frequency (Hz)

minus4

minus3

minus2

minus1

0

1

2

3

4



F(f

)








0 10 15 20 25 30 35 40Frequency (Hz)

minus5

0

5



5

F(f

)

times10minus8

(a) 0ndash40Hz

40 60 80 100 120 140 160 180minus10

minus8

minus6

minus4

minus2

0

2

4

6



F(f

)

times10minus6

Frequency (Hz)

(b) 40ndash180Hz

200 250 300 350 400 450minus6

minus4

minus2

0

2

4

F(f

)

times10minus3

Frequency (Hz)



(c) 180ndash450Hz









1


0

367

48

734

96

110

244

146

992

183

74

220

488

257

236

293

984

330

732

(a)

1


SMX = 853383

0

948

203

189

641

284

461

379

281

47410

2

568

922

663742

7585

62

853383

(b)

0

683

9

136

78

205

17

273

56

341

95

410

34

478

73

547

12

615

51

1


SMX = 61551

(c)

0

143

273

286

545

429

818

573

09

716

363

859

635

10029

1

114

618

12894

5

1


SMX = 128945

(d)

0

140

312

280

623

420

935

561

247

701

558

841

87

982

181

112

249

12628

1


SMX = 12628

(e)




minus30

minus20

minus10

0

10

20

30

40

50

60

410

019

629

238

848

458

067

677

286

896

410

6011

5612

5213

4814

44

Outer to inner

Inner to outer

(Hz)

Tran

smiss

ibili

ty (d

B)

14481 Hz12479 Hz

41519 Hz3776Hz

138886 Hz57294Hz32196Hz

7044Hz

1683 Hz


Figure 5

0

1000

2000

3000

4000

5000

6000

7000

0001 0002 0003 0004 0005

(Hz)

1st mode

2nd mode

3rd mode

4th mode

(a) Thickness

(Hz)

0

200

400

600

800

1000

1200

1400

0001 0006 0014 0019

1st mode

2nd mode

3rd mode

4th mode

(b) Inner radius

(Hz)

0

5000

10000

15000

20000

25000

30000

35000

40000

112 111 1 111 121

2nd mode1st mode

3rd mode

4th mode

(c) Radial span





0200400600800

100012001400160018002000

1062 2799 4540 7780 11600

PMMA

Al Ti STEEL Pb

Fre (

Hz)

1st mode

2nd mode

3rd mode

4th mode

Density (kgＧ3)

(a) Density

Fre (

Hz)

0

200

400

600

800

1000

1200

1400

032 408 721 117 2106

PMMA

PbAl

Ti STEEL

1st mode2nd mode

3rd mode

4th mode


(b) Elastic modulus





5 Conclusion





Appendix



120597119882120597119903 1 (119903119886) = 120597119882

120597119903 2 (119903119886)1198721 (119903119886) = 1198722 (119903119886)1198761 (119903119886) = 1198762 (119903119886)

(A1)



[[[[[[[[[


J2 Y2 I2 K2

119896311198691 (1198961119903119886) 119896311198841 (1198961119903119886) 119896311198681 (1198961119903119886) minus119896311198701 (1198961119903119886)

]]]]]]]]]

Ψ11

=[[[[[[[[


J3 Y3 I3 K3

119896321198691 (1198962119903119886) 119896321198841 (1198962119903119886) 119896321198681 (1198962119903119886) minus119896321198701 (1198962119903119886)

]]]]]]]]Ψ12

(A2)


J2 = 1198961119903119886 1198691 (1198961119903119886) minus 12059011198691 (1198961119903119886) minus 119896211198690 (1198961119903119886)

Y2 = 1198961119903119886 1198841 (1198961119903119886) minus 12059011198841 (1198961119903119886) minus 119896211198840 (1198961119903119886)

I2 = 1198961119903119886 12059011198681 (1198961119903119886) minus 1198681 (1198961119903119886) + 119896211198680 (1198961119903119886)

K2 = 1198961119903119886 1198701 (1198961119903119886) minus 12059011198701 (1198961119903119886) + 119896211198700 (1198961119903119886)

J3 = 1198962119903119886 1198691 (1198962119903119886) minus 12059021198691 (1198962119903119886) minus 119896221198690 (1198962119903119886)

Y3 = 1198962119903119886 1198841 (1198962119903119886) minus 12059021198841 (1198962119903119886) minus 119896221198840 (1198962119903119886)

I3 = 1198962119903119886 12059021198681 (1198962119903119886) minus 1198681 (1198962119903119886) + 119896221198680 (1198962119903119886) K3 = 1198962119903119886 1198701 (1198962119903119886) minus 12059021198701 (1198962119903119886) + 119896221198700 (1198962119903119886)

(A3)


H1Ψ11 = K1Ψ12 (A4)


1198822 (119903119887) = 1198821 (119903119887)120597119882120597119903 2 (119903119887) = 120597119882120597119903 1 (119903119887)1198722 (119903119887) = 1198721 (119903119887)1198762 (119903119887) = 1198761 (119903119887)

(A5)


[[[[[[[[


J4 Y4 I4 K4

119896321198691 (1198962119903119887) 119896321198841 (1198962119903119887) 119896321198681 (1198962119903119887) minus119896321198701 (1198962119903119887)

]]]]]]]]Ψ12

=[[[[[[[[[


J5 Y5 I5 K5

119896311198691 (1198961119903119887) 119896311198841 (1198961119903119887) 119896311198681 (1198961119903119887) minus119896311198701 (1198961119903119887)

]]]]]]]]]

Ψ13

(A6)




(A7)


K2Ψ12 = H2Ψ13 (A8)






Acknowledgments


References






















RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia









0 200 400 600 800 1000 1200 1400 1600Frequency (Hz)

minus4

minus3

minus2

minus1

0

1

2

3

4



F(f

)








0 10 15 20 25 30 35 40Frequency (Hz)

minus5

0

5



5

F(f

)

times10minus8

(a) 0ndash40Hz

40 60 80 100 120 140 160 180minus10

minus8

minus6

minus4

minus2

0

2

4

6



F(f

)

times10minus6

Frequency (Hz)

(b) 40ndash180Hz

200 250 300 350 400 450minus6

minus4

minus2

0

2

4

F(f

)

times10minus3

Frequency (Hz)



(c) 180ndash450Hz









1


0

367

48

734

96

110

244

146

992

183

74

220

488

257

236

293

984

330

732

(a)

1


SMX = 853383

0

948

203

189

641

284

461

379

281

47410

2

568

922

663742

7585

62

853383

(b)

0

683

9

136

78

205

17

273

56

341

95

410

34

478

73

547

12

615

51

1


SMX = 61551

(c)

0

143

273

286

545

429

818

573

09

716

363

859

635

10029

1

114

618

12894

5

1


SMX = 128945

(d)

0

140

312

280

623

420

935

561

247

701

558

841

87

982

181

112

249

12628

1


SMX = 12628

(e)




minus30

minus20

minus10

0

10

20

30

40

50

60

410

019

629

238

848

458

067

677

286

896

410

6011

5612

5213

4814

44

Outer to inner

Inner to outer

(Hz)

Tran

smiss

ibili

ty (d

B)

14481 Hz12479 Hz

41519 Hz3776Hz

138886 Hz57294Hz32196Hz

7044Hz

1683 Hz


Figure 5

0

1000

2000

3000

4000

5000

6000

7000

0001 0002 0003 0004 0005

(Hz)

1st mode

2nd mode

3rd mode

4th mode

(a) Thickness

(Hz)

0

200

400

600

800

1000

1200

1400

0001 0006 0014 0019

1st mode

2nd mode

3rd mode

4th mode

(b) Inner radius

(Hz)

0

5000

10000

15000

20000

25000

30000

35000

40000

112 111 1 111 121

2nd mode1st mode

3rd mode

4th mode

(c) Radial span





0200400600800

100012001400160018002000

1062 2799 4540 7780 11600

PMMA

Al Ti STEEL Pb

Fre (

Hz)

1st mode

2nd mode

3rd mode

4th mode

Density (kgＧ3)

(a) Density

Fre (

Hz)

0

200

400

600

800

1000

1200

1400

032 408 721 117 2106

PMMA

PbAl

Ti STEEL

1st mode2nd mode

3rd mode

4th mode


(b) Elastic modulus





5 Conclusion





Appendix



120597119882120597119903 1 (119903119886) = 120597119882

120597119903 2 (119903119886)1198721 (119903119886) = 1198722 (119903119886)1198761 (119903119886) = 1198762 (119903119886)

(A1)



[[[[[[[[[


J2 Y2 I2 K2

119896311198691 (1198961119903119886) 119896311198841 (1198961119903119886) 119896311198681 (1198961119903119886) minus119896311198701 (1198961119903119886)

]]]]]]]]]

Ψ11

=[[[[[[[[


J3 Y3 I3 K3

119896321198691 (1198962119903119886) 119896321198841 (1198962119903119886) 119896321198681 (1198962119903119886) minus119896321198701 (1198962119903119886)

]]]]]]]]Ψ12

(A2)


J2 = 1198961119903119886 1198691 (1198961119903119886) minus 12059011198691 (1198961119903119886) minus 119896211198690 (1198961119903119886)

Y2 = 1198961119903119886 1198841 (1198961119903119886) minus 12059011198841 (1198961119903119886) minus 119896211198840 (1198961119903119886)

I2 = 1198961119903119886 12059011198681 (1198961119903119886) minus 1198681 (1198961119903119886) + 119896211198680 (1198961119903119886)

K2 = 1198961119903119886 1198701 (1198961119903119886) minus 12059011198701 (1198961119903119886) + 119896211198700 (1198961119903119886)

J3 = 1198962119903119886 1198691 (1198962119903119886) minus 12059021198691 (1198962119903119886) minus 119896221198690 (1198962119903119886)

Y3 = 1198962119903119886 1198841 (1198962119903119886) minus 12059021198841 (1198962119903119886) minus 119896221198840 (1198962119903119886)

I3 = 1198962119903119886 12059021198681 (1198962119903119886) minus 1198681 (1198962119903119886) + 119896221198680 (1198962119903119886) K3 = 1198962119903119886 1198701 (1198962119903119886) minus 12059021198701 (1198962119903119886) + 119896221198700 (1198962119903119886)

(A3)


H1Ψ11 = K1Ψ12 (A4)


1198822 (119903119887) = 1198821 (119903119887)120597119882120597119903 2 (119903119887) = 120597119882120597119903 1 (119903119887)1198722 (119903119887) = 1198721 (119903119887)1198762 (119903119887) = 1198761 (119903119887)

(A5)


[[[[[[[[


J4 Y4 I4 K4

119896321198691 (1198962119903119887) 119896321198841 (1198962119903119887) 119896321198681 (1198962119903119887) minus119896321198701 (1198962119903119887)

]]]]]]]]Ψ12

=[[[[[[[[[


J5 Y5 I5 K5

119896311198691 (1198961119903119887) 119896311198841 (1198961119903119887) 119896311198681 (1198961119903119887) minus119896311198701 (1198961119903119887)

]]]]]]]]]

Ψ13

(A6)




(A7)


K2Ψ12 = H2Ψ13 (A8)






Acknowledgments


References






















RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



0 10 15 20 25 30 35 40Frequency (Hz)

minus5

0

5



5

F(f

)

times10minus8

(a) 0ndash40Hz

40 60 80 100 120 140 160 180minus10

minus8

minus6

minus4

minus2

0

2

4

6



F(f

)

times10minus6

Frequency (Hz)

(b) 40ndash180Hz

200 250 300 350 400 450minus6

minus4

minus2

0

2

4

F(f

)

times10minus3

Frequency (Hz)



(c) 180ndash450Hz









1


0

367

48

734

96

110

244

146

992

183

74

220

488

257

236

293

984

330

732

(a)

1


SMX = 853383

0

948

203

189

641

284

461

379

281

47410

2

568

922

663742

7585

62

853383

(b)

0

683

9

136

78

205

17

273

56

341

95

410

34

478

73

547

12

615

51

1


SMX = 61551

(c)

0

143

273

286

545

429

818

573

09

716

363

859

635

10029

1

114

618

12894

5

1


SMX = 128945

(d)

0

140

312

280

623

420

935

561

247

701

558

841

87

982

181

112

249

12628

1


SMX = 12628

(e)




minus30

minus20

minus10

0

10

20

30

40

50

60

410

019

629

238

848

458

067

677

286

896

410

6011

5612

5213

4814

44

Outer to inner

Inner to outer

(Hz)

Tran

smiss

ibili

ty (d

B)

14481 Hz12479 Hz

41519 Hz3776Hz

138886 Hz57294Hz32196Hz

7044Hz

1683 Hz


Figure 5

0

1000

2000

3000

4000

5000

6000

7000

0001 0002 0003 0004 0005

(Hz)

1st mode

2nd mode

3rd mode

4th mode

(a) Thickness

(Hz)

0

200

400

600

800

1000

1200

1400

0001 0006 0014 0019

1st mode

2nd mode

3rd mode

4th mode

(b) Inner radius

(Hz)

0

5000

10000

15000

20000

25000

30000

35000

40000

112 111 1 111 121

2nd mode1st mode

3rd mode

4th mode

(c) Radial span





0200400600800

100012001400160018002000

1062 2799 4540 7780 11600

PMMA

Al Ti STEEL Pb

Fre (

Hz)

1st mode

2nd mode

3rd mode

4th mode

Density (kgＧ3)

(a) Density

Fre (

Hz)

0

200

400

600

800

1000

1200

1400

032 408 721 117 2106

PMMA

PbAl

Ti STEEL

1st mode2nd mode

3rd mode

4th mode


(b) Elastic modulus





5 Conclusion





Appendix



120597119882120597119903 1 (119903119886) = 120597119882

120597119903 2 (119903119886)1198721 (119903119886) = 1198722 (119903119886)1198761 (119903119886) = 1198762 (119903119886)

(A1)



[[[[[[[[[


J2 Y2 I2 K2

119896311198691 (1198961119903119886) 119896311198841 (1198961119903119886) 119896311198681 (1198961119903119886) minus119896311198701 (1198961119903119886)

]]]]]]]]]

Ψ11

=[[[[[[[[


J3 Y3 I3 K3

119896321198691 (1198962119903119886) 119896321198841 (1198962119903119886) 119896321198681 (1198962119903119886) minus119896321198701 (1198962119903119886)

]]]]]]]]Ψ12

(A2)


J2 = 1198961119903119886 1198691 (1198961119903119886) minus 12059011198691 (1198961119903119886) minus 119896211198690 (1198961119903119886)

Y2 = 1198961119903119886 1198841 (1198961119903119886) minus 12059011198841 (1198961119903119886) minus 119896211198840 (1198961119903119886)

I2 = 1198961119903119886 12059011198681 (1198961119903119886) minus 1198681 (1198961119903119886) + 119896211198680 (1198961119903119886)

K2 = 1198961119903119886 1198701 (1198961119903119886) minus 12059011198701 (1198961119903119886) + 119896211198700 (1198961119903119886)

J3 = 1198962119903119886 1198691 (1198962119903119886) minus 12059021198691 (1198962119903119886) minus 119896221198690 (1198962119903119886)

Y3 = 1198962119903119886 1198841 (1198962119903119886) minus 12059021198841 (1198962119903119886) minus 119896221198840 (1198962119903119886)

I3 = 1198962119903119886 12059021198681 (1198962119903119886) minus 1198681 (1198962119903119886) + 119896221198680 (1198962119903119886) K3 = 1198962119903119886 1198701 (1198962119903119886) minus 12059021198701 (1198962119903119886) + 119896221198700 (1198962119903119886)

(A3)


H1Ψ11 = K1Ψ12 (A4)


1198822 (119903119887) = 1198821 (119903119887)120597119882120597119903 2 (119903119887) = 120597119882120597119903 1 (119903119887)1198722 (119903119887) = 1198721 (119903119887)1198762 (119903119887) = 1198761 (119903119887)

(A5)


[[[[[[[[


J4 Y4 I4 K4

119896321198691 (1198962119903119887) 119896321198841 (1198962119903119887) 119896321198681 (1198962119903119887) minus119896321198701 (1198962119903119887)

]]]]]]]]Ψ12

=[[[[[[[[[


J5 Y5 I5 K5

119896311198691 (1198961119903119887) 119896311198841 (1198961119903119887) 119896311198681 (1198961119903119887) minus119896311198701 (1198961119903119887)

]]]]]]]]]

Ψ13

(A6)




(A7)


K2Ψ12 = H2Ψ13 (A8)






Acknowledgments


References






















RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



1


0

367

48

734

96

110

244

146

992

183

74

220

488

257

236

293

984

330

732

(a)

1


SMX = 853383

0

948

203

189

641

284

461

379

281

47410

2

568

922

663742

7585

62

853383

(b)

0

683

9

136

78

205

17

273

56

341

95

410

34

478

73

547

12

615

51

1


SMX = 61551

(c)

0

143

273

286

545

429

818

573

09

716

363

859

635

10029

1

114

618

12894

5

1


SMX = 128945

(d)

0

140

312

280

623

420

935

561

247

701

558

841

87

982

181

112

249

12628

1


SMX = 12628

(e)




minus30

minus20

minus10

0

10

20

30

40

50

60

410

019

629

238

848

458

067

677

286

896

410

6011

5612

5213

4814

44

Outer to inner

Inner to outer

(Hz)

Tran

smiss

ibili

ty (d

B)

14481 Hz12479 Hz

41519 Hz3776Hz

138886 Hz57294Hz32196Hz

7044Hz

1683 Hz


Figure 5

0

1000

2000

3000

4000

5000

6000

7000

0001 0002 0003 0004 0005

(Hz)

1st mode

2nd mode

3rd mode

4th mode

(a) Thickness

(Hz)

0

200

400

600

800

1000

1200

1400

0001 0006 0014 0019

1st mode

2nd mode

3rd mode

4th mode

(b) Inner radius

(Hz)

0

5000

10000

15000

20000

25000

30000

35000

40000

112 111 1 111 121

2nd mode1st mode

3rd mode

4th mode

(c) Radial span





0200400600800

100012001400160018002000

1062 2799 4540 7780 11600

PMMA

Al Ti STEEL Pb

Fre (

Hz)

1st mode

2nd mode

3rd mode

4th mode

Density (kgＧ3)

(a) Density

Fre (

Hz)

0

200

400

600

800

1000

1200

1400

032 408 721 117 2106

PMMA

PbAl

Ti STEEL

1st mode2nd mode

3rd mode

4th mode


(b) Elastic modulus





5 Conclusion





Appendix



120597119882120597119903 1 (119903119886) = 120597119882

120597119903 2 (119903119886)1198721 (119903119886) = 1198722 (119903119886)1198761 (119903119886) = 1198762 (119903119886)

(A1)



[[[[[[[[[


J2 Y2 I2 K2

119896311198691 (1198961119903119886) 119896311198841 (1198961119903119886) 119896311198681 (1198961119903119886) minus119896311198701 (1198961119903119886)

]]]]]]]]]

Ψ11

=[[[[[[[[


J3 Y3 I3 K3

119896321198691 (1198962119903119886) 119896321198841 (1198962119903119886) 119896321198681 (1198962119903119886) minus119896321198701 (1198962119903119886)

]]]]]]]]Ψ12

(A2)


J2 = 1198961119903119886 1198691 (1198961119903119886) minus 12059011198691 (1198961119903119886) minus 119896211198690 (1198961119903119886)

Y2 = 1198961119903119886 1198841 (1198961119903119886) minus 12059011198841 (1198961119903119886) minus 119896211198840 (1198961119903119886)

I2 = 1198961119903119886 12059011198681 (1198961119903119886) minus 1198681 (1198961119903119886) + 119896211198680 (1198961119903119886)

K2 = 1198961119903119886 1198701 (1198961119903119886) minus 12059011198701 (1198961119903119886) + 119896211198700 (1198961119903119886)

J3 = 1198962119903119886 1198691 (1198962119903119886) minus 12059021198691 (1198962119903119886) minus 119896221198690 (1198962119903119886)

Y3 = 1198962119903119886 1198841 (1198962119903119886) minus 12059021198841 (1198962119903119886) minus 119896221198840 (1198962119903119886)

I3 = 1198962119903119886 12059021198681 (1198962119903119886) minus 1198681 (1198962119903119886) + 119896221198680 (1198962119903119886) K3 = 1198962119903119886 1198701 (1198962119903119886) minus 12059021198701 (1198962119903119886) + 119896221198700 (1198962119903119886)

(A3)


H1Ψ11 = K1Ψ12 (A4)


1198822 (119903119887) = 1198821 (119903119887)120597119882120597119903 2 (119903119887) = 120597119882120597119903 1 (119903119887)1198722 (119903119887) = 1198721 (119903119887)1198762 (119903119887) = 1198761 (119903119887)

(A5)


[[[[[[[[


J4 Y4 I4 K4

119896321198691 (1198962119903119887) 119896321198841 (1198962119903119887) 119896321198681 (1198962119903119887) minus119896321198701 (1198962119903119887)

]]]]]]]]Ψ12

=[[[[[[[[[


J5 Y5 I5 K5

119896311198691 (1198961119903119887) 119896311198841 (1198961119903119887) 119896311198681 (1198961119903119887) minus119896311198701 (1198961119903119887)

]]]]]]]]]

Ψ13

(A6)




(A7)


K2Ψ12 = H2Ψ13 (A8)






Acknowledgments


References






















RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




minus30

minus20

minus10

0

10

20

30

40

50

60

410

019

629

238

848

458

067

677

286

896

410

6011

5612

5213

4814

44

Outer to inner

Inner to outer

(Hz)

Tran

smiss

ibili

ty (d

B)

14481 Hz12479 Hz

41519 Hz3776Hz

138886 Hz57294Hz32196Hz

7044Hz

1683 Hz


Figure 5

0

1000

2000

3000

4000

5000

6000

7000

0001 0002 0003 0004 0005

(Hz)

1st mode

2nd mode

3rd mode

4th mode

(a) Thickness

(Hz)

0

200

400

600

800

1000

1200

1400

0001 0006 0014 0019

1st mode

2nd mode

3rd mode

4th mode

(b) Inner radius

(Hz)

0

5000

10000

15000

20000

25000

30000

35000

40000

112 111 1 111 121

2nd mode1st mode

3rd mode

4th mode

(c) Radial span





0200400600800

100012001400160018002000

1062 2799 4540 7780 11600

PMMA

Al Ti STEEL Pb

Fre (

Hz)

1st mode

2nd mode

3rd mode

4th mode

Density (kgＧ3)

(a) Density

Fre (

Hz)

0

200

400

600

800

1000

1200

1400

032 408 721 117 2106

PMMA

PbAl

Ti STEEL

1st mode2nd mode

3rd mode

4th mode


(b) Elastic modulus





5 Conclusion





Appendix



120597119882120597119903 1 (119903119886) = 120597119882

120597119903 2 (119903119886)1198721 (119903119886) = 1198722 (119903119886)1198761 (119903119886) = 1198762 (119903119886)

(A1)



[[[[[[[[[


J2 Y2 I2 K2

119896311198691 (1198961119903119886) 119896311198841 (1198961119903119886) 119896311198681 (1198961119903119886) minus119896311198701 (1198961119903119886)

]]]]]]]]]

Ψ11

=[[[[[[[[


J3 Y3 I3 K3

119896321198691 (1198962119903119886) 119896321198841 (1198962119903119886) 119896321198681 (1198962119903119886) minus119896321198701 (1198962119903119886)

]]]]]]]]Ψ12

(A2)


J2 = 1198961119903119886 1198691 (1198961119903119886) minus 12059011198691 (1198961119903119886) minus 119896211198690 (1198961119903119886)

Y2 = 1198961119903119886 1198841 (1198961119903119886) minus 12059011198841 (1198961119903119886) minus 119896211198840 (1198961119903119886)

I2 = 1198961119903119886 12059011198681 (1198961119903119886) minus 1198681 (1198961119903119886) + 119896211198680 (1198961119903119886)

K2 = 1198961119903119886 1198701 (1198961119903119886) minus 12059011198701 (1198961119903119886) + 119896211198700 (1198961119903119886)

J3 = 1198962119903119886 1198691 (1198962119903119886) minus 12059021198691 (1198962119903119886) minus 119896221198690 (1198962119903119886)

Y3 = 1198962119903119886 1198841 (1198962119903119886) minus 12059021198841 (1198962119903119886) minus 119896221198840 (1198962119903119886)

I3 = 1198962119903119886 12059021198681 (1198962119903119886) minus 1198681 (1198962119903119886) + 119896221198680 (1198962119903119886) K3 = 1198962119903119886 1198701 (1198962119903119886) minus 12059021198701 (1198962119903119886) + 119896221198700 (1198962119903119886)

(A3)


H1Ψ11 = K1Ψ12 (A4)


1198822 (119903119887) = 1198821 (119903119887)120597119882120597119903 2 (119903119887) = 120597119882120597119903 1 (119903119887)1198722 (119903119887) = 1198721 (119903119887)1198762 (119903119887) = 1198761 (119903119887)

(A5)


[[[[[[[[


J4 Y4 I4 K4

119896321198691 (1198962119903119887) 119896321198841 (1198962119903119887) 119896321198681 (1198962119903119887) minus119896321198701 (1198962119903119887)

]]]]]]]]Ψ12

=[[[[[[[[[


J5 Y5 I5 K5

119896311198691 (1198961119903119887) 119896311198841 (1198961119903119887) 119896311198681 (1198961119903119887) minus119896311198701 (1198961119903119887)

]]]]]]]]]

Ψ13

(A6)




(A7)


K2Ψ12 = H2Ψ13 (A8)






Acknowledgments


References






















RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





0200400600800

100012001400160018002000

1062 2799 4540 7780 11600

PMMA

Al Ti STEEL Pb

Fre (

Hz)

1st mode

2nd mode

3rd mode

4th mode

Density (kgＧ3)

(a) Density

Fre (

Hz)

0

200

400

600

800

1000

1200

1400

032 408 721 117 2106

PMMA

PbAl

Ti STEEL

1st mode2nd mode

3rd mode

4th mode


(b) Elastic modulus





5 Conclusion





Appendix



120597119882120597119903 1 (119903119886) = 120597119882

120597119903 2 (119903119886)1198721 (119903119886) = 1198722 (119903119886)1198761 (119903119886) = 1198762 (119903119886)

(A1)



[[[[[[[[[


J2 Y2 I2 K2

119896311198691 (1198961119903119886) 119896311198841 (1198961119903119886) 119896311198681 (1198961119903119886) minus119896311198701 (1198961119903119886)

]]]]]]]]]

Ψ11

=[[[[[[[[


J3 Y3 I3 K3

119896321198691 (1198962119903119886) 119896321198841 (1198962119903119886) 119896321198681 (1198962119903119886) minus119896321198701 (1198962119903119886)

]]]]]]]]Ψ12

(A2)


J2 = 1198961119903119886 1198691 (1198961119903119886) minus 12059011198691 (1198961119903119886) minus 119896211198690 (1198961119903119886)

Y2 = 1198961119903119886 1198841 (1198961119903119886) minus 12059011198841 (1198961119903119886) minus 119896211198840 (1198961119903119886)

I2 = 1198961119903119886 12059011198681 (1198961119903119886) minus 1198681 (1198961119903119886) + 119896211198680 (1198961119903119886)

K2 = 1198961119903119886 1198701 (1198961119903119886) minus 12059011198701 (1198961119903119886) + 119896211198700 (1198961119903119886)

J3 = 1198962119903119886 1198691 (1198962119903119886) minus 12059021198691 (1198962119903119886) minus 119896221198690 (1198962119903119886)

Y3 = 1198962119903119886 1198841 (1198962119903119886) minus 12059021198841 (1198962119903119886) minus 119896221198840 (1198962119903119886)

I3 = 1198962119903119886 12059021198681 (1198962119903119886) minus 1198681 (1198962119903119886) + 119896221198680 (1198962119903119886) K3 = 1198962119903119886 1198701 (1198962119903119886) minus 12059021198701 (1198962119903119886) + 119896221198700 (1198962119903119886)

(A3)


H1Ψ11 = K1Ψ12 (A4)


1198822 (119903119887) = 1198821 (119903119887)120597119882120597119903 2 (119903119887) = 120597119882120597119903 1 (119903119887)1198722 (119903119887) = 1198721 (119903119887)1198762 (119903119887) = 1198761 (119903119887)

(A5)


[[[[[[[[


J4 Y4 I4 K4

119896321198691 (1198962119903119887) 119896321198841 (1198962119903119887) 119896321198681 (1198962119903119887) minus119896321198701 (1198962119903119887)

]]]]]]]]Ψ12

=[[[[[[[[[


J5 Y5 I5 K5

119896311198691 (1198961119903119887) 119896311198841 (1198961119903119887) 119896311198681 (1198961119903119887) minus119896311198701 (1198961119903119887)

]]]]]]]]]

Ψ13

(A6)




(A7)


K2Ψ12 = H2Ψ13 (A8)






Acknowledgments


References






















RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




[[[[[[[[[


J2 Y2 I2 K2

119896311198691 (1198961119903119886) 119896311198841 (1198961119903119886) 119896311198681 (1198961119903119886) minus119896311198701 (1198961119903119886)

]]]]]]]]]

Ψ11

=[[[[[[[[


J3 Y3 I3 K3

119896321198691 (1198962119903119886) 119896321198841 (1198962119903119886) 119896321198681 (1198962119903119886) minus119896321198701 (1198962119903119886)

]]]]]]]]Ψ12

(A2)


J2 = 1198961119903119886 1198691 (1198961119903119886) minus 12059011198691 (1198961119903119886) minus 119896211198690 (1198961119903119886)

Y2 = 1198961119903119886 1198841 (1198961119903119886) minus 12059011198841 (1198961119903119886) minus 119896211198840 (1198961119903119886)

I2 = 1198961119903119886 12059011198681 (1198961119903119886) minus 1198681 (1198961119903119886) + 119896211198680 (1198961119903119886)

K2 = 1198961119903119886 1198701 (1198961119903119886) minus 12059011198701 (1198961119903119886) + 119896211198700 (1198961119903119886)

J3 = 1198962119903119886 1198691 (1198962119903119886) minus 12059021198691 (1198962119903119886) minus 119896221198690 (1198962119903119886)

Y3 = 1198962119903119886 1198841 (1198962119903119886) minus 12059021198841 (1198962119903119886) minus 119896221198840 (1198962119903119886)

I3 = 1198962119903119886 12059021198681 (1198962119903119886) minus 1198681 (1198962119903119886) + 119896221198680 (1198962119903119886) K3 = 1198962119903119886 1198701 (1198962119903119886) minus 12059021198701 (1198962119903119886) + 119896221198700 (1198962119903119886)

(A3)


H1Ψ11 = K1Ψ12 (A4)


1198822 (119903119887) = 1198821 (119903119887)120597119882120597119903 2 (119903119887) = 120597119882120597119903 1 (119903119887)1198722 (119903119887) = 1198721 (119903119887)1198762 (119903119887) = 1198761 (119903119887)

(A5)


[[[[[[[[


J4 Y4 I4 K4

119896321198691 (1198962119903119887) 119896321198841 (1198962119903119887) 119896321198681 (1198962119903119887) minus119896321198701 (1198962119903119887)

]]]]]]]]Ψ12

=[[[[[[[[[


J5 Y5 I5 K5

119896311198691 (1198961119903119887) 119896311198841 (1198961119903119887) 119896311198681 (1198961119903119887) minus119896311198701 (1198961119903119887)

]]]]]]]]]

Ψ13

(A6)




(A7)


K2Ψ12 = H2Ψ13 (A8)






Acknowledgments


References






















RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





(A7)


K2Ψ12 = H2Ψ13 (A8)






Acknowledgments


References






















RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


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