Research ArticleResearch on Radial Vibration of a Circular Plate

Wei Liu Donghua Wang Hanfeng Lu Yumeng Cao and Pengrong Zhang

College of Power and Energy Engineering Harbin Engineering University Harbin China

Correspondence should be addressed to Donghua Wang hitwdh163com

Received 25 May 2016 Revised 13 September 2016 Accepted 29 September 2016

Academic Editor Giorgio Dalpiaz

Copyright copy 2016 Wei Liu et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Radial vibration of the circular plate is presented using wave propagation approach and classical method containing Bessel solutionand Hankel solution for calculating the natural frequency theoretically In cylindrical coordinate system in order to obtain naturalfrequency propagation and reflection matrices are deduced at the boundaries of free-free fixed-fixed and fixed-free using wavepropagation approach Furthermore radial phononic crystal is constructed by connecting twomaterials periodically for the analysisof band phenomenon Also Finite Element Simulation (FEM) is adopted to verify the theoretical results Finally the radial andpiezoelectric effects on the band are also discussed

1 Introduction

Wave propagation approach is a usefulmethod on the aspectsof calculating natural frequency for the analysis of elasticstructures such as rod Euler beam curved beam andplate It benefits us with a better understanding to analyzewave when propagating inside structures In fact as early as1984 Mace analyzed the behaviors of wave by using wavepropagation approach through dividing them into prop-agation attenuation matrices And the reflection matricesunder three boundary conditions were deduced which builta theoretical foundation for wave method [1] Meanwhile thecharacteristics of propagation and reflection of Timoshenkobeam were also discussed with the discontinuous behaviorAnd the frequency curves were obtained [2] Harland et alstudied the motion of waves in tunable fluid-filled beams [3]Based on Love theory Kang et al deduced the propagationmatrices reflection matrices and coordinated matrices incurved beams [4] Different from Kang et al Lee made adetailed research on the behavior of waves of curved beambased on Fluggersquos theory [5 6] Based on the high-order platetheory Wang and Rose analyzed the properties of transientwave in an inhomogeneous beam And the results were alsoverified by the experiment [7] Tan and Kang consideredthe wave reflection and transmission in an axially strainedrotating Timoshenko shaft Meanwhile the influence ofcontinuous condition and cross-section on natural frequency

was also discussed [8] Li and Gan analyzed the free vibrationof cylindrical shell by usingwave propagation approach basedon Fluggersquos thin shell theory [9 10] Also Zhang calculatedthe frequencies of cross-ply laminated composite cylindricalshells and submerged cylindrical shells using wave approachbased on Loversquos theory [11 12] Using wave approach theabove scholars have done lots of analysis which exhibit aunique advantage for the analysis of structural vibration

In recent years people have paid great attention tostructure which consisted of two materials arraying peri-odically At present the research has been extended intoradial phononic crystal structures Shakeri et al analyzedthe behaviors of radial wave in functional graded radialperiodical structures [13] Torrent and Carbonell studiedthe propagation characteristic of acoustic in detail whenpropagating in radial periodical structure and found that italso had a low frequency band [14 15] After constructingand analyzing the two-dimensional radial phononic crystalXu et al found that these structures had low frequency bandthat the starting frequency was zero [16] Ma et al studiedthe band characteristics of Lamb wave in two-layer radialphononic crystal plate and it was found that crystal glidingin radial direction could open the lowest band gaps [17]The above researches indicate that radial phononic crystalplate has the low frequency band which brings an attractiveprospect of engineering application For example the naturalfrequency can be designed into the range of band gaps to

Hindawi Publishing CorporationShock and VibrationVolume 2016 Article ID 6758291 8 pageshttpdxdoiorg10115520166758291

2 Shock and Vibration

r0

rao

a+1

aminus1

b+1

bminus1

Figure 1 Model of single circular plate

isolate the vibration Also these structures can be regarded asa filter Additionally Shu et al researched torsional vibrationproblem of generalized phononic crystal finding that lowfrequency band was determined by radius However the highfrequency bandwas caused by the periodicity of circular plate[18] And Shursquos research shows that generalized phononiccrystal can generate a huge band which almost covers thewhole frequency range providing a practical application forthe rotating parts such as gear In the previous researchShu et al have studied the band gaps of flexural wave inthe radial phononic crystal Meanwhile numerical resultsare also verified by FEM [19] Herein this paper is mainlyfocused on radial vibration of composite structures In factother works on radial vibration of elastic solids under variousloadings can also be found [20ndash22]

The paper is organized into five sections In Section 1a brief introduction is given In Section 2 classic Besselsolution and Hankel solution of radial vibration are givenMeanwhile propagation and reflection matrices at threeboundary conditions are derived The results of naturalfrequency obtained bywave approach are also comparedwiththe classical method In Section 3 the band phenomenon isanalyzed and the effect of parameters is discussed Section 4is the conclusion

2 Theoretical Analysis for Natural Frequency

21 Solution of Radial Vibration Equation Single circularplate is shown in Figure 1 Young modulus density and Pois-sonrsquos coefficient are 1198641 1205881 ]1 The thickness of piezoelectricceramics is ℎ1 The inner and external radius are 1199030 and 119903119886

respectivelyWith respect to the axisymmetric circular platesradial equation of piezoelectric materials can be given by [23]

12058812059721199061199031205971199052 = 120597120590119903120597119903 + 120590119903 minus 120590120579119903 (1)

The expression of strain and radial displacement can bewritten as

120576119903 = 120597119906119903120597119903 (2)

120576120579 = 119906119903119903 (3)

Since piezoelectric material direction of polarization isalong z-axle piezoelectric equation can be expressed as

120576119903 = 11990411986411120590119903 + 11990411986412120590120579 + 11988931119864119885120576120579 = 11990411986412120590119903 + 11990411986411120590120579 + 11988931119864119885

119863119911 = 11988931120590119903 + 11988931120590120579 + 12058511987933119864119885(4)

where 119906119903 is radial displacement 120590119903 and 120590120579 are radial stressand tangential stress 120576119903 and 120576120579 are radial strain and tangentialstrain and 119903 is radius

Substituting (2)ndash(4) into (1) gives

12059721199061199031205971199032 + 1119903 120597119906119903120597119903 minus 1199061199031199032 = 11198882 12059721199061199031205971199052 (5)

where 1198882 = 111990411986411120588(1minus]2) is the square of radial wave velocityAfter simplifying (5) it can be obtained that

1205972119906119903120597 (119896119903)2 + 1119896119903 120597119906119903120597 (119896119903) + [1 minus 1(119896119903)2] 119906 = 0 (6)

Thus general solution of radial displacement can bewritten as

119906119903 = 119860+11198691 (119896119903) + 119860minus11198841 (119896119903) (7)

where 119896 = 119908119888 is the wave number 1198691(119896119903) is the Besselfunction of first kind 1198841(119896119903) is the Bessel function of secondkind And the constants 119860+1 and 119860minus1 can be determined byboundary conditions

Therefore radial stress can be obtained as

120590119903 = 1198641 minus ]2(120576119903 + ]120576120579)

= 1198641198961 minus ]2119860+1 [1198690 (119896119903) minus (1 minus ]) 1198691 (119896119903)119896119903 ]

+ 119860minus1 [1198840 (119896119903) minus (1 minus ]) 1198841 (119896119903)119896119903 ] (8)

22 Classical Method When both ends of circular plate arefixed it has

119906119903 (1199030) = 0119906119903 (119903119886) = 0 (9)

Shock and Vibration 3

Substituting (7) into (9) gives

10038161003816100381610038161003816100381610038161003816100381610038161198691 (1198961199030) 1198841 (1198961199030)1198691 (119896119903119886) 1198841 (119896119903119886)

1003816100381610038161003816100381610038161003816100381610038161003816 = 0 (10)

When both ends of circular plate are free it has

120590119903 (1199030) = 0120590119903 (119903119886) = 0 (11)

Substituting (8) into (11) gives

1198641198961 minus ]2

sdot100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161198690 (1198961199030) minus (1 minus ]) 1198691 (1198961199030)1198961199030 1198840 (1198961199030) minus (1 minus ]) 1198841 (1198961199030)11989611990301198690 (119896119903119886) minus (1 minus ]) 1198691 (119896119903119886)119896119903119886 1198690 (119896119903119886) minus (1 minus ]) 1198691 (119896119903119886)119896119903119886

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816= 0

(12)

Similarly when inner surface is fixed and outer surface isfree it has

119906119903 (1199030) = 0120590119903 (119903119886) = 0 (13)

Substituting (7) and (8) into (13) gives

1198641198961 minus ]2

sdot100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198690 (1198961199030) 1198840 (1198961199030)1198690 (119896119903119886) minus (1 minus ]) 1198691 (119896119903119886)119896119903119886 1198690 (119896119903119886) minus (1 minus ]) 1198691 (119896119903119886)119896119903119886

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816= 0

(14)

After solving roots of (10) (12) and (14) the naturalfrequencies of circular plate structure can be obtained withthe aforementioned boundary conditions

23 Wave Propagation Method One of the solutions ofcircular plate has been solved as shown in (7) And the othersolution can be written as

119906119903 = 119860+1119867(2)0 (119896119903) + 119860minus1119867(1)0 (119896119903) (15)

where 119867(1)0 (119896119903) and 119867(2)0 (119896119903) are the Hankel function of firstand second kinds which can be defined as

119867(1)0 (119896119903) = 1198690 (119896119903) + 1198941198840 (119896119903) 119867(2)0 (119896119903) = 1198690 (119896119903) minus 1198941198840 (119896119903) (16)

Considering (15) forward-traveling and negative-traveling waves can be written as

119892+1 = 119860+1119867(2)0 (119896119903) 119892minus1 = 119860minus1119867(1)0 (119896119903) (17)

Then (15) reduces to

119906119903 = 119892+1 + 119892minus1 (18)

When both ends of circular plate are fixed it gives

119906119903 (1199030) = 119892+1 (1199030) + 119892minus1 (1199030) = 0119906119903 (119903119886) = 119892+1 (119903119886) + 119892minus1 (119903119886) = 0 (19)

Thus the reflection matrices of both boundaries are

119877119860 = [minus1] (20a)

119877119862 = [minus1] (20b)

Substituting (2) (15) and (16) into (8) radial stress can beexpressed as

120590119903 = 1198641198961 minus ]2119860+1 [1198961198690 (119896119903) minus 1198691 (119896119903)119903 minus 1198941198961198840 (119896119903)

+ 1198941199031198841 (119896119903) + ]119903 1198691 (119896119903) minus ]119903 1198941198841 (119896119903)] + 119860minus1 [1198961198690 (119896119903)minus 1198691 (119896119903)119903 + 1198941198961198840 (119896119903) minus 1198941199031198841 (119896119903) + ]119903 1198691 (119896119903)+ ]119903 1198941198841 (119896119903)]

(21)

When both ends of circular plate are free it has

120590119903 (1199030) = 0120590119903 (119903119886) = 0 (22)

Thus reflection matrices of both boundaries are

119877119860 = minus [1198961198690 (1198961199030) minus 1198691 (1198961199030) 1199030 + 1198941198961198840 (1198961199030) minus (1198941199030) 1198841 (1198961199030) + (]1199030) 1198691 (1198961199030) + (]1199030) 1198941198841 (1198961199030)]119867(1)0 (1198961199030) minus1

times [1198961198690 (1198961199030) minus 1198691 (1198961199030) 1199030 minus 1198941198961198840 (1198961199030) + (1198941199030) 1198841 (1198961199030) + (]1199030) 1198691 (1198961199030) minus (]1199030) 1198941198841 (1198961199030)]119867(2)0 (1198961199030) (23)

4 Shock and Vibration

119877119862 = minus [1198961198690 (119896119903119886) minus 1198691 (119896119903119886) 119903119886 minus 1198941198961198840 (119896119903119886) + (119894119903119886) 1198841 (1198961199030) + (]119903119886) 1198691 (119896119903119886) minus (]119903119886) 1198941198841 (119896119903119886)]119867(2)0 (119896119903119886) minus1

sdot [1198961198690 (119896119903119886) minus 1198691 (119896119903119886) 119903119886 + 1198941198961198840 (119896119903119886) minus (119894119903119886) 1198841 (119896119903119886) + (]119903119886) 1198691 (119896119903119886) + (]119903119886) 1198941198841 (119896119903119886)]119867(1)0 (119896119903119886) (24)

Forward-traveling and negative-traveling waves can beshown clearly in Figure 1 And the expression can be writtenas follows

119887+1 = 119891+1 (119903119886 minus 1199030) 119886+1 119886minus1 = 119891minus1 (119903119886 minus 1199030) 119887minus1

(25)

Considering (17) forward-traveling and negative-traveling waves can be obtained as

119891+1 (119903119886 minus 1199030) = 119867(2)0 (119896119903119886)119867(2)0 (1198961199030)

119891minus1 (119903119886 minus 1199030) = 119867(1)0 (1198961199030)119867(1)0 (119896119903119886) (26)

Similarly the reflection matrices are (20a) and (24) forcase of inner surface fixed and outer surface free

24 Analysis and Discussion After combining propagationand reflection matrices derived above the natural charac-teristics of single circular plate can be analyzed effectivelyHence it gives

119887+1 = 119891+1 (119903119886 minus 1199030) 119886+1 119886+1 = 119877119860119886minus1 119886minus1 = 119891minus1 (119903119886 minus 1199030) 119887minus1 119887minus1 = 119877119862119887+1

(27)

And it yields

119865 (119891) = [119891+1 (119903119886 minus 1199030) 119877119860119891minus1 (119903119886 minus 1199030) 119877119862 minus 1] 119887+1 = 0 (28)

RESIN is selected as the material of circular plate Mate-rial and structural parameters are tabulated in Table 1

Substitute parameters listed in Table 1 into (10) (12) (14)and (28) and the natural frequency can be computedHerein red and black curves obtained by wave approachare real and imaginary wave solution respectively Also it isimportant to note that the intersection points of the curves arenatural frequency After solving the inherent characteristic ofcircular plate structure the characteristic curves calculated byclassicalmethod andwave approach are presented in Figure 2It can be seen that these natural frequencies intersect at thesame point in x-axis which verifies the correctness of thenumerical calculation

3 Radial Vibration Analysis ofPhononic Crystal

31 Model and Transfer Matrix Figure 3 is the schematicdiagram of radial phononic crystal circular plate Herein thewhite region represents RESIN materials and the gray regionrepresents piezoelectric ceramic PZT4 The inner and outerradius are 1199031 and 1199032 respectively The radial span ratios aredenoted with 1198861 and 1198862 And ℎ represents the thickness

Considering an eight-layer circular plate it correspondsto the model 119860119861119860119861119860119861119860119861 The interfaces between RESINand PZT4 material are 119886 119887 119888 119889 119890 119891 119898 119901 119902 (see Figure 3)With regard to RESIN Youngrsquosmodulus density and Poissoncoefficient are 1198641 1205881 ]1 For the PZT4 Youngrsquos modulusdensity and Poisson coefficient are 1198642 1205882 ]2 When radialwave propagates in RESIN layers 1 3 5 and 7 the velocityis 1198881 = [11986411205881(1 minus ]1

2)]12 and the wavenumber is 1198961 =1199081198881 = 21205871198911198881The velocity is 1198882 = [11986421205882(1minus]22)]12 and the

wavenumber is 1198962 = 1199081198882 = 21205871198911198882 when wave propagates inPZT4 layers 2 4 6 and 8

With regard to radial vibration there are two parameterswhich satisfy the continuous conditions at the interface ofdifferent materials Namely they are the radial displacementand radial stress So transfer matrices of radial vibration canbe derived as follows

At the interface b (119903119887 = 1199031 + 1198861) it gives1199061 = 11990621205901 = 1205902 (29)

Thus it can be arranged in a matrices form

1198671 [119860+11119860minus11] = 1198701 [119861+12119861minus12] (30)

At the interface c (119903119888 = 1199031 + 1198861 + 1198862) it gives1199062 = 11990631205902 = 1205903 (31)

Similarly one has

1198672 [119860+21119860minus21] = 1198702 [119861+12119861minus12] (32)

Combining (30) and (32) one has

[119860+21119860minus21] = 11987931 [119860+11119860minus11] (33)

Shock and Vibration 5

times104

minus2

minus1

0

1

2

3

4

5F

(f)

21 4 5 73 6 80

Frequency (Hz)

Classical Bessel solutionClassical Hankel solution

Wave solution (imag)Wave solution (real)

(a) Fixed-fixed boundary

times104

minus3

minus2

minus1

0

1

2

3

F (f

)

2 4 6 80

Frequency (Hz)

Classical Bessel solutionClassical Hankel solution

Wave solution (imag)Wave solution (real)

1 5 73

(b) Free-free boundary

times104

2 4 6 8 10 12 14 160

Frequency (Hz)

Classical Bessel solutionClassical Hankel solution

Wave solution (imag)Wave solution (real)

minus4

minus2

0

2

4

6

F (f

)

(c) Fixed-free boundary

Figure 2 Natural frequency calculated by classic method and wave approach

Table 1 Material and structural parameters

Material parameters 120588 (kgm3) 119864 (Pa) ]RESIN 1180 0435 times 1010 03679Structural parameters 1199030 (m) 119903119886 (m) ℎ (m)RESIN 008 016 0005

where 11987931 = 119867minus12 1198702119870minus11 1198671 is the transfer matrices of radialvibration between the 3rd and the 1st ring

As shown in Figure 3 replace 119903119887 and 119903119888 (in (30) and (32))with 119903119889 and 119903119890 Similarly with 11987931 we can obtain 11987953 which isthe transfer matrix between the fifth ring and the third ring

Similarly replace 119903119889 and 119903119890 with 119903119891 and 119903119898 and the matrix11987975 can be obtainedAt last submitting 119903119901 into (30) we can get 11987981 Therefore

with regard to the radial phononic crystal one has

11987981 = 119870minus18711986787119879751198795311987931 (34)

32 Numerical Result and Discussion The radial phononiccrystal circular plates are composed of twomaterials namelyRESIN and piezoelectric ceramic PZT4 Material parametersand structural parameters are consistent with Table 2

Table 2 Material parameters and structural parameters

Material parameters 120588 (kgm3) 119864 (Pa) ]I (RESIN) 1180 0435 times 1010 03679II (PZT4) 7500 813 times 1010 033Structural parameters 1198861 = 1198862 (m) 1199032 = 21199031 (m) ℎ (m)RESIN and PZT4 001 016 0003

Additionally piezoelectric parameters are set as 11990411986411 =123 times 10minus12m2N 120576119879331205760 = 1300 1205760 = 884 times 10minus12 cm and11988931 = minus123 times 10minus12 cNTaking advantage of transfer matrices and boundary

conditions the transmission characteristics are calculatedusing MATLAB numerically Figure 4 is the transmissioncurve of radial vibration with four periodic structures whichis composited by single material RESIN and RESINPZT4

Figure 4 is the comparison of numerical and FEM resultsIt indicates that there is no attenuation when radial wavepropagates in a single material while for the phononic crys-tal the transmission curve shows a big vibration attenuationband after introducing the periodicity However when radialwave propagates in the single material there is no band gapwhich attenuates weakly and only owns minus4 dB Furthermore

6 Shock and Vibration

r1o a1 a2 a1a2

IIIIII r2

I RESINII PZT4

h

12 3

45

6 78

qpmfedcba

Figure 3 Schematic diagram of radial phononic crystal circular plate

times104

minus40

minus30

minus20

minus10

0

10

20

30

40

50

60

Tran

smiss

ibili

ty (d

B)

1 2 3 4 5 60

Frequency (Hz)

Numerical resultFEM

(a) Single material (RESIN)

Numerical resultFEM

times104

minus100

minus80

minus60

minus40

minus20

0

20

40

60

Tran

smiss

ibili

ty (d

B)

1 2 3 4 5 60

Frequency (Hz)

(b) Phononic crystal (RESINPZT4)

Figure 4 Radial vibration curves of numerical and FEM results

ANSYS 145 is used to verify the radial propagation charac-teristics Using harmonic analysis and keeping the boundaryconditions of outer surface free radial displacement is excitedon the inner surface which is shown in Figure 4(b) Compar-ing numerical results with FEM results it can be found thatthe bands are in good agreement with the aspects of positionwidth and amplitude attenuation and other aspects whichindicates the correctness of theoretical calculations

33 Effects of Structural Parameters In this section the influ-ence of piezoelectric effect on radial wave has been discussedin detail Furthermore the influences of piezoelectric param-eter 11988931 and polar direction on the band have also been stud-ied in case of open and short circuit for piezoelectricmaterial

331 Open and Short Circuit With regard to piezoelectricmaterial the D-type equation of radial vibration can beexpressed as

120576119903 = 11990411986411120590119903 + 11990411986412120590120579 + 11988931119864119885

120576120579 = 11990411986412120590119903 + 11990411986411120590120579 + 11988931119864119885119863119911 = 11988931120590119903 + 11988931120590120579 + 12058511987933119864119885

(35)

It corresponds to short circuit situation when field inten-sity is 119864119885 = 0 Therefore (35) can be simplified as

120590119903 = 11990411986411(11990411986411)2 minus (11990411986412)2 120576119903 minus11990411986412(11990411986411)2 minus (11990411986412)2 120576120579

120590120579 = 11990411986411(11990411986411)2 minus (11990411986412)2 120576120579 minus 11990411986412(11990411986411)2 minus (11990411986412)2 120576119903119863119911 = 11988931 (120590119903 + 120590120579)

(36)

Then it has

]119864 = minus1199041198641211990411986411 1198841198640 = 111990411986411

(37)

Shock and Vibration 7

times104

minus100

minus80

minus60

minus40

minus20

0

20

40

60

Tran

smiss

ibili

ty (d

B)

1 2 3 4 5 6 7 8 9 100

Frequency (Hz)

Open circuitShort circuit

Figure 5 Comparison diagram with short and open circuits

where ]119864 denotes Poissonrsquos coefficient in the condition ofshort circuit 1198841198640 denotes Young modulus

Piezoelectric material is in the open circuit when thecharge is 119863119911 = 0 Thus (35) can be simplified as

120590119903 = 1198841198631 minus (]119863)2 (120576119903 + ]119863120576120579)

120590120579 = 1198841198631 minus (]119863)2 (120576120579 + ]119863120576119903)

(38)

Then it gives

]119863 = minus1205851198793311990411986412 minus 1198892311205851198793311990411986411 minus 119889231 1198841198630 = 111990411986411 minus 11988923112058511987933

(39)

where ]119863 denotes Poissonrsquos coefficient in the condition ofopen circuit 1198841198630 denotes Young modulus

By using transfer matrices derived in Section 21 vibra-tion characteristics curves have been calculated for the caseof short and open circuit As depicted in Figure 5 the selectedfrequency band is 0ndash10 kHz It indicates that the band changesobviously in the high frequency because of the piezoelectriceffect However the short circuit curve moves toward lowfrequency slightly From the above theoretical analysis bycomparing (37) and (39) it can be observed that the elasticitymodulus and Poissonrsquos coefficient of piezoelectric materialsare different when they are in short or open circuit For thecase of open circuit elasticity modulus is larger MeanwhilePoissonrsquos coefficient also changes It is important to note thatthese two parameters can influence the value of 120590119903 and 120590120579directly whichmeans that radial wavenumber changesThusit is the natural mechanism of band moving

332 Piezoelectric Parameters and Inner Radius As to theopen circuit case the effects of electrical parameter 11988931 andinner radius 1199030 on radial wave band are depicted in Figure 6Herein Figure 6(a) describes the effect of piezoelectric

times104

104 6 820 1 3 5 7 9

Frequency (Hz)

minus120

minus100

minus20

minus60

minus40

minus20

0

20

40

60

Tran

smiss

ibili

ty (d

B)

d31 = minus123 lowast 10minus12 cm2

d31 = minus200 lowast 10minus12 cm2

(a) Effect of piezoelectric parameter 11988931

times104Frequency (Hz)

minus100

minus80

minus60

minus40

minus20

0

20

40

60

Tran

smiss

ibili

ty (d

B)

1086421 3 5 7 90

r0 = 008mr0 = 004mr0 = 002m

(b) Effect of inner radius 1199030

Figure 6 Effect of piezoelectric modulus 11988931 and inner radius 1199030

parameter 11988931 However Figure 6(b) displays the effect ofinner radius 1199030

From (39) it can be easily observed that 11988931 determinesthe values of 1198841198630 and ]119863 which can influence the bandposition and attenuation Figure 6(a) indicates that radialband has been expanded with the increase of 11988931 (here thenegative sign only represents direction) However the changeis not obvious Figure 6(b) displays that inner radius 1199030 caninfluence the low frequency greatly It is also clear from thefigure that the band attenuates obviously at low frequencyHowever the effect on high frequency is much smaller

4 Conclusion

This paper presents the classical method and wave propaga-tion approach to obtain the natural frequency Based on radialvibration equation propagation and reflection matrices arederived It can be found that natural frequency of the singlecircular plate can be calculated conveniently by using wavepropagation approach Then the results obtained by usingwave approach are compared with the classical method at theboundary conditions of free-free fixed-fixed and fixed-free

8 Shock and Vibration

It can be seen that the results calculated by these twomethodscoincide with each other

Additionally the band phenomenon of radial phononcrystal circular plate is also analyzed for a deeper understand-ing of radial vibration FEM is used to verify the theoreticalresults It can be found that the inner radius of circular platehas great influence on the low frequency Piezoelectric coeffi-cient open circuit and short circuitmake the elasticmodulusand Poissonrsquos ratio change Then the radial wavenumber alsochanges which is the natural mechanism of band removingfor the case of open and short circuits

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work is financially supported by the National NaturalScience Foundation of China (no 51375105) and the Natu-ral Science Foundation of Heilongjiang Province of China(E201418)

References

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

[2] C Mei and B R Mace ldquoWave reflection and transmission inTimoshenko beams and wave analysis of Timoshenko beamstructuresrdquo Journal of Sound andAcoustics vol 127 pp 382ndash3942004

[3] N R Harland B R Mace and RW Jones ldquoWave propagationreflection and transmission in tunable fluid-filled beamsrdquo Jour-nal of Sound and Vibration vol 241 no 5 pp 735ndash754 2001

[4] 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

[5] 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

[6] S K Lee Wave reflection transmission and propagation instructural waveguides [PhD thesis] Southampton University2006

[7] CHWang and L R F Rose ldquoWave reflection and transmissionin beams containing delamination and inhomogeneityrdquo Journalof Sound and Vibration vol 264 no 4 pp 851ndash872 2003

[8] C A Tan and B Kang ldquoWave reflection and transmission in anaxially strained rotating Timoshenko shaftrdquo Journal of Soundand Vibration vol 213 no 3 pp 483ndash510 1998

[9] X B Li ldquoStudy on free vibration analysis of circular cylindricalshells using wave propagationrdquo Journal of Sound and Vibrationvol 311 no 3ndash5 pp 667ndash682 2008

[10] L Gan X B Li and Z Zhang ldquoFree vibration analysis of ring-stiffened cylindrical shells using wave propagation approachrdquoJournal of Sound and Vibration vol 326 no 3ndash5 pp 633ndash6462009

[11] X M Zhang ldquoVibration analysis of cross-ply laminated com-posite cylindrical shells using the wave propagation approachrdquoApplied Acoustics vol 62 no 11 pp 1221ndash1228 2001

[12] X M Zhang ldquoFrequency analysis of submerged cylindricalshells with the wave propagation approachrdquo International Jour-nal of Mechanical Sciences vol 44 no 7 pp 1259ndash1273 2002

[13] M Shakeri M Akhlaghi and S M Hoseini ldquoVibration andradial wave propagation velocity in functionally graded thickhollow cylinderrdquo Composite Structures vol 76 no 1-2 pp 174ndash181 2006

[14] D Torrent and J Sanchez-Dehesa ldquoAcoustic resonances in two-dimensional radial sonic crystal shellsrdquo New Journal of Physicsvol 12 Article ID 073034 2010

[15] J Carbonell D Torrent and J Sanchez-Dehesa ldquoRadial pho-tonic crystal shells and their application as resonant and radiat-ing elementsrdquo IEEE Transactions on Antennas and Propagationvol 61 no 2 pp 755ndash767 2013

[16] Z L Xu F G Wu and Z N Guo ldquoLow frequency phononicband structures in two-dimensional arc-shaped phononic crys-talsrdquo Physics Letters A vol 376 no 33 pp 2256ndash2263 2012

[17] T Ma T N Chen X P Wang Y Li and P Wang ldquoBandstructures of bilayer radial phononic crystal plate with crystalglidingrdquo Journal of Applied Physics vol 116 no 10 Article ID104505 pp 1ndash7 2014

[18] H S Shu L Q Dong S D Li et al ldquoPropagation of torsionalwaves in a thin circular plate of generalized phononic crystalsrdquoJournal of Physics D Applied Physics vol 47 no 29 Article ID295501 2014

[19] H S Shu W Liu S D Li et al ldquoResearch on flexural waveband gap of a thin circular plate of piezoelectric radial phononiccrystalsrdquo Journal of Vibration and Control vol 22 no 7 pp1777ndash1789 2016

[20] K Kiani ldquoElasto-dynamic analysis of spinning nanodisks viaa surface energy-based modelrdquo Journal of Physics D AppliedPhysics vol 49 no 27 Article ID 275306 2016

[21] K Kiani ldquoThermo-mechanical analysis of functionally gradedplate-like nanorotors a surface elasticity modelrdquo InternationalJournal of Mechanical Sciences vol 106 pp 39ndash49 2016

[22] K Kiani ldquoMagneto-elasto-dynamic analysis of an elasticallyconfined conducting nanowire due to an axial magnetic shockrdquoPhysics Letters A vol 376 no 20 pp 1679ndash1685 2012

[23] S Y Lin ldquoStudy on the step-type circular ring ultrasonicconcentrator in radial vibrationrdquo Ultrasonics vol 49 no 2 pp206ndash211 2009

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2 Shock and Vibration

r0

rao

a+1

aminus1

b+1

bminus1

Figure 1 Model of single circular plate

isolate the vibration Also these structures can be regarded asa filter Additionally Shu et al researched torsional vibrationproblem of generalized phononic crystal finding that lowfrequency band was determined by radius However the highfrequency bandwas caused by the periodicity of circular plate[18] And Shursquos research shows that generalized phononiccrystal can generate a huge band which almost covers thewhole frequency range providing a practical application forthe rotating parts such as gear In the previous researchShu et al have studied the band gaps of flexural wave inthe radial phononic crystal Meanwhile numerical resultsare also verified by FEM [19] Herein this paper is mainlyfocused on radial vibration of composite structures In factother works on radial vibration of elastic solids under variousloadings can also be found [20ndash22]

The paper is organized into five sections In Section 1a brief introduction is given In Section 2 classic Besselsolution and Hankel solution of radial vibration are givenMeanwhile propagation and reflection matrices at threeboundary conditions are derived The results of naturalfrequency obtained bywave approach are also comparedwiththe classical method In Section 3 the band phenomenon isanalyzed and the effect of parameters is discussed Section 4is the conclusion

2 Theoretical Analysis for Natural Frequency

21 Solution of Radial Vibration Equation Single circularplate is shown in Figure 1 Young modulus density and Pois-sonrsquos coefficient are 1198641 1205881 ]1 The thickness of piezoelectricceramics is ℎ1 The inner and external radius are 1199030 and 119903119886

respectivelyWith respect to the axisymmetric circular platesradial equation of piezoelectric materials can be given by [23]

12058812059721199061199031205971199052 = 120597120590119903120597119903 + 120590119903 minus 120590120579119903 (1)

The expression of strain and radial displacement can bewritten as

120576119903 = 120597119906119903120597119903 (2)

120576120579 = 119906119903119903 (3)

Since piezoelectric material direction of polarization isalong z-axle piezoelectric equation can be expressed as

120576119903 = 11990411986411120590119903 + 11990411986412120590120579 + 11988931119864119885120576120579 = 11990411986412120590119903 + 11990411986411120590120579 + 11988931119864119885

119863119911 = 11988931120590119903 + 11988931120590120579 + 12058511987933119864119885(4)

where 119906119903 is radial displacement 120590119903 and 120590120579 are radial stressand tangential stress 120576119903 and 120576120579 are radial strain and tangentialstrain and 119903 is radius

Substituting (2)ndash(4) into (1) gives

12059721199061199031205971199032 + 1119903 120597119906119903120597119903 minus 1199061199031199032 = 11198882 12059721199061199031205971199052 (5)

where 1198882 = 111990411986411120588(1minus]2) is the square of radial wave velocityAfter simplifying (5) it can be obtained that

1205972119906119903120597 (119896119903)2 + 1119896119903 120597119906119903120597 (119896119903) + [1 minus 1(119896119903)2] 119906 = 0 (6)

Thus general solution of radial displacement can bewritten as

119906119903 = 119860+11198691 (119896119903) + 119860minus11198841 (119896119903) (7)

where 119896 = 119908119888 is the wave number 1198691(119896119903) is the Besselfunction of first kind 1198841(119896119903) is the Bessel function of secondkind And the constants 119860+1 and 119860minus1 can be determined byboundary conditions

Therefore radial stress can be obtained as

120590119903 = 1198641 minus ]2(120576119903 + ]120576120579)

= 1198641198961 minus ]2119860+1 [1198690 (119896119903) minus (1 minus ]) 1198691 (119896119903)119896119903 ]

+ 119860minus1 [1198840 (119896119903) minus (1 minus ]) 1198841 (119896119903)119896119903 ] (8)

22 Classical Method When both ends of circular plate arefixed it has

119906119903 (1199030) = 0119906119903 (119903119886) = 0 (9)

Shock and Vibration 3

Substituting (7) into (9) gives

10038161003816100381610038161003816100381610038161003816100381610038161198691 (1198961199030) 1198841 (1198961199030)1198691 (119896119903119886) 1198841 (119896119903119886)

1003816100381610038161003816100381610038161003816100381610038161003816 = 0 (10)

When both ends of circular plate are free it has

120590119903 (1199030) = 0120590119903 (119903119886) = 0 (11)

Substituting (8) into (11) gives

1198641198961 minus ]2

sdot100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161198690 (1198961199030) minus (1 minus ]) 1198691 (1198961199030)1198961199030 1198840 (1198961199030) minus (1 minus ]) 1198841 (1198961199030)11989611990301198690 (119896119903119886) minus (1 minus ]) 1198691 (119896119903119886)119896119903119886 1198690 (119896119903119886) minus (1 minus ]) 1198691 (119896119903119886)119896119903119886

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816= 0

(12)

Similarly when inner surface is fixed and outer surface isfree it has

119906119903 (1199030) = 0120590119903 (119903119886) = 0 (13)

Substituting (7) and (8) into (13) gives

1198641198961 minus ]2

sdot100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198690 (1198961199030) 1198840 (1198961199030)1198690 (119896119903119886) minus (1 minus ]) 1198691 (119896119903119886)119896119903119886 1198690 (119896119903119886) minus (1 minus ]) 1198691 (119896119903119886)119896119903119886

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816= 0

(14)

After solving roots of (10) (12) and (14) the naturalfrequencies of circular plate structure can be obtained withthe aforementioned boundary conditions

23 Wave Propagation Method One of the solutions ofcircular plate has been solved as shown in (7) And the othersolution can be written as

119906119903 = 119860+1119867(2)0 (119896119903) + 119860minus1119867(1)0 (119896119903) (15)

where 119867(1)0 (119896119903) and 119867(2)0 (119896119903) are the Hankel function of firstand second kinds which can be defined as

119867(1)0 (119896119903) = 1198690 (119896119903) + 1198941198840 (119896119903) 119867(2)0 (119896119903) = 1198690 (119896119903) minus 1198941198840 (119896119903) (16)

Considering (15) forward-traveling and negative-traveling waves can be written as

119892+1 = 119860+1119867(2)0 (119896119903) 119892minus1 = 119860minus1119867(1)0 (119896119903) (17)

Then (15) reduces to

119906119903 = 119892+1 + 119892minus1 (18)

When both ends of circular plate are fixed it gives

119906119903 (1199030) = 119892+1 (1199030) + 119892minus1 (1199030) = 0119906119903 (119903119886) = 119892+1 (119903119886) + 119892minus1 (119903119886) = 0 (19)

Thus the reflection matrices of both boundaries are

119877119860 = [minus1] (20a)

119877119862 = [minus1] (20b)

Substituting (2) (15) and (16) into (8) radial stress can beexpressed as

120590119903 = 1198641198961 minus ]2119860+1 [1198961198690 (119896119903) minus 1198691 (119896119903)119903 minus 1198941198961198840 (119896119903)

+ 1198941199031198841 (119896119903) + ]119903 1198691 (119896119903) minus ]119903 1198941198841 (119896119903)] + 119860minus1 [1198961198690 (119896119903)minus 1198691 (119896119903)119903 + 1198941198961198840 (119896119903) minus 1198941199031198841 (119896119903) + ]119903 1198691 (119896119903)+ ]119903 1198941198841 (119896119903)]

(21)

When both ends of circular plate are free it has

120590119903 (1199030) = 0120590119903 (119903119886) = 0 (22)

Thus reflection matrices of both boundaries are

119877119860 = minus [1198961198690 (1198961199030) minus 1198691 (1198961199030) 1199030 + 1198941198961198840 (1198961199030) minus (1198941199030) 1198841 (1198961199030) + (]1199030) 1198691 (1198961199030) + (]1199030) 1198941198841 (1198961199030)]119867(1)0 (1198961199030) minus1

times [1198961198690 (1198961199030) minus 1198691 (1198961199030) 1199030 minus 1198941198961198840 (1198961199030) + (1198941199030) 1198841 (1198961199030) + (]1199030) 1198691 (1198961199030) minus (]1199030) 1198941198841 (1198961199030)]119867(2)0 (1198961199030) (23)

4 Shock and Vibration

119877119862 = minus [1198961198690 (119896119903119886) minus 1198691 (119896119903119886) 119903119886 minus 1198941198961198840 (119896119903119886) + (119894119903119886) 1198841 (1198961199030) + (]119903119886) 1198691 (119896119903119886) minus (]119903119886) 1198941198841 (119896119903119886)]119867(2)0 (119896119903119886) minus1

sdot [1198961198690 (119896119903119886) minus 1198691 (119896119903119886) 119903119886 + 1198941198961198840 (119896119903119886) minus (119894119903119886) 1198841 (119896119903119886) + (]119903119886) 1198691 (119896119903119886) + (]119903119886) 1198941198841 (119896119903119886)]119867(1)0 (119896119903119886) (24)

Forward-traveling and negative-traveling waves can beshown clearly in Figure 1 And the expression can be writtenas follows

119887+1 = 119891+1 (119903119886 minus 1199030) 119886+1 119886minus1 = 119891minus1 (119903119886 minus 1199030) 119887minus1

(25)

Considering (17) forward-traveling and negative-traveling waves can be obtained as

119891+1 (119903119886 minus 1199030) = 119867(2)0 (119896119903119886)119867(2)0 (1198961199030)

119891minus1 (119903119886 minus 1199030) = 119867(1)0 (1198961199030)119867(1)0 (119896119903119886) (26)

Similarly the reflection matrices are (20a) and (24) forcase of inner surface fixed and outer surface free

24 Analysis and Discussion After combining propagationand reflection matrices derived above the natural charac-teristics of single circular plate can be analyzed effectivelyHence it gives

119887+1 = 119891+1 (119903119886 minus 1199030) 119886+1 119886+1 = 119877119860119886minus1 119886minus1 = 119891minus1 (119903119886 minus 1199030) 119887minus1 119887minus1 = 119877119862119887+1

(27)

And it yields

119865 (119891) = [119891+1 (119903119886 minus 1199030) 119877119860119891minus1 (119903119886 minus 1199030) 119877119862 minus 1] 119887+1 = 0 (28)

RESIN is selected as the material of circular plate Mate-rial and structural parameters are tabulated in Table 1

Substitute parameters listed in Table 1 into (10) (12) (14)and (28) and the natural frequency can be computedHerein red and black curves obtained by wave approachare real and imaginary wave solution respectively Also it isimportant to note that the intersection points of the curves arenatural frequency After solving the inherent characteristic ofcircular plate structure the characteristic curves calculated byclassicalmethod andwave approach are presented in Figure 2It can be seen that these natural frequencies intersect at thesame point in x-axis which verifies the correctness of thenumerical calculation

3 Radial Vibration Analysis ofPhononic Crystal

31 Model and Transfer Matrix Figure 3 is the schematicdiagram of radial phononic crystal circular plate Herein thewhite region represents RESIN materials and the gray regionrepresents piezoelectric ceramic PZT4 The inner and outerradius are 1199031 and 1199032 respectively The radial span ratios aredenoted with 1198861 and 1198862 And ℎ represents the thickness

Considering an eight-layer circular plate it correspondsto the model 119860119861119860119861119860119861119860119861 The interfaces between RESINand PZT4 material are 119886 119887 119888 119889 119890 119891 119898 119901 119902 (see Figure 3)With regard to RESIN Youngrsquosmodulus density and Poissoncoefficient are 1198641 1205881 ]1 For the PZT4 Youngrsquos modulusdensity and Poisson coefficient are 1198642 1205882 ]2 When radialwave propagates in RESIN layers 1 3 5 and 7 the velocityis 1198881 = [11986411205881(1 minus ]1

2)]12 and the wavenumber is 1198961 =1199081198881 = 21205871198911198881The velocity is 1198882 = [11986421205882(1minus]22)]12 and the

wavenumber is 1198962 = 1199081198882 = 21205871198911198882 when wave propagates inPZT4 layers 2 4 6 and 8

With regard to radial vibration there are two parameterswhich satisfy the continuous conditions at the interface ofdifferent materials Namely they are the radial displacementand radial stress So transfer matrices of radial vibration canbe derived as follows

At the interface b (119903119887 = 1199031 + 1198861) it gives1199061 = 11990621205901 = 1205902 (29)

Thus it can be arranged in a matrices form

1198671 [119860+11119860minus11] = 1198701 [119861+12119861minus12] (30)

At the interface c (119903119888 = 1199031 + 1198861 + 1198862) it gives1199062 = 11990631205902 = 1205903 (31)

Similarly one has

1198672 [119860+21119860minus21] = 1198702 [119861+12119861minus12] (32)

Combining (30) and (32) one has

[119860+21119860minus21] = 11987931 [119860+11119860minus11] (33)

Shock and Vibration 5

times104

minus2

minus1

0

1

2

3

4

5F

(f)

21 4 5 73 6 80

Frequency (Hz)

Classical Bessel solutionClassical Hankel solution

Wave solution (imag)Wave solution (real)

(a) Fixed-fixed boundary

times104

minus3

minus2

minus1

0

1

2

3

F (f

)

2 4 6 80

Frequency (Hz)

Classical Bessel solutionClassical Hankel solution

Wave solution (imag)Wave solution (real)

1 5 73

(b) Free-free boundary

times104

2 4 6 8 10 12 14 160

Frequency (Hz)

Classical Bessel solutionClassical Hankel solution

Wave solution (imag)Wave solution (real)

minus4

minus2

0

2

4

6

F (f

)

(c) Fixed-free boundary

Figure 2 Natural frequency calculated by classic method and wave approach

Table 1 Material and structural parameters

Material parameters 120588 (kgm3) 119864 (Pa) ]RESIN 1180 0435 times 1010 03679Structural parameters 1199030 (m) 119903119886 (m) ℎ (m)RESIN 008 016 0005

where 11987931 = 119867minus12 1198702119870minus11 1198671 is the transfer matrices of radialvibration between the 3rd and the 1st ring

As shown in Figure 3 replace 119903119887 and 119903119888 (in (30) and (32))with 119903119889 and 119903119890 Similarly with 11987931 we can obtain 11987953 which isthe transfer matrix between the fifth ring and the third ring

Similarly replace 119903119889 and 119903119890 with 119903119891 and 119903119898 and the matrix11987975 can be obtainedAt last submitting 119903119901 into (30) we can get 11987981 Therefore

with regard to the radial phononic crystal one has

11987981 = 119870minus18711986787119879751198795311987931 (34)

32 Numerical Result and Discussion The radial phononiccrystal circular plates are composed of twomaterials namelyRESIN and piezoelectric ceramic PZT4 Material parametersand structural parameters are consistent with Table 2

Table 2 Material parameters and structural parameters

Material parameters 120588 (kgm3) 119864 (Pa) ]I (RESIN) 1180 0435 times 1010 03679II (PZT4) 7500 813 times 1010 033Structural parameters 1198861 = 1198862 (m) 1199032 = 21199031 (m) ℎ (m)RESIN and PZT4 001 016 0003

Additionally piezoelectric parameters are set as 11990411986411 =123 times 10minus12m2N 120576119879331205760 = 1300 1205760 = 884 times 10minus12 cm and11988931 = minus123 times 10minus12 cNTaking advantage of transfer matrices and boundary

conditions the transmission characteristics are calculatedusing MATLAB numerically Figure 4 is the transmissioncurve of radial vibration with four periodic structures whichis composited by single material RESIN and RESINPZT4

Figure 4 is the comparison of numerical and FEM resultsIt indicates that there is no attenuation when radial wavepropagates in a single material while for the phononic crys-tal the transmission curve shows a big vibration attenuationband after introducing the periodicity However when radialwave propagates in the single material there is no band gapwhich attenuates weakly and only owns minus4 dB Furthermore

6 Shock and Vibration

r1o a1 a2 a1a2

IIIIII r2

I RESINII PZT4

h

12 3

45

6 78

qpmfedcba

Figure 3 Schematic diagram of radial phononic crystal circular plate

times104

minus40

minus30

minus20

minus10

0

10

20

30

40

50

60

Tran

smiss

ibili

ty (d

B)

1 2 3 4 5 60

Frequency (Hz)

Numerical resultFEM

(a) Single material (RESIN)

Numerical resultFEM

times104

minus100

minus80

minus60

minus40

minus20

0

20

40

60

Tran

smiss

ibili

ty (d

B)

1 2 3 4 5 60

Frequency (Hz)

(b) Phononic crystal (RESINPZT4)

Figure 4 Radial vibration curves of numerical and FEM results

ANSYS 145 is used to verify the radial propagation charac-teristics Using harmonic analysis and keeping the boundaryconditions of outer surface free radial displacement is excitedon the inner surface which is shown in Figure 4(b) Compar-ing numerical results with FEM results it can be found thatthe bands are in good agreement with the aspects of positionwidth and amplitude attenuation and other aspects whichindicates the correctness of theoretical calculations

33 Effects of Structural Parameters In this section the influ-ence of piezoelectric effect on radial wave has been discussedin detail Furthermore the influences of piezoelectric param-eter 11988931 and polar direction on the band have also been stud-ied in case of open and short circuit for piezoelectricmaterial

331 Open and Short Circuit With regard to piezoelectricmaterial the D-type equation of radial vibration can beexpressed as

120576119903 = 11990411986411120590119903 + 11990411986412120590120579 + 11988931119864119885

120576120579 = 11990411986412120590119903 + 11990411986411120590120579 + 11988931119864119885119863119911 = 11988931120590119903 + 11988931120590120579 + 12058511987933119864119885

(35)

It corresponds to short circuit situation when field inten-sity is 119864119885 = 0 Therefore (35) can be simplified as

120590119903 = 11990411986411(11990411986411)2 minus (11990411986412)2 120576119903 minus11990411986412(11990411986411)2 minus (11990411986412)2 120576120579

120590120579 = 11990411986411(11990411986411)2 minus (11990411986412)2 120576120579 minus 11990411986412(11990411986411)2 minus (11990411986412)2 120576119903119863119911 = 11988931 (120590119903 + 120590120579)

(36)

Then it has

]119864 = minus1199041198641211990411986411 1198841198640 = 111990411986411

(37)

Shock and Vibration 7

times104

minus100

minus80

minus60

minus40

minus20

0

20

40

60

Tran

smiss

ibili

ty (d

B)

1 2 3 4 5 6 7 8 9 100

Frequency (Hz)

Open circuitShort circuit

Figure 5 Comparison diagram with short and open circuits

where ]119864 denotes Poissonrsquos coefficient in the condition ofshort circuit 1198841198640 denotes Young modulus

Piezoelectric material is in the open circuit when thecharge is 119863119911 = 0 Thus (35) can be simplified as

120590119903 = 1198841198631 minus (]119863)2 (120576119903 + ]119863120576120579)

120590120579 = 1198841198631 minus (]119863)2 (120576120579 + ]119863120576119903)

(38)

Then it gives

]119863 = minus1205851198793311990411986412 minus 1198892311205851198793311990411986411 minus 119889231 1198841198630 = 111990411986411 minus 11988923112058511987933

(39)

where ]119863 denotes Poissonrsquos coefficient in the condition ofopen circuit 1198841198630 denotes Young modulus

By using transfer matrices derived in Section 21 vibra-tion characteristics curves have been calculated for the caseof short and open circuit As depicted in Figure 5 the selectedfrequency band is 0ndash10 kHz It indicates that the band changesobviously in the high frequency because of the piezoelectriceffect However the short circuit curve moves toward lowfrequency slightly From the above theoretical analysis bycomparing (37) and (39) it can be observed that the elasticitymodulus and Poissonrsquos coefficient of piezoelectric materialsare different when they are in short or open circuit For thecase of open circuit elasticity modulus is larger MeanwhilePoissonrsquos coefficient also changes It is important to note thatthese two parameters can influence the value of 120590119903 and 120590120579directly whichmeans that radial wavenumber changesThusit is the natural mechanism of band moving

332 Piezoelectric Parameters and Inner Radius As to theopen circuit case the effects of electrical parameter 11988931 andinner radius 1199030 on radial wave band are depicted in Figure 6Herein Figure 6(a) describes the effect of piezoelectric

times104

104 6 820 1 3 5 7 9

Frequency (Hz)

minus120

minus100

minus20

minus60

minus40

minus20

0

20

40

60

Tran

smiss

ibili

ty (d

B)

d31 = minus123 lowast 10minus12 cm2

d31 = minus200 lowast 10minus12 cm2

(a) Effect of piezoelectric parameter 11988931

times104Frequency (Hz)

minus100

minus80

minus60

minus40

minus20

0

20

40

60

Tran

smiss

ibili

ty (d

B)

1086421 3 5 7 90

r0 = 008mr0 = 004mr0 = 002m

(b) Effect of inner radius 1199030

Figure 6 Effect of piezoelectric modulus 11988931 and inner radius 1199030

parameter 11988931 However Figure 6(b) displays the effect ofinner radius 1199030

From (39) it can be easily observed that 11988931 determinesthe values of 1198841198630 and ]119863 which can influence the bandposition and attenuation Figure 6(a) indicates that radialband has been expanded with the increase of 11988931 (here thenegative sign only represents direction) However the changeis not obvious Figure 6(b) displays that inner radius 1199030 caninfluence the low frequency greatly It is also clear from thefigure that the band attenuates obviously at low frequencyHowever the effect on high frequency is much smaller

4 Conclusion

This paper presents the classical method and wave propaga-tion approach to obtain the natural frequency Based on radialvibration equation propagation and reflection matrices arederived It can be found that natural frequency of the singlecircular plate can be calculated conveniently by using wavepropagation approach Then the results obtained by usingwave approach are compared with the classical method at theboundary conditions of free-free fixed-fixed and fixed-free

8 Shock and Vibration

It can be seen that the results calculated by these twomethodscoincide with each other

Additionally the band phenomenon of radial phononcrystal circular plate is also analyzed for a deeper understand-ing of radial vibration FEM is used to verify the theoreticalresults It can be found that the inner radius of circular platehas great influence on the low frequency Piezoelectric coeffi-cient open circuit and short circuitmake the elasticmodulusand Poissonrsquos ratio change Then the radial wavenumber alsochanges which is the natural mechanism of band removingfor the case of open and short circuits

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work is financially supported by the National NaturalScience Foundation of China (no 51375105) and the Natu-ral Science Foundation of Heilongjiang Province of China(E201418)

References

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

[2] C Mei and B R Mace ldquoWave reflection and transmission inTimoshenko beams and wave analysis of Timoshenko beamstructuresrdquo Journal of Sound andAcoustics vol 127 pp 382ndash3942004

[3] N R Harland B R Mace and RW Jones ldquoWave propagationreflection and transmission in tunable fluid-filled beamsrdquo Jour-nal of Sound and Vibration vol 241 no 5 pp 735ndash754 2001

[4] 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

[5] 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

[6] S K Lee Wave reflection transmission and propagation instructural waveguides [PhD thesis] Southampton University2006

[7] CHWang and L R F Rose ldquoWave reflection and transmissionin beams containing delamination and inhomogeneityrdquo Journalof Sound and Vibration vol 264 no 4 pp 851ndash872 2003

[8] C A Tan and B Kang ldquoWave reflection and transmission in anaxially strained rotating Timoshenko shaftrdquo Journal of Soundand Vibration vol 213 no 3 pp 483ndash510 1998

[9] X B Li ldquoStudy on free vibration analysis of circular cylindricalshells using wave propagationrdquo Journal of Sound and Vibrationvol 311 no 3ndash5 pp 667ndash682 2008

[10] L Gan X B Li and Z Zhang ldquoFree vibration analysis of ring-stiffened cylindrical shells using wave propagation approachrdquoJournal of Sound and Vibration vol 326 no 3ndash5 pp 633ndash6462009

[11] X M Zhang ldquoVibration analysis of cross-ply laminated com-posite cylindrical shells using the wave propagation approachrdquoApplied Acoustics vol 62 no 11 pp 1221ndash1228 2001

[12] X M Zhang ldquoFrequency analysis of submerged cylindricalshells with the wave propagation approachrdquo International Jour-nal of Mechanical Sciences vol 44 no 7 pp 1259ndash1273 2002

[13] M Shakeri M Akhlaghi and S M Hoseini ldquoVibration andradial wave propagation velocity in functionally graded thickhollow cylinderrdquo Composite Structures vol 76 no 1-2 pp 174ndash181 2006

[14] D Torrent and J Sanchez-Dehesa ldquoAcoustic resonances in two-dimensional radial sonic crystal shellsrdquo New Journal of Physicsvol 12 Article ID 073034 2010

[15] J Carbonell D Torrent and J Sanchez-Dehesa ldquoRadial pho-tonic crystal shells and their application as resonant and radiat-ing elementsrdquo IEEE Transactions on Antennas and Propagationvol 61 no 2 pp 755ndash767 2013

[16] Z L Xu F G Wu and Z N Guo ldquoLow frequency phononicband structures in two-dimensional arc-shaped phononic crys-talsrdquo Physics Letters A vol 376 no 33 pp 2256ndash2263 2012

[17] T Ma T N Chen X P Wang Y Li and P Wang ldquoBandstructures of bilayer radial phononic crystal plate with crystalglidingrdquo Journal of Applied Physics vol 116 no 10 Article ID104505 pp 1ndash7 2014

[18] H S Shu L Q Dong S D Li et al ldquoPropagation of torsionalwaves in a thin circular plate of generalized phononic crystalsrdquoJournal of Physics D Applied Physics vol 47 no 29 Article ID295501 2014

[19] H S Shu W Liu S D Li et al ldquoResearch on flexural waveband gap of a thin circular plate of piezoelectric radial phononiccrystalsrdquo Journal of Vibration and Control vol 22 no 7 pp1777ndash1789 2016

[20] K Kiani ldquoElasto-dynamic analysis of spinning nanodisks viaa surface energy-based modelrdquo Journal of Physics D AppliedPhysics vol 49 no 27 Article ID 275306 2016

[21] K Kiani ldquoThermo-mechanical analysis of functionally gradedplate-like nanorotors a surface elasticity modelrdquo InternationalJournal of Mechanical Sciences vol 106 pp 39ndash49 2016

[22] K Kiani ldquoMagneto-elasto-dynamic analysis of an elasticallyconfined conducting nanowire due to an axial magnetic shockrdquoPhysics Letters A vol 376 no 20 pp 1679ndash1685 2012

[23] S Y Lin ldquoStudy on the step-type circular ring ultrasonicconcentrator in radial vibrationrdquo Ultrasonics vol 49 no 2 pp206ndash211 2009

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Shock and Vibration 3

Substituting (7) into (9) gives

10038161003816100381610038161003816100381610038161003816100381610038161198691 (1198961199030) 1198841 (1198961199030)1198691 (119896119903119886) 1198841 (119896119903119886)

1003816100381610038161003816100381610038161003816100381610038161003816 = 0 (10)

When both ends of circular plate are free it has

120590119903 (1199030) = 0120590119903 (119903119886) = 0 (11)

Substituting (8) into (11) gives

1198641198961 minus ]2

sdot100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161198690 (1198961199030) minus (1 minus ]) 1198691 (1198961199030)1198961199030 1198840 (1198961199030) minus (1 minus ]) 1198841 (1198961199030)11989611990301198690 (119896119903119886) minus (1 minus ]) 1198691 (119896119903119886)119896119903119886 1198690 (119896119903119886) minus (1 minus ]) 1198691 (119896119903119886)119896119903119886

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816= 0

(12)

Similarly when inner surface is fixed and outer surface isfree it has

119906119903 (1199030) = 0120590119903 (119903119886) = 0 (13)

Substituting (7) and (8) into (13) gives

1198641198961 minus ]2

sdot100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198690 (1198961199030) 1198840 (1198961199030)1198690 (119896119903119886) minus (1 minus ]) 1198691 (119896119903119886)119896119903119886 1198690 (119896119903119886) minus (1 minus ]) 1198691 (119896119903119886)119896119903119886

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816= 0

(14)

After solving roots of (10) (12) and (14) the naturalfrequencies of circular plate structure can be obtained withthe aforementioned boundary conditions

23 Wave Propagation Method One of the solutions ofcircular plate has been solved as shown in (7) And the othersolution can be written as

119906119903 = 119860+1119867(2)0 (119896119903) + 119860minus1119867(1)0 (119896119903) (15)

where 119867(1)0 (119896119903) and 119867(2)0 (119896119903) are the Hankel function of firstand second kinds which can be defined as

119867(1)0 (119896119903) = 1198690 (119896119903) + 1198941198840 (119896119903) 119867(2)0 (119896119903) = 1198690 (119896119903) minus 1198941198840 (119896119903) (16)

Considering (15) forward-traveling and negative-traveling waves can be written as

119892+1 = 119860+1119867(2)0 (119896119903) 119892minus1 = 119860minus1119867(1)0 (119896119903) (17)

Then (15) reduces to

119906119903 = 119892+1 + 119892minus1 (18)

When both ends of circular plate are fixed it gives

119906119903 (1199030) = 119892+1 (1199030) + 119892minus1 (1199030) = 0119906119903 (119903119886) = 119892+1 (119903119886) + 119892minus1 (119903119886) = 0 (19)

Thus the reflection matrices of both boundaries are

119877119860 = [minus1] (20a)

119877119862 = [minus1] (20b)

Substituting (2) (15) and (16) into (8) radial stress can beexpressed as

120590119903 = 1198641198961 minus ]2119860+1 [1198961198690 (119896119903) minus 1198691 (119896119903)119903 minus 1198941198961198840 (119896119903)

+ 1198941199031198841 (119896119903) + ]119903 1198691 (119896119903) minus ]119903 1198941198841 (119896119903)] + 119860minus1 [1198961198690 (119896119903)minus 1198691 (119896119903)119903 + 1198941198961198840 (119896119903) minus 1198941199031198841 (119896119903) + ]119903 1198691 (119896119903)+ ]119903 1198941198841 (119896119903)]

(21)

When both ends of circular plate are free it has

120590119903 (1199030) = 0120590119903 (119903119886) = 0 (22)

Thus reflection matrices of both boundaries are

119877119860 = minus [1198961198690 (1198961199030) minus 1198691 (1198961199030) 1199030 + 1198941198961198840 (1198961199030) minus (1198941199030) 1198841 (1198961199030) + (]1199030) 1198691 (1198961199030) + (]1199030) 1198941198841 (1198961199030)]119867(1)0 (1198961199030) minus1

times [1198961198690 (1198961199030) minus 1198691 (1198961199030) 1199030 minus 1198941198961198840 (1198961199030) + (1198941199030) 1198841 (1198961199030) + (]1199030) 1198691 (1198961199030) minus (]1199030) 1198941198841 (1198961199030)]119867(2)0 (1198961199030) (23)

4 Shock and Vibration

119877119862 = minus [1198961198690 (119896119903119886) minus 1198691 (119896119903119886) 119903119886 minus 1198941198961198840 (119896119903119886) + (119894119903119886) 1198841 (1198961199030) + (]119903119886) 1198691 (119896119903119886) minus (]119903119886) 1198941198841 (119896119903119886)]119867(2)0 (119896119903119886) minus1

sdot [1198961198690 (119896119903119886) minus 1198691 (119896119903119886) 119903119886 + 1198941198961198840 (119896119903119886) minus (119894119903119886) 1198841 (119896119903119886) + (]119903119886) 1198691 (119896119903119886) + (]119903119886) 1198941198841 (119896119903119886)]119867(1)0 (119896119903119886) (24)

Forward-traveling and negative-traveling waves can beshown clearly in Figure 1 And the expression can be writtenas follows

119887+1 = 119891+1 (119903119886 minus 1199030) 119886+1 119886minus1 = 119891minus1 (119903119886 minus 1199030) 119887minus1

(25)

Considering (17) forward-traveling and negative-traveling waves can be obtained as

119891+1 (119903119886 minus 1199030) = 119867(2)0 (119896119903119886)119867(2)0 (1198961199030)

119891minus1 (119903119886 minus 1199030) = 119867(1)0 (1198961199030)119867(1)0 (119896119903119886) (26)

Similarly the reflection matrices are (20a) and (24) forcase of inner surface fixed and outer surface free

24 Analysis and Discussion After combining propagationand reflection matrices derived above the natural charac-teristics of single circular plate can be analyzed effectivelyHence it gives

119887+1 = 119891+1 (119903119886 minus 1199030) 119886+1 119886+1 = 119877119860119886minus1 119886minus1 = 119891minus1 (119903119886 minus 1199030) 119887minus1 119887minus1 = 119877119862119887+1

(27)

And it yields

119865 (119891) = [119891+1 (119903119886 minus 1199030) 119877119860119891minus1 (119903119886 minus 1199030) 119877119862 minus 1] 119887+1 = 0 (28)

RESIN is selected as the material of circular plate Mate-rial and structural parameters are tabulated in Table 1

Substitute parameters listed in Table 1 into (10) (12) (14)and (28) and the natural frequency can be computedHerein red and black curves obtained by wave approachare real and imaginary wave solution respectively Also it isimportant to note that the intersection points of the curves arenatural frequency After solving the inherent characteristic ofcircular plate structure the characteristic curves calculated byclassicalmethod andwave approach are presented in Figure 2It can be seen that these natural frequencies intersect at thesame point in x-axis which verifies the correctness of thenumerical calculation

3 Radial Vibration Analysis ofPhononic Crystal

31 Model and Transfer Matrix Figure 3 is the schematicdiagram of radial phononic crystal circular plate Herein thewhite region represents RESIN materials and the gray regionrepresents piezoelectric ceramic PZT4 The inner and outerradius are 1199031 and 1199032 respectively The radial span ratios aredenoted with 1198861 and 1198862 And ℎ represents the thickness

Considering an eight-layer circular plate it correspondsto the model 119860119861119860119861119860119861119860119861 The interfaces between RESINand PZT4 material are 119886 119887 119888 119889 119890 119891 119898 119901 119902 (see Figure 3)With regard to RESIN Youngrsquosmodulus density and Poissoncoefficient are 1198641 1205881 ]1 For the PZT4 Youngrsquos modulusdensity and Poisson coefficient are 1198642 1205882 ]2 When radialwave propagates in RESIN layers 1 3 5 and 7 the velocityis 1198881 = [11986411205881(1 minus ]1

2)]12 and the wavenumber is 1198961 =1199081198881 = 21205871198911198881The velocity is 1198882 = [11986421205882(1minus]22)]12 and the

wavenumber is 1198962 = 1199081198882 = 21205871198911198882 when wave propagates inPZT4 layers 2 4 6 and 8

With regard to radial vibration there are two parameterswhich satisfy the continuous conditions at the interface ofdifferent materials Namely they are the radial displacementand radial stress So transfer matrices of radial vibration canbe derived as follows

At the interface b (119903119887 = 1199031 + 1198861) it gives1199061 = 11990621205901 = 1205902 (29)

Thus it can be arranged in a matrices form

1198671 [119860+11119860minus11] = 1198701 [119861+12119861minus12] (30)

At the interface c (119903119888 = 1199031 + 1198861 + 1198862) it gives1199062 = 11990631205902 = 1205903 (31)

Similarly one has

1198672 [119860+21119860minus21] = 1198702 [119861+12119861minus12] (32)

Combining (30) and (32) one has

[119860+21119860minus21] = 11987931 [119860+11119860minus11] (33)

Shock and Vibration 5

times104

minus2

minus1

0

1

2

3

4

5F

(f)

21 4 5 73 6 80

Frequency (Hz)

Classical Bessel solutionClassical Hankel solution

Wave solution (imag)Wave solution (real)

(a) Fixed-fixed boundary

times104

minus3

minus2

minus1

0

1

2

3

F (f

)

2 4 6 80

Frequency (Hz)

Classical Bessel solutionClassical Hankel solution

Wave solution (imag)Wave solution (real)

1 5 73

(b) Free-free boundary

times104

2 4 6 8 10 12 14 160

Frequency (Hz)

Classical Bessel solutionClassical Hankel solution

Wave solution (imag)Wave solution (real)

minus4

minus2

0

2

4

6

F (f

)

(c) Fixed-free boundary

Figure 2 Natural frequency calculated by classic method and wave approach

Table 1 Material and structural parameters

Material parameters 120588 (kgm3) 119864 (Pa) ]RESIN 1180 0435 times 1010 03679Structural parameters 1199030 (m) 119903119886 (m) ℎ (m)RESIN 008 016 0005

where 11987931 = 119867minus12 1198702119870minus11 1198671 is the transfer matrices of radialvibration between the 3rd and the 1st ring

As shown in Figure 3 replace 119903119887 and 119903119888 (in (30) and (32))with 119903119889 and 119903119890 Similarly with 11987931 we can obtain 11987953 which isthe transfer matrix between the fifth ring and the third ring

Similarly replace 119903119889 and 119903119890 with 119903119891 and 119903119898 and the matrix11987975 can be obtainedAt last submitting 119903119901 into (30) we can get 11987981 Therefore

with regard to the radial phononic crystal one has

11987981 = 119870minus18711986787119879751198795311987931 (34)

32 Numerical Result and Discussion The radial phononiccrystal circular plates are composed of twomaterials namelyRESIN and piezoelectric ceramic PZT4 Material parametersand structural parameters are consistent with Table 2

Table 2 Material parameters and structural parameters

Material parameters 120588 (kgm3) 119864 (Pa) ]I (RESIN) 1180 0435 times 1010 03679II (PZT4) 7500 813 times 1010 033Structural parameters 1198861 = 1198862 (m) 1199032 = 21199031 (m) ℎ (m)RESIN and PZT4 001 016 0003

Additionally piezoelectric parameters are set as 11990411986411 =123 times 10minus12m2N 120576119879331205760 = 1300 1205760 = 884 times 10minus12 cm and11988931 = minus123 times 10minus12 cNTaking advantage of transfer matrices and boundary

conditions the transmission characteristics are calculatedusing MATLAB numerically Figure 4 is the transmissioncurve of radial vibration with four periodic structures whichis composited by single material RESIN and RESINPZT4

Figure 4 is the comparison of numerical and FEM resultsIt indicates that there is no attenuation when radial wavepropagates in a single material while for the phononic crys-tal the transmission curve shows a big vibration attenuationband after introducing the periodicity However when radialwave propagates in the single material there is no band gapwhich attenuates weakly and only owns minus4 dB Furthermore

6 Shock and Vibration

r1o a1 a2 a1a2

IIIIII r2

I RESINII PZT4

h

12 3

45

6 78

qpmfedcba

Figure 3 Schematic diagram of radial phononic crystal circular plate

times104

minus40

minus30

minus20

minus10

0

10

20

30

40

50

60

Tran

smiss

ibili

ty (d

B)

1 2 3 4 5 60

Frequency (Hz)

Numerical resultFEM

(a) Single material (RESIN)

Numerical resultFEM

times104

minus100

minus80

minus60

minus40

minus20

0

20

40

60

Tran

smiss

ibili

ty (d

B)

1 2 3 4 5 60

Frequency (Hz)

(b) Phononic crystal (RESINPZT4)

Figure 4 Radial vibration curves of numerical and FEM results

ANSYS 145 is used to verify the radial propagation charac-teristics Using harmonic analysis and keeping the boundaryconditions of outer surface free radial displacement is excitedon the inner surface which is shown in Figure 4(b) Compar-ing numerical results with FEM results it can be found thatthe bands are in good agreement with the aspects of positionwidth and amplitude attenuation and other aspects whichindicates the correctness of theoretical calculations

33 Effects of Structural Parameters In this section the influ-ence of piezoelectric effect on radial wave has been discussedin detail Furthermore the influences of piezoelectric param-eter 11988931 and polar direction on the band have also been stud-ied in case of open and short circuit for piezoelectricmaterial

331 Open and Short Circuit With regard to piezoelectricmaterial the D-type equation of radial vibration can beexpressed as

120576119903 = 11990411986411120590119903 + 11990411986412120590120579 + 11988931119864119885

120576120579 = 11990411986412120590119903 + 11990411986411120590120579 + 11988931119864119885119863119911 = 11988931120590119903 + 11988931120590120579 + 12058511987933119864119885

(35)

It corresponds to short circuit situation when field inten-sity is 119864119885 = 0 Therefore (35) can be simplified as

120590119903 = 11990411986411(11990411986411)2 minus (11990411986412)2 120576119903 minus11990411986412(11990411986411)2 minus (11990411986412)2 120576120579

120590120579 = 11990411986411(11990411986411)2 minus (11990411986412)2 120576120579 minus 11990411986412(11990411986411)2 minus (11990411986412)2 120576119903119863119911 = 11988931 (120590119903 + 120590120579)

(36)

Then it has

]119864 = minus1199041198641211990411986411 1198841198640 = 111990411986411

(37)

Shock and Vibration 7

times104

minus100

minus80

minus60

minus40

minus20

0

20

40

60

Tran

smiss

ibili

ty (d

B)

1 2 3 4 5 6 7 8 9 100

Frequency (Hz)

Open circuitShort circuit

Figure 5 Comparison diagram with short and open circuits

where ]119864 denotes Poissonrsquos coefficient in the condition ofshort circuit 1198841198640 denotes Young modulus

Piezoelectric material is in the open circuit when thecharge is 119863119911 = 0 Thus (35) can be simplified as

120590119903 = 1198841198631 minus (]119863)2 (120576119903 + ]119863120576120579)

120590120579 = 1198841198631 minus (]119863)2 (120576120579 + ]119863120576119903)

(38)

Then it gives

]119863 = minus1205851198793311990411986412 minus 1198892311205851198793311990411986411 minus 119889231 1198841198630 = 111990411986411 minus 11988923112058511987933

(39)

where ]119863 denotes Poissonrsquos coefficient in the condition ofopen circuit 1198841198630 denotes Young modulus

By using transfer matrices derived in Section 21 vibra-tion characteristics curves have been calculated for the caseof short and open circuit As depicted in Figure 5 the selectedfrequency band is 0ndash10 kHz It indicates that the band changesobviously in the high frequency because of the piezoelectriceffect However the short circuit curve moves toward lowfrequency slightly From the above theoretical analysis bycomparing (37) and (39) it can be observed that the elasticitymodulus and Poissonrsquos coefficient of piezoelectric materialsare different when they are in short or open circuit For thecase of open circuit elasticity modulus is larger MeanwhilePoissonrsquos coefficient also changes It is important to note thatthese two parameters can influence the value of 120590119903 and 120590120579directly whichmeans that radial wavenumber changesThusit is the natural mechanism of band moving

332 Piezoelectric Parameters and Inner Radius As to theopen circuit case the effects of electrical parameter 11988931 andinner radius 1199030 on radial wave band are depicted in Figure 6Herein Figure 6(a) describes the effect of piezoelectric

times104

104 6 820 1 3 5 7 9

Frequency (Hz)

minus120

minus100

minus20

minus60

minus40

minus20

0

20

40

60

Tran

smiss

ibili

ty (d

B)

d31 = minus123 lowast 10minus12 cm2

d31 = minus200 lowast 10minus12 cm2

(a) Effect of piezoelectric parameter 11988931

times104Frequency (Hz)

minus100

minus80

minus60

minus40

minus20

0

20

40

60

Tran

smiss

ibili

ty (d

B)

1086421 3 5 7 90

r0 = 008mr0 = 004mr0 = 002m

(b) Effect of inner radius 1199030

Figure 6 Effect of piezoelectric modulus 11988931 and inner radius 1199030

parameter 11988931 However Figure 6(b) displays the effect ofinner radius 1199030

From (39) it can be easily observed that 11988931 determinesthe values of 1198841198630 and ]119863 which can influence the bandposition and attenuation Figure 6(a) indicates that radialband has been expanded with the increase of 11988931 (here thenegative sign only represents direction) However the changeis not obvious Figure 6(b) displays that inner radius 1199030 caninfluence the low frequency greatly It is also clear from thefigure that the band attenuates obviously at low frequencyHowever the effect on high frequency is much smaller

4 Conclusion

This paper presents the classical method and wave propaga-tion approach to obtain the natural frequency Based on radialvibration equation propagation and reflection matrices arederived It can be found that natural frequency of the singlecircular plate can be calculated conveniently by using wavepropagation approach Then the results obtained by usingwave approach are compared with the classical method at theboundary conditions of free-free fixed-fixed and fixed-free

8 Shock and Vibration

It can be seen that the results calculated by these twomethodscoincide with each other

Additionally the band phenomenon of radial phononcrystal circular plate is also analyzed for a deeper understand-ing of radial vibration FEM is used to verify the theoreticalresults It can be found that the inner radius of circular platehas great influence on the low frequency Piezoelectric coeffi-cient open circuit and short circuitmake the elasticmodulusand Poissonrsquos ratio change Then the radial wavenumber alsochanges which is the natural mechanism of band removingfor the case of open and short circuits

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work is financially supported by the National NaturalScience Foundation of China (no 51375105) and the Natu-ral Science Foundation of Heilongjiang Province of China(E201418)

References

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

[2] C Mei and B R Mace ldquoWave reflection and transmission inTimoshenko beams and wave analysis of Timoshenko beamstructuresrdquo Journal of Sound andAcoustics vol 127 pp 382ndash3942004

[3] N R Harland B R Mace and RW Jones ldquoWave propagationreflection and transmission in tunable fluid-filled beamsrdquo Jour-nal of Sound and Vibration vol 241 no 5 pp 735ndash754 2001

[4] 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

[5] 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

[6] S K Lee Wave reflection transmission and propagation instructural waveguides [PhD thesis] Southampton University2006

[7] CHWang and L R F Rose ldquoWave reflection and transmissionin beams containing delamination and inhomogeneityrdquo Journalof Sound and Vibration vol 264 no 4 pp 851ndash872 2003

[8] C A Tan and B Kang ldquoWave reflection and transmission in anaxially strained rotating Timoshenko shaftrdquo Journal of Soundand Vibration vol 213 no 3 pp 483ndash510 1998

[9] X B Li ldquoStudy on free vibration analysis of circular cylindricalshells using wave propagationrdquo Journal of Sound and Vibrationvol 311 no 3ndash5 pp 667ndash682 2008

[10] L Gan X B Li and Z Zhang ldquoFree vibration analysis of ring-stiffened cylindrical shells using wave propagation approachrdquoJournal of Sound and Vibration vol 326 no 3ndash5 pp 633ndash6462009

[11] X M Zhang ldquoVibration analysis of cross-ply laminated com-posite cylindrical shells using the wave propagation approachrdquoApplied Acoustics vol 62 no 11 pp 1221ndash1228 2001

[12] X M Zhang ldquoFrequency analysis of submerged cylindricalshells with the wave propagation approachrdquo International Jour-nal of Mechanical Sciences vol 44 no 7 pp 1259ndash1273 2002

[13] M Shakeri M Akhlaghi and S M Hoseini ldquoVibration andradial wave propagation velocity in functionally graded thickhollow cylinderrdquo Composite Structures vol 76 no 1-2 pp 174ndash181 2006

[14] D Torrent and J Sanchez-Dehesa ldquoAcoustic resonances in two-dimensional radial sonic crystal shellsrdquo New Journal of Physicsvol 12 Article ID 073034 2010

[15] J Carbonell D Torrent and J Sanchez-Dehesa ldquoRadial pho-tonic crystal shells and their application as resonant and radiat-ing elementsrdquo IEEE Transactions on Antennas and Propagationvol 61 no 2 pp 755ndash767 2013

[16] Z L Xu F G Wu and Z N Guo ldquoLow frequency phononicband structures in two-dimensional arc-shaped phononic crys-talsrdquo Physics Letters A vol 376 no 33 pp 2256ndash2263 2012

[17] T Ma T N Chen X P Wang Y Li and P Wang ldquoBandstructures of bilayer radial phononic crystal plate with crystalglidingrdquo Journal of Applied Physics vol 116 no 10 Article ID104505 pp 1ndash7 2014

[18] H S Shu L Q Dong S D Li et al ldquoPropagation of torsionalwaves in a thin circular plate of generalized phononic crystalsrdquoJournal of Physics D Applied Physics vol 47 no 29 Article ID295501 2014

[19] H S Shu W Liu S D Li et al ldquoResearch on flexural waveband gap of a thin circular plate of piezoelectric radial phononiccrystalsrdquo Journal of Vibration and Control vol 22 no 7 pp1777ndash1789 2016

[20] K Kiani ldquoElasto-dynamic analysis of spinning nanodisks viaa surface energy-based modelrdquo Journal of Physics D AppliedPhysics vol 49 no 27 Article ID 275306 2016

[21] K Kiani ldquoThermo-mechanical analysis of functionally gradedplate-like nanorotors a surface elasticity modelrdquo InternationalJournal of Mechanical Sciences vol 106 pp 39ndash49 2016

[22] K Kiani ldquoMagneto-elasto-dynamic analysis of an elasticallyconfined conducting nanowire due to an axial magnetic shockrdquoPhysics Letters A vol 376 no 20 pp 1679ndash1685 2012

[23] S Y Lin ldquoStudy on the step-type circular ring ultrasonicconcentrator in radial vibrationrdquo Ultrasonics vol 49 no 2 pp206ndash211 2009

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

4 Shock and Vibration

119877119862 = minus [1198961198690 (119896119903119886) minus 1198691 (119896119903119886) 119903119886 minus 1198941198961198840 (119896119903119886) + (119894119903119886) 1198841 (1198961199030) + (]119903119886) 1198691 (119896119903119886) minus (]119903119886) 1198941198841 (119896119903119886)]119867(2)0 (119896119903119886) minus1

sdot [1198961198690 (119896119903119886) minus 1198691 (119896119903119886) 119903119886 + 1198941198961198840 (119896119903119886) minus (119894119903119886) 1198841 (119896119903119886) + (]119903119886) 1198691 (119896119903119886) + (]119903119886) 1198941198841 (119896119903119886)]119867(1)0 (119896119903119886) (24)

Forward-traveling and negative-traveling waves can beshown clearly in Figure 1 And the expression can be writtenas follows

119887+1 = 119891+1 (119903119886 minus 1199030) 119886+1 119886minus1 = 119891minus1 (119903119886 minus 1199030) 119887minus1

(25)

Considering (17) forward-traveling and negative-traveling waves can be obtained as

119891+1 (119903119886 minus 1199030) = 119867(2)0 (119896119903119886)119867(2)0 (1198961199030)

119891minus1 (119903119886 minus 1199030) = 119867(1)0 (1198961199030)119867(1)0 (119896119903119886) (26)

Similarly the reflection matrices are (20a) and (24) forcase of inner surface fixed and outer surface free

24 Analysis and Discussion After combining propagationand reflection matrices derived above the natural charac-teristics of single circular plate can be analyzed effectivelyHence it gives

119887+1 = 119891+1 (119903119886 minus 1199030) 119886+1 119886+1 = 119877119860119886minus1 119886minus1 = 119891minus1 (119903119886 minus 1199030) 119887minus1 119887minus1 = 119877119862119887+1

(27)

And it yields

119865 (119891) = [119891+1 (119903119886 minus 1199030) 119877119860119891minus1 (119903119886 minus 1199030) 119877119862 minus 1] 119887+1 = 0 (28)

RESIN is selected as the material of circular plate Mate-rial and structural parameters are tabulated in Table 1

Substitute parameters listed in Table 1 into (10) (12) (14)and (28) and the natural frequency can be computedHerein red and black curves obtained by wave approachare real and imaginary wave solution respectively Also it isimportant to note that the intersection points of the curves arenatural frequency After solving the inherent characteristic ofcircular plate structure the characteristic curves calculated byclassicalmethod andwave approach are presented in Figure 2It can be seen that these natural frequencies intersect at thesame point in x-axis which verifies the correctness of thenumerical calculation

3 Radial Vibration Analysis ofPhononic Crystal

31 Model and Transfer Matrix Figure 3 is the schematicdiagram of radial phononic crystal circular plate Herein thewhite region represents RESIN materials and the gray regionrepresents piezoelectric ceramic PZT4 The inner and outerradius are 1199031 and 1199032 respectively The radial span ratios aredenoted with 1198861 and 1198862 And ℎ represents the thickness

Considering an eight-layer circular plate it correspondsto the model 119860119861119860119861119860119861119860119861 The interfaces between RESINand PZT4 material are 119886 119887 119888 119889 119890 119891 119898 119901 119902 (see Figure 3)With regard to RESIN Youngrsquosmodulus density and Poissoncoefficient are 1198641 1205881 ]1 For the PZT4 Youngrsquos modulusdensity and Poisson coefficient are 1198642 1205882 ]2 When radialwave propagates in RESIN layers 1 3 5 and 7 the velocityis 1198881 = [11986411205881(1 minus ]1

2)]12 and the wavenumber is 1198961 =1199081198881 = 21205871198911198881The velocity is 1198882 = [11986421205882(1minus]22)]12 and the

wavenumber is 1198962 = 1199081198882 = 21205871198911198882 when wave propagates inPZT4 layers 2 4 6 and 8

With regard to radial vibration there are two parameterswhich satisfy the continuous conditions at the interface ofdifferent materials Namely they are the radial displacementand radial stress So transfer matrices of radial vibration canbe derived as follows

At the interface b (119903119887 = 1199031 + 1198861) it gives1199061 = 11990621205901 = 1205902 (29)

Thus it can be arranged in a matrices form

1198671 [119860+11119860minus11] = 1198701 [119861+12119861minus12] (30)

At the interface c (119903119888 = 1199031 + 1198861 + 1198862) it gives1199062 = 11990631205902 = 1205903 (31)

Similarly one has

1198672 [119860+21119860minus21] = 1198702 [119861+12119861minus12] (32)

Combining (30) and (32) one has

[119860+21119860minus21] = 11987931 [119860+11119860minus11] (33)

Shock and Vibration 5

times104

minus2

minus1

0

1

2

3

4

5F

(f)

21 4 5 73 6 80

Frequency (Hz)

Classical Bessel solutionClassical Hankel solution

Wave solution (imag)Wave solution (real)

(a) Fixed-fixed boundary

times104

minus3

minus2

minus1

0

1

2

3

F (f

)

2 4 6 80

Frequency (Hz)

Classical Bessel solutionClassical Hankel solution

Wave solution (imag)Wave solution (real)

1 5 73

(b) Free-free boundary

times104

2 4 6 8 10 12 14 160

Frequency (Hz)

Classical Bessel solutionClassical Hankel solution

Wave solution (imag)Wave solution (real)

minus4

minus2

0

2

4

6

F (f

)

(c) Fixed-free boundary

Figure 2 Natural frequency calculated by classic method and wave approach

Table 1 Material and structural parameters

Material parameters 120588 (kgm3) 119864 (Pa) ]RESIN 1180 0435 times 1010 03679Structural parameters 1199030 (m) 119903119886 (m) ℎ (m)RESIN 008 016 0005

where 11987931 = 119867minus12 1198702119870minus11 1198671 is the transfer matrices of radialvibration between the 3rd and the 1st ring

As shown in Figure 3 replace 119903119887 and 119903119888 (in (30) and (32))with 119903119889 and 119903119890 Similarly with 11987931 we can obtain 11987953 which isthe transfer matrix between the fifth ring and the third ring

Similarly replace 119903119889 and 119903119890 with 119903119891 and 119903119898 and the matrix11987975 can be obtainedAt last submitting 119903119901 into (30) we can get 11987981 Therefore

with regard to the radial phononic crystal one has

11987981 = 119870minus18711986787119879751198795311987931 (34)

32 Numerical Result and Discussion The radial phononiccrystal circular plates are composed of twomaterials namelyRESIN and piezoelectric ceramic PZT4 Material parametersand structural parameters are consistent with Table 2

Table 2 Material parameters and structural parameters

Material parameters 120588 (kgm3) 119864 (Pa) ]I (RESIN) 1180 0435 times 1010 03679II (PZT4) 7500 813 times 1010 033Structural parameters 1198861 = 1198862 (m) 1199032 = 21199031 (m) ℎ (m)RESIN and PZT4 001 016 0003

Additionally piezoelectric parameters are set as 11990411986411 =123 times 10minus12m2N 120576119879331205760 = 1300 1205760 = 884 times 10minus12 cm and11988931 = minus123 times 10minus12 cNTaking advantage of transfer matrices and boundary

conditions the transmission characteristics are calculatedusing MATLAB numerically Figure 4 is the transmissioncurve of radial vibration with four periodic structures whichis composited by single material RESIN and RESINPZT4

Figure 4 is the comparison of numerical and FEM resultsIt indicates that there is no attenuation when radial wavepropagates in a single material while for the phononic crys-tal the transmission curve shows a big vibration attenuationband after introducing the periodicity However when radialwave propagates in the single material there is no band gapwhich attenuates weakly and only owns minus4 dB Furthermore

6 Shock and Vibration

r1o a1 a2 a1a2

IIIIII r2

I RESINII PZT4

h

12 3

45

6 78

qpmfedcba

Figure 3 Schematic diagram of radial phononic crystal circular plate

times104

minus40

minus30

minus20

minus10

0

10

20

30

40

50

60

Tran

smiss

ibili

ty (d

B)

1 2 3 4 5 60

Frequency (Hz)

Numerical resultFEM

(a) Single material (RESIN)

Numerical resultFEM

times104

minus100

minus80

minus60

minus40

minus20

0

20

40

60

Tran

smiss

ibili

ty (d

B)

1 2 3 4 5 60

Frequency (Hz)

(b) Phononic crystal (RESINPZT4)

Figure 4 Radial vibration curves of numerical and FEM results

ANSYS 145 is used to verify the radial propagation charac-teristics Using harmonic analysis and keeping the boundaryconditions of outer surface free radial displacement is excitedon the inner surface which is shown in Figure 4(b) Compar-ing numerical results with FEM results it can be found thatthe bands are in good agreement with the aspects of positionwidth and amplitude attenuation and other aspects whichindicates the correctness of theoretical calculations

33 Effects of Structural Parameters In this section the influ-ence of piezoelectric effect on radial wave has been discussedin detail Furthermore the influences of piezoelectric param-eter 11988931 and polar direction on the band have also been stud-ied in case of open and short circuit for piezoelectricmaterial

331 Open and Short Circuit With regard to piezoelectricmaterial the D-type equation of radial vibration can beexpressed as

120576119903 = 11990411986411120590119903 + 11990411986412120590120579 + 11988931119864119885

120576120579 = 11990411986412120590119903 + 11990411986411120590120579 + 11988931119864119885119863119911 = 11988931120590119903 + 11988931120590120579 + 12058511987933119864119885

(35)

It corresponds to short circuit situation when field inten-sity is 119864119885 = 0 Therefore (35) can be simplified as

120590119903 = 11990411986411(11990411986411)2 minus (11990411986412)2 120576119903 minus11990411986412(11990411986411)2 minus (11990411986412)2 120576120579

120590120579 = 11990411986411(11990411986411)2 minus (11990411986412)2 120576120579 minus 11990411986412(11990411986411)2 minus (11990411986412)2 120576119903119863119911 = 11988931 (120590119903 + 120590120579)

(36)

Then it has

]119864 = minus1199041198641211990411986411 1198841198640 = 111990411986411

(37)

Shock and Vibration 7

times104

minus100

minus80

minus60

minus40

minus20

0

20

40

60

Tran

smiss

ibili

ty (d

B)

1 2 3 4 5 6 7 8 9 100

Frequency (Hz)

Open circuitShort circuit

Figure 5 Comparison diagram with short and open circuits

where ]119864 denotes Poissonrsquos coefficient in the condition ofshort circuit 1198841198640 denotes Young modulus

Piezoelectric material is in the open circuit when thecharge is 119863119911 = 0 Thus (35) can be simplified as

120590119903 = 1198841198631 minus (]119863)2 (120576119903 + ]119863120576120579)

120590120579 = 1198841198631 minus (]119863)2 (120576120579 + ]119863120576119903)

(38)

Then it gives

]119863 = minus1205851198793311990411986412 minus 1198892311205851198793311990411986411 minus 119889231 1198841198630 = 111990411986411 minus 11988923112058511987933

(39)

where ]119863 denotes Poissonrsquos coefficient in the condition ofopen circuit 1198841198630 denotes Young modulus

By using transfer matrices derived in Section 21 vibra-tion characteristics curves have been calculated for the caseof short and open circuit As depicted in Figure 5 the selectedfrequency band is 0ndash10 kHz It indicates that the band changesobviously in the high frequency because of the piezoelectriceffect However the short circuit curve moves toward lowfrequency slightly From the above theoretical analysis bycomparing (37) and (39) it can be observed that the elasticitymodulus and Poissonrsquos coefficient of piezoelectric materialsare different when they are in short or open circuit For thecase of open circuit elasticity modulus is larger MeanwhilePoissonrsquos coefficient also changes It is important to note thatthese two parameters can influence the value of 120590119903 and 120590120579directly whichmeans that radial wavenumber changesThusit is the natural mechanism of band moving

332 Piezoelectric Parameters and Inner Radius As to theopen circuit case the effects of electrical parameter 11988931 andinner radius 1199030 on radial wave band are depicted in Figure 6Herein Figure 6(a) describes the effect of piezoelectric

times104

104 6 820 1 3 5 7 9

Frequency (Hz)

minus120

minus100

minus20

minus60

minus40

minus20

0

20

40

60

Tran

smiss

ibili

ty (d

B)

d31 = minus123 lowast 10minus12 cm2

d31 = minus200 lowast 10minus12 cm2

(a) Effect of piezoelectric parameter 11988931

times104Frequency (Hz)

minus100

minus80

minus60

minus40

minus20

0

20

40

60

Tran

smiss

ibili

ty (d

B)

1086421 3 5 7 90

r0 = 008mr0 = 004mr0 = 002m

(b) Effect of inner radius 1199030

Figure 6 Effect of piezoelectric modulus 11988931 and inner radius 1199030

parameter 11988931 However Figure 6(b) displays the effect ofinner radius 1199030

From (39) it can be easily observed that 11988931 determinesthe values of 1198841198630 and ]119863 which can influence the bandposition and attenuation Figure 6(a) indicates that radialband has been expanded with the increase of 11988931 (here thenegative sign only represents direction) However the changeis not obvious Figure 6(b) displays that inner radius 1199030 caninfluence the low frequency greatly It is also clear from thefigure that the band attenuates obviously at low frequencyHowever the effect on high frequency is much smaller

4 Conclusion

This paper presents the classical method and wave propaga-tion approach to obtain the natural frequency Based on radialvibration equation propagation and reflection matrices arederived It can be found that natural frequency of the singlecircular plate can be calculated conveniently by using wavepropagation approach Then the results obtained by usingwave approach are compared with the classical method at theboundary conditions of free-free fixed-fixed and fixed-free

8 Shock and Vibration

It can be seen that the results calculated by these twomethodscoincide with each other

Additionally the band phenomenon of radial phononcrystal circular plate is also analyzed for a deeper understand-ing of radial vibration FEM is used to verify the theoreticalresults It can be found that the inner radius of circular platehas great influence on the low frequency Piezoelectric coeffi-cient open circuit and short circuitmake the elasticmodulusand Poissonrsquos ratio change Then the radial wavenumber alsochanges which is the natural mechanism of band removingfor the case of open and short circuits

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work is financially supported by the National NaturalScience Foundation of China (no 51375105) and the Natu-ral Science Foundation of Heilongjiang Province of China(E201418)

References

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

[2] C Mei and B R Mace ldquoWave reflection and transmission inTimoshenko beams and wave analysis of Timoshenko beamstructuresrdquo Journal of Sound andAcoustics vol 127 pp 382ndash3942004

[3] N R Harland B R Mace and RW Jones ldquoWave propagationreflection and transmission in tunable fluid-filled beamsrdquo Jour-nal of Sound and Vibration vol 241 no 5 pp 735ndash754 2001

[4] 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

[5] 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

[6] S K Lee Wave reflection transmission and propagation instructural waveguides [PhD thesis] Southampton University2006

[7] CHWang and L R F Rose ldquoWave reflection and transmissionin beams containing delamination and inhomogeneityrdquo Journalof Sound and Vibration vol 264 no 4 pp 851ndash872 2003

[8] C A Tan and B Kang ldquoWave reflection and transmission in anaxially strained rotating Timoshenko shaftrdquo Journal of Soundand Vibration vol 213 no 3 pp 483ndash510 1998

[9] X B Li ldquoStudy on free vibration analysis of circular cylindricalshells using wave propagationrdquo Journal of Sound and Vibrationvol 311 no 3ndash5 pp 667ndash682 2008

[10] L Gan X B Li and Z Zhang ldquoFree vibration analysis of ring-stiffened cylindrical shells using wave propagation approachrdquoJournal of Sound and Vibration vol 326 no 3ndash5 pp 633ndash6462009

[11] X M Zhang ldquoVibration analysis of cross-ply laminated com-posite cylindrical shells using the wave propagation approachrdquoApplied Acoustics vol 62 no 11 pp 1221ndash1228 2001

[12] X M Zhang ldquoFrequency analysis of submerged cylindricalshells with the wave propagation approachrdquo International Jour-nal of Mechanical Sciences vol 44 no 7 pp 1259ndash1273 2002

[13] M Shakeri M Akhlaghi and S M Hoseini ldquoVibration andradial wave propagation velocity in functionally graded thickhollow cylinderrdquo Composite Structures vol 76 no 1-2 pp 174ndash181 2006

[14] D Torrent and J Sanchez-Dehesa ldquoAcoustic resonances in two-dimensional radial sonic crystal shellsrdquo New Journal of Physicsvol 12 Article ID 073034 2010

[15] J Carbonell D Torrent and J Sanchez-Dehesa ldquoRadial pho-tonic crystal shells and their application as resonant and radiat-ing elementsrdquo IEEE Transactions on Antennas and Propagationvol 61 no 2 pp 755ndash767 2013

[16] Z L Xu F G Wu and Z N Guo ldquoLow frequency phononicband structures in two-dimensional arc-shaped phononic crys-talsrdquo Physics Letters A vol 376 no 33 pp 2256ndash2263 2012

[17] T Ma T N Chen X P Wang Y Li and P Wang ldquoBandstructures of bilayer radial phononic crystal plate with crystalglidingrdquo Journal of Applied Physics vol 116 no 10 Article ID104505 pp 1ndash7 2014

[18] H S Shu L Q Dong S D Li et al ldquoPropagation of torsionalwaves in a thin circular plate of generalized phononic crystalsrdquoJournal of Physics D Applied Physics vol 47 no 29 Article ID295501 2014

[19] H S Shu W Liu S D Li et al ldquoResearch on flexural waveband gap of a thin circular plate of piezoelectric radial phononiccrystalsrdquo Journal of Vibration and Control vol 22 no 7 pp1777ndash1789 2016

[20] K Kiani ldquoElasto-dynamic analysis of spinning nanodisks viaa surface energy-based modelrdquo Journal of Physics D AppliedPhysics vol 49 no 27 Article ID 275306 2016

[21] K Kiani ldquoThermo-mechanical analysis of functionally gradedplate-like nanorotors a surface elasticity modelrdquo InternationalJournal of Mechanical Sciences vol 106 pp 39ndash49 2016

[22] K Kiani ldquoMagneto-elasto-dynamic analysis of an elasticallyconfined conducting nanowire due to an axial magnetic shockrdquoPhysics Letters A vol 376 no 20 pp 1679ndash1685 2012

[23] S Y Lin ldquoStudy on the step-type circular ring ultrasonicconcentrator in radial vibrationrdquo Ultrasonics vol 49 no 2 pp206ndash211 2009

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 5

times104

minus2

minus1

0

1

2

3

4

5F

(f)

21 4 5 73 6 80

Frequency (Hz)

Classical Bessel solutionClassical Hankel solution

Wave solution (imag)Wave solution (real)

(a) Fixed-fixed boundary

times104

minus3

minus2

minus1

0

1

2

3

F (f

)

2 4 6 80

Frequency (Hz)

Classical Bessel solutionClassical Hankel solution

Wave solution (imag)Wave solution (real)

1 5 73

(b) Free-free boundary

times104

2 4 6 8 10 12 14 160

Frequency (Hz)

Classical Bessel solutionClassical Hankel solution

Wave solution (imag)Wave solution (real)

minus4

minus2

0

2

4

6

F (f

)

(c) Fixed-free boundary

Figure 2 Natural frequency calculated by classic method and wave approach

Table 1 Material and structural parameters

Material parameters 120588 (kgm3) 119864 (Pa) ]RESIN 1180 0435 times 1010 03679Structural parameters 1199030 (m) 119903119886 (m) ℎ (m)RESIN 008 016 0005

where 11987931 = 119867minus12 1198702119870minus11 1198671 is the transfer matrices of radialvibration between the 3rd and the 1st ring

As shown in Figure 3 replace 119903119887 and 119903119888 (in (30) and (32))with 119903119889 and 119903119890 Similarly with 11987931 we can obtain 11987953 which isthe transfer matrix between the fifth ring and the third ring

Similarly replace 119903119889 and 119903119890 with 119903119891 and 119903119898 and the matrix11987975 can be obtainedAt last submitting 119903119901 into (30) we can get 11987981 Therefore

with regard to the radial phononic crystal one has

11987981 = 119870minus18711986787119879751198795311987931 (34)

32 Numerical Result and Discussion The radial phononiccrystal circular plates are composed of twomaterials namelyRESIN and piezoelectric ceramic PZT4 Material parametersand structural parameters are consistent with Table 2

Table 2 Material parameters and structural parameters

Material parameters 120588 (kgm3) 119864 (Pa) ]I (RESIN) 1180 0435 times 1010 03679II (PZT4) 7500 813 times 1010 033Structural parameters 1198861 = 1198862 (m) 1199032 = 21199031 (m) ℎ (m)RESIN and PZT4 001 016 0003

Additionally piezoelectric parameters are set as 11990411986411 =123 times 10minus12m2N 120576119879331205760 = 1300 1205760 = 884 times 10minus12 cm and11988931 = minus123 times 10minus12 cNTaking advantage of transfer matrices and boundary

conditions the transmission characteristics are calculatedusing MATLAB numerically Figure 4 is the transmissioncurve of radial vibration with four periodic structures whichis composited by single material RESIN and RESINPZT4

Figure 4 is the comparison of numerical and FEM resultsIt indicates that there is no attenuation when radial wavepropagates in a single material while for the phononic crys-tal the transmission curve shows a big vibration attenuationband after introducing the periodicity However when radialwave propagates in the single material there is no band gapwhich attenuates weakly and only owns minus4 dB Furthermore

6 Shock and Vibration

r1o a1 a2 a1a2

IIIIII r2

I RESINII PZT4

h

12 3

45

6 78

qpmfedcba

Figure 3 Schematic diagram of radial phononic crystal circular plate

times104

minus40

minus30

minus20

minus10

0

10

20

30

40

50

60

Tran

smiss

ibili

ty (d

B)

1 2 3 4 5 60

Frequency (Hz)

Numerical resultFEM

(a) Single material (RESIN)

Numerical resultFEM

times104

minus100

minus80

minus60

minus40

minus20

0

20

40

60

Tran

smiss

ibili

ty (d

B)

1 2 3 4 5 60

Frequency (Hz)

(b) Phononic crystal (RESINPZT4)

Figure 4 Radial vibration curves of numerical and FEM results

ANSYS 145 is used to verify the radial propagation charac-teristics Using harmonic analysis and keeping the boundaryconditions of outer surface free radial displacement is excitedon the inner surface which is shown in Figure 4(b) Compar-ing numerical results with FEM results it can be found thatthe bands are in good agreement with the aspects of positionwidth and amplitude attenuation and other aspects whichindicates the correctness of theoretical calculations

33 Effects of Structural Parameters In this section the influ-ence of piezoelectric effect on radial wave has been discussedin detail Furthermore the influences of piezoelectric param-eter 11988931 and polar direction on the band have also been stud-ied in case of open and short circuit for piezoelectricmaterial

331 Open and Short Circuit With regard to piezoelectricmaterial the D-type equation of radial vibration can beexpressed as

120576119903 = 11990411986411120590119903 + 11990411986412120590120579 + 11988931119864119885

120576120579 = 11990411986412120590119903 + 11990411986411120590120579 + 11988931119864119885119863119911 = 11988931120590119903 + 11988931120590120579 + 12058511987933119864119885

(35)

It corresponds to short circuit situation when field inten-sity is 119864119885 = 0 Therefore (35) can be simplified as

120590119903 = 11990411986411(11990411986411)2 minus (11990411986412)2 120576119903 minus11990411986412(11990411986411)2 minus (11990411986412)2 120576120579

120590120579 = 11990411986411(11990411986411)2 minus (11990411986412)2 120576120579 minus 11990411986412(11990411986411)2 minus (11990411986412)2 120576119903119863119911 = 11988931 (120590119903 + 120590120579)

(36)

Then it has

]119864 = minus1199041198641211990411986411 1198841198640 = 111990411986411

(37)

Shock and Vibration 7

times104

minus100

minus80

minus60

minus40

minus20

0

20

40

60

Tran

smiss

ibili

ty (d

B)

1 2 3 4 5 6 7 8 9 100

Frequency (Hz)

Open circuitShort circuit

Figure 5 Comparison diagram with short and open circuits

where ]119864 denotes Poissonrsquos coefficient in the condition ofshort circuit 1198841198640 denotes Young modulus

Piezoelectric material is in the open circuit when thecharge is 119863119911 = 0 Thus (35) can be simplified as

120590119903 = 1198841198631 minus (]119863)2 (120576119903 + ]119863120576120579)

120590120579 = 1198841198631 minus (]119863)2 (120576120579 + ]119863120576119903)

(38)

Then it gives

]119863 = minus1205851198793311990411986412 minus 1198892311205851198793311990411986411 minus 119889231 1198841198630 = 111990411986411 minus 11988923112058511987933

(39)

where ]119863 denotes Poissonrsquos coefficient in the condition ofopen circuit 1198841198630 denotes Young modulus

By using transfer matrices derived in Section 21 vibra-tion characteristics curves have been calculated for the caseof short and open circuit As depicted in Figure 5 the selectedfrequency band is 0ndash10 kHz It indicates that the band changesobviously in the high frequency because of the piezoelectriceffect However the short circuit curve moves toward lowfrequency slightly From the above theoretical analysis bycomparing (37) and (39) it can be observed that the elasticitymodulus and Poissonrsquos coefficient of piezoelectric materialsare different when they are in short or open circuit For thecase of open circuit elasticity modulus is larger MeanwhilePoissonrsquos coefficient also changes It is important to note thatthese two parameters can influence the value of 120590119903 and 120590120579directly whichmeans that radial wavenumber changesThusit is the natural mechanism of band moving

332 Piezoelectric Parameters and Inner Radius As to theopen circuit case the effects of electrical parameter 11988931 andinner radius 1199030 on radial wave band are depicted in Figure 6Herein Figure 6(a) describes the effect of piezoelectric

times104

104 6 820 1 3 5 7 9

Frequency (Hz)

minus120

minus100

minus20

minus60

minus40

minus20

0

20

40

60

Tran

smiss

ibili

ty (d

B)

d31 = minus123 lowast 10minus12 cm2

d31 = minus200 lowast 10minus12 cm2

(a) Effect of piezoelectric parameter 11988931

times104Frequency (Hz)

minus100

minus80

minus60

minus40

minus20

0

20

40

60

Tran

smiss

ibili

ty (d

B)

1086421 3 5 7 90

r0 = 008mr0 = 004mr0 = 002m

(b) Effect of inner radius 1199030

Figure 6 Effect of piezoelectric modulus 11988931 and inner radius 1199030

parameter 11988931 However Figure 6(b) displays the effect ofinner radius 1199030

From (39) it can be easily observed that 11988931 determinesthe values of 1198841198630 and ]119863 which can influence the bandposition and attenuation Figure 6(a) indicates that radialband has been expanded with the increase of 11988931 (here thenegative sign only represents direction) However the changeis not obvious Figure 6(b) displays that inner radius 1199030 caninfluence the low frequency greatly It is also clear from thefigure that the band attenuates obviously at low frequencyHowever the effect on high frequency is much smaller

4 Conclusion

This paper presents the classical method and wave propaga-tion approach to obtain the natural frequency Based on radialvibration equation propagation and reflection matrices arederived It can be found that natural frequency of the singlecircular plate can be calculated conveniently by using wavepropagation approach Then the results obtained by usingwave approach are compared with the classical method at theboundary conditions of free-free fixed-fixed and fixed-free

8 Shock and Vibration

It can be seen that the results calculated by these twomethodscoincide with each other

Additionally the band phenomenon of radial phononcrystal circular plate is also analyzed for a deeper understand-ing of radial vibration FEM is used to verify the theoreticalresults It can be found that the inner radius of circular platehas great influence on the low frequency Piezoelectric coeffi-cient open circuit and short circuitmake the elasticmodulusand Poissonrsquos ratio change Then the radial wavenumber alsochanges which is the natural mechanism of band removingfor the case of open and short circuits

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work is financially supported by the National NaturalScience Foundation of China (no 51375105) and the Natu-ral Science Foundation of Heilongjiang Province of China(E201418)

References

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

[2] C Mei and B R Mace ldquoWave reflection and transmission inTimoshenko beams and wave analysis of Timoshenko beamstructuresrdquo Journal of Sound andAcoustics vol 127 pp 382ndash3942004

[3] N R Harland B R Mace and RW Jones ldquoWave propagationreflection and transmission in tunable fluid-filled beamsrdquo Jour-nal of Sound and Vibration vol 241 no 5 pp 735ndash754 2001

[4] 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

[5] 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

[6] S K Lee Wave reflection transmission and propagation instructural waveguides [PhD thesis] Southampton University2006

[7] CHWang and L R F Rose ldquoWave reflection and transmissionin beams containing delamination and inhomogeneityrdquo Journalof Sound and Vibration vol 264 no 4 pp 851ndash872 2003

[8] C A Tan and B Kang ldquoWave reflection and transmission in anaxially strained rotating Timoshenko shaftrdquo Journal of Soundand Vibration vol 213 no 3 pp 483ndash510 1998

[9] X B Li ldquoStudy on free vibration analysis of circular cylindricalshells using wave propagationrdquo Journal of Sound and Vibrationvol 311 no 3ndash5 pp 667ndash682 2008

[10] L Gan X B Li and Z Zhang ldquoFree vibration analysis of ring-stiffened cylindrical shells using wave propagation approachrdquoJournal of Sound and Vibration vol 326 no 3ndash5 pp 633ndash6462009

[11] X M Zhang ldquoVibration analysis of cross-ply laminated com-posite cylindrical shells using the wave propagation approachrdquoApplied Acoustics vol 62 no 11 pp 1221ndash1228 2001

[12] X M Zhang ldquoFrequency analysis of submerged cylindricalshells with the wave propagation approachrdquo International Jour-nal of Mechanical Sciences vol 44 no 7 pp 1259ndash1273 2002

[13] M Shakeri M Akhlaghi and S M Hoseini ldquoVibration andradial wave propagation velocity in functionally graded thickhollow cylinderrdquo Composite Structures vol 76 no 1-2 pp 174ndash181 2006

[14] D Torrent and J Sanchez-Dehesa ldquoAcoustic resonances in two-dimensional radial sonic crystal shellsrdquo New Journal of Physicsvol 12 Article ID 073034 2010

[15] J Carbonell D Torrent and J Sanchez-Dehesa ldquoRadial pho-tonic crystal shells and their application as resonant and radiat-ing elementsrdquo IEEE Transactions on Antennas and Propagationvol 61 no 2 pp 755ndash767 2013

[16] Z L Xu F G Wu and Z N Guo ldquoLow frequency phononicband structures in two-dimensional arc-shaped phononic crys-talsrdquo Physics Letters A vol 376 no 33 pp 2256ndash2263 2012

[17] T Ma T N Chen X P Wang Y Li and P Wang ldquoBandstructures of bilayer radial phononic crystal plate with crystalglidingrdquo Journal of Applied Physics vol 116 no 10 Article ID104505 pp 1ndash7 2014

[18] H S Shu L Q Dong S D Li et al ldquoPropagation of torsionalwaves in a thin circular plate of generalized phononic crystalsrdquoJournal of Physics D Applied Physics vol 47 no 29 Article ID295501 2014

[19] H S Shu W Liu S D Li et al ldquoResearch on flexural waveband gap of a thin circular plate of piezoelectric radial phononiccrystalsrdquo Journal of Vibration and Control vol 22 no 7 pp1777ndash1789 2016

[20] K Kiani ldquoElasto-dynamic analysis of spinning nanodisks viaa surface energy-based modelrdquo Journal of Physics D AppliedPhysics vol 49 no 27 Article ID 275306 2016

[21] K Kiani ldquoThermo-mechanical analysis of functionally gradedplate-like nanorotors a surface elasticity modelrdquo InternationalJournal of Mechanical Sciences vol 106 pp 39ndash49 2016

[22] K Kiani ldquoMagneto-elasto-dynamic analysis of an elasticallyconfined conducting nanowire due to an axial magnetic shockrdquoPhysics Letters A vol 376 no 20 pp 1679ndash1685 2012

[23] S Y Lin ldquoStudy on the step-type circular ring ultrasonicconcentrator in radial vibrationrdquo Ultrasonics vol 49 no 2 pp206ndash211 2009

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

6 Shock and Vibration

r1o a1 a2 a1a2

IIIIII r2

I RESINII PZT4

h

12 3

45

6 78

qpmfedcba

Figure 3 Schematic diagram of radial phononic crystal circular plate

times104

minus40

minus30

minus20

minus10

0

10

20

30

40

50

60

Tran

smiss

ibili

ty (d

B)

1 2 3 4 5 60

Frequency (Hz)

Numerical resultFEM

(a) Single material (RESIN)

Numerical resultFEM

times104

minus100

minus80

minus60

minus40

minus20

0

20

40

60

Tran

smiss

ibili

ty (d

B)

1 2 3 4 5 60

Frequency (Hz)

(b) Phononic crystal (RESINPZT4)

Figure 4 Radial vibration curves of numerical and FEM results

ANSYS 145 is used to verify the radial propagation charac-teristics Using harmonic analysis and keeping the boundaryconditions of outer surface free radial displacement is excitedon the inner surface which is shown in Figure 4(b) Compar-ing numerical results with FEM results it can be found thatthe bands are in good agreement with the aspects of positionwidth and amplitude attenuation and other aspects whichindicates the correctness of theoretical calculations

33 Effects of Structural Parameters In this section the influ-ence of piezoelectric effect on radial wave has been discussedin detail Furthermore the influences of piezoelectric param-eter 11988931 and polar direction on the band have also been stud-ied in case of open and short circuit for piezoelectricmaterial

331 Open and Short Circuit With regard to piezoelectricmaterial the D-type equation of radial vibration can beexpressed as

120576119903 = 11990411986411120590119903 + 11990411986412120590120579 + 11988931119864119885

120576120579 = 11990411986412120590119903 + 11990411986411120590120579 + 11988931119864119885119863119911 = 11988931120590119903 + 11988931120590120579 + 12058511987933119864119885

(35)

It corresponds to short circuit situation when field inten-sity is 119864119885 = 0 Therefore (35) can be simplified as

120590119903 = 11990411986411(11990411986411)2 minus (11990411986412)2 120576119903 minus11990411986412(11990411986411)2 minus (11990411986412)2 120576120579

120590120579 = 11990411986411(11990411986411)2 minus (11990411986412)2 120576120579 minus 11990411986412(11990411986411)2 minus (11990411986412)2 120576119903119863119911 = 11988931 (120590119903 + 120590120579)

(36)

Then it has

]119864 = minus1199041198641211990411986411 1198841198640 = 111990411986411

(37)

Shock and Vibration 7

times104

minus100

minus80

minus60

minus40

minus20

0

20

40

60

Tran

smiss

ibili

ty (d

B)

1 2 3 4 5 6 7 8 9 100

Frequency (Hz)

Open circuitShort circuit

Figure 5 Comparison diagram with short and open circuits

where ]119864 denotes Poissonrsquos coefficient in the condition ofshort circuit 1198841198640 denotes Young modulus

Piezoelectric material is in the open circuit when thecharge is 119863119911 = 0 Thus (35) can be simplified as

120590119903 = 1198841198631 minus (]119863)2 (120576119903 + ]119863120576120579)

120590120579 = 1198841198631 minus (]119863)2 (120576120579 + ]119863120576119903)

(38)

Then it gives

]119863 = minus1205851198793311990411986412 minus 1198892311205851198793311990411986411 minus 119889231 1198841198630 = 111990411986411 minus 11988923112058511987933

(39)

where ]119863 denotes Poissonrsquos coefficient in the condition ofopen circuit 1198841198630 denotes Young modulus

By using transfer matrices derived in Section 21 vibra-tion characteristics curves have been calculated for the caseof short and open circuit As depicted in Figure 5 the selectedfrequency band is 0ndash10 kHz It indicates that the band changesobviously in the high frequency because of the piezoelectriceffect However the short circuit curve moves toward lowfrequency slightly From the above theoretical analysis bycomparing (37) and (39) it can be observed that the elasticitymodulus and Poissonrsquos coefficient of piezoelectric materialsare different when they are in short or open circuit For thecase of open circuit elasticity modulus is larger MeanwhilePoissonrsquos coefficient also changes It is important to note thatthese two parameters can influence the value of 120590119903 and 120590120579directly whichmeans that radial wavenumber changesThusit is the natural mechanism of band moving

332 Piezoelectric Parameters and Inner Radius As to theopen circuit case the effects of electrical parameter 11988931 andinner radius 1199030 on radial wave band are depicted in Figure 6Herein Figure 6(a) describes the effect of piezoelectric

times104

104 6 820 1 3 5 7 9

Frequency (Hz)

minus120

minus100

minus20

minus60

minus40

minus20

0

20

40

60

Tran

smiss

ibili

ty (d

B)

d31 = minus123 lowast 10minus12 cm2

d31 = minus200 lowast 10minus12 cm2

(a) Effect of piezoelectric parameter 11988931

times104Frequency (Hz)

minus100

minus80

minus60

minus40

minus20

0

20

40

60

Tran

smiss

ibili

ty (d

B)

1086421 3 5 7 90

r0 = 008mr0 = 004mr0 = 002m

(b) Effect of inner radius 1199030

Figure 6 Effect of piezoelectric modulus 11988931 and inner radius 1199030

parameter 11988931 However Figure 6(b) displays the effect ofinner radius 1199030

From (39) it can be easily observed that 11988931 determinesthe values of 1198841198630 and ]119863 which can influence the bandposition and attenuation Figure 6(a) indicates that radialband has been expanded with the increase of 11988931 (here thenegative sign only represents direction) However the changeis not obvious Figure 6(b) displays that inner radius 1199030 caninfluence the low frequency greatly It is also clear from thefigure that the band attenuates obviously at low frequencyHowever the effect on high frequency is much smaller

4 Conclusion

This paper presents the classical method and wave propaga-tion approach to obtain the natural frequency Based on radialvibration equation propagation and reflection matrices arederived It can be found that natural frequency of the singlecircular plate can be calculated conveniently by using wavepropagation approach Then the results obtained by usingwave approach are compared with the classical method at theboundary conditions of free-free fixed-fixed and fixed-free

8 Shock and Vibration

It can be seen that the results calculated by these twomethodscoincide with each other

Additionally the band phenomenon of radial phononcrystal circular plate is also analyzed for a deeper understand-ing of radial vibration FEM is used to verify the theoreticalresults It can be found that the inner radius of circular platehas great influence on the low frequency Piezoelectric coeffi-cient open circuit and short circuitmake the elasticmodulusand Poissonrsquos ratio change Then the radial wavenumber alsochanges which is the natural mechanism of band removingfor the case of open and short circuits

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work is financially supported by the National NaturalScience Foundation of China (no 51375105) and the Natu-ral Science Foundation of Heilongjiang Province of China(E201418)

References

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

[2] C Mei and B R Mace ldquoWave reflection and transmission inTimoshenko beams and wave analysis of Timoshenko beamstructuresrdquo Journal of Sound andAcoustics vol 127 pp 382ndash3942004

[3] N R Harland B R Mace and RW Jones ldquoWave propagationreflection and transmission in tunable fluid-filled beamsrdquo Jour-nal of Sound and Vibration vol 241 no 5 pp 735ndash754 2001

[4] 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

[5] 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

[6] S K Lee Wave reflection transmission and propagation instructural waveguides [PhD thesis] Southampton University2006

[7] CHWang and L R F Rose ldquoWave reflection and transmissionin beams containing delamination and inhomogeneityrdquo Journalof Sound and Vibration vol 264 no 4 pp 851ndash872 2003

[8] C A Tan and B Kang ldquoWave reflection and transmission in anaxially strained rotating Timoshenko shaftrdquo Journal of Soundand Vibration vol 213 no 3 pp 483ndash510 1998

[9] X B Li ldquoStudy on free vibration analysis of circular cylindricalshells using wave propagationrdquo Journal of Sound and Vibrationvol 311 no 3ndash5 pp 667ndash682 2008

[10] L Gan X B Li and Z Zhang ldquoFree vibration analysis of ring-stiffened cylindrical shells using wave propagation approachrdquoJournal of Sound and Vibration vol 326 no 3ndash5 pp 633ndash6462009

[11] X M Zhang ldquoVibration analysis of cross-ply laminated com-posite cylindrical shells using the wave propagation approachrdquoApplied Acoustics vol 62 no 11 pp 1221ndash1228 2001

[12] X M Zhang ldquoFrequency analysis of submerged cylindricalshells with the wave propagation approachrdquo International Jour-nal of Mechanical Sciences vol 44 no 7 pp 1259ndash1273 2002

[13] M Shakeri M Akhlaghi and S M Hoseini ldquoVibration andradial wave propagation velocity in functionally graded thickhollow cylinderrdquo Composite Structures vol 76 no 1-2 pp 174ndash181 2006

[14] D Torrent and J Sanchez-Dehesa ldquoAcoustic resonances in two-dimensional radial sonic crystal shellsrdquo New Journal of Physicsvol 12 Article ID 073034 2010

[15] J Carbonell D Torrent and J Sanchez-Dehesa ldquoRadial pho-tonic crystal shells and their application as resonant and radiat-ing elementsrdquo IEEE Transactions on Antennas and Propagationvol 61 no 2 pp 755ndash767 2013

[16] Z L Xu F G Wu and Z N Guo ldquoLow frequency phononicband structures in two-dimensional arc-shaped phononic crys-talsrdquo Physics Letters A vol 376 no 33 pp 2256ndash2263 2012

[17] T Ma T N Chen X P Wang Y Li and P Wang ldquoBandstructures of bilayer radial phononic crystal plate with crystalglidingrdquo Journal of Applied Physics vol 116 no 10 Article ID104505 pp 1ndash7 2014

[18] H S Shu L Q Dong S D Li et al ldquoPropagation of torsionalwaves in a thin circular plate of generalized phononic crystalsrdquoJournal of Physics D Applied Physics vol 47 no 29 Article ID295501 2014

[19] H S Shu W Liu S D Li et al ldquoResearch on flexural waveband gap of a thin circular plate of piezoelectric radial phononiccrystalsrdquo Journal of Vibration and Control vol 22 no 7 pp1777ndash1789 2016

[20] K Kiani ldquoElasto-dynamic analysis of spinning nanodisks viaa surface energy-based modelrdquo Journal of Physics D AppliedPhysics vol 49 no 27 Article ID 275306 2016

[21] K Kiani ldquoThermo-mechanical analysis of functionally gradedplate-like nanorotors a surface elasticity modelrdquo InternationalJournal of Mechanical Sciences vol 106 pp 39ndash49 2016

[22] K Kiani ldquoMagneto-elasto-dynamic analysis of an elasticallyconfined conducting nanowire due to an axial magnetic shockrdquoPhysics Letters A vol 376 no 20 pp 1679ndash1685 2012

[23] S Y Lin ldquoStudy on the step-type circular ring ultrasonicconcentrator in radial vibrationrdquo Ultrasonics vol 49 no 2 pp206ndash211 2009

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 7

times104

minus100

minus80

minus60

minus40

minus20

0

20

40

60

Tran

smiss

ibili

ty (d

B)

1 2 3 4 5 6 7 8 9 100

Frequency (Hz)

Open circuitShort circuit

Figure 5 Comparison diagram with short and open circuits

where ]119864 denotes Poissonrsquos coefficient in the condition ofshort circuit 1198841198640 denotes Young modulus

Piezoelectric material is in the open circuit when thecharge is 119863119911 = 0 Thus (35) can be simplified as

120590119903 = 1198841198631 minus (]119863)2 (120576119903 + ]119863120576120579)

120590120579 = 1198841198631 minus (]119863)2 (120576120579 + ]119863120576119903)

(38)

Then it gives

]119863 = minus1205851198793311990411986412 minus 1198892311205851198793311990411986411 minus 119889231 1198841198630 = 111990411986411 minus 11988923112058511987933

(39)

where ]119863 denotes Poissonrsquos coefficient in the condition ofopen circuit 1198841198630 denotes Young modulus

By using transfer matrices derived in Section 21 vibra-tion characteristics curves have been calculated for the caseof short and open circuit As depicted in Figure 5 the selectedfrequency band is 0ndash10 kHz It indicates that the band changesobviously in the high frequency because of the piezoelectriceffect However the short circuit curve moves toward lowfrequency slightly From the above theoretical analysis bycomparing (37) and (39) it can be observed that the elasticitymodulus and Poissonrsquos coefficient of piezoelectric materialsare different when they are in short or open circuit For thecase of open circuit elasticity modulus is larger MeanwhilePoissonrsquos coefficient also changes It is important to note thatthese two parameters can influence the value of 120590119903 and 120590120579directly whichmeans that radial wavenumber changesThusit is the natural mechanism of band moving

332 Piezoelectric Parameters and Inner Radius As to theopen circuit case the effects of electrical parameter 11988931 andinner radius 1199030 on radial wave band are depicted in Figure 6Herein Figure 6(a) describes the effect of piezoelectric

times104

104 6 820 1 3 5 7 9

Frequency (Hz)

minus120

minus100

minus20

minus60

minus40

minus20

0

20

40

60

Tran

smiss

ibili

ty (d

B)

d31 = minus123 lowast 10minus12 cm2

d31 = minus200 lowast 10minus12 cm2

(a) Effect of piezoelectric parameter 11988931

times104Frequency (Hz)

minus100

minus80

minus60

minus40

minus20

0

20

40

60

Tran

smiss

ibili

ty (d

B)

1086421 3 5 7 90

r0 = 008mr0 = 004mr0 = 002m

(b) Effect of inner radius 1199030

Figure 6 Effect of piezoelectric modulus 11988931 and inner radius 1199030

parameter 11988931 However Figure 6(b) displays the effect ofinner radius 1199030

From (39) it can be easily observed that 11988931 determinesthe values of 1198841198630 and ]119863 which can influence the bandposition and attenuation Figure 6(a) indicates that radialband has been expanded with the increase of 11988931 (here thenegative sign only represents direction) However the changeis not obvious Figure 6(b) displays that inner radius 1199030 caninfluence the low frequency greatly It is also clear from thefigure that the band attenuates obviously at low frequencyHowever the effect on high frequency is much smaller

4 Conclusion

This paper presents the classical method and wave propaga-tion approach to obtain the natural frequency Based on radialvibration equation propagation and reflection matrices arederived It can be found that natural frequency of the singlecircular plate can be calculated conveniently by using wavepropagation approach Then the results obtained by usingwave approach are compared with the classical method at theboundary conditions of free-free fixed-fixed and fixed-free

8 Shock and Vibration

It can be seen that the results calculated by these twomethodscoincide with each other

Additionally the band phenomenon of radial phononcrystal circular plate is also analyzed for a deeper understand-ing of radial vibration FEM is used to verify the theoreticalresults It can be found that the inner radius of circular platehas great influence on the low frequency Piezoelectric coeffi-cient open circuit and short circuitmake the elasticmodulusand Poissonrsquos ratio change Then the radial wavenumber alsochanges which is the natural mechanism of band removingfor the case of open and short circuits

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work is financially supported by the National NaturalScience Foundation of China (no 51375105) and the Natu-ral Science Foundation of Heilongjiang Province of China(E201418)

References

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

[2] C Mei and B R Mace ldquoWave reflection and transmission inTimoshenko beams and wave analysis of Timoshenko beamstructuresrdquo Journal of Sound andAcoustics vol 127 pp 382ndash3942004

[3] N R Harland B R Mace and RW Jones ldquoWave propagationreflection and transmission in tunable fluid-filled beamsrdquo Jour-nal of Sound and Vibration vol 241 no 5 pp 735ndash754 2001

[4] 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

[5] 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

[6] S K Lee Wave reflection transmission and propagation instructural waveguides [PhD thesis] Southampton University2006

[7] CHWang and L R F Rose ldquoWave reflection and transmissionin beams containing delamination and inhomogeneityrdquo Journalof Sound and Vibration vol 264 no 4 pp 851ndash872 2003

[8] C A Tan and B Kang ldquoWave reflection and transmission in anaxially strained rotating Timoshenko shaftrdquo Journal of Soundand Vibration vol 213 no 3 pp 483ndash510 1998

[9] X B Li ldquoStudy on free vibration analysis of circular cylindricalshells using wave propagationrdquo Journal of Sound and Vibrationvol 311 no 3ndash5 pp 667ndash682 2008

[10] L Gan X B Li and Z Zhang ldquoFree vibration analysis of ring-stiffened cylindrical shells using wave propagation approachrdquoJournal of Sound and Vibration vol 326 no 3ndash5 pp 633ndash6462009

[11] X M Zhang ldquoVibration analysis of cross-ply laminated com-posite cylindrical shells using the wave propagation approachrdquoApplied Acoustics vol 62 no 11 pp 1221ndash1228 2001

[12] X M Zhang ldquoFrequency analysis of submerged cylindricalshells with the wave propagation approachrdquo International Jour-nal of Mechanical Sciences vol 44 no 7 pp 1259ndash1273 2002

[13] M Shakeri M Akhlaghi and S M Hoseini ldquoVibration andradial wave propagation velocity in functionally graded thickhollow cylinderrdquo Composite Structures vol 76 no 1-2 pp 174ndash181 2006

[14] D Torrent and J Sanchez-Dehesa ldquoAcoustic resonances in two-dimensional radial sonic crystal shellsrdquo New Journal of Physicsvol 12 Article ID 073034 2010

[15] J Carbonell D Torrent and J Sanchez-Dehesa ldquoRadial pho-tonic crystal shells and their application as resonant and radiat-ing elementsrdquo IEEE Transactions on Antennas and Propagationvol 61 no 2 pp 755ndash767 2013

[16] Z L Xu F G Wu and Z N Guo ldquoLow frequency phononicband structures in two-dimensional arc-shaped phononic crys-talsrdquo Physics Letters A vol 376 no 33 pp 2256ndash2263 2012

[17] T Ma T N Chen X P Wang Y Li and P Wang ldquoBandstructures of bilayer radial phononic crystal plate with crystalglidingrdquo Journal of Applied Physics vol 116 no 10 Article ID104505 pp 1ndash7 2014

[18] H S Shu L Q Dong S D Li et al ldquoPropagation of torsionalwaves in a thin circular plate of generalized phononic crystalsrdquoJournal of Physics D Applied Physics vol 47 no 29 Article ID295501 2014

[19] H S Shu W Liu S D Li et al ldquoResearch on flexural waveband gap of a thin circular plate of piezoelectric radial phononiccrystalsrdquo Journal of Vibration and Control vol 22 no 7 pp1777ndash1789 2016

[20] K Kiani ldquoElasto-dynamic analysis of spinning nanodisks viaa surface energy-based modelrdquo Journal of Physics D AppliedPhysics vol 49 no 27 Article ID 275306 2016

[21] K Kiani ldquoThermo-mechanical analysis of functionally gradedplate-like nanorotors a surface elasticity modelrdquo InternationalJournal of Mechanical Sciences vol 106 pp 39ndash49 2016

[22] K Kiani ldquoMagneto-elasto-dynamic analysis of an elasticallyconfined conducting nanowire due to an axial magnetic shockrdquoPhysics Letters A vol 376 no 20 pp 1679ndash1685 2012

[23] S Y Lin ldquoStudy on the step-type circular ring ultrasonicconcentrator in radial vibrationrdquo Ultrasonics vol 49 no 2 pp206ndash211 2009

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

8 Shock and Vibration

It can be seen that the results calculated by these twomethodscoincide with each other

Additionally the band phenomenon of radial phononcrystal circular plate is also analyzed for a deeper understand-ing of radial vibration FEM is used to verify the theoreticalresults It can be found that the inner radius of circular platehas great influence on the low frequency Piezoelectric coeffi-cient open circuit and short circuitmake the elasticmodulusand Poissonrsquos ratio change Then the radial wavenumber alsochanges which is the natural mechanism of band removingfor the case of open and short circuits

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work is financially supported by the National NaturalScience Foundation of China (no 51375105) and the Natu-ral Science Foundation of Heilongjiang Province of China(E201418)

References

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

[2] C Mei and B R Mace ldquoWave reflection and transmission inTimoshenko beams and wave analysis of Timoshenko beamstructuresrdquo Journal of Sound andAcoustics vol 127 pp 382ndash3942004

[3] N R Harland B R Mace and RW Jones ldquoWave propagationreflection and transmission in tunable fluid-filled beamsrdquo Jour-nal of Sound and Vibration vol 241 no 5 pp 735ndash754 2001

[4] 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

[5] 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

[6] S K Lee Wave reflection transmission and propagation instructural waveguides [PhD thesis] Southampton University2006

[7] CHWang and L R F Rose ldquoWave reflection and transmissionin beams containing delamination and inhomogeneityrdquo Journalof Sound and Vibration vol 264 no 4 pp 851ndash872 2003

[8] C A Tan and B Kang ldquoWave reflection and transmission in anaxially strained rotating Timoshenko shaftrdquo Journal of Soundand Vibration vol 213 no 3 pp 483ndash510 1998

[9] X B Li ldquoStudy on free vibration analysis of circular cylindricalshells using wave propagationrdquo Journal of Sound and Vibrationvol 311 no 3ndash5 pp 667ndash682 2008

[10] L Gan X B Li and Z Zhang ldquoFree vibration analysis of ring-stiffened cylindrical shells using wave propagation approachrdquoJournal of Sound and Vibration vol 326 no 3ndash5 pp 633ndash6462009

[11] X M Zhang ldquoVibration analysis of cross-ply laminated com-posite cylindrical shells using the wave propagation approachrdquoApplied Acoustics vol 62 no 11 pp 1221ndash1228 2001

[12] X M Zhang ldquoFrequency analysis of submerged cylindricalshells with the wave propagation approachrdquo International Jour-nal of Mechanical Sciences vol 44 no 7 pp 1259ndash1273 2002

[13] M Shakeri M Akhlaghi and S M Hoseini ldquoVibration andradial wave propagation velocity in functionally graded thickhollow cylinderrdquo Composite Structures vol 76 no 1-2 pp 174ndash181 2006

[14] D Torrent and J Sanchez-Dehesa ldquoAcoustic resonances in two-dimensional radial sonic crystal shellsrdquo New Journal of Physicsvol 12 Article ID 073034 2010

[15] J Carbonell D Torrent and J Sanchez-Dehesa ldquoRadial pho-tonic crystal shells and their application as resonant and radiat-ing elementsrdquo IEEE Transactions on Antennas and Propagationvol 61 no 2 pp 755ndash767 2013

[16] Z L Xu F G Wu and Z N Guo ldquoLow frequency phononicband structures in two-dimensional arc-shaped phononic crys-talsrdquo Physics Letters A vol 376 no 33 pp 2256ndash2263 2012

[17] T Ma T N Chen X P Wang Y Li and P Wang ldquoBandstructures of bilayer radial phononic crystal plate with crystalglidingrdquo Journal of Applied Physics vol 116 no 10 Article ID104505 pp 1ndash7 2014

[18] H S Shu L Q Dong S D Li et al ldquoPropagation of torsionalwaves in a thin circular plate of generalized phononic crystalsrdquoJournal of Physics D Applied Physics vol 47 no 29 Article ID295501 2014

[19] H S Shu W Liu S D Li et al ldquoResearch on flexural waveband gap of a thin circular plate of piezoelectric radial phononiccrystalsrdquo Journal of Vibration and Control vol 22 no 7 pp1777ndash1789 2016

[20] K Kiani ldquoElasto-dynamic analysis of spinning nanodisks viaa surface energy-based modelrdquo Journal of Physics D AppliedPhysics vol 49 no 27 Article ID 275306 2016

[21] K Kiani ldquoThermo-mechanical analysis of functionally gradedplate-like nanorotors a surface elasticity modelrdquo InternationalJournal of Mechanical Sciences vol 106 pp 39ndash49 2016

[22] K Kiani ldquoMagneto-elasto-dynamic analysis of an elasticallyconfined conducting nanowire due to an axial magnetic shockrdquoPhysics Letters A vol 376 no 20 pp 1679ndash1685 2012

[23] S Y Lin ldquoStudy on the step-type circular ring ultrasonicconcentrator in radial vibrationrdquo Ultrasonics vol 49 no 2 pp206ndash211 2009

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of