Research on Radial Vibration of a Circular Plate

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


Introduction
Wave propagation approach is a useful method on the aspects of calculating natural frequency for the analysis of elastic structures, such as rod, Euler beam, curved beam, and plate.It benefits us with a better understanding to analyze wave when propagating inside structures.In fact, as early as 1984, Mace analyzed the behaviors of wave by using wave propagation approach through dividing them into propagation attenuation matrices.And the reflection matrices under three boundary conditions were deduced, which built a theoretical foundation for wave method [1].Meanwhile, the characteristics of propagation and reflection of Timoshenko beam were also discussed with the discontinuous behavior.And the frequency curves were obtained [2].Harland et al. studied the motion of waves in tunable fluid-filled beams [3].Based on Love theory, Kang et al. deduced the propagation matrices, reflection matrices, and coordinated matrices in curved beams [4].Different from Kang et al., Lee made a detailed research on the behavior of waves of curved beam based on Flugge's theory [5,6].Based on the high-order plate theory, Wang and Rose analyzed the properties of transient wave in an inhomogeneous beam.And the results were also verified by the experiment [7].Tan and Kang considered the wave reflection and transmission in an axially strained, rotating Timoshenko shaft.Meanwhile, the influence of continuous condition and cross-section on natural frequency was also discussed [8].Li and Gan analyzed the free vibration of cylindrical shell by using wave propagation approach based on Flugge's thin shell theory [9,10].Also, Zhang calculated the frequencies of cross-ply laminated composite cylindrical shells and submerged cylindrical shells using wave approach based on Love's theory [11,12].Using wave approach, the above scholars have done lots of analysis, which exhibit a unique advantage for the analysis of structural vibration.
In recent years, people have paid great attention to structure which consisted of two materials arraying periodically.At present, the research has been extended into radial phononic crystal structures.Shakeri et al. analyzed the behaviors of radial wave in functional graded radial periodical structures [13].Torrent and Carbonell studied the propagation characteristic of acoustic in detail when propagating in radial periodical structure and found that it also had a low frequency band [14,15].After constructing and analyzing the two-dimensional radial phononic crystal, Xu et al. found that these structures had low frequency band that the starting frequency was zero [16].Ma et al. studied the band characteristics of Lamb wave in two-layer radial phononic crystal plate, and it was found that crystal gliding in radial direction could open the lowest band gaps [17].The above researches indicate that radial phononic crystal plate has the low frequency band which brings an attractive prospect of engineering application.For example, the natural frequency can be designed into the range of band gaps to isolate the vibration.Also, these structures can be regarded as a filter.Additionally, Shu et al. researched torsional vibration problem of generalized phononic crystal, finding that low frequency band was determined by radius.However, the high frequency band was caused by the periodicity of circular plate [18].And Shu's research shows that generalized phononic crystal can generate a huge band which almost covers the whole frequency range, providing a practical application for the rotating parts, such as gear.In the previous research, Shu et al. have studied the band gaps of flexural wave in the radial phononic crystal.Meanwhile, numerical results are also verified by FEM [19].Herein, this paper is mainly focused on radial vibration of composite structures.In fact, other works on radial vibration of elastic solids under various loadings can also be found [20][21][22].
The paper is organized into five sections.In Section 1, a brief introduction is given.In Section 2, classic Bessel solution and Hankel solution of radial vibration are given.Meanwhile, propagation and reflection matrices at three boundary conditions are derived.The results of natural frequency obtained by wave approach are also compared with the classical method.In Section 3, the band phenomenon is analyzed and the effect of parameters is discussed.Section 4 is the conclusion.

Theoretical Analysis for Natural Frequency
The expression of strain and radial displacement can be written as Since piezoelectric material direction of polarization is along z-axle, piezoelectric equation can be expressed as where   is radial displacement,   and   are radial stress and tangential stress,   and   are radial strain and tangential strain, and  is radius.Substituting ( 2)-( 4) into (1) gives where ) is the square of radial wave velocity.After simplifying (5), it can be obtained that Thus, general solution of radial displacement can be written as where  = / is the wave number. 1 () is the Bessel function of first kind;  1 () is the Bessel function of second kind.And the constants  + 1 and  − 1 can be determined by boundary conditions.
Therefore, radial stress can be obtained as 2.2.Classical Method.When both ends of circular plate are fixed, it has Substituting ( 7) into (9) gives When both ends of circular plate are free, it has Substituting ( 8) into (11) gives Similarly, when inner surface is fixed and outer surface is free, it has Substituting ( 7) and ( 8) into (13) gives After solving roots of ( 10), (12), and ( 14), the natural frequencies of circular plate structure can be obtained with the aforementioned boundary conditions.
Similarly, the reflection matrices are (20a) and (24) for case of inner surface fixed and outer surface free.

Analysis and Discussion
. After combining propagation and reflection matrices derived above, the natural characteristics of single circular plate can be analyzed effectively.Hence, it gives And it yields RESIN is selected as the material of circular plate.Material 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 computed.Herein, red and black curves obtained by wave approach are real and imaginary wave solution, respectively.Also, it is important to note that the intersection points of the curves are natural frequency.After solving the inherent characteristic of circular plate structure, the characteristic curves calculated by classical method and wave approach are presented in Figure 2. It can be seen that these natural frequencies intersect at the same point in x-axis, which verifies the correctness of the numerical calculation.
With regard to radial vibration, there are two parameters which satisfy the continuous conditions at the interface of different materials.Namely, they are the radial displacement and radial stress.So transfer matrices of radial vibration can be derived as follows.
At the interface b (  =  1 +  1 ), it gives Thus, it can be arranged in a matrices form At the interface c (  =  1 +  1 +  2 ), it gives Similarly, one has Combining ( 30) and ( 32), one has  where  31 =  −1 2  2  −1 1  1 is the transfer matrices of radial vibration between the 3rd and the 1st ring.
As shown in Figure 3, replace   and   (in (30) and ( 32)) with   and   .Similarly with  31 , we can obtain  53 which is the transfer matrix between the fifth ring and the third ring.
Similarly, replace   and   with   and   , and the matrix  75 can be obtained.
At last, submitting   into (30), we can get  81 .Therefore, with regard to the radial phononic crystal, one has   Taking advantage of transfer matrices and boundary conditions, the transmission characteristics are calculated using MATLAB numerically.Figure 4 is the transmission curve of radial vibration with four periodic structures which is composited by single material RESIN and RESIN/PZT4. Figure 4 is the comparison of numerical and FEM results.It indicates that there is no attenuation when radial wave propagates in a single material, while, for the phononic crystal, the transmission curve shows a big vibration attenuation band after introducing the periodicity.However, when radial wave propagates in the single material, there is no band gap which attenuates weakly and only owns −4 dB.Furthermore, ANSYS 14.5 is used to verify the radial propagation characteristics.Using harmonic analysis and keeping the boundary conditions of outer surface free, radial displacement is excited on the inner surface which is shown in Figure 4(b).Comparing numerical results with FEM results, it can be found that the bands are in good agreement with the aspects of position, width, and amplitude attenuation and other aspects, which indicates the correctness of theoretical calculations.

Effects of Structural Parameters.
In this section, the influence of piezoelectric effect on radial wave has been discussed in detail.Furthermore, the influences of piezoelectric parameter  31 and polar direction on the band have also been studied in case of open and short circuit for piezoelectric material.

Open and Short
Circuit.With regard to piezoelectric material, the D-type equation of radial vibration can be expressed as where ]  denotes Poisson's coefficient in the condition of short circuit.  0 denotes Young modulus.Piezoelectric material is in the open circuit when the charge is   = 0. Thus, (35) can be simplified as where ]  denotes Poisson's coefficient in the condition of open circuit.  0 denotes Young modulus.By using transfer matrices derived in Section 2.1, vibration characteristics curves have been calculated for the case of short and open circuit.As depicted in Figure 5, the selected frequency band is 0-10 kHz.It indicates that the band changes obviously in the high frequency because of the piezoelectric effect.However, the short circuit curve moves toward low frequency slightly.From the above theoretical analysis by comparing (37) and (39), it can be observed that the elasticity modulus and Poisson's coefficient of piezoelectric materials are different when they are in short or open circuit.For the case of open circuit, elasticity modulus is larger.Meanwhile, Poisson's coefficient also changes.It is important to note that these two parameters can influence the value of   and   directly, which means that radial wavenumber changes.Thus, it is the natural mechanism of band moving.

Piezoelectric Parameters and Inner Radius.
As to the open circuit case, the effects of electrical parameter  31 and inner radius  0 on radial wave band are depicted in Figure 6.Herein, Figure 6(a) describes the effect of piezoelectric  From (39), it can be easily observed that  31 determines the values of   0 and ]  which can influence the band position and attenuation.Figure 6(a) indicates that radial band has been expanded with the increase of  31 (here, the negative sign only represents direction).However, the change is not obvious.Figure 6(b) displays that inner radius  0 can influence the low frequency greatly.It is also clear from the figure that the band attenuates obviously at low frequency.However, the effect on high frequency is much smaller.

Conclusion
This paper presents the classical method and wave propagation approach to obtain the natural frequency.Based on radial vibration equation, propagation and reflection matrices are derived.It can be found that natural frequency of the single circular plate can be calculated conveniently by using wave propagation approach.Then, the results obtained by using wave approach are compared with the classical method at the boundary conditions of free-free, fixed-fixed, and fixed-free.
It can be seen that the results calculated by these two methods coincide with each other.
Additionally, the band phenomenon of radial phonon crystal circular plate is also analyzed for a deeper understanding of radial vibration.FEM is used to verify the theoretical results.It can be found that the inner radius of circular plate has great influence on the low frequency.Piezoelectric coefficient, open circuit, and short circuit make the elastic modulus and Poisson's ratio change.Then the radial wavenumber also changes, which is the natural mechanism of band removing for the case of open and short circuits.

Figure 2 :
Figure 2: Natural frequency calculated by classic method and wave approach.

Figure 5 :
Figure 5: Comparison diagram with short and open circuits.
Effect of inner radius  0

Table 1 :
Material and structural parameters.
(34) 3.2.Numerical Result and Discussion.The radial phononic crystal circular plates are composed of two materials, namely, RESIN and piezoelectric ceramic PZT4.Material parameters and structural parameters are consistent with Table 2.

Table 2 :
Material parameters and structural parameters.