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 two materials 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.

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 [

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 [

The paper is organized into five sections. In Section

Single circular plate is shown in Figure

Model of single circular plate.

The expression of strain and radial displacement can be written as

Since piezoelectric material direction of polarization is along

Substituting (

After simplifying (

Thus, general solution of radial displacement can be written as

Therefore, radial stress can be obtained as

When both ends of circular plate are fixed, it has

Substituting (

When both ends of circular plate are free, it has

Substituting (

Similarly, when inner surface is fixed and outer surface is free, it has

Substituting (

After solving roots of (

One of the solutions of circular plate has been solved as shown in (

Considering (

Then, (

When both ends of circular plate are fixed, it gives

Thus, the reflection matrices of both boundaries are

Substituting (

When both ends of circular plate are free, it has

Thus, reflection matrices of both boundaries are

Forward-traveling and negative-traveling waves can be shown clearly in Figure

Considering (

Similarly, the reflection matrices are (

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

Material and structural parameters.

Material parameters | ^{3}) | | |
---|---|---|---|

RESIN | 1180 | | 0.3679 |

Structural parameters | | | |

RESIN | 0.08 | 0.16 | 0.005 |

Substitute parameters listed in Table

Natural frequency calculated by classic method and wave approach.

Fixed-fixed boundary

Free-free boundary

Fixed-free boundary

Figure

Schematic diagram of radial phononic crystal circular plate.

Considering an eight-layer circular plate, it corresponds to the model

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

Thus, it can be arranged in a matrices form

At the interface

Similarly, one has

Combining (

As shown in Figure

Similarly, replace

At last, submitting

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

Material parameters and structural parameters.

Material parameters | ^{3}) | | |
---|---|---|---|

I (RESIN) | 1180 | | 0.3679 |

II (PZT4) | 7500 | | 0.33 |

| |||

Structural parameters | | | |

RESIN and PZT4 | 0.01 | 0.16 | 0.003 |

Additionally, piezoelectric parameters are set as ^{2}/N,

Taking advantage of transfer matrices and boundary conditions, the transmission characteristics are calculated using MATLAB numerically. Figure

Radial vibration curves of numerical and FEM results.

Single material (RESIN)

Phononic crystal (RESIN/PZT4)

Figure

In this section, the influence of piezoelectric effect on radial wave has been discussed in detail. Furthermore, the influences of piezoelectric parameter

With regard to piezoelectric material, the

It corresponds to short circuit situation when field intensity is

Piezoelectric material is in the open circuit when the charge is

Then, it gives

By using transfer matrices derived in Section

Comparison diagram with short and open circuits.

As to the open circuit case, the effects of electrical parameter

Effect of piezoelectric modulus

Effect of piezoelectric parameter

Effect of inner radius

From (

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.

The authors declare that they have no competing interests.

This work is financially supported by the National Natural Science Foundation of China (no. 51375105) and the Natural Science Foundation of Heilongjiang Province of China (E201418).