Relationship between geometric features and vibration characteristics in functionally graded piezoelectric annular plates: An extended double Legendre polynomial method

Abstract

The utilization of flexural, longitudinal, and torsional modes in functionally graded piezoelectric (FGP) annular plates depends on their geometric features. The investigation of the relationship between geometric features and vibrational characteristics offers valuable insights for optimal design. In this paper, the double Legendre polynomial method (DLPM) is extended for the first time to study the three-dimensional (3D) flexural vibration of FGP annular plates from two-dimensional (2D) axisymmetric vibration based on 3D elasticity theory. The validity of the proposed method is verified by literature results and finite element method (FEM) results. The influence of the relationship between the radial wall thickness and the height on the flexural, longitudinal and torsional vibrations of FGP annular plates is systematically revealed. The results provide comprehensive guidance for the design and application of FGP annular plates in domains requiring high precision, high stability, and high sensitivity.

Keywords

double Legendre polynomial method geometric features vibration characteristics functionally graded piezoelectric materials annular plates

1. Highlights

1. The DLPM is extended for the first time to investigate the 3D vibration of FGP annular plates from 2D vibration.

2. The effects of the radial wall thickness and the height on the vibration characteristics are revealed.

3. This paper provides comprehensive and detailed insights for the design and application of FGP annular plates.

2. Introduction

Functionally graded materials (FGMs) have attracted extensive attention from mechanics scholars due to their exceptional mechanical performance such as weak stress concentration and minimized residual stresses (Birman and Byrd, 2007). With the rapid development of electromechanical systems, there is a greater demand for high-performance intelligent materials. Functionally graded piezoelectric materials (FGPMs) have great research value in the fields of aerospace, structural health detection, and precision electronic devices due to their excellent mechanical properties and electromechanical coupling characteristics (Feri et al., 2021; Ghemari et al., 2023; Jian et al., 2023; Wang et al., 2022).

The functionally graded piezoelectric (FGP) annular plate exhibits flexural, longitudinal and torsional vibration modes, and has good application prospects in the fields of energy harvesting (Panda et al., 2022; Sezer and Koç, 2021), structural health monitoring (Aabid et al., 2021; Huan et al., 2019), and acoustics (Wang et al., 2023). Choosing the appropriate vibration mode according to different requirements and conditions is an important way to achieve the best design effect. The flexural mode which is sensitive to external changes is suitable for high sensitivity environment (Belardinelli et al., 2020; Tu et al., 2022); the longitudinal mode is suitable for occasions with high transmission and conversion efficiency (Cao et al., 2021; Piao et al., 2020); the torsional mode is suitable for monitoring changes in the rotating system (Pal and Das, 2020; Zafarani et al., 2021).

The geometric parameters of the FGP annular plates significantly influence their vibration characteristics. Yang et al. (2024) studied the symmetric and asymmetric free vibration of rotating eccentric annular plates in a uniform thermal environment based on the first-order shear deformation theory. Liu et al. (2021) investigated the influence of the inner radius and outer radius of an annular plate on the torsional vibration and transmission effects by using the wave method. Sharma and Ahlawat (2024) proposed a semi-analytical technique based on Taylor’s series method to solve the axisymmetric natural frequency of FGM annular plates with different radius ratios. Zhong et al. (2021) employed a semi-analytical model to analyze the in-plane vibration of porous annular plates with different radius ratios. Yang et al. (2023) discussed the influence of the height and inner radius of rotating graphene sheet reinforced composite annular plates on symmetric and asymmetric free and forced vibration in a thermal environment. Roshanbakhsh et al. (2020) proposed a simple and effective analytical method to solve the non-dimensional frequencies of FGM circular plates at different heights. The influence of geometric parameters on vibration characteristics is also investigated in foam metal plates (Arshid et al., 2023b, 2024b; Givi et al., 2024; Khoddami Maraghi et al., 2024; Yao and Arshid, 2024). Arshid et al. (2023a) explored the influence of the relationship between the core thickness and the total thickness on the natural frequency characteristics of the FG GNPs-RPN annular plates with piezoelectric/metallic coatings. Arshid et al. (2024a) analyzed the influence of radius ratio on the free vibration characteristics of annular sector microplates based on the first-order shear deformation theory.

Heshmati and Jalali (2019) focused on the longitudinal vibration of sandwich circular and annular plates made of FGM porosity materials at different heights and inner radii based on the first-order shear deformation theory. Guru and Jain (2020) applied the Chebyshev collocation technique to analyze the influence of core thickness, surface thickness, and radius ratio on the free axisymmetric vibration frequency of annular sandwich plates based on Reddy’s high-order shear deformation theory. Huang et al. (2005) determined the flexural, longitudinal, and tangential vibration modes of the piezoelectric disk by theoretical analysis, numerical simulation and experimental measurement, and analyzed the influence of the radius ratio on the natural frequencies of the three modes. Huang et al. (2021) applied Mindlin and Kirchhoff plate models to study the transverse and in-plane vibration characteristics of piezoelectric ceramic disks with different radius-to-height ratios. The generalized differential quadrature method is a traditional method to study mechanical problems (Alhaifi et al., 2023; Arshid et al., 2025; Babaei et al., 2024; Khoddami Maraghi and Arshid, 2024; Rasooli Jazi et al., 2024). The advantage of the double Legendre polynomial method (DLPM) with good orthogonality for solving FGP structural mechanics problems is that the model does not need to be layered, the calculation efficiency is high, and the convergence speed is good. However, this method has the disadvantage of long calculation time for multi-layer structures. It can be obtained from the above literature analysis that the influence of geometric features on vibration characteristics is often concentrated on the individual change of height and radius ratio in a small range. However, the influence of the geometric relationship between height and radial wall thickness on flexural, longitudinal and torsional vibration characteristics is not considered. The geometric dimensions and vibration mode determine the natural frequency and affect the frequency stability and anti-interference ability. Appropriate geometric parameters and vibration modes are essential for the development of high-performance piezoelectric resonators.

The methods of the three-dimensional (3D) analytical solution for FGP annular plates also need to be further enriched. At present, Legendre polynomial method (LPM) has already been used in the field of axisymmetric vibration (Khalfi et al., 2023; Zhou et al., 2023) and wave propagation (Liu et al., 2024; Wang et al., 2021a; Yu et al., 2022; Zhang et al., 2025). However, the flexural vibration characteristics of FGP annular plates have not been investigated by this method. The flexural mode shows higher sensitivity in the fields of sensors, transducers and acoustics. In this paper, DLPM is extended to study the flexural, longitudinal and torsional vibrations of FGP annular plates. This method can efficiently obtain the analytical solutions of flexural, longitudinal, and torsional modes in FGP annular plates by combining with analytical integration. The influence of the geometric relationship between the radial wall thickness and the height on the flexural, longitudinal, and torsional vibration of the FGP annular plate is first revealed.

3. Mathematics and formulations of the problem

3.1. Description of the model

The research object of this paper is a finite-size FGP annular plate with height H, outer radius a, inner radius b, and radial wall thickness a-b, as shown in Figure 1. The perfect conductive electrode with zero thickness completely covers the upper and lower surfaces of the FGP annular plate. These electrodes are connected to a time-dependent electrical excitation source $V_{0} e^{i (N θ - ω t)}$ .

Figure 1.

FGP annular plate structure.

Based on the 3D elastic theory, there are three types of parameters with different aspects in the constitutive equation. The constitutive equation can be expressed as (Zhou et al., 2023):

\begin{array}{l} T_{i j} = C_{i j k l} ε_{k l} - e_{k i j} E_{k} \\ D_{i} = e_{i k l} ε_{k l} + δ_{i k} E_{k} \end{array}

(1)

where

i, j, k, l = 1, 2, 3

;

T_{i j}

and

D_{i}

are defined as the stress and electric displacements, respectively.

C_{i j k l}

e_{k i j}

, and

δ_{i k}

represent the elastic, piezoelectric and dielectric coefficients, respectively.

ε_{k l}

and

E_{k}

denote the strains and electric field, respectively.

In the cylindrical coordinate system, the relationship between strain and displacement can be expressed as (Zhou et al., 2023):

ε_{θ θ} = \frac{1}{r} \frac{\partial v}{\partial θ} + \frac{u}{r}, ε_{z z} = \frac{\partial w}{\partial z}, ε_{r r} = \frac{\partial u}{\partial r}, ε_{r z} = \frac{1}{2} (\frac{\partial u}{\partial z} + \frac{\partial w}{\partial r}) ε_{θ z} = \frac{1}{2} (\frac{\partial v}{\partial z} + \frac{\partial w}{r \partial θ}), ε_{r θ} = \frac{1}{2} (\frac{1}{r} \frac{\partial u}{\partial θ} + \frac{\partial v}{\partial r} - \frac{v}{r})

(2)

The electric field is defined as (Zhou et al., 2023):

E_{θ} = - \frac{1}{r} \frac{\partial φ}{\partial θ}, E_{z} = - \frac{\partial φ}{\partial z}, E_{r} = - \frac{\partial φ}{\partial r}

(3)

In this paper, the inner and outer edges of FGP annular plates should meet the following boundary conditions when r = b and a:

(a) Free boundary condition (FF):

{T_{r r} |}_{r = b, a} = {T_{r θ} |}_{r = b, a} = {T_{r z} |}_{r = b, a} = 0

(4a)

(b) Clamped boundary condition (CC):

{u |}_{r = b, a} = {v |}_{r = b, a} = {w |}_{r = b, a} = 0

(4b)

When z = 0 and H, the top and bottom edges of FGP annular plate should satisfy the following boundary conditions:

{T_{z z} |}_{z = 0, H} = {T_{r z} |}_{z = 0, H} = {T_{θ z} |}_{z = 0, H} = 0

(4c)

The electrical boundary conditions are as follows:

\begin{array}{l} \begin{array}{l} {D_{r} |}_{r = b, a} = 0 & open circuit & {φ |}_{z = 0, H} = 0 \end{array} & closed circuit \end{array}

(4d)

The vibration control equation automatically incorporates the boundary conditions by introducing the functions $π_{1}$ and $π_{3}$ . The effectiveness of this method has been proved in Datta and Hunsinger (1978).

π_{1} = {\begin{cases} 1, & b \leq r \leq a \\ 0, & others \end{cases} \begin{array}{l} , & π_{3} = {\begin{cases} 1, & 0 \leq z \leq H \\ 0, & others \end{cases} \end{array}

(5)

The expressions of stress and electric displacement are obtained by substituting equations (2)–(5) into equation (1), respectively:

\begin{array}{l} \begin{array}{l} T_{r r} = [C_{11} \frac{\partial u}{\partial r} + C_{12} (\frac{1}{r} \frac{\partial v}{\partial θ} + \frac{u}{r}) + C_{13} \frac{\partial w}{\partial z} + e_{31} \frac{\partial φ}{\partial z}] π_{1} \\ T_{θ θ} = C_{12} \frac{\partial u}{\partial r} + C_{11} (\frac{1}{r} \frac{\partial v}{\partial θ} + \frac{u}{r}) + C_{13} \frac{\partial w}{\partial z} + e_{31} \frac{\partial φ}{\partial z} \\ T_{z z} = [C_{13} \frac{\partial u}{\partial r} + C_{13} (\frac{1}{r} \frac{\partial v}{\partial θ} + \frac{u}{r}) + C_{33} \frac{\partial w}{\partial z} + e_{33} \frac{\partial φ}{\partial z}] π_{3} \end{array} \\ T_{θ z} = [C_{44} (\frac{\partial v}{\partial z} + \frac{\partial w}{r \partial θ}) + e_{15} (\frac{1}{r} \frac{\partial φ}{\partial θ})] π_{3} \\ T_{r z} = [C_{44} (\frac{\partial u}{\partial z} + \frac{\partial w}{\partial r}) + e_{15} \frac{\partial φ}{\partial r}] π_{1} π_{3} T_{r θ} = [\frac{1}{2} (C_{11} - C_{12}) (\frac{1}{r} \frac{\partial u}{\partial θ} + \frac{\partial v}{\partial r} - \frac{v}{r})] π_{1} \end{array}

(6)

\begin{array}{l} D_{r} = [e_{15} (\frac{\partial u}{\partial z} + \frac{\partial w}{\partial r}) - δ_{11} \frac{\partial φ}{\partial r}] π_{1} \\ D_{θ} = e_{15} (\frac{\partial v}{\partial z} + \frac{\partial w}{r \partial θ}) - δ_{11} (\frac{1}{r} \frac{\partial φ}{\partial θ}) \\ D_{z} = e_{31} \frac{\partial u}{\partial r} + e_{31} (\frac{1}{r} \frac{\partial v}{\partial θ} + \frac{u}{r}) + e_{33} \frac{\partial w}{\partial z} - δ_{33} \frac{\partial φ}{\partial z} \end{array}

(7)

The Hamilton’s principle is used to discuss the motion equations of various engineering structures (Amir et al., 2020; Arshid et al., 2024c; Kaveh et al., 2024; Mousavi et al., 2021) according to kinetic energy and potential energy. In this paper, the DLPM is extended to study the 3D flexural vibration from 2D axisymmetric vibration. The flexural vibration control equation of FGP annular plate is as follows:

\begin{array}{l} \begin{array}{l} \frac{\partial T_{r r}}{\partial r} + \frac{1}{r} \frac{\partial T_{r θ}}{\partial θ} + \frac{\partial T_{r z}}{\partial z} + \frac{T_{r r} - T_{θ θ}}{r} = ρ \frac{\partial^{2} u}{\partial t^{2}} \\ \frac{\partial T_{r θ}}{\partial r} + \frac{1}{r} \frac{\partial T_{θ θ}}{\partial θ} + \frac{\partial T_{θ z}}{\partial z} + \frac{2 T_{r θ}}{r} = ρ \frac{\partial^{2} v}{\partial t^{2}} \\ \frac{\partial T_{r z}}{\partial r} + \frac{1}{r} \frac{\partial T_{θ z}}{\partial θ} + \frac{\partial T_{z z}}{\partial z} + \frac{T_{r z}}{r} = ρ \frac{\partial^{2} w}{\partial t^{2}} \end{array} \\ \frac{\partial D_{r}}{\partial r} + \frac{1}{r} \frac{\partial D_{θ}}{\partial θ} + \frac{\partial D_{z}}{\partial z} + \frac{D_{r}}{r} = 0 \end{array}

(8)

The harmonic solution expression is as follows:

\begin{array}{l} \begin{array}{l} u (r, θ, z, t) = U (r, z) e^{i (N θ - ω t)} \\ v (r, θ, z, t) = V (r, z) e^{i (N θ - ω t)} \\ w (r, θ, z, t) = W (r, z) e^{i (N θ - ω t)} \end{array} \\ φ (r, θ, z, t) = [V_{0} z / H + z (z - H) M (r, z)] e^{i (N θ - ω t)} \end{array}

(9)

3.2. Material properties

The Voigt mixing rule is used to describe the volume fraction and material properties:

P (z) = P_{1} + (P_{2} - P_{1}) V_{2} (z)

(10)

where

P_{i}

and

V_{i}

represent the material properties and volume fraction of the ith material, respectively.

The material constants of the FGP annular plate can be defined as a function of the variable z, which is then fitted as an expression of the sum of the power function:

C (z) = {\sum_{g = 0}^{s} {C_{i j}}^{(g)} (\frac{2 z - H}{H})}^{g} = \sum_{m = 0}^{s} {C_{i j}}^{(g)} {(t)}^{g}

(11)

3.3. Method of solutions

The following equations can be derived by substituting equations (6), (7), (9), and (11) into equation (8):

\begin{array}{l} \begin{array}{l} \begin{array}{l} \begin{array}{l} \begin{array}{l} {\sum_{g = 0}^{l} (\frac{2 z - H}{H})}^{g} {C_{11}^{(g)} r^{2} \frac{\partial U}{\partial r} \frac{\partial π_{1}}{\partial r} - C_{11}^{(g)} U + C_{11}^{(g)} r \frac{\partial U}{\partial r} π_{1} \\ + C_{11}^{(g)} r^{2} \frac{\partial^{2} U}{\partial r^{2}} π_{1} + C_{12}^{(g)} r U \frac{\partial π_{1}}{\partial r} + C_{44}^{(g)} r^{2} \frac{\partial U}{\partial z} \frac{\partial π_{3}}{\partial z} π_{1} \\ + C_{44}^{(g)} r^{2} \frac{\partial^{2} U}{\partial z^{2}} π_{1} π_{3} - N^{2} C_{66}^{(g)} U π_{1} + i N (- C_{11}^{(g)} V + C_{12}^{(g)} r V \frac{\partial π_{1}}{\partial r} \end{array} \\ + C_{12}^{(g)} r \frac{\partial V}{\partial r} π_{1} - C_{66}^{(g)} V π_{1} + C_{66}^{(g)} r \frac{\partial V}{\partial r} π_{1}) + C_{13}^{(g)} r^{2} \frac{\partial W}{\partial z} \frac{\partial π_{1}}{\partial r} \\ + C_{13}^{(g)} r^{2} \frac{\partial^{2} W}{\partial z \partial r} π_{1} + C_{44}^{(g)} r^{2} \frac{\partial W}{\partial r} \frac{\partial π_{3}}{\partial z} π_{1} + C_{44}^{(g)} r^{2} \frac{\partial^{2} W}{\partial r \partial z} π_{1} π_{3} \end{array} \\ + e_{31}^{(g)} r^{2} [(2 z - H) \frac{\partial M}{\partial r} + (z^{2} - z H) \frac{\partial^{2} M}{\partial z \partial r}] π_{1} \\ + e_{31}^{(g)} r^{2} [(2 z - H) M + (z^{2} - z H) \frac{\partial M}{\partial z}] \frac{\partial π_{1}}{\partial r} \end{array} \\ + e_{15}^{(g)} r^{2} [(2 z - H) \frac{\partial M}{\partial r} + (z^{2} - z H) \frac{\partial^{2} M}{\partial z \partial r}] π_{1} π_{3} \\ + e_{15}^{(g)} r^{2} (z^{2} - z H) \frac{\partial M}{\partial r} \frac{\partial π_{3}}{\partial z} π_{1}} \end{array} \\ + {\sum_{g = 1}^{l} \frac{2 g}{H} (\frac{2 z - H}{H})}^{g - 1} {C_{44}^{(g)} r^{2} \frac{\partial U}{\partial z} π_{3} π_{1} + C_{44}^{(g)} r^{2} \frac{\partial W}{\partial r} π_{3} π_{1} + e_{15}^{(g)} r^{2} (z^{2} - z H) \frac{\partial M}{\partial r} π_{3} π_{1}} = - {\sum_{g = 0}^{l} (\frac{2 z - H}{H})}^{g} ω^{2} r^{2} ρ^{(g)} U \end{array}

(12a)

\begin{array}{l} \begin{array}{l} \begin{array}{l} {\sum_{g = 0}^{l} (\frac{2 z - H}{H})}^{g} {i N (C_{11}^{(g)} U + C_{12}^{(g)} r \frac{\partial U}{\partial r} + C_{66}^{(g)} r U \frac{\partial π_{1}}{\partial r} \\ + C_{66}^{(g)} U π_{1} + C_{66}^{(g)} r \frac{\partial U}{\partial r} π_{1}) - N^{2} C_{11}^{(g)} V + C_{44}^{(g)} r^{2} \frac{\partial V}{\partial z} \frac{\partial π_{3}}{\partial z} \\ + C_{44}^{(g)} r^{2} \frac{\partial^{2} V}{\partial z^{2}} π_{3} + C_{66}^{(g)} r^{2} \frac{\partial V}{\partial r} \frac{\partial π_{1}}{\partial r} - C_{66}^{(g)} r V \frac{\partial π_{1}}{\partial r} - C_{66}^{(g)} V π_{1} \end{array} \\ + C_{66}^{(g)} r \frac{\partial V}{\partial r} π_{1} + C_{66}^{(g)} r^{2} \frac{\partial^{2} V}{\partial r^{2}} π_{1} + i N (C_{13}^{(g)} r \frac{\partial W}{\partial z} + C_{44}^{(g)} r \frac{\partial W}{\partial z} π_{3} \\ + C_{44}^{(g)} r W \frac{\partial π_{3}}{\partial z}) + i N {e_{31}^{(g)} r [(2 z - H) M + (z^{2} - z H) \frac{\partial M}{\partial z}] \end{array} \\ + e_{15}^{(g)} r [(2 z - H) M + (z^{2} - z H) \frac{\partial M}{\partial z}] π_{3} + e_{15}^{(g)} r (z^{2} - z H) M \frac{\partial π_{3}}{\partial z}}} \\ + {\sum_{g = 1}^{l} \frac{2 g}{H} (\frac{2 z - H}{H})}^{g - 1} {C_{44}^{(g)} r^{2} \frac{\partial V}{\partial z} π_{3} + i N C_{44}^{(g)} r W π_{3} + i N e_{15}^{(g)} r (z^{2} - z H) M π_{3}} = - {\sum_{g = 0}^{l} (\frac{2 z - H}{H})}^{g} ω^{2} r^{2} ρ^{(g)} V \end{array}

(12b)

\begin{array}{l} \begin{array}{l} \begin{array}{l} \begin{array}{l} \begin{array}{l} {\sum_{g = 0}^{l} (\frac{2 z - H}{H})}^{g} {C_{13}^{(g)} r^{2} \frac{\partial U}{\partial r} \frac{\partial π_{3}}{\partial z} + C_{13}^{(g)} r U \frac{\partial π_{3}}{\partial z} + C_{13}^{(g)} r \frac{\partial U}{\partial z} π_{3} \\ + C_{13}^{(g)} r^{2} \frac{\partial^{2} U}{\partial r \partial z} π_{3} + C_{44}^{(g)} r^{2} \frac{\partial U}{\partial z} \frac{\partial π_{1}}{\partial r} π_{3} + C_{44}^{(g)} r \frac{\partial U}{\partial z} π_{1} π_{3} + C_{44}^{(g)} r^{2} \frac{\partial^{2} U}{\partial z \partial r} π_{1} π_{3} \\ + i N (C_{44}^{(g)} r \frac{\partial V}{\partial z} π_{3} + C_{13}^{(g)} r \frac{\partial V}{\partial z} π_{3} + C_{13}^{(g)} r V \frac{\partial π_{3}}{\partial z}) + C_{33}^{(g)} r^{2} \frac{\partial W}{\partial z} \frac{\partial π_{3}}{\partial z} \end{array} \\ + C_{33}^{(g)} r^{2} \frac{\partial^{2} W}{\partial z^{2}} π_{3} + C_{44}^{(g)} r^{2} \frac{\partial W}{\partial r} \frac{\partial π_{1}}{\partial r} π_{3} - N^{2} C_{44}^{(g)} W π_{3} + C_{44}^{(g)} r \frac{\partial W}{\partial r} π_{1} π_{3} \\ + C_{44}^{(g)} r^{2} \frac{\partial^{2} W}{\partial r^{2}} π_{1} π_{3} + e_{15}^{(g)} r^{2} (z^{2} - z H) \frac{\partial^{2} M}{\partial r^{2}} π_{1} π_{3} + e_{15}^{(g)} r^{2} (z^{2} - z H) \frac{\partial M}{\partial r} \frac{\partial π_{1}}{\partial r} π_{3} \end{array} \\ - N^{2} e_{15}^{(g)} (z^{2} - z H) M π_{3} + e_{15}^{(g)} r (z^{2} - z H) \frac{\partial M}{\partial r} π_{1} π_{3} \\ + e_{33}^{(g)} r^{2} [2 M + (4 z - 2 H) \frac{\partial M}{\partial z} + (z^{2} - z H) \frac{\partial^{2} M}{\partial z^{2}}] π_{3} \end{array} \\ + e_{33}^{(g)} r^{2} [(2 z - H) M + (z^{2} - z H) \frac{\partial M}{\partial z}] \frac{\partial π_{3}}{\partial z}} \\ + {\sum_{g = 1}^{l} \frac{2 g}{H} (\frac{2 z - H}{H})}^{g - 1} {C_{13}^{(g)} r^{2} \frac{\partial U}{\partial r} π_{3} + C_{13}^{(g)} r U π_{3} + i N C_{13}^{(g)} r V π_{3} \end{array} \\ + C_{33}^{(g)} r^{2} \frac{\partial W}{\partial z} π_{3} + e_{33}^{(g)} r^{2} [(2 z - H) M + (z^{2} - z H) \frac{\partial M}{\partial z}] π_{3}} = - {\sum_{g = 0}^{l} (\frac{2 z - H}{H})}^{g} ω^{2} r^{2} ρ^{(g)} W \end{array}

(12c)

\begin{array}{l} \begin{array}{l} \begin{array}{l} \begin{array}{l} {\sum_{g = 0}^{l} (\frac{2 z - H}{H})}^{g} {e_{15}^{(g)} r^{2} \frac{\partial U}{\partial z} \frac{\partial π_{1}}{\partial r} + e_{15}^{(g)} r \frac{\partial U}{\partial z} π_{1} + e_{15}^{(g)} r^{2} \frac{\partial^{2} U}{\partial z \partial r} π_{1} \\ + e_{31}^{(g)} r \frac{\partial U}{\partial z} + e_{31}^{(g)} r^{2} \frac{\partial^{2} U}{\partial r \partial z} + i N (e_{15}^{(g)} r \frac{\partial V}{\partial z} + e_{31}^{(g)} r \frac{\partial V}{\partial z}) \\ + e_{15}^{(g)} r^{2} \frac{\partial W}{\partial r} \frac{\partial π_{1}}{\partial r} - N^{2} e_{15}^{(g)} W + e_{15}^{(g)} r \frac{\partial W}{\partial r} π_{1} + e_{15}^{(g)} r^{2} \frac{\partial^{2} W}{\partial r^{2}} π_{1} \end{array} \\ + e_{33}^{(g)} r^{2} \frac{\partial^{2} W}{\partial z^{2}} - δ_{11}^{(g)} r^{2} (z^{2} - z H) \frac{\partial M}{\partial r} \frac{\partial π_{1}}{\partial r} + N^{2} δ_{11}^{(g)} (z^{2} - z H) M \\ - δ_{11}^{(g)} r (z^{2} - z H) \frac{\partial M}{\partial r} π_{1} - δ_{11}^{(g)} r^{2} (z^{2} - z H) \frac{\partial^{2} M}{\partial r^{2}} π_{1} \end{array} \\ - δ_{33}^{(g)} r^{2} [2 M + (4 z - 2 H) \frac{\partial M}{\partial z} + (z^{2} - z H) \frac{\partial^{2} M}{\partial z^{2}}]} \\ + {\sum_{g = 1}^{l} \frac{2 g}{H} (\frac{2 z - H}{H})}^{g - 1} {e_{31}^{(g)} r^{2} \frac{\partial U}{\partial r} + e_{31}^{(g)} r U + i N e_{31}^{(g)} r V \end{array} \\ + e_{33}^{(g)} r^{2} \frac{\partial W}{\partial z} - δ_{33}^{(g)} r^{2} [(2 z - H) M + (z^{2} - z H) \frac{\partial M}{\partial z}]} = 0 \end{array}

(12d)

The displacement amplitude in equations (12a)–(12d) can be expanded into the form of double Legendre orthogonal polynomial series:

\begin{array}{l} U (r, z) = \sum_{m, n = 0}^{\infty} p_{m, n}^{1} Q_{m} (r) K_{n} (z) \\ V (r, z) = \sum_{m, n = 0}^{\infty} p_{m, n}^{2} Q_{m} (r) K_{n} (z) \\ W (r, z) = \sum_{m, n = 0}^{\infty} p_{m, n}^{3} Q_{m} (r) K_{n} (z) M (r, z) = \sum_{m, n = 0}^{\infty} p_{m, n}^{4} Q_{m} (r) K_{n} (z) \end{array}

(13)

where

p_{m, n}^{1}

(i = 1, 2, 3,4) is the unknown coefficient of polynomials.

The series of $Q_{m} (r)$ and $K_{n} (z)$ are defined as:

Q_{m} (r) = \sqrt{\frac{2 m + 1}{a - b}} p_{m} (\frac{2 r - (a + b)}{a - b}), K_{n} (z) = \sqrt{\frac{2 n + 1}{H}} p_{n} (\frac{2 z - H}{H})

(14)

where

Q_{m} (r)

and

K_{n} (z)

are Legendre polynomials of mth and nth, respectively. When m and n are set as M and N, respectively, the sum of polynomials converges, and the terms of higher order are small enough to be ignored.

Multiply both sides of equations (12a)–(12d) by $Q_{m 1} (r) K_{n 1} (z)$ . Here, m1 and n1 are taken from 0 to M and from 0 to N, respectively. Then integrate the variable r from b to a, and the variable z from 0 to H. Based on the orthogonality of Legendre polynomials, the following equality can be obtained:

\begin{array}{l} A_{11}^{m 1, n 1, m, n} p_{m, n}^{1} + A_{12}^{m 1, n 1, m, n} p_{m, n}^{2} + A_{13}^{m 1, n 1, m, n} p_{m, n}^{3} + A_{14}^{m 1, n 1, m, n} p_{m, n}^{4} = - ω^{2} M_{11}^{m 1, n 1, m, n} p_{m, n}^{1} \\ A_{21}^{m 1, n 1, m, n} p_{m, n}^{1} + A_{22}^{m 1, n 1, m, n} p_{m, n}^{2} + A_{23}^{m 1, n 1, m, n} p_{m, n}^{3} + A_{24}^{m 1, n 1, m, n} p_{m, n}^{4} = - ω^{2} M_{22}^{m 1, n 1, m, n} p_{m, n}^{2} \\ A_{31}^{m 1, n 1, m, n} p_{m, n}^{1} + A_{32}^{m 1, n 1, m, n} p_{m, n}^{2} + A_{33}^{m 1, n 1, m, n} p_{m, n}^{3} + A_{34}^{m 1, n 1, m, n} p_{m, n}^{4} = - ω^{2} M_{33}^{m 1, n 1, m, n} p_{m, n}^{3} \\ A_{41}^{m 1, n 1, m, n} p_{m, n}^{1} + A_{42}^{m 1, n 1, m, n} p_{m, n}^{2} + A_{43}^{m 1, n 1, m, n} p_{m, n}^{3} + A_{44}^{m 1, n 1, m, n} p_{m, n}^{4} = 0 \end{array}

(15)

The following expression is obtained by substituting the $p_{m, n}^{4}$ into equation (15):

B p = - ω^{2} M p

(16)

where:

B = [\begin{array}{l} B_{11}^{m 1, n 1, m, n} & B_{12}^{m 1, n 1, m, n} & B_{13}^{m 1, n 1, m, n} \\ B_{21}^{m 1, n 1, m, n} & B_{22}^{m 1, n 1, m, n} & B_{23}^{m 1, n 1, m, n} \\ B_{31}^{m 1, n 1, m, n} & B_{32}^{m 1, n 1, m, n} & B_{33}^{m 1, n 1, m, n} \end{array}]

(17a)

M = [\begin{array}{c} M_{11}^{m 1, n 1, m, n} & 0 & 0 \\ 0 & M_{22}^{m 1, n 1, m, n} & 0 \\ 0 & 0 & M_{33}^{m 1, n 1, m, n} \end{array}]

(17b)

p = {[\begin{array}{l} p_{m, n}^{1} & p_{m, n}^{2} & p_{m, n}^{3} \end{array}]}^{T}

(17c)

The expression of matrix A and the relationship between matrix A and B are shown in Appendix A.

In this paper, the traditional analytical integral (Wang et al., 2021b) is extended, and the improved analytical integral method is applied to the study of the flexural vibration of FGP annular plate, which greatly reduces the computational complexity and shortens the calculation time. The expression of the analytic integral is shown in Appendix B.

4. Numerical results and analyses

In this paper, the flexural, longitudinal, and torsional vibration characteristics of FGP annular plates with PZT4 material on the top surface and ALN material on the bottom surface are studied. The material parameters are shown in Table 1.

Table 1.

Material properties.

Material	$C_{11}$	$C_{12}$	$C_{13}$	$C_{33}$	$C_{44}$	$C_{66}$	$e_{31}$	$e_{33}$	$e_{15}$	$δ_{11}$	$δ_{33}$	$ρ$
PZT-4	139	77.8	74.3	115	25.6	30.6	−5.2	15.1	12.7	646	562	7.5
AlN	345	125	120	395	118	110	−0.58	1.55	−0.48	8.0	9.5	3.26
PSI-5A4E	80.4	30.02	21.05	60.03	15.94	25.19	−12.77	15.41	8.765	1112	507.5	7.8
PIC-151	107.6	63.13	63.86	100.4	19.62	22.235	−9.52	15.14	11.97	983.6794	818.995	7.8

Unit: $C_{i j} (GPa)$ , $e_{i j} (C / m^{2})$ , $δ_{i j} (1 0^{- 11} F / m)$ , $ρ (1 0^{3} kg / m^{3})$ .

4.1. Convergence analysis

In this section, the natural frequencies of flexural, longitudinal, and torsional modes of FGP annular plates under different cut-off orders M and N are solved. A physical quantity that can represent the relative error is defined as $Δ = \frac{(c_{(M + 2) (N + 2)} - c_{M N})}{c_{M N}}$ . When the relative error is less than 0.001, it indicates that the model has converged. The geometric parameters of the FGP annular plate are considered as the height H = 0.6 mm, the outer radius a = 15 mm, and the inner radius b = 7.5 mm.

Table 2 shows that the flexural, longitudinal, and torsional modes of the FGP annular plates reach the convergence state when the cut-off order is M = N = 11. The convergence of torsional mode is the best, followed by longitudinal mode. The coupling vibration characteristics of longitudinal mode and torsional mode are complex, so the convergence of flexural mode is slightly worse than that of longitudinal mode.

Table 2.

Convergence of natural frequencies of flexural, longitudinal and torsional modes for the FF FGP annular plates.

Modes		M = N = 5	M = N = 7	M = N = 9	M = N = 11	M = N = 13
Flexural	F (2, 0)	3.07815	3.07175	3.06814	3.06696	3.0667
	$Δ$		−0.00208	−0.00118	−0.00038	−8.5E − 05
	F (0, 1)	6.81981	6.80176	6.80123	6.80102	6.80094
	$Δ$		−0.00265	−7.8E − 05	−3.1E − 05	−1.2E − 05
	F (3, 0)	8.23118	12.1934	2.14246	2.14126	2.14074
	$Δ$		0.481367	−0.82429	−0.00056	−0.00024
	F (1, 1)	12.3023	12.1934	12.1452	12.1293	12.1259
	$Δ$		−0.00885	−0.00395	−0.00131	−0.00028
	F (4, 0)	15.1501	15.1158	15.0997	15.0947	15.0937
	$Δ$		−0.00226	−0.00107	−0.00033	−6.6E − 05
Longitudinal	L (0, 0)	86.351	86.3496	86.3496	86.3496	86.3496
	$Δ$		−1.6E − 05	0	0	0
	L (0, 1)	423.588	423.588	423.587	423.586	423.586
	$Δ$		0	−2.4E − 06	−2.4E − 06	0
	L (0, 2)	833.891	828.03	827.972	827.966	827.963
	$Δ$		−0.00703	−7E − 05	−7.2E − 06	−3.6E − 06
Torsional	T (0, 0)	261.045	261.03	261.03	261.03	261.03
	$Δ$		−5.7E − 05	0	0	0
	T (0, 1)	493.35	491.001	490.979	490.979	490.979
	$Δ$		−0.00476	−4.5E − 05	0	0
	T (0, 2)	970.319	731.134	723.805	723.716	723.715
	$Δ$		−0.2465	−0.01002	−0.00012	−1.4E − 06

4.2. Validation analyses

Due to the lack of analysis of the flexural vibration of FGP annular plates, the piezoelectric circular plate is analyzed (Section 4.2.1). In addition, the natural frequencies of flexural, longitudinal, and torsional modes (Section 4.2.2) are solved for the FF FGP annular plates by theoretical and finite element methods, respectively.

4.2.1. Flexural vibration of piezoelectric circular plate

In this paper, the natural frequency of flexural mode is studied for the FF piezoelectric circular plate based on 3D elastic theory. The piezoelectric circular plate is made of piezoelectric material PSI-5A4E, and the material constants are shown in Table 1. Table 3 shows that the error1 between the 3D results and Mindlin’s model results (Huang et al., 2021) is less than 0.27%, and the error2 between the 3D results and Kirchhoff’s model results (Huang et al., 2021) is less than 1.25%, which shows the correctness of the proposed method. In addition, from the size of the two error values, the Mindlin’s model has higher accuracy than the Kirchhoff’s model.

Table 3.

Comparison of natural frequencies of flexural mode for the FF piezoelectric circular plates (kHz).

R/H	Modes	Results^a			Present 3D model results	Error (%)
R/H	Modes	FEM	Mindlin’s model	Kirchhoff’s model	Present 3D model results	Error1	Error 2
40	(2, 0)	0.662	0.662	0.664	0.662	0.0671	0.3694
	(0, 1)	1.406	1.407	1.409	1.407	0.0114	0.1535
	(3, 0)	1.560	1.560	1.570	1.560	0.0019	0.6430
	(4, 0)	2.766	2.768	2.792	2.766	0.0680	0.9356
	(1, 1)	3.069	3.072	3.083	3.064	0.2627	0.6217
	(5, 0)	4.272	4.275	4.325	4.272	0.0787	1.2492
	(2, 1)	5.172	5.178	5.214	5.176	0.0346	0.7301
	(0, 2)	5.729	5.744	5.780	5.741	0.0603	0.6874

^aNumbers represent the results proposed by Huang et al. (2021).

4.2.2. Vibration characteristics of FGP annular plates

In this paper, the flexural, longitudinal and torsional vibrations of FGP annular plates with gradient indices g = 1, 3, and 5 are studied based on 3D elastic theory. The geometric parameters of the FGP annular plate are considered as the height H = 1 mm, the outer radius a = 20 mm, and the inner radius b = 10 mm. The natural frequencies of flexural, longitudinal, and torsional modes are solved for the FF FGP annular plates by FEM and DLPM, as shown in Table 4. The results are in good agreement with the FEM results, and the theoretical method has the advantages of fast solving speed, high precision, and a low requirement for basic knowledge of finite elements. Therefore, the DLPM proposed in this paper is an effective method to solve the natural frequency efficiently in engineering applications.

Table 4.

Comparison of natural frequencies of flexural, longitudinal, and torsional modes for the FF FGP annular plates (kHz).

Modes		Results	g = 1	g = 3	g = 5
Flexural	F (2, 0)	Present results	2.8687	3.5253	3.8295
		FEM	2.8729	3.5301	3.8428
		Error (%)	0.1478	0.1350	0.3468
	F (0, 1)	Present results	6.3603	7.7641	8.4156
		FEM	6.3618	7.7653	8.4332
		Error (%)	0.0242	0.0157	0.2090
	F (3, 0)	Present results	7.6667	9.4180	10.2289
		FEM	7.6793	9.4324	10.2650
		Error (%)	0.1645	0.1531	0.3529
	F (1, 1)	Present results	11.2766	13.8638	15.0555
		FEM	11.3470	13.9470	15.1790
		Error (%)	0.6243	0.6001	0.8203
	F (4, 0)	Present results	14.0847	17.3006	18.7885
		FEM	14.1160	17.3370	18.8650
		Error (%)	0.2222	0.2104	0.4072
Longitudinal	L (0, 0)	Present results	64.7619	81.3338	87.8085
		FEM	65.5830	81.7280	87.9960
		Error (%)	1.2679	0.4847	0.2135
	L (0, 1)	Present results	317.637	397.816	429.238
		FEM	324.3100	402.8800	432.4500
		Error (%)	2.1008	1.2730	0.7483
	L (0, 2)	Present results	620.543	776.949	838.371
		FEM	620.62	776.08	837.03
		Error (%)	0.0124	−0.1119	−0.1600
Torsional	T (0, 0)	Present results	195.641	247.353	267.572
		FEM	195.67	247.37	267.58
		Error (%)	0.0148	0.0069	0.0030
	T (0, 1)	Present results	367.356	465.002	503.647
		FEM	367.57	465.15	503.78
		Error (%)	0.0583	0.0318	0.0264
	T (0, 2)	Present results	539.955	684.762	743.232
		FEM	540.01	685.46	743.88
		Error (%)	0.0102	0.1019	0.0872

4.3. Computational efficiency

The time spent by the traditional method and the analytical method under different cut-off orders M and N is shown in Table 5. The geometric parameters of the piezoelectric circular plate are considered as H = 1 mm and a = 15 mm.

Table 5.

Comparison of the calculation time of the two methods.

M = N	Mode (2, 0)	Mode (2, 1)	Methods		Efficiency (%)
M = N	Mode (2, 0)	Mode (2, 1)	Analytical (s)	Traditional (s)	Efficiency (%)
2	2.75138	43.5638	0.062	166.422	99.963
3	3.210	28.1884	0.359	1880.44	99.981
4	3.20291	26.0644	1.11	9282.27	99.988
5	3.19845	25.2354	3.218	33,644.6	99.990
6	3.19506	25.1129	6.547	86,138.3	99.992
7	3.19268	25.0891	13.656	207,515	99.993
Results^a	3.19	25.087

^aNumbers represent the results proposed by Liu et al. (2008).

The natural frequency of the flexural mode gradually converges with the increase of M and N, and the theoretical results after convergence are in good agreement with the results of Liu et al. (2008). The calculation efficiency of the analytical method can be improved by more than 99%.

4.4. Effects of height-to-diameter ratio on the vibration characteristics

In this section, the effects of the height-to-diameter ratio H/2a on the non-dimensional natural frequencies of the flexural, longitudinal, and torsional modes are studied for the FF and CC FGP annular plates, as shown in Figure 2. The geometric parameters of the FGP annular plate with the gradient index g = 1 are considered as the outer radius a = 20 mm and the inner radius b = 10 mm. The non-dimensional natural frequency is defined as $Ω_{1} = H ω \sqrt{\frac{ρ_{P Z T 4}}{C_{11 P Z T 4}}}$ .

Figure 2.

Variation of non-dimensional natural frequency Ω₁ with the height-to-diameter ratio H/2a: (a) flexural modes, (b) longitudinal modes, and (c) torsional modes.

Figure 2 demonstrates that Ω₁ of flexural mode increases as H/2a increases under the FF boundary condition. Ω₁ of the longitudinal mode increases first and then tends to be stable as H/2a increases. When H/2a exceeds 1.4, 5, and 9, Ω₁ of modes L (0, 0), L (0, 1), and L (0, 2) remain constant at 4.6, 15.5, and 25.6, respectively. When H/2a exceeds 0.25, Ω₁ of mode T (0, 0) remains constant at 2.53. When H/2a is in the range of (0.13, 0.25) or greater than 0.6, Ω₁ of mode T (0, 1) remains constant at 2.53 and 4.96, respectively. When H/2a is in the range of (0.1, 0.13), (0.35, 0.6) or greater than 1, Ω₁ of mode T (0, 2) remains constant at 2.53, 4.96, and 7.41, respectively.

Ω₁ of the flexural, longitudinal and torsional modes under the CC boundary condition increases as H/2a increases. When H/2a = 0.25, Ω₁ of modes L (0, 0) and L (0, 1) are equal, and Ω₁ of modes T (0, 1) and T (0, 2) are also equal.

4.5. Effects of radius ratio a/b on the vibration characteristics

4.5.1. Flexural modes

The geometric parameters are the inner radius b = 10 mm and the heights H = 2, 6, and 12 mm, respectively. The non-dimensional natural frequency is defined as $Ω_{2} = a ω \sqrt{\frac{ρ_{P Z T 4}}{C_{11 P Z T 4}}}$ . The influence of a on Ω₂ for flexural mode is shown in Figure 3. Ω₂ initially increases and then decreases as a/b increases, indicating that Ω₂ reaches its maximum value in this range for the FF FGP annular plate. When H = 2, 6, and 12 mm, the maximums of Ω₂ occur at a/b = 1.2, 1.6, and 2.2, respectively. When a reaches the condition where the radial wall thickness a − b is equal to the height H, Ω₂ of flexural vibration reaches its maximum value.

Figure 3.

Variation of non-dimensional natural frequency Ω₂ of the flexural mode with the radius ratio a/b: (a) H = 2 mm, (b) H = 6 mm, and (c) H = 12 mm.

The increase of the outer radius a has a weakening effect on Ω₂ of flexural vibration for the CC FGP annular plate, and the weakening effect presents a trend of first sharp and then gentle as a/b increases.

4.5.2. Longitudinal modes

The variation of the longitudinal as the radius ratio a/b increases for the FF and CC FGP annular plates is shown in Figure 4. The longitudinal mode for the FF FGP annular plate first increases, then decreases, and finally stabilizes as the radius ratio a/b increases. When the heights H = 2, 6, and 12 mm, Ω₂ of modes L (0, 1) and L (0, 2) are equal at a/b = 1.2, 1.6, and 2.2, respectively. When a reaches the condition where the radial wall thickness a-b and height H are equal, Ω₂ of the modes L (0, 1) and L (0, 2) are approximately equal at that point.

Figure 4.

Variation of non-dimensional natural frequency Ω₂ of the longitudinal mode with the radius ratio a/b: (a) H = 2 mm, (b) H = 6 mm, and (c) H = 12 mm.

Ω₂ of longitudinal mode decreases sharply as a/b increases in the range of a − b < H for the CC FGP annular plate. Ω₂ of longitudinal mode changes slightly as a/b increases in the range of a − b > H.

4.5.3. Torsional modes

The influence of the radius ratio a/b on the torsional mode with the heights H = 2, 6, and 12 mm is shown in Figure 5. Ω₂ of modes T (0, 0) and T (0, 1) are approximately equal at a/b = 1.2, 1.6, and 2.2, and Ω₂ of modes T (0, 1) and T (0, 2) are very close at a/b = 1.4, 2, and 3 for the FF FGP annular plate.

Figure 5.

Variation of non-dimensional natural frequency Ω₂ of the torsional mode with the radius ratio a/b: (a) H = 2 mm, (b) H = 6 mm, and (c) H = 12 mm.

Ω₂ of the torsional mode decreases significantly as a/b increases in the range of a − b < H for the CC FGP annular plate; Ω₂ of the torsional mode remains stable as a/b increases in the range of a − b > H. When a − b = H, Ω₂ of modes T (0, 1) and T (0, 2) are approximately equal.

4.6. Effects of inner radius b on the vibration characteristics

4.6.1. Flexural modes

The influence of inner radius b on the non-dimensional natural frequencies of flexural modes is shown in Figure 6. The geometric parameters are considered as the outer radius a = 40 mm and the heights H = 10, 13 and 16 mm, respectively. The non-dimensional natural frequency is defined as $Ω_{3} = b ω \sqrt{\frac{ρ_{P Z T 4}}{C_{11 P Z T 4}}}$ . Figure 6 shows that Ω₃ increases first and then decreases as the inner radius b increases for the FF FGP annular plate, indicating that Ω₃ has a maximum value in this range. When the heights H = 10, 13, and 16 mm, the maximum value of Ω₃ for the flexural mode occurs at b = 30, 27, and 24 mm. In summary, Ω₃ of the flexural mode reaches its maximum value at a − b = H. It can be observed from Figures 3 and 6 that when a − b equals H, the coupling effect between longitudinal and torsional vibrations is enhanced, resulting in a maximum value of the natural frequency of flexural vibration under this condition.

Figure 6.

Variation of non-dimensional natural frequency Ω₃ of the flexural mode with the inner radius b: (a) H = 10 mm, (b) H = 13 mm, and (c) H = 16 mm.

4.6.2. Longitudinal modes

The influence of inner radius b on Ω₃ of longitudinal modes for the FF and CC FGP annular plates is shown in Figure 7. When the heights H = 10, 13, and 16 mm, Ω₃ of the modes L (0, 1) and L (0, 2) are nearly identical at b = 30, 27, and 24 mm for the FF FGP annular plate. Ω₃ of modes L (0, 1) and L (0, 2) are approximately equal at a – b = H. The influence of inner radius b on Ω₃ of mode L (0, 0) is very weak. Figures 4 and 7 indicate that when a − b equals H, the geometric symmetry is enhanced, the adjacent modes of longitudinal vibration show similar stiffness, and the values of the natural frequencies are close.

Figure 7.

Variation of non-dimensional natural frequency Ω₃ of the longitudinal mode with the inner radius b: (a) H = 10 mm, (b) H = 13 mm, and (c) H = 16 mm.

Ω₃ of the longitudinal mode increases as b increases for the CC FGP annular plate. Ω₃ of the longitudinal mode increases slightly as b increases in the range of a − b > H. Ω₃ of the longitudinal mode increases sharply as b increases in the range of a − b < H.

4.6.3. Torsional modes

The influence of the inner radius b on Ω₃ for torsional mode with outer radius a = 40 mm is shown in Figure 8. When the heights H = 10, 13, and 16 mm, Ω₃ of modes T (0, 0) and T (0, 1) are nearly identical at inner radius b = 30, 27, and 24 mm for the FF FGP annular plate, respectively. When a-b = H, the Ω₃ of modes T (0, 0) and T (0, 1) are nearly identical. Similarly, Ω₃ of modes T (0, 1) and T (0, 2) are nearly identical when a − b = H/2. It can be observed from Figures 5 and 8 that when a − b equals H, the geometric symmetry is enhanced, the adjacent modes of torsional vibration exhibit similar stiffness, and the values of the natural frequencies are close.

Figure 8.

Variation of non-dimensional natural frequency Ω₃ of the torsional mode with the inner radius b: (a) H = 10 mm, (b) H = 13 mm, and (c) H = 16 mm.

Ω₃ of the torsional mode increases slightly as b increases in the range of a − b > H for the CC FGP annular plate. Ω₃ of the torsional mode increases sharply as b increases in the range of a − b < H. Ω₃ of the modes T (0, 1) and T (0, 2) are nearly identical at a – b = H.

4.7. Mode shape

The analysis of vibration modes is an important way to understand the flexural, longitudinal and torsional modes of the FGP annular plate. In this section, the vibration modes at the interface of FGP annular plate are analyzed.

The geometric relationship of the FGP annular plate is considered as 2a/H = 20, where the outer radius a = 30 mm and the inner radius b = 6 mm. The radial (u), circumferential (v), and axial (w) displacements of modes F (2, 0) under FF and CC boundary conditions are shown in Figure 9.

Figure 9.

Mode shapes of mode F (2, 0) of FGP annular plate at 2a/H = 20: (a) u displacement, (b) v displacement, and (c) w displacement.

5. Conclusion

The flexural, longitudinal, and torsional vibrations of FGP annular plates exhibit distinct advantages and disadvantages in terms of sensitivity, stability, and precision of piezoelectric elements. There exists a close correlation between vibration characteristics and geometric features. In this paper, the double Legendre polynomial method (DLPM) is extended to study the flexural vibration of FGP annular plates from axisymmetric vibration. In addition, the influence of the relationship between radial wall thickness a-b and axial height H on the flexural, longitudinal and torsional vibrations is revealed for the first time. The main conclusions are as follows:

(1) The DLPM is extended for the first time to study the flexural vibration of FGP annular plates from axisymmetric vibration.

(2) Under the FF boundary condition, when the outer radius a and the inner radius b reach the condition where the radial wall thickness a-b equals the height H, the non-dimensional natural frequency of flexural vibration attains its maximum value.

(3) Under the FF boundary condition, when the outer radius a and the inner radius b respectively reach the condition where the radial wall thickness a-b equals the height H, the non-dimensional natural frequencies of the modes L (0, 1) and L (0, 2) are approximately equal, and the non-dimensional natural frequencies of the modes T (0, 0) and T (0, 1) are also approximately equal.

(4) Under the CC boundary condition, when the outer radius a and inner radius b reach the condition where the radial wall thickness a-b is less than the height H, the non-dimensional natural frequencies of the longitudinal mode and the torsional mode decrease sharply as the radial wall thickness a-b increases.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors gratefully acknowledge the support by the Henan University Science and Technology Innovation Team Support Plan (No. 23IRTSTHN016), Natural Science Foundation of Henan Province (242300420248) and Henan Polytechnic University double first-class discipline creation project (AQ20240744).

ORCID iDs

Kangle Han

Hongmei Zhou

Jiangong Yu

Appendix

References

Aabid

Parveez

Raheman

, et al. (2021) A review of piezoelectric material-based structural control and health monitoring techniques for engineering structures: challenges and opportunities. Actuators 10(5): 101.

Alhaifi

Khorshidvand

Al-Masoudy

, et al. (2023) A shooting method for buckling and post-buckling analyses of FGSP circular plates considering various patterns of pores’ placement. Structural Engineering & Mechanics 85(3): 419–432.

Amir

Arshid

Maraghi

(2020) Free vibration analysis of magneto-rheological smart annular three-layered plates subjected to magnetic field in viscoelastic medium. Smart Structures and Systems 25(5): 581–592.

Arshid

Amir

Loghman

(2023a) On the vibrations of FG GNPs-RPN annular plates with piezoelectric/metallic coatings on Kerr elastic substrate considering size dependency and surface stress effects. Acta Mechanica 234(9): 4035–4076.

Arshid

Amir

Loghman

(2023b) Thermoelastic vibration characteristics of asymmetric annular porous reinforced with nano-fillers microplates embedded in an elastic medium: CNTs Vs. GNPs. Archives of Civil and Mechanical Engineering 23(2): 100.

Arshid

Amir

Loghman

(2024a) Aero-hygro-thermoelastic size-dependent analysis of NCMF-reinforced GNPs sector microplates located between piezoelectric patches in supersonic flow considering surface stress effects. Mechanics Based Design of Structures and Machines 52(9): 6911–6972.

Arshid

Amir

Loghman

, et al. (2024b) Aerodynamic stability and free vibration of FGP-reinforced nano-fillers annular sector microplates exposed to supersonic flow. Thin-Walled Structures 197: 111610.

Arshid

Azimi

Moradi

, et al. (2024c) Mathematical solution for vibrational response of shear and normal deformable advanced metal foam CMBs treated with nanocomposite actuators. International Journal of Structural Stability and Dynamics 2550118. DOI: 10.1142/S0219455425501184.

Arshid

Maraghi

Civalek

(2025) Variable-thickness higher-order sandwich beams with FG cellular core and CNT-RC patches: vibrational analysis in thermal environment. Archive of Applied Mechanics 95(1): 2–24.

10.

Babaei

Zavari

Kaveh

, et al. (2024) Dynamic response of advanced lightweight porous plates integrated with nanocomposite face sheets resting on elastic substrate. International Journal of Structural Stability and Dynamics 2550132. DOI: 10.1142/S0219455425501329.

11.

Belardinelli

Souza

SNFD

Verlinden

, et al. (2020) Second flexural and torsional modes of vibration in suspended microfluidic resonator for liquid density measurements. Journal of Micromechanics and Microengineering 30(5): 055003.

12.

Birman

Byrd

(2007) Modeling and analysis of functionally graded materials and structures. American Society of Mechanical Engineers 60(05): 195–216.

13.

Cao

Zhu

, et al. (2021) Elastic bound state in the continuum with perfect mode conversion. Journal of the Mechanics and Physics of Solids 154: 104502.

14.

Datta

Hunsinger

(1978) Analysis of surface waves using orthogonal functions. Journal of Applied Physics 49(2): 475–479.

15.

Feri

Krommer

Alibeigloo

(2021) Three-dimensional static analysis of a viscoelastic rectangular functionally graded material plate embedded between piezoelectric sensor and actuator layers. Mechanics Based Design of Structures and Machines 51(7): 3843–3867.

16.

Ghemari

Belkhiri

Saad

(2023) Improvement of the relative sensitivity for obtaining a high performance piezoelectric sensor. IEEE Instrumentation and Measurement Magazine 26(4): 49–56.

17.

Givi

Ghorbanpour Arani

Khoddami Maraghi

, et al. (2024) Free vibration and supersonic flutter analyses of a sandwich cylindrical shell with CNT-reinforced honeycomb core integrated with piezoelectric layers. Mechanics Based Design of Structures and Machines 1–29. DOI: 10.1080/15397734.2024.2422413.

18.

Guru

Jain

(2020) Free axisymmetric vibrations of composite annular sandwich plates by higher-order theory using Chebyshev collocation technique. Thin-Walled Structures 155: 106823.

19.

Heshmati

Jalali

(2019) Effect of radially graded porosity on the free vibration behavior of circular and annular sandwich plates. European Journal of Mechanics - A: Solids 74: 417–430.

20.

Huan

Chen

(2019) A practical omni-directional SH wave transducer for structural health monitoring based on two thickness-poled piezoelectric half-rings. Ultrasonics 94: 342–349.

21.

Huang

C-H

C-C

Lin

Y-C

(2005) Theoretical, numerical, and experimental investigation on resonant vibrations of piezoceramic annular disks. IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control 52(8): 1204–1216.

22.

Huang

C-H

C-C

Y-C

, et al. (2021) Theoretical analysis of transverse and planar vibrations for the piezoceramic disk based on mindlin plate theory. Applied Mathematical Modelling 97: 568–587.

23.

Jian

Tang

, et al. (2023) Analytical and experimental study of a metamaterial beam with grading piezoelectric transducers for vibration attenuation band widening. Engineering Structures 275: 115091.

24.

Kaveh

Babaei

Zavari

, et al. (2024) Vibrational response of a sandwich microplate considering the impact of flexoelectricity and based on a novel porous-FGM formulation. Mechanics Based Design of Structures and Machines 52: 9122–9143.

25.

Khalfi

Naciri

Raghib

, et al. (2023) Axisymmetric free vibration modeling of a functionally graded piezoelectric resonator by a double Legendre polynomial method. Acta Mechanica 235(2): 615–631.

26.

Khoddami Maraghi

Arshid

(2024) On the vibrational behavior of variable thickness FG porous beams with graphene-reinforced nanocomposite facesheets. Acta Mechanica 235: 5161–5185.

27.

Khoddami Maraghi

Amir

Arshid

(2024) On the natural frequencies of smart circular plates with magnetorheological fluid core embedded between magnetostrictive patches on Kerr elastic substance. Mechanics Based Design of Structures and Machines 52(3): 1651–1668.

28.

Liu

C-F

Chen

T-J

Chen

Y-J

(2008) A modified axisymmetric finite element for the 3-D vibration analysis of piezoelectric laminated circular and annular plates. Journal of Sound and Vibration 309(3–5): 794–804.

29.

Liu

Sun

Zhao

(2021) Free vibration and transmission response analysis for torsional vibration of circular annular plate. Iranian Journal of Science and Technology, Transactions of Mechanical Engineering 45(3): 631–638.

30.

Liu

Zhang

, et al. (2024) Size parameter calibration of nonlocal strain gradient theory based on molecular dynamics simulation of guided wave propagation in aluminum plates. Thin-Walled Structures 198: 111659.

31.

Mousavi

Amir

Jafari

, et al. (2021) Analytical solution for analyzing initial curvature effect on vibrational behavior of PM beams integrated with FGP layers based on trigonometric theories. Advances in nano research 10(3): 235–251.

32.

Pal

Das

(2020) Free vibration behavior of rotating bidirectional functionally-graded micro-disk for flexural and torsional modes in thermal environment. International Journal of Mechanical Sciences 179: 105635.

33.

Panda

Hajra

Mistewicz

, et al. (2022) Piezoelectric energy harvesting systems for biomedical applications. Nano Energy 100: 107514.

34.

Piao

Yang

Kweun

, et al. (2020) Ultrasonic flow measurement using a high-efficiency longitudinal-to-shear wave mode-converting meta-slab wedge. Sensors and Actuators A: Physical 310: 112080.

35.

Rasooli Jazi

Amir

Arshid

(2024) Vibration analysis of asymmetric sandwich rotating FG porous discs coated with agglomerated nanocomposite facesheets. Archives of Civil and Mechanical Engineering 24(4): 201.

36.

Roshanbakhsh

Tavakkoli

Navayi Neya

(2020) Free vibration of functionally graded thick circular plates: an exact and three-dimensional solution. International Journal of Mechanical Sciences 188: 105967.

37.

Sezer

Koç

(2021) A comprehensive review on the state-of-the-art of piezoelectric energy harvesting. Nano Energy 80: 105567.

38.

Sharma

Ahlawat

(2024) Free axisymmetric vibrations of functionally graded material annular plates via DTM. International Journal of Structural Stability and Dynamics 24(3): 2450027.

39.

Yang

M-H

Zhang

Z-Q

, et al. (2022) Highly sensitive temperature sensor based on coupled-beam AlN-on-Si MEMS resonators operating in out-of-plane flexural vibration modes. Research: Ideas for Today’s Investors 2022: 9865926.

40.

Wang

Zhang

, et al. (2021a) Wave propagation in thermoelastic inhomogeneous hollow cylinders by analytical integration orthogonal polynomial approach. Applied Mathematical Modelling 99: 57–80.

41.

Wang

Zhang

, et al. (2021b) Thermoelastic guided wave in fractional order functionally graded plates: an analytical integration legendre polynomial approach. Composite Structures 256: 112997.

42.

Wang

Chen

, et al. (2022) Spherical piezoelectric transducers of functionally graded materials. Journal of the Acoustical Society of America 152(1): 193–200.

43.

Wang

Cheng

, et al. (2023) Electro-mechanical vibro-acoustic characteristics of submerged functionally graded piezoelectric plates with general boundary conditions. Composite Structures 322: 117411.

44.

Yang

J-A

Chen

, et al. (2023) Symmetric and asymmetric vibrations of rotating GPLRC annular plate. International Journal of Mechanical Sciences 250: 108282.

45.

Yang

Liu

J-A

, et al. (2024) Symmetric and asymmetric free vibrations of rotating eccentric annular plate. Journal of Sound and Vibration 576: 118302.

46.

Yao

Arshid

(2024) Effects of folding degree and mass fraction on the static and natural frequency characteristics of functionally graded graphene origami-enabled auxetic metamaterials annular plates. International Journal of Structural Stability and Dynamics 2650128. DOI: 10.1142/S0219455426501282.

47.

Wang

Zhang

, et al. (2022) An analytical integration legendre polynomial series approach for lamb waves in fractional order thermoelastic multilayered plates. Mathematical Methods in the Applied Sciences 45(12): 7631–7651.

48.

Zafarani

Jafari

Akin

(2021) Lateral and torsional vibration monitoring of multistack rotor induction motors. IEEE Transactions on Industrial Electronics 68(4): 3494–3505.

49.

Zhang

Shen

(2025) Dispersion characteristics of lamb wave in a multilayered magnetoelectroelastic plate with an imperfect interface. Journal of Intelligent Material Systems and Structures 36: 1045389X241306779.

50.

Zhong

Qin

Wang

, et al. (2021) Prediction of the in-plane vibration behavior of porous annular plate with porosity distributions in the thickness and radial directions. Mechanics of Advanced Materials and Structures 29(25): 4206–4223.

51.

Zhou

Han

Elmaimouni

, et al. (2023) Double Legendre polynomial quadrature-free method for axisymmetric vibration of functionally graded piezoelectric circular plates. Journal of Vibration and Control 30(3–4): 598–615.