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Axisymmetric free vibration modeling of a functionally graded piezoelectric resonator by a double Legendre polynomial method

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Abstract

The paper presents a semi-analytical method, named the Double Legendre Polynomial Method (DLPM), which is extended for modeling the axisymmetric free vibration of a functionally graded piezoelectric (FGP) hollow cylinder resonator. The FGP resonator is composed of two-phase graded piezoelectric materials, Lead zirconate titanate (PZT4) and aluminum nitrid (AlN), on the top and bottom surfaces, respectively, where the material properties gradually change along the thickness direction. The integration of FGP materials with piezoelectric materials is an innovative aspect in material science, and this paper explores their dynamic behavior, providing insights into their mechanical and electrical properties and contributes to expand our understanding of the axisymmetric free vibration of FGP hollow cylinder resonators as a crucial factor for optimizing their performance across a wide range of applications, including sensors, actuators, energy harvesting, and structural health monitoring systems. The DLPM is applied to solve the wave governing differential equations of motion, offering a new tool for studying the dynamic behavior of FGP hollow cylinders under axisymmetric vibration. The study investigates the effects of the diameter-thickness ratio on the effective electromechanical coupling coefficient and the resonant and anti-resonant frequencies of the FGP hollow cylinder. The results show that as the diameter-thickness ratio increases, the frequency-diameter product appears to become more stable. This observation is a valuable insight for optimizing the design of these resonators. Furthermore, the study presents the electrical input admittance, the natural frequencies as well as the mechanical displacement and electric potential profiles of the resonator. All the provided results are based on the calculation of the resonant frequencies. The validity of the presented results was verified by the comparison with the existing ones reported in the literature.

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Authors and Affiliations

  1. Lab PETI-ERMAM, Polydisciplinary Faculty of Ouarzazate, Ibnou Zohr University, BP.638, 45000, Ouarzazate, Morocco

    Hassna Khalfi, Ismail Naciri, Rabab Raghib & Lahoucine Elmaimouni

  2. LAPAUF, Fianarantsoa University, Fianarantsoa, Madagascar

    Joli Randrianarivelo & Faniry Emilson Ratolojanahary

  3. School of Mechanical and Power Engineering, Henan Polytechnic University, Jiaozuo, 454003, China

    Jiangong Yu

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  1. Hassna Khalfi
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  2. Ismail Naciri
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Appendix

Appendix

The integral expression of the matrix in the above equation is:

$${A1}_{mnjk}={\overline{C} }_{11}^{l}4{a}^{2}\left({\left({q}_{1}+b\right)}^{2}{q}_{3}^{l}\frac{{\partial }^{2}\overline{u}}{\partial {q }_{1}^{2}}{\pi }_{r}{\pi }_{z}+{\left({q}_{1}+b\right)}^{2}{q}_{3}^{l}\frac{\partial \overline{u}}{\partial {q }_{1}}\frac{\partial {\pi }_{r}}{\partial {q}_{1}}{\pi }_{z}-{q}_{3}^{l}\overline{u}{\pi }_{r}{\pi }_{z}+\left({q}_{1}+b\right){q}_{3}^{l}\frac{\partial \overline{u}}{\partial {q }_{1}}{\pi }_{r}{\pi }_{z}\right)+{\overline{C} }_{12}^{l}4{a}^{2}\left({q}_{1}+b\right){q}_{3}^{l}\overline{u}\frac{\partial {\pi }_{r}}{\partial {q}_{1}}{\pi }_{z}+{\overline{C} }_{55}^{l}4\left({\left({q}_{1}+b\right)}^{2}{q}_{3}^{l}\frac{{\partial }^{2}\overline{u}}{\partial {q }_{3}^{2}}{\pi }_{r}{\pi }_{z}+{\left({q}_{1}+b\right)}^{2}{q}_{3}^{l}\frac{\partial \overline{u}}{\partial {q }_{3}}\frac{\partial {\pi }_{z}}{\partial {q}_{3}}{\pi }_{r}+{\left({q}_{1}+b\right)}^{2}\frac{\partial \overline{u}}{\partial {q }_{3}}\frac{\partial {q}_{3}^{l}}{\partial {q}_{3}}{\pi }_{r}{\pi }_{z}\right)$$
$${A2}_{mnjk}={\overline{C} }_{13}^{l}4a\left({\left({q}_{1}+b\right)}^{2}{q}_{3}^{l}\frac{{\partial }^{2}\overline{w}}{\partial {q }_{1}{\partial q}_{3}}{\pi }_{r}{\pi }_{z}+{\left({q}_{1}+b\right)}^{2}{q}_{3}^{l}\frac{\partial \overline{w}}{\partial {q }_{3}}\frac{\partial {\pi }_{r}}{\partial {q}_{1}}{\pi }_{z}\right)+{\overline{C} }_{55}^{l}4a\left({\left({q}_{1}+b\right)}^{2}{q}_{3}^{l}\frac{{\partial }^{2}\overline{w}}{\partial {q }_{1}\partial {q}_{3}}{\pi }_{r}{\pi }_{z}+{\left({q}_{1}+b\right)}^{2}{q}_{3}^{l}\frac{\partial \overline{w}}{\partial {q }_{1}}\frac{\partial {\pi }_{z}}{\partial {q}_{3}}{\pi }_{r}+{\left({q}_{1}+b\right)}^{2}\frac{\partial \overline{w}}{\partial {q }_{1}}\frac{\partial {q}_{3}^{l}}{\partial {q}_{3}}{\pi }_{r}{\pi }_{z}\right)$$
$${A3}_{mnjk}=\frac{{\overline{e} }_{31}^{l}}{{\beta }^{l}}4a\left({\left({q}_{1}+b\right)}^{2}{q}_{3}^{l}\frac{{\partial }^{2}\varphi }{\partial {q}_{1}\partial {q}_{3}}{\pi }_{r}{\pi }_{z}+{\left({q}_{1}+b\right)}^{2}{q}_{3}^{l}\frac{\partial \varphi }{\partial {q}_{3}}\frac{\partial {\pi }_{r}}{\partial {q}_{1}}{\pi }_{z}\right)+\frac{{\overline{e} }_{15}^{l}}{{\beta }^{l}}4a\left({\left({q}_{1}+b\right)}^{2}{q}_{3}^{l}\frac{{\partial }^{2}\varphi }{\partial {q}_{1}\partial {q}_{3}}{\pi }_{r}{\pi }_{z}+{\left({q}_{1}+b\right)}^{2}{q}_{3}^{l}\frac{\partial \varphi }{\partial {q}_{1}}\frac{\partial {\pi }_{z}}{\partial {q}_{3}}{\pi }_{r}+{\left({q}_{1}+b\right)}^{2}\frac{\partial \varphi }{\partial {q}_{1}}\frac{\partial {q}_{3}^{l}}{\partial {q}_{3}}{\pi }_{r}{\pi }_{z}\right)$$
$$\begin{aligned}{B1}_{mnjk}&={\overline{C} }_{55}^{l}4a\left(\left({q}_{1}+b\right){q}_{3}^{l}\frac{{\partial }^{2}\overline{u}}{\partial {q }_{1}\partial {q}_{3}}{\pi }_{r}{\pi }_{z}+\left({q}_{1}+b\right){q}_{3}^{l}\frac{\partial \overline{u}}{\partial {q }_{3}}\frac{\partial {\pi }_{r}}{\partial {q}_{1}}{\pi }_{z}+{q}_{3}^{l}\frac{\partial \overline{u}}{\partial {q }_{3}}{\pi }_{r}{\pi }_{z}\right)\\ &\quad +{\overline{C} }_{13}^{l}4a\left(\left({q}_{1}+b\right){q}_{3}^{l}\frac{{\partial }^{2}\overline{u}}{\partial {q }_{1}{\partial q}_{3}}{\pi }_{r}{\pi }_{z}+\left({q}_{1}+b\right){q}_{3}^{l}\frac{\partial \overline{u}}{\partial {q }_{1}}\frac{\partial {\pi }_{z}}{\partial {q}_{3}}{\pi }_{r}+\left({q}_{1}+b\right)\frac{\partial \overline{u}}{\partial {q }_{1}}\frac{\partial {q}_{3}^{l}}{\partial {q}_{3}}{\pi }_{r}{\pi }_{z}\right. \\ &\quad \left. +{q}_{3}^{l}\frac{\partial \overline{u}}{\partial {q }_{3}}{\pi }_{r}{\pi }_{z}+{q}_{3}^{l}\overline{u}\frac{\partial {\pi }_{z}}{\partial {q}_{3}}{\pi }_{r}+\overline{u}\frac{\partial {q }_{3}^{l}}{\partial {q}_{3}}{\pi }_{r}{\pi }_{z}\right)\end{aligned}$$
$${B2}_{mnjk}={\overline{C} }_{55}^{l}4{a}^{2}\left(\left({q}_{1}+b\right){q}_{3}^{l}\frac{{\partial }^{2}\overline{w}}{\partial {q }_{1}^{2}}{\pi }_{r}{\pi }_{z}+\left({q}_{1}+b\right){q}_{3}^{l}\frac{\partial \overline{w}}{\partial {q }_{1}}\frac{\partial {\pi }_{r}}{\partial {q}_{1}}{\pi }_{z}+{q}_{3}^{l}\frac{\partial \overline{w}}{\partial {q }_{1}}{\pi }_{r}{\pi }_{z}\right)+{\overline{C} }_{33}^{l}4\left(\left({q}_{1}+b\right){q}_{3}^{l}\frac{{\partial }^{2}\overline{w}}{\partial {q }_{3}^{2}}{\pi }_{r}{\pi }_{z}+\left({q}_{1}+b\right){q}_{3}^{l}\frac{\partial \overline{w}}{\partial {q }_{3}}\frac{\partial {\pi }_{z}}{\partial {q}_{3}}{\pi }_{r}+\left({q}_{1}+b\right)\frac{\partial \overline{w}}{\partial {q }_{3}}\frac{\partial {q}_{3}^{l}}{\partial {q}_{3}}{\pi }_{r}{\pi }_{z}\right)$$
$${B3}_{mnjk}=\frac{{\overline{e} }_{15}^{l}}{{\beta }^{l}}4{a}^{2}\left(\left({q}_{1}+b\right){q}_{3}^{l}\frac{{\partial }^{2}\varphi }{\partial {q}_{1}^{2}}{\pi }_{r}{\pi }_{z}+\left({q}_{1}+b\right){q}_{3}^{l}\frac{\partial \varphi }{\partial {q}_{1}}\frac{\partial {\pi }_{r}}{\partial {q}_{1}}{\pi }_{z}+{q}_{3}^{l}\frac{\partial \varphi }{\partial {q}_{1}}{\pi }_{r}{\pi }_{z}\right)+\frac{{\overline{e} }_{33}^{l}}{{\beta }^{l}}4\left(\left({q}_{1}+b\right){q}_{3}^{l}\frac{{\partial }^{2}\varphi }{\partial {q}_{3}^{2}}{\pi }_{r}{\pi }_{z}+\left({q}_{1}+b\right){q}_{3}^{l}\frac{\partial \varphi }{\partial {q}_{3}}\frac{\partial {\pi }_{z}}{\partial {q}_{3}}{\pi }_{r}+\left({q}_{1}+b\right)\frac{\partial \varphi }{\partial {q}_{3}}\frac{\partial {q}_{3}^{l}}{\partial {q}_{3}}{\pi }_{r}{\pi }_{z}\right)$$
$${C1}_{mnjk}={\overline{e} }_{15}^{l}4a{\beta }^{l}\left({q}_{3}^{l}\frac{\partial \overline{u}}{\partial {q }_{3}}{\pi }_{r}{\pi }_{z}+\left({q}_{1}+b\right){q}_{3}^{l}\frac{{\partial }^{2}\overline{u}}{\partial {q }_{1}\partial {q}_{3}}{\pi }_{r}{\pi }_{z}+\left({q}_{1}+b\right){q}_{3}^{l}\frac{\partial \overline{u}}{\partial {q }_{3}}\frac{\partial {\pi }_{r}}{\partial {q}_{1}}{\pi }_{z}\right)+{\overline{e} }_{31}^{l}4a{\beta }^{l}\left(\left({q}_{1}+b\right){q}_{3}^{l}\frac{{\partial }^{2}\overline{u}}{\partial {q }_{1}\partial {q}_{3}}{\pi }_{r}{\pi }_{z}+{q}_{3}^{l}\frac{\partial \overline{u}}{\partial {q }_{3}}{{\pi }_{r}\pi }_{z}\right)$$
$${C2}_{mnjk}={\overline{e} }_{15}^{l}4{a}^{2}{\beta }^{l}\left({q}_{3}^{l}\frac{\partial \overline{w}}{\partial {q }_{1}}{\pi }_{r}{\pi }_{z}+\left({q}_{1}+b\right){q}_{3}^{l}\frac{{\partial }^{2}\overline{w}}{\partial {q }_{1}^{2}}{\pi }_{r}{\pi }_{z}+{\left({q}_{1}+b\right)q}_{3}^{l}\frac{\partial \overline{w}}{\partial {q }_{1}}\frac{\partial {\pi }_{r}}{\partial {q}_{1}}{\pi }_{z}\right)+{\overline{e} }_{33}^{l}4{\beta }^{l}\left(\left({q}_{1}+b\right){q}_{3}^{l}\frac{{\partial }^{2}\overline{w}}{\partial {q }_{3}^{2}}{\pi }_{r}{\pi }_{z}+\left({q}_{1}+b\right){q}_{3}^{l}\frac{\partial \overline{w}}{\partial {q }_{3}}\frac{\partial {\pi }_{z}}{\partial {q}_{3}}{\pi }_{r}+\left({q}_{1}+b\right)\frac{\partial \overline{w}}{\partial {q }_{3}}\frac{\partial {q}_{3}^{l}}{\partial {q}_{3}}{\pi }_{r}{\pi }_{z}\right)$$
$${C3}_{mnjk}=-{\overline{\varepsilon }}_{11}^{l}4{a}^{2}\left({q}_{3}^{l}\frac{\partial \varphi }{\partial {q}_{1}}{\pi }_{r}{\pi }_{z}+\left({q}_{1}+b\right){q}_{3}^{l}\frac{{\partial }^{2}\varphi }{\partial {q}_{1}^{2}}{\pi }_{r}{\pi }_{z}+\left({q}_{1}+b\right){q}_{3}^{l}\frac{\partial \varphi }{\partial {q}_{1}}\frac{\partial {\pi }_{r}}{\partial {q}_{1}}{\pi }_{z}\right)-{\overline{\varepsilon }}_{33}^{l}4\left(\left({q}_{1}+b\right){q}_{3}^{l}\frac{{\partial }^{2}\varphi }{\partial {q}_{3}^{2}}{\pi }_{r}{\pi }_{z}+\left({q}_{1}+b\right){q}_{3}^{l}\frac{\partial \varphi }{\partial {q}_{3}}\frac{\partial {\pi }_{z}}{\partial {q}_{3}}{\pi }_{r}+\left({q}_{1}+b\right)\frac{\partial \varphi }{\partial {q}_{3}}\frac{\partial {q}_{3}^{l}}{\partial {q}_{3}}{\pi }_{r}{\pi }_{z}\right)$$
$${M1}_{mnjk}=-{\Omega }^{2}{\pi }^{2}{\left({q}_{1}+b\right)}^{2}{q}_{3}^{l}\overline{u}{\pi }_{r}{\pi }_{z}; {M2}_{mnjk}=-{\Omega }^{2}{\pi }^{2}\left({q}_{1}+b\right){q}_{3}^{l}\overline{w}{\pi }_{r}{\pi }_{z}$$
$${{g}_{1}}_{jk}=12{\overline{\varepsilon }}_{31}^{l}\frac{a}{{\beta }^{l}}{\left({q}_{1}+b\right)}^{2}{Q}_{j}^{*}\left({q}_{1}\right){Q}_{0}\left({q}_{1}\right){Q}_{k}^{*}\left({q}_{3}\right){q}_{3}^{l+2}{Q}_{0}\left({q}_{3}\right)\frac{\partial {\pi }_{r}}{\partial {q}_{1}}{\pi }_{z}$$
$${{g}_{2}}_{jk}=12{\overline{\varepsilon }}_{33}^{l}\frac{1}{{\beta }^{l}}\left({q}_{1}+b\right)\left({Q}_{j}^{*}\left({q}_{1}\right){Q}_{0}\left({q}_{1}\right){Q}_{k}^{*}\left({q}_{3}\right){q}_{3}^{l+2}{Q}_{0}\left({q}_{3}\right)\frac{\partial {\pi }_{z}}{\partial {q}_{3}}{\pi }_{r}+{Q}_{j}^{*}\left({q}_{1}\right){Q}_{0}\left({q}_{1}\right){Q}_{k}^{*}\left({q}_{3}\right)\frac{\partial {q}_{3}^{l}}{\partial {q}_{3}}{Q}_{0}\left({q}_{3}\right){\pi }_{r}{\pi }_{z}+2{Q}_{j}^{*}\left({q}_{1}\right){Q}_{0}\left({q}_{1}\right){Q}_{k}^{*}\left({q}_{3}\right){q}_{3}^{l}{Q}_{0}\left({q}_{3}\right){\pi }_{r}{\pi }_{z}\right)$$
$${{g}_{3}}_{jk}=-12{\overline{\varepsilon }}_{33}^{l}\left({q}_{1}+b\right)\left({Q}_{j}^{*}\left({q}_{1}\right){Q}_{0}\left({q}_{1}\right){Q}_{k}^{*}\left({q}_{3}\right){q}_{3}^{l+2}{Q}_{0}\left({q}_{3}\right)\frac{\partial {\pi }_{z}}{\partial {q}_{3}}{\pi }_{r}+{Q}_{j}^{*}\left({q}_{1}\right){Q}_{0}\left({q}_{1}\right){Q}_{k}^{*}\left({q}_{3}\right)\frac{\partial {q}_{3}^{l}}{\partial {q}_{3}}{Q}_{0}\left({q}_{3}\right){\pi }_{r}{\pi }_{z}+2{Q}_{j}^{*}\left({q}_{1}\right){Q}_{0}\left({q}_{1}\right){Q}_{k}^{*}\left({q}_{3}\right){q}_{3}^{l+2}{Q}_{0}\left({q}_{3}\right){\pi }_{r}{\pi }_{z}\right)$$
$$\begin{aligned} {k12}_{m,n}^{\left(1\right)} &={\overline{e} }_{15}^{l}{\beta }^{l}a\left(\left(\frac{1}{b}{\int }_{-1}^{1}{Q}_{0}^{*}\left({q}_{1}\right){q}_{1}\frac{\partial {Q}_{m}\left({q}_{1}\right)}{\partial {q}_{1}}d{q}_{1}+{\int }_{-1}^{1}{Q}_{0}^{*}\left({q}_{1}\right)\frac{\partial {Q}_{m}\left({q}_{1}\right)}{\partial {q}_{1}}d{q}_{1}\right)\right. \\ &\quad +\left. \frac{1}{b}{\int }_{-1}^{1}{Q}_{0}^{*}\left({q}_{1}\right){Q}_{m}\left({q}_{1}\right)d{q}_{1}\right){\int }_{-1}^{1}{Q}_{0}^{*}\left({q}_{3}\right){q}_{3}^{l}{Q}_{n}\left({q}_{3}\right)d{q}_{3}\end{aligned}$$
$${k12}_{m,n}^{(2)}={\overline{e} }_{33}^{l}{\beta }^{l}\left(\left(\frac{1}{b}{\int }_{-1}^{1}{Q}_{0}^{*}\left({q}_{1}\right){q}_{1}{Q}_{m}\left({q}_{1}\right)d{q}_{1}+{\int }_{-1}^{1}{Q}_{0}^{*}\left({q}_{1}\right){Q}_{m}\left({q}_{1}\right)d{q}_{1}\right)\right){\int }_{-1}^{1}{Q}_{0}^{*}\left({q}_{3}\right){q}_{3}^{l}\frac{\partial {Q}_{n}\left({q}_{3}\right)}{\partial {q}_{3}}d{q}_{3}$$

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Khalfi, H., Naciri, I., Raghib, R. et al. Axisymmetric free vibration modeling of a functionally graded piezoelectric resonator by a double Legendre polynomial method. Acta Mech 235, 615–631 (2024). https://doi.org/10.1007/s00707-023-03766-1

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