Advanced Dynamic Vibration Control Algorithms of Materials Terfenol-D Si₃N₄ and SUS304 Plates/Cylindrical Shells with Velocity Feedback Control Law

Abstract

A numerical, generalized differential quadrature (GDQ) method is presented on applied heat vibration for a thick-thickness magnetostrictive functionally graded material (FGM) plate coupled with a cylindrical shell. A nonlinear c₁ term in the z axis direction of a third-order shear deformation theory (TSDT) displacement model is applied into an advanced shear factor and equation of motions, respectively. The equilibrium partial differential equation used for the thick-thickness magnetostrictive FGM layer plate coupled with the cylindrical shell under thermal and magnetostrictive loads can be implemented into the dynamic GDQ discrete equations. Parametric effects including nonlinear term coefficient of TSDT displacement field, advanced nonlinear varied shear coefficient, environment temperature, index of FGM power law and control gain on displacement, and stress of the thick magnetostrictive FGM plate coupled with cylindrical shell are studied. The vibrations of displacement and stress can be controlled by the control gain algorithms in velocity feedback control law.

1. Introduction

Structure-controlled studies of magnetostrictive composites were presented. In 2023, Al-Furjan et al. [1] used the refined zigzag theory of structure modeling to study wave propagation of a functionally graded material (FGM) magnetostrictive porous nanocomposite. In 2023, Chen et al. [2] reviewed some applications of magneto-optical sensor devices. In 2023, El-Shahrany and Zenkour [3] used higher-order shear-deformation theory (HSDT) and constant feedback control gain to provide a vibration control in magnetostrictive visco-Pasternak-medium FGM beams. In 2023, Wang et al. [4] presented computational generalized differential quadrature (GDQ) with a third-order shear-deformation theory (TSDT) model to study a primary resonant of a magnetostrictive truncated–conical FGM micro shell. In 2022, Gao et al. [5] reviewed the biomedical applications of magnetostrictive alloys. In 2022, Huang et al. [6] studied and controlled high-frequency-excitation properties on the magnetostrictive layers of Terfenol-D, Fe-Ga, and Fe-Co materials under temperature effects. In 2022, Yan et al. [7] studied the magnetic and elastic parameters under temperature effects for three types of giant magnetostrictive materials. In 2022, Qin et al. [8] used large deformation, Villari effect of thin magnetostrictive film, and piston theory to study the FGM beam under an aerodynamic supersonic airflow and heat condition. In 2020, Zhan and Lin [9] presented the nonlinear constitutive model for a magneto-strictive material under thermal environments.

The following are some useful recent works from the affinity literature in terms of topic and theoretical approaches: In 2024, Tornabene et al. [10] presented the GDQ results for the coupling hygrometric–thermal–magnetic–electric–mechanical properties of doubly curved shells. In 2024, Tornabene et al. [11] presented the GDQ results for the coupling thermal–mechanical properties of doubly curved shells. To thoroughly discuss and review the existing literature for FGM plates, cylindrical shells, and beams with velocity feedback control law, more literature references are listed as follows: In 2016, Sheng and Wang [12] presented the feedback control voltage from the sensor’s inner-layer electric potential and control gain to control the vibration of an outer-layer piezoelectric actuator for a cylindrical FGM shell. In 2020, Mohammadrezazadeh and Jafari [13] presented the two magnetostrictive outer layers of control law to control the vibration for the rotating cylindrical shell. In 2020, Dong et al. [14] presented the feedback control voltage from the piezoelectric sensor inner layer to control the vibration of a piezoelectric actuator’s outer layer for a cylindrical shell. In 2020, Wang et al. [15] presented the feedback control voltage for the piezoelectric actuator’s outer layer to control a cylindrical-FGM-shell-filling and -conveying fluid. In 2022, Rostami and Mohammadimehr [16] presented the magneto-electro-elastic layers of control law to control the vibration for a rotating FGM cylindrical shell. In 2011, Gupta et al. [17] presented the feedback control voltage from the piezoelectric sensor layer to control the vibration of a piezoelectric actuator layer for a plate. In 2010, Fakhari and Ohadi [18] presented the feedback control voltage from a piezoelectric sensor layer to control the vibration of a piezoelectric actuator layer for an FGM plate. In 2024, Ebrahimi and Ahari [19] presented a magnetostrictive central layer of control law to control the vibration for an FGM nanobeam.

Numerical GDQ computations in FGM compose the plate and shell by including the effect of applied heat. In 2022, Hong [20] presented the thermal vibration of stress and displacement results for plate/cylindrical shell by including TSDT and advanced nonlinear shear. Some numerical GDQ solutions had been studied for the layer of magneto-strictive material by including first-order shear deformation theory (FSDT) model, with or without constant shear factor, respectively. It is novel to study the thermal displacement and stress of the TSDT vibration approach and advanced nonlinear shear correction in magneto-strictive FGM plates and cylindrical shells. Effected important analyses of power law index and environmental temperature on the displacement and thermal stress of plate/cylindrical shell are presented. And the nonlinear effect of the advanced shear factor with magnetostrictive control gain are also included and controlled. The vibrations of displacement and stress can be controlled into small values by using the high-gain algorithms in velocity feedback control law.

2. Formulations

In the magnetostrictive material and two-material FGM plate coupled with cylindrical shell, as shown in Figure 1, by using velocity feedback control law, ${\mathsf{\theta }}_{\mathrm{L}}$ is left-side angle of intersection, ${\mathsf{\theta }}_{\mathrm{R}}$ is right-side angle of intersection, $h_3$ is magneto-strictive thickness, $h_1$ is inner-layer thickness, $h_2$ is outer-layer thickness, $L$ is axial length, h* is total thickness of magnetostrictive FGM plate/cylindrical shell, i.e., $h^{*}$ = $h_1$ + $h_2$ + $h_3$. Power-law material properties functions of the FGM plate/cylindrical shell are used, e.g., Young’s modulus; $E_{fgm}$ is in terms of power-law index $R_n$ [21].

Displacements $u$, $v$, and $w$ in functions of time for the thick-thickness FGM plate are used in TSDT model and expressed in the following [22]:

$$\left\{\begin{array}{c}u\\ v\\ w\end{array}\right\}=\left\{\begin{array}{c}{u}^0\left(x,y,t\right)\\ v^0\left(x,y,t\right)\\ w\left(x,y,t\right)\end{array}\right\}+z\left\{\begin{array}{c}{\psi }_x\left(x,y,t\right)\\ {\psi }_y\left(x,y,t\right)\\ 0\end{array}\right\}-c_1{z}^3\left\{\begin{array}{c}{\psi }_x+{\displaystyle \frac{\partial w}{\partial x}}\\ {\psi }_y+{\displaystyle \frac{\partial w}{\partial y}}\\ 0\end{array}\right\}$$

(1)

where $u^0$, $v^0$, and $w$ are displacements in $x$, $y$, and $z$ directions, respectively. ${\psi }_x$ and ${\psi }_y$ are shear rotations. $t$ is time. $c_1=4$/[3(h*)²] is the coefficient term used in TSDT.

Also, $u$, $v$, and $w$ for the thick-thickness FGM circular cylindrical shell are used in TSDT as follows [20]:

$$\left\{\begin{array}{c}u\\ v\\ w\end{array}\right\}=\left\{\begin{array}{c}{u}_0\left(x,\theta ,t\right)\\ v_0\left(x,\theta ,t\right)\\ w\left(x,\theta ,t\right)\end{array}\right\}+z\left\{\begin{array}{c}{\varphi }_x\left(x,\theta ,t\right)\\ {\varphi }_{\theta }\left(x,\theta ,t\right)\\ 0\end{array}\right\}-c_1{z}^3\left\{\begin{array}{c}{\varphi }_x+{\displaystyle \frac{\partial w}{\partial x}}\\ {\varphi }_{\theta }+{\displaystyle \frac{\partial w}{R\partial \theta }}\\ 0\end{array}\right\}$$

(2)

where $u_0$, $v_0$, $\mathrm{and}$ $w$ are displacements in the $x$, $\theta$, and $z$ directions, respectively. $R$ denotes the middle-surface radius. ${\varphi }_x$ and ${\varphi }_{\theta }$ denote the shear rotations.

For the (k)-th in-plane stresses and inter-laminar shear stresses, the physical assumptions are made considering the generally orthotropic multi-layers and without considering the angle effects of layer directions. In general, the strains can be in nonlinear functions of $c_1$z² term and coupled with temperature and thermal expansion coefficients together with the stiffness and the actuated magnetostrictive loads to induce stress. The normal stress ${\sigma }_x$, ${\sigma }_y$ and shear stress ${\sigma }_{xy}$, ${\sigma }_{yz}$, ${\sigma }_{xz}$ for the thick-thickness FGM plate in (k)-th layer subjected to applied-heat temperature difference $\Delta T$ are obtained by including the magnetostrictive material loads. Also, the normal stress ${\sigma }_x$, ${\sigma }_{\theta }$ and shear stress ${\sigma }_{x\theta }$, ${\sigma }_{\theta z}$, ${\sigma }_{xz}$ for the thick-thickness FGM shell in the (k)-th layer under applied heat $\Delta T$ include the magnetostrictive material loads and are used as follows:

$$\begin{array}{l}{\left\{\begin{array}{c}{\sigma }_x\\ {\sigma }_{\theta }\\ {\sigma }_{x\theta }\end{array}\right\}}_{(k)}={\left[\begin{array}{ccc}{\overline{Q}}_{11}& {\overline{Q}}_{12}& {\overline{Q}}_{16}\\ {\overline{Q}}_{12}& {\overline{Q}}_{22}& {\overline{Q}}_{26}\\ {\overline{Q}}_{16}& {\overline{Q}}_{26}& {\overline{Q}}_{66}\end{array}\right]}_{(k)}{\left\{\begin{array}{c}{\epsilon }_x-{\alpha }_x\Delta T\\ {\epsilon }_{\theta }-{\alpha }_{\theta }\Delta T\\ {\epsilon }_{x\theta }-{\alpha }_{x\theta }\Delta T\end{array}\right\}}_{(k)}-{\left[\begin{array}{ccc}0& 0& {\tilde{e}}_{31}\\ 0& 0& {\tilde{e}}_{32}\\ 0& 0& {\tilde{e}}_{36}\end{array}\right]}_{(k)}{\left\{\begin{array}{c}0\\ 0\\ {\tilde{H}}_z\end{array}\right\}}_{(k)}\\ {\left\{\begin{array}{c}{\sigma }_{\theta z}\\ {\sigma }_{xz}\end{array}\right\}}_{(k)}={\left[\begin{array}{cc}{\overline{Q}}_{44}& {\overline{Q}}_{45}\\ {\overline{Q}}_{45}& {\overline{Q}}_{55}\end{array}\right]}_{(k)}{\left\{\begin{array}{c}{\epsilon }_{\theta z}\\ {\epsilon }_{xz}\end{array}\right\}}_{(k)}-{\left[\begin{array}{ccc}{\tilde{e}}_{14}& {\tilde{e}}_{24}& 0\\ {\tilde{e}}_{15}& {\tilde{e}}_{25}& 0\end{array}\right]}_{(k)}{\left\{\begin{array}{c}0\\ 0\\ {\tilde{H}}_z\end{array}\right\}}_{(k)}\end{array}$$

(3)

in which ${\alpha }_{x\theta }$ is thermal shear coefficient. ${\alpha }_x$, ${\alpha }_{\theta }$ denote the thermal expansion coefficients. ${\overline{Q}}_{ij}$ is stiffness. ${\epsilon }_x$, ${\epsilon }_{\theta }$, ${\epsilon }_{x\theta }$ denote the in-plane strains. ${\epsilon }_{\theta z}$, ${\epsilon }_{xz}$ denote the shear strains. ${\tilde{e}}_{ij}$ denotes the magnetostrictive moduli. ${\tilde{H}}_z$ denotes the magnetic intensity actuated by velocity feedback ${\displaystyle \frac{\partial w}{\partial t}}$ and control gain $k_c$$c(t)$ of control law. There is a velocity sensor to feedback the velocity signal and suitable control gains for the vibration control algorithms. The (k)-th layer is denoted as the 1st layer for FGM 1, 2nd layer for FGM 2, and 3rd layer for the magnetostrictive material.

The $\Delta T$ expression and the heat conduction equation on the magnetostrictive FGM plate can be obtained. Also, $\Delta T$ expression and heat conduction expression on magnetostrictive FGM cylindrical shell and curing area in cylindrical coordinates are given in the following equations:

$$\Delta T={\displaystyle \frac{z}{h^{*}}}{\overline{T}}_1\left(x,\theta ,t\right)$$

(4)

where ${\overline{T}}_1$ denotes the applied heat temperature amplitude.

$$K({\displaystyle \frac{{\partial }^2\Delta T}{\partial x^2}}+{\displaystyle \frac{{\partial }^2\Delta T}{R^2\partial {\theta }^2}}+{\displaystyle \frac{{\partial }^2\Delta T}{\partial z^2}})={\displaystyle \frac{\partial \Delta T}{\partial t}}$$

(5)

in which $K={\kappa }_{fgm}/({\rho }_{fgm}{C}_{vfgm})$. ${\kappa }_{fgm}$ is the thermal conductivity. ${\rho }_{fgm}$ is the density. $C_{vfgm}$ is the specific heat.

The algorithm for using the velocity feedback control law is when the velocity sensor feeds the signal ${\displaystyle \frac{\partial w}{\partial t}}$ backs and the controller implements a suitable control gain $k_{c}c\left(t\right)$; thus, the actuated magnetostrictive layer is used to control the thermal vibrations of FGM plates-shells. So, the actuated magnetostrictive loads need to be included as the external loads onto an infinite small volume in FGM plate-shell. Dynamic partial differential equations of motion in infinite small volume dV for a thick-thickness magnetostrictive FGM plate can be used with TSDT. Also, dynamic partial differential motion equations in dV of the thick-thickness magnetostrictive FGM cylindrical shell with TSDT are given in the following:

$$\begin{array}{l}{\displaystyle \frac{\partial N_{xx}}{\partial x}}+{\displaystyle \frac{1}{R}}{\displaystyle \frac{\partial N_{x\theta }}{\partial \theta }}=I_0{\displaystyle \frac{{\partial }^2{u}_0}{\partial t^2}}+J_1{\displaystyle \frac{{\partial }^2{\phi }_x}{\partial t^2}}-c_1{I}_3{\displaystyle \frac{{\partial }^2}{\partial t^2}}({\displaystyle \frac{\partial w}{\partial x}})\\ {\displaystyle \frac{\partial N_{x\theta }}{\partial x}}+{\displaystyle \frac{1}{R}}{\displaystyle \frac{\partial N_{\theta \theta }}{\partial \theta }}=I_0{\displaystyle \frac{{\partial }^2{v}_0}{\partial t^2}}+J_1{\displaystyle \frac{{\partial }^2{\phi }_{\theta }}{\partial t^2}}-c_1{I}_3{\displaystyle \frac{{\partial }^2}{\partial t^2}}({\displaystyle \frac{1}{R}}{\displaystyle \frac{\partial w}{\partial \theta }})\\ {\displaystyle \frac{\partial {\overline{Q}}_x}{\partial x}}+{\displaystyle \frac{1}{R}}{\displaystyle \frac{\partial {\overline{Q}}_{\theta }}{\partial \theta }}+c_1({\displaystyle \frac{{\partial }^2{P}_{xx}}{\partial x^2}}+{\displaystyle \frac{2}{R}}{\displaystyle \frac{{\partial }^2{P}_{x\theta }}{\partial x\partial \theta }}+{\displaystyle \frac{1}{R^2}}{\displaystyle \frac{{\partial }^2{P}_{\theta \theta }}{\partial {\theta }^2}})+q=I_0{\displaystyle \frac{{\partial }^{2}w}{\partial t^2}}-c_1^{ 2}{I}_6{\displaystyle \frac{{\partial }^2}{\partial t^2}}({\displaystyle \frac{{\partial }^{2}w}{\partial x^2}}+{\displaystyle \frac{1}{R^2}}{\displaystyle \frac{{\partial }^{2}w}{\partial {\theta }^2}})\\ \hspace{1em}+c_1[I_3{\displaystyle \frac{{\partial }^2}{\partial t^2}}({\displaystyle \frac{\partial u_0}{\partial x}}+{\displaystyle \frac{1}{R}}{\displaystyle \frac{\partial v_0}{\partial \theta }})+J_4{\displaystyle \frac{{\partial }^2}{\partial t^2}}({\displaystyle \frac{\partial {\phi }_x}{\partial x}}+{\displaystyle \frac{1}{R}}{\displaystyle \frac{\partial {\phi }_{\theta }}{\partial \theta }})],\\ {\displaystyle \frac{\partial {\overline{M}}_{xx}}{\partial x}}+{\displaystyle \frac{1}{R}}{\displaystyle \frac{\partial {\overline{M}}_{x\theta }}{\partial \theta }}-{\overline{Q}}_x={\displaystyle \frac{{\partial }^2}{\partial t^2}}(J_1{u}_0+K_2{\phi }_x-c_1{J}_4{\displaystyle \frac{\partial w}{\partial x}})\\ {\displaystyle \frac{\partial {\overline{M}}_{x\theta }}{\partial x}}+{\displaystyle \frac{1}{R}}{\displaystyle \frac{\partial {\overline{M}}_{\theta \theta }}{\partial \theta }}-{\overline{Q}}_{\theta }={\displaystyle \frac{{\partial }^2}{\partial t^2}}(J_1{v}_0+K_2{\phi }_{\theta }-c_1{J}_4{\displaystyle \frac{1}{R}}{\displaystyle \frac{\partial w}{\partial \theta }})\end{array}$$

(6)

where ${\overline{M}}_{\alpha \beta }=M_{\alpha \beta }-c_1{P}_{\alpha \beta }$, ${\overline{Q}}_{\alpha }=Q_{\alpha }-3c_1{R}_{\alpha }$, $(\alpha ,\beta =x,\theta )$; the stress resultants $N_{xx}$, $N_{\theta \theta }$, $N_{x\theta }$, $M_{xx}$, $M_{\theta \theta }$, $M_{x\theta }$, $P_{xx}$, $P_{\theta \theta }$, $P_{x\theta }$,$R_{\theta }$,$R_x$, $Q_{\theta }$, and $Q_x$ are due to three effects of the stiffness with strains, the stiffness coupled with temperature and thermal expansion coefficients, and the actuated magnetostrictive loads, which can be listed as follows:

$$\begin{array}{l}\left\{\begin{array}{c}{N}_{xx}\\ N_{\theta \theta }\\ N_{x\theta }\end{array}\right\}={{\displaystyle \int }}_{-{\displaystyle \frac{h^{*}}{2}}}^{{\displaystyle \frac{h^{*}}{2}}}\left\{\begin{array}{c}{\sigma }_x\\ {\sigma }_{\theta }\\ {\sigma }_{x\theta }\end{array}\right\}dz, \left\{\begin{array}{c}{M}_{xx}\\ M_{\theta \theta }\\ M_{x\theta }\end{array}\right\}={{\displaystyle \int }}_{-{\displaystyle \frac{h^{*}}{2}}}^{{\displaystyle \frac{h^{*}}{2}}}\left\{\begin{array}{c}{\sigma }_x\\ {\sigma }_{\theta }\\ {\sigma }_{x\theta }\end{array}\right\}zdz, \left\{\begin{array}{c}{P}_{xx}\\ P_{\theta \theta }\\ P_{x\theta }\end{array}\right\}={{\displaystyle \int }}_{-{\displaystyle \frac{h^{*}}{2}}}^{{\displaystyle \frac{h^{*}}{2}}}\left\{\begin{array}{c}{\sigma }_x\\ {\sigma }_{\theta }\\ {\sigma }_{x\theta }\end{array}\right\}{z}^{3}dz,\\ \left\{\begin{array}{c}{R}_{\theta }\\ R_x\end{array}\right\}={{\displaystyle \int }}_{-{\displaystyle \frac{h^{*}}{2}}}^{{\displaystyle \frac{h^{*}}{2}}}\left\{\begin{array}{c}{\sigma }_{\theta z}\\ {\sigma }_{xz}\end{array}\right\}{z}^{2}dz, \left\{\begin{array}{c}{Q}_{\theta }\\ Q_x\end{array}\right\}={{\displaystyle \int }}_{-{\displaystyle \frac{h^{*}}{2}}}^{{\displaystyle \frac{h^{*}}{2}}}\left\{\begin{array}{c}{\sigma }_{\theta z}\\ {\sigma }_{xz}\end{array}\right\}dz.\end{array}$$

The integral density parameters $I_i={\displaystyle \sum _{k=1}^{N^{*}}{\displaystyle {\int }_k^{k+1}{\rho }^{(k)}}}{z}^{i}dz$, $(i=0,1,2,\cdots ,6)$, where $N^{*}$ denotes the layers total number, and ${\rho }^{(k)}$ denotes the density in (k)-th ply. $J_i=I_i-c_1{I}_{i+2}$, with subscript i = 1,4. $K_2=I_2-2c_1{I}_4+c_1^{ 2}{I}_6$. $q$ is the applied external pressure load.

Strain–displacement expressions with ${\displaystyle \frac{\partial v_0}{\partial z}}={\displaystyle \frac{-v_0}{R}}$, ${\displaystyle \frac{\partial u_0}{\partial z}}={\displaystyle \frac{-u_0}{R}}$ and ${\displaystyle \frac{\partial w}{\partial z}}={\displaystyle \frac{\partial {\phi }_x}{\partial z}}={\displaystyle \frac{\partial {\phi }_{\theta }}{\partial z}}=0$ can be expressed in the following:

$$\begin{array}{l}{\epsilon }_x={\displaystyle \frac{\partial u_0}{\partial x}}+z{\displaystyle \frac{\partial {\phi }_x}{\partial x}}-c_1{z}^3({\displaystyle \frac{\partial {\phi }_x}{\partial x}}+{\displaystyle \frac{{\partial }^{2}w}{\partial x^2}})+{\displaystyle \frac{1}{2}}{({\displaystyle \frac{\partial w}{\partial x}})}^2,\\ {\epsilon }_{\theta }={\displaystyle \frac{1}{R}}{\displaystyle \frac{\partial v_0}{\partial \theta }}+z{\displaystyle \frac{1}{R}}{\displaystyle \frac{\partial {\phi }_{\theta }}{\partial \theta }}-c_1{z}^3({\displaystyle \frac{1}{R}}{\displaystyle \frac{\partial {\phi }_{\theta }}{\partial \theta }}+{\displaystyle \frac{1}{R^2}}{\displaystyle \frac{{\partial }^{2}w}{\partial {\theta }^2}})+{\displaystyle \frac{1}{2}}{\displaystyle \frac{1}{R^2}}{({\displaystyle \frac{\partial w}{\partial \theta }})}^2,\\ {\epsilon }_{zz}=0,\\ {\epsilon }_{x\theta }={\displaystyle \frac{1}{R}}[{\displaystyle \frac{\partial u_0}{\partial \theta }}+z{\displaystyle \frac{\partial {\phi }_x}{\partial \theta }}-c_1{z}^3({\displaystyle \frac{\partial {\phi }_x}{\partial \theta }}+{\displaystyle \frac{{\partial }^{2}w}{\partial x\partial \theta }})]\\ +{\displaystyle \frac{\partial v_0}{\partial x}}+z{\displaystyle \frac{\partial {\phi }_{\theta }}{\partial x}}-c_1{z}^3({\displaystyle \frac{\partial {\phi }_{\theta }}{\partial x}}+{\displaystyle \frac{1}{R}}{\displaystyle \frac{{\partial }^{2}w}{\partial x\partial \theta }})+({\displaystyle \frac{\partial w}{\partial x}})({\displaystyle \frac{1}{R}}{\displaystyle \frac{\partial w}{\partial \theta }}),\\ {\epsilon }_{\theta z}={\displaystyle \frac{-v_0}{R}}+{\phi }_{\theta }-3c_1{z}^2({\phi }_{\theta }+{\displaystyle \frac{1}{R}}{\displaystyle \frac{\partial w}{\partial \theta }})+{\displaystyle \frac{1}{R}}{\displaystyle \frac{\partial w}{\partial \theta }},\\ {\epsilon }_{xz}={\displaystyle \frac{-u_0}{R}}+{\phi }_x-3c_1{z}^2({\phi }_x+{\displaystyle \frac{\partial w}{\partial x}})+{\displaystyle \frac{\partial w}{\partial x}}.\end{array}$$

(7)

After substituting (3) and (7) into (6), dynamic partial differential equilibrium equations in TSDT for magneto-strictive FGM cylindrical shells can be used. The ${\overline{Q}}_{i^s{j}^s}$ and ${\overline{Q}}_{i^{*}{j}^{*}}$ stiffness integrals are used in the following parameter forms:

$$\begin{array}{c}(A_{i^s{j}^s},B_{i^s{j}^s},D_{i^s{j}^s},E_{i^s{j}^s},F_{i^s{j}^s},H_{i^s{j}^s})={\displaystyle {\int }_{\frac{-h^{*}}{2}}^{\frac{h^{*}}{2}}{\overline{Q}}_{i^s{j}^s}}(1,z,z^2,z^3,z^4,z^6)dz, \mathrm{with}\\ \mathrm{subscripts} (i^s,j^s=1,2,6),\\ (A_{i^{*}{j}^{*}},B_{i^{*}{j}^{*}},D_{i^{*}{j}^{*}},E_{i^{*}{j}^{*}},F_{i^{*}{j}^{*}},H_{i^{*}{j}^{*}})={\displaystyle {\int }_{\frac{-h^{*}}{2}}^{\frac{h^{*}}{2}}{k}_{\alpha }{\overline{Q}}_{i^{*}{j}^{*}}}(1,z,z^2,z^3,z^4,z^5)dz, \mathrm{with} \\ \mathrm{subscripts} (i^{*},j^{*}=4,5),\end{array}$$

(8)

where $k_{\alpha }$ denotes the advanced nonlinear shear correction, which is in function of $T$, $h_3$, $h^{*}$, $c_1$, and $R_n$.

Stiffness ${\overline{Q}}_{i^s{j}^s}$ and ${\overline{Q}}_{i^{*}{j}^{*}}$ in the magnetostrictive FGM thick-thickness shell with non-neglected term $z/R$ can be used in simpler expressions. Thus, simpler ${\overline{Q}}_{i^s{j}^s}$, ${\overline{Q}}_{i^{*}{j}^{*}}$ expressions are then used to compute stress, stiffness integrals parameters $A_{i^s{j}^s},B_{i^s{j}^s},D_{i^s{j}^s},E_{i^s{j}^s},F_{i^s{j}^s},H_{i^s{j}^s}$ and $A_{i^{*}{j}^{*}},B_{i^{*}{j}^{*}},D_{i^{*}{j}^{*}},E_{i^{*}{j}^{*}},F_{i^{*}{j}^{*}},H_{i^{*}{j}^{*}}$, e.g., the $A_{11}$, $E_{11}$, $F_{11}$, $H_{11}$, $H_{44}$, $A_{12}$, $B_{12}$, $D_{12}$, $E_{12}$ $F_{12}$, $H_{12}$, $A_{55}$, $B_{55}$, $D_{55}$, $E_{55}$, $F_{55}$, and $H_{55}$ can be obtained.

The GDQ method was presented to solve dynamic partial differential equations into dynamic discrete equations in matrix form as follows for a grid point (i, j) of magnetostrictive FGM cylindrical shell [20,23,24]:

$$[A]\left\{W^{*}\right\}=\left\{B\right\},$$

(9)

in which $[A]$ is the matrix in $N^{**}$-th by $N^{**}$-th with dimension of $N^{**}$ = 5(N − 2)(M − 2), which contains stiffness integrals parameters, density integrals parameters, $c_1$ term, and GDQ weighting parameters. $\left\{W^{*}\right\}$ is $N^{**}$-th unknown displacements and shear rotations. $\left\{B\right\}$ is $N^{**}$-th applied heat loads and magnetostrictive loads.

The Lahey–Fujitsu Fortran programs implemented to simulate the GDQ dynamic discrete equations for the magnetostrictive FGM plate coupled with the shell under applied heat in time-sinusoidal vibrations and actuated magnetostrictive loads as follows, e.g.,:

$$\begin{gathered} u_0(x,\theta ,t)=u_0(x,\theta )\sin ({\omega }_{mn}t), v_0(x,\theta ,t)=v_0(x,\theta )\sin ({\omega }_{mn}t), \\ w(x,\theta ,t)=w(x,\theta )\sin ({\omega }_{mn}t), \\ {\phi }_x(x,\theta ,t)={\phi }_x(x,\theta )\sin ({\omega }_{mn}t), {\phi }_{\theta }(x,\theta ,t)={\phi }_{\theta }(x,\theta )\sin ({\omega }_{mn}t), \\ \Delta T={\displaystyle \frac{z}{h^{*}}}{\overline{T}}_1\sin \left({\displaystyle \frac{\pi x}{L}}\right)\sin \left(\pi \theta \right)\sin \left(\gamma t\right), \\ {\tilde{H}}_z\left(x,\theta ,t\right)=k_{c}c\left(t\right)w\left(x,\theta \right){\omega }_{mn}cos\left({\omega }_{mn}t\right), \end{gathered}$$

(10)

in which ${\omega }_{mn}$ denotes the natural frequency. $\gamma$ denotes the applied heat flux frequency.

3. Numerical Results

It is novel and interesting to study the nonlinear TSDT thick-thickness magnetostrictive FGM plate/cylindrical shell. The materials used for the studies are as follows: the magnetostrictive layer is Terfenol-D, FGM 1 is SUS304, and FGM 2 is Si₃N₄. With $L/R$ = 1, $h_1$ = $h_2$, $h^{*}$ = 1.2 mm, $k_{c}c\left(t\right)$ = 0, $q$ = 0, ${\omega }_{11}$ and $c_1$ = 0.925925/mm² for basic data to obtain advanced $k_{\alpha }$ values vs. $R_n$ and $h_3$ = 0.1 mm–1.0 mm under T = 100 K, which are presented in

Table 1. Advanced nonlinear $k_{\alpha }$ vs.$R_n$, $h_3$ and T = 100 K.

${\mathit{h}}_3$ (mm)	${\mathit{k}}_{\mathit{\alpha }}$
	${\mathit{R}}_{\mathit{n}}$: 0.1	0.2	0.5	1	2	5	10
0.1	0.037104	0.039685	0.049168	0.069195	0.117301	0.245510	0.334065
0.2	0.041771	0.045295	0.058288	0.086731	0.161186	0.384678	0.521903
0.3	0.049420	0.054397	0.072819	0.114345	0.229530	0.566865	0.694518
0.4	0.062453	0.069776	0.096835	0.158129	0.323746	0.677580	0.722486
0.5	0.084918	0.095833	0.134965	0.216903	0.390003	0.571030	0.560417
0.6	0.120294	0.134674	0.180651	0.254273	0.344935	0.381031	0.368428
0.7	0.157532	0.169341	0.199188	0.230172	0.251058	0.251351	0.245202
0.8	0.167441	0.171462	0.179415	0.184980	0.186852	0.184625	0.182127
0.9	0.155595	0.155993	0.156575	0.156684	0.156272	0.155241	0.154367
1.0	0.147206	0.147117	0.146916	0.146709	0.146496	0.146213	0.145987

. The maximum value of $k_{\alpha }$ = 0.722486 is found on $h_3$ = 0.4 mm and $R_n$ = 10. The advanced nonlinear $k_{\alpha }$ values are usually smaller than theoretical constant value 5/6 = 0.833333 of traditional study.

3.1. Dynamic Convergence

The w(a/2,b/2)(mm) is studied for ${\mathsf{\theta }}_{\mathrm{L}}={30}^{°}$ and ${60}^{°}$, respectively, in the TSDT of $k_{c}c\left(t\right)$ = 0, $c_1$ = 0.925925/mm², $h_3$ = 0.1 mm, advanced $k_{\alpha }$, $L/h^{*}$ = 5, and T = 100 K at t = 6 s under applied heat flux ${\overline{T}}_1$ = 100 K are presented in

Table 2. Convergence in magnetostrictive FGM plates/cylindrical shells for advanced $k_{\alpha }$ at T = 6 s.

$\mathit{L}/{\mathit{h}}^{*}$	${\mathsf{\theta }}_{\mathbf{L}}$	Grids	w(a/2,b/2)(mm)
		$\mathit{N}\times \mathit{M}$	${\mathit{R}}_{\mathit{n}}$ = 0.5	${\mathit{R}}_{\mathit{n}}$ = 1	${\mathit{R}}_{\mathit{n}}$ = 2
5	${30}^{°}$	7 × 7	0.039876	0.048409	0.066685
		9 × 9	0.065312	0.047408	0.066193
		11 × 11	0.032934	0.042817	0.058517
		13 × 13	0.031425	0.043521	0.055847
	${60}^{°}$	7 × 7	−0.004903	0.081978	0.122267
		9 × 9	0.009086	0.019592	0.031389
		11 × 11	0.015974	0.023372	0.032928
		13 × 13	0.016464	0.022055	0.030752

. The error accuracy is 1.6176 × 10⁻² for the w(a/2,b/2), ${\mathsf{\theta }}_{\mathrm{L}}={30}^{°},$ and $R_n$ = 1. The $N\times M$ = 13 × 13 grids can be used in the GDQ computations for the thermal vibration of magnetostrictive FGM plate coupled with cylindrical shell by using advanced $k_{\alpha }$.

3.2. Time Responses of Results

Figure 2 presents the time responses of w(a/2,b/2) and ${\sigma }_x$ for t = 0.1 s–3.0 s under the TSDT with $c_1$ = 0.925925/mm², advanced $k_{\alpha }$, $L/h^{*}$ = 5, $h_3$ = 0.1 mm, $k_{c}c\left(t\right)$ = 0,$R_n$ = 1, T = 600 K, and ${\overline{T}}_1$ = 100 K. The w(a/2,b/2) vs. t of ${\mathsf{\theta }}_{\mathrm{L}}={30}^{°}$ and ${60}^{°}$ are presented. The greatest value at t = 0.1 s for w(a/2,b/2) is −9.459201 mm under ${\mathsf{\theta }}_{\mathrm{L}}={30}^{°}$. The values of w(a/2,b/2) rapidly converge to 0.108490 mm for ${\mathsf{\theta }}_{\mathrm{L}}={30}^{°}$ and converge to 0.066303 mm for ${\mathsf{\theta }}_{\mathrm{L}}={60}^{°}$. Also, the ${\sigma }_x$ versus t on z = 0.5$h^{*}$ are presented. The greatest value of ${\sigma }_x$ is −1.3909 × 10⁻³ GPa, found at t = 0.1 s under ${\mathsf{\theta }}_{\mathrm{L}}={30}^{°}$ Values of ${\sigma }_x$ have small oscillation and converge to −7.9112 × 10⁻⁴ GPa for ${\mathsf{\theta }}_{\mathrm{L}}={30}^{°}$ and to −8.1968 × 10⁻⁴ GPa for ${\mathsf{\theta }}_{\mathrm{L}}={60}^{°}$.

3.3. Values for w(a/2,b/2), ${\sigma }_x$ vs. T and $R_n$

Figure 3 presents the w(a/2,b/2) values at t = 0.1 s versus T for 100 K, 600 K, 1000 K, and $R_n$ for 0.1–10 under TSDT with $c_1$ = 0.925925/mm², $k_{c}c\left(t\right)$ = 0, $L/h^{*}$ = 5, ${\overline{T}}_1$ = 100 K, ${\mathsf{\theta }}_{\mathrm{L}}={30}^{°}$, and ${60}^{°}$, respectively. The w(a/2,b/2) values of ${\mathsf{\theta }}_{\mathrm{L}}={30}^{°}$ are presented, the greatest value on T = 1000 K, $R_n$ = 0.2 for w(a/2,b/2), is −28.84869 mm. Values of w(a/2,b/2) decrease vs. T from 600 K to 1000 K under $R_n$ = 0.5 and 10; also, the w(a/2,b/2) can withstand heating deformation under T = 1000 K. The w(a/2,b/2) values of ${\mathsf{\theta }}_{\mathrm{L}}={60}^{°}$ are also presented, the greatest value on T = 1000 K, $R_n$ = 0.2 for w(a/2,b/2) is −106.298721 mm. Values of w(a/2,b/2) decrease vs. T from 600 K to 1000 K under $R_n$ = 0.1; also, the w(a/2,b/2) can withstand heating deformation under higher T = 1000 K.

Figure 4 shows ${\sigma }_x$ on z = 0.5$h^{*}$ vs. T, $R_n$ at t = 0.1 s under the TSDT with $c_1$ = 0.925925/mm², $k_{c}c\left(t\right)$ = 0, ${\overline{T}}_1$ = 100 K, $L/h^{*}$ = 5, ${\mathsf{\theta }}_{\mathrm{L}}={30}^{°}$, and ${60}^{°}$, respectively. The ${\sigma }_x$ values of ${\mathsf{\theta }}_{\mathrm{L}}={30}^{°}$ are presented, the greatest value on T = 1000 K, $R_n$ = 0.2 for ${\sigma }_x$, is −2.9435 × 10⁻³ GPa. Values of ${\sigma }_x$ decrease vs. T from 600 K to 1000 K under $R_n$ = 0.1, 0.5, 5; also, the values of ${\sigma }_x$ can withstand heating deformation under T = 1000 K. The ${\sigma }_x$ values of ${\mathsf{\theta }}_{\mathrm{L}}={60}^{°}$ are also presented, and the greatest value on T = 1000 K, $R_n$ = 0.2 for ${\sigma }_x$, is −6.4179 × 10⁻³ GPa. Values of ${\sigma }_x$ decrease vs. T from 600 K to 1000 K under $R_n$ = 0.1; also, the values of ${\sigma }_x$ can withstand heating deformation under higher T = 1000 K.

3.4. Compared Transient Responses of w(a/2,b/2) and ${\sigma }_x$

Transient of w(a/2,b/2) response vs. t and ${\sigma }_x$ response vs. t for t = 0.001 s–0.025 s are obtained under the TSDT with $c_1$ = 0.925925/mm², $L/h^{*}$ = 5, and $k_{c}c\left(t\right)$ = 0, ${\omega }_{11}$ = 0.008380/s, $\gamma$ = 15.707963/s,${\mathsf{\theta }}_{\mathrm{L}}={30}^{°}$, and ${60}^{°}$, respectively. Also, it used $h_1$ = $h_2$ = 0.55 mm, $h_3$ = 0.1 mm, $R_n$ = 1, advanced $k_{\alpha }$ = −3.535402, T = 600 K, and ${\overline{T}}_1$ = 100 K. Figure 5 shows compared transient w(a/2,b/2) for ${\mathsf{\theta }}_{\mathrm{L}}={30}^{°}$ and ${60}^{°}$. The w(a/2,b/2) transient values are in greater oscillations under ${\mathsf{\theta }}_{\mathrm{L}}={60}^{°}$ are presented. The compared ${\sigma }_x$ on z = 0.5$h^{*}$ for ${\mathsf{\theta }}_{\mathrm{L}}={30}^{°}$ and ${60}^{°}$ are also presented. The transient ${\sigma }_x$ values are in greater oscillations under ${\mathsf{\theta }}_{\mathrm{L}}={60}^{°}$. The compared transient w(a/2,b/2) vs. t and ${\sigma }_x$ vs. t on z = 0.5$h^{*}$, ${\mathsf{\theta }}_{\mathrm{L}}={60}^{°}$ are also obtained for $k_{c}c\left(t\right)$ = 10⁹ and $k_{c}c\left(t\right)$ = 0, respectively, in Figure 6. Thus, the transient oscillating values can be reduced into smaller under adjusted control value with $k_{c}c\left(t\right)$ = 10⁹ of control law.

4. Conclusions

Numerical displacement values and stress data are studied and subjected to the applied heat vibration on the thick-thickness magnetostrictive FGM plate/cylindrical shell by advanced shear correction and TSDT. The algorithm for using velocity feedback control law is when the velocity sensor feeds the signal ${\displaystyle \frac{\partial w}{\partial t}}$ back and the controller implements the suitable control gain $k_{c}c\left(t\right)$; thus, the actuated magnetostrictive layer is used to control the thermal vibrations of FGM plates/shells. Values of stress and displacement all converge with time for $L/h^{*}$ = 5 and ${\mathsf{\theta }}_{\mathrm{L}}={30}^{°}$, ${60}^{°}$. Displacement values for ${\mathsf{\theta }}_{\mathrm{L}}={30}^{°}$ and $R_n$ = 0.5, 10 can withstand heating deformation under T = 1000 K. Values of stress for ${\mathsf{\theta }}_{\mathrm{L}}={60}^{°}$ and $R_n$ = 0.1 can withstand heating deformation under T = 1000 K. The transient oscillating values of displacement and stress can be reduced under the control algorithms gain, $k_{c}c\left(t\right)$ = 10⁹, of the velocity feedback control law. The experimental data with the results would be expanded if possible in a future work.