Advanced Dynamic Vibration of Terfenol-D Control Law on Functionally Graded Material Plates/Cylindrical Shells in Unsteady Supersonic Flow

Abstract

The thermal vibration of thick Terfenol-D control law on functionally graded material (FGM) plates/cylindrical shells in nonlinear unsteady supersonic flow with third-order shear deformation theory (TSDT) is investigated by using the generalized differential quadrature (GDQ) method. The effects of the coefficient term of TSDT displacement models on the thermal stress and center displacement of Terfenol-D control law on FGM plates/cylindrical shells in nonlinear unsteady supersonic flow are investigated. The coefficient term of TSDT models of thick Terfenol-D control law on FGM plates/cylindrical shells provide an additional effect on the values of displacements and stresses.

1. Introduction

There are many studies about the aerodynamic, thermal, and elastic coupled structures of panels, plates, and shells in supersonic airflow. In 2022, Ben-Youssef et al. [1] used the first-order piston theory and nonlinear strain–displacement relationships of Novozhilov’s theory to study the vibrations of thin cylindrical shells in supersonic airflow. The classical finite element method (FEM) is used to find the numerical results by considering the effects of nonlinear coupling and circumferential wave number. In 2022, Khalafi and Fazilati [2] used the variational approach of energy method to study the flutter of cracked functionally graded material (FGM) plates in yawed supersonic airflow. The numerical solutions are presented by considering the first-order shear deformation theory (FSDT) model, thermo-elastic theory, and the first-order piston theory. In 2022, Tian et al. [3] presented the vibration and supersonic flutter for the stiffened plate by using the Hamilton principle, FSDT model, and supersonic piston aerodynamic theory. In 2022, Tian et al. [4] presented the vibration and supersonic flutter for the FGM plate by using the Hamilton principle, von Karman large deformation theory, and supersonic piston aerodynamic theory. In 2021, Zhong et al. [5] used the Rayleigh-Ritz method, FSDT model, and supersonic piston theory to study the flutter instability of the magnetic-electric FGM plates in yawed supersonic airflow. In 2020, Lin et al. [6] used the nonlinear third-order piston theory for the aerodynamic pressure load in the supersonic flow to study the nonlinear thermal flutter for the shape memory alloy (SMA) panel. In 2020, Zhang et al. [7] used the Hamilton principle and large deformation theory to study the piezoelectric FGM beams in supersonic airflow by using MATLAB numerical simulation. In 2019, Ghorbanpour Arani and Soleymani [8] used Hamilton’s principle for the equations of motion, the Euler–Bernoulli theory for the displacement field, and the Galerkin method for the numerical solution to study the vibration of the sandwich magnetorheological (MR) FGM beams in yawed supersonic airflow. In 2019, Song et al. [9] presented the mode localization of flutter phenomenon for the FGM beams in supersonic airflow. In 2019, Yang et al. [10] used Hamilton’s principle for the equations of motion, the von Karman nonlinear geometric assumptions, and FSDT for the displacement field and the Galerkin method for the numerical solution to study the vibration of the truncated FGM conical shells under internal supersonic aerodynamic pressure. In 2017, Barati and Shahverdi [11] used Hamilton’s principle for the equations of motion, higher-order refined shear deformable theory for the displacement field, and the Galerkin method for the numerical solution to study the stability of the porous FGM panels in supersonic airflow. In 2015, Lee and Kim [12] presented FSDT model for the displacement field, the homogenization methods for the effective properties of FGM panels, and the Newmark time integration method for the numerical solution to study the macroscopic behaviors of the FGM panels in supersonic airflow. In 2012, Prakash et al. [13] presented the FSDT model and von Kármán’s assumptions for large displacement field, the Newmark time integration method, and the finite element model for the numerical solution to study the vibrations of the FGM plates in supersonic airflow. In 2010, Lee and Kim [14] presented the FSDT model and von Kármán’s assumptions for the displacement field to study the post-buckling behaviors and linear flutters of the FGM panels in supersonic airflow. In 2009, Lee and Kim [15] also presented the FSDT model and von Kármán’s assumptions for the displacement field to study the post-buckling and limit-cycle oscillation of the FGM panels in supersonic airflow.

The major solution to the aeroelasticity effect with control law on the vibration reduction of structures in supersonic flow is presented. More topical studies including the thermal load and aerodynamic force on the structure are also important to understand the detailed solutions for Terfenol-D and FGM. Some generalized differential quadrature (GDQ) experiences in the composited FGM shells and plates are presented. In 2019, Hong [16] presented the thermal vibration of numerical GDQ results with a fully homogeneous equation for thick FGM plates. In 2016, Hong [17] presented the thermal vibration and transient response of Terfenol-D FGM circular cylindrical shells by considering the FSDT model. In 2017, Hong [18] presented the linear effects of varied shear correction on the thermal vibration of FGM shells in linear unsteady supersonic flow. It is interesting to investigate the thermal stresses and center displacement of the nonlinear TSDT vibration with the advanced varied effects of shear correction coefficient of magnetostrictive material Terfenol-D control law on FGM plates/cylindrical shells in nonlinear unsteady supersonic flow. The effects of the coefficient term of TSDT models on the thermal stress and center displacement of magnetostrictive FGM plates/cylindrical shells in nonlinear unsteady supersonic flow are investigated.

2. Formulations

For the nonlinear unsteady supersonic flow over the outer surface of the magnetostrictive material Terfenol-D control law and two-material FGM plates/cylindrical shells in which ${\theta }_L$ is the angle of the left side intersection, ${\theta }_R$ is the angle of the right side intersection, as shown in Figure 1, with thickness $h_3$ of the magnetostrictive layer, thickness $h_1$ of inner layer FGM 1, and thickness $h_2$ of outer layer FGM 2. $L$ is the axial length of FGM shells and $h^{\ast }$ is the total thickness of magnetostrictive FGM plates/cylindrical shells. The material properties of the power law function of FGM plates/cylindrical shells are considered in power law index $R_n$ and functions of environment temperature $T$ [19].

The details of algorithms in the control process can be described as follows. A velocity is sensed by the sensor, then the velocity feeds back to the control law with a high gain controller and the displacements actuated by the Terfenol-D layer to control the thermal vibrations for FGM plates and shells. The displacements of thick FGM plates are assumed functions of time t and coefficient $c_1=4/[3{(h^{\ast })}^2]$ term of TSDT equations [16]. Also, the displacements of thick FGM circular cylindrical shells are assumed functions of time and coefficient $c_1$ term of TSDT equations [20]. The stresses in the thick FGM plate under temperature difference $\Delta T$ for the ($k$)-th constituent material can be obtained. Also the stresses in the thick FGM circular cylindrical shells under $\Delta T$ for the ($k$)-th constituent material can be obtained in terms of the stiffness ${\overline{Q}}_{ij}$, the corresponding strains, and the magnetostrictive coupling effect controlled by velocity feedback ${\displaystyle \frac{\partial w}{\partial t}}$ and controlled gain $k_c$$c(t)$ value for the Terfenol-D control law [21,22]. The $\Delta T$ equation can be assumed in linear form with z and temperature parameter ${\overline{T}}_1$ as a function of time. Also, $\Delta T$ can be assumed in the simple form of the heat conduction equation [16,17].

The dynamic equation of motion of magnetostrictive FGM plates/cylindrical shells could be obtained by using the generalized Hamilton total energy principle for the differential volume dV in the plate and cylindrical shell, respectively [23]. The dynamic equilibrium differential equations in the cylindrical coordinates with TSDT of magnetostrictive FGM shells in terms of partial derivatives of displacements and shear rotations subjected to partial derivatives of thermal loads, mechanical loads, and inertia terms can be obtained in matrix forms. It is similar to the expression in the magnetostrictive FGM plates [16], only replacing the parameter $\partial y$ with $R\partial \theta$ and with some different term of elements, e.g., $u_0/R$ and $v_0/R$, in which $R$ is the middle surface radius of magnetostrictive FGM shell and $u_0$ and $v_0$ are tangential displacements in the in-surface coordinates x and $\theta$ axes direction, respectively. The external loads in column vector ${\{f}_1,f_2,f_3,f_4,f_5{\}}^t$ are the derivative expressions of terms in thermal loads $(\overline{N}, \tilde{\overline{M}},\overline{P})$ which are functions of $\Delta T$ and mechanical loads $(p_1, p_2, q)$ and magnetostrictive loads $(\tilde{N}, \tilde{M})$ as follows

$$\begin{gathered} f_1={\displaystyle \frac{\partial {\overline{N}}_{xx}}{\partial x}}+{\displaystyle \frac{1}{R}}{\displaystyle \frac{\partial {\overline{N}}_{x\theta }}{\partial \theta }}+p_1+{\displaystyle \frac{\partial {\tilde{N}}_{xx}}{\partial x}}+{\displaystyle \frac{1}{R}}{\displaystyle \frac{\partial {\tilde{N}}_{x\theta }}{\partial \theta }}, \\ {f}_2={\displaystyle \frac{\partial {\overline{N}}_{x\theta }}{\partial x}}+{\displaystyle \frac{1}{R}}{\displaystyle \frac{\partial {\overline{N}}_{\theta \theta }}{\partial \theta }}+p_2+{\displaystyle \frac{\partial {\tilde{N}}_{x\theta }}{\partial x}}+{\displaystyle \frac{1}{R}}{\displaystyle \frac{\partial {\tilde{N}}_{\theta \theta }}{\partial \theta }}, \\ {f}_3=-q+c_1({\displaystyle \frac{{\partial }^2{\overline{P}}_{xx}}{\partial x^2}}+{\displaystyle \frac{2}{R}}{\displaystyle \frac{{\partial }^2{\overline{P}}_{x\theta }}{\partial x\partial \theta }}+{\displaystyle \frac{1}{R^2}}{\displaystyle \frac{{\partial }^2{\overline{P}}_{\theta \theta }}{\partial {\theta }^2}}), \\ {f}_4={\displaystyle \frac{\partial {\tilde{\overline{M}}}_{xx}}{\partial x}}+{\displaystyle \frac{1}{R}}{\displaystyle \frac{\partial {\tilde{\overline{M}}}_{x\theta }}{\partial \theta }}+{\displaystyle \frac{\partial {\tilde{M}}_{xx}}{\partial x}}+{\displaystyle \frac{1}{R}}{\displaystyle \frac{\partial {\tilde{M}}_{x\theta }}{\partial \theta }}-c_1({\displaystyle \frac{\partial {\overline{P}}_{xx}}{\partial x}}+{\displaystyle \frac{1}{R}}{\displaystyle \frac{\partial {\overline{P}}_{x\theta }}{\partial \theta }}), \\ {f}_5={\displaystyle \frac{\partial {\tilde{\overline{M}}}_{x\theta }}{\partial x}}+{\displaystyle \frac{1}{R}}{\displaystyle \frac{\partial {\tilde{\overline{M}}}_{\theta \theta }}{\partial \theta }}+{\displaystyle \frac{\partial {\tilde{M}}_{x\theta }}{\partial x}}+{\displaystyle \frac{1}{R}}{\displaystyle \frac{\partial {\tilde{M}}_{\theta \theta }}{\partial \theta }}-c_1({\displaystyle \frac{\partial {\overline{P}}_{x\theta }}{\partial x}}+{\displaystyle \frac{1}{R}}{\displaystyle \frac{\partial {\overline{P}}_{\theta \theta }}{\partial \theta }}), \end{gathered}$$

(1)

where

$$\begin{gathered} ({\overline{N}}_{xx},{\tilde{\overline{M}}}_{xx},{\overline{P}}_{xx})={\displaystyle {\int }_{-\frac{h^{\ast }}{2}}^{\frac{h^{\ast }}{2}}({\overline{Q}}_{11}{\alpha }_x}+{\overline{Q}}_{12}{\alpha }_{\theta }+{\overline{Q}}_{16}{\alpha }_{x\theta })\Delta T(1,z,z^3)dz, \\ ({\overline{N}}_{\theta \theta },{\tilde{\overline{M}}}_{\theta \theta },{\overline{P}}_{\theta \theta })={\displaystyle {\int }_{\frac{-h^{\ast }}{2}}^{\frac{h^{\ast }}{2}}({\overline{Q}}_{12}}{\alpha }_x+{\overline{Q}}_{22}{\alpha }_{\theta }+{\overline{Q}}_{26}{\alpha }_{x\theta })\Delta T(1,z,z^3)dz, \\ ({\overline{N}}_{x\theta },{\tilde{\overline{M}}}_{x\theta },{\overline{P}}_{x\theta })={\displaystyle {\int }_{\frac{-h^{\ast }}{2}}^{\frac{h^{\ast }}{2}}({\overline{Q}}_{16}}{\alpha }_x+{\overline{Q}}_{26}{\alpha }_{\theta }+{\overline{Q}}_{66}{\alpha }_{x\theta })\Delta T(1,z,z^3)dz, \\ ({\tilde{N}}_{xx},{\tilde{M}}_{xx})={\int }_{-{\displaystyle \frac{h^{*}}{2}}}^{{\displaystyle \frac{h^{*}}{2}}}{\tilde{e}}_{31}{\tilde{H}}_z(1,z^2)dz, \\ ({\tilde{N}}_{\theta \theta },{\tilde{M}}_{\theta \theta })={\int }_{-{\displaystyle \frac{h^{*}}{2}}}^{{\displaystyle \frac{h^{*}}{2}}}{\tilde{e}}_{32}{\tilde{H}}_z(1,z^2)dz, \\ ({\tilde{N}}_{x\theta },{\tilde{M}}_{x\theta })={\int }_{-{\displaystyle \frac{h^{*}}{2}}}^{{\displaystyle \frac{h^{*}}{2}}}{\tilde{e}}_{36}{\tilde{H}}_z(1,z^2)dz, \\ q={\displaystyle \frac{{\rho }_{\infty }{U_{\infty }}^2}{M_{\infty }}}[({\displaystyle \frac{\partial w\left(x,\theta ,t\right)}{\partial x}}+{\displaystyle \frac{1}{U_{\infty }}}{\displaystyle \frac{\partial w\left(x,\theta ,t\right)}{\partial t}})+{\displaystyle \frac{\left(C_{\gamma }+1\right)M_{\infty }}{4}}({\displaystyle \frac{\partial w\left(x,\theta ,t\right)}{\partial x}}+{\displaystyle \frac{1}{U_{\infty }}}{\displaystyle \frac{\partial w\left(x,\theta ,t\right)}{\partial t}}{)}^2+ \\ {\displaystyle \frac{\left(C_{\gamma }+1\right){M_{\infty }}^2}{12}}({\displaystyle \frac{\partial w\left(x,\theta ,t\right)}{\partial x}}+{\displaystyle \frac{1}{U_{\infty }}}{\displaystyle \frac{\partial w\left(x,\theta ,t\right)}{\partial t}}{)}^3], \\ (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^{\ast }}{2}}^{\frac{h^{\ast }}{2}}{\overline{Q}}_{i^s{j}^s}}(1,z,z^2,z^3,z^4,z^6)dz, \\ (i^s,j^s=1,2,6), \\ (A_{i^{\ast }{j}^{\ast }},B_{i^{\ast }{j}^{\ast }},D_{i^{\ast }{j}^{\ast }},E_{i^{\ast }{j}^{\ast }},F_{i^{\ast }{j}^{\ast }},H_{i^{\ast }{j}^{\ast }})={\displaystyle {\int }_{\frac{-h^{\ast }}{2}}^{\frac{h^{\ast }}{2}}{k}_{\alpha }{\overline{Q}}_{i^{\ast }{j}^{\ast }}}(1,z,z^2,z^3,z^4,z^5)dz, \\ (i^{\ast },j^{\ast }=4,5), \end{gathered}$$

in which $w$ is transverse displacement in coordinate z axis direction of the magnetostrictive FGM middle-plane and $p_1$ and $p_2$ are external in-plane distributed forces in x and $\theta$ direction, respectively. ${\tilde{H}}_z$ is the magnetic intensity of Terfenol-D control law. ${\tilde{e}}_{31}$, ${\tilde{e}}_{32}$, and ${\tilde{e}}_{36}$ are the magnetostrictive moduli. $q$ is the aerodynamic pressure load expressed in the nonlinear third-order piston theory introduced by Lin et al. [6] in 2020 for the supersonic ($M_{\infty }>>1$) in unsteady, in-viscid fluid flow over the outer surface of thick magnetostrictive FGM plates/cylindrical shells with free stream density ${\rho }_{\infty }$, velocity $U_{\infty }$, Mach number $M_{\infty }$, and specific heat ratio $C_{\gamma }$. The linear case of aerodynamic pressure load with first-order transverse displacement derivatives are referenced by Dowell et al. in 1978 [24]. $k_{\alpha }$ is the advanced nonlinear shear coefficient, the advanced $k_{\alpha }$ values are usually functions of $h^{\ast }$, $h_3$, $c_1$, $R_n$, and $T$.

In 1990, the GDQ method was presented by Shu and Richards, then applied by Hong in 2017 [18] and by Shu and Du in 1997 [25]. The differential quadrature (DQ) method was presented by Bert et al. in 1989 [26]. The GDQ method approximates the derivative of function $f(x,\theta )$ at the coordinate of arbitrarily grid point $(x_i,{\theta }_j)$, i = 1, 2, …, N, $j=1,2,\dots ,M$, in the two-dimensional formulation can be given, e.g.,

$$\begin{gathered} {{\displaystyle \frac{\partial f}{\partial x}}}_{\left|i,j\right.}\approx {\displaystyle \sum _{l=1}^N{A}_{i,l}^{(1)}}{f}_{l,j}, {{\displaystyle \frac{\partial f}{\partial \theta }}}_{\left|i,j\right.}\approx {\displaystyle \sum _{l=1}^M{B}_{j,m}^{(1)}}{f}_{i,m}, \dots , \\ {{\displaystyle \frac{{\partial }^{4}f}{\partial x^2\partial {\theta }^2}}}_{\left|i,j\right.}\approx {\displaystyle \sum _{l=1}^N{A}_{i,l}^{(2)}}{\displaystyle \sum _{m=1}^M{B}_{j,m}^{(2)}}{f}_{l,m}, \end{gathered}$$

(2)

where $A_{i,l}^{(m)}$ and $B_{j,m}^{(m)}$ denote the weighting coefficients for the (m)-th order derivative of the function $f(x,\theta )$ with respect to the x and $\theta$ directions. Take the function $f(x,\theta )$ for the component of middle-plane displacements $u^0$, $v^0$, $w$ for the plate and $u_0$, $v_0$, $w$ for the shell and shear rotations, ${\psi }_x$ and ${\psi }_y$ for the plate, and ${\varphi }_x$ and ${\varphi }_{\theta }$ for the shell in the places for the (m)-th order derivative, and the non-dimensional parameters are used in the GDQ approaches, e.g., for the cylindrical shells, used as follows:

$$X=x/L, U=u_0/L, V=v_0/R, W=w/h^{\ast }.$$

(3)

The time sinusoidal displacement, shear rotations, temperature of thermal vibrations, and actuated magnetostrictive Terfenol-D control law are assumed in the simulation examples, e.g., for the shell as follows [17]:

$$\begin{gathered} u_0(x,\theta ,t)=u_0(x,\theta )\sin ({\omega }_{mn}t), v_0(x,\theta ,t)=v_0(x,\theta )\sin ({\omega }_{mn}t), \\ w(x,\theta ,t)=w(x,\theta )\sin ({\omega }_{mn}t), {\varphi }_x(x,\theta ,t)={\varphi }_x(x,\theta )\sin ({\omega }_{mn}t), \\ {\varphi }_{\theta }(x,\theta ,t)={\varphi }_{\theta }(x,\theta )\sin ({\omega }_{mn}t), ∆T={\displaystyle \frac{\mathrm{z}}{h^{\mathrm{\ast }}}}{\overline{T}}_1\mathrm{s}\mathrm{i}\mathrm{n}(\pi x/L)\mathrm{s}\mathrm{i}\mathrm{n}(\pi \theta )\mathrm{s}\mathrm{i}\mathrm{n}(\gamma t), \\ {\tilde{H}}_z\left(x,\theta ,t\right)=k_{c}c\left(t\right)w(x,\theta ){\omega }_{mn}cos({\omega }_{mn}t), \end{gathered}$$

(4)

in which ${\omega }_{mn}$ is the natural frequency in mode shape subscript numbers m and n of the shells, $\gamma$ is the frequency of applied heat flux, ${\overline{T}}_1$ is the amplitude of temperature, and t is time.

For a typical grid point (i, j), the dynamic GDQ discrete equations can be rewritten into the matrix form for the magnetostrictive FGM plate and cylindrical shell, respectively, as follows:

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

(5)

where $[A]$ is a dimension of $N^{\ast \ast }$ by $N^{\ast \ast }$ coefficient matrix ($N^{\ast \ast }=(N-2)\times (M-2)\times 5$) containing nonlinear term $c_1$, vibration term ${\omega }_{mn}$t, inertia terms, the stiffness coefficients ($A_{i^s{j}^s},B_{i^s{j}^s},D_{i^s{j}^s},E_{i^s{j}^s},F_{i^s{j}^s},H_{i^s{j}^s}$), and weighting parameters ($A_{i,l}^{(m)},B_{j,m}^{(m)}$).

$\left\{W^{\ast }\right\}$ is an $N^{\ast \ast }$-th order unknown column and vector can be expressed for the shell as follows:

$$\begin{gathered} \{U_{\mathrm{2,2}},U_{\mathrm{2,3}},\cdots ,U_{N-1,M-1},V_{\mathrm{2,2}},V_{\mathrm{2,3}},\cdots ,V_{N-1,M-1}{W}_{\mathrm{2,2}},W_{\mathrm{2,3}},\cdots ,W_{N-1,M-1},{{\varphi }_x}_{\mathrm{2,2}},{{\varphi }_x}_{\mathrm{2,3}},\cdots ,{{\varphi }_x}_{N-1,M-1} \\ ,{{\varphi }_x}_{\mathrm{2,2}},{{\varphi }_x}_{\mathrm{2,3}},\cdots ,{{\varphi }_x}_{N-1,M-1}{\mathsf{\}}}^t. \end{gathered}$$

$\left\{B\right\}$ is an $N^{\ast \ast }$-th order row external loads vector and can be written as follows, e.g., for the shell, $\left\{B\right\}={\{F_1\dots F_1,F_2\dots F_2,F_3\dots F_3,F_4\dots F_4,F_5\dots F_5\}}^t$, in which

$$\begin{gathered} F_1=\left({\displaystyle \frac{1}{L}}\sum _{l=1}^N{A}_{i,l}^{\left(1\right)}{\overline{N}}_{{xx}_{l,j}}+{\displaystyle \frac{1}{R}}\sum _{m=1}^M{B}_{j,m}^{\left(1\right)}{\overline{N}}_{{x\theta }_{i,m}}\right)\sin \left(\gamma t\right)+p_{1_{i,j}} \\ {F}_2=\left({\displaystyle \frac{1}{L}}\sum _{l=1}^N{A}_{i,l}^{\left(1\right)}{\overline{N}}_{{x\theta }_{l,j}}+{\displaystyle \frac{1}{R}}\sum _{m=1}^M{B}_{j,m}^{\left(1\right)}{\overline{N}}_{{\theta \theta }_{i,m}}\right)\sin \left(\gamma t\right)+p_{2_{i,j}} \\ {F}_3=c_1\left({\displaystyle \frac{1}{L^2}}\sum _{l=1}^N{A}_{i,l}^{\left(2\right)}{\overline{P}}_{{xx}_{l,j}}+{\displaystyle \frac{2}{LR}}\sum _{l=1}^N{A}_{i,l}^{\left(1\right)}\sum _{m=1}^M{B}_{j,m}^{\left(1\right)}{\overline{P}}_{x\theta l,m}+{\displaystyle \frac{1}{R^2}}\sum _{m=1}^M{B}_{j,m}^{\left(2\right)}{\overline{P}}_{{\theta \theta }_{i,m}}\right)\sin \left(\gamma t\right)-q_{i,j} \\ {F}_4=\left({\displaystyle \frac{1}{L}}\sum _{l=1}^N{A}_{i,l}^{\left(1\right)}{\tilde{\overline{M}}}_{{xx}_{l,j}}+{\displaystyle \frac{1}{R}}\sum _{m=1}^M{B}_{j,m}^{\left(1\right)}{\tilde{\overline{M}}}_{{x\theta }_{i,m}}\right)\sin \left(\gamma t\right) \\ {+c}_1\left({\displaystyle \frac{1}{L}}\sum _{l=1}^N{A}_{i,l}^{\left(1\right)}{\overline{P}}_{{xx}_{l,j}}+{\displaystyle \frac{1}{R}}\sum _{m=1}^M{B}_{j,m}^{\left(1\right)}{\overline{P}}_{{x\theta }_{i,m}}\right)\sin \left(\gamma t\right) \\ {F}_5=\left({\displaystyle \frac{1}{L}}\sum _{l=1}^N{A}_{i,l}^{\left(1\right)}{\tilde{\overline{M}}}_{{x\theta }_{l,j}}+{\displaystyle \frac{1}{R}}\sum _{m=1}^M{B}_{j,m}^{\left(1\right)}{\tilde{\overline{M}}}_{{\theta \theta }_{i,m}}\right)\sin \left(\gamma t\right) \\ {+c}_1\left({\displaystyle \frac{1}{L}}\sum _{l=1}^N{A}_{i,l}^{\left(1\right)}{\overline{P}}_{{x\theta }_{l,j}}+{\displaystyle \frac{1}{R}}\sum _{m=1}^M{B}_{j,m}^{\left(1\right)}{\overline{P}}_{{\theta \theta }_{i,m}}\right)\sin \left(\gamma t\right) \end{gathered}$$

3. Numerical Results

The software name Lahey–Fujitsu Fortran (Lahey-Fujitsu Fortran 7.8 Microsoft Visual Studio 2015) was used to simulate the data. The frequency $\gamma$ of applied heat flux for the thermal loads of magnetostrictive FGM plates and circular cylindrical shells can be obtained. Also the vibration frequency ${\omega }_{mn}$ for the $u_0$, $v_0$, $w$, ${\varphi }_x$, and ${\varphi }_{\theta }$ of magnetostrictive FGM under simply supported boundary conditions can be obtained, in which subscripts for m and n are the mode shape in the two directions of vibrations [16,17]. The cosine form of coordinates for the grid points numbers $N$ and $M$ of thick magnetostrictive FGM plates and circular cylindrical shells, respectively, are used to study the GDQ results under applied heat with in-plane distributed forces $p_1=p_2=0$ and external aerodynamic pressure load $q$ due to nonlinear supersonic air flow.

3.1. Dynamic Convergence

The magnetostrictive material is Terfenol-D, FGM 1 is SUS304, and FGM 2 is Si₃N₄. The nonlinear supersonic air flow $M_{\infty }=2$, i.e., $U_{\infty }=\mathrm{23,304} in/s$, over the outer surface of Terfenol-D control law on FGM plates/cylindrical shells with $C_{\gamma }$ = 1.4, ${\rho }_{\infty }=0.00000678 lb/i{n}^3$, at altitude 50,000 ft for the numerical GDQ computations including the effect of the advanced nonlinear varied shear correction coefficient. It is very interesting and novel of the nonlinear TSDT thick magnetostrictive FGM plates/cylindrical shells in nonlinear unsteady supersonic flow with a = L, $h^{\ast }$ = 1.2 mm, $h_1$ = $h_2$, and $h_3$ = 0.1 mm. At t = 6 s, $M_{\infty }=2$, $k_c$$c(t)$ = 0, advanced $k_{\alpha }$ values, $L/h^{\ast }$ = 5, $L/R=1$, T = 100 K, ${\overline{T}}_1$ = 100 K in ${{\theta }_L=60}^{°}$, i.e., $a/b=0.5773502$, $R_n$ = 2 are used for the convergence tendencies of middle-plane displacement $w(a/2,b/2)$ (mm) vs. N = M shown in Figure 2 for $c_1$ = 0.925925/mm² that are compared with $c_1$ = 0.0/mm² case. The effect of $c_1$ value on the nonlinear unsteady supersonic flow model is much more significant. Thus, $N\times M$ = 13 × 13 grid points can be used in the GDQ calculations.

3.2. Time Responses

Figure 3 shows the middle-plane displacement $w(a/2,b/2)$ (mm) and normal stress ${\sigma }_x$ (GPa) at center position of outer surface z = 0.5 $h^{\ast }$ for the thermal vibration of nonlinear TSDT of FGM thick plates/cylindrical shells during t = 0.1−3.0 s, in unsteady supersonic flow $M_{\infty }=2$, $k_c$$c(t)$ = 0 under ${{\theta }_L=60}^{°}$ with advanced $k_{\alpha }$, $L/h^{\ast }$ = 5, $L/R=1$, $R_n$ = 2, T = 600 K and ${\overline{T}}_1$ = 100 K. Figure 3a shows the compared response values of $w(a/2,b/2)$ versus t for $c_1$ = 0.925925/mm² and $c_1$ = 0.0/mm², respectively. The maximum value of center displacement amplitude is 16.588001 mm for $c_1$ = 0.925925/mm² and occurs at t = 0.1 s under ${{\theta }_L=60}^{°}$. The values of center displacement amplitudes have the converging tendency with time in both cases of $c_1$ = 0.925925/mm² that is converging to 0.144461 mm and $c_1$ = 0.0/mm² that is converging to −3.63 × 10⁻¹⁰ mm. Figure 3b shows the compared time responses of ${\sigma }_x$ under ${{\theta }_L=60}^{°}$ for $c_1$ = 0.925925/mm² and $c_1$ = 0.0/mm², respectively. The maximum absolute value of ${\sigma }_x$ is −2.1244 × 10⁻³ GPa for $c_1$ = 0.925925/mm² which occurs at t = 2.3 s. The values of ${\sigma }_x$ have the converging tendency firstly with time from t = 0.1 s to 1.8 s then oscillating with time from t = 1.9 s to 2.8 s in the case of $c_1$ = 0.925925/mm². The ${\sigma }_x$ values have a constant tendency firstly with time from t = 0.2 s to 2.5 s then oscillating with time from t = 2.6 s to 3.0 s in the case $c_1$ = 0.0/mm².

3.3. Comparisons of $w(a/2,b/2)$, ${\sigma }_x$ versus T and $R_n$

The comparisons of $w(a/2,b/2)$ versus T for $c_1$ = 0.925925/mm² and $c_1$ = 0.0/mm², respectively, under ${{\theta }_L=60}^{°}$ are shown in Figure 4 at t = 0.1 s, $M_{\infty }=2$, $k_c$$c(t)$ = 0, $L/h^{\ast }$ = 5, $L/R=1$, and ${\overline{T}}_1$ = 100 K. Figure 4a shows the curves of $w(a/2,b/2)$ vs. T and $R_n$ = 1 case; the maximum value of $w(a/2,b/2)$ is 32.773151 mm for $c_1$ = 0.925925/mm², which occurs at T = 1000 K. The $w(a/2,b/2)$ values are firstly decreasing versus T from T = 100 K to T = 600 K, then increasing versus T from T = 600 K to T = 1000 K for $c_1$ = 0.925925/mm². Thus, amplitude $w(a/2,b/2)$ of the $R_n$ = 1 under ${{\theta }_L=60}^{°}$ cannot withstand the higher temperature T = 1000 K of environment for $c_1$ = 0.925925/mm². The values of $w(a/2,b/2)$ versus T are located in the constant line for $c_1$ = 0.0/mm². Figure 4b shows the curves of $w(a/2,b/2)$ vs. T and $R_n$ = 2 case, the maximum value of $w(w(a/2,b/2)$ is 1377.64099 mm occurs at T = 1000 K for $R_n$ = 2. The $w(a/2,b/2)$ values are slightly decreasing versus T from T = 100 K to T = 600 K then increasing versus T from T = 600 K to T = 1000 K for $R_n$ = 2. Thus, amplitude $w(a/2,b/2)$ of the $R_n$ = 2 under ${{\theta }_L=60}^{°}$ cannot withstand the higher temperature T = 1000 K of environment for $c_1$ = 0.925925/mm². The values of $w(a/2,b/2)$ versus T are located in the constant line for $c_1$ = 0.0/mm². Figure 4c shows the curves of $w(a/2,b/2)$ vs. T and $R_n$ = 10 case; the maximum value of $w(a/2,b/2)$ is 14.614171 mm, which occurs at T = 600 K for $R_n$ = 10. The $w(a/2,b/2)$ values are firstly increasing versus T from T = 100 K to T = 600 K, then decreasing versus T from T = 600 K to T = 1000 K for $c_1$ = 0.925925/mm². Thus, the amplitude $w(a/2,b/2)$ of the $R_n$ = 10 under ${{\theta }_L=60}^{°}$ can withstand a higher temperature T = 1000 K of environment for $c_1$ = 0.925925/mm². The values of $w(a/2,b/2)$ versus T are located in the constant line for $c_1$ = 0.0/mm².

The comparisons of ${\sigma }_x$ versus T for $c_1$ = 0.925925/mm² and $c_1$ = 0.0/mm², respectively, are shown in Figure 5 at t = 0.1 s, $k_c$$c(t)$ = 0, $L/R=1$, and ${\overline{T}}_1$ = 100 K. Figure 5 shows the normal stresses ${\sigma }_x$ (GPa) vs. T at the center position of outer surface z = 0.5 $h^{\ast }$ for $M_{\infty }=2$ with $L/h^{\ast }$ = 5 and ${{\theta }_L=60}^{°}$. Figure 5a shows the curves of ${\sigma }_x$ vs. T and $R_n$ = 1 case; the maximum absolute value of ${\sigma }_x$ is −2.8882 × 10⁻³ GPa, which occurs at T = 600 K fo r $R_n$ = 1. The ${\sigma }_x$ absolute values are firstly increasing versus T from T = 100 K to T = 600 K, then decreasing versus T from T = 600 K to T = 1000 K for $c_1$ = 0.925925/mm². Thus, stress ${\sigma }_x$ of the $R_n$= 1 under ${{\theta }_L=60}^{°}$ can withstand the higher temperature T = 1000 K of environment for $c_1$ = 0.925925/mm². The absolute values of ${\sigma }_x$ versus T are located in the straight line for $c_1$ = 0.0/mm². Figure 5b shows the curves of ${\sigma }_x$ vs. T and $R_n$ = 2 case; the maximum value of ${\sigma }_x$ is 1.4213 × 10⁻¹ GPa, which occurs at T = 1000 K for $R_n$ = 2. The absolute values of ${\sigma }_x$ are increasing from T = 100 K to T = 1000 K for $R_n$ = 2. Thus, stress ${\sigma }_x$ of the $R_n$ = 2 under ${{\theta }_L=60}^{°}$ cannot withstand the higher temperature T = 1000 K of environment for $c_1$ = 0.925925/mm². The values of ${\sigma }_x$ versus T are almost located in the constant line for $c_1$ = 0.0/mm². Figure 5c shows the curves of ${\sigma }_x$ vs. T and $R_n$ = 10 case; the maximum absolute value of ${\sigma }_x$ is −8.4455 × 10⁻⁴ GPa, which occurs at T = 1000 K for $R_n$ = 10. The absolute values of ${\sigma }_x$ are increasing from T = 100 K to T = 1000 K for $R_n$ = 10. Thus, stress ${\sigma }_x$ of $R_n$ = 10 under ${{\theta }_L=60}^{°}$ cannot withstand the higher temperature T = 1000 K of environment for $c_1$ = 0.925925/mm². The absolute values of ${\sigma }_x$ are in increasing versus T from T = 100 K to T = 600 K, then decreasing versus T from T = 600 K to T = 1000 K for $c_1$ = 0.0/mm². Thus, absolute stress ${\sigma }_x$ of the $R_n$ = 10 under ${{\theta }_L=60}^{°}$ can withstand the higher temperature T = 1000 K of environment for $c_1$ = 0.0/mm².

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

The transient responses of amplitude $w(a/2,b/2)$ (mm) and stress ${\sigma }_x$ (GPa) at the center position of outer surface z = 0.5 $h^{\ast }$ during t = 0.001–0.025s are presented in Figure 6 for $L/h^{\ast }$ = 5, $M_{\infty }=2$, and $k_c$$c(t)$= 0 under ${{\theta }_L=60}^{°}$, respectively, with $c_1$ = 0.925925/mm² and $c_1$= 0.0/mm², by fixing the values of ${\omega }_{11}$ = 0.008380/s and $\gamma$ = 15.707963/s. Also, it used $L/R=1$, $h^{\ast }$ = 1.2 mm, $R_n$ = 2, T = 600 K, and ${\overline{T}}_1$ = 100 K. Figure 6a shows the comparisons of advanced transient response $w(a/2,b/2)$ under $c_1$ = 0.925925/mm² and $c_1$ = 0.0/mm². The absolute transient values of center displacements for $c_1$ = 0.925925/mm² have greater values of firstly oscillating and then decreasing than that for $c_1$ = 0.0/mm². Figure 6b shows the comparisons of advanced transient response ${\sigma }_x$ under $c_1$ = 0.925925/mm² and $c_1$ = 0.0/mm². The absolute transient values of stresses are greater in an increasing straight line for $c_1$ = 0.0/mm² than for $c_1$ = 0.925925/mm².

Furthermore, the compared transient responses of $w(a/2,b/2)$ (mm) and ${\sigma }_x$ (GPa) in $M_{\infty }=2$ are presented, respectively, in Figure 7 by using the controlled value $k_c$$c(t)$ and uncontrolled value $k_c$$c(t)$ = 0, respectively, during t = 0.001–0.025 s, $c_1$ = 0.925925/mm² for the FGM plates/cylindrical shells of $L/h^{\ast }$ = 5, $R_n$ = 2, T = 600 K, and ${\overline{T}}_1$ = 100 K under ${{\theta }_L=60}^{°}$. Figure 7a shows the compared transient response values of $w(a/2,b/2)$ (mm). The $w(a/2,b/2)$ values in $k_c$$c(t)$ = 1 × 10⁹ case can be controlled into smaller than that in $k_c$$c(t)$ = 0 uncontrolled case. Figure 7b shows the compared transient response values of stress ${\sigma }_x$ (GPa). The stress ${\sigma }_x$ in $k_c$$c(t)$ = 1 × 10⁹ case are in positive tensional values and the different signs with that in $k_c$$c(t)$ = 0 uncontrolled case are in negative compression values. The trend and comparison of the transient curve due to the algorithms in the control process can be described as follows. When an open-loop system is used, the $w(a/2,b/2)$ vibrations with $k_c$$c(t)$ = 0 uncontrolled case are actuated. When a close-loop system is used, the $w(a/2,b/2)$ vibrations with $k_c$$c(t)$ = 1 × 10⁹ controlled case are actuated by a sensor, control-law controller, and Terfenol-D layer to reduce the displacements for FGM plates and shells in supersonic airflow.

A graphical chart explaining the calculated results due to $c_1$, $k_c$$c(t)$, and $M_{\infty }$ is shown in Figure 8.

4. Conclusions

The GDQ solutions are calculated and investigated for the displacements and stresses in the advanced thermal vibration of Terfenol-D control law on FGM plates/cylindrical shells in nonlinear unsteady supersonic flow. The amplitudes of center displacements and stresses, respectively, have a converging tendency with time for the thick $L/h^{\ast }$ = 5. $c_1$ has a significant effect on the values of center displacements and stresses. The amplitude of center displacements for $c_1$ = 0.925925/mm² and $k_c$$c(t)$ = 0 in $R_n$ = 10 under ${{\theta }_L=60}^{°}$ case can withstand a higher temperature T = 1000 K of environment at t = 0.1s. The amplitude of absolute stresses for $c_1$ = 0.0/mm² and $k_c$$c(t)$ = 0 in $R_n$ = 10 under ${{\theta }_L=60}^{°}$ case can withstand a higher temperature T = 1000 K of environment at t = 0.1 s. The amplitude of center displacements in $k_c$$c(t)$ = 1 × 10⁹ case can be controlled to be smaller than that in $k_c$$c(t)$ = 0 uncontrolled case. The nonlinear term value of TSDT and high gain value of Terfenol-D are accomplished in detail more studies on FGM structure. Additionally, the high gain value of the control law algorithm is very important in the reduction of vibration to provide a safe way to enable the manipulation ability of a structure in the supersonic flow. The experimental design has its rationality and would be in the future work if possible.