## 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, \(\theta_{\text{L}}\) is left-side angle of intersection, \(\theta_{\text{R}}\) 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, \(h^{\star}\) is total thickness of magnetostrictive FGM plate/cylindrical shell, i.e., \(h^{\star}=h_{1}+h_{2}+h_{3}\). 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]:  
\[ \begin{split}\begin{cases}u\\ v\\ w\end{cases}=\begin{cases}u^{0}(x,y,t)\\ v^{0}(x,y,t)\\ w(x,y,t)\end{cases}+z\begin{cases}\psi_{x}(x,y,t)\\ \psi_{y}(x,y,t)\\ 0\end{cases}-c_{1}z^{3}\begin{cases}\psi_{x}+\frac{\partial w}{\partial x} \\ \psi_{y}+\frac{\partial w}{\partial y}\\ 0\end{cases}\end{split}  \tag{1}\]  
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. \(c_{1}=4/[3(h^{\star})^{2}]\) 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]:  
\[\left\{ \begin{aligned} & u\\ v\\ w\end{aligned} \right\}=\left\{ \begin{aligned} & u_{0}(x,\theta,t)\\ v_{0}(x,\theta,t)\\ w(x,\theta,t)\end{aligned} \right\}+z\left\{ \begin{aligned} &\phi_{x}(x,\theta,t)\\ \phi_{\theta}(x,\theta,t)\\ 0\end{aligned} \right\}-c_{1}z^{3}\left\{ \begin{aligned} & \phi_{x}+\frac{\partial w}{\partial x}\\ &\phi_{\theta}+\frac{\partial w}{\partial x\theta}\\ 0\end{aligned} \right\} \tag{2}\]  
where \(u_{0}\), \(v_{0}\), and \(w\) are displacements in the \(x\), \(\theta\), and \(z\) directions, respectively. \(R\) denotes the middle-surface radius. \(\phi_{x}\) and \(\phi_{\theta}\) 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 \(c_{1}z^{2}\) 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 \(\sigma_{x\theta}\), \(\sigma_{yz}\), \(\sigma_{xz}\) 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:  
\[\left\{ \begin{aligned} &\sigma_{x}\\ &\sigma_{\theta}\\ &\sigma_{x\theta}\end{aligned} \right\}_{(k)}=\left[ \begin{array} []{cc}\overline{Q}_{11}&\overline{Q}_{12}&\overline{Q}_{16}\\ \overline{Q}_{12}&\overline{Q}_{22}&\overline{Q}_{26}\\ \overline{Q}_{16}&\overline{Q}_{26}&\overline{Q}_{66}\end{array} \right]_ {(k)}\left\{ \begin{aligned} &\varepsilon_{x}-\alpha_{x}\Delta T\\ \varepsilon_{\theta}-\alpha_{\theta}\Delta T\\ \varepsilon_{x\theta}-\alpha_{x\theta}\Delta T\\ 0\end{aligned} \right\}_{(k)}-\left[ \begin{array}{cc}0&0& \widetilde{c}_{31}\\ 0&0&\widetilde{c}_{32}\\ 0&0&\widetilde{c}_{36}\end{array} \right]_{(k)}\left\{ \begin{aligned} & 0\\ 0\\ \widetilde{H}_{2}\end{aligned} \right\}_{(k)} \tag{3}\] \[\left\{ \begin{aligned} &\sigma_{\theta z}\\ &\sigma_{xz}\end{aligned} \right\}_{(k)}=\left[