## 4 Conclusions  
The dynamic responses of deflection and stress computed by the GDQ method in TSDT thick FGM plates under sinusoidal applied heat loads by considering the parameter effects of advanced \(k_{\alpha}\), \(T\) and \(R_{n}\). By using the simply homogeneous equation to calculate the natural frequency. The advanced nonlinear \(k_{\alpha}\) are usually in functions of \(c_{1}\), \(R_{n}\) and \(T\). The maximum absolute value of \(w(a/2,b/2)\) is 0.006138 mm at \(t=0.1\) s found for \(a/h^{*}=5\) with advanced \(k_{\alpha}=-3.535402\) and \(\gamma=15.707964/\)s. The \(w(a/2,b/2)\) amplitudes in simply homogeneous equation are found in greater values and overestimated in the approach. The transient responses of \(w(a/2,b/2)\) are compared with the cases of simply homogeneous equation and fully homogeneous equation. Also, the transient responses of \(w(a/2,b/2)\) are compared with the cases of nonlinear and linear varied \(k_{\alpha}\). The funding of this paper is by a grant MOST 111-2221-E-164-001 from National Science and Technology Council (NSTC), Taiwan, ROC. Not applicable for studies not involving humans or animals. Not applicable for studies not involving humans. The manuscript completely presented the data used to generate the figures and tables. Data are all available on request. All the data are generated by the author; data are openly available. The author has declared no conflict of interest. There is no funder for this study. There is only the author stated in the manuscripts. The boundary equations of displacements and shear rotations with amplitudes: \(a_{mn}\), \(b_{mn}\), \(c_{mn}\), \(d_{mn}\), and \(e_{mn}\) for 4-sided simply supported plates are given explicitly as follows,  
\[ \begin{array}{l}u^{0}(x,y)=a_{mn}\cos(m\pi x/a)\sin(n\pi y/b)\\ v^{0}(x,y)=b_{mn}\sin(m\pi x/a)\cos(n\pi y/b)\\ w(x,y)=c_{mn}\sin(m\pi x/a)\sin(n\pi y/b)\\ \psi_{x}(x,y)=d_{mn}\cos(m\pi x/a)\sin(n\pi y/b)\\ \psi_{y}(x,y)=e_{mn}\sin(m\pi x/a)\cos(n\pi y/b)\\ \end{array}  \tag{1}\]  
The fully homogeneous equation is expressed as follows,\[\left[ \begin{array}{ccccc}FH_{11}&FH_{12}&FH_{13}&FH_{14}&FH_{15}\\ -\frac{I_{0}\lambda_{nm}}{l_{0}}&&+\frac{c_{1}l_{3}(\frac{m\pi}{l})\lambda_{nm}} {l_{0}}&-\frac{l_{1}\lambda_{nm}}{l_{0}}&\\ FH_{12}&FH_{22}-\frac{I_{0}\lambda_{nm}}{l_{0}}&FH_{23}&FH_{24}&FH_{25}\\ &&+\frac{c_{1}l_{3}(\frac{m\pi}{l})\lambda_{nm}}{l_{0}}&-\frac{I_{1}\lambda_{ nm}}{l_{0}}\\ FH_{13}&FH_{23}&FH_{33}&FH_{34}&FH_{35}\\ +\frac{c_{1}l_{3}(\frac{m\pi}{a})\lambda_{nm}}{l_{0}}&+\frac{c_{1}l_{3}(\frac{ m\pi}{a})\lambda_{nm}}{l_{0}}&-\left[I_{0}+{c_{1}}^{2}I_{6}(\frac{m\pi}{a})^{2} \right.+\frac{c_{1}l_{4}(\frac{m\pi}{a})\lambda_{nm}}{l_{0}}&+\frac{c_{1}l_{4} (\frac{m\pi}{b})\lambda_{nm}}{l_{0}}\\ &&\left.+{c_{1}}^{2}I_{6}\left(\frac{m\pi}{b}\right)^{2}\right]\lambda_{nm}/l_{ 0}&\\ FH_{14}&FH_{24}&FH_{34}&FH_{44}&FH_{45}\\ -\frac{l_{1}\lambda_{nm}}{l_{0}}&&+\frac{c_{1}l_{4}(\frac{m\pi}{a})\lambda_{nm} }{l_{0}}&-\frac{K_{2}\lambda_{nm}}{l_{0}}&\\ FH_{15}&FH_{25}&FH_{35}&FH_{45}&FH_{55}\\ &&-\frac{l_{1}\lambda_{nm}}{l_{0}}&+\frac{c_{1}l_{4}(\frac{m\pi}{a})\lambda_{nm }}{l_{0}}&-\frac{K_{2}\lambda_{nm}}{l_{0}}\end{array} \right] \tag{12}\]  
where \(\lambda_{nm}=I_{0}\omega_{nm}\)2, \(I_{i}=\sum\limits_{k=1}^{N^{*}}\int_{k}^{k+1}\rho^{(k)}z^{i}dz\) (\(i=0\), \(1\), \(2\),..., \(6\)), \(\rho^{(k)}\) is the density of \(k\) th constituent ply, \(FH_{11}=A_{11}(m\pi/a)^{2}+A_{66}(n\pi/b)^{2}\), etc. all of the parameters and coefficients can be referred [10]. The simply homogeneous equation is expressed as follows,  
\[\left[ \begin{array}{ccccc}FH_{11}-\lambda_{mn}&FH_{12}&0&0&0\\ FH_{12}&FH_{22}-\lambda_{nm}&0&0&0\\ 0&0&FH_{33}-\lambda_{mn}&FH_{34}&FH_{35}\\ 0&0&FH_{34}&FH_{44}-\frac{K_{2}}{l_{0}}\lambda_{nm}&FH_{45}\\ 0&0&FH_{35}&FH_{45}&FH_{55}-\frac{K_{2}}{l_{0}}\lambda_{nm}\end{array} \right] \left[ \begin{array}{c}a_{nm}\\ b_{mn}\\ c_{mn}\\ d_{mn}\\ e_{mn}\end{array} \right]=\left[ \begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\end{array}