## 4 Numerical Examples and Illustrations
### Bifurcation and Stability Analysis
#### 4.2.1 In-Plane Excitations  
Firstly, it is assumed that the external excitation is applied in the in-plane direction, and the force-response curves with thermal effects are discussed. Setting two detuning \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \(\Delta T/^{\circ}\)C & \(\chi_{\Delta T}\) & \(\chi_{\Delta T}\) & \(\chi_{\Delta T}\) & \(\chi_{\Delta T}\) & \(\chi_{\Delta T}\) & \(\chi_{\Delta T}\) & \(\chi_{\Delta T}\) \\ \hline \(-40\) & 1.097 & 6.8913 & \(-5295\) & \(-5295\) & \(-196,781\) & \(-590,337\) & \(-10,590\) & \(-590,337\) & \(-1,770,990\) \\ \(0\) & 1.000 & 6.2832 & \(-4402\) & \(-4402\) & \(-163,579\) & \(-490,736\) & \(-8803\) & \(-490,736\) & \(-1,472,200\) \\ \(40\) & 0.886 & 5.5644 & \(-3452\) & \(-3452\) & \(-128,293\) & \(-384,875\) & \(-6904\) & \(-384,875\) & \(-1,154,610\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Linear and non-linear coefficients of in-plane and out-of-plane motions with thermal effects. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(\Delta T/^{\circ}\)C & \(\chi_{\Delta T}\) & \(\chi_{\Delta T}\) & \(\chi_{\Delta T}\) & \(\chi_{\Delta T}\) & \(\chi_{\Delta T}\) & \(\chi_{\Delta T}\) \\ \hline \(-40\) & 1.097 & 1.378 & \(-3.345\) & 1.378 & 2.755 \\ \(0\) & 1.000 & 1.145 & \(-2.781\) & 1.145 & 2.290 \\ \(40\) & 0.886 & 0.898 & \(-2.181\) & 0.898 & 1.796 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Effective interaction coefficients with thermal effects at the first crossover point (CP\({}_{1}\)) between in-plane and out-of-plane mode shapes. Figure 2: The first two in-plane (symmetric) and out-of-plane mode frequencies of suspended cables in thermal environments. parameters \(\sigma_{1}=0.15\) and \(\sigma_{2}=0\), the influences of the in-plane excitation amplitudes \((f_{v})\) on the in-plane \((a_{v})\) and out-of-plane \((a_{w})\) responses in warming and cooling conditions are shown in Figure 3.  
As the excitation amplitude \((f_{v})\) increases from zero, only a single-mode solution \((a_{v})\) is exhibited and increases monotonously from 0 to 0.005. However, provided that the excitation amplitude \((f_{v})\) is greater than 0.002 approximately, another coupled branches appear for some selected initial conditions. As to these two-mode solutions, the non-planar coupled motions are exhibited, and the out-of-plane responses are greater than the in-plane ones. As the excitation amplitude \((f_{v})\) decreases from 0.005, the steady-state responses decrease monotonously until the saddle-node bifurcation SN\({}_{1}\). It is known that SN\({}_{1}\) is a typical static bifurcation and corresponds to the jump phenomenon, and the stability and number of periodic motion states are changed suddenly. As to these two types of solutions, two response amplitudes \((a_{v}\) and \(a_{w})\) in the warming condition are larger than the one in the cooling condition. The saddle-node bifurcation SN\({}_{1}\) is exhibited at a smaller excitation amplitude in the warming condition. Furthermore, some numerical integration solutions are presented by using the fourth-order Runge-Kutta method in Equation (6) directly. As shown in Figure 3, for the single-mode solutions, the perturbation ones are in excellent agreement with the numerical results. For the two-mode solutions, the numerical results seem a little larger than the perturbation ones as to the directly excited amplitudes \(a_{v}\). On the contrary, these numerical results seem a little smaller than the perturbation ones as to internally excited amplitudes \(a_{w}\). Although a little bit of difference is observed, the discrepancies induced by temperature effects are confirmed quantitatively by numerical simulations. Figure 4 illustrates the force-response curves with thermal effects for the second group of detuning parameters, with \(\sigma_{1}=-0.15\) and \(\sigma_{2}=0\). Two types of solutions are presented: single-mode \((a_{v})\) and two-mode \((a_{v}\) and \(a_{w})\). For the single-mode solution, only in-plane motions are exhibited.