# A Semi-Analytical Approach for the Linearized Vibration of
## 4 Numerical Results and Discussions
### Results and Discussion
#### 4.2.3 The Effect of Combined Static and Thermal Load  
To investigate the effect of a combination of static and thermal loads on the dynamic behavior of the structure, different sets of combined loads are listed in Table 8. The deflections of beams are illustrated in Figure 11. The deflections of the structure increase non-linearly as the combined loads increase, with the maximum deflections of 1.14 mm, 1.99 mm, 2.14 mm, and 2.99 mm for cases _c1_, _c2_, _c3_, and _c4_ respectively. The thermal effect causes structural stiffness to decrease, resulting in a significant increase in static  
Figure 10: RMS of acceleration in the z-direction under different thermal loads. Figure 9: PSD curves of acceleration in the z-direction under different thermal loads at various locations: (**a**) PSD curves for a point located at \(x=150\) mm (**b**) PSD curves for a point located at \(x=100\) mm; (**c**) PSD curves for a point located at \(x=75\) mm. deformation when exposed to combined loads. The analysis indicates that the deflection curve under the same static load is similarly affected by a uniform thermal load. The effect of the combined load is more complex in modal analysis than in static analysis. The first five-mode frequencies are presented in Table 9. In the majority of conditions, the decrease in structural stiffness due to the thermal load is more pronounced than the stiffness increase resulting from the geometric nonlinear effect of the static deformation, leading to a significant reduction in the mode frequencies. Under the same temperature load, the increase in static deformation caused by the increase in static load leads to a slight increase in structural mode frequencies. Compared with the single load case, the effects of static and thermal loads on the dynamic characteristics of the structure cannot be simulated by linear superposition. The coupling effect between thermal and static deformation needs to be considered in the modal analysis. As the static deformation increases, the 1 st and 2 nd mode shapes remain relatively unchanged and can still be approximated using triangular functions. However, the 3 rd, 4 th, and 5 th mode shapes undergo significant changes which are shown in Figure 12. The peak values of the mode shapes near the center of the beam are significantly lower than those on the two sides. Figure 11: Deflections of beams under different combined loads. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Case** & **Ref** & \(c1\) & \(c2\) & \(c3\) & \(c4\) \\ \hline \(P_{0}\) (N/mm) & 0 & 0.25 & 0.5 & 0.25 & 0.5 \\ Temperature (°C) & 20 & 50 & 50 & 100 & 100 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Different cases set of combined loads. As depicted in Figure 13, the PSD contour of acceleration of the beam under different static-thermal combined loads is analyzed. The analysis reveals that, with the static-thermal load increment, the structural nephogram undergoes minor changes. And the peak frequency of the nephogram shifts with the fluctuation of mode frequencies. Figure 12: The mode shapes under different combined loads: (**a**) the 3 rd mode shape; (**b**) the 4 th mode shape; (**c**) the 5 th mode shape. Figure 13: PSD contour of acceleration in the z-direction under different combined loads (\(\text{g}^{2}/\text{Hz}\)): (**a**) case \(c1\); (**b**) case \(c2\); (**c**) case \(c3\); (**d**) case \(c4\). The PSD curves of acceleration under different combined loads for points located at \(x=150\) mm, \(x=100\) mm, and \(x=75\) mm are shown in Figure 14. After analyzing the load range, the effect of thermal loads on beam response is greater than that of static loads. The peak frequency of structural response decreases due to the influence of thermal loads and increases due to the influence of static loads. By comparing \(c1\) with \(c3\) and \(c2\) with \(c4\), it can be concluded that temperature has a greater impact than static loads.