# A Semi-Analytical Approach for the Linearized Vibration of
## 4 Numerical Results and Discussions
### Results and Discussion  
4.2.1. The Effect of Static Load  
The influence of the deflections of the beam caused by different static loads on the structural response at room temperature is investigated. Table 4 displays the set of static loads, and Figure 3 illustrates the corresponding deflections. It is observed that the deflection increases in a nonlinear fashion with increasing static loads. The maximum deflections for cases _al_, \(a\)2, and \(a\)3 are 0.17 mm, 0.83 mm, and 1.56 mm respectively. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Case** & **Ref** & _a_**1** & _a_**2** & _a_**3** \\ \hline \(P_{0}\) (N/mm) & 0 & 0.05 & 0.25 & 0.5 \\ Temperature (°C) & & 20 & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Different cases set of static loads. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Model Order** & **1** & **2** & **3** & **4** & **5** \\ \hline Takabatake & 302.7 & 822.4 & 1602.1 & 2640.5 & 3938.1 \\ \(n\) = 5 & 299.5 & 825.4 & 1600.8 & 2640.5 & 3938.3 \\ \(n\) = 10 & 298.1 & 826.2 & 1604.8 & 2642.6 & 3939.9 \\ \(n\) = 30 & 297.8 & 826.5 & 1605.2 & 2642.9 & 3940.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The first five-mode frequencies of beams under static load (Hz). Figure 2: Deflection of a clamped-clamped beam under static load. Table 5 presents the first five-mode frequencies, which demonstrate that the mode frequencies increase with the increase of pressure load. Furthermore, the growth rate increases with the increase of deflection. The deflection under uniform pressure load is comparable to the fundamental mode shapes. Additionally, the lower-order mode frequencies are more sensitive to pressure load than higher-order ones. Moreover, there is a slight variation in the 3 rd and 4 th mode shapes with the increase of static load. Figure 4 illustrates that the wave peak near the center of the beam is lower than the sides. Following the dynamic characteristic analysis, the mode-superposition method is employed to compute the random response analysis of beams while taking into account the effects of static and thermal loads. Figure 5 presents the z-direction acceleration power spectral density nephogram of the beam under the influence of different static loads. It can be observed that the symmetrical mode accounts for a significant portion of the structural response. Moreover, the 1 st, 3 rd, and 5 th-order modal responses are the primary responses in the nephogram. Furthermore, an increase in the static load does not significantly alter the distribution of the structural acceleration power spectral density nephogram. This is due to only a slight change in the structural mode shapes and frequencies. Figure 4: The mode shapes under different static loads: (**a**) the 3 rd mode shape; (**b**) the 4 th mode shape. Figure 3: Deflections of beams under different static loads. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Case** & **1** & **2** & **3** & **4** & **5** \\ \hline ref & 293.9 & 810.6 & 1589.0 & 2626.9 & 3924.1 \\ \(a1\) & 294.1 & 811.2 & 1589.8 & 2627.5 & 3924.8 \\ \(a2\) & 297.8 & 826.5 & 1605.2 & 2642.9 & 3940.0 \\ \(a3\) & 306.8 & 864.4 & 1644.4 & 2682.2 & 3979.4 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: The first five-mode frequencies under different static loads (Hz). The PSD curves of acceleration for points located at \(x=150\) mm, \(100\) mm, and \(75\) mm are shown in Figure 6. With an increase in the static force, the initial deformation of the structure increases, causing a shift in the peak frequency of the structural response to a higher frequency. Additionally, the amplitude of the higher-order response becomes greater than that of the fundamental frequency. It is observed that the 1 st mode frequency remains relatively unaffected by the initial deformation. However, the 3 rd mode frequency is highly impacted by it. It is noteworthy that the anti-resonance peak after the third-order mode frequency is significantly shifted to a higher frequency due to the alteration of mode frequency.