## 3 Sensitivity Analysis of the Parameter
### Effect of External Force on Repeated Strain  
Finally, the impact of changing the size of the external force (Q\({}_{0}\)) on the simulated displacement of the mass-spring-damping system is discussed. For this type, the external force is reduced using the calculation result Q\({}_{0}\) = 10.1 sin2\(\pi\)t kg as the original data. Ten times and ten times magnification, two different sizes of external forces Q\({}_{0}\) can be obtained, Q\({}_{0}\) = 1.01 sin2\(\pi\)t kg and Q\({}_{0}\) = 101 sin2\(\pi\)t kg, respectively, I1 to I3, as shown in Table 8, to discuss the mass-spring-damping system when simulating the effect of the strain.  
Substitute the above three sets of parameters into calculating the mass-spring-damping system to explore the impact of reducing the external force (Q\({}_{0}\)) by ten times and amplifying it by ten times on the displacement. After calculation with Mathematica software, the displacements from I1 to I3 and the time chart are shown in Figure 9.  
Figure 8: Strain versus cyclic number for different damping coefficient c.  
Figure 9: Strain versus cyclic number for different external force Q\({}_{0}\).  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter in m-k-c** & \multicolumn{3}{c}{**Value Q\({}_{0}\)**} \\ \hline No. & I1 & I2 & I3 \\ \hline Q\({}_{0}\) (kg) & 10.1 sin2\(\pi\)t & 1.01 sin2\(\pi\)t & 101 sin2\(\pi\)t \\ \hline k (kgf/cm) & \multicolumn{3}{c}{244\(\mathrm{e}^{-0.1\mathrm{t}}\)} \\ \hline m (kg) & \multicolumn{3}{c}{1} \\ \hline c (kgf-s/cm) & \multicolumn{3}{c}{0.739\(\mathrm{e}^{-0.257\mathrm{t}}\)} \\ \hline \hline \end{tabular}
\end{table}
Table 8: The parameter value for different external force Q\({}_{0}\).  
From Figure 9 stated above, we know that based on the original data of I1 Q\({}_{0}\) = 10.1 sin2\(\pi\)t kg, the strain will be affected by the increase or decrease in the external force (Q\({}_{0}\)), and the two are proportional and seem not to be multiples.  
From the preceding content, the summary of parameter analysis can be summarized as follows.  
1. Mass (m) is a constant. According to the comparison result of changing the size of mass m, it can be found that it is inversely proportional to the displacement and is not a multiple. When the mass is reduced ten times, the displacement will be enlarged, and when the mass is reduced by ten times, the displacement will be enlarged. When the mass is enlarged ten times, the displacement will be slightly reduced.
2. The spring coefficient (k) is a variable that changes with time. According to the comparison results of changing the spring coefficient k, it can be found that the displacement will be affected by the increase or decrease in the spring coefficient in an inverse proportion and not a multiple relation.
3. The damping coefficient (c) is a variable that changes with time. According to the comparison results of changing the damping coefficient c, the impact of the damping coefficient on displacement could be more apparent.
4. The external force (Q\({}_{0}\)) is the cyclic load multiplied by Sin2\(\pi\)t. According to the comparison results of changing the magnitude of the external force Q\({}_{0}\), it can be found that the displacement will be affected by the increase or decrease in the external force in a proportional and non-multiple relationship.
5. After reducing and enlarging the original data of spring coefficient k and damping coefficient c ten times and discussing, respectively, the influence of each parameter on the displacement can be obtained, as shown in Table 9.