## 5 Result and Discussion
### Dynamic Stiffness Analysis  
The dynamic stiffness of each spring was further calculated using the FE models. The six DoF for the nodes on one side were fixed to measure the dynamic stiffness; the degree of freedom in the vertical direction was not constrained and left free for the other end, and all other degrees were placed under constraints. A harmonic vertical force with an amplitude of a unit was exerted for every frequency on the end that remained free, and its distribution was on all the nodes present on the surface. The range of the excitation frequency range was from 0 Hz to 200 Hz and had an interval of 0.2 Hz. With excitation at the other end, this process was then repeated. The axial displacement for the end that was excited, as well as the total force of reaction at the end that remained fixed, could be determined. Then, from these two values, the dynamic point stiffness values of k\({}_{11}\) and k\({}_{22}\) could be found; in addition, the dynamic transfer stiffness of k\({}_{12}\) was equal to k\({}_{21}\).  
Figure 9a,b illustrates the magnitude of the transfer and point dynamic stiffnesses for the two individual springs. A material damping loss factor of 0.001 was assumed. It can be observed that the dynamic stiffness tended to a constant at low frequencies: 115.74 kN/m and 205.14 kN/m for inner and outer springs, respectively. The point stiffness decreased above about 10 Hz, whereas there was an increase in the transfer stiffness. In the point stiffness of inner as well as outer springs, the first dips occurred at 29.15 Hz and 29.42 Hz. These frequencies correspond to the first natural frequencies of the springs when they were under the fixed-free conditions for the boundary. Moreover, for fixed-fixed boundary conditions, the fist peaks for the point stiffness values occurred at the frequencies of 58.2 Hz (inner) and 51.9 Hz (outer). There were a large number of other modes which were excited at higher frequencies, and the values of stiffness varied from 1 kN/m to more than 1e5 kN/m.  
Similarly, Figure 10a,b illustrates the value of the transfer and point dynamic stiffnesses for the spring set and spring set with rubber, respectively. It can again be observed that the dynamic stiffness tended to a constant at low frequencies. However, at higher frequencies, this changed significantly, and the values differed by multiples of 100. By contrasting the effects of the spring collection alone, the stiffness limit at low frequencies was reduced only slightly, from 320 kN/m to 310 kN/m. Due to the resilience of the rubber pad, the peak values of dynamic stiffness were removed. High-frequency dynamics were amplified, with peak values of about 30 MN/m. This illustrates how the inclusion of the rubber pad can reduce the stiffening effect of the coil springs at high frequencies.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Mode** & **Innerspring (Hz)** & **Outer Spring (Hz)** &
\begin{tabular}{c} **Vertical** \\ **Mode Shape** \\ \end{tabular} \\ \hline
1 & 59.32 & 53.14 & First \\
2 & 116.92 & 100.24 & Second \\
3 & 168.68 & 135.49 & Third \\ \hline \hline \end{tabular}
\end{table}
Table 2: Modal analysis for the first three vertical mode shapes for inner and outer springs.