# Error Caused by Damping Formulating in Multiple Support Excitation Problems
## 5 Numerical Simulation
### Cascaded Two Supports Multiple Degrees of Freedom Structure  
Firstly, cascaded two supports multiple degrees of freedom structures are used to demonstrate the errors exist in the modal frequency response function. In this part, concentrated multiple degrees of freedom mass-spring-damper systems are used in the numerical simulation to show the features of the virtual damping error and the damping truncation error in the FRF in modal coordinates. The error prediction formulas derived in the previous sections are testified. A sketch drawing of the structure is shown in Figure 6. In this model, n masses and n+1 springs are used. In Figure 6, \(m=1\) kg, the stiffnesses of the springs are 1 N/m. Rayleigh damping matrix is calculated with the first and the penultimate mode. The damping ratio of the structure is assumed to be 0.05. In the equation of motions derived in the previous section, the excitation coefficients are exactly the same. As a result, when the modal frequency response functions are studied, the form of excitation can only amplify the absolute value of the frequency response function but not the shape of it along the frequency axis. In this simulation, the excitations at the supports are assumed to be identical. Different numbers of masses are used to study the influence of \(\
u\) on the damping truncation error. The structure parameters of structures having different numbers of masses are shown in Table 6. For each structure, the mode with the smallest damping ratio is used to calculate \(\mu\) and \(\
u\). These modes are also used for the calculation of the errors in the following context. Figure 6: Sketch drawing of the numerical model. The Fundamental frequency of the structures decreases with the increase of the mass number; whereas \(\mu\) and \(\
u\) increases with it. Figure 7 shows the absolute errors and error ratios in the modal FRF of the 4 th mode, when \(n=20\), and when LMM-R and RMM-R are use. The Rayleigh damping matrices are calculated with the 1 st and the 19 th mode. As shown in Figure 7a, the input of this frequency response function is acceleration. The value of both the dashed line and the dotted line close to 0 Hz is significant showing the influence of the second term at the right-hand side of Equation (15). The error in this part is the main component of the virtual damping error in LMM-R. One peak of the error and error ratio appears at the resonance point. The shape of the dotted lines around the resonance point in Figure 7a,b are similar but with small differences, showing that the error in LMM-R near the resonance point contains little virtual damping error and is mainly composed of the damping truncation error. The varying trend of the error ratio is the same as predicted in the previous section. The largest value of the error ratio can be as large as 94% at the resonance point. Figure 8 shows the errors and error ratios of ADM-R and ADM-M when \(n=20\). \begin{table}
\begin{tabular}{c c c c} \hline \hline \(n\) & \(\omega_{1}\) (Rad) & \(\mu\) & \(\
u\) \\ \hline
5 & 0.518 & 1.93 & 3.35 \\
10 & 0.285 & 2.92 & 6.74 \\
15 & 0.196 & 2.96 & 10.01 \\
20 & 0.149 & 3.94 & 13.23 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Parameters of the numerical simulations for structures with different number of degree of freedom. Figure 7: Frequency response function of the 4 th mode when n = 20, error and error ratio for different modeling methods, (**a**) large mass method with the Rayleigh damping method (LMM-R) (**b**) relative motion method with the Rayleigh damping method (RMM-R). As shown in Figure 8a, the varying trend of the dotted line can be approximated by a linear function which is predicted in Equation (33). The peak value of the dashed line, which is the absolute value of the error, also appears at the resonance point. In Figure 8b, the behavior of the error ratio is more complex than the other three models, multiple peaks appear in the dashed line; but the largest error ratio appears at the resonance point.