## 4 Sub-Optimal STMD with Clipped Viscous Damping-Stiffness Control Force
### Numerical Analysis of Sub-Optimal STMD  
Figure 7 presents the frequency responses calculated for the sub-optimal STMD with \(\mu=1\%\), for six different values of \(K\) from 2 to 20, in comparison to the frequency responses obtained for the TMD. These results clearly indicate that the performance of the sub-optimal STMD is noticeably worse than that of the ATMD (Figure 2). This is attributed to the described increase in damping and reduction in stiffness as the result of clipping.  
For normalized frequencies \(\omega/\omega_{1}\) close to \(1\), the control force (33) is minimally affected by clipping. As a result, in this frequency range, the sub-optimal STMD operates as intended, and the resulting vibration amplitude \(X_{1}\) is consistent with that for the TMD with an enlarged mass ratio \(K\mu\) (Figure 7a). Specifically, when \(\omega=\omega_{1}/\sqrt{\mu+1}\), the stiffness \(k_{a}\) becomes zero, making the control force of the STMD purely dissipative. At this central frequency, the performance of the STMD is equivalent to that of the ATMD.  
Figure 6: Force parameters before clipping and their equivalent counterparts after clipping versus normalized frequency of vibration for the sub-optimal STMD with \(\mu=1\%\) and \(K=3\): (**a**) Viscous damping coefficient \(c_{a}\) and equivalent viscous damping coefficient \(c_{eq}\); (**b**) stiffness \(k_{a}\) and equivalent stiffness \(k_{eq}\).  
Figure 7: Frequency responses of the primary structure with the sub-optimal STMD with \(\mu=1\%\), for six different \(K\) ranges from 2 to 20, in comparison to the frequency responses due to the TMD with \(\mu=1\%\), \(2\%\) and \(4\%\): (**a**) Dynamic amplification of primary structure displacement; (**b**) dynamic amplification of relative displacement.  
The numerical analysis revealed that the sub-optimal STMD has a clear limit in effectiveness, determined as the smallest achievable value of maximum dynamic amplification factor (\(DAF_{max}\)). For \(\mu=1\%\), the calculations show that the minimum \(DAF_{max}\) of 10.27 is achieved for \(K=4.04\), as indicated by the red line in Figure 7a. Increasing \(K\) beyond 4.04 can only result in an increase in \(DAF_{max}\), with a slightly higher right peak.  
Additional significant limitations of the sub-optimal STMD are evident in Figure 7b. The maximum relative displacement for the sub-optimal STMD far exceeds that of a TMD with the same mass. Such performance from STMD is considered unacceptable. In practice, constraining the relative motion is crucial due to spatial limitations at the installation site of the mass damper. Active and semi-active solutions are expected to fit within the same space as a TMD with the same mass. This consideration is also taken into account in the subsequent numerical optimization of the STMD.  
Note that the frequency responses shown in Figure 7 were calculated using (16) and (17), as previously in Section 3.5. However, in these calculations, \(c_{a}\) and \(k_{a}\) were substituted with the equivalent model \(c_{eq}\) (44) and \(k_{eq}\) (45), respectively. To assess the accuracy of the equivalent model, analogous steady-state frequency responses were also determined through simulations in MATLAB/Simulink. Based on this comparison, it was determined that regardless of the amplitude of force excitation, the maximum discrepancy between the calculated and simulated amplitude \(X_{1}\) in the steady state does not exceed 0.58% for \(K=3\), 0.93% for \(K=4\), and 2.7% for \(K=20\). Therefore, the equivalent model (44) and (45) is assumed to be sufficiently accurate and useful in predicting the frequency responses of the primary structure equipped with the STMD.  
Appendix A provides the MATLAB code for calculating the frequency responses for the sub-optimal STMD.