## 4 Sub-Optimal STMD with Clipped Viscous Damping-Stiffness Control Force
### Energy Dissipation Resulting from Clipping  
Another aspect in the analysis of the clipping operation concerns its impact on the energy dissipation during a vibration cycle. As illustrated by the force-displacement loops in Figures 3b and 4b, the clipping augments cycle energy dissipation. This is evident from the enlargement of the area enclosed by the force-displacement plot. This area is quantitatively equal to the cycle energy. The total cycle energy resulting from the clipped force (33) is represented by the dashed area in Figure 3b and can be expressed as:  
\[E_{clipped}=E_{unclipped}+\Delta E, \tag{38}\]  
where \(E_{unclipped}=\pi\omega c_{a}X_{d}^{2}\) is the cycle energy due to the active control force \(f_{a}\), and \(\Delta E\) denotes the increase in cycle energy due to the clipping. This additional energy \(\Delta E\) is represented by the shaded areas in Figure 3b and, for \(k_{a}>0\), can be calculated as:  
\[\Delta E=2\int_{x_{d0}}^{X_{d}}f_{c2}dx_{d}. \tag{39}\]  
where \(f_{c2}\) and \(x_{d0}\) are given by (35) and (36), respectively. Evaluating the integral in (39) yields an analytical expression for the increase in cycle energy due to the clipping that is valid for both positive and negative stiffness \(k_{a}\):  
Figure 5: Clipping coefficient versus normalized frequency of vibration calculated for the sub-optimal STMD for different \(K\): (**a**) For \(\mu=1\%\); (**b**) for \(\mu=2\%\). \[\Delta E=X_{d}^{2}\Big{(}c_{a}\omega\Big{(}\alpha\sqrt{1-\alpha^{2}}+arcsin(\alpha )-\frac{\pi}{2}\Big{)}-|k_{a}|\Big{(}\alpha^{2}-1\Big{)}\Big{)}. \tag{40}\]  
After substituting (40) into (38), the total cycle energy resulting from the clipped force (33) is expressed as ((40) with the sign before the \(\pi/2\) term changed):  
\[E_{clipped}=X_{d}^{2}\Big{(}c_{a}\omega\Big{(}\alpha\sqrt{1-\alpha^{2}}+arcsin( \alpha)+\frac{\pi}{2}\Big{)}-|k_{a}|\Big{(}\alpha^{2}-1\Big{)}\Big{)}. \tag{41}\]  
The above expression can be employed to derive the energy-equivalent viscous damping coefficient, \(c_{eq}\), by equalizing \(E_{clipped}=\pi\omega c_{eq}X_{d}^{2}\). However, to establish a full equivalent model for the clipped force (33), an adequate equivalent linearization method must be employed to also derive an equivalent stiffness. Calculation of Equivalent Viscous Damping and Equivalent Stiffness for the Clipped Viscous Damping-Stiffness Force Using Harmonic Balance Method  
The method of equivalent linearization by [PERSON] and [PERSON] (2010) is a well-established analytical technique based on the harmonic balance method for linearizing non-linear systems. It can be used to replace a non-linear function of strongly non-linear systems, see e.g., ([PERSON] and [PERSON], 2010), with an equivalent linear function. In the context of semi-active vibration control systems, a good example can be found in ([PERSON] and [PERSON], 2010). Using this method, the equivalent damping and stiffness are determined by multiplying the force (33) by \(cos(\omega t)\) and \(sin(\omega t)\), respectively, and then integrating the resulting expressions over a full vibration period:  
\[c_{eq}=\frac{1}{X_{d}\pi}\int_{0}^{2\pi/\omega}f_{a-clipped}cos(\omega t)dt, \tag{42}\]  
\[k_{eq}=\frac{\omega}{X_{d}\pi}\int_{0}^{2\pi/\omega}f_{a-clipped}sin(\omega t)dt. \tag{43}\]  
Detailed analytical calculations for integrals (42) and (43) are provided in Appendix B. As an outcome of these calculations, the following expressions are derived:  
\[c_{eq}=\frac{c_{a}}{2}+\frac{c_{a}arcsin(\alpha)}{\pi}+\frac{c_{a}sin(2 arcsin (\alpha))}{2\pi}-\frac{|k_{a}|\big{(}\alpha^{2}-1\big{)}}{\pi\omega}, \tag{44}\]  
\[k_{eq}=\frac{k_{a}}{2}+\frac{k_{a}arcsin(\alpha)}{\pi}-\frac{k_{a}sin(2 arcsin (\alpha))}{2\pi}-\frac{sgn(k_{a})c_{a}\omega(\alpha^{2}-1\big{)}}{\pi}. \tag{45}\]  
It should be noted that although the amplitude of relative motion \(X_{d}\) appears in (42) and (43), the resulting equivalent model is independent of the amplitude \(X_{d}\). Consequently, the equivalent model is independent of the amplitude of force excitation.