## 4 Advantages and Limitations of MSLP
### Parameter Dependency of the Operating Range  
For engineering purposes, it might be interesting to know the maximum amplitude of the system \(x_{max}=\hat{u}_{max}+q_{max}\) to set the parameters respective to the available space. The maximum difference amplitude can be calculated by looking at the square root of the frequency-response Equation (36) in question. Since the maximum point is the point where the expression containing the negative square root goes over into the expression with the positive square root, the maximum point is where both expressions are equal. This means that the square root is zero. Thereby  
\[\sqrt{\frac{q_{i,j}^{2}q^{*2}}{4\hat{u}_{max}^{2}}-\frac{{d^{*}}^{2}}{4\omega^ {2}}}=\sqrt{\frac{q_{i,j}^{2}{q^{*}}^{2}}{4\hat{u}_{max}^{2}}-\frac{{d^{*}}^{2 }}{4\left(\omega_{0}^{2}+\frac{3}{2}\alpha^{*}\hat{u}_{max}^{2}\right)}}\ =0 \tag{62}\]  
solves to  
\[\hat{u}_{max}=\sqrt{\frac{\frac{4\omega_{0}^{2}}{{d^{*}}^{2}}}{\left(\frac{4} {\omega_{i,j}^{2}{q^{*}}^{2}}-\frac{3\alpha^{*}}{{d^{*}}^{2}}\right)}} \tag{63}\]  
wherein only the positive square roots are being used. An additional friction element yields to a much larger expression for the maximum amplitude:  
\[\hat{u}_{max}=\sqrt{-\frac{B-D}{2(A-E)}+\sqrt{\left(\frac{B-D}{2(A-E)}\right) ^{2}-\frac{C}{A-E}}} \tag{64}\]  
with  
\[\begin{array}{c}A=\left(\frac{3}{4}\alpha^{*}\varphi_{i,j}q^{*}\pi\alpha \omega_{0}\right)^{2}\\ B=\frac{3}{2}\alpha^{*}\varphi_{i,j}q^{*}\pi\omega_{0}\left(\pi\varphi_{i,j}q^ {*}\omega_{0}^{3}-4\mu^{*}\right)\\ C=\left(\pi\varphi_{i,j}q^{*}\omega_{0}^{3}-4\mu^{*}\right)^{2}\\ D=\left(\pi d^{*}\omega_{0}^{2}\right)^{2}\\ E=\frac{3}{4}\alpha^{*}\left(\pi d^{*}\omega_{0}\right)^{2}\end{array} \tag{65}\]  
which is again obtained by analyzing the square root expression of the frequency-response equation in question. Using these techniques, a three-dimensional plot can be created, showing how the damping parameter \(d^{*}\) and the coefficient of the nonlinearity \(\alpha^{*}\) affected the maximum amplitude \(\hat{u}_{max}\) for a given excitation.  
As the point of the maximum amplitude is characterized by the square root in Equation (36) being zero, the frequency of the maximum amplitude can be described by  
\[f(\hat{u}_{max})=\omega\left(1+\frac{3{\alpha^{*}}^{2}\hat{u}_{max}{}^{4}}{256 \omega^{4}}\right);\ \omega=\sqrt{\omega_{0}^{2}+\frac{3{\alpha^{*}}^{4}\hat{u}_{max}{}^{2}}{4}} \tag{66}\]  
which, by the use of Equation (45), can be further simplified to  
\[f(\hat{u}_{max})=\omega_{0}\frac{256{d^{*}}^{4}+3{\alpha^{*}}^{2}\left(\varphi _{i,j}q^{*}\right)^{4}}{128{d^{3}}\sqrt{4{d^{*}}^{2}-3{\alpha^{*}}\varphi_{i,j }^{2}{q^{*}}^{2}}} \tag{67}\]or  
\[\partial f=f(\hat{u}_{max})-f(\hat{u}_{0})=\omega_{0}\left(\frac{256d^{*^{4}}+3 \kappa^{*^{2}}\left(\varphi_{i,j}q^{*}\right)^{4}}{128d^{3}\sqrt{4d^{*^{2}}-3 \alpha^{*}\left(\varphi_{i,j}q^{*}\right)^{2}}}-1\right) \tag{68}\]  
as an expression for the frequency shift \(\partial f\) of the viscous damper (see Figure 4a).  
Figure 4b shows again that a higher nonlinearity coefficient and a lower damping coefficient result in a higher maximum amplitude. According to the imagination, a higher nonlinearity coefficient \(\alpha\) leads to a higher frequency shift. However, an increase of the damping coefficient \(d^{*}\) decreases this frequency shift as it decreases the maximum amplitude. Thereby, the combination of a very small maximum amplitude and a very large frequency shift can be a mutual contradiction.