## 4 Numerical Validation
### Example 3: A Nonlinear Absorber System
It is easy to show that system (43) has the normal canonical modal representation form (4) with  
\[\varphi_{1}=r_{1}^{4}\Big{[}k_{2}+k_{4}(f_{1}+1)(f_{1}-3)^{3}\Big{]},\qquad \varphi_{2}=3r_{1}^{3}r_{2}\Big{[}k_{2}+k_{4}(f_{1}-1)^{2}(f_{1}+1)(f_{2}-1) \Big{]}, \tag{44}\]  
\[\varphi_{3}=3r_{1}^{2}r_{2}^{2}\Big{[}k_{2}+k_{4}\Big{(}f_{1}^{2}-1\Big{)}(f_{ 2}-1)^{2}\Big{]},\qquad\varphi_{4}=r_{1}^{3}r_{2}\Big{[}k_{2}+k_{4}(f_{1}+1)(f _{1}-3)^{3}\Big{]}, \tag{45}\]  
\[\varphi_{5}=r_{1}^{3}r_{2}\Big{[}k_{2}+k_{4}(f_{1}-1)^{3}(f_{2}+1)\Big{]}, \qquad\varphi_{6}=3r_{1}^{2}r_{2}^{2}\Big{[}k_{2}+k_{4}(f_{1}-1)^{2}\Big{(}f_ {2}^{2}-1\Big{)}^{2}\Big{]}, \tag{46}\]  
\[\varphi_{7}=3r_{1}r_{2}^{3}\Big{[}k_{2}+k_{4}(f_{1}-1)(f_{2}-1)^{2}(f_{2}+1) \Big{]},\qquad\varphi_{8}=r_{2}^{4}\Big{[}k_{2}+k_{4}(f_{2}-1)^{3}(f_{2}+1) \Big{]}, \tag{47}\]  
\[\
u_{1}=r_{1}^{2}\Big{[}c_{1}+c_{2}(f_{1}-1)^{2}\Big{]},\qquad\
u_{2}=r_{1}r_ {2}[c_{1}+c_{2}(f_{1}-1)(f_{2}-1)],\qquad\
u_{3}=r_{2}^{2}\Big{[}c_{1}+c_{2}(f _{1}-1)^{2}\Big{]}, \tag{48}\]  
\[\omega_{n1}^{2}=1/(2m_{1}m_{2})\Big{\{}k_{1}m_{2}+k_{3}(m_{1}+m_{2})-\sqrt{[k _{1}m_{2}+k_{3}(m_{1}+m_{2})]^{2}-4k_{1}k_{3}m_{1}m_{2}}\Big{\}} \tag{49}\]  
\[\omega_{n2}^{2}=1/(2m_{1}m_{2})\Big{\{}k_{1}m_{2}+k_{3}(m_{1}+m_{2})+\sqrt{[k _{1}m_{2}+k_{3}(m_{1}+m_{2})]^{2}-4k_{1}k_{3}m_{1}m_{2}}\Big{\}} \tag{50}\]  
\[f_{1}=\Big{(}k_{1}+k_{3}-\omega_{n1}^{2}m_{1}\Big{)}/k_{3},\qquad f_{2}=\Big{(} k_{1}+k_{3}-\omega_{n2}^{2}m_{1}\Big{)}/k_{3},\qquad r_{1}^{2}=1/\Big{(}m_{1}+f_ {1}^{2}m_{2}\Big{)} \tag{51}\]  
Figure 7:0  
To obtain the frequency-amplitude response curves of the nonlinear absorber system, the parameter values of \(m_{1}=10\) kg, \(m_{2}=0.6\) kg, \(c_{1}=0.1\) Ns/m, \(c_{2}=0.08\) Ns/m, \(k_{1}=44\) N/m, \(k_{2}=8\) N/m, \(k_{3}=2\) N/m, \(k_{4}=-0.15\) N/m\({}^{3}\), and \(f_{0}=0.37\) N were considered, which are similar to those used in [50]. Figure 8 illustrates the frequency-amplitude response curves obtained from the numerical integrations of Equations (29) and (43), in which the equivalent damping coefficients, \(\mu_{1}\) and \(\mu_{2}\), are bigger than zero. Figure 8 shows that the decoupled solutions describe the dynamic behavior of the original equations of motion well. The LLE numerically-calculated curves are illustrated in Figure 7:1 As before, these computed LLE curves agree well along the driving frequency range, since the estimated RMSE value is lower than 0.0017. Next, the following system parameter values are used to plot the frequency-amplitude curves shown in Figure 7:2 \(m_{1}=10\) kg, \(m_{2}=0.8\) kg, \(c_{1}=0.1\) Ns/m, \(c_{2}=0.08\) Ns/m, \(k_{1}=44\) N/m, \(k_{3}=2\) N/m, \(k_{4}=-0.65\) N/m\({}^{3}\), \(f_{0}=0.37\) N, and \(k_{2}=0\) and \(8\) N/m\({}^{3}\). As can be seen from Figure 10, the frequency-amplitude curves obtained using the corresponding decoupled expressions follow closely, in spite of having an absorber device that can have linear or nonlinear stiffness effects; these curves are obtained by numerically integrating the original expressions (43). In this case, the corresponding LLE-computed curves for the original and equivalent equations in the sense of Lyapunov are shown in Figure 7:3  
To further assess the accuracy of the proposed approach, the values of \(k_{3}=9.016\) N/m, and \(20.828\) N/m with \(m_{2}=0.6\) kg were used to excite internal resonances of the types, \(\omega_{n2}=2\omega_{n1}\) and \(\omega_{n2}=3\omega_{n1}\), which are second-order and third-order internal resonance relationships, respectively. Figure 12 illustrates the frequency-amplitude curves obtained from the numerical integration solutions of Equations (29) and (43). It is evident that the equivalent equations in the sense of Lyapunov capture both types of internal resonances without any additional assumptions or simplifications in the method.