## 4 Numerical Validation
### Example 1: Dynamic System with Cubic Nonlinearities
Therefore, it is concluded that the replacement of the original equation of motion for two equivalent equations of the Duffing-type describes the qualitative and quantitative dynamic system behaviors well, as shown in Figure 2, in which the frequency-amplitude backbone curves for several values of \(k\) and \(\varepsilon\) have been plotted. Next, the damped nonlinear case was considered with parameter values of \(m=1\), \(c=0.3\), \(k=1\) and \(\varepsilon=0.5\) to compare the numerical integration of Equation (30) with respect to those given by expressions (29). The results of the simulations in physical coordinates are depicted in Figure 3 in which the amplitude-time curves have been plotted by using the system's initial conditions of \(\big{(}x_{1},\ x_{2},\ \dot{x}_{1},\ \dot{x}_{2}\big{)}=(0,\ 0,\ 2,\ 0)\). It is evident from Figure 3, that the decoupled Equation (29) is in agreement with the numerical integration solutions of the original equations of motion (30). In fact, the LCEs of the original system have mean values of \(\lambda_{1}=-0.0756\), \(\lambda_{2}=-0.0802\), \(\lambda_{3}=-0.3698\), and \(\lambda_{4}=-0.3743\ \text{bits}/\text{second}\), while the LCEs computed from the equivalent models are \(\lambda_{1\varepsilon}=-0.0749\), \(\lambda_{2\varepsilon}=-0.0750\), \(\lambda_{3\varepsilon}=-0.3726\), and \(\lambda_{4\varepsilon}=-0.3773\ \text{bits}/\text{second}\). In spite of having some discrepancies in the computed numerical values, the order of magnitude of the LCEs of the equivalent expressions tend to be similar to those computed from the original equations of motion with root mean square error (RMSE) values of \(0.1117\) and \(0.0788\) for the first and second mode, respectively. To further assess the accuracy of the proposed nonlinear decoupling procedure, the frequency-amplitude steady-state response curves of the forced, damped nonlinear system (30) were plotted with parameter values of \(m=1\), \(k=1\), \(c=0.3\), \(\varepsilon=0.5\) and \(f_{1}=0.25\). Figure 1: Modal amplitudes: (**a**) \(u_{1}\) vs \(\omega_{1}\), (**b**) \(u_{2}\) vs \(\omega_{2}\). The backbone curves were computed from Equations (33) to (35). The parameter values used to obtain these plots were \(m_{1}=m_{2}=1\), \(k=1\), \(\varepsilon=0.5\) with \(c=0\) and \(f_{1}=0\). Here, the black solid lines describe the solutions obtained from Equations (34) and (35), while the blue dashed lines represent the backbone curves computed from Equation (33). Figure 3: Time-amplitude response curves (**a**) \(x_{1}\) vs \(t\), (**b**) \(x_{2}\) vs \(t\). These were computed from the numerical integration solutions of Equations (29) and (30). The parameter values used to obtain these plots were \(m_{1}=m_{2}=1\), \(k=1\), \(\varepsilon=0.5\) with \(c=0.3\), with initial conditions given by \(\left(x_{1}\text{, }x_{2}\text{, }\dot{x}_{1}\text{, }\dot{x}_{2}\right)= \left(0\text{, }0\text{, }2\text{, }0\right)\). Here, the black lines describe the numerical integration solutions of Equation (30), while the blue, dashed lines represent the numerical solutions obtained from the nonlinear equations of motion (29). Figure 2: Modal amplitudes: (**a**) \(u_{1}\) vs \(\omega_{1}\), (**b**) \(u_{2}\) vs \(\omega_{2}\). The backbone curves were computed from Equations (33) to (35). The parameter values used to obtain these plots for different values of \(k\) and \(\varepsilon\) were \(m_{1}=m_{2}=1\), with \(c=0\) and \(f_{1}=0\). Here, the black solid lines describe the solutions obtained from Equations (34) and (35), while the colored, dashed lines represent the backbone curves computed from Equation (33). It can be seen from Figure 4 that the numerical integration solutions of the decoupled Equation (29) provide curves that slightly differ close to the resonant region to those obtained from the numerical integration solutions of the original equations of motion, but its qualitative and quantitative system dynamic predictions are, in general, good.