## 3 Determination of \(a_{i}\), \(b_{i}\), and \(\mu_{i}\)  
To determine the coefficients \(a_{i}\), \(b_{i}\), and \(\mu_{i}\), a minimization procedure analogous to the one followed by [PERSON] in [2] is assumed. Here, it is assumed that the square of the difference between the acceleration of the originally coupled system and that of the simplified cubic one, given by Expressions (5) and (6), could be minimized for each system by considering that the weighted mean square errors, \(U_{1}\) and \(U_{2}\), can be determined by using the following expressions [17]:  
\[ \begin{array}{c}U_{1}=\min\int\limits_{0}^{V_{12}}\int\limits_{0}^{V_{11}} \int\limits_{0}^{V_{12}}\int\limits_{0}^{\eta_{12}}(\omega_{n1}^{2}u_{1}+v_{1 }\dot{u}_{1}+\
u_{2}\dot{u}_{2}+\varphi_{1}u_{1}^{3}+\varphi_{2}u_{1}^{2}u_{2 }+\varphi_{3}u_{1}u_{2}^{2}+\varphi_{4}u_{2}^{3}-a_{1}u_{1}\\ -a_{2}u_{1}^{2}-a_{3}u_{1}^{3}-2\mu_{1}\dot{u}_{1}-\cdots)^{2}\ du_{1}\ du_{2}\ d\dot{u}_{1}\ d\dot{u}_{2},\end{array}  \tag{7}\]  
\[ \begin{array}{c}U_{2}=\min\int\limits_{0}^{V_{11}}\int\limits_{0}^{V_{22}} \int\limits_{0}^{\eta_{11}}\int\limits_{0}^{\eta_{22}}(\omega_{n2}^{2}u_{2}+2 \
u_{2}\dot{u}_{1}+\
u_{3}\dot{u}_{2}+\varphi_{5}u_{1}^{3}+\varphi_{6}u_{1}^{ 2}u_{2}+\varphi_{7}u_{1}u_{2}^{2}+\varphi_{8}u_{2}^{3}\\ -b_{1}u_{2}-b_{2}u_{2}^{2}-b_{3}u_{2}^{3}-2\mu_{2}\dot{u}_{2}-\cdots)^{2}\ du_{2}\ du_{1}\ d\dot{u}_{2}\ d\dot{u}_{1}.\end{array}  \tag{8}\]  
Furthermore, the minimization of \(U_{1}\) and \(U_{2}\) with respect to the coefficients, \(a_{i}\), \(b_{i}\) and \(\mu_{i}\), can be achieved from [9]  
\[\frac{\partial U_{1}}{\partial a_{i}}=0\quad\text{with }i=1,\ 2,\ 3,\quad\frac{ \partial U_{1}}{\partial\mu_{1}}=0, \tag{9}\]  
\[\frac{\partial U_{2}}{\partial b_{i}}=0\quad\text{with }i=1,\ 2,\ 3,\quad\frac{ \partial U_{2}}{\partial\mu_{2}}=0. \tag{10}\]  
Equations (9) and (10) yield, after performing some algebraic computer calculations with the help of Mathematica 11.1.1.0 symbolic computer package, the following expressions:  
\[a_{1}=\omega_{n1}^{2}+\frac{\varphi_{3}\eta_{2}^{2}}{3}+\frac{85}{74\eta_{1}} \Big{(}\varphi_{4}\eta_{2}^{3}+2\
u_{2}V_{2}\Big{)},\quad a_{2}=\frac{37\varphi _{2}\eta_{1}^{2}\eta_{2}-320\big{(}\varphi_{4}\eta_{2}^{3}+\
u_{2}V_{2}\big{)} }{37\eta_{1}^{2}}, \tag{11}\]  
\[a_{3}=\frac{111\varphi_{1}\eta_{1}^{3}+560\big{(}\varphi_{4}\eta_{2}^{3}+\
u _{2}V_{2}\big{)}}{111\eta_{1}^{3}},\quad\mu_{1}=\frac{48\varphi_{4}\eta_{2}^{ 3}+37\
u_{1}V_{1}+48\
u_{2}V_{2}}{74V_{1}}, \tag{12}\]  
\[b_{1}=-\frac{-222\eta_{22}\omega_{n2}^{2}-255\varphi_{5}\eta_{11}^{3}-74 \varphi_{6}\eta_{11}^{2}\eta_{22}-510\
u_{2}V_{11}}{22\eta_{22}\eta_{22}}, \tag{13}\]  
\[b_{2}=-\frac{160\varphi_{5}\eta_{11}^{3}-37\varphi_{7}\eta_{11}\eta_{22}^{2}+ 320\
u_{2}V_{11}}{74\eta_{22}^{2}}, \tag{14}\]  
\[b_{3}=-\frac{-140\varphi_{5}\eta_{11}^{3}-111\eta_{8}\eta_{22}^{3}-280\
u_{2} V_{11}}{11\eta_{22}^{2}}, \tag{15}\]  
\[\mu_{2}=-\frac{-12\varphi_{5}\eta_{11}^{3}-24\
u_{2}V_{11}-37\
u_{3}V_{22}}{74 \
u_{22}}. \tag{16}\]  
Here, \(\eta_{i}\), \(\eta_{ii}\), \(V_{i}\), and \(V_{ii}\) describe integration constants whose values must minimize Expressions (7) and (8). The general procedure to compute the integration constants \(\eta_{i}\), \(\eta_{ii}\), \(V_{i}\), and \(V_{ii}\) is discussed in the following section. Computation of \(\eta_{i}\), \(\eta_{ii}\), \(V_{i}\), and \(V_{ii}\) Values  
To determine the above integration constants, Equations (11) and (12) are substituted into Equations (7), and (13)-(16) are substituted into Equation (8). This step yields, after performing the corresponding integrations, the following expressions:  
\[S_{1}=\frac{121\eta_{1}\eta_{2}V_{1}V_{2}}{258741}\left(12\varphi_{4}^{2}\eta_ {2}^{6}+21\varphi_{4}\eta_{2}^{3}v_{2}V_{2}+28\imath_{2}^{2}V_{2}^{2}\right), \tag{17}\]  
\[