## 3 Approximate Solution Based on the Multi-Scale Mixed Harmonic Balance Method
### Approximate Analytical Solutions of the Nonlinear Equations  
The factor \(\sigma\) is introduced to represent the difference between the external excitation frequency and the natural frequency of the pipe, namely, \(\Omega=\omega+\varepsilon\sigma\), where \(\varepsilon\) is a small parameter. Considering the time scale \(t_{0}=t\), \(t_{1}=\varepsilon t\), the analytical solution of Equation (3) is set as:  
\[\left\{ \begin{aligned} & x_{1}=x_{11}(t_{0},t_{1})+\varepsilon x_{12}(t_{0},t_{1}) \\ & z=z_{11}(t_{0},t_{1})+\varepsilon z_{12}(t_{0},t_{1})\end{aligned} \right. \tag{4}\]  
We can substitute Equation (4) into Equation (3). According to the coefficients of \(\varepsilon\) with the same order of two sides of the equation being equal, the following equations are obtained:  
\[{D_{0}}^{2}x_{11}+\omega^{2}x_{11}=0 \tag{5}\]  
\[{D_{0}}^{2}x_{12}+\omega^{2}x_{12}=\kappa{z_{11}}^{3}+\xi_{2}D_{0}z_{11}-\xi_{ 1}D_{0}x_{11}-2D_{0}D_{1}x_{11}+F\cos(\Omega t) \tag{6}\]  
\[\rho_{2}{D_{0}}^{2}x_{11}+\rho_{2}{D_{0}}^{2}z_{11}+\kappa{z_{11}}^{3}+\xi_{2} D_{0}z_{11}=0 \tag{7}\]\[{D_{1}}^{2}x_{11}+2D_{0}D_{1}x_{12}+\xi_{1}D_{0}x_{12}+\xi_{1}D_{1}x_{11}-\xi_{2} D_{0}z_{12}-\xi_{2}D_{1}z_{11}-3\kappa{z_{11}}^{2}z_{12}=0 \tag{8}\] \[\rho_{2}{D_{0}}^{2}x_{12}+\rho_{2}{D_{0}}^{2}z_{12}+2\rho_{2}D_{0}D _{1}x_{12}+2\rho_{2}D_{0}D_{1}z_{11}\] (9) \[+\xi_{2}D_{12}z_{11}+\xi_{2}D_{0}z_{12}+3\kappa{z_{11}}^{2}z_{12}=0\]  
where \(D_{i}(i=1,2)\) is the differential operator [21]. Higher-order terms, such as \(3\Omega\) and \(5\Omega\), have little influence on the results. To reduce the complexity of the calculation, the higher-order terms are ignored when the solution of the multi-scale mixed harmonic balance method is assumed. According to Equation (5), the solution of \(x_{11}\) is set as:  
\[x_{11}(t_{0},t_{1})=A(t_{1})e^{i\omega t_{0}}+cc \tag{10}\]  
where \(cc\) is the conjugate complex and \(A\) is the undetermined complex function of the variable \(t_{1}\). The multi-scale mixed harmonic balance method is used to set the solution of \(z_{11}\) as follows:  
\[z_{11}(t_{0},t_{1})=B_{1}(t_{1})e^{i\Omega t_{0}}+cc=B_{1}(t_{1})e^{i(\sigma t _{1}+\omega t_{0})}+cc \tag{11}\]  
where \(B\) is the undetermined complex function of the variable \(t_{1}\). Equations (10) and (11) are substituted into Equation (7), and the secular term is eliminated. The first-order perturbed solution can be represented as the slow-invariant manifold equation, as follows:  
\[3\kappa{B_{1}}^{2}(t_{1})\overline{B}_{1}(t_{1})e^{i\sigma t_{1}}+i\xi_{2}B_{ 1}(t_{1})\omega e^{i\sigma t_{1}}-\rho_{2}B_{1}(t_{1})\omega^{2}e^{i\sigma t_{ 1}}-\rho_{2}A(t_{1})\omega^{2}=0 \tag{12}\]  
Equation (12) represents the relationship between the amplitudes of the pipe (main system) and an NES (subsystem), which is called the slow-invariant manifold. Equations (10) and (11) are substituted into Equation (6), and the first solvability condition is obtained by eliminating the secular term:  
\[D_{1}A(t_{1})=-\frac{3\kappa{B_{1}}(t_{1})^{2}\overline{B}_{1}(t_{1})}{2\omega }e^{i\omega t_{1}}+\frac{\xi_{2}B_{1}(t_{1})}{2}e^{i\sigma t_{1}}-\frac{\xi_{ 1}A(t_{1})}{2}-\frac{iF}{4\omega}e^{i\omega t_{1}} \tag{13}\]  
Equations (10)-(12) are substituted into Equation (6) to obtain \(x_{12}\), and then, the multi-scale mixed harmonic balance method is used to set the solution of \(z_{12}\) as follows:  
\[ \begin{array}{l}x_{12}=-\frac{1}{8}\kappa A(t_{1})^{3}e^{3i\omega t_{1}}+cc \\ z_{12}=B_{2}(t_{1})e^{i(\sigma t_{1}+\omega t_{0})}+cc\end{array}  \tag{14}\]  
where \(B_{2}\) is the undetermined complex function of \(z_{12}\) when using the multi-scale mixed harmonic balance method and \(z_{12}\) is the second-order small value of Equation (14). Equations (10), (11) and (14) are substituted into Equation (9), and the secular term is eliminated by using the following formula:  
\[