# Nonlinear Vibration and Post-Buckling Behaviors of Metal and FGM Pipes Transporting Heavy Crude Oil
## 4 Galerkin Approach  
Equation (20) is converted to a nonlinear ordinary differential equation using the [PERSON] method [32]. In this study, a first-order Galerkin approximation is applied, where the transverse displacement \(w(x,t)\) is assumed in the following form:  
\[w(x,t)=W(t)\zeta(x) \tag{21}\]  
where \(\zeta(x)\) represents the shape function, and for simply supported FGM pipes, the mode shape is assumed to follow \(\zeta(x)=\sin\left(\frac{mx}{L}\right)\). By substituting Equation (21) into Equation (20) and applying the [PERSON] approach, the discretized NGE is obtained as follows:\[\begin{split}\frac{d^{4}W(t)}{dt^{4}}+\left(\frac{2m_{f}\Gamma}{m_{f}+m_{ p}}\left[\frac{\int\limits_{0}^{L}\xi^{\prime}(x)\zeta(x)dx}{\int\limits_{0}^{L} \xi^{\prime}(x)\zeta(x)dx}\right]\right)\frac{d^{4}W(t)}{dt^{3}}\\ +\left(\frac{K_{g}\zeta\overline{A}}{m_{f}+m_{p}}+\left[\frac{m_{ f}\Gamma^{2}-K_{g}\zeta\overline{A}-P_{p}}{m_{f}+m_{p}}-\frac{\overline{EI}}{ \overline{P}(r)^{\prime}}\right]\left\{\frac{\int\limits_{0}^{L}\xi^{\prime \prime}(x)\zeta(x)dx}{\int\limits_{0}^{L}\xi^{\prime}(x)dx}\right\}\right)\frac{ d^{2}W(t)}{dt^{2}}\\ +\left(\frac{2m_{f}\Gamma K_{g}\overline{C}\overline{A}}{ \overline{\rho}(r)^{\prime}(1m_{f}+m_{p})}\left\{\frac{\int\limits_{0}^{L} \xi^{\prime\prime}(x)\zeta(x)dx}{\int\limits_{0}^{L}\xi^{\prime\prime}(x)dx} \right\}-\frac{2m_{f}\Gamma\overline{EI}}{\overline{\rho}(r)^{\prime}(1m_{f}+ m_{p})}\left\{\frac{\int\limits_{0}^{L}\xi^{\prime\prime}(x)\zeta(x)dx}{ \int\limits_{0}^{L}\xi^{\prime\prime}(x)dx}\right\}\right)\frac{dW(t)}{dt^{2}} \\ +\left(\frac{\overline{EI}}{2\overline{\rho}(r)^{\prime}LL(m_{f}+m_ {p})}\left\{\frac{\int\limits_{0}^{L}\xi^{\prime\prime}(x)\xi^{\prime\prime}(x) dx}{\int\limits_{0}^{L}\xi^{\prime}(x)dx}\right\}-\frac{\overline{EI}K_{g} \overline{C}\overline{A}}{2\overline{\rho}(r)^{\prime}LL(m_{f}+m_{p})}\left\{ \frac{\int\limits_{0}^{L}\xi^{\prime\prime}(x)\zeta(x)dx}{\int\limits_{0}^{L }\xi^{\prime\prime}(x)dx}\right\}\right)W^{3}(t)\\ -\left(\frac{\overline{EA}}{2\overline{L}\left(m_{f}+m_{p} \right)}\left\{\frac{\int\limits_{0}^{L}\left(\int\limits_{0}^{L}\xi^{\prime \prime}(x)dx\right)\xi^{\prime\prime}(x)\zeta(x)dx}{\int\limits_{0}^{L}\xi^{ \prime}(x)dx}\right\}\right)W^{2}(t)\frac{d^{4}W(t)}{dt^{2}}=0\end{split} \tag{22}\]  
where \((\ )^{\prime}=\frac{d(\ )}{dx}\). Therefore, the nonlinear vibration response of FGM pipes transporting heavy crude oil is determined by numerically solving Equation (22) through the application of the fourth-order Runge-Kutta method. This explicit time integration technique is widely used for solving initial value problems due to its balance between computational efficiency and accuracy. In the current analysis, the time step (\(\Delta t\)) was chosen based on convergence testing, with a typical value of \(\Delta t=0.001\) s, ensuring stability and precise tracking of nonlinear dynamic responses over extended time intervals. The fourth-order Runge-Kutta algorithm updates the system state iteratively by evaluating the ordinary differential equation at four intermediate stages within each time step and combining them to compute the solution with fourth-order accuracy. This approach is particularly suitable for capturing the complex vibration behavior observed in FGM pipes under nonlinear excitation, including transient and steady-state oscillations.