Nonlinear dynamics of inclined viscoelastic pipes subjected to pulsatile flow and multi-harmonic excitations

Abstract

Theoretical modeling and nonlinear responses of inclined simply supported pipes conveying fluid are presented, with focus on the interaction between nonlinear forced and parametric oscillations. Based on the Euler–Bernoulli beam theory, the new equations of motion accounting for the effects of stretching, curvature, and material nonlinearities are established. The Kelvin–Voigt viscoelastic model is adopted to model the system damping. Afterward, the Galerkin scheme is performed on the three-order approximation model and the resultant discretized equations are solved via two numerical techniques, including the pseudo-arclength technique and a direct time integration method. Analyses on static stability, linear natural frequency, and nonlinear dynamic are conducted to showcase the impacts of the inclination angle, mean flow velocity, pulsatile flow amplitude, and multi-harmonic external excitations on the static and dynamic characteristics, mainly developing to explore the appearance of primary and principal parametric resonances, the transition between nonlinear hardening and softening responses, and the significance of nonlinear curvature strains.

Appendices

Appendix A. Coefficient matrices in the Galerkin discretization

The coefficient matrices of the equations of motion, i.e., Eqs. (26) and (27), are

$$\begin{aligned} {{\textbf {M}}}_{ij}^1&=\int _{0}^{1}\varphi _i\varphi _j\text {d}\xi , \end{aligned}$$

(A.1)

$$\begin{aligned} {{\textbf {C}}}_{ij}^1&=2\sqrt{\beta }v\int _{0}^{1}\varphi _i\varphi _j^{\prime }\text {d}\xi -a\kappa ^2\int _{0}^{1}\varphi _i\varphi _j^{\prime \prime }\text {d}\xi , \end{aligned}$$

(A.2)

$$\begin{aligned} {{\textbf {K}}}_{ij}^1&=\left( v^2-\kappa ^2\right) \int _{0}^{1}\varphi _i\varphi _j^{\prime \prime }\text {d}\xi +\sqrt{\beta }{\dot{v}}\int _{0}^{1}\varphi _i\varphi _j^{\prime }\text {d}\xi , \end{aligned}$$

(A.3)

$$\begin{aligned} {{\textbf {A}}}_{ijk}^1&=\left( \Gamma -\Pi \right) \int _{0}^{1}\varphi _i\phi _j^{\prime }\phi _k^{\prime \prime }\text {d}\xi -\int _{0}^{1}\varphi _i\phi _j^{\prime \prime }\phi _k^{\prime \prime \prime }\text {d}\xi \nonumber \\&\quad -\int _{0}^{1}\varphi _i\phi _j^{\prime }\phi _k^{\prime \prime \prime \prime }\text {d}\xi -\kappa ^2\int _{0}^{1}\varphi _i\phi _j^{\prime }\phi _k^{\prime \prime }\text {d}\xi \nonumber \\&\quad +\left( \sqrt{\beta }{\dot{v}}-\gamma \text {cos}\chi \right) \nonumber \\&\quad \quad \int _{0}^{1}\varphi _i\left( \frac{1}{2}\phi _j^{\prime }\phi _k^{\prime }-\phi _j^{\prime }\phi _k^{\prime \prime }+\xi \phi _j^{\prime }\phi _k^{\prime \prime }\right) \text {d}\xi , \end{aligned}$$

(A.4)

$$\begin{aligned} {{\textbf {B}}}_{ijk}^1&=-a\kappa ^2\int _{0}^{1}\varphi _i\left( \phi _j^{\prime }\phi _k^{\prime \prime }+\phi _j^{\prime \prime }\phi _k^{\prime }\right) \text {d}\xi \nonumber \\&\quad -a\int _{0}^{1}\varphi _i\left( \phi _j^{\prime \prime }\phi _k^{\prime \prime \prime }+\phi _j^{\prime \prime \prime }\phi _k^{\prime \prime }+\phi _j^{\prime }\phi _k^{\prime \prime \prime \prime }+\phi _j^{\prime \prime \prime \prime }\phi _k^{\prime }\right) \text {d}\xi , \end{aligned}$$

(A.5)

$$\begin{aligned} {{\textbf {D}}}_{ijkl}^1&=\int _{0}^{1}\varphi _i\left( 4\varphi _j^{\prime \prime \prime }\phi _k^{\prime }\phi _l^{\prime \prime }+5\varphi _j^{\prime \prime }\phi _k^{\prime }\phi _l^{\prime \prime \prime }+3\varphi _j^{\prime }\phi _k^{\prime \prime }\phi _l^{\prime \prime \prime }\right. \nonumber \\&\quad \left. +\varphi _j^{\prime \prime \prime \prime }\phi _k^{\prime }\phi _l^{\prime }+3\varphi _j^{\prime }\phi _k^{\prime }\phi _l^{\prime \prime \prime \prime }+2\varphi _j^{\prime \prime }\phi _k^{\prime \prime }\phi _l^{\prime \prime }\right) \text {d}\xi \nonumber \\&\quad -\left( \Gamma -\Pi \right) \int _{0}^{1}\varphi _i\left( \varphi _j^{\prime \prime }\phi _k^{\prime }\phi _l^{\prime }+2\varphi _j^{\prime }\phi _k^{\prime }\phi _l^{\prime \prime }\right) \text {d}\xi \nonumber \\&\quad +\kappa ^2\int _{0}^{1}\varphi _i\left( \varphi _j^{\prime \prime }\phi _k^{\prime }\phi _l^{\prime }+2\varphi _j^{\prime }\phi _k^{\prime }\phi _l^{\prime \prime }\right) \text {d}\xi \nonumber \\&\quad +\left( \sqrt{\beta }{\dot{v}}-\gamma \text {cos}\chi \right) \nonumber \\&\quad \quad \int _{0}^{1}\varphi _i\left( -\varphi _j^{\prime }\phi _k^{\prime }\phi _l^{\prime }+\varphi _j^{\prime \prime }\phi _k^{\prime }\phi _l^{\prime }-\xi \varphi _j^{\prime \prime }\phi _k^{\prime }\phi _l^{\prime }\right. \nonumber \\&\quad \left. +2\varphi _j^{\prime }\phi _k^{\prime }\phi _l^{\prime \prime }-2\xi \varphi _j^{\prime }\phi _k^{\prime }\phi _l^{\prime \prime }\right) \text {d}\xi , \end{aligned}$$

(A.6)

$$\begin{aligned} {{\textbf {E}}}_{ijkl}^{11}&=a\kappa ^2\int _{0}^{1}\varphi _i\left( 2\varphi _j^{\prime }\phi _k^{\prime }\phi _l^{\prime \prime }+\varphi _j^{\prime \prime }\phi _k^{\prime }\phi _l^{\prime }\right) \text {d}\xi \nonumber \\&\quad +a\int _{0}^{1}\varphi _i\left( 2\varphi _j^{\prime \prime }\phi _k^{\prime \prime }\phi _l^{\prime \prime }+3\varphi _j^{\prime }\phi _k^{\prime \prime }\phi _l^{\prime \prime \prime }+5\varphi _j^{\prime \prime }\phi _k^{\prime }\phi _l^{\prime \prime \prime }\right. \nonumber \\&\quad \left. +4\varphi _j^{\prime \prime \prime }\phi _k^{\prime }\phi _l^{\prime \prime }+3\varphi _j^{\prime }\phi _k^{\prime }\phi _l^{\prime \prime \prime \prime }+\varphi _j^{\prime \prime \prime \prime }\phi _k^{\prime }\phi _l^{\prime }\right) \text {d}\xi , \end{aligned}$$

(A.7)

$$\begin{aligned} {{\textbf {E}}}_{ijkl}^{12}&=a\kappa ^2\int _{0}^{1}\varphi _i\left( 2\varphi _j^{\prime \prime }\phi _k^{\prime }\phi _l^{\prime }+2\varphi _j^{\prime }\phi _k^{\prime }\phi _l^{\prime \prime }+2\varphi _j^{\prime }\phi _k^{\prime \prime }\phi _l^{\prime }\right) \text {d}\xi \nonumber \\&\quad +a\int _{0}^{1}\varphi _i\left( 4\varphi _j^{\prime \prime }\phi _k^{\prime \prime }\phi _l^{\prime \prime }+4\varphi _j^{\prime \prime \prime }\phi _k^{\prime }\phi _l^{\prime \prime }\right. \nonumber \\&\quad +4\varphi _j^{\prime \prime \prime }\phi _k^{\prime \prime }\phi _l^{\prime }+5\varphi _j^{\prime \prime }\phi _k^{\prime }\phi _l^{\prime \prime \prime }\nonumber \\&\quad +3\varphi _j^{\prime }\phi _k^{\prime \prime }\phi _l^{\prime \prime \prime }+5\varphi _j^{\prime \prime }\phi _k^{\prime \prime \prime }\phi _l^{\prime }+3\varphi _j^{\prime }\phi _k^{\prime \prime \prime }\phi _l^{\prime \prime }+2\varphi _j^{\prime \prime \prime \prime }\phi _k^{\prime }\phi _l^{\prime }\nonumber \\&\quad \left. +3\varphi _j^{\prime }\phi _k^{\prime }\phi _l^{\prime \prime \prime \prime } +3\varphi _j^{\prime }\phi _k^{\prime \prime \prime \prime }\phi _l^{\prime }\right) \text {d}\xi , \end{aligned}$$

(A.8)

$$\begin{aligned} {{\textbf {M}}}_{ij}^2&=\int _{0}^{1}\phi _i\phi _j\text {d}\xi , \end{aligned}$$

(A.9)

$$\begin{aligned} {{\textbf {C}}}_{ij}^2&=2\sqrt{\beta }v\int _{0}^{1}\phi _i\phi _j^{\prime }\text {d}\xi +a\int _{0}^{1}\phi _i\phi _j^{\prime \prime \prime \prime }\text {d}\xi , \end{aligned}$$

(A.10)

$$\begin{aligned} {{\textbf {K}}}_{ij}^2&=\int _{0}^{1}\phi _i\phi _j^{\prime \prime \prime \prime }\text {d}\xi +v^2\int _{0}^{1}\phi _i\phi _j^{\prime \prime }\text {d}\xi -\left( \Gamma -\Pi \right) \nonumber \\&\quad \int _{0}^{1}\phi _i\phi _j^{\prime \prime }\text {d}\xi +\left( \sqrt{\beta }{\dot{v}}-\gamma \text {cos}\chi \right) \nonumber \\&\quad \int _{0}^{1}\phi _i\left( \phi _j^{\prime \prime }-\xi \phi _j^{\prime \prime }\right) \text {d}\xi \nonumber \\&\quad +\gamma \text {cos}\chi \int _{0}^{1}\left( \phi _i\phi _j^{\prime }-\phi _i\phi _j^{\prime \prime }+\phi _i \xi \phi _j^{\prime \prime }\right) \text {d}\xi , \end{aligned}$$

(A.11)

$$\begin{aligned} {{\textbf {A}}}_{ijk}^2&=\left( \Gamma -\Pi \right) \int _{0}^{1}\phi _i\left( \varphi _j^{\prime \prime }\phi _k^{\prime }+\varphi _j^{\prime }\phi _k^{\prime \prime }\right) \text {d}\xi \nonumber \\&\quad -\int _{0}^{1}\phi _i\left( 3\varphi _j^{\prime \prime \prime }\phi _k^{\prime \prime }+4\varphi _j^{\prime \prime }\phi _k^{\prime \prime \prime }+\varphi _j^{\prime \prime \prime \prime }\phi _k^{\prime }+2\varphi _j^{\prime }\phi _k^{\prime \prime \prime \prime }\right) \text {d}\xi \nonumber \\&\quad -\kappa ^2\int _{0}^{1}\phi _i\left( \varphi _j^{\prime \prime }\phi _k^{\prime }+\varphi _j^{\prime }\phi _k^{\prime \prime }\right) \text {d}\xi +\left( \sqrt{\beta }{\dot{v}}-\gamma \text {cos}\chi \right) \nonumber \\&\quad \int _{0}^{1}\phi _i\left( \varphi _j^{\prime }\phi _k^{\prime }-\varphi _j^{\prime \prime }\phi _k^{\prime }+\xi \varphi _j^{\prime \prime }\phi _k^{\prime } -\varphi _j^{\prime }\phi _k^{\prime \prime }+\xi \varphi _j^{\prime }\phi _k^{\prime \prime }\right) \text {d}\xi , \end{aligned}$$

(A.12)

$$\begin{aligned} {{\textbf {B}}}_{ijk}^{21}&=-a\kappa ^2\int _{0}^{1}\phi _i\left( \varphi _j^{\prime \prime }\phi _k^{\prime }+\varphi _j^{\prime }\phi _k^{\prime \prime }\right) \text {d}\xi \nonumber \\&\quad -a\int _{0}^{1}\phi _i\left( 4\varphi _j^{\prime \prime }\phi _k^{\prime \prime \prime }+3\varphi _j^{\prime \prime \prime }\phi _k^{\prime \prime }+2\varphi _j^{\prime }\phi _k^{\prime \prime \prime \prime }+\varphi _j^{\prime \prime \prime \prime }\phi _k^{\prime }\right) \text {d}\xi , \end{aligned}$$

(A.13)

$$\begin{aligned} {{\textbf {B}}}_{ijk}^{22}&=-a\kappa ^2\int _{0}^{1}\phi _i\left( \varphi _j^{\prime \prime }\phi _k^{\prime }+\varphi _j^{\prime }\phi _k^{\prime \prime }\right) \text {d}\xi \nonumber \\&\quad -a\int _{0}^{1}\phi _i\left( 3\varphi _j^{\prime \prime \prime }\phi _k^{\prime \prime }+4\varphi _j^{\prime \prime }\phi _k^{\prime \prime \prime }+\varphi _j^{\prime \prime \prime \prime }\phi _k^{\prime }+2\varphi _j^{\prime }\phi _k^{\prime \prime \prime \prime }\right) \text {d}\xi , \end{aligned}$$

(A.14)

$$\begin{aligned} {{\textbf {D}}}_{ijkl}^{21}&=-\left( \Gamma -\Pi \right) \int _{0}^{1}\phi _i\left( 2\varphi _j^{\prime }\varphi _k^{\prime \prime }\phi _l^{\prime }+\varphi _j^{\prime }\varphi _k^{\prime }\phi _l^{\prime \prime }\right) \text {d}\xi \nonumber \\&\quad +\int _{0}^{1}\phi _i\left( 8\varphi _j^{\prime \prime }\varphi _k^{\prime \prime }\phi _l^{\prime \prime }+7\varphi _j^{\prime \prime }\varphi _k^{\prime \prime \prime }\phi _l^{\prime }+9\varphi _j^{\prime }\varphi _k^{\prime \prime \prime }\phi _l^{\prime \prime }\right. \nonumber \\&\quad \left. +12\varphi _j^{\prime }\varphi _k^{\prime \prime }\phi _l^{\prime \prime \prime }+3\varphi _j^{\prime }\varphi _k^{\prime \prime \prime \prime }\phi _l^{\prime }+3\varphi _j^{\prime }\varphi _k^{\prime }\phi _{l}^{\prime \prime \prime \prime }\right) \text {d}\xi \nonumber \\&\quad +\kappa ^2\int _{0}^{1}\phi _i\left( 2\varphi _j^{\prime }\varphi _k^{\prime \prime }\phi _l^{\prime }+\varphi _j^{\prime }\varphi _k^{\prime }\phi _l^{\prime \prime }\right) \text {d}\xi \nonumber \\&\quad +\left( \sqrt{\beta }{\dot{v}}-\gamma \text {cos}\chi \right) \int _{0}^{1}\phi _i\left( -\varphi _j^{\prime }\varphi _k^{\prime }\phi _l^{\prime }+2\varphi _j^{\prime }\varphi _k^{\prime \prime }\phi _l^{\prime }\right. \nonumber \\&\quad \left. -2\xi \varphi _j^{\prime }\varphi _k^{\prime \prime }\phi _l^{\prime }+\varphi _j^{\prime }\varphi _k^{\prime }\phi _l^{\prime \prime }-\xi \varphi _j^{\prime }\varphi _k^{\prime }\phi _l^{\prime \prime }\right) \text {d}\xi , \end{aligned}$$

(A.15)

$$\begin{aligned} {{\textbf {D}}}_{ijkl}^{22}&=\frac{3}{2}\left( \Gamma -\Pi \right) \int _{0}^{1}\phi _i\phi _j^{\prime }\phi _k^{\prime }\phi _l^{\prime \prime }\text {d}\xi \nonumber \\&\quad -2\int _{0}^{1}\phi _i\left( \phi _j^{\prime \prime }\phi _k^{\prime \prime }\phi _l^{\prime \prime }+4\phi _j^{\prime }\phi _k^{\prime \prime }\phi _l^{\prime \prime \prime }+\phi _j^{\prime }\phi _k^{\prime }\phi _l^{\prime \prime \prime \prime }\right) \text {d}\xi \nonumber \\&\quad -\frac{3}{2}\kappa ^2\int _{0}^{1}\phi _i\phi _j^{\prime }\phi _k^{\prime }\phi _l^{\prime \prime }\text {d}\xi +\frac{1}{2}\left( \sqrt{\beta }{\dot{v}}-\gamma \text {cos}\chi \right) \nonumber \\&\quad \int _{0}^{1}\phi _i\left( \phi _j^{\prime }\phi _k^{\prime }\phi _l^{\prime }-3\phi _j^{\prime }\phi _k^{\prime }\phi _l^{\prime \prime }+3\xi \phi _j^{\prime }\phi _k^{\prime }\phi _l^{\prime \prime }\right) \text {d}\xi , \end{aligned}$$

(A.16)

$$\begin{aligned} {{\textbf {E}}}_{ijkl}^{21}&=2a\kappa ^2\int _{0}^{1}\phi _i\left( \varphi _j^{\prime }\varphi _k^{\prime \prime }\phi _l^{\prime }+\varphi _j^{\prime }\varphi _k^{\prime }\phi _l^{\prime \prime }+\varphi _j^{\prime \prime }\varphi _k^{\prime }\phi _l^{\prime }\right) \text {d}\xi \nonumber \\&\quad +a\int _{0}^{1}\phi _i\left( 16\varphi _j^{\prime \prime }\varphi _k^{\prime \prime }\phi _l^{\prime \prime }+9\varphi _j^{\prime }\varphi _k^{\prime \prime \prime }\phi _l^{\prime \prime }\right. \nonumber \\&\quad +7\varphi _j^{\prime \prime }\varphi _k^{\prime \prime \prime }\phi _l^{\prime }+12\varphi _j^{\prime }\varphi _k^{\prime \prime }\phi _j^{\prime \prime \prime }+12\varphi _j^{\prime \prime }\varphi _k^{\prime }\phi _l^{\prime \prime \prime }\nonumber \\&\quad +7\varphi _j^{\prime \prime \prime }\varphi _k^{\prime \prime }\phi _l^{\prime }+9\varphi _j^{\prime \prime \prime }\varphi _k^{\prime }\phi _l^{\prime \prime }+3\varphi _j^{\prime }\varphi _k^{\prime \prime \prime \prime }\phi _l^{\prime }\nonumber \\&\quad \left. +6\varphi _j^{\prime }\varphi _k^{\prime }\phi _l^{\prime \prime \prime \prime }+3\varphi _j^{\prime \prime \prime \prime }\varphi _k^{\prime }\phi _l^{\prime }\right) \text {d}\xi , \end{aligned}$$

(A.17)

$$\begin{aligned} {{\textbf {E}}}_{ijkl}^{22}&=a\kappa ^2\int _{0}^{1}\phi _i\left( 2\varphi _j^{\prime }\varphi _k^{\prime \prime }\phi _l^{\prime }+\varphi _j^{\prime }\varphi _k^{\prime }\phi _l^{\prime \prime }\right) \text {d}\xi \nonumber \\&\quad +a\int _{0}^{1}\phi _i\left( 8\varphi _j^{\prime \prime }\varphi _k^{\prime \prime }\phi _l^{\prime \prime }+7\varphi _j^{\prime \prime }\varphi _k^{\prime \prime \prime }\phi _l^{\prime }+9\varphi _j^{\prime }\varphi _k^{\prime \prime \prime }\phi _l^{\prime \prime }\right. \nonumber \\&\quad \left. +12\varphi _j^{\prime }\varphi _k^{\prime \prime }\phi _l^{\prime \prime \prime }+3\varphi _j^{\prime }\varphi _k^{\prime \prime \prime \prime }\phi _l^{\prime }+3\varphi _j^{\prime }\varphi _k^{\prime }\phi _l^{\prime \prime \prime \prime }\right) \text {d}\xi , \end{aligned}$$

(A.18)

$$\begin{aligned} {{\textbf {E}}}_{ijkl}^{23}&=-a\kappa ^2\int _{0}^{1}\phi _i\left( 3\phi _j^{\prime }\phi _k^{\prime }\phi _l^{\prime \prime }+\frac{3}{2}\phi _j^{\prime \prime }\phi _k^{\prime }\phi _l^{\prime }\right) \text {d}\xi \nonumber \\&\quad -2a\int _{0}^{1}\phi _i\left( 3\phi _j^{\prime \prime }\phi _k^{\prime \prime }\phi _l^{\prime \prime }+4\phi _j^{\prime }\phi _k^{\prime \prime }\phi _l^{\prime \prime \prime }+4\phi _j^{\prime \prime }\phi _k^{\prime }\phi _l^{\prime \prime \prime }\right. \nonumber \\&\quad \left. +4\phi _j^{\prime \prime \prime }\phi _k^{\prime }\phi _l^{\prime \prime }+2\phi _j^{\prime }\phi _k^{\prime }\phi _l^{\prime \prime \prime \prime }+\phi _j^{\prime \prime \prime \prime }\phi _k^{\prime }\phi _l^{\prime }\right) \text {d}\xi , \end{aligned}$$

(A.19)

$$\begin{aligned} {{\textbf {g}}}_i^2&=\gamma \text {sin}\chi \int _{0}^{1}\phi _i\text {d}\xi , \end{aligned}$$

(A.20)

$$\begin{aligned} {{\textbf {f}}}_i^2&=f_0\left[ \text {cos}\left( \omega _f\tau \right) +\sum _{j=2}^{J}\left( f_{j,c}\text {cos}\left( j\omega _f\tau \right) \right. \right. \nonumber \\&\quad \left. \left. +f_{j,s}\text {sin}\left( j\omega _f\tau \right) \right) \right] \int _{0}^{1}\phi _i\text {d}\xi . \end{aligned}$$

(A.21)

Appendix B. Convergence study

To obtain accurate and convergence results of the pipe’s response, the truncated number, N, and time step, $\triangle t$, utilized in the BDF technique should be determined. Since the pipe system has rich dynamic responses when conveying high flow velocity, the mean flow velocity $v_0=7$ is chosen in the present study. As shown in Fig. 21, the nonlinear responses predicted using $\triangle t=0.01$ and 0.001 are almost the same for both cases of $\chi =0$ and $\frac{\pi }{4}$, which means taking $\triangle t=0.01$ in the Galerkin discretization process is acceptable. As to the chosen of the truncated number, by comparing three values of N, it is seen that the truncated number needs to be at least at $N=6$. In addition, by comparing the frequency-response curves and stability boundaries of the vertical pipe, it is clearly seen that the numerical results predicted using the truncated number $N=4$ are acceptable (Fig. 22). In summary, the truncated number and time step are selected as $N=6$ and $\triangle t=0.01$, respectively, in all numerical calculations.

Fig. 21

Nonlinear responses of the midpoint transverse displacement of the pipe with different truncated numbers and time steps

Fig. 22

Frequency-response curves and stability boundaries of the vertical pipe with different truncated numbers

Appendix C. Derivation of the simplified model

For moderate rotations in the fluid-conveying pipe, the higher-order curvature terms can be neglected; so, the von K$\acute{\text {a}}$rm$\acute{\text {a}}$n geometric nonlinearity is adopted, for which the following expression for the axial strain and curvature can be obtained:

$$\begin{aligned}&\varepsilon _0\left( x, t\right) =\frac{\partial u}{\partial x}+\frac{1}{2}\left( \frac{\partial w}{\partial x}\right) ^2, \end{aligned}$$

(C.1)

$$\begin{aligned}&k\left( x, t\right) =\frac{\partial ^2 w}{\partial x^2}. \end{aligned}$$

(C.2)

Substituting Eqs. (C.1) and (C.2) into Eqs. (17) and (18), the nonlinear equations for an inclined fluid-conveying pipe with moderate rotations can be expressed as

$$\begin{aligned}&\left( M+m\right) \frac{\partial ^2 u}{\partial t^2}+2MU\frac{\partial ^2 u}{\partial x\partial t}+MU^2\frac{\partial ^2 u}{\partial x^2}\nonumber \\&\quad +M\frac{\partial U}{\partial t}\frac{\partial u}{\partial x}-EA\left( \frac{\partial ^2 u}{\partial x^2}+\frac{\partial w}{\partial x}\frac{\partial ^2 w}{\partial x^2}\right) \nonumber \\&\quad -\alpha EA\left( \frac{\partial ^3 u}{\partial x^2\partial t}+\frac{\partial ^2 w}{\partial x\partial t}\frac{\partial ^2 w}{\partial x^2}+\frac{\partial w}{\partial x}\frac{\partial ^3 w}{\partial x^2\partial t}\right) =0, \end{aligned}$$

(C.3)

$$\begin{aligned}&\left( M+m\right) \frac{\partial ^2 w}{\partial t^2}+2MU\frac{\partial ^2 w}{\partial x\partial t}+MU^2\frac{\partial ^2 w}{\partial x^2}\nonumber \\&\quad +M\frac{\partial U}{\partial t}\frac{\partial ^2 w}{\partial x^2}\left( L-x\right) +EI\frac{\partial ^4 w}{\partial x^4}+\alpha EI\frac{\partial ^5 w}{\partial x^4\partial t}\nonumber \\&\quad -EA\left[ \frac{\partial u}{\partial x}\frac{\partial ^2 w}{\partial x^2}+\frac{\partial ^2 u}{\partial x^2}\frac{\partial w}{\partial x}+\frac{3}{2}\left( \frac{\partial w}{\partial x}\right) ^2\frac{\partial ^2 w}{\partial x^2}\right] \nonumber \\&\quad -\alpha EA\left[ \frac{\partial ^2 u}{\partial x\partial t}\frac{\partial ^2 w}{\partial x^2}+\frac{\partial ^3 u}{\partial x^2\partial t}\frac{\partial w}{\partial x}+\frac{\partial u}{\partial x}\frac{\partial ^3 w}{\partial x^2\partial t}\right. \nonumber \\&\quad \left. +2\frac{\partial w}{\partial x}\frac{\partial ^2 w}{\partial x\partial t}\frac{\partial ^2 w}{\partial x^2}+\frac{3}{2}\left( \frac{\partial w}{\partial x}\right) ^2\frac{\partial ^3 w}{\partial x^2\partial t}\right] \nonumber \\&\quad -\left( T_0-P\right) |_{x=L}\frac{\partial ^2 w}{\partial x^2}+\left( M+m\right) \text {g}\text {sin}\chi \nonumber \\&\quad +\left( M+m\right) \text {g}\text {cos}\chi \left[ \frac{\partial w}{\partial x}+\frac{\partial ^2 w}{\partial x^2}\left( L-x\right) \right] =F. \end{aligned}$$

(C.4)