Parametric analysis of an axially moving beam with time-dependent velocity, longitudinally varying tension and subjected to internal resonance

Abstract

The current study aims to analyze the dynamic characteristics of the nonlinear system excited parametrically in the presence of internal resonance. The method of multiple time scales (MMS) is directly adopted to simplify the higher-order integro-partial differential equation of motion to get an approximate solution that leads to a set of first-order partial differential equations. To develop a suitable model for a moving beam, the parameters incorporated are viscoelasticity and viscous damping, geometric nonlinearity, Coriolis acceleration, harmonically varying velocity, and axially varying tension. The stability and bifurcations of the steady-state solution are examined under a subcritical speed regime. The investigation focuses on the changes in the stability and bifurcation features of steady-state solutions accounting for the effects of variations in the system parameters like internal and parametric frequency detuning parameters, the amplitude of fluctuating speed, and axial stiffness. The outcomes of this investigation are unique, interesting, and not available in the existing literature, which may provide theoretical insight in designing a traveling system.

Appendix

$$ \varsigma _{1} = \left[ { - 2\,(i\omega _{1} \phi _{1} + V_{0} \phi ^{\prime})A^{\prime}_{1} - 2i\alpha \,\omega _{1} A_{1} \phi _{1}^{{\prime \prime \prime \prime }} - 2i\mu \,\omega _{1} A_{1} \phi _{1} } \right] $$

$$ \varsigma _{2} = \frac{1}{2}v_{l}^{2} \left\{ {A_{1}^{2} \bar{A}_{1} \left( {2\phi _{1}^{{\prime \prime }} \int\limits_{0}^{1} {\phi _{1}^{\prime } \bar{\phi }_{1}^{\prime } \,dx + \bar{\phi }^{{\prime \prime }} _{1} } \int\limits_{0}^{1} {\phi _{1}^{{\prime 2}} {\text{d}}x} } \right) + 2A_{1} A_{2} \bar{A}_{2} \left( {\bar{\phi }_{2}^{{\prime \prime }} \int\limits_{0}^{1} {\phi _{1}^{\prime } \phi _{2}^{\prime } dx + 2A_{1} A_{2} \bar{A}_{2} \phi _{1}^{{\prime \prime }} \int\limits_{0}^{1} {\phi _{2}^{\prime } \bar{\phi }_{2}^{\prime } dx} + 2A_{1} A_{2} \bar{A}_{2} \phi _{2}^{{\prime \prime }} \int\limits_{0}^{1} {\phi _{1}^{\prime } \bar{\phi }_{2}^{\prime } dx} } } \right)} \right\} $$

$$ \varsigma _{3} = \frac{1}{2}v_{l} ^{2} \left\{ {2\bar{A}_{1} ^{2} A_{2} \bar{\phi }_{1}^{{\prime \prime }} \int\limits_{0}^{1} {\phi ^{\prime } _{2} \bar{\phi }^{\prime } _{1} x} + \bar{A}_{1} ^{2} A_{2} \phi ^{{\prime \prime }} _{2} \int\limits_{0}^{1} {\bar{\phi ^{\prime 2}}_{1} \,dx} } \right\} $$

$$ \varsigma _{4} = \left[ { - 2\left( {i\omega _{2} \phi _{2} + V_{0} A_{2}^{\prime } \phi _{2}^{\prime } } \right)A_{2}^{\prime } \, - 2\alpha i\omega _{2} A_{2} \phi _{2}^{{\prime \prime \prime \prime }} - 2\mu \,i\omega _{2} A_{2} \phi _{2} } \right] $$

$$ \varsigma _{5} = \frac{1}{2}v_{l} ^{2} \left\{ {A_{2}^{2} \bar{A}_{2} \left( {2\phi _{2}^{{\prime \prime }} \int\limits_{0}^{1} {\phi _{2}^{\prime } \bar{\phi }_{2}^{\prime } {\text{d}}x} + \bar{\phi }_{2}^{{\prime \prime }} \int\limits_{0}^{1} {\phi _{2}^{{'2}} {\text{d}}x} } \right) + 2A_{1} \bar{A}_{1} A_{2} \left( {\phi _{1}^{{\prime \prime }} \int\limits_{0}^{1} {\phi ^{\prime}_{2} \bar{\phi }_{1}^{\prime } {\text{d}}x} + \phi _{2}^{{\prime \prime }} \int\limits_{0}^{1} {\phi _{1}^{\prime } \bar{\phi }_{1}^{\prime } {\text{d}}x} + \bar{\phi }^{\prime \prime}_{1} \int\limits_{0}^{1} {\phi _{1}^{\prime } \phi _{2}^{\prime } {\text{d}}x} } \right)} \right\}$$

$$ \varsigma _{6} = \frac{1}{2}v_{l} ^{2} \left\{ {A_{1}^{3} \left( {\phi _{1}^{{\prime \prime }} \int\limits_{0}^{1} {\phi ^{{\prime 2}} _{1} dx} } \right)} \right\} $$

$$ \zeta_{7} = \overline{A}_{2} \left\{ {\,V_{1} \omega_{2} \overline{\phi^{\prime}}_{2} - \frac{{\,V_{1} \Omega }}{2}\left( {1 - x} \right)\overline{\phi^{\prime\prime}}_{2} + i\,kV_{0} V_{1} \overline{\phi^{\prime\prime}}_{2} } \right\} $$

$S_{1} = \frac{{\frac{1}{16}v_{l}^{2} \left\{ {2\int\limits_{0}^{1} {\phi_{1}^{^{\prime\prime}} \overline{\phi }_{1} dx} \int\limits_{0}^{1} {\phi_{1}^{^{\prime}} \overline{\phi }_{1}^{^{\prime}} dx} + \int\limits_{0}^{1} {\overline{\phi }_{1}^{^{\prime\prime}} \overline{\phi }_{1} dx} \int\limits_{0}^{1} {\phi_{1}^{^{\prime}2} dx} } \right\}}}{{ - \left\{ {i\omega_{1} \int\limits_{0}^{1} {\phi_{1} \overline{\phi }_{1} dx} + V_{0} \int\limits_{0}^{1} {\phi_{1}^{^{\prime}} \overline{\phi }_{1} dx} } \right\}}}$;$S_{2} = \frac{{\frac{1}{8}v_{l}^{2} \left\{ {\int\limits_{0}^{1} {\overline{\phi }^{\prime\prime}_{2} \overline{\phi }_{1} dx} \int\limits_{0}^{1} {\phi^{\prime}_{1} \phi^{\prime}_{2} dx} + \int\limits_{0}^{1} {\phi^{\prime\prime}_{1} \overline{\phi }_{1} dx} \int\limits_{0}^{1} {\phi^{\prime}_{2} \overline{\phi }^{\prime}_{2} dx} + \int\limits_{0}^{1} {\phi^{\prime\prime}_{2} \overline{\phi }_{1} dx} \int\limits_{0}^{1} {\phi^{\prime}_{1} \overline{\phi }^{\prime}_{2} dx} } \right\}}}{{ - \left\{ {i\omega_{1} \int\limits_{0}^{1} {\phi_{1} \overline{\phi }_{1} dx} + \,V_{0} \int\limits_{0}^{1} {\phi^{\prime}_{1} \overline{\phi }_{1} dx} } \right\}}}$

$S_{3} = \frac{{\frac{1}{8}v_{l}^{2} \left\{ {\int\limits_{0}^{1} {\phi^{\prime\prime}_{2} \overline{\phi }_{2} dx} \int\limits_{0}^{1} {\phi^{\prime}_{1} \overline{\phi }^{\prime}_{1} dx} + \int\limits_{0}^{1} {\overline{\phi }^{\prime\prime}_{1} \overline{\phi }_{2} dx} \int\limits_{0}^{1} {\phi^{\prime}_{1} \phi^{\prime}_{2} dx} + \int\limits_{0}^{1} {\phi^{\prime\prime}_{1} \overline{\phi }_{2} dx} \int\limits_{0}^{1} {\phi^{\prime}_{2} \overline{\phi }^{\prime}_{1} dx} } \right\}}}{{ - \left\{ {i\omega_{2} \int\limits_{0}^{1} {\phi_{2} \overline{\phi }_{2} dx} + \,V_{0} \int\limits_{0}^{1} {\phi^{\prime}_{2} \overline{\phi }_{2} dx} } \right\}}}$; $S_{4} = \frac{{\frac{1}{{16}}v_{l}^{2} \left\{ {2\int\limits_{0}^{1} {\phi _{2}^{{\prime \prime }} \bar{\phi }_{2} dx} \int\limits_{0}^{1} {\phi _{2}^{\prime } \bar{\phi }_{2}^{\prime } dx} + \int\limits_{0}^{1} {\bar{\phi }_{2}^{{\prime \prime }} \bar{\phi }_{2} dx} \int\limits_{0}^{1} {\phi ^{{\prime _{2}^{2} }} dx} } \right\}}}{{ - \left\{ {i\omega _{2} \int\limits_{0}^{1} {\phi _{2} \bar{\phi }_{2} dx} + {\mkern 1mu} V_{0} \int\limits_{0}^{1} {\phi _{2}^{\prime } \bar{\phi }_{2} dx} } \right\}}}$.

$K_{4} = \frac{{\frac{1}{2}\left\{ {V_{1} \omega_{2} \int\limits_{0}^{1} {\overline{\phi }_{2}^{\prime } \overline{\phi }_{2} dx} - \frac{{V_{1} \Omega }}{2}\int\limits_{0}^{1} {\left( {1 - x} \right)\overline{\phi }_{2}^{\prime \prime } } \overline{\phi }_{2} dx + ikV_{0} V_{1} \int\limits_{0}^{1} {\overline{\phi }_{2}^{\prime \prime } \overline{\phi }_{2} dx} } \right\}}}{{ - \left\{ {i\omega_{2} \int\limits_{0}^{1} {\phi_{2} \overline{\phi }_{2} dx} + V_{0} \int\limits_{0}^{1} {\phi^{\prime}_{2} \overline{\phi }_{2} dx} } \right\}}}$_;$C_{1} = \frac{{ - i\omega_{1} \int\limits_{0}^{1} {\phi_{1} \overline{\phi }_{1} dx} }}{{ - \left\{ {i\omega_{1} \int\limits_{0}^{1} {\phi_{1} \overline{\phi }_{1} dx} + \,V_{0} \int\limits_{0}^{1} {\phi^{\prime}_{1} \overline{\phi }_{1} dx} } \right\}}}$.

$C_{2} = \frac{{ - i\omega_{2} \int\limits_{0}^{1} {\phi_{2} \overline{\phi }_{2} dx} }}{{ - \left\{ {i\omega_{2} \int\limits_{0}^{1} {\phi_{2} \overline{\phi }_{2} dx} + V_{0} \int\limits_{0}^{1} {\phi^{\prime}_{2} \overline{\phi }_{2} dx} } \right\}}}$;$e_{1} = \frac{{ - i\omega _{1} \int\limits_{0}^{1} {\phi ^{{\prime \prime \prime \prime _{1} }} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} \bar{\phi }_{1} dx} }}{{ - \left\{ {i\omega _{1} \int\limits_{0}^{1} {\phi _{1} \bar{\phi }_{1} dx} + V_{0} \int\limits_{0}^{1} {\phi _{1}^{\prime } \bar{\phi }_{1} dx} } \right\}}}$;$e_{2} = \frac{{ - i\omega _{2} \int\limits_{0}^{1} {\phi ^{{\prime \prime \prime \prime _{2} }} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} \bar{\phi }_{2} dx} }}{{ - \left\{ {i\omega _{2} \int\limits_{0}^{1} {\phi _{2} \bar{\phi }_{2} dx} + V_{0} \int\limits_{0}^{1} {\phi _{2}^{\prime } \bar{\phi }_{2} dx} } \right\}}}$.$g_{1} = \frac{{\frac{1}{{16}}v_{l}^{2} \left\{ {2\int\limits_{0}^{1} {\bar{\phi }_{1}^{{\prime \prime }} \bar{\phi }_{1} dx} \int\limits_{0}^{1} {\phi _{2}^{\prime } \bar{\phi }_{1}^{\prime } dx} + \int\limits_{0}^{1} {\phi _{2}^{{\prime \prime }} \bar{\phi }_{1} dx} \int\limits_{0}^{1} {\bar{\phi }^{{\prime _{1}^{2} }} dx} } \right\}}}{{ - \left\{ {i\omega _{1} \int\limits_{0}^{1} {\phi _{1} \bar{\phi }_{1} dx} + V_{0} \int\limits_{0}^{1} {\phi _{1}^{\prime } \bar{\phi }_{1} dx} } \right\}}}$; $g_{2} = \frac{{\frac{1}{{16}}v_{l}^{2} \left\{ {\int\limits_{0}^{1} {\phi _{1}^{\prime } \bar{\phi }_{2} dx} \int\limits_{0}^{1} {\phi ^{{\prime _{1}^{2} }} dx} } \right\}}}{{ - \left\{ {i\omega _{2} \int\limits_{0}^{1} {\phi _{2} \bar{\phi }_{2} dx} + V_{0} \int\limits_{0}^{1} {\phi _{2}^{\prime } \bar{\phi }_{2} dx} } \right\}}}$