Nonlinear dynamics of a viscoelastic beam traveling with pulsating speed, variable axial tension under two-frequency parametric excitations and internal resonance

Abstract

This paper investigates nonlinear combined parametric transverse vibrations of a traveling viscoelastic beam. The combined parametric excitations originate from the time dependency of axial velocity as well as axial tension. Two parametric excitations are enforced into the system amid the internal resonance. Two-frequency parametric resonance is assumed to be comprised of combination parametric resonance of first two modes due to the time dependency of axial velocity, and the principal parametric resonance of first mode due to the variable tension in the axial direction in the presence of internal resonance for viscoelastic beam is considered for the first time. The higher-order integro-partial differential equation of motion is solved through direct method of multiple scales. Continuation algorithm is employed to explore the stability and various bifurcations of the nonlinear dynamic system. Focus has been made to study the effect of variations of fluctuating tension component, fluctuating velocity component independently and when combined, internal and parametric frequency detuning parameters and damping on the system response. Frequency response equilibrium curves are complex and unique in shapes which are embodied with various bifurcations. Such steady-state behavior is not seen in the existent literature. With variation in fluctuating velocity component, the number of steady-state nontrivial equilibrium curves increases to three and with variation in fluctuating axial tension, they become four. In this process, significant changes in stability, number and position of various bifurcations like supercritical and subcritical pitchfork, Hopf and saddle node are observed. Unlike the previous study, the shape, stability and bifurcations of equilibrium curves under the combined effect of axial velocity and tension closely match with the case of fluctuating axial tension component. The effect of variation in internal and parametric frequency detuning parameter is more realized for second mode compared to first mode. A comparison of the present work with a previous one where axial tension is variable reveals many qualitative and quantitative similarities and dissimilarities. But when compared with earlier work where axial velocity is constant, significant dissimilarities are surfaced. The system displays a wide ranging dynamic behavior including stable periodic, quasiperiodic and unstable chaotic behavior. The numerical computation depicts various nonlinear characteristics and oscillatory behaviors which are not found so far in the existent literature.

Appendix

$$\begin{aligned} \varGamma _1= & {} -2 i\omega _1 {A}'_1 \phi _1 -2 v_0 {A}'_1 {\phi }^{{\prime }}_1 -2i\mu \omega _1 A_1 \phi _1 \\&-2i\alpha \omega _1 A_1 {\phi }^{{\prime }{\prime }{\prime }{\prime }}_1 +\frac{1}{2}v_\mathrm{l} ^{2} \left\{ 2A_1^2 \bar{{A}}_1 {\phi }^{{\prime }{\prime }}_1 \int \limits _0^1 {\phi }^{{\prime }}_1 {\bar{{\phi }}}'_1 \hbox {d}x\right. \\&+A_1^2 \bar{{A}}_1 {\bar{{\phi }}}^{{\prime }{\prime }}_1 \int \limits _0^1 {{\phi }_1^{{\prime }2}\hbox {d}x} +2A_1 A_2 \bar{{A}}_2 {\bar{{\phi }}}^{{\prime }{\prime }}_2 \int \limits _0^1 {{\phi }^{{\prime }}_1 {\phi }^{{\prime }}_2 \hbox {d}x} \\&+\left. {2A_1 A_2 \bar{{A}}_2 {\phi }^{{\prime }{\prime }}_1 \int \limits _0^1 {{\phi }^{{\prime }}_2 {\bar{{\phi }}}'_2 \hbox {d}x} +2A_1 A_2 \bar{{A}}_2 {\phi }^{{\prime }{\prime }}_2 \int \limits _0^1 {{\phi }^{{\prime }}_1 {\bar{{\phi }}}'_2 \hbox {d}x} } \right\} \\ \varGamma _2= & {} \frac{1}{2}v_\mathrm{l} ^{2}\left\{ {2\bar{{A}}_1 ^{2}A_2 {\bar{{\phi }}}^{{\prime }{\prime }}_1 \int \limits _0^1 {{\phi }^{{\prime }}_2 {\bar{{\phi }}}'_1 \hbox {d}x} +\bar{{A}}_1 ^{2}A_2 {\phi }^{{\prime }{\prime }}_2 \int \limits _0^1 \bar{\phi }^{\prime {2}}_1 \hbox {d}x} \right\} \ \\ \varGamma _3= & {} \bar{{A}}_1 \left\{ { v_1 \omega _1 \bar{{\phi }}_1 ^{\prime }-\frac{ v_1 \varOmega }{2} {\bar{{\phi }}}'_1 +i v_0 v_1 {\bar{{\phi }}}^{{\prime }{\prime }}_1 } \right\} \\ \varGamma _4= & {} \bar{{A}}_1 \left\{ {-i\frac{p_1 }{2}\bar{{\phi }}_1 ^{\prime \prime }} \right\} \ \\ \end{aligned}$$

$$\begin{aligned} \varGamma _5= & {} -2 i\omega _2 {A}'_2 \phi _2 -2 v_0 {A}'_2 {\phi }^{{\prime }}_2 -2\mu i\omega _2 A_2 \phi _2 \\&\quad -2\alpha i\omega _2 A_2 {\phi }^{{\prime }{\prime }{\prime }{\prime }}_2 +\frac{1}{2}v_\mathrm{l}^{2}\left\{ A_2^2 \bar{{A}}_2 {{\bar{{\phi }}}^{{\prime }{\prime }}}_2 \int \limits _0^1 {\phi }^{{\prime {2}}}_2 \hbox {d}x\right. \\&\quad +2A_1 \bar{{A}}_1 A_2 {\phi }^{{\prime }{\prime }}_2 \int \limits _{0}^1 {{\phi }^{{\prime }}_1 }{\bar{{\phi }}}'_1 \hbox {d}x+2A_1 \bar{{A}}_1 A_2 {\bar{{\phi }}}^{{\prime }{\prime }}_1 \int \limits _0^1 {{\phi }^{{\prime }}_1 {\phi }^{{\prime }}_2 \hbox {d}x}\\&\quad \left. {+2A_2^2 \bar{{A}}_2 {\phi }^{{\prime }{\prime }}_2 \int \limits _0^1 {{\phi }^{{\prime }}_2 {\bar{{\phi }}}'_2 \hbox {d}x} +2A_1 \bar{{A}}_1 A_2 {\phi }^{{\prime }{\prime }}_1 \int \limits _0^1 {{\phi }^{{\prime }}_2 {\bar{{\phi }}}'_1 \hbox {d}x} } \right\} \\ \varGamma _6= & {} \frac{1}{2}v_\mathrm{l} ^{2}\left\{ {A_1^3 {\phi }^{{\prime }{\prime }}_1 \int \limits _0^1 {{\phi }^{{\prime {2}}}_1\hbox {d}x} } \right\} \\ \varGamma _7= & {} A_2 \left\{ {-i\frac{p_1 }{2}\phi _2^{\prime \prime }} \right\} \\ \varGamma _8= & {} \bar{{A}}_2 \left\{ { v_1 \omega _2 {\bar{{\phi }}}'_2 -\frac{ v_1 \varOmega }{2}{\bar{{\phi }}}'_2 +i v_0 v_1 {\bar{{\phi }}}^{{\prime }{\prime }}_2 } \right\} \\ \varGamma _9= & {} A_1 \left\{ {-i\frac{p_1 }{2}\phi _1 ^{\prime \prime }} \right\} \\ S_1= & {} \frac{\frac{1}{16}v_\mathrm{l} ^{2}\left\{ {2\int \limits _0^1 {{\phi }^{{\prime }{\prime }}_1 \bar{{\phi }}_1 \hbox {d}x} \int \limits _0^1 {{\phi }^{{\prime }}_1 {\bar{{\phi }}}'_1 \hbox {d}x} +\int \limits _0^1 {{\bar{{\phi }}}^{{\prime }{\prime }}_1 \bar{{\phi }}_1 \hbox {d}x} \int \limits _0^1 {{\phi }^{{\prime {2}}}_1 \hbox {d}x} } \right\} }{-\left\{ {i\omega _1 \int \limits _0^1 {\phi _1 \bar{{\phi }}_1 \hbox {d}x} +v_0 \int \limits _0^1{{\phi }^{{\prime }}_1 \bar{{\phi }}_1 \hbox {d}x} } \right\} }\\ \end{aligned}$$

$$\begin{aligned} S_2= & {} \frac{\frac{1}{8}v_\mathrm{l}^{2}\left\{ {\int \limits _0^1 {{\bar{{\phi }}}^{{\prime }{\prime }}_2 \bar{{\phi }}_1 \hbox {d}x} \int \limits _0^1 {{\phi }^{{\prime }}_1 {\phi }^{{\prime }}_2 \hbox {d}x} +\int \limits _0^1 {{\phi }^{{\prime }{\prime }}_1 \bar{{\phi }}_1 \hbox {d}x} \int \limits _0^1 {{\phi }^{{\prime }}_2 {\bar{{\phi }}}'_2 \hbox {d}x} +\int \limits _0^1 {{\phi }^{{\prime }{\prime }}_2 \bar{{\phi }}_1 \hbox {d}x} \int \limits _0^1 {{\phi }^{{\prime }}_1 {\bar{{\phi }}}'_2 \hbox {d}x} } \right\} }{-\left\{ {i\omega _1 \int \limits _0^1 {\phi _1 \bar{{\phi }}_1 \hbox {d}x} + v_0 \int \limits _0^1 {{\phi }^{{\prime }}_1 \bar{{\phi }}_1 \hbox {d}x} } \right\} } \\ S_3= & {} \frac{\frac{1}{8}v_\mathrm{l} ^{2}\left\{ {\int \limits _0^1 {{\phi }^{{\prime }{\prime }}_2 \bar{{\phi }}_2 \hbox {d}x} \int \limits _0^1 {{\phi }^{{\prime }}_1 {\bar{{\phi }}}'_1 \hbox {d}x} +\int \limits _0^1 {{\bar{{\phi }}}^{{\prime }{\prime }}_1 \bar{{\phi }}_2 \hbox {d}x} \int \limits _0^1 {{\phi }^{{\prime }}_1 {\phi }^{{\prime }}_2 \hbox {d}x} +\int \limits _0^1 {{\phi }^{{\prime }{\prime }}_1 \bar{{\phi }}_2 \hbox {d}x} \int \limits _0^1 {{\phi }^{{\prime }}_2 {\bar{{\phi }}}'_2 \hbox {d}x} } \right\} }{-\left\{ {i\omega _2 \int \limits _0^1 {\phi _2 \bar{{\phi }}_2 \hbox {d}x} + v_0 \int \limits _0^1 {{\phi }^{{\prime }}_2 \bar{{\phi }}_2 \hbox {d}x} } \right\} } \\ S_4= & {} \frac{\frac{1}{16}v_\mathrm{l} ^{2}\left\{ {2\int \limits _0^1 {{\phi }^{{\prime }{\prime }}_2 \bar{{\phi }}_2 \hbox {d}x} \int \limits _0^1 {{\phi }^{{\prime }}_2 {\bar{{\phi }}}'_2 \hbox {d}x} +\int \limits _0^1 {{\bar{{\phi }}}^{{\prime }{\prime }}_2 \bar{{\phi }}_2 \hbox {d}x} \int \limits _0^1 {{\phi }^{{\prime {2}}}_2 \hbox {d}x} } \right\} }{-\left\{ {i\omega _2 \int \limits _0^1 {\phi _2 \bar{{\phi }}_2 \hbox {d}x} + v_0 \int \limits _0^1 {{\phi }^{{\prime }}_2 \bar{{\phi }}_2 \hbox {d}x} } \right\} } \ \\ C_1= & {} \frac{-i\omega _1 \int \limits _0^1 {\phi _1 \bar{{\phi }}_1 \hbox {d}x} }{-\left\{ {i\omega _1 \int \limits _0^1 {\phi _1 \bar{{\phi }}_1 \hbox {d}x} + v_0 \int \limits _0^1 {{\phi }^{{\prime }}_1 \bar{{\phi }}_1 \hbox {d}x} } \right\} } \\ C_2= & {} \frac{-i\omega _2 \int \limits _0^1 {\phi _2 \bar{{\phi }}_2 \hbox {d}x} }{-\left\{ {i\omega _2 \int \limits _0^1 {\phi _2 \bar{{\phi }}_2 \hbox {d}x} +v_0 \int \limits _0^1 {{\phi }^{{\prime }}_2 \bar{{\phi }}_2 \hbox {d}x} } \right\} } \\ e_1= & {} \frac{-i\omega _1 \int \limits _0^1 {\phi }^{{\prime }{\prime }{\prime }{\prime }}_1 \bar{{\phi }}_1 \hbox {d}x}{-\left\{ {i\omega _1 \int \limits _0^1 {\phi _1 \bar{{\phi }}_1 \hbox {d}x} +v_0 \int \limits _0^1 {{\phi }^{{\prime }}_1 \bar{{\phi }}_1 \hbox {d}x} } \right\} } \ \\ e_2= & {} \frac{-i\omega _2 \int \limits _0^1 {\phi }^{{\prime }{\prime }{\prime }{\prime }}_2 \bar{{\phi }}_2 \hbox {d}x}{-\left\{ {i\omega _2 \int \limits _0^1 {\phi _2 \bar{{\phi }}_2 \hbox {d}x} +v_0 \int \limits _0^1 {{\phi }^{{\prime }}_2 \bar{{\phi }}_2 \hbox {d}x} } \right\} } \\ \end{aligned}$$

$$\begin{aligned} g_1= & {} \frac{\frac{1}{16}v_\mathrm{l} ^{2}\left\{ {2\int \limits _0^1 {{\bar{{\phi }}}^{{\prime }{\prime }}_1 \bar{{\phi }}_1 \hbox {d}x} \int \limits _0^1 {{\phi }^{{\prime }}_2 {\bar{{\phi }}}'_1 \hbox {d}x} +\int \limits _0^1 {{\phi }^{{\prime }{\prime }}_2 \bar{{\phi }}_1 \hbox {d}x} \int \limits _0^1 {{\bar{{\phi }}}^{{\prime }2}_1 \hbox {d}x} } \right\} }{-\left\{ {i\omega _1 \int \limits _0^1 {\phi _1 \bar{{\phi }}_1 \hbox {d}x} +v_0 \int \limits _0^1 {{\phi }^{{\prime }}_1 \bar{{\phi }}_1 \hbox {d}x} }\right\} } \\ g_2= & {} \frac{\frac{1}{16}v_\mathrm{l} ^{2}\left\{ {\int \limits _0^1 {{\phi }^{{\prime }{\prime }}_1 \bar{{\phi }}_2 \hbox {d}x} \int \limits _0^1 {{\phi }^{{\prime {2}}}_1 \hbox {d}x} } \right\} }{-\left\{ {i\omega _2 \int \limits _0^1 {\phi _2 \bar{{\phi }}_2 \hbox {d}x} +v_0 \int \limits _0^1 {{\phi }^{{\prime }}_2 \bar{{\phi }}_2 \hbox {d}x} } \right\} } \ \\ K_4= & {} \frac{\frac{1}{2}\left\{ {v_1 \omega _2 \int \limits _0^1 {{\bar{{\phi }}}'_2 \bar{{\phi }}_1 \hbox {d}x} -\frac{v_1 \varOmega }{2}\int \limits _0^1 {\bar{{\phi }}_2 ^{\prime }} \bar{{\phi }}_1 \hbox {d}x+i v_0 v_1 \int \limits _0^1 {{\bar{{\phi }}}^{{\prime }{\prime }}_2 \bar{{\phi }}_1 \hbox {d}x} } \right\} }{-\left\{ {i\omega _1 \int \limits _0^1 {\phi _1 \bar{{\phi }}_1 \hbox {d}x} +v_0 \int \limits _0^1 {{\phi }^{{\prime }}_1 \bar{{\phi }}_1 \hbox {d}x} } \right\} } \\ K_5= & {} \frac{\frac{1}{2}\left\{ {v_1 \omega _1 \int \limits _0^1 {{\bar{{\phi }}}'_1 \bar{{\phi }}_2 \hbox {d}x} -\frac{v_1 \varOmega }{2}\int \limits _0^1 {{\bar{{\phi }}}'_1 \ } \bar{{\phi }}_2 \hbox {d}x+i v_0 v_1 \int \limits _0^1 {\bar{{\phi }}_1 ^{\prime \prime }\bar{{\phi }}_2 \hbox {d}x} } \right\} }{-\left\{ {i\omega _2 \int \limits _0^1 {\phi _2 \bar{{\phi }}_2 \hbox {d}x} +v_0 \int \limits _0^1 {{\phi }^{{\prime }}_2 \bar{{\phi }}_2 \hbox {d}x} } \right\} } \ \\ K_6= & {} \frac{ \frac{1}{2}\left\{ {-i\frac{p_1 }{2}\int \limits _0^1 {\bar{{\phi }}_1 ^{\prime \prime }\bar{{\phi }}_1 \hbox {d}x} } \right\} }{-\left\{ {i\omega _1 \int \limits _0^1 {\phi _1 \bar{{\phi }}_1 \hbox {d}x} +v_0 \int \limits _0^1 {{\phi }^{{\prime }}_1 \bar{{\phi }}_1 \hbox {d}x} } \right\} }\\ K_7= & {} \frac{ \frac{1}{2}\left\{ {-i\frac{p_1 }{2}\int \limits _0^1 {\phi _2 ^{\prime \prime }\bar{{\phi }}_1 \hbox {d}x} } \right\} }{-\left\{ {i\omega _1 \int \limits _0^1 {\phi _1 \bar{{\phi }}_1 \hbox {d}x} +v_0 \int \limits _0^1 {{\phi }^{{\prime }}_1 \bar{{\phi }}_1 \hbox {d}x} } \right\} }\\ K_8= & {} \frac{\frac{1}{2} \left\{ {-i\frac{p_1 }{2}\int \limits _0^1 {\phi _1 ^{\prime \prime }\bar{{\phi }}_2 \hbox {d}x} } \right\} }{-\left\{ {i\omega _2 \int \limits _0^1 {\phi _2 \bar{{\phi }}_2 \hbox {d}x} +v_0 \int \limits _0^1 {{\phi }^{{\prime }}_2 \bar{{\phi }}_2 \hbox {d}x} } \right\} } \\ \end{aligned}$$