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Global dynamics of an autoparametric beam structure

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Abstract

Global dynamics of an autoparametric beam structure derived from a flexible L-shaped beam subjected to base excitation with one-to-two internal resonance and principal resonance are investigated. Hamilton’s principle is employed to obtain the nonlinear partial differential governing equations of the multi-beam structure. A linear theoretical analysis is implemented to derive the modal functions, and the orthogonality conditions are established. The analytical modal functions obtained are then adopted to truncate the partial differential governing equations into a set of coupled nonlinear ordinary differential equations via the Galerkin’s procedure. The method of multiple scales is applied to yield a set of autonomous equations of the first-order approximations to the response of the dynamical system. The Energy-Phase method is used to study the global bifurcation and multi-pulse chaotic dynamics of such autoparametric system. The present analysis indicates that the chaotic dynamics results from the existence of Šilnikov’s type of homoclinic orbits and the parameter set for which the system may exhibit chaotic motions in the sense of Smale horseshoes are predicted analytically. Numerical simulations are performed to validate the theoretical results.

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Acknowledgements

This research is supported by National Natural Science Foundation of China (NNSFC) through Grant Nos. 11290150, 11290152, 11290154, 11322214 and 11427801, the Funding Project for Academic Human Resources Development in Institutions of Higher Learning under the Jurisdiction of Beijing Municipality (PHRIHLB).

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Authors and Affiliations

  1. Beijing Key Laboratory of Nonlinear Vibrations and Strength of Mechanical Structures, College of Mechanical Engineering, Beijing University of Technology, Beijing, 100124, P. R. China

    Tian-Jun Yu, Wei Zhang & Xiao-Dong Yang

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  1. Tian-Jun Yu
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  2. Wei Zhang
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  3. Xiao-Dong Yang
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Correspondence to Wei Zhang.

Appendix

Appendix

The coefficients of two-degree-of-freedom equations of motions are expressed as follows

$$\begin{aligned} \mu _1= & {} \frac{\mu K_{11} }{2M_{11} },\quad \omega _1^2 = \frac{K_{11} }{M_{11} },\quad p_{11} = \frac{P_{11} }{M_{11} },\\ p_{12}= & {} \frac{P_{12} }{M_{11} },\quad g_{11} = \frac{G_{11} }{M_{11} },\quad g_{12} = \frac{G_{12} }{M_{11} },\\ g_{22}= & {} \frac{G_{22} }{M_{11} },s_{11} = \frac{S_{11} }{M_{11} },\quad s_{12} = \frac{S_{12} }{M_{11} },\\ s_{21}= & {} \frac{S_{21} }{M_{11} },\quad s_{22} = \frac{S_{22} }{M_{11} }, \quad e_{11} = \frac{E_{11} }{M_{11} }\\ M_{11}= & {} \int _0^1 {\left( {S_{\eta _{11} } } \right) } ^{2}\hbox {d}\xi _1 +\alpha ^{3}\beta \int _0^1 {\left( {S_{\eta _{21} } } \right) } ^{2}\hbox {d}\xi _2\\&+\,\alpha \beta \left( {S_{\eta _{11} } \left( 1 \right) } \right) ^{2},\\ K_{11}= & {} \int _0^1 {\left( {{S}''_{\eta _{11} } } \right) } ^{2}\hbox {d}\xi _1 +\frac{\gamma }{\alpha }\int _0^1 {\left( {{S}''_{\eta _{21} } } \right) } ^{2}\hbox {d}\xi _2\\ P_{11}= & {} \alpha ^{2}\beta \left[ \int _0^1 {S_{\eta _{21} } \left( {-{S}'_{\eta _{21} } } \right) } \hbox {d}\xi _2\right. \\&\left. + \,\int _0^1 {\left( {1-\xi _2 } \right) S_{\eta _{21} } {S}''_{\eta _{21} } } \hbox {d}\xi _2 \right] \\ P_{12}= & {} \alpha ^{2}\beta \left[ \int _0^1 {S_{\eta _{21} } \left( {-{S}'_{\eta _{22} } } \right) } \hbox {d}\xi _2 \right. \\&\left. + \,\int _0^1 {\left( {1-\xi _2 } \right) S_{\eta _{21} } {S}''_{\eta _{22} } } \hbox {d}\xi _2 \right] \\ G_{11}= & {} \alpha ^{2}\beta \int _0^1 {S_{\eta _{21} } } \int _0^1 {\left( {{S}'_{\eta _{11} } } \right) } ^{2}\hbox {d}\xi _1 \hbox {d}\xi _2 ,\\ G_{12}= & {} 2\alpha ^{2}\beta \int _0^1 {S_{\eta _{21} } } \int _0^1 {S}'_{\eta _{11} } {S}'_{\eta _{12} } \hbox {d}\xi _1 \hbox {d}\xi _2\\ G_{22}= & {} \alpha ^{2}\beta \int _0^1 {S_{\eta _{21} } } \int _0^1 {\left( {{S}'_{\eta _{12} } } \right) } ^{2}\hbox {d}\xi _1 \hbox {d}\xi _2\\ S_{11}= & {} -\alpha ^{2}\beta \left[ \int _0^1 S_{\eta _{21} } \hbox {d}\xi _2 \int _0^1 {S_{\eta _{11} } {S}''_{\eta _{11} } } \hbox {d}\xi _1\right. \\&+ \,\frac{1}{2} {S}'_{\eta _{11} } \left( 1 \right) \int _0^1 {S_{\eta _{11} } {S}''_{\eta _{11} } } \hbox {d}\xi _1 +S_{\eta _{11} } \left( 1 \right) \end{aligned}$$
$$\begin{aligned}&\left. \int _0^1 {S_{\eta _{21} } {S}'_{\eta _{21} } } \hbox {d}\xi _2 \right] \\&- \, S_{\eta _{11} } \left( 1 \right) \int _0^1 \left( {1-\xi _2 } \right) S_{\eta _{21}} {S}''_{\eta _{21} } \hbox {d}\xi _2 \\&-\,\int _0^1 {S_{\eta _{21} } } \int _0^1 {\left( {{S}'_{\eta _{11} } } \right) } ^{2}\hbox {d}\xi _1 \hbox {d}\xi _2\\ S_{12}= & {} -\alpha ^{2}\beta \left[ \int _0^1 S_{\eta _{21} } \hbox {d}\xi _2 \int _0^1 {S_{\eta _{11} } {S}''_{\eta _{12} } } \hbox {d}\xi _1 \right. \\&+ \,\frac{1}{2} {S}'_{\eta _{11} } \left( 1 \right) \int _0^1 {S_{\eta _{11} } {S}''_{\eta _{12} } } \hbox {d}\xi _1 +S_{\eta _{11} } \left( 1 \right) \\&\left. \times \int _0^1 {S_{\eta _{21} } {S}'_{\eta _{22} } } \hbox {d}\xi _2 \right] \\&- \, S_{\eta _{11} } \left( 1 \right) \int _0^1 {\left( {1-\xi _2 } \right) S_{\eta _{21} } {S}''_{\eta _{22} } } \hbox {d}\xi _2\\&-\,\int _0^1 {S_{\eta _{21} } } \int _0^1 {{S}'_{\eta _{11} } {S}'_{\eta _{12} } } \hbox {d}\xi _1 \hbox {d}\xi _2\\ S_{21}= & {} -\alpha ^{2}\beta \left[ \int _0^1 S_{\eta _{22} } \hbox {d}\xi _2 \int _0^1 {S_{\eta _{11} } {S}''_{\eta _{11} } } \hbox {d}\xi _1 \right. \\&+ \,\frac{1}{2} {S}'_{\eta _{12} } \left( 1 \right) \int _0^1 {S_{\eta _{11} } {S}''_{\eta _{11} } } \hbox {d}\xi _1 +S_{\eta _{12} } \left( 1 \right) \\&\left. \times \int _0^1 {S_{\eta _{21} } {S}'_{\eta _{21} } } \hbox {d}\xi _2 \right] \\&- \, S_{\eta _{12} } \left( 1 \right) \int _0^1 {\left( {1-\xi _2 } \right) S_{\eta _{21} } {S}''_{\eta _{21} } } \hbox {d}\xi _2 \\&-\,\int _0^1 {S_{\eta _{21} } } \int _0^1 {{S}'_{\eta _{11} } {S}'_{\eta _{12} } } \hbox {d}\xi _1 \hbox {d}\xi _2\\ S_{22}= & {} -\alpha ^{2}\beta \left[ \int _0^1 S_{\eta _{22} } \hbox {d}\xi _2 \int _0^1 {S_{\eta _{11} } {S}''_{\eta _{12} } } \hbox {d}\xi _1\right. \\&+ \,\frac{1}{2} {S}'_{\eta _{12} } \left( 1 \right) \int _0^1 {S_{\eta _{11} } {S}''_{\eta _{12} } } \hbox {d}\xi _1\\&\left. + \, S_{\eta _{12} } \left( 1 \right) \int _0^1 {S_{\eta _{21} } {S}'_{\eta _{22} } } \hbox {d}\xi _2 \right] \\&- \, S_{\eta _{12} } \left( 1 \right) \int _0^1 {\left( {1-\xi _2 } \right) S_{\eta _{21} } {S}''_{\eta _{22} } } \hbox {d}\xi _2\\&-\,\int _0^1 {S_{\eta _{21} } } \int _0^1 {\left( {{S}'_{\eta _{12} } } \right) } ^{2}\hbox {d}\xi _1 \hbox {d}\xi _2\\ E_{11}= & {} \int _0^1 {S_{\eta _{11} } } \hbox {d}\xi _1\mu _2 = \frac{\mu K_{22} }{2M_{22} },\\ \omega _2^2= & {} \frac{K_{22} }{M_{22} }, \quad p_{21} = \frac{P_{21} }{M_{22} },p_{22} = \frac{P_{22} }{M_{22} },\\ \bar{{g}}_{11}= & {} \frac{\bar{{G}}_{11} }{M_{22} },\quad \bar{{g}}_{12} = \frac{\bar{{G}}_{12} }{M_{22} } \end{aligned}$$
$$\begin{aligned} \bar{{g}}_{22}= & {} \frac{\bar{{G}}_{22} }{M_{22} },\quad \bar{{s}}_{11} = \frac{\bar{{S}}_{11} }{M_{22} },\quad \bar{{s}}_{12} = \frac{\bar{{S}}_{12} }{M_{22} },\\ \bar{{s}}_{21}= & {} \frac{\bar{{S}}_{21} }{M_{22} },\bar{{s}}_{22} = \frac{\bar{{S}}_{22} }{M_{22} },e_{12} = \,\frac{E_{12} }{M_{22} }\\ M_{22}= & {} \int _0^1 {\left( {S_{\eta _{12} } } \right) } ^{2}\hbox {d}\xi _1 +\,\alpha ^{3}\beta \int _0^1 {\left( {S_{\eta _{22} } } \right) } ^{2}\hbox {d}\xi _2 \\&+\,\alpha \beta \left( {S_{\eta _{12} } \left( 1 \right) } \right) ^{2},\\ K_{22}= & {} \int _0^1 {\left( {{S}''_{\eta _{12} } } \right) } ^{2}\hbox {d}\xi _1 +\,\frac{\gamma }{\alpha }\int _0^1 {\left( {{S}''_{\eta _{22} } } \right) } ^{2}\hbox {d}\xi _2\\ P_{21}= & {} \alpha ^{2}\beta \left[ \int _0^1 {S_{\eta _{22} } \left( {-{S}'_{\eta _{21} } } \right) } \hbox {d}\xi _2 \right. \\&\left. +\int _0^1 {\left( {1-\xi _2 } \right) S_{\eta _{22} } {S}''_{\eta _{21} } } \hbox {d}\xi _2 \right] \\ P_{22}= & {} \alpha ^{2}\beta \left[ \int _0^1 {S_{\eta _{22} } \left( {-{S}'_{\eta _{22} } } \right) } \hbox {d}\xi _2\right. \\&\left. +\int _0^1 {\left( {1-\xi _2 } \right) S_{\eta _{22} } {S}''_{\eta _{22} } } \hbox {d}\xi _2 \right] \\ \bar{{G}}_{11}= & {} \alpha ^{2}\beta \int _0^1 {S_{\eta _{22} } } \int _0^1 {\left( {{S}'_{\eta _{11} } } \right) } ^{2}\hbox {d}\xi _1 \hbox {d}\xi _2 ,\\ \bar{{G}}_{12}= & {} \alpha ^{2}\beta \left[ \int _0^1 {S_{\eta _{22} } } \int _0^1 {S}'_{\eta _{11} } {S}'_{\eta _{12} } \hbox {d}\xi _1 \hbox {d}\xi _2 \right. \\&\left. + \,\int _0^1 {{S}'_{\eta _{22} } } \int _0^1 {S}'_{\eta _{11} } {S}'_{\eta _{12} } \hbox {d}\xi _1 \hbox {d}\xi _2 \right] \\ \bar{{G}}_{22}= & {} \alpha ^{2}\beta \int _0^1 {S_{\eta _{22} } } \int _0^1 {\left( {{S}'_{\eta _{12} } } \right) } ^{2}\hbox {d}\xi _1 \hbox {d}\xi _2\\ \bar{{S}}_{11}= & {} -\alpha ^{2}\beta \left[ \int _0^1 S_{\eta _{21} } \hbox {d}\xi _2 \int _0^1 {S_{\eta _{12} } {S}''_{\eta _{11} } } \hbox {d}\xi _1\right. \\&+\,\frac{1}{2} {S}'_{\eta _{11} } \left( 1 \right) \int _0^1 {S_{\eta _{12} } {S}''_{\eta _{11} } } \hbox {d}\xi _1\\&\left. + \, S_{\eta _{11} } \left( 1 \right) \int _0^1 {S_{\eta _{22} } {S}'_{\eta _{21} } } \hbox {d}\xi _2 \right] \\&- \, S_{\eta _{11} } \left( 1 \right) \int _0^1 {\left( {1-\xi _2 } \right) S_{\eta _{22} } {S}''_{\eta _{22} } } \hbox {d}\xi _2 \\&- \, \int _0^1 {S_{\eta _{22} } } \int _0^1 {\left( {{S}'_{\eta _{11} } } \right) } ^{2}\hbox {d}\xi _1 \hbox {d}\xi _2\\ \bar{{S}}_{12}= & {} -\alpha ^{2}\beta \left[ \int _0^1 S_{\eta _{21} } \hbox {d}\xi _2 \int _0^1 {S_{\eta _{12} } {S}''_{\eta _{12} } } \hbox {d}\xi _1\right. \\&\left. + \,\frac{1}{2} {S}'_{\eta _{11} } \left( 1 \right) \int _0^1 {S_{\eta _{12} } {S}''_{\eta _{12} } } \hbox {d}\xi _1 +S_{\eta _{11} } \left( 1 \right) \right. \\&\left. \times \int _0^1 S_{\eta _{22} } {S}'_{\eta _{22} } \hbox {d}\xi _2 \right] \end{aligned}$$
$$\begin{aligned}&- \, S_{\eta _{11} } \left( 1 \right) \int _0^1 {\left( {1-\xi _2 } \right) S_{\eta _{22} } {S}''_{\eta _{22} } } \hbox {d}\xi _2 \\&- \, \int _0^1 {S_{\eta _{22} } } \int _0^1 {{S}'_{\eta _{11} } {S}'_{\eta _{12} } } \hbox {d}\xi _1 \hbox {d}\xi _2\\ \bar{{S}}_{21}= & {} -\alpha ^{2}\beta \left[ \int _0^1 S_{\eta _{22} } \hbox {d}\xi _2 \int _0^1 {S_{\eta _{12} } {S}''_{\eta _{11} } } \hbox {d}\xi _1 \right. \\&\left. + \,\frac{1}{2} {S}'_{\eta _{12} } \left( 1 \right) \int _0^1 {S_{\eta _{12} } {S}''_{\eta _{11} } } \hbox {d}\xi _1 +S_{\eta _{12} } \left( 1 \right) \right. \\&\times \left. \int _0^1 {S_{\eta _{22} } {S}'_{\eta _{21} } } \hbox {d}\xi _2 \right] \\&- \, S_{\eta _{12} } \left( 1 \right) \int _0^1 {\left( {1-\xi _2 } \right) S_{\eta _{22} } {S}''_{\eta _{21} } } \hbox {d}\xi _2\\&- \, \int _0^1 {S_{\eta _{22} } } \int _0^1 {{S}'_{\eta _{11} } {S}'_{\eta _{12} } } \hbox {d}\xi _1 \hbox {d}\xi _2\\ \bar{{S}}_{22}= & {} -\alpha ^{2}\beta \left[ \int _0^1 S_{\eta _{22} } \hbox {d}\xi _2 \int _0^1 {S_{\eta _{12} } {S}''_{\eta _{12} } } \hbox {d}\xi _1 \right. \\&\left. + \,\frac{1}{2} {S}'_{\eta _{12} } \left( 1 \right) \int _0^1 {S_{\eta _{12} } {S}''_{\eta _{12} } } \hbox {d}\xi _1 +S_{\eta _{12} } \left( 1 \right) \right. \\&\times \left. \int _0^1 {S_{\eta _{22} } {S}'_{\eta _{22} } } \hbox {d}\xi _2 \right] \\&- \, S_{\eta _{12} } \left( 1 \right) \int _0^1 {\left( {1-\xi _2 } \right) S_{\eta _{22} } {S}''_{\eta _{22} } } \hbox {d}\xi _2 \\&- \, \int _0^1 {S_{\eta _{22} } } \int _0^1 {\left( {{S}'_{\eta _{12} } } \right) } ^{2}\hbox {d}\xi _1 \hbox {d}\xi _2\\ E_{12}= & {} \int _0^1 {S_{\eta _{12} } } \hbox {d}\xi _1 \end{aligned}$$

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Yu, TJ., Zhang, W. & Yang, XD. Global dynamics of an autoparametric beam structure. Nonlinear Dyn 88, 1329–1343 (2017). https://doi.org/10.1007/s11071-016-3313-0

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