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Magnetoelastic axisymmetric multi-modal resonance and Hopf bifurcation of a rotating circular plate under aerodynamic load

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Abstract

In this article, an investigation of magnetoelastic axisymmetric multi-mode interaction and Hopf bifurcations of a circular plate rotating in air and uniform transverse magnetic fields is presented. The expressions of electromagnetic forces and an empirical aerodynamic model are applied in the derivation of the dynamical equations, through which a set of nonlinear differential equations for axisymmetric forced oscillation of the clamped circular plate are deduced. The method of multiple scales combined with the polar coordinate transformation is employed to solve the differential equations and achieve the phase–amplitude modulation equations for the interaction among the first three modes under primary resonance. Then, the frequency response equation for the single-mode vibration, the steady-state response equations for three-mode resonance and the corresponding Jacobian matrix are obtained by means of the modulation equations. Numerical examples are presented to show the dependence of amplitude solutions as a function of different parameters in the cases of single mode and three-mode response. Furthermore, a Hopf bifurcation can be found in three-mode equilibrium by choosing appropriate parameters, where a limit cycle occurs and then evolves into chaos after undergoing a series of period-doubling bifurcations.

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Acknowledgements

This project was supported by the National Natural Science Foundation of China (No. 11472239), Hebei Provincial Natural Science Foundation of China (No. A2015203023), Hebei Provincial Graduate Innovation Foundation of China (No. CXZZBS2018058) and Hebei Provincial Yanshan University Graduate Innovation Foundation of China (No. CXZS201908).

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Authors and Affiliations

  1. School of Civil Engineering and Mechanics, Yanshan University, Qinhuangdao, 066004, China

    Y. D. Hu & W. Q. Li

  2. Hebei Key Laboratory of Mechanical Reliability for Heavy Equipments and Large Structures, Yanshan University, Qinhuangdao, 066004, China

    Y. D. Hu & W. Q. Li

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  1. Y. D. Hu
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  2. W. Q. Li
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Appendix

Appendix

$$\begin{aligned}& \hat{{\alpha }}_{e0} =\frac{3}{2}\left( {3{\phi }'_{e0} {\phi }'_{e0} {\phi }''_{e0} +\frac{1}{r}{\phi }'_{e0} {\phi }'_{e0} {\phi }'_{e0} } \right) \\& \hat{{\alpha }}_{x0e0} =\left( {3{\phi }'_{e0} {\phi }'_{x0} {\phi }''_{x0} +\frac{1}{r}{\phi }'_{e0} {\phi }'_{x0} {\phi }'_{x0} } \right) \\&\, +\left( {\frac{3}{2}{\phi }'_{x0} {\phi }'_{x0} {\phi }''_{e0} +\frac{1}{2r}{\phi }'_{x0} {\phi }'_{x0} {\phi }'_{e0} } \right) \\& \hat{{\alpha }}_3 =\left( {\frac{3}{2}{\phi }'_{10} {\phi }'_{10} {\phi }''_{00} +\frac{1}{2r}{\phi }'_{10} {\phi }'_{10} {\phi }'_{00} } \right) \\&\, +\left( {3{\phi }'_{10} {\phi }'_{00} {\phi }''_{10} +\frac{1}{r}{\phi }'_{10} {\phi }'_{00} {\phi }'_{10} } \right) \\& \hat{{\alpha }}_4 =\left( {3{\phi }'_{20} {\phi }'_{10} {\phi }''_{10} +\frac{1}{r}{\phi }'_{20} {\phi }'_{10} {\phi }'_{10} } \right) \\&\, +\left( {\frac{3}{2}{\phi }'_{10} {\phi }'_{10} {\phi }''_{20} +\frac{1}{2r}{\phi }'_{10} {\phi }'_{10} {\phi }'_{20} } \right) \\& \hat{{\alpha }}_5 =\left( {3{\phi }'_{00} {\phi }'_{10} {\phi }''_{20} +\frac{1}{r}{\phi }'_{00} {\phi }'_{10} {\phi }'_{20} } \right) \\&\, +\left( {3{\phi }'_{10} {\phi }'_{20} {\phi }''_{00} +\frac{1}{r}{\phi }'_{10} {\phi }'_{20} {\phi }'_{00} } \right) \\& +\left( {3{\phi }'_{20} {\phi }'_{00} {\phi }''_{10} +\frac{1}{r}{\phi }'_{20} {\phi }'_{00} {\phi }'_{10} } \right) \end{aligned}$$

\(\delta _{je} \) is the Kronecker delta which satisfies \(\delta _{je} =\left\{ {\begin{array}{ll} 0 &{}\,\, j\ne e \\ 1 &{}\,\, j=e \\ \end{array}} \right. \), \(e=0\), 1, 2

$$\begin{aligned} \vartheta _{e0}= & {} \int _0^1 {\frac{1}{r}} \left( {{\phi }'_{e0} +r{\phi }''_{e0} } \right) \phi _{e0} r\hbox {d}r\\ Q_{e0}= & {} \int _0^1 {q_0 \phi _{e0} r\hbox {d}r}\\ \varGamma _{e0}= & {} \int _0^1 {\hat{{\alpha }}_{e0} \phi _{e0} r\hbox {d}r}\\ \varGamma _{x0e0}= & {} \int _0^1 {\hat{{\alpha }}_{x0e0} \phi _{e0} r\hbox {d}r}\\ \varGamma _{3e0}= & {} \int _0^1 {\hat{{\alpha }}_3 \delta _{e2} \phi _{e0} r\hbox {d}r}\\ \varGamma _{4e0}= & {} \int _0^1 {\hat{{\alpha }}_4 \delta _{e0} \phi _{e0} r\hbox {d}r}\\ \varGamma _{5e0}= & {} \int _0^1 {\hat{{\alpha }}_5 \delta _{e1} \phi _{e0} r\hbox {d}r}\\ \mu _{00}= & {} \frac{1}{2}C_d -\frac{\sigma B_{0z}^2 }{24}\vartheta _{00}\\ \mu _{10}= & {} \frac{1}{2}C_d -\frac{\sigma B_{0z}^2 }{24}\vartheta _{10}\\ \mu _{20}= & {} \frac{1}{2}C_d -\frac{\sigma B_{0z}^2 }{24}\vartheta _{20}\\ \hat{{B}}_1= & {} \frac{1}{288}\left( \frac{\omega _{00}^2 +\varOmega ^{2}}{\omega _{00} }\vartheta _{00} \right. \\&\left. +2\frac{\omega _{10}^2 +\varOmega ^{2}}{\omega _{10} }\vartheta _{10} -\frac{\omega _{20}^2 +\varOmega ^{2}}{\omega _{20} }\vartheta _{20} \right) \\ \hat{{C}}_1= & {} \frac{1}{8}\left( {\frac{\varGamma _{00} }{\omega _{00} }+\frac{4\varGamma _{0010} }{\omega _{10} }-\frac{2\varGamma _{0020} }{\omega _{20} }} \right) \\ \hat{{D}}_1= & {} \frac{1}{8}\left( {\frac{2\varGamma _{10} }{\omega _{10} }+\frac{2\varGamma _{1000} }{\omega _{00} }-\frac{2\varGamma _{1020} }{\omega _{20} }} \right) \\ \hat{{E}}_1= & {} \frac{1}{8}\left( {\frac{\varGamma _{20} }{\omega _{20} }-\frac{2\varGamma _{2000} }{\omega _{00} }-\frac{4\varGamma _{2010} }{\omega _{10} }} \right) \\ \hat{{B}}_2= & {} \frac{1}{288}\frac{\omega _{20}^2 +\varOmega ^{2}}{\omega _{20} }\vartheta _{20}\\ \hat{{C}}_2= & {} \frac{\varGamma _{0020} }{4\omega _{20} }\\ \hat{{D}}_2= & {} \frac{\varGamma _{1020} }{4\omega _{20} }\\ \hat{{E}}_2= & {} \frac{\varGamma _{20} }{8\omega _{20} }\\ G_1= & {} \frac{1}{{X}_{00} }\left[ \sigma _1 -\sigma \right. \\&+\frac{1}{288}\left( {\frac{\omega _{00}^2 +\varOmega ^{2}}{\omega _{00} }\vartheta _{00} +2\frac{\omega _{10}^2 +\varOmega ^{2}}{\omega _{10} }\vartheta _{10} } \right) \\&-\frac{3}{8}{X}_{00}^2 \left( {\frac{\varGamma _{00} }{\omega _{00} }+\frac{4\varGamma _{0010} }{\omega _{10} }} \right) \\&-\frac{1}{8}{X}_{10}^2 \left( {\frac{2\varGamma _{10} }{\omega _{10} }+\frac{2\varGamma _{1000} }{\omega _{00} }} \right) \\&- \frac{1}{8}{X}_{20}^2 \left( {\frac{2\varGamma _{2000} }{\omega _{00} }+\frac{4\varGamma _{2010} }{\omega _{10} }} \right) \\&\left. -\frac{4}{8}\frac{\varGamma _{510} }{\omega _{10} }{X}_{00} {X}_{20} \cos {H}_1 \right] \\&+\frac{1}{{X}_{20} }\left[ \frac{1}{8\omega _{20} }\left( 4{X}_{00} {X}_{20} \varGamma _{0020} \right. \right. \\&\left. \left. +{X}_{10}^2 \varGamma _{320} \cos {H}_1 \right) \right] \\ G_2= & {} \frac{1}{{X}_{00} }\left[ -\frac{1}{8}2{X}_{10} {X}_{00} \left( {\frac{2\varGamma _{10} }{\omega _{10} }+\frac{2\varGamma _{1000} }{\omega _{00} }} \right) \right. \\&\left. -\frac{1}{8}2{X}_{10} {X}_{20} \frac{2\varGamma _{400} }{\omega _{00} }\cos {H}_1 \right] \\&+\frac{1}{{X}_{20} }\left[ \frac{1}{8\omega _{20} }\left( 4{X}_{10} {X}_{20} \varGamma _{1020} \right. \right. \\&\left. \left. +2\varGamma _{320} {X}_{00} {X}_{10} \cos {H}_1 \right) \right] \\ G_3= & {} \frac{1}{{X}_{00} }\left[ -\frac{1}{8}\left( {\frac{2\varGamma _{2000} }{\omega _{00} }+\frac{4\varGamma _{2010} }{\omega _{10} }} \right) 2{X}_{20} {X}_{00}\right. \\&\left. -\frac{1}{8}\left( {\frac{\varGamma _{400} }{\omega _{00} }{X}_{10}^2 +\frac{2\varGamma _{510} }{\omega _{10} }{X}_{00}^2 } \right) \cos {H}_1 \right] \\&+\frac{1}{{X}_{20} }\left[ \sigma -\frac{1}{288}\frac{\omega _{20}^2 +\varOmega ^{2}}{\omega _{20} }\vartheta _{20}\right. \\&+\frac{1}{8\omega _{20} }\left( 2{X}_{00}^2 \varGamma _{0020}\right. \\&\left. \left. +2{X}_{10}^2 \varGamma _{1020} +3{X}_{20}^2 \varGamma _{20} \right) \right] \\ G_4= & {} \left\{ \frac{1}{{X}_{00} }\left[ {\frac{1}{8}\left( {\frac{\varGamma _{400} }{\omega _{00} }{X}_{10}^2 {X}_{20} +2\frac{\varGamma _{510} }{\omega _{10} }{X}_{00}^2 {X}_{20} } \right) } \right] \right. \\&\left. +\frac{1}{{X}_{20} }\left[ {-\frac{1}{8}\frac{\varGamma _{320} }{\omega _{20} }{X}_{00} {X}_{10}^2 } \right] \right\} \sin {H}_1\\ G_5= & {} -\frac{1}{{X}_{20} }\frac{Q_{20} }{\omega _{20} }\sin {H}_2\\ J_1= & {} \frac{1}{8{X}_{20} \omega _{20} }\left( {4{X}_{00} {X}_{20} \varGamma _{0020} +{X}_{10}^2 \varGamma _{320} \cos {H}_1 } \right) \\ J_2= & {} \frac{1}{8{X}_{20} \omega _{20} }\left( 4\varGamma _{1020} {X}_{10} {X}_{20} \right. \\&\left. +2\varGamma _{320} {X}_{00} {X}_{10} \cos {H}_1 \right) \\ J_3= & {} \frac{1}{{X}_{20} }\left[ \sigma -\frac{1}{288}\frac{\omega _{20}^2 +\varOmega ^{2}}{\omega _{20} }\vartheta _{20} \right. \\&+\frac{1}{8\omega _{20} }\left( 2{X}_{00}^2 \varGamma _{0020} \right. \\&\left. \left. +2{X}_{10}^2 \varGamma _{1020} +3{X}_{20}^2 \varGamma _{20} \right) \right] \\ J_4= & {} -\frac{1}{8{X}_{20} }\frac{\varGamma _{320} }{\omega _{20} }{X}_{00} {X}_{10}^2 \sin {H}_1\\ J_5= & {} -\frac{1}{{X}_{20} }\frac{Q_{20} }{\omega _{20} }\sin {H}_2\\ \end{aligned}$$

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Hu, Y.D., Li, W.Q. Magnetoelastic axisymmetric multi-modal resonance and Hopf bifurcation of a rotating circular plate under aerodynamic load. Nonlinear Dyn 97, 1295–1311 (2019). https://doi.org/10.1007/s11071-019-05049-8

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