Nonlinear resonant response of a buckled beam coupled with a boundary massive oscillator

Abstract

This study focuses on nonlinear modal resonant dynamics of a buckled beam coupled with a boundary massive oscillator. To reveal buckled beam–boundary oscillator coupling effect, extended Hamilton principle is employed to derive a dynamic model with geometric nonlinearity included, and direct multiple-scale method (i.e., attacking directly partial differential equations) is then applied to reduce the original infinite-dimensional beam–support coupled system, leading to nonlinear modulation equations characterizing reduced slow dynamics of the coupled system, by focusing on beam’s one-to-one internally resonant dynamics around its first buckled shape. Time history responses, frequency responses, and Poincaré mapping are employed to investigate stability/bifurcation of nonlinear forced coupled dynamics, with one-to-one internal resonance activated or not.

Appendices

Appendix A

The variational process of kinetic energy is as follows:

$$ \begin{aligned} \int_{{t_{1} }}^{{t_{2} }} {\delta T{\text{d}}\hat{t}} = & \int_{{t_{1} }}^{{t_{2} }} {\left\{ {\hat{m}\int_{0}^{l} {[\dot{\hat{w}}\delta \dot{\hat{w}}_{{}} } + \dot{\hat{u}}\delta \dot{\hat{u}}]d\hat{x} + \hat{M}\dot{\hat{g}}\delta \dot{\hat{g}}} \right\}} {\text{d}}\hat{t} = \int_{{t_{1} }}^{{t_{2} }} {\left\{ {\hat{m}\int_{0}^{l} {[\dot{\hat{w}}_{{}} {\text{d}}} \delta \hat{w}_{{}} + \dot{\hat{u}}{\text{d}}\delta \hat{u}]d\hat{x} + \hat{M}\dot{\hat{g}}{\text{d}}\delta \hat{g}} \right\}} \\ = & \hat{m}\int_{0}^{l} {[\left. {\dot{\hat{u}}\delta \hat{u}} \right|}_{{t_{1} }}^{{t_{2} }} - \int_{{t_{1} }}^{{t_{2} }} {\ddot{\hat{u}}} \delta \hat{u}{\text{d}}\hat{t}]{\text{d}}\hat{x} + \hat{m}\int_{0}^{l} {[\left. {\dot{\hat{w}}_{{}} \delta \hat{w}_{{}} } \right|}_{{t_{1} }}^{{t_{2} }} - \int_{{t_{1} }}^{{t_{2} }} {\ddot{\hat{w}}_{{}} } \delta \hat{w}_{{}} {\text{d}}\hat{t}]d\hat{x} + \left. {\hat{M}\dot{\hat{g}}{\text{d}}\delta \hat{g}} \right|_{{t_{1} }}^{{t_{2} }} - \int_{{t_{1} }}^{{t_{2} }} {\hat{M}\ddot{\hat{g}}\delta \hat{g}{\text{d}}\hat{t}} \\ = & - \int_{{t_{1} }}^{{t_{2} }} {\left\{ {m\int_{0}^{l} {(\ddot{\hat{w}}_{{}} } \delta \hat{w}_{{}} + \ddot{\hat{u}}\delta \hat{u})d\hat{x} + \hat{M}\ddot{\hat{g}}\delta \hat{g}} \right\}} {\text{d}}\hat{t} \\ \end{aligned} $$

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Also, the variational process of potential energy is as follows:

$$ \begin{aligned} \int_{{t_{1} }}^{{t_{2} }} {\delta U{\text{d}}\hat{t}} = & \int_{{t_{1} }}^{{t_{2} }} {[EA\int_{0}^{l} {(\hat{u^{\prime}}} } + \frac{1}{2}\hat{w^{\prime}}_{{}}^{2} )(\delta \hat{u}^{\prime } + \hat{w^{\prime}}_{{}} \delta \hat{w^{\prime}}_{{}} ){\text{d}}\hat{x} + EI\int_{0}^{l} {\hat{w^{\prime\prime}}_{{}} } \delta \hat{w^{\prime\prime}}_{{}} {\text{d}}x - \hat{P}\hat{u}(0,\hat{t}) + \hat{K}\hat{g}\delta \hat{g}]{\text{d}}\hat{t} \\ = & \underbrace {{EA\int_{{t_{1} }}^{{t_{2} }} {\int_{0}^{l} {(\hat{u^{\prime}}} } + \frac{1}{2}\hat{w^{\prime}}_{{}}^{2} )\delta \hat{u}^{\prime } {\text{d}}\hat{x}{\text{d}}\hat{t}}}_{(1)} \, + \underbrace {{EA\int_{{t_{1} }}^{{t_{2} }} {\int_{0}^{L} {(\hat{u}^{\prime } } } + \frac{1}{2}w^{\prime 2} )\hat{w^{\prime}}\delta \hat{w^{\prime}}{\text{d}}\hat{x}{\text{d}}\hat{t}}}_{(2)} \, \\ & + \underbrace {{EI\int_{{t_{1} }}^{{t_{2} }} {\int_{0}^{l} {\hat{w^{\prime\prime}}_{{}} } } \delta \hat{w^{\prime\prime}}_{{}} {\text{d}}\hat{x}{\text{d}}\hat{t}}}_{(3)} \, - \int_{{t_{1} }}^{{t_{2} }} {\hat{P}\delta \hat{u}(0,\hat{t})} {\text{d}}\hat{t} + \int_{{t_{1} }}^{{t_{2} }} {\hat{K}\hat{g}\delta \hat{g}} {\text{d}}\hat{t} \\ \end{aligned} $$

(47)

The details above are given below

$$ \begin{aligned} (1){ = } & EA\int_{{t_{1} }}^{{t_{2} }} {\int_{0}^{l} {(\hat{u}^{\prime } } } + \frac{1}{2}\hat{w^{\prime}}_{{}}^{2} ){\text{d}}\delta \hat{u}{\text{d}}\hat{t} \\ = & EA\int_{{t_{1} }}^{{t_{2} }} [ (\hat{u^{\prime}} + \frac{1}{2}\hat{w^{\prime}}_{{}}^{2} )\left. {\delta \hat{u}} \right|_{0}^{l} - \int_{0}^{l} {(\hat{u^{\prime}} + \frac{1}{2}\hat{w^{\prime}}_{{}}^{2} )^{\prime}} \delta \hat{u}{\text{d}}\hat{x}]{\text{d}}\hat{t} \\ = & EA\int_{{t_{1} }}^{{t_{2} }} [ (\hat{u^{\prime}} + \frac{1}{2}\hat{w^{\prime}}_{{}}^{2} )\left. {\delta \hat{u}} \right|_{0}^{l} {\text{d}}\hat{t} - EA\int_{{t_{1} }}^{{t_{2} }} {\int_{0}^{l} {(\hat{u^{\prime}}} } + \frac{1}{2}\hat{w^{\prime}}_{{}}^{2} )^{\prime}\delta \hat{u}{\text{d}}\hat{x}{\text{d}}\hat{t} \\ = & \left. {EA\int_{{t_{1} }}^{{t_{2} }} [ (\hat{u^{\prime}} + \frac{1}{2}\hat{w^{\prime}}_{{}}^{2} )} \right|_{{\hat{x} = l}} \delta \hat{u}(l,\hat{t}){\text{d}}\hat{t} - \left. {EA\int_{{t_{1} }}^{{t_{2} }} [ (\hat{u^{\prime}} + \frac{1}{2}\hat{w^{\prime}}_{{}}^{2} )} \right|_{{\hat{x} = 0}} \delta \hat{u}(0,\hat{t}){\text{d}}\hat{t} \\ & - EA\int_{{t_{1} }}^{{t_{2} }} {\int_{0}^{l} {(\hat{u^{\prime}}} } + \frac{1}{2}\hat{w^{\prime}}_{{}}^{2} )^{\prime}\delta \hat{u}{\text{d}}\hat{x}{\text{d}}\hat{t} \\ \end{aligned} $$

(48)

$$ \begin{aligned} (2){ = } & EA\int_{{t_{1} }}^{{t_{2} }} {\int_{0}^{l} {(\hat{u^{\prime}}} } + \frac{1}{2}\hat{w}^{\prime 2} )\hat{w^{\prime}}_{{}} {\text{d}}\delta \hat{w}{\text{d}}\hat{t} \\ = & EA\left. {\int_{{t_{1} }}^{{t_{2} }} {[(\hat{u}^{\prime } + \frac{1}{2}\hat{w}^{\prime 2} \hat{w^{\prime}}_{{}}^{2} )} \hat{w^{\prime}}_{{}} \delta \hat{w}_{{}} } \right|_{0}^{l} - \int_{0}^{l} {[(\hat{u^{\prime}} + \frac{1}{2}\hat{w}^{\prime 2} )} \hat{w}^{\prime } ]^{\prime}\delta \hat{w}{\text{d}}\hat{x}]{\text{d}}\hat{t} \\ = & - EA\int_{{t_{1} }}^{{t_{2} }} {\int_{0}^{l} {[(\hat{u}^{\prime } } } + \frac{1}{2}\hat{w}^{\prime 2} )\hat{w^{\prime}}_{{}} ]^{\prime}\delta \hat{w}{\text{d}}\hat{x}{\text{d}}\hat{t} \\ \end{aligned} $$

(49)

$$ \begin{aligned} (3){ = } & EI\int_{{{\text{t}}_{1} }}^{{t_{2} }} {\int_{0}^{l} {\hat{w}^{\prime \prime } } d\delta \hat{w}^{\prime } {\text{d}}\hat{t}} = EI\left. {\int_{{t_{1} }}^{{t_{2} }} {[\hat{w}^{\prime \prime } } \delta \hat{w}^{\prime } } \right|_{0}^{l} - \int_{0}^{l} {\hat{w}^{\prime \prime \prime } } \delta \hat{w}^{\prime } {\text{d}}\hat{x}]{\text{d}}\hat{t} = - EI\int_{{t_{1} }}^{{t_{2} }} {\int_{0}^{l} {\hat{w}^{\prime \prime \prime } } \delta \hat{w}^{\prime } {\text{d}}\hat{x}{\text{d}}\hat{t}} \\ = & - EI\int_{{t_{1} }}^{{t_{2} }} {\int_{0}^{l} {\hat{w}^{\prime \prime \prime } } {\text{d}}\delta \hat{w}} {\text{d}}\hat{t} = - EI\left. {\int_{{t_{1} }}^{{t_{2} }} {[\hat{w}^{\prime \prime \prime } } \delta \hat{w}} \right|_{0}^{l} - \int_{0}^{l} {\hat{w}^{\prime \prime \prime \prime } } \, \delta \hat{w}{\text{d}}\hat{x}]{\text{d}}\hat{t} \\ = & EI\int_{{t_{1} }}^{{t_{2} }} {\int_{0}^{l} {\hat{w}^{\prime \prime \prime \prime } } \, \delta \hat{w}} {\text{d}}\hat{x}{\text{d}}\hat{t} \\ \end{aligned} $$

(50)

In a similar way, the variational process of virtual work is as follows:

$$ \begin{aligned} \int_{{t_{1} }}^{{t_{2} }} {\delta W{\text{d}}\hat{t}} = & \int_{{t_{1} }}^{{t_{2} }} {\left\{ {\int_{0}^{l} {(\hat{f}(\hat{x})} \cos \hat{\Omega }\hat{t} - 2\hat{c}\dot{\hat{w}}_{{}} )\delta \hat{w}_{{}} {\text{d}}\hat{x} - 2\hat{\nu }\dot{\hat{g}}\delta \hat{g}} \right\}{\text{d}}\hat{t}} \\ = & \int_{{t_{1} }}^{{t_{2} }} {\int_{0}^{l} {(\hat{f}(\hat{x})} \cos \hat{\Omega }\hat{t} - 2\hat{c}\dot{\hat{w}}_{{}} )\delta \hat{w}_{{}} {\text{d}}\hat{x}{\text{d}}\hat{t}} - \int_{{t_{1} }}^{{t_{2} }} {2\hat{\nu }\dot{\hat{g}}\delta \hat{g}{\text{d}}\hat{t}} \\ \end{aligned} $$

(51)

Note that one key constraint above is $\delta \hat{g}\left( {\hat{t}} \right) = \delta \hat{u}\left( {l,\hat{t}} \right)$, which is due to the beam-support interface connection condition $\hat{u}\left( {l,\hat{t}} \right) = \hat{g}\left( {\hat{t}} \right)$.

Appendix B

The buckled beam’s linear mode shapes $\phi_{k} (x)$, around the first buckled mode $y(x) = 0.5 \times b(1 - \cos 2\pi x),\;\;b = {{2\sqrt {P - 4\pi^{2} } } \mathord{\left/ {\vphantom {{2\sqrt {P - 4\pi^{2} } } \pi }} \right. \kern-0pt} \pi }$, are derived by solving the following boundary-value problem [7]

$$ \begin{gathered} \ddot{w} + w^{iv} + \lambda^{2} w^{\prime\prime} - y^{\prime\prime}\int_{0}^{1} {y^{\prime } w^{\prime } } {\text{d}}x = 0,\quad \lambda^{2} = 4\pi^{2} \hfill \\ x = 0:w = w^{\prime } = 0; \, x = 1:w = w^{\prime } = 0 \hfill \\ \end{gathered} $$

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Explicitly, one has

$$ \phi_{k} (x) = c_{1} \sin \lambda x + c_{2} cos\lambda x + c_{3} \sinh \mu x + c_{4} \cosh \mu x + c_{5} \cos 2\pi x $$

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where

$$ \Gamma = \sqrt {2\pi^{2} + \sqrt {4\pi^{4} + \omega_{k}^{2} } } ,\quad \mu = \sqrt { - 2\pi^{2} + \sqrt {4\pi^{4} + \omega_{k}^{2} } } $$

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the $c_{i}$ and $\omega_{k}$ in the mode shapes are solutions to the following algebraic eigenvalue problem [7]:

$$ \begin{gathered} c_{2} + c_{4} + c_{5} = 0 \hfill \\ c_{1} \sin \Gamma + c_{2} cos\Gamma + c_{3} \sinh \mu + c_{4} \cosh \mu + c_{5} = 0 \hfill \\ c_{1} \Gamma + c_{3} \mu = 0 \hfill \\ c_{1} \Gamma \cos \Gamma - c_{2} \Gamma \sin \Gamma + c_{3} \mu \cosh \mu + c_{4} \mu \sinh \mu = 0 \hfill \\ 4b^{2} \Gamma^{3} \pi^{4} (2c_{1} \sin \frac{{\Gamma^{2} }}{4} + c_{2} \sin \Gamma ) + 4b^{2} \mu^{3} \pi^{4} (2c_{3} \sinh \frac{{\mu^{2} }}{4} + c_{4} \sinh \mu ) \hfill \\ + c_{5} \omega_{k}^{2} (2b^{2} \pi^{4} - \omega_{k}^{2} ) = 0 \hfill \\ \end{gathered} $$

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Typical mode shapes are depicted in Fig. 15

Fig. 15

An illustration of typical modal shape functions (the lowest four modes) of the clamped–clamped buckled beam when b = 6.21

.

Appendix C

The shape functions used in Eq. (31) are governed by the following linear boundary-value problems (BVP)

$$ \begin{gathered} - 4\omega_{m}^{2} \Psi_{mm} + L[\Psi_{mm} ] = G_{2} [\phi_{m} ,\phi_{m} ] \hfill \\ - 4\omega_{n}^{2} \Psi_{nn} + L[\Psi_{nn} ] = G_{2} [\phi_{n} ,\phi_{n} ] \hfill \\ - (\omega_{m} + \omega_{n} )^{2} \Psi_{mn} + L[\Psi_{mn} ] = G_{2} [\phi_{m} ,\phi_{n} ] + G_{2} [\phi_{n} ,\phi_{m} ] \hfill \\ L[\chi_{mm} ] = G_{2} [\phi_{m} ,\phi_{m} ] \hfill \\ L[\chi_{nn} ] = G_{2} [\phi_{n} ,\phi_{n} ] \hfill \\ - (\omega_{m} - \omega_{n} )^{2} \chi_{mn} + L[\chi_{mn} ] = G_{2} [\phi_{m} ,\phi_{n} ] + G_{2} [\phi_{n} ,\phi_{m} ] \hfill \\ \end{gathered} $$

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with boundary conditions $\Psi_{ij} (0) = \Psi_{ij} (1) = 0, \, \chi_{ij} (0) = \chi_{ij} (1) = 0$. Recall that the linear structural operator is defined as $L[w] = w^{iv} + 4\pi^{2} w^{\prime \prime } - y^{\prime\prime}\int_{0}^{1} {y^{\prime } w^{\prime } } {\text{d}}x$.

Appendix D

$$ \begin{aligned} \Gamma_{mm} (x) = & 3G_{3} [\phi_{m} ,\phi_{m} ,\phi_{m} ] + G_{2} [\psi_{mm} ,\phi_{m} ] + G_{2} [\phi_{m} ,\psi_{mm} ] + 2G_{2} [\phi_{m} ,\chi_{mm} ] + 2G_{2} [\chi_{mm} ,\phi_{m} ] \\ \Gamma_{nn} (x) = & 3G_{3} [\phi_{n} ,\phi_{n} ,\phi_{n} ] + G_{2} [\psi_{nn} ,\phi_{n} ] + G_{2} [\phi_{n} ,\psi_{nn} ] + 2G_{2} [\phi_{n} ,\chi_{nn} ] + 2G_{2} [\chi_{nn} ,\phi_{n} ] \\ \Gamma_{mn} (x) = & 2G_{3} [\phi_{m} ,\phi_{m} ,\phi_{n} ] + 2G_{3} [\phi_{n} ,\phi_{m} ,\phi_{m} ] + 2G_{3} [\phi_{m} ,\phi_{n} ,\phi_{m} ] + 2G_{2} [\chi_{mm} ,\phi_{n} ] \\ & + 2G_{2} [\phi_{n} ,\chi_{mm} ] + G_{2} [\chi_{mn} ,\phi_{m} ] + G_{2} [\phi_{m} ,\chi_{mn} ] + G_{2} [\psi_{mn} ,\phi_{m} ] + G_{2} [\phi_{m} ,\psi_{mn} ] \\ \Gamma_{nm} (x) = & 2G_{3} [\phi_{m} ,\phi_{n} ,\phi_{n} ] + 2G_{3} [\phi_{n} ,\phi_{m} ,\phi_{n} ] + 2G_{3} [\phi_{n} ,\phi_{n} ,\phi_{m} ] + 2G_{2} [\chi_{nn} ,\phi_{m} ] \\ & + 2G_{2} [\phi_{m} ,\chi_{nn} ] + G_{2} [\chi_{mn} ,\phi_{n} ] + G_{2} [\phi_{n} ,\chi_{mn} ] + G_{2} [\psi_{mn} ,\phi_{n} ] + G_{2} [\phi_{n} ,\psi_{mn} ] \\ \Gamma_{m} (x) = & G_{3} [\phi_{n} ,\phi_{n} ,\phi_{m} ] + G_{3} [\phi_{m} ,\phi_{n} ,\phi_{n} ] + G_{3} [\phi_{n} ,\phi_{m} ,\phi_{n} ] + G_{2} [\phi_{n} ,\chi_{mn} ] \\ & + G_{2} [\chi_{mn} ,\phi_{n} ] + G_{2} [\phi_{m} ,\chi_{nn} ] + G_{2} [\chi_{nn} ,\phi_{m} ] + G_{2} [\phi_{n} ,\psi_{mn} ] + G_{2} [\psi_{mn} ,\phi_{n} ] \\ \Gamma_{n} (x) = & G_{3} [\phi_{n} ,\phi_{m} ,\phi_{m} ] + G_{3} [\phi_{m} ,\phi_{n} ,\phi_{m} ] + G_{3} [\phi_{m} ,\phi_{m} ,\phi_{n} ] + G_{2} [\phi_{m} ,\chi_{mn} ] \\ & + G_{2} [\chi_{mn} ,\phi_{m} ] + G_{2} [\phi_{n} ,\chi_{mm} ] + G_{2} [\chi_{mm} ,\phi_{n} ] + G_{2} [\phi_{m} ,\psi_{mn} ] + G_{2} [\psi_{mn} ,\phi_{m} ] \\ \end{aligned} $$

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