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Dynamics of excited piecewise linear oscillators

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Abstract

The current work is devoted to the analytical study of the dynamics of piecewise linear (PWL) oscillators subjected to various types of excitations. Straightforward analytical study of the response of this class of strongly nonlinear oscillators is rather complex, as it requires the computation of time instants of transitions from one linear state to another. This difficulty is commonly overcome by using the averaging procedure. In the present study, we devise an averaging method which naturally fits to the analysis of a general class of resonantly forced PWL systems with zero offset. This method is based on introduction of non-analytic PWL basis functions with corresponding algebra thereby enabling the application of direct averaging method to the resonantly forced, strongly nonlinear PWL oscillators (PWLOs). We demonstrate the efficiency of the considered averaging method for the three different cases of forced PWLOs. We first consider the PWL Mathieu equation and asymptotically obtain the relatively simple analytical expressions for the transition curves corresponding to the most significant family of m:1 sub-harmonic resonances for both damped and undamped cases. The second dynamical system considered herein is a PWL Van der Pol oscillator. In this example, we use the derived averaged model to explore the effect of asymmetry parameter on the limit cycle oscillations. As a final example, we consider an externally forced PWL Duffing oscillator to illustrate the effect of stiffness asymmetry parameter on its resonance curves. Results of the analysis for all the three considered models show a fairly good correspondence with the numerical simulations of the model within the limit of asymptotic validity.

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Acknowledgements

KRJ and YS would like to acknowledge the support received through the SPARC project (Project Code: P483) funded by the MHRD, Government of India. YS would also like to acknowledge the support received by Israel Science Foundation (Grant number 1079/16).

Funding

SPARC project (Project Code: P483) funded by the MHRD, Government of India; Israel Science Foundation (Grant number 1079/16).

Author information

Authors and Affiliations

  1. Discipline of Mechanical Engineering, IIT Gandhianagar, Palaj, Gandhinagar, India

    K. R. Jayaprakash & Vaibhav Tandel

  2. Faculty of Mechanical Engineering, Israel Institute of Technology-Technion, Haifa, Israel

    Yuli Starosvetsky

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  1. K. R. Jayaprakash
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Correspondence to K. R. Jayaprakash.

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Appendices

Appendix A

The PWL function can be defined in terms of the Heaviside function in the form

$$ K\left( \psi \right) = H\left( {\left| \psi \right|} \right) + \left( {\delta - 1} \right) H\left( {\left| \psi \right| - \pi } \right) $$
(A1)

The generalized derivative of the function \(K\left( \psi \right)\) thus takes the form

$$ \begin{aligned} \frac{{{\text{d}}K\left( \psi \right)}}{{{\text{d}}\psi }} & = \left\{ {{\Delta }\left( {\left| \psi \right|} \right) + \left( {\delta - 1} \right){\Delta }\left( {\left| \psi \right| - \pi } \right)} \right\}\frac{{{\text{d}}\left| \psi \right|}}{{{\text{d}}\psi }} \\ & = \left\{ {{\Delta }\left( {\left| \psi \right|} \right) + \left( {\delta - 1} \right){\Delta }\left( {\left| \psi \right| - \pi } \right)} \right\}\\ & \times \left\{ {1 - T\mathop \sum \limits_{n = 1}^{\infty } {\Delta }\left( {\psi - nT} \right)} \right\} \\ & = {\Delta }\left( {\left| \psi \right|} \right) + \left( {\delta - 1} \right){\Delta }\left( {\left| \psi \right| - \pi } \right) \\ \end{aligned} $$
(A2)

where the function \(\left| \psi \right|\) and its derivative \(\left| \psi \right|\) is defined in the form

$$ \begin{aligned} \left| \psi \right| & = \bmod \left( {\psi ,T} \right) = \psi - T\mathop \sum \limits_{n = 1}^{\infty } H\left( {\psi - nT} \right) \\ \frac{{{\text{d}}\left| \psi \right|}}{{{\text{d}}\psi }} & = 1 - T\mathop \sum \limits_{n = 1}^{\infty } {\Delta }\left( {\psi - nT} \right) \\ \end{aligned} $$
(A3)

Appendix B

The averaging of the vector field (Eq. 10) reads,

$$ \begin{aligned} \dot{A}_{a} & = - \frac{\varepsilon A_{a}B}{{2T}}\left\{ {\mathop \int \limits_{0}^{\pi } \sin \left( {m\omega_{{{\text{pwl}}}} \left( {\psi - \theta_{{\text{a}}} } \right) + \varepsilon \sigma t} \right)\sin \left( {2\psi } \right){\text{d}}\psi } \right. \\ & \quad \left. { + \,\frac{1}{\delta }\mathop \int \limits_{\pi }^{T} \sin \left( {m\omega_{{{\text{pwl}}}} \left( {\psi - \theta_{{\text{a}}} } \right) + \varepsilon \sigma t} \right)\sin \left( {2\delta \psi + 2\rho } \right){\text{d}}\psi } \right\} \\ & \quad - \,\frac{\varepsilon \mu A_{a}}{T}\left\{ {\mathop \int \limits_{0}^{\pi } \cos^{2} \left( \psi \right){\text{d}}\psi + \mathop \int \limits_{\pi }^{T} \cos^{2} \left( {\delta \psi + \rho } \right){\text{d}}\psi } \right\} \\ & \quad = \, - \frac{\varepsilon A_{a}B}{{2T}}\left\{ {\mathop \int \limits_{0}^{\pi } \left\{ {\sin \left( {m\omega_{{{\text{pwl}}}} \left( {\psi - \theta_{{\text{a}}} } \right)} \right)\cos \left( {\varepsilon \sigma t} \right)} \right.} \right. \\ & \quad \left. { + \,\cos \left( {m\omega_{{{\text{pwl}}}} \left( {\psi - \theta_{{\text{a}}} } \right)} \right)\sin \left( {\varepsilon \sigma t} \right)} \right\}\sin \left( {2\psi } \right){\text{d}}\psi \\ & \quad + \,\frac{1}{\delta }\mathop \int \limits_{\pi }^{T} \left\{ {\sin \left( {m\omega_{{{\text{pwl}}}} \left( {\psi - \theta_{{\text{a}}} } \right)} \right)\cos \left( {\varepsilon \sigma t} \right)} \right. \\ & \quad \left. { + \,\cos \left( {m\omega_{{{\text{pwl}}}} \left( {\psi - \theta_{{\text{a}}} } \right)} \right)\sin \left( {\varepsilon \sigma t} \right)} \right\}\sin \left( {2\delta \psi + 2\rho } \right){\text{d}}\psi \\ & \quad - \,\frac{\varepsilon \mu A_{a}}{{2T}}\pi \left( {1 + \frac{1}{\delta }} \right) = \frac{\varepsilon A_{a}B}{T}\left\{ {\frac{{\sin \left( {m\omega_{{{\text{pwl}}}} \left( {\theta_{a} - \pi } \right) - \varepsilon \sigma t} \right) - \sin \left( {m\omega_{{{\text{pwl}}}} \theta_{a} - \varepsilon \sigma t} \right)}}{{m^{2} \omega_{{{\text{pwl}}}}^{2} - 4}}} \right. \\ & \quad \left. { + \,\frac{{\sin \left( {m\omega_{{{\text{pwl}}}} \left( {\theta_{a} - T} \right) - \varepsilon \sigma t} \right) - \sin \left( {m\omega_{{{\text{pwl}}}} \left( {\theta_{a} - \pi } \right) - \varepsilon \sigma t} \right)}}{{m^{2} \omega_{{{\text{pwl}}}}^{2} - 4\delta^{2} }}} \right\} - \frac{\varepsilon \mu A_{a}}{2} \\ \end{aligned} $$
(B1)
$$ \begin{aligned} \dot{\theta }_{a} & = \frac{\varepsilon B}{T}\left\{ {\mathop \int \limits_{0}^{\pi } \sin \left( {m\omega_{{{\text{pwl}}}} \left( {\psi - \theta_{{\text{a}}} } \right) + \varepsilon \sigma t} \right)\sin^{2} \left( \psi \right){\text{d}}\psi } \right. \\ & \quad \left. { + \,\frac{1}{{\delta^{2} }}\mathop \int \limits_{\pi }^{T} \sin \left( {m\omega_{{{\text{pwl}}}} \left( {\psi - \theta_{{\text{a}}} } \right) + \varepsilon \sigma t} \right)\sin^{2} \left( {\delta \psi + \rho } \right){\text{d}}\psi } \right\} \\ & \quad + \,\frac{\varepsilon \mu }{T}\left\{ {\mathop \int \limits_{0}^{\pi } \sin \left( \psi \right)\cos \left( \psi \right){\text{d}}\psi + \frac{1}{\delta }\mathop \int \limits_{\pi }^{T} \sin \left( {\delta \psi + \rho } \right)\cos \left( {\delta \psi + \rho } \right){\text{d}}\psi } \right\} \\ & \quad = \,\frac{\varepsilon B}{T}\left\{ {\mathop \int \limits_{0}^{\pi } \left\{ {\sin \left( {m\omega_{{{\text{pwl}}}} \left( {\psi - \theta_{{\text{a}}} } \right)} \right)\cos \left( {\varepsilon \sigma t} \right)} \right.} \right. \\ & \quad \left. { + \,\cos \left( {m\omega_{{{\text{pwl}}}} \left( {\psi - \theta_{{\text{a}}} } \right)} \right)\sin \left( {\varepsilon \sigma t} \right)} \right\}\sin^{2} \left( \psi \right){\text{d}}\psi \\ & \quad + \,\frac{1}{{\delta^{2} }}\mathop \int \limits_{\pi }^{T} \left\{ {\sin \left( {m\omega_{{{\text{pwl}}}} \left( {\psi - \theta_{{\text{a}}} } \right)} \right)\cos \left( {\varepsilon \sigma t} \right)} \right. \\ & \quad \left. { + \,\cos \left( {m\omega_{{{\text{pwl}}}} \left( {\psi - \theta_{{\text{a}}} } \right)} \right)\sin \left( {\varepsilon \sigma t} \right)} \right\}\left. {\sin^{2} \left( {\delta \psi + \rho } \right){\text{d}}\psi } \right\} \\ & \quad + \,\frac{\varepsilon \mu }{{2T}}\left\{ {\mathop \int \limits_{0}^{\pi } \sin \left( {2\psi } \right){\text{d}}\psi + \frac{1}{\delta }\mathop \int \limits_{\pi }^{T} \sin \left( {2\delta \psi + 2\rho } \right){\text{d}}\psi } \right\} \\ & \quad = \,\frac{2\varepsilon B}{{Tm\omega_{{{\text{pwl}}}} }}\left\{ {\frac{{\cos \left( {m\omega_{{{\text{pwl}}}} \left( {\theta_{a} - \pi } \right) - \varepsilon \sigma t} \right) - \cos \left( {m\theta_{a} \omega_{{{\text{pwl}}}} - \varepsilon \sigma t} \right)}}{{m^{2} \omega_{{{\text{pwl}}}}^{2} - 4}}} \right. \\ & \quad \left. { + \,\frac{{\cos \left( {m\omega_{{{\text{pwl}}}} \left( {\theta_{a} - T} \right) - \varepsilon \sigma t} \right) - \cos \left( {m\omega_{{{\text{pwl}}}} \left( {\theta_{a} - \pi } \right) - \varepsilon \sigma t} \right)}}{{m^{2} \omega_{{{\text{pwl}}}}^{2} - 4\delta^{2} }}} \right\} \\ \end{aligned} $$
(B2)

Appendix C

The averaging of the vector field (Eq. 17) reads

$$ \begin{aligned} \dot{A}_{a} & = - \frac{\varepsilon \mu A_{a}}{T}\left\{ {A_{a}^{2} \left( {\mathop \int \limits_{0}^{\pi } \sin^{2} \left( \psi \right)\cos^{2} \left( \psi \right){\text{d}}\psi + \frac{1}{{\delta^{2} }}\mathop \int \limits_{\pi }^{T} \sin^{2} \left( {\delta \psi + \rho } \right)\cos^{2} \left( {\delta \psi + \rho } \right){\text{d}}\psi } \right)} \right. \\ & \quad \left. { - \left( {\mathop \int \limits_{0}^{\pi } \cos^{2} \left( \psi \right){\text{d}}\psi + \frac{1}{{\delta^{2} }}\mathop \int \limits_{\pi }^{T} \cos^{2} \left( {\delta \psi + \rho } \right){\text{d}}\psi } \right)} \right\} \\ & \quad = - \frac{\varepsilon \mu A_{a}}{T}\left\{ {\frac{{\pi A_{a}^{2} \left( {1 + \delta^{3} } \right)}}{{8\delta^{3} }} - \frac{{\pi \left( {1 + \delta } \right)}}{2\delta }} \right\} \\ \end{aligned} $$
(C1)
$$ \begin{aligned} \dot{\theta }_{a} & = - \frac{\varepsilon \mu }{T}\left\{ {A_{a}^{2} \left( {\mathop \int \limits_{0}^{\pi } \sin^{3} \left( \psi \right)\cos \left( \psi \right){\text{d}}\psi + \frac{1}{{\delta^{3} }}\mathop \int \limits_{\pi }^{T} \sin^{3} \left( {\delta \psi + \rho } \right)\cos \left( {\delta \psi + \rho } \right){\text{d}}\psi } \right)} \right. \\ & \quad \left. { - \left( {\mathop \int \limits_{0}^{\pi } \sin \left( \psi \right)\cos \left( \psi \right){\text{d}}\psi + \frac{1}{\delta }\mathop \int \limits_{\pi }^{T} \sin \left( {\delta \psi + \rho } \right)\cos \left( {\delta \psi + \rho } \right){\text{d}}\psi } \right)} \right\} = 0 \\ \end{aligned} $$
(C2)

Appendix D

The averaging of the vector field (Eq. 21) reads

$$ \begin{aligned} \dot{A}_{a} & = \frac{\varepsilon k}{T}\left\{ {\mathop \int \limits_{0}^{\pi } \sin \left( {\omega_{{{\text{pwl}}}} \left( {\psi - \theta_{a} } \right) + \varepsilon \sigma t} \right)\cos \left( \psi \right){\text{d}}\psi + \mathop \int \limits_{\pi }^{T} \sin \left( {\omega_{{{\text{pwl}}}} \left( {\psi - \theta_{a} } \right) + \varepsilon \sigma t} \right)\cos \left( {\delta \psi + \rho } \right){\text{d}}\psi } \right\} \\ & \quad - \frac{\varepsilon }{T}\left\{ {\mu A_{a}\left( {\mathop \int \limits_{0}^{\pi } \cos^{2} \left( \psi \right){\text{d}}\psi + \mathop \int \limits_{\pi }^{T} \cos^{2} \left( {\delta \psi + \rho } \right){\text{d}}\psi } \right)} \right. \\ & \quad \left. { + \,\alpha A_{a}^{3} \left( { \mathop \int \limits_{0}^{\pi } \sin^{3} \left( \psi \right)\cos \left( \psi \right){\text{d}}\psi + \frac{1}{{\delta^{3} }}\mathop \int \limits_{\pi }^{T} \sin^{3} \left( {\delta \psi + \rho } \right)\cos \left( {\delta \psi + \rho } \right){\text{d}}\psi } \right)} \right\} \\ & \quad = \frac{\varepsilon k}{T}\left\{ {\mathop \int \limits_{0}^{\pi } \left( {\sin \left( {\omega_{{{\text{pwl}}}} \left( {\psi - \theta_{a} } \right)} \right)\cos \left( {\varepsilon \sigma t} \right) + \cos \left( {\omega_{{{\text{pwl}}}} \left( {\psi - \theta_{a} } \right)} \right)\sin \left( {\varepsilon \sigma t} \right)} \right)\cos \left( \psi \right){\text{d}}\psi } \right. \\ & \quad \left. { + \mathop \int \limits_{\pi }^{T} \left( {\sin \left( {\omega_{{{\text{pwl}}}} \left( {\psi - \theta_{a} } \right)} \right)\cos \left( {\varepsilon \sigma t} \right) + \cos \left( {\omega_{{{\text{pwl}}}} \left( {\psi - \theta_{a} } \right)} \right)\sin \left( {\varepsilon \sigma t} \right)} \right)\cos \left( {\delta \psi + \rho } \right){\text{d}}\psi } \right\} \\ & \quad - \frac{\varepsilon }{T}\left\{ {\mu A_{a}\left( {\mathop \int \limits_{0}^{\pi } \cos^{2} \left( \psi \right){\text{d}}\psi + \mathop \int \limits_{\pi }^{T} \cos^{2} \left( {\delta \psi + \rho } \right){\text{d}}\psi } \right)} \right. \\ & \quad \left. { + \,\alpha A_{a}^{3} \left( { \mathop \int \limits_{0}^{\pi } \sin^{3} \left( \psi \right)\cos \left( \psi \right){\text{d}}\psi + \frac{1}{{\delta^{3} }}\mathop \int \limits_{\pi }^{T} \sin^{3} \left( {\delta \psi + \rho } \right)\cos \left( {\delta \psi + \rho } \right){\text{d}}\psi } \right)} \right\} \\ & \quad = \frac{{\varepsilon k\omega_{{{\text{pwl}}}} }}{T}\left\{ {\frac{{\cos \left( {\omega_{{{\text{pwl}}}} \left( {\theta_{a} - \pi } \right) - \varepsilon \sigma t} \right) + \cos \left( {\omega_{{{\text{pwl}}}} {\theta_{a}} - \varepsilon \sigma t} \right)}}{{\omega_{{{\text{pwl}}}}^{2} - 1}}} \right. \\ & \quad \left. { - \frac{{\cos \left( {\omega_{{{\text{pwl}}}} \left( {\theta_{a} - T} \right) - \varepsilon \sigma t} \right) + \cos \left( {\omega_{{{\text{pwl}}}} \left( {\theta_{a} - \pi } \right) - \varepsilon \sigma t} \right)}}{{\omega_{{{\text{pwl}}}}^{2} - \delta^{2} }}} \right\} - \frac{\varepsilon }{T}\left\{ {\mu A_{a}\frac{{\pi \left( {1 + \delta } \right)}}{2\delta }} \right\} \\ \end{aligned} $$
(D1)
$$ \begin{aligned} \dot{\theta}_{a} & = - \frac{\varepsilon k}{{A_{a}T}}\left\{ {\mathop \int \limits_{0}^{\pi } \sin \left( {\omega_{{{\text{pwl}}}} \left( {\psi - \theta_{a} } \right) + \varepsilon \sigma t} \right)\sin \left( \psi \right){\text{d}}\psi + \frac{1}{\delta }\mathop \int \limits_{\pi }^{T} \sin \left( {\omega_{{{\text{pwl}}}} \left( {\psi - \theta_{a} } \right) + \varepsilon \sigma t} \right)\sin \left( {\delta \psi + \rho } \right){\text{d}}\psi } \right\} \\ & \quad + \varepsilon \left\{ {\mu \left( {\mathop \int \limits_{0}^{\pi } \sin \left( \psi \right)\cos \left( \psi \right){\text{d}}\psi + \frac{1}{\delta }\mathop \int \limits_{\pi }^{T} \sin \left( {\delta \psi + \rho } \right)\cos \left( {\delta \psi + \rho } \right){\text{d}}\psi } \right)} \right. \\ & \quad \left. { + \,\alpha A_{a}^{2} \left( { \mathop \int \limits_{0}^{\pi } \sin^{4} \left( \psi \right){\text{d}}\psi + \frac{1}{{\delta^{4} }}\mathop \int \limits_{\pi }^{T} \sin^{4} \left( {\delta \psi + \rho } \right){\text{d}}\psi } \right)} \right\} \\ & \quad = - \frac{\varepsilon k}{{A_{a}T}}\left\{ {\mathop \int \limits_{0}^{\pi } \left( {\sin \left( {\omega_{{{\text{pwl}}}} \left( {\psi - \theta_{a} } \right)} \right)\cos \left( {\varepsilon \sigma t} \right) + \cos \left( {\omega_{{{\text{pwl}}}} \left( {\psi - \theta_{a} } \right)} \right)\sin \left( {\varepsilon \sigma t} \right)} \right)\sin \left( \psi \right){\text{d}}\psi } \right. \\ & \quad \left. { + \frac{1}{\delta }\mathop \int \limits_{\pi }^{T} \left( {\sin \left( {\omega_{{{\text{pwl}}}} \left( {\psi - \theta_{a} } \right)} \right)\cos \left( {\varepsilon \sigma t} \right) + \cos \left( {\omega_{{{\text{pwl}}}} \left( {\psi - \theta_{a} } \right)} \right)\sin \left( {\varepsilon \sigma t} \right)} \right)\sin \left( {\delta \psi + \rho } \right){\text{d}}\psi } \right\} \\ & \quad + \varepsilon \left\{ {\mu \left( {\mathop \int \limits_{0}^{\pi } \sin \left( \psi \right)\cos \left( \psi \right){\text{d}}\psi + \frac{1}{\delta }\mathop \int \limits_{\pi }^{T} \sin \left( {\delta \psi + \rho } \right)\cos \left( {\delta \psi + \rho } \right){\text{d}}\psi } \right)} \right. \\ & \quad \left. { + \,\alpha A_{a}^{2} \left( { \mathop \int \limits_{0}^{\pi } \sin^{4} \left( \psi \right){\text{d}}\psi + \frac{1}{{\delta^{4} }}\mathop \int \limits_{\pi }^{T} \sin^{4} \left( {\delta \psi + \rho } \right){\text{d}}\psi } \right)} \right\} \\ & \quad = - \frac{\varepsilon k}{{A_{a}T}}\left\{ {\frac{{\sin \left( {\omega_{{{\text{pwl}}}} \left( {\theta_{a} - \pi } \right) - \varepsilon \sigma t} \right) + \sin \left( {\omega_{{{\text{pwl}}}} \theta_{a} - \varepsilon \sigma t} \right)}}{{\omega_{{{\text{pwl}}}}^{2} - 1}}} \right. \\ & \quad \left. { - \frac{{\sin \left( {\omega_{{{\text{pwl}}}} \left( {\theta_{a} - T} \right) - \varepsilon \sigma t} \right) + \sin \left( {\omega_{{{\text{pwl}}}} \left( {\theta_{a} - \pi } \right) - \varepsilon \sigma t} \right)}}{{\omega_{{{\text{pwl}}}}^{2} - \delta^{2} }}} \right\} + \frac{\varepsilon }{T}\left\{ {\alpha A_{a}^{2} \frac{{3\pi \left( {1 + \delta^{5} } \right)}}{{8\delta^{5} }}} \right\} \\ \end{aligned} $$
(D2)

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Jayaprakash, K.R., Tandel, V. & Starosvetsky, Y. Dynamics of excited piecewise linear oscillators. Nonlinear Dyn 111, 5513–5532 (2023). https://doi.org/10.1007/s11071-022-08108-9

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