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Harmonically excited generalized van der Pol oscillators: Entrainment phenomenon

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Abstract

Harmonically excited generalized van der Pol oscillators with power-form non-linearities in the restoring and damping-like force are investigated from the viewpoint of the occurrence of harmonic entrainment. Locked periodic motion is obtained by adjusting the averaging method. The influence of the powers of the restoring and damping-like force on the occurrence of this phenomenon is examined.

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Acknowledgements

This study has partially been supported by the Ministry of Education and Science (Grant ON174028).

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Authors and Affiliations

  1. Department of Mechanics, Faculty of Technical Sciences, University of Novi Sad, 21215, Novi Sad, Serbia

    Ivana Kovacic

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  1. Ivana Kovacic
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Correspondence to Ivana Kovacic.

Appendix

Appendix

For free conservative oscillators with a power-form restoring force governed by Eq. (1a) with ε=0

$$ \ddot{x} + \operatorname{sgn} (x)\vert x \vert ^{\alpha} = 0, $$
(A.1)

the energy conservation has the form

$$ \frac{1}{2} ( \dot{x} )^{2} + \frac{\vert x \vert ^{\alpha + 1}}{\alpha+ 1} = E_{0}, $$
(A.2)

where E 0 is the initial total mechanical energy. The expression for the period of oscillations can be found by following the procedure [12], where it is given for non-zero initial amplitude and zero initial velocity, unlike here where it is presented here for arbitrary initial conditions. When the motion starts with non-zero initial amplitude and zero initial velocity, the initial amplitude is the maximal displacement during time. When neither the initial displacement A 0, nor the initial velocity is zero, the maximal potential energy corresponding to the amplitude A is higher than the one corresponding to A 0 due to the initial kinetic energy.

Thus, one fourth of the exact period T ex of the oscillations of (A.1) can be defined as

$$ \frac{T_{\mathrm{ex}}}{4} = \int_{0}^{A} \frac{\mathrm{d}x}{\vert \dot{x} \vert }, $$
(A.3)

which, after using the energy conservation law (A2), becomes

(A.4)

By expressing the solution of the integral in terms of a hypergeometric function 2 F 1, one follows

(A.5)

The maximal displacement, i.e. the amplitude A can be calculation by going back to the energy conservation law, knowing that the velocity corresponding to this position is zero, which gives

$$ A = \bigl[ E_{0} ( \alpha+ 1 ) \bigr]^{\frac{1}{1 + \alpha }}. $$
(A.6)

Equation (A.5) now becomes

$$ T_{\mathrm{ex}} = 4\sqrt{\frac{\alpha+ 1}{2}} A^{\frac{1 - \alpha}{2}} \cdot {}_{2}F_{1} \biggl[ 1,\frac{1}{\alpha+ 1},1 + \frac{1}{\alpha+ 1},1 \biggr]. $$
(A.7)

By using

$$ _{2}F_{1} [ p,q,r,1 ] = \frac{\varGamma ( r )\varGamma ( r - p - q )}{\varGamma ( r - p )\varGamma ( r - q )}, $$
(A.8)

and performing some transformations, one obtains

$$ T_{\mathrm{ex}} = \sqrt{\frac{8\pi}{\alpha+ 1}} \frac{\varGamma ( \frac {1}{\alpha+ 1} )}{\varGamma ( \frac{\alpha+ 3}{2 ( \alpha+ 1 )} )}A^{\frac{1 - \alpha}{2}}. $$
(A.9)

If one assumes that the time response of the oscillators (A.1) corresponds to harmonic periodic oscillations with the period 2π, the angular frequency is ω=2π/T ex. This implies that the angular frequency of conservative oscillators is proportional to their amplitude defined by Eqs. (A.6) in the following way

$$ \omega ( A ) = cA^{\frac{\alpha- 1}{2}}, $$
(A.10)

where the coefficient of proportionality depends on the power α

$$ c = \sqrt{\frac{\pi ( \alpha+ 1 )}{2}} \frac{\varGamma ( \frac{\alpha+ 3}{2 ( \alpha+ 1 )} )}{\varGamma ( \frac{1}{\alpha+ 1} )}. $$
(A.11)

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Kovacic, I. Harmonically excited generalized van der Pol oscillators: Entrainment phenomenon. Meccanica 48, 2415–2425 (2013). https://doi.org/10.1007/s11012-013-9757-0

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