Abstract
We develop here the method for obtaining approximate stability boundaries in the space of parameters for systems with parametric excitation. The monodromy (Floquet) matrix of linearized system is found by averaging method. For system with two degrees of freedom (DOF) we derive general approximate stability conditions. We study domains of stability with the use of fourth order approximations of monodromy matrix on example of inverted position of a pendulum with vertically oscillating pivot. Addition of small damping shifts the stability boundaries upwards, thus resulting in both stabilization and destabilization effects.
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Notes
- 1.
Lyapunov regularity condition of the linear system holds when its matrix is periodic.
- 2.
Periodic linear system is asymptotically stable if, and only if, it is exponentially stable. Exponential stability of \(\dot {x}(t)=\mathbf J(t)\, x(t)\) results in exponential stability, and hence asymptotic stability, of the nonlinear system solution y ≡ 0.
- 3.
The expressions are derived by differentiating (5) w.r.t. time
$$\displaystyle \begin{aligned}\dot{\mathbf Y}(t) = & \left(\dot{\mathbf U}_1(t) + \dot{\mathbf U}_2(t)+\cdots \right)\cdot \mathbf Z(t) + \left(\mathbf I + \mathbf U_1(t) + \mathbf U_2(t)+\cdots \right)\cdot \dot{\mathbf Z}(t), \end{aligned} $$substituting there expressions for time derivatives from (3) and (6)
$$\displaystyle \begin{aligned} & \left(\mathbf H_1(t) + \mathbf H_2(t)+\cdots \right)\cdot \left(\mathbf I + \mathbf U_1(t) + \mathbf U_2(t)+\cdots \right)\cdot \mathbf Z \\ &\quad = \left(\dot{\mathbf U}_1(t) + \dot{\mathbf U}_2(t)+\cdots\right)\cdot\mathbf Z + \left(\mathbf I + \mathbf U_1(t) + \mathbf U_2(t)+\cdots \right) \cdot \left(\mathbf A_1 + \mathbf A_2+\cdots \right)\cdot\mathbf Z, \end{aligned} $$collecting there terms of the same order, and canceling non-degenerate matrix Z, which yield the following equalities:
First order: \( \mathbf H_1(t) = \dot {\mathbf U}_1(t) + \mathbf A_1. \)
Second order: \( \mathbf H_2(t) + \mathbf H_1(t)\cdot \mathbf U_1(t) = \dot {\mathbf U}_2(t) + \mathbf U_1(t)\cdot \mathbf A_1 + \mathbf A_2, \)
Third order: \( \mathbf H_3(t) + \mathbf H_1(t)\cdot \mathbf U_2(t) + \mathbf H_2(t)\cdot \mathbf U_1(t) = \dot {\mathbf U}_3(t) + \mathbf U_2(t)\cdot \mathbf A_1 + \mathbf U_1(t)\cdot \mathbf A_2 + \mathbf A_3, \)
Fourth order: \( \mathbf H_4(t) + \mathbf H_1(t)\cdot \mathbf U_3(t) + \mathbf H_2(t)\cdot \mathbf U_2(t)+ \mathbf H_3(t)\cdot \mathbf U_1(t) = \dot {\mathbf U}_4(t) + \mathbf U_3(t)\cdot \mathbf A_1 + \mathbf U_2(t)\cdot \mathbf A_2 + \mathbf U_1(t)\cdot \mathbf A_3 + \mathbf A_4 \), and so on.
References
Floquet, G.: Sur les équations différentielles linéaires à coefficients périodiques. Ann. Sci. Ecol. Norm. Super. 12, 47–88 (1883)
Xu, X., Wiercigroch, M.: Approximate analytical solutions for oscillatory and rotational motion of a parametric pendulum. Nonlinear Dyn. 47(1–3), 311–320 (2007)
Butikov, E.I.: A physically meaningful new approach to parametric excitation and attenuation of oscillations in nonlinear systems. Nonlinear Dyn. 88(4), 2609–2627 (2017)
Bogoliubov, N.N., Mitropolsky, Y.A.: Asymptotic Methods in the Theory of Non-Linear Oscillations. Gordon and Breach, New York (1961)
Seyranian, A.P., Mailybaev, A.A.: Multiparameter Stability Theory with Mechanical Applications. World Scientific, New Jersey (2003)
Seyranian, A.A., Seyranian, A.P.: The stability of an inverted pendulum with a vibrating suspension point. J. Appl. Math. Mech. 70, 754–761 (2006)
Arkhipova, I.M., Luongo, A.: On the effect of damping on the stabilization of mechanical systems via parametric excitation. Zeitschrift für angewandte Mathematik und Physik 67(3), 69 (2016). https://doi.org/10.1007/s00033-016-0659-6
Belyakov, A.O.: Another mechanical model of parametrically excited pendulum and stabilization of its inverted equilibrium position. In: Proceedings of 8th European Nonlinear Dynamics Conference, ENOC 2014
Kuznetsov, S.P.: Dynamical Chaos. Physmatlit, Moscow (2006) (in Russian)
Yakubovich, V.A., Starzhinskii, V.M.: Parametric Resonance in Linear Systems. Nauka, Moscow (1987)
Arnold, V.I.: Ordinary Differential Equations. Springer, New York (1992)
Meissner, E.: Ueber Schüttelerscheinungen in Systemen mit periodisch veränderlicher Elastizität. Schweizerische Bauzeitung 72(11), 95–98 (1918)
Acknowledgement
A.O. Belyakov received funding from the Russian Science Foundation grant 19-11-00223.
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Belyakov, A.O., Seyranian, A.P. (2020). Stability Boundary Approximation of Periodic Dynamics. In: Lacarbonara, W., Balachandran, B., Ma, J., Tenreiro Machado, J., Stepan, G. (eds) Nonlinear Dynamics of Structures, Systems and Devices. Springer, Cham. https://doi.org/10.1007/978-3-030-34713-0_2
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