86 research outputs found
An Application of the Melnikov Method to a Piecewise Oscillator
In this paper we present a new application of the Melnikov method to a class of periodically perturbed Duffing equations where the nonlinearity is non-smooth as otherwise required in the classical applications. Extensions of the Melnikov method to these situations is a topic with growing interests from the researchers in the past decade. Our model, motivated by the study of mechanical vibrations for systems with āstopsā, considers a case of a nonlinear equation with piecewise linear components. This allows us to provide a precise analytical representation of the homoclinic orbit for the associated autonomous planar system and thus obtain simply computable conditions for the zeros of the associated Melnikov function
Invariant tori and boundedness of solutions of non-smooth oscillators with Lebesgue integrable forcing term
Since Littlewood works in the 1960's, the boundedness of solutions of
Duffing-type equations has been extensively investigated.
More recently, some researches have focused on the family of non-smooth forced
oscillators , mainly because it represents a
simple limit scenario of Duffing-type equations for when is bounded. Here,
we provide a simple proof for the boundedness of solutions of the non-smooth
forced oscillator in the case that the forcing term is a -periodic
Lebesgue integrable function with vanishing average. We reach this result by
constructing a sequence of invariant tori whose union of their interiors covers
all the -space,
Parameter switching in a generalized Duffing system: Finding the stable attractors
This paper presents a simple periodic parameter-switching method which can
find any stable limit cycle that can be numerically approximated in a
generalized Duffing system. In this method, the initial value problem of the
system is numerically integrated and the control parameter is switched
periodically within a chosen set of parameter values. The resulted attractor
matches with the attractor obtained by using the average of the switched
values. The accurate match is verified by phase plots and Hausdorff distance
measure in extensive simulations
Vibration energy flow transmission in systems with Coulomb friction
Acknowledgements This research was supported by the National Natural Science Foundation of China under Grant numbers 12172185 & 51605233 and by the Zhejiang Provincial Natural Science Foundation of China under Grant number LY22A020006.Peer reviewedPostprin
Analysis and Control of Nonlinear Attitude Motion of Gravity-Gradient Stabilized Spacecraft via Lyapunov-Floquet Transformation and Normal Forms
This chapter demonstrates analysis and control of the attitude motion of a gravity-gradient stabilized spacecraft in eccentric orbit. The attitude motion is modeled by nonlinear planar pitch dynamics with periodic coefficients and additionally subjected to external periodic excitation. Consequently, using system state augmentation, Lyapunov-Floquet (L-F) transformation, and normal form simplification, we convert the unwieldy attitude dynamics into relatively more amenable schemes for motion analysis and control law development. We analyze the dynamical systemās periodicity, stability, resonance, and chaos via numerous nonlinear dynamic theory techniques facilitated by intuitive system state augmentation and Lyapunov-Floquet transformation. Versal deformation of the normal forms is constructed to investigate the bifurcation behavior of the dynamical system. Outcome from the analysis indicates that the motion is quasi-periodic, chaotic, librational, and undergoing a Hopf bifurcation in the small neighborhood of the critical point-engendering locally stable limit cycles. Consequently, we demonstrate the implementation of linear and nonlinear control laws (i.e., bifurcation and sliding mode control laws) on the relatively acquiescent transformed attitude dynamics. By employing a two-pronged approach, the quasiperiodic planar motion is independently shown to be stabilizable via the nonlinear control approaches
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