2007
DOI: 10.1007/s11071-007-9244-z
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Stability properties of equilibrium sets of non-linear mechanical systems with dry friction and impact

Abstract: In this paper, we will give conditions under which the equilibrium set of multi-degree-of-freedom non-linear mechanical systems with an arbitrary number of frictional unilateral constraints is attractive. The theorems for attractivity are proved by using the framework of measure differential inclusions together with a Lyapunov-type stability analysis and a generalisation of LaSalle's invariance principle for non-smooth systems. The special structure of mechanical multi-body systems allows for a natural Lyapuno… Show more

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Cited by 50 publications
(58 citation statements)
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“…Since the functions f , g,f , andg are smooth and satisfy Assumption 1, the inverse function theorem implies that for each ϵ 1 > 0 there exists an ϵ > 0 such that (11) implies that for each p satisfying Therefore a functionΨ can be constructed for the equilibrium setẼ analogously to the function Ψ for equilibrium set E when ϵ > 0 is sufficiently small, such that the equilibrium setẼ can be parameterised as follows: …”
Section: Appendix a Existence Of An Equilibrium Setmentioning
confidence: 99%
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“…Since the functions f , g,f , andg are smooth and satisfy Assumption 1, the inverse function theorem implies that for each ϵ 1 > 0 there exists an ϵ > 0 such that (11) implies that for each p satisfying Therefore a functionΨ can be constructed for the equilibrium setẼ analogously to the function Ψ for equilibrium set E when ϵ > 0 is sufficiently small, such that the equilibrium setẼ can be parameterised as follows: …”
Section: Appendix a Existence Of An Equilibrium Setmentioning
confidence: 99%
“…Letà be an arbitrary system satisfying (11) for ϵ sufficiently small. By Lemma 7, we may assume that the equilibrium sets of A andà coincide.…”
Section: Lemma 12mentioning
confidence: 99%
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“…In [4,[11][12][13]] the authors considered this problem and developed an algorithm for the compensation of kinematic backlash based on an adaptive controller. This kind of nonlinear dynamic phenomenon has been an active area of research but well established conclusions are still lacking, mainly due to the considerable randomness and diversity of reasons underlying the dynamic effects [7,10]. This paper investigates the dynamics of systems that contain backlash and impacts through the describing function (DF) method.…”
Section: Introductionmentioning
confidence: 99%