Low-velocity impacts of quasiperiodic oscillations

Published on Aug 1, 2002in Chaos Solitons & Fractals5.944
· DOI :10.1016/S0960-0779(01)00230-2
Harry Dankowicz25
Estimated H-index: 25
(VT: Virginia Tech),
Petri T. Piiroinen17
Estimated H-index: 17
(KTH: Royal Institute of Technology),
Arne Nordmark22
Estimated H-index: 22
(KTH: Royal Institute of Technology)
Sources
Abstract
Abstract A method based on the idea of a discontinuity mapping is derived for predicting the characteristics of system attractors that occur following a grazing intersection of a two-frequency, quasiperiodic oscillation with a two-dimensional impact surface in a three-dimensional state space. Within certain restrictions, the correction to the non-impacting flow afforded by the discontinuity mapping is computable using quantities determined solely by the non-impacting flow and the properties of the impact surface and the associated impact mapping in the immediate vicinity of the initial grazing contact. A model example is discussed to illustrate the quantitative predictive power of the discontinuity-mapping approach even relatively far away in parameter space from the original grazing intersection. Finally, constraints on the applicability of the methodology are described in detail with suggestions for suitable modifications.
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References20
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#1M. di Bernardo (UoB: University of Bristol)H-Index: 27
#2Chris Budd (University of Bath)H-Index: 31
Last. Alan R Champneys (UoB: University of Bristol)H-Index: 55
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Abstract This paper presents a unified framework for performing local analysis of grazing bifurcations in n-dimensional piecewise-smooth systems of ODEs. These occur when a periodic orbit has a point of tangency with a smooth (n−1)-dimensional boundary dividing distinct regions in phase space where the vector field is smooth. It is shown under quite general circumstances that this leads to a normal-form map that contains to lowest order either a square-root or a (3/2)-type singularity according ...
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#1J Jaap Molenaar (TU/e: Eindhoven University of Technology)H-Index: 14
#2John de Weger (TU/e: Eindhoven University of Technology)H-Index: 3
Last. Willem van de Water (TU/e: Eindhoven University of Technology)H-Index: 22
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Impacting systems are found in a great variety of mechanical constructions and they are intrinsically nonlinear. In this paper it is shown how near-grazing systems, i.e. systems in which the impacts take place at low speed, can be described by discrete mappings. The derivation of this mapping for a harmonic oscillator with a stop is dealt with in detail. It is found that the resulting mapping for rigid obstacles is somewhat different from those presented earlier in the literature. The derivation...
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#1Harry Dankowicz (KTH: Royal Institute of Technology)H-Index: 25
#2Arne Nordmark (KTH: Royal Institute of Technology)H-Index: 22
A recently proposed model of macroscopic friction is investigated using methods of dynamical systems analysis. Particular emphasis is put on the bifurcations associated with the appearance of stick ...
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#1Mats H. Fredriksson (KTH: Royal Institute of Technology)H-Index: 4
#2Arne Nordmark (KTH: Royal Institute of Technology)H-Index: 22
Normal form calculations are useful for analysing the dynamics close to bifurcations. However, the application to non-smooth systems is a topic for current research. Here we consider a class of imp ...
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#1Mats H. Fredriksson (KTH: Royal Institute of Technology)H-Index: 4
#2Dan Borglund (KTH: Royal Institute of Technology)H-Index: 11
Last. Arne Nordmark (KTH: Royal Institute of Technology)H-Index: 22
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The transition from stable periodic nonimpacting motion to impacting motion, due to variations of parameters, is observable in a wide range of vibro-impact systems. Recent theoretical studies suggest a possible scenario for this type of transition. A key element in the proposed scenario is fulfilled if the oscillatory motion involved in the transition is born in a supercritical Hopf bifurcation. If the onset of impacting motion is close to the Hopf bifurcation, the impacting motion is likely to ...
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A low-dimensional dynamic model of frictional interactions is proposed by considering microscopic interactions between nominally flat surfaces. Coupling is introduced between the tangential motion and the separation of the surfaces due to the asperity contact between the surfaces. The numerical and analytical predictions of the resulting nonlinear model are found to qualitatively agree with a number of experimentally observed properties of friction. The paper suggests a means for arriving at a f...
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#1Bernard BrogliatoH-Index: 46
1 Distributional model of impacts.- 1.1 External percussions.- 1.2 Measure differential equations.- 1.2.1 Some properties.- 1.2.2 Additional comments.- 1.3 Systems subject to unilateral constraints.- 1.3.1 General considerations.- 1.3.2 Flows with collisions.- 1.3.3 A system theoretical geometric approach.- 1.3.4 Descriptor variable systems.- 1.4 Changes of coordinates in MDEs.- 1.4.1 From measure to Caratheodory systems.- 1.4.2 Decoupling of the impulsive effects (commutativity conditions).- 1....
#1Harry Dankowicz (KTH: Royal Institute of Technology)H-Index: 25
A five-dimensional dynamical model of friction is proposed based on a macroscopic description of the interactions between nominally flat surfaces. In particular, dynamical contacts between surface in homogeneities result in coupling between the tangential motion and the separation of the surfaces and corresponding variations in interfacial forces. The model is shown to exhibit qualitative agreement with experimentally observed properties of dynamical friction. For example, an apparent dependence...
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#1Mats H. Fredriksson (KTH: Royal Institute of Technology)H-Index: 4
#2Arne Nordmark (KTH: Royal Institute of Technology)H-Index: 22
The transition from stable periodic non-impacting motion to impacting motion is analysed for a mechanical oscillator. By using local methods, it is shown that a grazing impact leads to an almost one-dimensional stretching in state space. A condition can then be formulated, such that a grazing trajectory will be stable if the condition is fulfilled. If this is the case, the bifurcation will be continuous and the motion after the bifurcation can be understood by a one-dimensional mapping. This map...
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#1Wai ChinH-Index: 3
#2Edward OttH-Index: 119
Last. Celso GrebogiH-Index: 87
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Impact oscillators demonstrate interesting dynamical features. In particular, new types of bifurcations take place as such systems evolve from a nonimpacting to an impacting state (or vice versa), as a system parameter varies smoothly. These bifurcations are called grazing bifurcations. In this paper we analyze the different types of grazing bifurcations that can occur in a simple sinusoidally forced oscillator system in the presence of friction and a hard wall with which the impacts take place....
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#1Eoghan J. Staunton (National University of Ireland, Galway)H-Index: 2
#2Petri T. Piiroinen (National University of Ireland, Galway)H-Index: 17
Abstract For stability and bifurcation analysis involving recurrent behaviour such as periodic orbits, it is important to be able to quantify how nearby trajectories behave by means of a local mapping. In smooth systems these mappings can be computed using the system’s variational equations. For piecewise-smooth or hybrid systems the same technique cannot be used without some corrections. This is due to the fact that nearby trajectories can be topologically distinct because they can undergo diff...
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#1Jinqian Feng (Xi'an Polytechnic University)H-Index: 1
The chaos controls of a Duffing system with impacts are investigated using appropriate random phase. Based on the discontinuous map and Wedig’s algorithm, a procedure of calculating top Lyapunov exponent is presented for stochastic Duffing system with impacts. As the random phase changes, two kinds of chaos control are achieved for Duffing system with impacts by the criterion of the top Lyapunov exponent: one is chaos suppression, and the other is chaos generation. In addition, the obtained resu...
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#1Tingting Yi (Sichuan University)H-Index: 1
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Degenerate grazing bifurcation occurring in a simple bilinear oscillator, namely the limit discontinuous case of the smooth and discontinuous (SD) oscillator, is investigated by numerical simulations. The unperturbed system has a saddle-like singularity at the origin with two periodic orbits grazing at the same point. The matrix in the leading-order truncation of the Poincare map at a grazing bifurcation for either of the two periodic orbits has a zero eigenvalue. Our numerical experiments sugge...
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Abstract Homoclinic bifurcation for a nonlinear inverted pendulum impacting between two rigid walls under external periodic excitation is analyzed under the hypothesis that the unperturbed system has a homoclinc orbit tangent to the wall. Consequently, the impact surface cannot be chosen as the Poincare section to measure the distance between the perturbed stable and unstable manifolds. Furthermore, compared to the case that the unperturbed homoclinic orbit intersects the wall transversally, mor...
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#1Harry Dankowicz (UIUC: University of Illinois at Urbana–Champaign)H-Index: 25
#2Michael Katzenbach (UIUC: University of Illinois at Urbana–Champaign)H-Index: 1
Abstract This paper collects four distinct instances of grazing contact of a periodic trajectory in a hybrid dynamical system under a common abstract framework and establishes selected general properties of the associated near-grazing dynamics. In particular, it is shown that for critical choices of parameter values, commonly used physical models of rigid or compliant mechanical contact, capillary adhesion, and cell division satisfy the conditions required by the general framework. The paper rel...
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#1Sergey Kryzhevich (SPbU: Saint Petersburg State University)H-Index: 6
#2Marian Wiercigroch (Aberd.: University of Aberdeen)H-Index: 51
Abstract The grazing bifurcation is considered for the Newtonian model of vibro-impact systems. A brief review on the conditions, sufficient for the existence of a grazing family of periodic solutions, is given. The properties of these periodic solutions are discussed. A plenty of results on the topological structure of attractors of vibro-impact systems is known. However, since the considered system is strongly nonlinear, these attractors must be very sensitive to changes of parameters of the s...
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This paper investigates the onset of low-velocity, near-grazing collisions in an example vibro-impacting system with dry friction with particular emphasis on feedback control strategies that regulate the grazing-induced bifurcation behaviour. The example system is characterized by a twofold degeneracy of grazing contact along an extremal stick solution that is shown to result in a locally one-dimensional and piecewise-linear description of the near-grazing dynamics. Explicit control strategies a...
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