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On the stability and control of piecewise-smooth dynamical systems with impacts and friction
KTH, School of Engineering Sciences (SCI), Aeronautical and Vehicle Engineering, Vehicle Dynamics. (KTH Vehicle Dynamics)
2009 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis concerns the analysis of dynamical systems suitable to be modelled by piecewise-smooth differential equations. In such systems the continuous-in-time dynamics is interrupted by discrete-in-time jumps in the state or governing equations of motion. Not only can this framework be used to describe existing systems with strong nonlinear behaviour such as impacts and friction, but the non-smooth properties can be exploited to design new mechanical devices. As suggested in this work it opens up the possibility of, for example, fast limit switches and energy transfer mechanisms.

Particularly, the dynamics at the onset of low-velocity impacts in systems with recurrent dynamics, so called grazing bifurcations in impact-oscillators, are investigated. As previous work has shown, low-velocity impacts is a strong source of instability to the dynamics, and efforts to control the behaviour is of importance. This problem is approached in two ways in this work. One is to investigate the influence of parameter variations on the dynamic behaviour of the system. The other is to implement low-cost control strategies to regulate the dynamics at the grazing bifurcation. The control inputs are of impulsive nature, and utilizes the natural dynamics of the system to the greatest extent.

The scientific contributions of this work is collected in five appended papers. The first paper consists of an experimental verification of a map that captures the correction to the smooth dynamics induced by an impact, known in the literature as the discontinuity map. It is shown that the lowest order expansion of the map accurately captures the transient growth rate of impact velocities. The second paper presents a constructive proof of a control algorithm for a rather large class of impact oscillators. The proof is constructive in the sense that it gives control parameters which stabilizes the dynamics at the onset of low-velocity impacts. In the third paper a piecewise-smooth quarter-car model is derived, and the control strategy is implemented to reduce impact velocities in the suspension system. In the fourth and fifth papers the grazing bifurcation of an impact oscillator with dry friction type damping is investigated. It turns out that the bifurcation is triggered by the disappearance of an interval of stable stick solutions. A condition on the parameters of the system is derived which differentiates between stable and unstable types of bifurcation scenarios. Additionally, a low-cost control strategy is proposed, similar to the one previously mentioned, to regulate the bifurcation scenario.

Place, publisher, year, edition, pages
Stockholm: KTH , 2009. , viii, 53 p.
Series
Trita-AVE, ISSN 1651-7660 ; 2009:66
Keyword [en]
non-smooth dynamics, nonlinear control, low-velocity impacts, friction, bifurcations
National Category
Applied Mechanics
Identifiers
URN: urn:nbn:se:kth:diva-11079ISBN: 978-91-7415-437-5 (print)OAI: oai:DiVA.org:kth-11079DiVA: diva2:235399
Public defence
2009-10-09, D2, Lindstedtsvägen 5, Kungliga Tekniska Högskolan, Stockholm, 10:00 (English)
Opponent
Supervisors
Note
QC 20100811Available from: 2009-09-21 Created: 2009-09-15 Last updated: 2010-08-11Bibliographically approved
List of papers
1. Transient growth rates of near-grazing impact velocities: Theory and experiments
Open this publication in new window or tab >>Transient growth rates of near-grazing impact velocities: Theory and experiments
2009 (English)In: Journal of Sound and Vibration, ISSN 0022-460X, E-ISSN 1095-8568, Vol. 325, no 4-5, 950-958 p.Article in journal (Refereed) Published
Abstract [en]

In this paper, nonsmooth fold bifurcations associated with the onset of low-relative- velocity (near-grazing) impacts in an oscillatory mechanical system are proposed as a potential operating principle for high-speed limit switches. Specifically, analytical, numerical, and experimental methods are employed to investigate the near-grazing transient behavior in a representative system. It is shown that the rate of growth of successive impact velocities increases beyond all bounds as the threshold parameter value is approached. A limit switch based on the proposed nonsmooth fold scenario would thus be expected to outperform one that relies on a smooth bifurcation, such as the cyclic-fold bifurcation, in terms of switching speed and sensitivity.

Place, publisher, year, edition, pages
London: Academic Press, 2009
Keyword
nonlinear dynamics, non-smooth dynamics, impacts
National Category
Applied Mechanics
Identifiers
urn:nbn:se:kth:diva-11069 (URN)10.1016/j.jsv.2009.04.012 (DOI)000268515900019 ()2-s2.0-67649506227 (Scopus ID)
Note

QC 20100811

Available from: 2009-09-15 Created: 2009-09-15 Last updated: 2017-12-13Bibliographically approved
2. On the stabilizability of near-grazing dynamics in impact oscillators
Open this publication in new window or tab >>On the stabilizability of near-grazing dynamics in impact oscillators
2007 (English)In: International Journal of Robust and Nonlinear Control, ISSN 1049-8923, E-ISSN 1099-1239, Vol. 17, no 15, 1405-1429 p.Article in journal (Refereed) Published
Abstract [en]

A constructive proof is presented for the existence of event-driven control strategies that guarantee the local persistence of system attractors with at most low-velocity contact in vibro-impacting oscillators. In particular, sufficient conditions are formulated on the linearization of the control strategies along a grazing periodic trajectory, i.e. an oscillating motion that achieves zero-relative-velocity contact with a mechanical obstacle, to ensure the asymptotic stability of the grazing trajectory and, consequently, sustained dynamics in the vicinity of the grazing trajectory even under small changes in system parameters. The implications of the methodology are illustrated with linear and nonlinear, single- and multiple-degree-of-freedom examples of vibro-impact oscillators.

Keyword
grazing bifurcations, vibro-impact oscillators, stabilizability, control
National Category
Applied Mechanics
Identifiers
urn:nbn:se:kth:diva-7835 (URN)10.1002/rnc.1252 (DOI)000250093000004 ()2-s2.0-34848816213 (Scopus ID)
Note
QC 20100811Available from: 2007-12-14 Created: 2007-12-14 Last updated: 2017-12-14Bibliographically approved
3. Suppression of bumpstop instabilities in a quarter-car model
Open this publication in new window or tab >>Suppression of bumpstop instabilities in a quarter-car model
2009 (English)In: Non-smooth Problems in Vehicle Systems Dynamics: Proceedings of the Euromech Colloquium, Berlin Heidelberg: Springer-Verlag , 2009Conference paper, Published paper (Refereed)
Abstract [en]

Vehicle manufacturers are constantly pushed to reduce the aerodynamic drag of vehicles, for example by constructing lower vehicles with less road clearance. This, however, reduces the available margin for oscillations within the suspension. If the oscillation amplitude exceeds a critical value, the suspension will impact a bumpstop. Under periodic excitation, the onset of low-velocity impacts is associated with a strong instability in favor of high-velocity impacts. Such impacts reduce comfort and could be damaging to the vehicle. Efforts should therefore be made to limit impact velocities with the bumpstop, for example by suppressing the instability associated with low-velocity impacts. This paper proposes a low-cost feedback-control strategy, based on making small adjustments to the position of the bumpstop, that serve to suppress the transition to high-velocity impacts with the bumpstop in the case of periodic excitation. The control law is derived from the theory of discontinuity maps. The results demonstrate that the feedback strategy works even when wheel-hop is present.

Place, publisher, year, edition, pages
Berlin Heidelberg: Springer-Verlag, 2009
Series
Proceedings of the Euromech Colloquium
Keyword
non-smooth dynamics, active suspension, impacts
National Category
Vehicle Engineering
Identifiers
urn:nbn:se:kth:diva-11076 (URN)10.1007/978-3-642-01356-0_12 (DOI)000283746100012 ()978-3-642-01355-3 (ISBN)
Conference
Euromech Colloquium,Non-smooth Problems in Vehicle Systems Dynamics
Note

QC 20100811

Available from: 2009-09-15 Created: 2009-09-15 Last updated: 2016-09-16Bibliographically approved
4. Energy Transfer in Vibratory Systems with Friction Exhibiting Low-velocity Collisions
Open this publication in new window or tab >>Energy Transfer in Vibratory Systems with Friction Exhibiting Low-velocity Collisions
2008 (English)In: Journal of Vibration and Control, ISSN 1077-5463, E-ISSN 1741-2986, no 1-2, 255-284 p.Article in journal (Refereed) Published
Abstract [en]

This paper investigates the dynamic response of an initially stationary part of a mechanism in the presence of a restoring force and dry friction to low-velocity collisions with a relatively more massive oscillating element. Of particular interest is the persistence of a local attractor in the motion of the less massive part as the path of the oscillating element grows to encompass the entire set of possible equilibrium positions in the absence of contact. It is argued that loss of a local attractor and the associated large-amplitude oscillations of the less massive part affords a means for energy transfer through the mechanism and a means for energy damping. The paper contains a rigorous derivation of conditions that appear sufficient for the persistence of a local attractor in the case where the massive oscillating element is replaced by an oscillating rigid unilateral constraint corresponding to an infinite mass ratio. Numerical simulations are subsequently used to investigate the response in the case where the mass ratio is assumed finite.

Place, publisher, year, edition, pages
Thousand Oaks: SAGE Publications Inc., 2008
Keyword
nonlinear dynamics, non-smooth dynamics, impacts, friction
National Category
Applied Mechanics
Identifiers
urn:nbn:se:kth:diva-11077 (URN)10.1177/1077546307079390 (DOI)000252947600013 ()2-s2.0-38149128709 (Scopus ID)
Note
QC 20100811Available from: 2009-09-15 Created: 2009-09-15 Last updated: 2017-12-13Bibliographically approved
5. Controlled onset of low-velocity collisions in a vibro-impacting system with friction
Open this publication in new window or tab >>Controlled onset of low-velocity collisions in a vibro-impacting system with friction
2009 (English)In: Proceedings of the Royal Society. Mathematical, Physical and Engineering Sciences, ISSN 1364-5021, E-ISSN 1471-2946, Vol. 465, 3647-3665 p.Article in journal (Refereed) Published
Abstract [en]

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 are derived that ensure a persistent, low-impact-velocity, steady-state response across the critical parameter value corresponding to grazing contact even in instances where the dynamics in the absence ofcontrol exhibit a sudden transition to a high-impact-velocity response.

Place, publisher, year, edition, pages
London: Royal Society Publishing, 2009
Keyword
non-smooth dynamics, nonlinear control, impacts, friction
National Category
Applied Mechanics
Identifiers
urn:nbn:se:kth:diva-11078 (URN)10.1098/rspa.2009.0207 (DOI)000271274700004 ()2-s2.0-73149087013 (Scopus ID)
Note
QC 20100811Available from: 2009-09-15 Created: 2009-09-15 Last updated: 2017-12-13Bibliographically approved

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