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On the stabilizability of near-grazing dynamics in impact oscillators
Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign.
KTH, Skolan för teknikvetenskap (SCI), Farkost och flyg, Fordonsdynamik.
2007 (Engelska)Ingår i: International Journal of Robust and Nonlinear Control, ISSN 1049-8923, E-ISSN 1099-1239, Vol. 17, nr 15, s. 1405-1429Artikel i tidskrift (Refereegranskat) 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.

Ort, förlag, år, upplaga, sidor
2007. Vol. 17, nr 15, s. 1405-1429
Nyckelord [en]
grazing bifurcations, vibro-impact oscillators, stabilizability, control
Nationell ämneskategori
Teknisk mekanik
Identifikatorer
URN: urn:nbn:se:kth:diva-7835DOI: 10.1002/rnc.1252ISI: 000250093000004Scopus ID: 2-s2.0-34848816213OAI: oai:DiVA.org:kth-7835DiVA, id: diva2:12973
Anmärkning
QC 20100811Tillgänglig från: 2007-12-14 Skapad: 2007-12-14 Senast uppdaterad: 2017-12-14Bibliografiskt granskad
Ingår i avhandling
1. Low-cost control of discontinuous systems including impacts and friction
Öppna denna publikation i ny flik eller fönster >>Low-cost control of discontinuous systems including impacts and friction
2007 (Engelska)Licentiatavhandling, sammanläggning (Övrigt vetenskapligt)
Abstract [en]

For a successful design of an engineering system it is essential to pay careful attention to its dynamic response. This is particularly true, in the case of nonlinear systems, since they can exhibit very complex dynamic behaviour, including multiple co-existing stable solutions and chaotic motions, characterized by large sensitivity to initial conditions. In some systems nonlinear characteristics are desired and designed for, but in other cases they are unwanted and can cause fatigue and failure. A type of dynamical system which is highly nonlinear is discontinuous or non-smooth systems. In this work, systems with impacts are primarily investigated, and this is a typical example of a discontinuous system. To enhance or optimize the performance of dynamical systems, some kind of control can be implemented. This thesis concerns implementation of low-cost control strategies for discontinuous systems. Low-cost control means that a minimum amount of energy is used when performing the control actions, which is a desirable situation regardless of the application. The disadvantage of such a method is that the performance might be limited as compared with a control strategy with no restrictions on energy consumption. In this work, the control objective is to enforce a continuous or discontinuous grazing bifurcation of the system, whichever is desirable. In Paper A, the dynamic response and bifurcation behaviour of an impactoscillator with dry friction is investigated. For a one-degree-of-freedom model of the system, analytical solutions are found in separate regions of state space. These are then used to perform a perturbation analysis around a grazing trajectory. Through the analysis, a condition on the parameters of the system is derived, which assures a continuous grazing bifurcation. It is also shown that the result has bearing on the dynamic response of a two-degree-of-freedom model of the system. A low-cost active control strategy for a class of impact oscillators is proposed in Paper B. The idea of the control method is to introduce small adjustments in the position of the impact surface, at discrete moments in time, to assure a continuous bifurcation. A proof is given for what control parameters assures the stabilization. In Paper C, the proposed low-cost control method is implemented in a quarter-car model of a vehicle suspension, in order to minimize impact velocities with the bumpstop in case of high amplitude excitation. It is shown that the control method is effective for harmonic road excitation.

Ort, förlag, år, upplaga, sidor
Stockholm: KTH, 2007. s. viii, 48
Serie
Trita-AVE, ISSN 1651-7660 ; 2007:79
Nyckelord
nonlinear dynamics, discontinuities, nonlinear control, impacts, friction, vehicle suspensions
Nationell ämneskategori
Farkostteknik
Identifikatorer
urn:nbn:se:kth:diva-4581 (URN)978-91-7178-830-6 (ISBN)
Presentation
2007-12-18, D3, D-huset, Lindstedtsvägen 5, Stockholm, 10:15
Opponent
Handledare
Anmärkning
QC 20101118Tillgänglig från: 2007-12-14 Skapad: 2007-12-14 Senast uppdaterad: 2010-11-18Bibliografiskt granskad
2. On the stability and control of piecewise-smooth dynamical systems with impacts and friction
Öppna denna publikation i ny flik eller fönster >>On the stability and control of piecewise-smooth dynamical systems with impacts and friction
2009 (Engelska)Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)
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.

Ort, förlag, år, upplaga, sidor
Stockholm: KTH, 2009. s. viii, 53
Serie
Trita-AVE, ISSN 1651-7660 ; 2009:66
Nyckelord
non-smooth dynamics, nonlinear control, low-velocity impacts, friction, bifurcations
Nationell ämneskategori
Teknisk mekanik
Identifikatorer
urn:nbn:se:kth:diva-11079 (URN)978-91-7415-437-5 (ISBN)
Disputation
2009-10-09, D2, Lindstedtsvägen 5, Kungliga Tekniska Högskolan, Stockholm, 10:00 (Engelska)
Opponent
Handledare
Anmärkning
QC 20100811Tillgänglig från: 2009-09-21 Skapad: 2009-09-15 Senast uppdaterad: 2010-08-11Bibliografiskt granskad

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