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Energy Transfer in Vibratory Systems with Friction Exhibiting Low-velocity Collisions
KTH, School of Engineering Sciences (SCI), Aeronautical and Vehicle Engineering, Vehicle Dynamics. (Vehicle Dynamics)
(Department of Mechanical Science and Engineering)
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. no 1-2, 255-284 p.
Keyword [en]
nonlinear dynamics, non-smooth dynamics, impacts, friction
National Category
Applied Mechanics
Identifiers
URN: urn:nbn:se:kth:diva-11077DOI: 10.1177/1077546307079390ISI: 000252947600013Scopus ID: 2-s2.0-38149128709OAI: oai:DiVA.org:kth-11077DiVA: diva2:235383
Note
QC 20100811Available from: 2009-09-15 Created: 2009-09-15 Last updated: 2010-11-18Bibliographically approved
In thesis
1. On the stability and control of piecewise-smooth dynamical systems with impacts and friction
Open this publication in new window or tab >>On the stability and control of piecewise-smooth dynamical systems with impacts and friction
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
non-smooth dynamics, nonlinear control, low-velocity impacts, friction, bifurcations
National Category
Applied Mechanics
Identifiers
urn:nbn:se:kth:diva-11079 (URN)978-91-7415-437-5 (ISBN)
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
2. Low-cost control of discontinuous systems including impacts and friction
Open this publication in new window or tab >>Low-cost control of discontinuous systems including impacts and friction
2007 (English)Licentiate thesis, comprehensive summary (Other scientific)
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.

Place, publisher, year, edition, pages
Stockholm: KTH, 2007. viii, 48 p.
Series
Trita-AVE, ISSN 1651-7660 ; 2007:79
Keyword
nonlinear dynamics, discontinuities, nonlinear control, impacts, friction, vehicle suspensions
National Category
Vehicle Engineering
Identifiers
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
Supervisors
Note
QC 20101118Available from: 2007-12-14 Created: 2007-12-14 Last updated: 2010-11-18Bibliographically approved

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