On periodic steady state response and stability of Filippov-type mechanical models

Abstract

In the first part of this study, the basic steps of a methodology are presented, leading to a long time response of a class of periodically excited mechanical models with contact and dry friction. In particular, the models examined belong to the special class of Filippov-type dynamical systems, which possess continuous displacements and velocities, but exhibit discontinuities in their accelerations. The direct determination of periodic steady state response of this class of models is achieved by combining suitable numerical integration of the equations of motion with an appropriate technique yielding the corresponding monodromy matrix. This matrix, which arises from a linearization of the motion around a located periodic solution, involves saltations (jumps) and is also useful in predicting its stability properties. The analytical part is complemented by a suitable continuation procedure, enabling evaluation of complete branches of periodic motions. In the second part of the study, the effectiveness of the methodology developed is confirmed by presenting representative sets of numerical results obtained for selected examples. The first two of them are single degree of freedom oscillators. Besides investigating some interesting aspects of regular periodic response, some cases involving rich dynamics of the C. Theodosiou · A. Pournaras · S. Natsiavas ( ) class of the system examined are also studied in a systematic way. The last example is a more involved and challenging model, related to the function of an engine valve and characterized by large numerical stiffness.

A complementarity approach for the computation of periodic oscillations in piecewise linear systems

“…The most popular time-domain technique is the so-called shooting method [7][8][9]. A drawback of this method is the cost of calculating the sensitivity matrix, which is a crucial issue [10][11][12]. For instance, in [13,14], in order to compute periodic stick-slip vibrations belonging to mechanical systems with discontinuous dry friction, the shooting method is applied only after the approximation of the discontinuous characteristic by a suitable smooth function.…”

Computing period and shape of oscillations in piecewise linear Lur'e systems: A complementarity approach

“…Time-domain approaches are based mainly on the so-called shooting method which determines the initial condition and the period for the periodic solution by solving a sequence of nonlinear initial value problems with the Newton-Raphson method, [1], [2], [3]. The main drawback of this method is the evaluation of the sensitivity matrix, which is often computationally expensive and becomes even more complicated for nonsmooth systems, [4], [5]. In frequency-domain, harmonic balance is the classical technique used for determining the steady-state behaviour of nonlinear autonomous systems that exhibit a single periodic attractor, [6].…”

Editorial: Special issue on stability of non-linear dynamic structures and systems

“…In this special issue, the concept stability is also frequently encountered with respect to periodic solutions caused by periodic or pure harmonic excitation [1][2][3][4][5][6][7]. The local stability of periodic solutions is evaluated using Floquet theory, which can be extended for systems of Filippov-type by introducing saltation matrices, as reviewed in [6].…”

“…The local stability of periodic solutions is evaluated using Floquet theory, which can be extended for systems of Filippov-type by introducing saltation matrices, as reviewed in [6]. The global stability of a periodic solution can be determined by investigating its domain of attraction; see, e.g.…”

“…In the first part of their study, Theodosiou et al [6] present the basic steps of a methodology, leading to the periodic steady state response (and its stability) of periodically excited mechanical models of Filippovtype with contact and dry friction. The analytical part is complemented by a suitable continuation procedure, enabling evaluation of complete branches of periodic motions.…”