The complicated bifurcation of an archetypal self-excited SD oscillator with dry friction

In this paper, we investigate the local and global bifurcation behaviors of an archetypal self-excited smooth and discontinuous oscillator driven by moving belt friction. The belt friction is described in the sense of Stribeck characteristic to formulate the mathematical model of the proposed system. For such a friction characteristic, the complicated bifurcation behaviors of the system are discussed. The bifurcation of the multiple sliding segments for this self-excited system is exhibited by analytically exploring the collision of tangent points. The Hopf bifurcation of this self-excited system with viscous damping is analyzed by making the examination of the eigenvalues at the steady state and discussing the stability of the limit cycles. The bifurcation diagrams and the corresponding phase portraits are depicted to demonstrate the complicated dynamical behaviors of double tangency bifurcation, the bifurcation of sliding homoclinic orbit to a saddle, subcritical Hopf bifurcation and grazing bifurcation for this system.     

Stick-slip vibrations of a self-excited SD oscillator with Coulomb friction

Nonlinear Dynamics - In this paper, the stick-slip vibrations of an archetypal self-excited smooth and discontinuous (SD) oscillator are investigated. The mathematical model of the self-excited SD...     

Dynamics and impact in a system with unilateral constraints the relevance of dry friction

Summary The aim of the present work is to investigate the impact and the dynamics of a rigid system, subject to unilateral constraints. The problem is particularly interesting to understand the mechanical behaviour of stone structures subject to external forces and to give informations about the dynamical evolution; such informations are basic for a stability study.

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Sommario Scopo del presente lavoro è quello di indagare sulla dinamica e sull'urto di un sistema rigido, soggetto a vincoli unilaterali. Tale problema è particolarmente interessante per la comprensione del comportamento meccanico delle strutture monumentali costituite da blocchi lapidei, in presenza di forze esterne variabili. L 'analisi fornisce informazioni sulla evoluzione dinamica del sistema; tali informazioni sono indispensabili per uno studio sulla stabilità.

     

Analysis of the non-periodic oscillations of a self-excited friction-damped system with closely spaced modes

Nonlinear Dynamics - It is widely known that dry friction damping can bound the self-excited vibrations induced by negative damping. The vibrations typically take the form of (periodic) limit cycle...     

Criterion of multiple collisions in a simple mechanical system with viscous damping and dry friction

A criterion has been formulated for single and multiple central collisions of two rigid bodies of a simple mechanical system that can be either conservative or non-conservative. Analytical discussion has been confined to investigations of the possibility of appearance of four successive collisions between the bodies with damping neglected and two collisions when viscous damping and Coulomb dry friction are considered.     

On self-excited vibrations in friction force measurement systems

Problems of friction force measurement under conditions of cosmic experiment on the orbit are considered. To increase the measurement accuracy, some arrangements should be made for preventing the onset of a self-vibration mode in mechanical systems with movable contact. Two self-vibration modes are studied. One of them occurs in the case of moderate relative slip velocities of the moving element and the indenter, and the other, in the case of small velocities. The existence and stability conditions are obtained in terms of the mass-elastic parameters of the contact pair.     

On the brachystochrone with dry friction

The problem of the brachystochrone with dry friction is governed by an autonomous set of highly non-linear ordinary differential equations, which contain a small parameter μ (the coefficient of friction). Using perturbation technique an approximate solution is constructed.     

Forced nonlinear oscillator with nonsymmetric dry friction

In this paper we discuss an approximately steady motion of an oscillator as a single whole with a constant “on the average” velocity. For that purpose we analyze the position and stability of some special points of the phase portrait. In the presence of internal excitation and nonsymmetric Coulomb dry friction, a motion of the oscillator with a constant “on the average” velocity is possible. The algebraic equation for this constant velocity is found. For different parameters of the model there exist at most three regimes of motion with a constant velocity, but only one or two of them are stable. The theoretical results obtained can be used for the design of worm-like moving robots.     

On chaotic motions of systems with dry friction

The principles of formation of closed and quasiperiodic orbitally stable trajectories in systems with dry friction are formulated. A frictional oscillator with periodic forcing is considered. It is established that random and stable oscillations exist. It is discovered that orbital instability and irregular oscillations are associated with the instability domain near a singular point Translated from Prikladnaya Mekhanika, Vol. 44, No. 9, pp. 123–132, September 2008.     

On models of dynamic systems with dry friction

An approach to the design of models of dynamical systems with high dry friction in the kinematic pair is developed. The members of the kinematic pair are represented by parts of rigid bodies. The system as a whole is considered to have a variable structure. According to this assumption, two modes of motion with different dissipative characteristics are possible. The states in which the modes exchange and the motion switches over into critical modes with dynamic self-locking are established. A system with a variable transfer function between members that form a nonideal constraint is described __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 5, pp. 97–105, May 2007.     

Self-excited vibration of the shell-liquid coupled system induced by dry friction

The nonlinear vibration theory and the experimental modal analysis are used in this paper to study the self-excited vibration of the shell-liquid coupled system induced by dry friction. The effect of dry friction stick-slip coefficients and rubbing velocity on self-excited vibration, and the limit cycle and Hopf bifurcation solution of the system are obtained. In particular, it is shown that the phenomenon of 4 point (or 6 point) water droplet spurting of the Chinese cultural relic Dragon Washbasin is the result of the perfect combination of the self-excited vibration induced by dry friction and its special modes, which indicates the significant scientific value of the Chinese cultural relic Dragon Washbasin.     

Periodic and chaotic dynamics of a sliding driven oscillator with dry friction

We have performed a numerical study of the dynamics of a harmonically forced sliding oscillator with two degrees of freedom and dry friction. The study of the four-dimensional dynamical system corresponding to the two non-linear motion equations can be reduced, in this case, to the study of a three-dimensional Poincaré map. The behaviour of the system has been investigated calculating bifurcation diagrams, time series, periodic and chaotic attractors and basins of attraction. Furthermore, a systematic study of the stability of periodic solutions and their bifurcations has been carried out applying the Floquet theory. The results show rich dynamics being very sensitive to the changes in forcing amplitudes (control parameter), where periodic and chaotic states alternatively appear. It is shown how the system exhibits different types of bifurcational phenomena (saddle-node, symmetry-breaking, period-doubling cascades and intermittent transitions to chaos) into relatively narrow intervals of the control parameter. Moreover, a collection of chaotic attractors was computed to show the evolution of the chaotic regime. Finally, basins of attraction were calculated. In all the cases studied, the basins exhibit fractal structure boundaries and, when more of two attractors are coexisting, we have found Wada basin boundaries.     

An improved quasi-zero stiffness vibration isolation system utilizing dry friction damping

Nonlinear Dynamics - Quasi-zero stiffness (QZS) isolators, a nonlinear vibration isolation technique, enhance the isolation performance, by lowering the natural frequencies of an isolation system...