Stick-Slip Vibrations Induced by Alternate Friction Models

In the present paper a simple and efficient alternate friction model is presented to simulate stick-slip vibrations. The alternate friction model consists of a set of ordinary non-stiff differential equations and has the advantage that the system can be integrated with any standard ODE-solver. Comparison with a smoothing method reveals that the alternate friction model is more efficient from a computational point of view. A shooting method for calculating limit cycles, based on the alternate friction model, is presented. Time-dependent static friction is studied as well as application in a system with 2-DOF.     

Analytical approximations for stick-slip vibration amplitudes

The classical “mass-on-moving-belt” model for describing friction-induced vibrations is considered, with a friction law describing friction forces that first decreases and then increases smoothly with relative interface speed. Approximate analytical expressions are derived for the conditions, the amplitudes, and the base frequencies of friction-induced stick-slip and pure-slip oscillations. For stick-slip oscillations, this is accomplished by using perturbation analysis for the finite time interval of the stick phase, which is linked to the subsequent slip phase through conditions of continuity and periodicity. The results are illustrated and tested by time-series, phase plots and amplitude response diagrams, which compare very favorably with results obtained by numerical simulation of the equation of motion, as long as the difference in static and kinetic friction is not too large.     

Types of Bifurcations of FitzHugh–Nagumo Maps

The FitzHugh–Nagumo-like systems are of fundamental importance to the understanding of the qualitative nature of nerve impulse propagation. Our work provides a numerical investigation of bifurcations associated with a family of piecewise differentiable canonical maps for a planar FitzHugh–Nagumo system. We describe the bifurcation structure of the maps with the variation of the parameters.     

An Approximate Analysis of Dry-Friction-Induced Stick-Slip Vibrations by a Smoothing Procedure

This paper deals with a systematic procedure to find both stable and unstable periodic stick-slip vibrations of autonomous dynamic systems with dry friction. In this procedure, the discontinuous friction forces are approximated by smooth functions. Using the simple shooting method with a stiff-ODE solver, in combination with a path following algorithm, branches of periodic solutions are computed for a changing design variable. For testing purposes, both 1 and 2-DOF autonomous block-on-belt models and a 1-DOF autonomous drill string model from literature are investigated. Comparison of the results shows that the smoothing procedure accurately describes the behavior of the discontinuous systems. The proposed procedure can also easily be applied to more complex MDOF models, as well as to nonautonomous dynamic systems.     

Synchronisation and Chaos in a Parametrically and Self-Excited System with Two Degrees of Freedom

Vibration analysis of a non-linear parametrically andself-excited system of two degrees of freedom was carried out. The modelcontains two van der Pol oscillators coupled by a linear spring with a aperiodically changing stiffness of the Mathieu type. By means of amultiple-scales method, the existence and stability of periodicsolutions close to the main parametric resonances have beeninvestigated. Bifurcations of the system and regions of chaoticsolutions have been found. The possibility of the appearance ofhyperchaos has also been discussed and an example of such solution hasbeen shown.     

Nonlinear vibrations of structures induced by dry friction

The chattering of machine tools, the squealing noise generated by tram wheels in narrow curves and the noise of band saws are examples of physical processes in which elastic structures exhibit self-sustained stick-slip vibrations. The nonlinear contact forces are often due to dry friction. Periodic, multiperiodic, and chaotic motions can occur, depending on the parameters.Because the governing equations of motion are non-integrable, solutions can only be determined by numerical integration methods. The numerical investigations of continuous structures requires themodal approach to reduce the number of degrees of freedom.As an example, a beam system has been investigated numerically and experimentally in this paper. The nonlinear motion of a point of the continuous structure has been measured by a specially developedlaser vibrometer.The friction characteristic has been measured directly and identified from a measured time series by means of amodal state observer. The correlation dimension, which represents a lower bound of thefractal dimension, has been calculated using thecorrelation integral method from a measured time series of the beam system.     

Performances of Unknown Input Observers for Chaotic LPV Maps in a Stochastic Context

Jointly modeling chaotic maps as Linear Parameter Varying (LPV) systems and using Unknown Input Observers (UIOs) for retrieving the information in a secure communication scheme has previously been motivated in a deterministic context [Millerioux, G. and Daafouz, J. International Journal of Bifurcation and Chaos 14(4), 2004]. In this paper, some new theoretical results from a control theory point of view, concerning the design in a stochastic and so more realistic context of UIOs for chaotic LPV systems is provided. The design of such observers is expressed in terms of the resolution of a finite set of Matrix Inequalities constraints and guarantees some prescribed performances on the state reconstruction error.     

Subharmonic Response of a Quasi-Isochronous Vibroimpact System to a Randomly Disordered Periodic Excitation

A quasi-isochronous vibroimpact system is considered, i.e. a linear system with a rigid one-sided barrier, which is slightly offset from the system's static equilibrium position. The system is excited by a sinusoidal force with disorder, or random phase modulation. The mean excitation frequency corresponds to a simple or subharmonic resonance, i.e. the value of its ratio to the natural frequency of the system without a barrier is close to some even integer. Influence of white-noise fluctuations of the instantaneous excitation frequency around its mean on the response is studied in this paper. The analysis is based on a special Zhuravlev transformation, which reduces the system to one without impacts, or velocity jumps, thereby permitting the application of asymptotic averaging over the period for slowly varying inphase and quadrature responses. The averaged stochastic equations are solved exactly by the method of moments for the mean square response amplitude for the case of zero offset. A perturbation-based moment closure scheme is proposed for the case of nonzero offset and small random variations of amplitude. Therefore, the analytical results may be expected to be adequate for small values of excitation/system bandwidth ratio or for small intensities of the excitation frequency variations. However, at very large values of the parameter the results are approaching those predicted by a stochastic averaging method. Moreover, Monte-Carlo simulation has shown the moment closure results to be sufficiently accurate in general for any arbitrary bandwidth ratio. The basic conclusion, both of analytical and numerical simulation studies, is a sort of smearing of the amplitude frequency response curves owing to disorder, or random phase modulation: peak amplitudes may be strongly reduced, whereas somewhat increased response may be expected at large detunings, where response amplitudes to perfectly periodic excitation are relatively small.     

Topological Invariants in a Model of a Time-Delayed Chua's Circuit

In the last 30 years, some authors have been studying several classes of boundary value problems (BVP) for partial differential equations (PDE) using the method of reduction to obtain a difference equation with continuous argument which behavior is determined by the iteration of a one-dimensional (1D) map (see, for example, Romanenko, E. Yu. and Sharkovsky, A. N., International Journal of Bifurcation and Chaos 9(7), 1999, 1285–1306; Sharkovsky, A. N., International Journal of Bifurcation and Chaos 5(5), 1995, 1419–1425; Sharkovsky, A. N., Analysis Mathematica Sil 13, 1999, 243–255; Sharkovsky, A. N., in “New Progress in Difference Equations”, Proceedings of the ICDEA'2001, Taylor and Francis, 2003, pp. 3–22; Sharkovsky, A. N., Deregel, Ph., and Chua, L. O., International Journal of Bifurcation and Chaos 5(5), 1995, 1283–1302; Sharkovsky, A. N., Maistrenko, Yu. L., and Romanenko, E. Yu., Difference Equations and Their Applications, Kluwer, Dordrecht, 1993.). In this paper we consider the time-delayed Chua's circuit introduced in (Sharkovsky, A. N., International Journal of Bifurcation and Chaos 4(5), 1994, 303–309; Sharkovsky, A. N., Maistrenko, Yu. L., Deregel, Ph., and Chua, L. O., Journal of Circuits, Systems and Computers 3(2), 1993, 645–668.) which behavior is determined by properties of one-dimensional map, see Sharkovsky, A. N., Deregel, Ph., and Chua, L. O., International Journal of Bifurcation and Chaos 5(5), 1995, 1283–1302; Maistrenko, Yu. L., Maistrenko, V. L., Vikul, S. I., and Chua, L. O., International Journal of Bifurcation and Chaos 5(3), 1995, 653–671; Sharkovsky, A. N., International Journal of Bifurcation and Chaos 4(5), 1994, 303–309; Sharkovsky, A. N., Maistrenko, Yu. L., Deregel, Ph., and Chua, L. O., Journal of Circuits, Systems and Computers 3(2), 1993, 645–668. To characterize the time-evolution of these circuits we can compute the topological entropy and to distinguish systems with equal topological entropy we introduce a second topological invariant.     

A Perturbation Method for Nonlinear Two-Dimensional Maps

A new perturbation method for a weakly nonlinear two-dimensional discrete-time dynamical system is presented. The proposed technique generalizes the asymptotic perturbation method that is valid for continuous-time systems and detects periodic or almost-periodic orbits and their stability. Two equations for the amplitude and the phase of solutions are derived and their fixed points correspond to limit cycles for the starting nonlinear map. The method is applied to various nonlinear (autonomous or not) two-dimensional maps. For the autonomous maps we derive the conditions for the appearance of a supercritical Hopf bifurcation and predict the characteristics of the corresponding limit cycle. For the nonautonomous maps we show amplitude-response and frequency-response curves. Under appropriate conditions, we demonstrate the occurrence of saddle-node bifurcations of cycles and of jumps and hysteresis effects in the system response (cusp catastrophe). Modulated motion can be observed for very low values of the external excitation and an infinite-period bifurcation occurs if the external excitation increases. Analytic approximate solutions are in good agreement with numerically obtained solutions.     

Motion types and chaos of multi-body systems vibrating with impacts

In this paper, the limit sets theory for an autonomous dynamical system is generalized to a multi-body system vibrating with impacts. We discover that if every motion of the system is bounded, it has only four different types: periodic motion γ1, non·periodic recurrent motion γ2, and non-Poisson stable motions γ3 and γ4 approaching γ1 and γ2, respectively. γ2 is the source of chaos. It is very interesting that chaotic motions seem stochastic but possess the character of recurrence. By way of example, we discuss chaotic motions of a small ball bouncing vertically on a massive vibrating table. The result obtained by us is different from that obtained by Holmes. The project supported by National Natural Science Foundation of China     

Bifurcation and stability analysis for a non-smooth friction oscillator

Summary Friction-induced self-sustained oscillations, also known as stick-slip vibrations, occur in mechanical systems as well as in everyday life. On the basis of a one-dimensional map, the bifurcation behaviour including unstable branches is investigated for a friction oscillator with simultaneous self-and external excitation. The chosen way of mapping also allows a simple determination of Lyapunov exponents.Dedicated to Prof. Dr.-Ing. Dr.-Ing. E.h. Dr. h.c. mult. Erwin Stein on the occasion of his 65th birthday.     

Pseudoelastic effect in autoparametric non-ideal vibrating system with SMA spring

In this paper a three degrees of freedom autoparametric system with limited power supply is investigated numerically. The system consists of the body, which is hung on a spring and a damper, and two pendulums connected by shape memory alloy (SMA) spring. Shape memory alloys have ability to change their material properties with temperature. A polynomial constitutive model is assumed to describe the behavior of the SMA spring. The non-ideal source of power adds one degree of freedom, so the system has four degrees of freedom. The equations of motion have been solved numerically and pseudoelastic effects associated with the martensitic phase transformation are studied. It is shown that in this type system one mode of vibrations might excite or damp another mode, and that except different kinds of periodic vibrations there may also appear chaotic vibrations. For the identification of the responses of the system's various techniques, including chaos techniques such as bifurcation diagrams and time histories, power spectral densities, Poincarè maps and exponents of Lyapunov may be used.     

Dynamics of a Gear System with Faults in Meshing Stiffness

Gear box dynamics is characterised by a periodically changing stiffness. In real gear systems, a backlash also exists that can lead to a loss in contact between the teeth. Due to this loss of contact the gear has piecewise linear stiffness characteristics, and the gears can vibrate regularly and chaotically. In this paper we examine the effect of tooth shape imperfections and defects. Using standard methods for nonlinear systems we examine the dynamics of gear systems with various faults in meshing stiffness.     

Transition from Planar to Whirling Oscillations in a Certain Nonlinear System

A single-mass two-degrees-of-freedom system is considered, witha radially oriented nonlinear restoring force. The latter is smooth andbecomes infinite at a certain value of a radial displacement. Stabilityanalysis is made for planar oscillation, or motion along a givendirection. As long as this motion is periodic, the nonlinearity in therestoring force provides a periodic parametric excitation in thetransverse direction. The linearized stability analysis is reduced tostudy of the Mathieu equation for the (infinitesimal) motions in thetransverse direction. For the case of free oscillations in the givendirection an exact solution is obtained, since a specific analyticalform is used for the (strongly nonlinear) restoring force, which permitsexplicit integration of the equation of motion. Stability of the planarmotion in this case is shown to be very sensitive to even slightdeviations from polar symmetry in the restoring force (as well as to theamplitude of oscillations in the given direction). Numerical integrationof the original equations of motion shows the resulting motion to be awhirling type indeed in case of the transversal instability. For thecase of a sinusoidal forcing in the given direction solution for the(periodic) response is obtained by Krylov–Bogoliubov averaging. Thisresults in the transmitted Ince–Strutt chart – namely, stabilitychart for transverse direction on the amplitude-frequency plane of theexcitation in the original direction.     

Global Wave Maps on Robertson–Walker Spacetimes

We prove the global existence and uniqueness of wave maps onexpanding universes of dimension three or four, that is Robertson–Walkerspacetimes whose inverse radius is integrable with respect to the cosmictime. A result is obtained for small initial data by using the first andsecond energy estimates.     

Life Expectancy Calculations of Transient Chaotic Behaviour Using Approximate 1D Maps

In engineering practice, chaotic oscillations are often observed which disappear suddenly. This phenomenon is often referred to as transient chaos. The duration of these oscillations varies stochastically. In this work two methods are presented for the simple estimation of the expected length of the chaotic behaviour. As an example, the Lorenz system is considered at some specific parameter values.     

Dynamics of a Simple Damped Oscillator Undergoing Stick-Slip Vibrations

The dynamics of a simple dynamical system subjected to an elastic restoring force, viscous damping and dry friction forces is investigated. Self-sustained oscillations occur with non-standard attracting properties. Discontinuity of the governing equations leads to non-standard bifurcations, which are studied here, with analytical and numerical tools. This revised version was published online in July 2006 with corrections to the Cover Date.     

Stability of the Interface of a Liquid-Suspension System under High-Frequency Nonlinearly Polarized Vibration

The stability of the state of a two-layer system consisting of a homogeneous liquid and a solid particle suspension in the same liquid with a plane interface between them is investigated. The system performs both horizontal and vertical high-frequency vibration with an arbitrary phase shift. It is shown that a mean flow develops under the simultaneous action of both horizontal and vertical vibration; quantitative flow stability characteristics are determined numerically using the differential sweep method. It is shown that a traveling wave relief exists on the liquid-suspension interface. Transverse oscillations in phase with longitudinal oscillations destabilize the entire system. The presence of an oscillation phase shift can lead to an increase in the stability limit; the direction of motion of the wave relief can differ depending on the value of the phase shift. Instability of the system in the presence of strictly vertical vibration is observed, the crisis being associated with longwave monotonic perturbations.__________Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, 2005, pp. 3–13.Original Russian Text Copyright © 2005 by Lobov, Lyubimov, and Lyubimova.