Oscillation pattern of stick-slip vibrations

This paper studies the stick-slip oscillations of discrete systems interacting with translating energy source through a non-linear smooth friction curve. The stick-slip limit cycle oscillations of a single degree-of-freedom model are examined by means of numerical time-integration and analytical methods. Similar approaches are also applied to the model of the coupled friction oscillator. Particularly, it is found that the steady-state response of the coupled oscillator can be divided into two different forms of oscillation (mode-merged and mode-separated oscillations) according to the frequency separation of two modes. The oscillation pattern of the steady-state response is shown to depend on system parameters such as detuning factor, energy source speed, and normal contact load.     

Non-linear control of friction-induced self-excited vibration

The present paper introduces a new method of controlling friction-driven self-excited vibration. The control law is primarily derived using the Lyapunov's second method. A single degree-of-freedom oscillator on a moving belt represents the primary model of the system. The control action is achieved by modulating the normal load at the frictional interface based on the state of the oscillatory system. The basic mechanism of the control action utilises subcritical Hopf bifurcation of the equilibrium followed by cyclic-fold bifurcation (of limit cycle oscillations) to globally stabilise the equilibrium. The basic mechanism is qualitatively independent of the exact model of friction. Different models of friction, like, algebraic model, LuGre model and switch model with time-dependent static friction are considered to substantiate the above claim. An approximate method for estimating the critical value of the control parameter that ensures global stability of the equilibrium is also proposed.     

The stick-slip problem for a round jet

Summary The stick-slip problem for a round jet studied in Part I gives a good approximation for the swell of a low speed jet when the surface tension is large but it fails when the surface tension is small. In this paper a new stick-slip problem (II) is defined and solved using matched eigenfunction expansions. The new problem reduces to that solved in Part I when the surface tension is large and gives good results in the case of zero and small surface tension.With 18 figures     

The integrated singular basis function method for the stick-slip and the die-swell problems

We further develop a new singular finite element method, the integrated singular basis function method (ISBFM), for the solution of Newtonian flow problems with stress singularities. The ISBFM is based on the direct subtraction of the leading local solution terms from the governing equations and boundary conditions of the original problem, followed by a double integration by parts applied to those integrals with singular contributions. The method is applied to the stick-slip and the die-swell problems and improves the accuracy of the numerical results in both cases. In the case of the die-swell problem it considerably accelerates the convergence of the free surface profile with mesh refinement. The advantages and disadvantages of the ISBFM when compared to other singular methods are also discussed.     

Stick-slip instability analysis

A theoretical investigation of friction-induced self-excited oscillations for systems with one degree of freedom is proposed. The friction force is assumed as an odd function of the relative sliding velocity with a jump discontinuity at a value of zero for the relative sliding velocity. The friction characteristic is approximated with a piecewise linear function, i.e. straight line segments with a suitable slope. For the generic system belonging to the class in question, the stick-slip instability region is located on a suitable dimensionless map.

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Sommario Viene proposta un'indagine teorica sulle oscillazioni autoeccitate indotte dall'attrito per sistemi ad un grado di libertà. La forza d'attrito viene assunta come funzione dispari della velocità relativa tra le superfici accoppiate, con una discontinuità di prima specie in corrispondenza del valore nullo della velocità. La caratteristica d'attrito viene approssimata mediante una funzione lineare a tratti con segmenti di opportuna pendenza. Per il generico sistema appartenente alla classe in esame, si perviene all'individuazione, su opportuna mappa adimensionale, della regione di instabilità da stick-slip.

     

Events Maps in a Stick-Slip System

This paper describes a one-dimensional map generated by a two degree-of-freedom mechanical system that undergoes self-sustained oscillations induced by dry friction. The iterated map allows a much simpler representation and a better understanding of some dynamic features of the system. Some applications of the map are illustrated and its behaviour is simulated by means of an analytically defined one-dimensional map. A method of reconstructing one-dimensional maps from experimental data from the system is introduced. The method uses cubic splines to approximate the iterated mappings. From a sequence of such time series the parameter dependent bifurcation behaviour is analysed by interpolating between the defined mappings. Similarities and differences between the bifurcation behaviour of the exact iterated mapping and the reconstructed mapping are discussed.     

Frictional Phenomena in Dynamical Systemwith Two-Frequency Excitation

The paper deals with investigation of a self-excited vibrating system with dry friction. The system is composed of a mass connected by viscoelastic element with the referring frame and interacting with a moving belt by means of dry friction. An experimentally identified, multi-parametric dry friction model for the pair composed of soft and hard elements like steel–polyester pair, describing both the case of stick-slip and quasi-harmonic vibration, has been applied. Additionally, the system is influenced by external, two-frequency kinematic excitation. The results of computer simulation for different excitation conditions are submitted in the present paper.     

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.     

On the Analytical Approach to Galin’s Stick-Slip Problem. A Survey

Galin’s classical work (PMM J Appl Math Mech 9:413–424, 1945) on the contact of a rigid flat-ended indenter with an elastic half-plane with partial slip was the first successful attempt to take into account friction in the problem of normal contact. As Galin was unable to find an exact solution of the formulated problem, the problem of contact with partial slip of a rigid punch with an elastic half-plane was challenged by many researchers. At the same time Galin’s seminal work stimulated development of solutions for other contact problems with friction that feature different punch geometries and different material responses. This paper presents an overview of the developments in the area of elastic contact with partial slip. In the spirit of Galin’s work the focus is placed on contributions with substantial analytical merit.      

Free vibration of two coupled nonlinear oscillators

An analytical investigation is carried out on the free vibration of a two degree of freedom weakly nonlinear oscillator. Namely, the method of multiple time scales is first applied in deriving modulation equations for a van der Pol oscillator coupled with a Duffing oscillator. For the case of non-resonant oscillations, these equations are in standard normal form of a codimension two (Hopf-Hopf) bifurcation, which permits a complete analysis to be performed. Three different types of asymptotic states-corresponding to trivial, periodic and quasiperiodic motions of the original system-are obtained and their stability is analyzed. Transitions between these different solutions are also identified and analyzed in terms of two appropriate parameters. Then, effects of a coupling, a detuning, a nonlinear stiffness and a damping parameter are investigated numerically in a systematic manner. The results are interpreted in terms of classical engineering terminology and are related to some relatively new findings in the area of nonlinear dynamical systems.     

Analytical investigation of stick-slip motions in coupled brake-driveline systems

In this paper, we investigate the brake creep-groan problem by formulating the issues in terms of two dynamic sub-systems that are coupled at the friction interface, and thus, experience stick-slip motions. Thorough examination of the nature of discontinuous solutions using numerical methods is a useful prelude to analytical studies. We examine such dynamics through parametric studies for magnitude and rate of brake release, where the vehicle is initially at rest and under low constant drive torque. Dependency on the initial conditions and solution flow before reaching the orbit will thus be illustrated. Four types of motions (one steady sliding and three stick-slip) are found based on extensive studies. Our formulation and analysis should lead to a better understanding of the brake groan phenomenon and systems coupled by interfacial friction.     

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.     

Understanding root cause of stick–slip vibrations in deep drilling with drag bits

In search for the root cause of stick–slip, a mode of torsional vibrations of a drilling assembly, a linear stability analysis of coupled axial–torsional vibrations has been carried out. It has been shown that in a rotary drilling system with axial and torsional degree of freedom two distinct modes of self-excited vibrations are present: axial and torsional. These axial (torsional) modes of vibrations are due to resonance between the cutting forces acting at the bit and the axial (torsional) natural modes of drillstring vibrations. It has been demonstrated that although axial and torsional modes of vibrations do affect each other the underlying mechanisms driving these modes of vibrations are completely different. In particular, the only driving mechanism of the axial vibrations is the regenerative effect, while there are two distinct mechanisms that drive the torsional vibrations: (i) the cutting action of the bit, and (ii) the wearflat/rock interaction. Moreover, in the case of the torsional vibrations the regenerative effect plays only a secondary role. The results of the present study indicate that the axial compliance can play a stabilizing role. In particular, the stabilizing role of the axial compliance increases as the ratio of the torsional to the axial natural frequency of the drillstring vibrations decreases.     

Multi-frequency oscillations in self-excited systems

A self-excited three-mass chain system is considered here. For a self-excitation of van der Pol type, the possibility of multi-frequency oscillations is investigated. Both analytical approximate solutions and numerical simulation are used. The averaging method is used to establish existence and stability of the normal modes, the two-frequency modes as well as the three-frequency oscillations solutions. We found at first that the single mode seems to prevail. However a three-frequency solution can be stabilised by adapting the system slightly. A generic bifurcation diagram is given where all the possible phase portraits are sketched. The flow turns out to be quite predictable. There is no “room” for chaos or strange attractors. This behaviour is not typical for systems of coupled oscillators but turns out to be partly related to the involved symmetries as well as the particular choice of the system parameters.     

Analytical Solution for a Free Vibration of a Thermoelastic Hollow Sphere

For a free vibration problem of a thermoelastic hollow sphere into the context of the generalized thermoelasticity theory with one relaxation time, exact analytic solutions are obtained with the use of eigenvalue approach. Both the inner and outer curved surfaces of the sphere are considered stress-free and isothermal surfaces. The dispersion relations for the existence of various types of possible modes of vibrations in the considered hollow sphere are derived. The numerical results have been presented graphically in respect of natural frequencies, thermoelastic damping, and frequency shift.     

Discontinuous fold bifurcations in mechanical systems

Summary  This paper treats discontinuous fold bifurcations of periodic solutions of discontinuous systems. It is shown how jumps in the fundamental solution matrix lead to jumps of the Floquet multipliers of periodic solutions. A Floquet multiplier of a discontinuous system can jump through the unit circle, causing a discontinuous bifurcation. Numerical examples are treated, which show discontinuous fold bifurcations. A discontinuous fold bifurcation can connect stable branches to branches with infinitely unstable solutions. Received 20 September 2000; accepted for publication 26 June 2001     

The bouncing motion appearing in a robotic system with unilateral constraint

The hopping or bouncing motion can be observed when robotic manipulators are sliding on a rough surface. Making clear the reason of generating such phenomenon is important for the control and dynamical analysis for mechanical systems. In particular, such phenomenon may be related to the problem of Painlevé paradox. By using LCP theory, a general criterion for identifying the bouncing motion appearing in a planar multibody system subject to single unilateral constraint is established, and found its application to a two-link robotic manipulator that comes in contact with a rough constantly moving belt. The admissible set in state space that can assure the manipulator keeping contact with the rough surface is investigated, and found which is influenced by the value of the friction coefficient and the configuration of the system. Painlevé paradox can cause either multiple solutions or non-existence of solutions in calculating contact force. Developing some methods to fill in the flaw is also important for perfecting the theory of rigid-body dynamics. The properties of the tangential impact relating to the inconsistent case of Painlevé paradox have been discovered in this paper, and a jump rule for determining the post-states after the tangential impact finishes is developed. Finally, the comprehensively numerical simulation for the two-link robotic manipulator is carried out, and its dynamical behaviors such as stick-slip, the bouncing motion due to the tangential impact at contact point or the external forces, are exhibited.     

Nonlinear dynamic analysis of a structure with a friction-based seismic base isolation system

Many dynamical systems are subject to some form of non-smooth or discontinuous nonlinearity. One eminent example of such a nonlinearity is friction. This is caused by the fact that friction always opposes the direction of movement, thus changing sign when the sliding velocity changes sign. In this paper, a structure with friction-based seismic base isolation is regarded. Seismic base isolation can be employed to decouple a superstructure from the potentially hazardous surrounding ground motion. As a result, the seismic resistance of the superstructure can be improved. In this case study, the base isolation system is composed of linear laminated rubber bearings and viscous dampers and nonlinear friction elements. The nonlinear dynamic modelling of the base-isolated structure with the aid of constraint equations, is elaborated. Furthermore, the influence of the dynamic characteristics of the superstructure and the nonlinear modelling of the isolation system, on the total system’s dynamic response, is examined. Hereto, the effects of various modelling approaches are considered. Furthermore, the dynamic performance of the system is studied in both nonlinear transient and steady-state analyses. It is shown that, next to (and in correlation with) transient analyses, steady-state analyses can provide valuable insight in the discontinuous dynamic behaviour of the system. This case study illustrates the importance and development of nonlinear modelling and nonlinear analysis tools for non-smooth dynamical systems.     

Nonlinear vibration analysis by an extended averaged equation approach

The paper presents an extended averaged equation approach to the investigation of nonlinear vibration problems. The proposed method is applied to some free and self-excited oscillators, the Duffing's forced oscillators including main resonance, subharmonic resonance and super harmonic resonance. The results in analyzing the vibration systems with arbitrary non-linearity show advantages of the method.