Instability of systems with a frictional point contact. Part 2: model extensions (original) (raw)
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Instability of systems with a frictional point contact. Part 1: basic modelling
Journal of Sound and Vibration, 2004
In a companion paper, a theory was presented which allows the study of the linear stability of a class of systems consisting of two subsystems coupled through a frictional contact point. A stability criterion in terms of transfer functions was derived and used to simulate the behaviour of generic systems. In the present paper, this approach was pursued and generalized by relaxing in turn certain of the assumptions made earlier. By doing this, it is possible to catalogue systematically all the routes to instability conceivable within the scope of linearity for the class of systems considered. The additional routes to instability identified are as follows. First, the contact point was made compliant by adding a linear contact spring at the interface between the two subsystems. This feature proved to have a significant influence on stability when the contact spring stiffness takes values of the same order of magnitude or lower than that of the average structural stiffness of the system. Second, a route to instability is possible if the system structural damping possesses a slight non-proportional component. The last and most elaborate extension consisted in allowing the coefficient of friction to vary linearly with the sliding speed. Simulation results suggest that a coefficient falling with increasing sliding speed can destabilize an otherwise stable system or can make it even more unstable. In accordance with previous results, a coefficient of friction rising with the sliding speed tends to make a system more stable, although this is not systematic. The theory presented here allows these possible routes to instability to be combined, so that data from vibration measurements or modelling and from frictional measurements can be used directly to predict the region of instability in parameter space. r
Instability of systems with a frictional point contact—Part 3: Experimental tests
Journal of Sound and Vibration, 2007
In two earlier papers, a new formalism was derived which led to the prediction of instability for two linear systems in sliding contact at a single point. In Duffour and Woodhouse [Instability of systems with a sliding point contact. Part 1: basic modelling 271 (2004) 365-390], predictions were obtained using a friction law featuring a constant coefficient of friction. This formalism was generalised in Duffour and Woodhouse [Instability of systems with a sliding point contact. Part 2: model extensions 271 (2004) 391-410] to include all possible linear routes to instability. This paper presents results from the experimental investigation carried out to test the validity of the theoretical results obtained in the first of the two papers. The rig is of the pin-on-disc type. An instrumented pin was specially designed so that the quantities necessary for the prediction could be measured. It emerged that incipient squeal frequencies observed experimentally could be predicted in 75% of the cases using the simplest formalism presented in Duffour and Woodhouse (2004). The presence of unpredicted squeal frequencies points towards the importance of other effects, such as the disc nominal rotation speed and the value of the normal preload. This study also reveals that the ever-changing nature of friction-induced noises can, to a good extent, be explained by the slight structural variations undergone by any mechanical system.
Extension of the mode coupling instability to a single deformable body with contact.Energy flows analysis during frictional dynamic instability.Identification of dissipative terms: contact dissipation and material damping dissipation.Key role of the material damping for a reliable numerical results.Effects of friction coefficient and rotational speed on the transient response.Mechanical systems present several contact surfaces between deformable bodies. The contact interface can be either static (joints) or in sliding (active interfaces). The sliding interfaces can have several roles and according to their application they can be developed either for maximizing the friction coefficient and the energy dissipation (e.g. brakes) or rather to allow the relative displacement at joints with a maximum efficiency. In both cases the coupling between system and local contact dynamics can bring to system dynamics instabilities (e.g. brake squeal or squeaking of hip prostheses). This results in unstable vibrations of the system, induced by the oscillation of the contact forces.In the literature, a large number of works deal with such kind of instabilities and are mainly focused on applied problems such as brake squeal noise. This paper shows a more general numerical analysis of a simple system constituted by two bodies in sliding contact: a rigid cylinder rotating inside a deformable one. The parametrical Complex Eigenvalue Analysis and the transient numerical simulations show how the friction forces can give rise to in-plane dynamic instabilities due to the interaction between two system modes, even for such a simple system characterized by one deformable body. Results from transient simulations highlight the key role of realistic values of the material damping to have convergence of the model and, consequently, reliable physical results. To this aim an experimental estimation of the material damping has been carried out. Moreover, the simplicity of the system allows for a deeper analysis of the contact instability and a balance of the energy flux among friction, system vibrations and damping. The numerical results have been validated by comparison with experimental ones, obtained by a specific test bench developed to reproduce and analyze the contact friction instabilities.
Linear stability of steady sliding in point contacts with velocity dependent and LuGre type friction
Journal of Sound and Vibration, 2007
In the present work the influence of a LuGre type friction law [cf. to C. Canudas de Wit, H. Olsson, K.J. Astro¨m, P. Lischinsky, A new model for control of systems with friction, IEEE Transactions on Automatic Control 40 (1995) 419] on the fundamental mechanisms resulting in linear instability of steady sliding in point contacts is investigated. Both a velocity-dependent kinetic friction coefficient as well as mode-coupling are considered. It turns out that the destabilizing effect of a kinetic friction coefficient decreasing with relative sliding velocity reduces when the rate-dependent effects of LuGre type friction become marked. Mode-coupling instability however seems to remain largely unaffected.
Instability Analysis of Coupled Friction Oscillators With Uncertainties in Contact Conditions
Although brake squeal is a significant noise, vibration and harshness (NVH) issue which incurs significant cost in the automotive industry, its prediction is still difficult. This is because brake squeal is essentially a nonlinear phenomenon and traditional complex eigenvalue analysis (CEA) is a linear method. In addition, there are many uncertainties in a brake system such as material properties, operating and contact conditions which cannot be determined accurately with confidence. Here, the influence of uncertainties in contact conditions on the instability of an analytical model consisting of 3x3 coupled oscillators in point contact with a sliding rigid plate is analysed. The uncertainties in contact conditions considered are: percentage of contact, stiffness and friction laws for the contact (Amonton-Coulomb, relative velocity dependent and LuGre law). The instability is analysed in the frequency domain by randomising these three uncertainty parameters. The results will be disc...
Analytical Approaches for Friction Induced Vibration and Stability Analysis
2011
The traditional mass on a moving belt model without external force excitation is considered. The displacement and velocity amplitudes and the period of the friction induced vibrations can be predicted using a friction force modelled by the mean of friction characteristics. A more precise look at the non-smooth transition points of the trajectories reveals that an extended friction model is looked-for. In present job, two so-called polynomial and exponential friction functions are investigated. Both of these friction laws describe a friction force that first drops off and then raises with relative interface velocity. An analytical approximation is applied in order to derive relations for the vibration amplitudes and base frequency and in parallel a stability analysis is performed. Moreover, results and phase plots are illustrated for both analytical and numerical approaches.
Dynamic and energy analysis of frictional contact instabilities on a lumped system
Meccanica, 2014
When dealing with complex mechanical systems, the frictional contact is at the origin of significant changes in their dynamic behavior. The presence of frictional contact can give rise to modecoupling instabilities that produce harmonic friction induced vibrations. Unstable vibrations can reach large amplitude that could compromise the structural and surface integrity of the system and are often associated with annoying noise emission. The study of this kind of dynamic instability has been the subject of many studies ranging from both theoretical and numerical analysis of simple lumped models to numerical and experimental investigation on real mechanical systems, such as automotive brakes, typically affected by such issue. In this paper the numerical analysis of a lumped system constituted by several degrees of freedom in frictional contact with a slider is presented, where the introduction of friction can give rise to an unstable dynamic behavior. Two different approaches are used to investigate the effects of friction forces. The first approach, the Complex Eigenvalues Analysis, allows for calculating the complex eigenvalues of the linear system that can be characterized by a positive real part (i.e. negative modal damping). The complex eigenvalues and eigenvectors of the system are investigated with respect to friction. In the second approach a non linear model has been developed accounting for the stickslip-detachment behavior at the interface to solve the time history solution and analyze the unstable vibration. The effects of boundary conditions and of system parameters are investigated. Results comparison between the two different approaches highlights how nonlinearities affect the time history solution. The lumped model allows for a detailed analysis of the energy flows between the boundary and the system during self-excited vibrations, which are at the origin of the selection between the predicted unstable mode. Keywords Frictional contact Á Mode coupling instability Á Unstable induced vibration 1 Introduction Complex mechanical systems are always subjected to vibrations induced by the frictional contacts [15, 16,
Instability scenarios between elastic media under frictional contact
Mechanical Systems and Signal Processing, 2013
This article presents the results of a numerical dynamic analysis of two bodies in sliding contact. The 2D model consists of two finite elastic media separated by a contact interface, governed by classical Coulomb friction law. The aim of this work is to investigate the instability scenarios occurring when friction forces excite the mechanical systems during the relative motion; simulation results show that the coupling between the frictional behaviour at the contact and the global dynamic of the system can bring to either stickslip phenomena, or mode coupling instability. Complex eigenvalue analysis and transient non-linear simulations highlight how system parameters, like structural damping, affect the macroscopic frictional behaviour, switching from stick-slip phenomena to harmonic vibrations (due to mode coupling instability), up to the stable sliding state. The presented results allow for generalizing the instabilities due to mode coupling, named in brake squeal literature "lock-in" instability, to any mechanical system with frictional contact. åThe analyses show how maps of the instability scenarios can be drawn as a function of different parameters to help the design of systems in frictional contact.
European Journal of Mechanics A-solids, 2007
Friction-induced vibrations due to coupling modes can cause severe damage and are recognized as one of the most serious problems in industry. In order to avoid these problems, engineers must find a design to reduce or to eliminate mode coupling instabilities in braking systems. Though many researchers have studied the problem of friction-induced vibrations with experimental, analytical and numerical approaches, the effects of system parameters, and more particularly damping, on changes in stable-unstable regions and limit cycle amplitudes are not yet fully understood.The goal of this study is to propose a simple non-linear two-degree-of-freedom system with friction in order to examine the effects of damping on mode coupling instability. By determining eigenvalues of the linearized system and by obtaining the analytical expressions of the Routh–Hurwitz criterion, we will study the stability of the mechanical system's static solution and the evolution of the Hopf bifurcation point as functions of the structural damping and system parameters. It will be demonstrated that the effects of damping on mode coupling instability must be taken into account to avoid design errors. The results indicate that there exists, in some cases, an optimal structural damping ratio between the stable and unstable modes which decreases the unstable region. We also compare the evolution of the limit cycle amplitudes with structural damping and demonstrate that the stable or unstable dynamic behaviour of the coupled modes are completely dependent on structural damping.
Perturbation Stability of Frictional Sliding With Varying Normal Force
Journal of Vibration and Acoustics, 1996
In many systems, the normal force at friction contacts is not constant, but is instead a function of the system’s state variables. Examples include machine tools, friction dampers, brake systems and robotic contact with the environment. Friction at these contacts has been shown to possess dynamics associated with changes in normal force. In an earlier paper, the authors derived a critical value of system stiffness for stability based on a linearized analysis of constant velocity sliding (Dupont and Bapna, 1994). In this paper, the domain of attraction for the steady sliding equilibrium point is characterized for a system in which normal force is coupled to tangential displacement. Perturbations consisting of sudden changes in the displacement and velocity of the loading point are considered. These perturbations can be viewed as either actuator disturbances or changes in control input. The effect and interaction of the frictional and geometric parameters are elucidated. The results a...