T. Butlin

Sensitivity studies of friction-induced vibration

Friction-induced vibration is notoriously twitchy. This paper examines the origin of the sensitivity, using a model with two linear systems coupled at a single-point sliding contact where a general linearised model for dynamic frictional force is allowed. Sensitivity and convergence studies show that system uncertainty is significant enough to affect the stability of predictions and that modes neglected from the model can sensitively affect predictions. Some key results from a large-scale experimental study are presented. The integration of the uncertainty and sensitivity analysis with data-processing techniques to extract reliable data allows critical evaluation of the modelling details.

Studies of the sensitivity of brake squeal

The problem of ‘brake squeal’ in the automotive industry remains despite over 70 years of research: the phenomenon is still surprisingly unpredictable and poorly understood. The literature has moved from very simple lumped parameter models to ever more sophisticated finite element models, but testing theory against measurements has been hindered by the difficulty in obtaining repeatable results. It would seem the phenomenon is extremely sensitive to changes in parameters beyond an experimenter’s control. This paper describes recent results from a project to identify and quantify the sources of uncertainty within sliding contact systems and to determine the sensitivity of the friction-coupled system to uncertain parameters. The theoretical approach taken is to use a linear analysis based on the uncoupled transfer functions of two general subsystems to predict stability when they are coupled by a sliding point contact. The model is tested using a pin-on-disc rig whose uncoupled transfer functions can be measured. Using a stability criterion based on the roots of the characteristic equation of the system, the sensitivity of model predictions to parameter variations is investigated numerically. It is shown that using a realistic range of parameters the root locations change considerably and enough to change stability predictions. As the complexity of the model is increased reliable results become harder to achieve as the characteristic equation becomes more ill-conditioned. This is not simply a result of the high order of the system, but is thought to be a result of particular mode combinations. Experimental work shows uncoupled transfer functions vary over time and by enough to significantly affect squeal predictions. These results suggest reasons for the difficulty in obtaining repeatable measurements and for the unreliability of squeal prediction theories developed so far. If the reasons for the sensitivity of squeal can be understood it may be possible to design sliding contact systems that are more robust.

Anti-Optimisation Applied to the Analysis of Rotor/Stator Interaction

There is a drive towards minimising operating clearances within turbomachines in order to limit reverse leakage flows and hence improve their efficiency. This increases the likelihood of contact occurring between the blade and the casing, which can give rise to high amplitude vibration. Modelling this in- teraction represents a significant computational challenge. The non-linear contact precludes the use of well-established linear methods, and is also subject to uncertainties: the contact law is imprecisely known and the exact geometry of imperfections that trigger contactmay be unknown. In this paper a novel approach is presented that aims to account for the uncertainties associated with the non-linearity in a non-probabilistic way. The worst case is sought, by fram- ing the systemas a constrained anti-optimisation problem. The target to be maximised represents a metric of the output of in- terest. The degrees of freedom of the anti-optimisation are the non-linear input forces (considered as external loads), and the constraints are designed to capturewhat is thought to be known about the non-linear contact law and geometry. A realistic three-dimensional model of a turbine blade is used to explore the approach, with contact considered at the leading and trailing edge. The blade dynamics are described in terms of a linear transfer function matrix and the target metric of interest is chosen to be the peak displacement of the contact Address all correspondence to this author: tb267@cam.ac.uk points. The non-linearity is taken to result from an offset shaft, giving a sinusoidal clearance variation. The blade is driven at constant frequency and the scope of the study is limited to find- ing bounds on periodic solutions. A variety of constraint condi- tions are explored that describe aspects of the non-linearity. For example, only compressive forces are permitted (no tension from the contact), and the displacement must not exceed the clear- ance. The method yields encouraging initial results: constraints can be identified that give efficient estimates of the upper bound response of the system as a function of drive frequency. The re- sults are compared with a benchmark time-domain simulation and are found to correctly over-predict the response without be- ing overly conservative. Broad trends are also in agreementwith the benchmark solution. The proposed method appears to be a promising approach for efficiently accounting for uncertainties associated with the non-linearity and thus improving blade de- sign. Copyright

An efficient model of drill-string dynamics with localised non-linearities

High amplitude vibration of drill-strings can lead to damage or loss of down-hole equipment costing the industry hundreds of millions of dollars per year. Theoretical models of drill-string dynamics are often limited to lumped parameter models of subsections of the drilling assembly that are difficult to correlate with real systems, or high resolution finite elements models that are more realistic but computationally demanding and do not provide clear insight into the fundamental mechanisms at play. In addition, drill-strings are subject to many uncertainties: to model these using Monte-Carlo simulations can be prohibitively slow. This paper presents a digital filter implementation of the transfer matrix method that allows time-domain coupling of the full length of a periodic drill-string to other systems or arbitrary, localised non-linearities. In this way the dynamics of the complete drill-string are included and the computational effort is focussed on those parts of the system that are harder to model while retaining the efficiency of the linear model where it applies. The efficiency makes the use of Monte-Carlo simulations to explore uncertainties a feasible approach.

The landscape of nonlinear structural dynamics: an introduction

Nonlinear behaviour is ever-present in vibrations and other dynamical motions of engineering structures. Manifestations of nonlinearity include amplitude-dependent natural frequencies, buzz, squeak and rattle, self-excited oscillation and non-repeatability. This article primarily serves as an extended introduction to a theme issue in which such nonlinear phenomena are highlighted through diverse case studies. More ambitiously though, there is another goal. Both the engineering context and the mathematical techniques that can be used to identify, analyse, control or exploit these phenomena in practice are placed in the context of a mind-map, which has been created through expert elicitation. This map, which is available in software through the electronic supplementary material, attempts to provide a practitioner’s guide to what hitherto might seem like a vast and complex research landscape.

What kind of friction model is needed to predict brake squeal

Predictive models of friction-induced vibration have proved elusive despite decades of research. There are many mechanisms that can cause brake squeal; friction coupled systems can be highly sensitive to small perturbations; and the dynamic properties of friction at the contact zone seem to be poorly understood. This paper describes experimental and theoretical work aimed at identifying the key ingredients of a predictive model. A large-scale experiment was carried out to identify squeal initiations using a pin-on-disc test rig: approximately 30,000 squeal initiations were recorded, covering a very wide range of frequencies. The theoretical model allows for completely general linear systems coupled at a single sliding point by friction: squeal is predicted using a linearised stability analysis. Results will be presented that show that almost all observed squeal events can be predicted within this model framework, but that some subsets require innovative friction modelling: predictions are highly dependent on the particular choice of friction model and its associated parameters. Copyright

Does a Predictive Minimal Model of Friction-Induced Vibration Exist?

Highly idealised models of friction-induced vibration have been motivated by an attempt to capture what is essential to the phenomenon. This approach has resulted in a few simple mechanisms that are thought to capture common routes to instability. This paper aims to determine how well these perform as approximations to a more complex system, and whether the essential ingredients needed for a minimal model can be identified. We take a reduced-order model that exemplifies ‘mode-coupling’ and explore the extent to which it can approximate predictions based on an experimentally identified test-system. For the particular test system under study, two-mode ‘mode-coupling’ is rarely a good approximation and three modes are usually required to model a limited frequency range. We then compare predictions with results from an extensive program of sliding contact tests on a pin-on-disc rig in order to identify which ingredients are needed to explain observed squeal events. The results suggest that several minimal models would be needed to describe all observed squeal initiations, but the ‘negative-damping’ route to instability, which requires a velocity-dependent friction law, convincingly accounts for one cluster.Copyright