Slip-stick motion in harmonic oscillator chains subject to Coulomb friction

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Abstract

Slip-stick motion is an extremely common phenomenon that exists in systems as diverse as tectonic plates, door hinges, and AFM probes. An examination of a simple harmonic oscillator subject to Coulomb friction has uncovered an important non-dimensional number that governs slip-stick motion. In this simple model, the boundary between systems that undergo slip-stick motion and steady sliding has been identified by a critical value of this non-dimensional number as a function of damping ratio. The non-dimensional number was found using a non-dimensionalisation of a simple one degree-of-freedom system, however, it has been found to govern the slip-stick boundaries of more complex systems with any number of degrees-of-freedom.

Introduction

The term slip-stick refers to the motion of objects where self-excited vibrations occur due to successive slipping and sticking at a frictional interface. Slip-stick is present in a huge variety of mechanical systems from micro to geological length scales. The phenomenon has been studied from a variety of perspectives over the past 75 years [1]. The usual ingredients of slip-stick are a component of compliance, and a frictional interaction that decreases with increasing velocity. Slip-stick cycles represent a transfer of energy between the potential energy stored by the compliance and the kinetic energy of the object. The requirement that the frictional coefficient reduces with increasing velocity is strictly not necessary, however, this is only true for certain systems [3].

A simple mechanical system that is useful for studying slip-stick is shown in Fig. 1. In this system, a mass is being pulled along by a constant driving velocity, which acts through a compliant element. This simple system (or something similar) has been studied many times and is the usual starting point for an examination of the slip-stick phenomenon. It is a lumped system that is commonly used to represent a variety of rigid body systems that fall victim to slip-stick motion. In this paper, we will examine slip-stick in this simple system, and build upon these results into more complex systems with multiple degrees-of-freedom. The particular focus of this paper will be on identifying the boundaries between systems that slip-stick and those that do not.

Section snippets

Governing equation

The equation of motion for the system shown in Fig. 1 can be written as Eq. (1)ẍ=ω2(vtx)+2ζω(vx˙)ffwhere

    x(t)

    is the displacement of the mass at time t,

    ω=k/M

    is the natural frequency of the system,

    ζ=(C/2)

    is the damping ratio, and

    ff

    represents the force of friction that acts to oppose the motion of the mass (divided by the mass of the object).

The behaviour of friction during slip-stick is rather complex and has been studied by various authors [1], [2], [4], [5], [6], [7], [8]. In its simplest

The limits of slip-stick

Obviously, not all systems of the type shown in Fig. 1 will show slip-stick motion. Fig. 2 shows two solutions to Eq. (2) for different values of the driving velocity. Clearly, Fig. 2a is a system displaying slip-stick cycles, whereas Fig. 2b decays to the stationary solution of x˙=1. By examining Eq. (2), it is possible to determine the boundaries between systems that slip-stick and those that do not.

The discontinuity in the frictional force presents a serious analytical difficulty. This

Slip-stick in multiple degrees-of-freedom

Building upon the previous discovery, we form a chain of any number (n) of masses connected together by the same springs and dampers and subject to the same driving mechanism (Fig. 4). By examining the dynamic equilibrium of the ith mass in the chain we can form the system of governing Eq. (9) (zero matrix elements are blank):x˜̈=[2112112111](x˜+2ζx˜˙)f˜f+[t+2ζ00]

The frictional forces, ffi, are calculated in a similar way to before:ff1={min(t2x1+x2+2ζ(12x˙1+x˙2),μsgωv)forx˙1=0μkgωvforx˙

Conclusions

Analysis of the slip-stick phenomenon in simple one-dimensional discrete systems has uncovered a very important non-dimensional number that defines the boundary between slip-stick and smooth sliding as a function of the damping ratio. This number originated through an investigation of the motion of a single mass, however, it was found to play an important role in the occurrence of slip-stick in more complex systems with multiple degrees-of-freedom. In multiple degrees-of-freedom, critical

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