Advances in Acoustics and Vibration

Volume 2016, Article ID 9275147, 9 pages

http://dx.doi.org/10.1155/2016/9275147

## Vibrational Interaction of Two Rotors with Friction Coupling

Department of Mechanical Engineering and Energy Processes, Southern Illinois University Carbondale, Carbondale, IL 62901, USA

Received 1 May 2016; Accepted 30 August 2016

Academic Editor: Sven Johansson

Copyright © 2016 H. Larsson and K. Farhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A lumped parameter model is presented for studying the dynamic interaction between two disks in relative rotational motion and in friction contact. The contact elastic and dissipative characteristics are represented by equivalent stiffness and damping coefficient in the axial as well as torsional direction. The formulation accounts for the coupling between the axial and angular motions by viewing the contact normal force a result of axial behavior of the system. The model is used to investigate stick-slip behavior of a two-disk friction system. In this effort the friction coefficient is represented as an exponentially decaying function of relative angular velocity, varying from its static value at zero relative velocity to its kinetic value at very high velocities. This investigation results in the establishment of critical curve defining two-parameter regions: one in which stick-slip occurs and that in which stick-slip does not occur. Moreover, the onset and termination of stick-slip, when it occurs, are related to the highest component frequency in the system. It is found that stick-slip starts at a period nearly equal to that of the highest component frequency and terminates at a period almost three times that of the highest component frequency.

#### 1. Introduction

It has been known that mechanical systems with friction exhibit a phenomenon known as stick-slip. For example, in a brake system, a stator and a rotor in friction contact are in slip state initially at the start of braking. If the intent is to bring the rotor to a complete stop, then a permanent state of stick (zero relative velocity between the rotor and stator) will exist at the end of braking. In addition to the states of slip and stick, a stick-slip transition region may exist which is generally characterized by high frequency oscillations. In the stick-slip transition region, the system displays a series of transient states, that is, slip to stick, and vice versa. With progression of time, the states corresponding to stick gain increasingly larger duration up to the point at which a steady state of stick is achieved.

The existence of stick-slip gives rise to friction forces which are complex in nature and have been known to be responsible for adverse dynamic characteristics. The effects of stick-slip friction are manifested through the development of excessive vibration and noise. Friction-induced vibration is undesirable due to its detrimental effects on the operation and acceptable performance of mechanical systems. Frequent vibration leads to accelerated wear of components, surface damage, fatigue, and noise.

Many aspects of friction-induced vibration are found in the literature. For a comprehensive review of friction the reader may refer to Ibrahim [1, 2], Armstrong-Hélouvry et al. [3–6], Crolla and Lang [7], and Tabor [8]. Specifically, Oden and Martins [9] include a review on frictional contact of metallic surfaces and in a more recent publication by Martins et al. [10] some aspects of low speed frictional sliding phenomena are reviewed.

A simple system which has been used to examine the stick-slip phenomenon is that of a mass sliding on a moving belt [11, 12]. Another typical system is the pin-on-disk apparatus that has been utilized to study dynamic instabilities [13]. Analytical studies of a pin-on-disk apparatus [14] show that the coupling between rotational and normal modes of vibration is the primary mechanism of the resulting self-exited oscillations. Soom and Kim [15, 16] as well as Dweib and D’Souza [17] came to similar conclusions, showing that the force oscillations are primarily associated with the normal and tangential contact vibrations. This coupling effect was considered in the steady sliding point characterization of a simple system [18]. A dynamic stability study [19], considering the Schallamach waves (growing surface oscillations which propagate from front to rear at the friction surface), showed that, for large coefficients of friction and sufficiently large Poisson’s ratio, steady sliding is unstable.

The most widely accepted cause of stick-slip motion is the speed dependence of kinetic friction [3, 4, 20–22]. The velocity term is often included in analytical models [5, 23]. Bengisu and Akay [24] modeled the friction force as an exponential function of the relative sliding velocity. The friction forces in a model of an automotive disk brake are defined in terms of relative sliding speed, pressure, and temperature at the friction interface [25]. Based on experimental data on stick-slip motion, Bo and Pavelescu [26] developed a friction model consisting of two exponential functions of relative speed. Linear stability theory has shown that there is a critical value of the normal load and the average coefficient of friction, above which the steady state sliding motion becomes unstable [27]. Similar results were found by [28] by studying the stability of a simple model where the friction force was assumed to be steady. It has been observed that the amplitude of the stick-slip motion decreases when the driving velocity, damping coefficient, and spring stiffness increase and the mass of the sliding body decreases [9]. Sherif and Kossa [29] measured and modeled the normal and tangential contact stiffness of two rough surfaces in contact. Sherif [30] analyzed the effect of contact stiffness on the instability condition of frictional vibration. Gao et al. [31] focus on the amplitude of stick-slip motion as a function of humidity, speed, and applied load. It is found that the stick-slip amplitude decreases with increasing substrate speed. In lubricated contacts, the cause of stick-slip motion is attributed to local region of instability where shear stress falls with respect to shear rate [32]. A set of lubricated contact experiments are presented by Polycarpou and Soom [33]. The transitions between sliding and sticking are measured and discussed.

In contrast to the speed dependence of the kinetic friction, Adams [34] presents a model for the dynamic interaction of two sliding surfaces, consisting of a series of moving linear springs at the friction interface. The springs are used to account for the elastic properties of asperities on one of the surfaces. The dynamic analysis indicates that the system is dynamically unstable for any finite speed, even though the coefficient of friction is constant. The idea of representing the friction interaction of two bodies using multiple elastic members such as springs or “bristles” is also presented by [35]. The authors assume that the friction between the two surfaces is caused by a large number of bristles, each contributing a small portion to the total load. The model is numerically inefficient and the authors therefore present a refined model.

In this paper a lumped parameter vibration model for a two-disk friction system is presented. A friction coefficient function that expresses the kinetic coefficient as a function of relative velocity is introduced. The interfacial force between the two plates is calculated based on the dynamic behavior of the disks in the axial direction. This is an important feature in that it allows direct coupling between the axial and rotational motions. That is, the rotational motion must be influenced by the actuating axial force and material properties of the friction material (stiffness and damping) in compression. Therefore, in addition to the properties of the friction material in torsional direction (shear), the model accounts for the properties in axial direction and how the combination of the mechanical properties can influence the dynamic behavior of the system. This capability enables simulations to accurately demonstrate effects of the mechanical properties on dynamic and vibration behavior of the friction system.

#### 2. The Two-Disk Model

The interaction between two disks in frictional contact is represented using a lumped parameter model shown in Figures 1–3. The two disks are engaged by the actuating force . The compression of the disks depends on , (mass of the rotor), (equivalent axial stiffness of the disk pair), and (axial damping of the disk pair). The axial behavior of the disks determines the normal interface force between the disk surfaces. Figure 1 shows the schematic of the axial vibration model.