Friction-induced vibration of an elastic disc and a moving slider with separation and reattachment

The transverse vibration of an elastic disc, excited by a preloaded mass-damperspring slider dragged around on the disc surface at a constant rotating speed and undergoing in-plane stick-slip oscillation due to friction, is studied. As the vertical vibration of the slider grows at certain conditions, it can separate from the disc and then reattach to the disc. Numerical simulation results show that separation and reattachment between the slider and the disc could occur in a low speed range well below the critical disc speed in the context of a rotating load. Rich nonlinear dynamic behaviour is discovered. Time-frequency analysis reveals the time varying properties of this system and the contributions of separation and in-plane stick-slip vibration to the system frequencies. One major finding is that ignoring separation, as is usually done, often leads to very different dynamic behaviour and possibly misleading results.


Introduction
Elastic discs are key components in a wide range of mechanical applications such as rotors and stators in some engines, brakes and clutches, computer hard disc drives, and saws. During the operation of these mechanical devices, dry friction plays an important role on the dynamic behaviour. Generally speaking, dry friction dissipates energy and thus reduces vibration, but it can also sustain selfexcited oscillation and even cause vibration to grow under certain conditions. For example, brake squeal is a well-known friction-induced vibration phenomenon in car brakes. The annoying noise can cause customers to doubt the quality of their automobiles. Friction-induced vibration has been generally accepted as the main reason for brake squeal [1][2][3][4]. Another consequence of friction-induced disc vibration is data losses of a computer hard disc drive because of its undesirable vibration.
Several physical mechanisms that attempt to explain unstable friction-induced vibration have been proposed in the literature and were reviewed in [5]: the negative friction slope [6], sprag-slip instability [7] , mode-coupling instability [8], and stick-slip instability [9]. However, there has been no universal acceptance of an explanation for brake squeal [10] and the dynamic behaviour of frictioninduced vibration is not fully understood.
Stick-slip vibration occurs when the static friction coefficient is greater than the kinetic friction coefficient [9]. Numerous investigations have focused on dry friction-induced stick-slip instability [11][12][13][14]. Popp and Stelter [9] studied the chaotic behaviour of several simple systems, which provided an insight into stickslip instability. In [15], the critical speed for the initiation of stick-slip oscillation from pure sliding oscillation was derived by an analytical method. The results indicated that stick-slip motion took place in a wide speed range of the moving belt. Kinkaid et al. [16] examined the dynamics behaviour of a four-degree-offreedom model with friction force in two orthogonal directions at the contact interface. Since the friction in [16] followed the Amontons-Coulomb's law of friction, a new mechanism due to the combination of the stick-slip instability in both directions was presented. Stelter [17] investigated the nonlinear stick-slip behaviour of a cantilever beam excited by dry friction via numerical analyses and experiments. In [18], the influences of the non-smooth Coulomb's law of friction on the stability of the self-excited vibration of a one-degree-of-freedom model with negative damping were studied. Pascal [19] explored the sticking and nonsticking orbits of a two-degree-of-freedom model with dry friction under harmonic excitation. Feeny et al. [20] presented a very interesting review of stickslip vibration.
Research on the vibration of an elastic disc excited by a rotating slider system has been reported in [21][22][23][24][25]. Mottershead [21] reviewed vibration of stationary and rotating discs under various loads, including friction. In [26], parametric resonances in a disc with a rotating mass-spring-damper system were studied in the subcritical speed range, in which friction force was treated as a follower force.
Ouyang et al. [23] examined the transverse instability of an elastic disc under the action of a rotating friction slider with stick-slip vibration. The influence of system parameters on the disc's transverse vibration and the slider's horizontal stick-slip vibration was investigated through numerical simulations. In a later paper [27], a model consisting of an elastic disc with two rotating oscillators acting on each side of the disc was developed. In that model, a bending couple was produced by the unbalanced friction forces on the lower and upper surfaces of the disc. The instability of the disc due to the friction couple was studied. The rotating speed of the mass-spring-damper slider system studied in these papers is in the subcritical range.
Spelsberg-Korspeter et al. [29] proposed a new model containing a rotating Kirchhoff plate and an idealised elastic pad, which was in friction contact with the rotating plate. In that paper, both the in-plane and bending vibration of the rotating plate due to distributed friction forces were investigated. In [32], the wave pattern and the limit cycle of the stick-slip motion of a rotating disc, which was in the frictional contact with a pad under uniform pressure, in a simplified brake system were analysed.
Loss of contact at the friction interface of the disc has been neglected in most of the studies mentioned above. Sinou [33] investigated the transient and stationary dynamics of a nonlinear automotive disc brake model due to friction. He showed that more unstable modes took part in the transient vibration because of the nonlinearity and loss of contact at the friction surface. However, the specific roles of separation and its importance to the friction-induced vibration have not been studied. The main purpose of the current paper is to investigate the frictioninduced transverse vibration of a disc subjected to a rotating slider undergoing vertical vibration and in-plane stick-slip vibration.
In the present paper, a model containing an elastic disc in friction contact with a rotating oscillator is developed. Stick-slip motion of the slider takes place on the disc surface due to friction governed by the Coulomb's law of friction, which leads to the coupling between the transverse vibration of the disc and the horizontal (in-plane) vibration of the rotating slider. Theoretical formulations of the system in stick and slip states are derived, and the conditions for staying in individual motion states are discussed in Section 2. In Section 3, the conditions and equations of motion for separation and reattachment are given, meanwhile impact at the instant of reattachment is considered. In Section 4, dynamic behaviour of the model is analysed and numerical results show that separation can happen during unstable vibration at a low rotating speed level. Moreover, comparisons between the dynamic behaviour of the disc considering and ignoring separation indicate the importance of considering separation. Then, the effects of key system parameters on the friction-induced vibration of the system are examined via a numerical parametric analysis. Finally, the evolutions of the frequencies of the system with time are studied through the short time Fourier transform that reveals the time varying nature of the whole system due to the transverse separation-reattachment and in-plane stick-slip events. Fig. 1 presents the mechanical model studied in this paper. The system contains an elastic annular disc, which is clamped at its inner radius a and free at its outer radius b, and a slider system in friction contact with the disc. The annular disc is a Kirchhoff plate which exhibits only transverse motion. The mass (slider) has a vertical branch and a horizontal (in-plane) branch, each having a spring and a damper in parallel. A vertical displacement Δ is applied on the top of the vertical branch and is kept constant throughout the subsequent vibration, so that a vertical pre-load is generated and is always present. The horizontal branch is connected with a drive point that moves around on the surface of the elastic annular plate at a constant rotating speed Ω. In this paper, the Coulomb's law of friction is used with a static friction coefficient s and kinetic friction coefficient k . The slider is capable of stick-slip oscillation in the horizontal direction. Such a system was studied in [22] in which loss of contact and subsequent reattachment were excluded.

In-plane stick-slip motion of the slider
As the friction coefficient s is assumed to be greater than k in this work, the slider can undergo stick-slip oscillation in the horizontal direction. When the slider is sliding, its in-plane equation of motion is expressed in Eq. (1): in which denotes the circumferential angular position of the slider relative to the drive point and is the absolute circumferential angular position of the slider, r 0 is the radial position of the slider; p is the in-plane damping coefficient of the slider, p is the in-plane stiffness of the slider, and P is the (total) contact force between the disc and the slider.
The sliding motion can be maintained if the following conditions are satisfied: The relation between the relative motion represented by and absolute motion is: where is time.
Otherwise, the slider sticks to the plate. In this motion phase, the slider's absolute circumferential velocity ̇ and its acceleration ̈ equal to zero, and its circumferential position is referred to as stick . The relative motion of the slider is given by Eq. (5): The condition for the slider staying in the stick phase is:

Transverse vibration of the disc
The equation of transverse motion of the disc under the action of the moving slider is given by Eq. (7): where w denotes the transverse displacement of the plate, and  are the radial and circumferential coordinates in the cylindrical coordinate system respectively; D* is the damping coefficient of the disc, ρ is the mass-density of the disc, is the flexural rigidity of the disc; and δ(•) is the Dirac delta function.
P can be obtained from the equation of vertical motion of slider m: where and ̇ are the vertical motion and vertical velocity of the slider, 0 is the initial vertical displacement of the slider, c is the damping coefficient and k is the stiffness of the vertical branch of the slider, N is the pre-load as a result of the vertical displacement Δ applied.
In this paper, contact force P is defined as positive when there is contact (so that P is a compressive force). Thus the condition for maintaining contact is: If there is contact between the slider and the plate, the relationship between the transverse displacement of the plate and the vertical displacement of the slider is [35]: and therefore where w 0 is initial transverse displacement of the disc as a result of applying  to the vertical branch of the slider.
Although Eq. (13) is applicable to both stick and slip motion states, as ̇ and ̈ are zero in the stick phase, Eq. (13) is simplified to Eq. (14) which represents the equation of motion when the slider sticks to the disc.

Coupled equations of motion of the whole system in modal coordinates
The transverse displacement of the disc can be expressed as an infinite series in modal coordinates: (15) where ( , ) is the mode shape of the plate given by Eq. (16): (16) in which The ortho-normality conditions of modal functions are: 17) in which ̅ is the complex conjugate of .
Then by multiplying ̅ on both sides of Eqs. (13) and (14), then integrating them over the whole disc surface, and by using the ortho-normality conditions shown in Eq. (17), Eqs. (13) and (14) are rewritten in terms of modal coordinates ( ) shown below.
In the stick phase, the equation of transverse motion of the disc in terms of modal coordinates is expressed as: (18) in which = stick (19) and the relative motion of the slider in the stick phase has been given by Eq. (6).
The condition for remaining in stick phase given by Eq. (6) is transformed into Eq. (20): During the sliding motion, the equations of transverse motion of the disc and the equation of horizontal motion of the slider are given by Eqs. (21) and (22): and And because of the axial symmetry of the annular disc, the relationships in Eq.
(23) are satisfied [25]: The conditions for staying in the slip phase (Eqs. (2) and (3)) can be expressed in modal coordinates as:

Separation and reattachment
In this paper, separation takes place when contact force P(t) drops to zero. During the numerical calculations, it is important to monitor P(t) at each time step, because if separation happens, a new set of equations of motion of the slider and disc needs to be used. When P(t) becomes negative, the bisection method is used to find the critical point at which P(t) satisfying |P(t)| ≤ ε, in which ε is a small tolerance defined in the Matlab codes. During separation, the contact force is zero and the sliding friction force vanishes.
The transverse motion of the disc and the vertical motion of the slider during separation are governed by Eqs. (25) and (26) respectively: The new equation of horizontal motion of the slider is expressed in Eq. (27): Separation can be maintained when Eq. (28) is satisfied: After separation, the slider may get into contact with the disc again. Reattachment which was presented in [36] for a moving-mass-on-beam problem, is derived for the present problems below.
For simplification, a simple perfectly plastic impact is assumed, and slider sticks onto the disc after the impact. Thus the slider takes the displacement and the velocity of the disc at time r + . Suppose the impulse at r is p, the equation of motion of the disc is: By using the same modal expansion process described in Section 2.3, Eq. (29) can be converted to Eq. (30) in modal coordinates: The velocity jump as a result of the impact can be solved from Eq. (30) and given by Eq. (31): Similarly, the velocity jump of the slider can be acquired: The combination of Eq. (31) and (32) gives: For perfectly plastic impact, the slider takes the displacement and the velocity of the disc at time r + . ̇( r + ) can be expressed as Eq. (34): Because the transverse displacement is continuous and in-plane motion of the slider does not change by the vertical impact, one gets: By substituting Eq. (34) into (33), and combining Eq. (35), then modal velocity ̇ and vertical velocity ̇ at time r + are derived as:

Numerical study
As the state of the system switches between stick and slip phases, and between separation and contact phases, the dynamic behaviour of the system needs to be obtained by solving three different sets of governing equations, which brings about some difficulties in the numerical computations. In this paper, Runge-Kutta method appropriate for the second-order differential equations [37] is used to solve this non-smooth dynamic problem. The states of the disc and the slider during vibration, including the contact force, the absolute circumferential speed of the slider and the force in the horizontal spring and damper, are monitored at each time step. If the results at the end of a time step do not satisfy the conditions for the system to stay in the same motion phase as at the start of the time step, then the bisection method is used to find the critical point where the dynamics switches from one phase to another phase. After getting the critical point, the current set of equations of motion changes to another set. Rich dynamic behaviour, some of which has not been seen in the literature, is found. Due to the limited space, however, only some distinct and interesting results are presented in this paper. The basic parameter values used in the numerical examples are listed in Table 1.

Separation during vibration
Firstly, the occurrence of separation is illustrated by a numerical example. The time response of the contact force and transverse vibration of the disc are shown in Figs. 2-3. In this example, the rotating speed of the driving point is Ω=20 rad/s, and the pre-load is N=200N. A long time calculation is run. Fig. 2 (a) shows the time response of the contact force during the entire calculation time. Although details of the vibration cannot be observed easily from Fig. 2 (a), it can be observed that the oscillating range of the contact force grows, and the contact force can drop to zero, which means that separation can occur during the vibration. Then for a clearer observation, the zoom-in view of Fig. 2 (a) within a short time interval is given in Fig. 2 (b). It shows that when the contact force decreases to zero, separation takes place, then contact force remains zero during separation. Moreover, multiple separation events can happen. The results of transverse vibration of the disc and the vertical vibration of the slider during one full event of the separation and reattachment process are shown in Fig. 3. As shown in Fig. 3, separation happens while the disc moves upward, therefore the growing vibration of the disc is bounded due to loss of contact. It shows that the duration of the separation is very short, which can be explained as follows: the pre-load acts on the slider all the time even during separation, so the slider quickly gets into contact with the disc again under these parameter values.  for this can be explained. Because of separation, the disc cannot get further excitation from the slider (note that the rotating slider is the source of excitation), unlike the cases when contact is assumed to be always maintained even though the contact force has dropped to a negative value. Therefore, separation serves to contain the vibration in a smaller range of magnitude.
(a)Separation is ignored (b)Separation is considered  Ω=15 rad/s and Ω=15.1 rad/s. Fig.11 clearly indicates that separation does not happen at Ω=15 rad/s, and the disc vibration does not grow and only oscillates in a small constant range. However, at a slightly higher rotating speed of Ω=15.1 rad/s, the oscillation range of the contact force grows, as shown in Fig. 13 (a), and then several separation events take place. In this case, the disc vibrates in a larger range in Fig. 13 (b). The Poincare maps of these two cases shown in Figs. 12 and 14 indicate that the dynamic behaviour of the system can be very different when the system becomes unstable. This rotating speed is referred to as the critical speed for separation.  In order to study the critical speed range of this system, numerical calculations for various values of initial pre-load and rotating speed are carried out. Fig. 15 shows the changes of critical rotating speed Ω c for the occurrence of separation with preload N. When the rotating speed is smaller than the critical speed, the contact is always maintained during vibration. Otherwise, when the rotating speed is greater than Ω c , the slider can lose contact with the disc along with growing vibration. It can be seen that the critical speed for the loss of contact of this system can be low, which is much lower than the conventional critical speed (defined as the speed value of a rotating constant load which causes the resonance of the disc).

The critical speed for separation
Moreover, with the initial increase of pre-load N, the system becomes unstable and separation occurs at a lower rotating speed; from a certain value of N, with further increase of N, the system becomes unstable and separation takes place at a higher rotating speed. The other extreme situation is: when N is extremely large, the slider can hardly move, which means that the slider sticks to the disc within the time duration of observation and the system is also stable. Between the two extreme situations, horizontal stick-slip motion appears and is affected by the value of the normal force N; as the horizontal motion of the slider is coupled with the vertical motion of the slider and the transverse motion of the disc, the whole system dynamics is affected by the normal force in a complicated way.   The influences of the vertical damping coefficient are also studied. When there is vertical damping, as shown in Fig. 20, the slider's in-plane motion is a periodic stable stick-slip motion; and the vibration of the disc is also stable and it oscillates within a small range around its static equilibrium position. Therefore, vertical damping coefficient appears as a stabilising factor to the system.  When k becomes smaller, the system becomes unstable sooner and separation takes place more easily. The system becomes stable when k is large enough (i.e. The role of the in-plane stiffness of the slider on the vibration of the system is complex. When k p =2×10 3 N/m, the vibration of the disc initially vibrates quasiperiodically. However, after separation occurs, the points, shown by green dots, on the Poincare section of the disc wander within a certain range and become unpredictable, shown in Fig. 22. When k p =2×10 4 N/m, the unstable vibration grows faster and separation takes place earlier. However, a large enough k p (2×10 5 N/m) then appears to stabilise the system. Finally, the value of the slider's mass is found to affect the separation location in the vertical direction. In all the results shown above, separation happens while the disc is moving upward. In Fig. 23, however, the position of separation is changed if the mass is small (m=0.01). In this example, separation happens when the mass reaches its lowest vertical position. This information is not available from Poincare maps and can only be obtained from the time response of vibration. destabilising. However, when the normal pre-load is large enough, the in-plane damping then appears as a destabilising factor to the system.

4.4Nonstationary Dynamic Behaviour
As the system actually experiences distinct motion states during vibration, the vibration frequencies in these motion states can be different, and thus the system is nonstationary and FFT analysis is no longer suitable. In this sub-section, timefrequency analysis through the short time Fourier transform is carried out to explore evolution of the vibration frequency of the system studied in this paper.
In   Table 2. Among these frequencies, f Ω is the predominant frequency, which comes from the rotating driving point, and its superharmonic components 2f Ω and 3f Ω also take part in the vibration. Additionally, frequencies f h1 and f h2 are associated with the inplane vibration of the slider whose frequency is 70Hz and splits into the two frequencies due to the rotation of the slider. f 1 to f 6 in Fig. 25 and in Table 2 are close to but not the same as some natural frequencies of the static system (135.42, 237.55 and 288.85 Hz). This is due to the effect of the in-plane rotation of the slider.  The dynamic response of the third example is shown in Fig. 29. The parameter values used are: E=100 GPa, N=200 N, Ω=11 rad/s. In this case, there is no separation during the vibration which has been illustrated in Fig. 21. Although the vibration magnitude of the disc, in Fig. 29 (a), is bounded due to the nonlinearity of the in-plane stick-slip vibration, how the limit cycle of the vibration evolving to is different from those cases in which the transverse disc vibration is non-smooth because of repeated events of separation and reattachment. Consequently, the time-frequency response in this case does not show any high frequency arising above the maximum natural frequency (4383.04 rad/s) of the system with slider being stationary, during steady-state vibration, after the transient phase of vibration (marked by t in Fig. 29(a)). The vibration of the disc in this case is quite erratic as its frequency spectrum shows several prominent incommensurate frequencies and many low-amplitude frequencies emerge, vanish or shift with time.

Conclusions
In this paper, the dynamic behaviour of a disc modelled as a thin elastic annular plate excited by a rotating oscillator which has a vertical branch normal to the disc and a horizontal branch in the plane of the disc is studied. Because of the nonsmooth nature of friction between the slider and the disc, the slider undergoes stick-slip vibration in the circumferential direction on the disc. The variable inplane location of the slider leads to a varying contact force at the interface between the disc and the slider, which affects the transverse vibration of the disc, and makes the in-plane stick-slip vibration and vertical vibration of the slider system coupled and complicated. During vibration the slider can lose contact (separation) with the disc and then reattach to the disc again.
The equations of motion of this discrete-continuous system at three motion states (stick motion, slip motion and separation) are derived. The conditions for staying in each state are established, and impact at the moment of the reattachment is formulated. Then, numerical study is carried out at various values of the key parameters. The following conclusions can be drawn: (1) Separation can happen during the unstable vibration of the system caused by friction. The time duration of separation is very short. Reattachment naturally occurs following separation.
(2) The system become unstable and separation occurs in low speed range of the driving point, which is much smaller than the critical speed of the disc in the corresponding moving load problem. The most important conclusion of this paper is that separation should be taken into account in many friction-induced vibration problems.   A1 shows that the horizontal motion of the slider lies on a regular stick-slip limit cycle, during the steady state, when separation is ignored, which is periodic vibration. However, the actual horizontal vibration, when separation is considered, is quasi-periodic, as the regular stick-slip limit cycle breaks out, and an intricate phase portrait can be observed in Fig. A2.