Friction induced vibration of a moving slider on an elastic disc with separation and
reattachment
Zilin Li1, 2, H. Ouyang1a, Zhenqun Guan2
1 School
2
of Engineering, University of Liverpool, Liverpool L69 3GH, England, UK
State key Laboratory of Structural Analysis for Industrial Equipment, Department of Engineering Mechanics,
Dalian University of Technology, Dalian 116024, China
a
mehou@liverpool.ac.uk
Abstract. The vertical vibration of an elastic disc excited by a rotating frictional slider pressed on its surface
as well as in-plane vibration of the slider is studied. The vertical mass-spring-damping system (slider) is
compressed on the plate through a normal force applied on its top. In the circumferential direction, the slider
is connected with a driving point through a spring and damper, meanwhile the driving point moves at a constant
speed around the disc modelled as an annular elastic plate. The elastic plate is clamped at the inner radius
and free at the outer radius. Dry friction between the slider and the disc induces in-plane stick-slip vibration of
the slider because the kinetic friction coefficient is smaller than the static friction coefficient. Thus friction affects
vertical vibration of a disc through the changing circumferential location of the moving slider undergoing stickslip oscillation. While the vertical vibration of the plate excited by the moving slider changes the contact force
between them, it affects in-plane vibration of the slider as well, so that the two kinds of vibration are coupled.
In recent research on friction-induced vibration, it is usually assumed that there always is contact between a
slider and a structure underneath during vibration. In this paper, separation and subsequent reattachment
during vibration are considered. Firstly coupled equations of in-plane vibration of the slider and out-of-plane
vibration of the disc are derived based on the modal expansion method, and then conditions for the in-plane
stick-slip vibration and loss of contact and reattachment are worked out. A numerical method is used to
calculate the results during contact and an analytical method is used during separation. Because of the
discontinuity in this problem, the values of the contact force and the conditions for staying in stick or slip state
must be checked during each time step. Results show that separation and reattachment between the slider
and the disc could occur within the low speed range which is much smaller than the critical speed of the disc
in the context of a rotating load. Vibration of the system firstly grows because of in-plane friction, while after
separation it fluctuates within a certain range instead of growing without bound.
Introduction
Dry friction always acts as a resistance to a relative motion; however, under certain conditions, it can cause
self-excited oscillations in engineering as well as in daily life. Various examples include automotive brake
squeal and flutter of computer hard disc drives. Different types of friction-induced vibration have been studied.
Popp [1] studied chaotic behaviour of several models of vibration induced by dry friction through numerical and
experimental analyses to reveal the excitation mechanism of violins. Ouyang [2] analysed the instability of a
classical annular disc with a rotating mass-spring-damper system with dry friction. However, separation and
reattachment have not been considered in this kind of friction induced vibration in the past.
Theoretical Derivations
Fig .1 Annular plate and slider system in cylindrical coordinate system
The system under investigation is shown in Fig. 1 below. The slider system undergoes stick-slip oscillations in
the circumferential (horizontal) direction. When the slider is sticking to the disc, its in-plane position remains
the same and its in-plane speed and acceleration are zero. During the slipping state, the in-plane equation of
motion of the slider system is expressed as Eq.1:
𝑟0 (𝑚𝜓̈ + 𝑐p 𝜓̇ + 𝑘p 𝜓) = 𝜇d sgn(𝜑̇ )𝑃
(1)
in which ψ means circumferential position of the slider relative to the drive point, φ is its absolute circumferential
position, r0 is slider’s radial position, cp and kp are the in-plane damping and stiffness respectively which are
shown in Fig. 1, μd is kinetic friction coefficient, P is the (total) contact force between the disc and the slider.
For transverse vibration of a disc excited by a moving slider, the equation of motion is expressed as Eq.2:
𝜌ℎ
(2)
𝜕2 𝑤
𝜕𝑡 2
1
+ 𝐷 ∗ 𝛻 4 𝑤̇ + 𝐷𝛻 4 𝑤 = − 𝛿(𝑟 − 𝑟0 )𝛿(𝜃 − 𝜑)𝑃
𝑟
where w means the transverse displacement, r and θ are the radial and circumferential coordinates in the
cylindrical coordinate system respectively, D* is damping coefficient of the disc, ρ is mass-density of the disc,
D is flexural rigidity of the disc, δ (●) is the Dirac delta function. P is expressed as Eq.3:
𝑃 = 𝑁 + 𝑚𝑢̈ + 𝑐𝑢̇ + 𝑘(𝑢 − 𝑢0 )
(3)
where u is the vertical motion of the slider which equals to w (r0, φ(t), t) as long as the slider stays in contact
with the disc, and u0 is the initial position of the slider.
When the contact force drops to zero during vibration, separation occurs and then both the slider and plate
undergoes free vibration until they reattach to each other. The free vibration solutions are determined using
an analytic approach.
Examples and numerical results
Fig .2 Contact force (driving speed of 13 rad/s)
Fig .3 Vertical displacement (driving speed of 50 rad/s)
Numerical results indicates that the critical rotating speed for separation is 13 rad/s, well below the critical disc
speed of 608 rad/s. Fig.2 shows that the contact force during vibration becomes zero at times due to growing
oscillation of the system. Fig.3 shows the vertical displacements of the slider and the disc within a tiny time
period. Multiple separation events are found to happen.
Conclusions
In this paper, the vertical vibration of an elastic disc with a moving oscillator compressed on its surface and
the in-plane stick-slip oscillation of the slider considering separation and reattachment is studied. Results show
that loss of contact and reattachment can happen during vibration and the critical rotating speed for separation
is well below the critical speed of the moving load. The system initially becomes unstable due to dry friction,
but finally oscillates, because of separation, within a certain range rather than grows unbounded.
References
[1]
Popp, K. and P. Stelter, Stick-Slip Vibrations and Chaos. Philosophical Transactions of the Royal Society A: Mathematical, Physical
and Engineering Sciences, 1990. 332(1624): p. 89-105.
[2]
H. Ouyang, J.E.M., M.P. Cartmell, D. J. Brookfield, Friction-induced vibration of an elastic slider on a vibrating disc. International
Journal of Mechanical Sciences, 1999. 41: p. 325-336.