Nonlinear dynamics of system with combined rolling–sliding contact and clearance

Abstract

Rolling–sliding contacts are found in a variety of systems, such as gears, drum brakes, and tire pavements. Such systems inherently have multiple nonlinearities such as kinematic, contact, and friction nonlinearities. Further, most of these systems have clearance between components which causes excessive vibration and noise. The combination of clearance with other nonlinearities makes the system dynamically interesting. It is important to understand the dynamic behavior of such systems under various operating conditions. Hence, this article focuses on theoretically investigating the transient and steady-state responses of a cam–follower system with rolling–sliding contact and clearance as an exemplary case. A contact mechanics-based model for the same has been developed for this purpose. The transient behavior of the system is examined based on energy and time-varying frequency contents. The domain of attractors, frequency response plots, and phase portraits are deployed to analyze the effect of initial conditions and excitation speed on the steady-state behavior, quantitatively and qualitatively. The steady-state solutions were classified into various branches based on periodicity and phase portraits. In addition, parametric analyses of the effects of loading, damping, and friction on the response have been conducted. Finally, the system is examined for start-up and shut-down characteristics with a constant rate of change in excitation frequency.

Appendices

Appendix A: Nomenclature

$\alpha$	Angle made by the follower with $\hat{e}_x$
	(measured counter-clockwise)
$\zeta$	Damping ratio
$\theta$	Angular displacement of cam $\overrightarrow{EG}_c$ with $\hat{e}_x$
	(measured ccw)
$\mu$	Coefficient of friction
$\phi$	Eccentricity of the cam, $|\overrightarrow{EG_c}|$

$\kappa$	Stiffness of torsional spring
$\chi$	Distance of contact point $O_b$ from pivot (P)
	along follower length
$\omega$	Natural frequency (in contact)
$\Omega$	Cam speed
$\psi (i,j)$	Moving coordinate system on the follower at Q
b	Backlash defined between the cam and follower
C	Viscous damping coefficient
E	Pivot point of the cam
F	Force
g	Acceleration due to gravity
G	Geometric center
I	Mass moment of inertia of follower about P
K	Linear contact stiffness
l	Length of the follower
m	Mass of the follower
n	Periodicity of the solutions ($T/T_c$)
O	Contact point
P	Pivot point of the follower
Q	Location of contact point in kinematic zero
	state [21]
r	Radius of the cam
t	Time
T	Time period
w	Width of the follower
Subscripts
0	Initial conditions
a	Alternating
b	Follower
c	Cam
f	Friction
i	Component along $\hat{i}$
j	Component along $\hat{j}$
m	Mean
N	Normal
n	Natural
p	Preload
ss	Steady-state
z	0-to-peak amplitude
Superscript
0	Kinematic Zero state

1. Note: Overbar denotes normalized parameters

Appendix B: System parameters

Table 1 Parameters used to generate the results

Appendix C: Largest Lyapunov exponent

Table 2 Largest Lyapunov exponent at each cam speed

Fig. 21

Effect of change in damping when $F_m=0.1,\hat{F}=2, \mu =0$ ( key - - -o- : n=1 No-impact, -x- : n=1 impact, -*- : n=2, -$\triangle$- : n=3, -$\square$- : n=4, $\diamond$ : outliers )

Appendix D: Effect of damping