Subcritical bifurcation in a self-excited single-degree-of-freedom system with velocity weakening–strengthening friction law: analytical results and comparison with experiments
Nonlinear Dyn
Subcritical bifurcation in a self-excited single-degree-of-freedom system with velocity weakening-strengthening friction law: analytical results and comparison with experiments
A. Papangelo 0 1 2
M. Ciavarella 0 1 2
N. Hoffmann 0 1 2
0 N. Hoffmann Imperial College London , Exhibition Road, London SW7 2AZ , UK
1 M. Ciavarella Department of Mechanical Engineering, Center of Excellence in Computational Mechanics, Politecnico di BARI , Viale Gentile 182, 70126 Bari , Italy
2 A. Papangelo (
The dynamical behavior of a single-degreeof-freedom system that experiences friction-induced vibrations is studied with particular interest on the possibility of the so-called hard effect of a subcritical Hopf bifurcation, using a velocity weakening-strengthening friction law. The bifurcation diagram of the system is numerically evaluated using as bifurcation parameter the velocity of the belt. Analytical results are provided using standard linear stability analysis and nonlinear stability analysis to large perturbations. The former permits to identify the lowest belt velocity (vlw) at which the full sliding solution is stable, the latter allows to estimate a priori the highest belt velocity at which large amplitude stick-slip vibrations exist. Together the two boundaries [vlw, vup] define the range where two equilibrium solutions coexist, i.e., a stable full sliding solution and a stable stick-slip limit cycle. The model is used to fit recent experimental observations.
Mass-on-moving-belt model; Exponential decaying; Weakening-strengthening friction law; Bistable equilibrium; Subcritical bifurcation
1 Introduction
Subcritical as well as supercritical Hopf bifurcations
are often encountered in different engineering
applications, e.g., aeroelastic response of airfoils with
structural nonlinearities [
1, 2
], dynamics of ball joints [3],
brake squeal [
4
]. Engineers are generally more
concerned about subcritical (hard) bifurcations as a small
perturbation around the equilibrium position can lead
the system to large amplitude vibration states, which
the structure may not tolerate [
5
]. A number of authors
have studied the “Mass-on-moving-Belt” model (“MB
model” in the following), Tondl [
6
], Hetzler et al. [
7
],
Hetzler [
8
], Won and Chung [
9
], Nayfeh and Mook
[
10
], Mitropolskii and Van Dao [
11
], Popp [
12
], Popp
et al. [
13
], Hinrichs et al. [
14
], Andreaus and Casini
[
15
], Awrejcewicz and Holicke [
16
], Awrejcewicz et
al. [
17
], which present various types of analysis of
a mass-on-belt system with various kinds of friction
laws, and provide in some cases, analytical
expressions for the change between stick–slip and pure-slip
oscillations. Many authors have attempted to use fast
vibrations which in some respects seems to transform
classical Coulomb friction into viscous-like
damping ([
5, 18
]). Most often, supercritical bifurcations are
found, namely where the system undergoes a smooth
transition to a limit cycle (generally involving stick–
slip) when the control parameter is varied.
In [
19
] Hoffmann studied the effect of LuGre type
friction law [
20
] on the stability of the classical MB
model. It was shown that rate-dependent effects act
against the destabilizing effect of the velocity
decaying friction characteristic. The reader is referred to the
review by Awrejcewicz and Olejnik [
21
] where the
dynamical behavior of different lumped mechanical
systems (see also [
22
]) with various friction laws has
been investigated.
Hetzler et al. [
7
] (see also [
23
]) studied the dynamic
behavior of the MB model using different friction
characteristics, (exponential and polynomial
decaying). They assumed a weakly nonlinear behavior and
used a first-order averaging method to find approximate
solutions. It was shown that the exponential
decaying leads to subcritical Hopf bifurcation while, using a
cubic polynomial friction law, the dynamical behavior
(subcritical/supercritical) depends on the friction law
parameters [
7
].
Also in [
8
] Hetzler showed that adding a Coulomb
frictional damping to the self-excited MB model leads
to an “imperfect” Hopf bifurcation scenario where it
does not make sense to ask for stability of the steady
state but rather one should seek for stability to a certain
level of perturbation.
Recently, Papangelo et al. [
24
] have found
localized vibration states in a self-excited chain of
mechanical oscillators weakly elastically coupled, which lead
to the so-called snaking bifurcations in the bifurcation
diagram. A key feature of the system was that, if
isolated from the structure, each nonlinear oscillator
experiences a subcritical Hopf bifurcation in a certain range
of the control parameter (yielding bistability1).
However, Papangelo et al. [
24
], adopted a polynomial
nonlinearity quite remote from a real friction law. Here,
perhaps with an eye to the classical Stribeck curve, for
the MB model we propose an exponen (...truncated)