Passive vibration absorber with dry friction
A. Hartung, H. Schmieg, P. Vielsack
Summary The properties of a passive vibration absorber with dry friction signi®cantly differ
from those of the classical linear absorber. The exceptional phenomenon is the possibility of
suppressing all excited modes. This effect is in¯uenced to a small extent by a special shape of
the friction characteristic, but mainly by an appropriately adjusted threshold of the static
friction. The theoretical predictions are con®rmed by experimental investigations.
Keywords Nonlinear Absorber, Dry Friction, Experiment
1
Introduction
A passive vibration absorber is a mass-spring subsystem coupled to a superstructure to control
its oscillations under the action of periodic excitation. A simple form of this arrangement is
shown in Fig. 1 where M
1
is a mass emulating the superstructure and K
1
is its mounting spring.
The second mass, M
2
, the coupling spring K
2
and a viscous damper d constitute the absorber
system. The superstructure is driven by a harmonic base motion with amplitude A and angular
frequency X.Letx
1
be the displacement of M
1
and x
2
the displacement of M
2
, respectively. So
far, the problem is well known from elementary textbooks on linear vibration theory.
Now, a friction device is added to the substructure which turns the problem into a strongly
nonlinear mechanical system. The law for the friction force R must be de®ned in a way that R is
an active force if the device slides, and a passive one if the device sticks. This gives strict
separation between stick and slip phases during motion.
Classical investigations on motions of mechanical systems with dry friction are mostly based
on deterministic laws which are de®ned by the product of a dynamic friction coef®cient,
depending on the relative velocity at the contact area, and the normal pressure, generally
depending on time, [1]. In the following, the normal force is assumed to be constant during
motion. Then, the dynamic friction force can be reduced to a simple expression
R R
d
sgn
_
x
2
a
_
x
2
: Introducing the threshold value R
s
for the static friction force, three
possibilities will be investigated as plotted in Fig. 2.
The simplest possibility is Coulomb's law (Fig. 2a). Here, R
s
is equal to R
d
, and the dynamic
force R depends only on the direction of sliding and not on the value of the relative velocity
_
x
2
.
In the case of a decreasing characteristic (Fig. 2b), the equality R
s
R
d
still holds, but the
friction force depends linearly on the relative velocity
_
x
2
with a negative slope a < 0. In the
third case (Fig. 2c), the value R
d
of dynamic friction remains constant for
_
x 6 0, but the static
friction coef®cient R
s
is larger than R
d
.
The ®rst question is whether or not different laws lead to signi®cantly different responses
and phenomena of the vibration absorber. Secondly, the total behaviour of the mechanical
system is of interest, compared with the well-known ef®ciency of the classical linear vibration
absorber. And ®nally, experimental investigations should con®rm the theoretical results.
Archive of Applied Mechanics 71 (2001) 463±472 Ó Springer-Verlag 2001
463
Received 10 January 2000; accepted for publication 26 September
2000
A. Hartung, H. Schmieg (&), P. Vielsack
Institut fu
È
r Mechanik, Universita
È
t Karlsruhe,
D-76128 Karlsruhe, Germany
Fax: (0721) 608 7990
E-mail: Mechanik@bau-verm.uni-karlsruhe.de
First published in:
EVA-STAR (Elektronisches Volltextarchiv – Scientific Articles Repository)
http://digbib.ubka.uni-karlsruhe.de/volltexte/8382001
2
Equation of motion and integration procedure
Comprehensive literature on the subject of nonsmooth dynamical systems has been made
available in the last decade, [2]. The motion of the nonsmooth dynamical system considered
can consist of three states at the friction device, i.e. M
2
slides to the right
_
x
2
> 0, M
2
slides to
the left
_
x
2
< 0, and M
2
sticks
_
x
2
0. In the last case, the 2-DOF system degenerates into a
1-DOF system, and the active friction force turns over to a passive contact force.
For the computation of the time response, a dimensionless representation of all quantities is
recommendable. A dash indicates differentiation with respect to a dimensionless time
s t
K
2
=M
2
p
: The coordinates n
i
x
i
=A; i 1; 2 are referred to the amplitude A of the ex-
citation. The parameters of the system are given by the friction coef®cients q
s
R
s
=AK
2
and
q
d
R
d
=AK
2
, the mass ratio m M
1
=M
2
, the stiffness ratio k K
1
=K
2
, the viscous damping
coef®cient D d=
K
2
M
2
p
and the slope of the characteristic D
a=
K
2
M
2
p
. Then, the
property of the drive is simply given by the excitation frequency ratio g X
M
2
=K
2
p
.
In a state of sliding, the equations of motion read
mn
00
1
Dn
0
1
Dn
0
2
1 kn
1
n
2
k cosgs;
n
00
2
Dn
0
1
D D
n
0
2
n
1
n
2
q
d
sgn n
0
2
;
1
The validity of these equations must be controlled by the condition n
0
2
6 0. In the state of
sticking, only one equation
mn
00
1
Dn
0
1
1 kn
1
n
2
k cosgs ; 2
exists. Its validity is controlled by the fact that the passive contact force must be smaller than
the threshold value q
s
which reads jn
2
n
1
Dn
0
1
j < q
s
. The constant displacement n
2
is
known from the end of the preceding state of sliding.
The total motion consists of a sequence of intermittent states described by Eqs. (1) or (2).
Each state is valid during a certain time interval which depends on the history of motion. The
transition points between two successive states are called switching times. They are determined
by switching conditions. If the velocity of the friction device in a state of sliding reaches the
value
n
0
2
0 ; 3
sliding in one direction is terminated. Sticking is terminated if the condition
jn
2
n
1
Dn
0
1
jq
s
; 4
holds.
At the end of each state, there must be a decision about the following state for times larger
than the last switching time s
0
. The transition is controlled by the switching decision
qs
0
n
2
n
1
s
0
Dn
0
1
s
0
; 5
Fig. 1. Mechanical model
Fig. 2a±c. Friction laws. a Coulomb's
law, b decreasing characteristic,
c static friction larger than dynamic
friction
464
which corresponds to the value of the passive contact force at time s
0
. If a state of sliding is
terminated, two possibilities have to be taken into account: jqs
0
j < q
s
indicates a transition
from sliding to sticking, and jqs
0
j > q
s
gives a sudden reversal of the direction of sliding with
sgn n
0
2
s
0
0sgn n
0
2
s
0
0. At the end of a state of sticking, only sliding is possible.
The direction of sliding equals the direction of the contact force at s
0
, which leads to
sgn n
0
2
s
0
0sgn qs
0
0.
The values of variables n
1
; n
0
1
; n
2
; n
0
2
at the end of a certain state give the initial conditions
for the equation(s) of motion of the following state. The total solution is pieced together.
The process is strongly history-dependent. Procedures for integrating nonsmooth dynamical
systems, therefore, consist of two tasks. Firstly, the integration on the linear equations
of motion (1) or (2) within two successive switching times. In the case of sticking (Eq. 2),
an explicit analytical solution can be given very easily. In the state of sliding, the equations
of motion (1) contain unsymmetrical damping. No analytical solution is known. Therefore,
a numerical integration of both cases is chosen. This will be done with a Runge±Kutta
formula, as described in [3]. This procedure has already been used successfully in [4].
Secondly, the numerical determination of the switching times by considering the switching
conditions (3) and (4). Problems arising from this procedure have been discussed in [5]
and [6].
3
Theoretical results
A numerical calculation needs numbers for the parameters m, k and D of the mechanical
system. Classical passive vibration absorbers are designed in a way that m and k are as large as
possible, which means that the absorber itself should be a small vibratory subsystem. With view
on the experimental investigations, this demand cannot be ful®lled in the present case, because
the friction device needs a certain geometrical dimension to ensure safe and reproducible
experimental results. Therefore, in the following, the values m 4 and k 1 are chosen for
both numerical and experimental investigations. Viscous damping D 0:005 is assumed to be
very small. This value agrees with the experimentally measured damping of the experimental
arrangement in the case of vanishing dry friction (2-DOF system).
The optimum design of a vibration absorber system is normally discussed by frequency
response curves in the case of the classical linear problem. Because the principle of superpo-
sition does no longer exist for the nonlinear system with dry friction under consideration, all
following results are valid for ®xed parameters only. But one can hope that they can be
extended in general, at least in a qualitative sense.
The following graphs do not provide any statements on the periodicity or non-periodicity
and uniqueness of stationary solutions, [7]. Therefore, the value
A
i
A
max n
i
min n
i
2
; i 1; 2 ; 6
is de®ned to be the amplitude of the response. In the following plots (Figs. 3±5) two limit cases
exist: at the top, the frequency response curves for the 2-DOF system without friction showing
resonance peaks at g g
1
and g g
2
; at the bottom, the frequency response curves for the
1-DOF system with one resonance peak at g g
0
, for the case when the friction force tends to
in®nity. Both graphs, well known from the linear vibration absorber, will serve as a reference in
the discussion about the in¯uence of friction within the range 0 < q
s
< 1.
Figure 3 shows a family of frequency response curves for Coulomb's law. Starting with
the linear system without friction (Fig. 3a), ®rst the antiresonance A
1
=A 0atg 1:0
disappears even for very low values of q
s
(Fig. 3b). A further increase of q
s
leads to a
removal of the resonance peak at the second natural frequency g
2
(Fig. 3c). Exceeding an
exciting frequency higher than the second natural frequency g
2
, the vibration absorber does
no longer move. Within a small increase from q
s
0:33 (Fig. 3d) to q
s
0:375 (Fig. 3e), the
resonance peak at the ®rst natural frequency g
1
also vanishes suddenly. From now on,
increasing values of q
s
produce standstill of the vibration absorber for a wide range of
exciting frequencies. In addition, the resonance peak at the natural frequency g
0
of the
1-DOF system appears. The most interesting phenomenon in the evolution of frequency
response curves under consideration is the existence of a certain q
s
-range, with small
amplitudes, independent of the value of the exciting frequency (Fig. 3e±f ). Properly chosen
friction can suppress high amplitude motions, which leads to a global stabilization in the
465
entire frequency range. Such a saturation phenomenon is known from active nonlinear
absorbers, [8].
Figure 4 shows the corresponding evolution of frequency response curves for the case of a
decreasing characteristic. The values q
s
of the static friction are the same as before. The second
frictional parameter D
0:04 is intentionally chosen to be relatively large, to emphasize
possible differences in comparison to Coulomb's law. On the other hand, the friction force
cannot become too small or even negative in reality. The computation is, therefore, interrupted
when the friction force becomes less than half the statical value q
s
. This is the case in Fig. 4b
and c at the resonance peaks.
Comparing the shape of the family of frequency response curves with the preceding ones
shows no qualitative differences between Coulomb friction and a decreasing characteristic.
Even the amplitudes are of the same order of magnitude. Quantitatively, there exists a small but
negligible additional peak in the vicinity of g
0
. The differences caused by both friction laws are
insigni®cant. Even the elimination of the resonance peaks occur for the same values q
s
as
before.
The same statements are valid for the third law considered, Fig. 5. Here again, an unrealistic
value q
d
q
s
=2 is chosen to emphasize the difference in comparison to Coulomb's law. The
in¯uence of q
d
results in the fact that the suppression of high amplitude motions is shifted to
Fig. 3a±g. Frequency response curves
for Coulomb friction
466
larger values of q
s
, Fig. 5f. If q
d
tends to q
s
, the system behaviour tends to the case of Coulomb
friction.
Comparing the results for the three different friction laws shows that the system's response is
in¯uenced only to a small extent by the special shape of the friction characteristic for velocities
j
_
x
2
j > 0, but signi®cantly by the static friction coef®cient q
s
at
_
x
2
0, which is responsible for a
transition from sticking to sliding. A similar statement can be found in [9] for stick-slip
motions induced by decelerative sliding.
4
Experimental investigations
The aim of the experimental investigation is to con®rm the frequency response relationship for
different values of q
s
, to prove the existence of stick phases and to identify the corresponding
friction laws. The latter task causes severe uncertainties because friction laws are in¯uenced not
only by mechanical parameters, such as relative velocity and normal force at the contact area,
but also by a change of the material properties at the interface, temperature, surface lubrica-
tion, wear, etc., [1]. To exclude the above-mentioned effects as much as possible the investi-
gation is restricted to a single frictional material which is used in the engineering practice for
pads of car-disc brakes, [10].
Fig. 4a±g. Frequency response curves
for a decreasing characteristic
D
0:04
467