case increasing amplitudes of self-excited oscillations cannot be limited by
increased friction, but only by the hard stops. Then the mass gains energy during
slipping and dissipates it during impacts. If the energy obtained during slipping
exceeds what is lost during impact, the total energy will increase. But the oscillation
amplitudes cannot exceed the fixed clearance width 2D, so increased system energy
can only go into increased velocity, and correspondingly increased oscillation
frequency. Thus, when friction-induced oscillations are first initiated, their amplitude will build up until the mass starts hitting the stops, whereafter the frequency of
oscillations increase until a balance between gained and dissipated energy is
attained.
To calculate the frequency of stationary oscillations we first determine the
corresponding stationary energy ~
E 1 as a solution of qð ~
E 1 Þ ¼ 0: Then the corresponding oscillation period ~
T (equal to twice the free-flight time between two
successive impacts) and frequency ~
x ¼ 2p= ~
T is calculated from (3.304), which is
valid in-between two impacts, z 2 ] – p/2; p/2 [:
dz
dt
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
2 ~
E À VðzÞ
À
Á
q
) ~
T ¼ 2
Z z¼ p
2
z¼À p
2
dt ¼ 2
Z p
2
À p
2
dz
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 ~
E À VðzÞ
À
Á
q
;
ð3:310Þ
which, by inserting (3.300) and (3.274) and integrating, gives:
~
x ¼
2p
~
T
¼
p=2
arcsin
p=2
ffiffiffiffi
2 ~
E
p
:
ð3:311Þ
Fig. 3.36 illustrates how the stationary oscillation frequency computed by this
expression increases with the input energy parameter (–h 1 ) for parameters as given
in the legend. As appears the approximate results (in solid line) agree asymptotically with numerical simulation (circles) of the original system (3.296) for small
values of |h 1 |, i.e. as the assumptions (3.299) are better fulfilled.
3.9.7 Second-Order Analysis
It is a common experience with nonlinear systems, demonstrated also by the
examples in Sects. 3.9.3–3.9.6, that essential system behavior is revealed by
approximate analysis to lowest (i.e. first) order. However, higher order analysis may
be necessary when better numerical accuracy is required, or if parameters that are
assumed small are not really so, or if certain phenomena are only revealing
themselves at higher order. The general averaging method for vibro-impact analysis
196
3 Nonlinear Vibrations: Classical Local Theory
increased friction, but only by the hard stops. Then the mass gains energy during
slipping and dissipates it during impacts. If the energy obtained during slipping
exceeds what is lost during impact, the total energy will increase. But the oscillation
amplitudes cannot exceed the fixed clearance width 2D, so increased system energy
can only go into increased velocity, and correspondingly increased oscillation
frequency. Thus, when friction-induced oscillations are first initiated, their amplitude will build up until the mass starts hitting the stops, whereafter the frequency of
oscillations increase until a balance between gained and dissipated energy is
attained.
To calculate the frequency of stationary oscillations we first determine the
corresponding stationary energy ~
E 1 as a solution of qð ~
E 1 Þ ¼ 0: Then the corresponding oscillation period ~
T (equal to twice the free-flight time between two
successive impacts) and frequency ~
x ¼ 2p= ~
T is calculated from (3.304), which is
valid in-between two impacts, z 2 ] – p/2; p/2 [:
dz
dt
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
2 ~
E À VðzÞ
À
Á
q
) ~
T ¼ 2
Z z¼ p
2
z¼À p
2
dt ¼ 2
Z p
2
À p
2
dz
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 ~
E À VðzÞ
À
Á
q
;
ð3:310Þ
which, by inserting (3.300) and (3.274) and integrating, gives:
~
x ¼
2p
~
T
¼
p=2
arcsin
p=2
ffiffiffiffi
2 ~
E
p
:
ð3:311Þ
Fig. 3.36 illustrates how the stationary oscillation frequency computed by this
expression increases with the input energy parameter (–h 1 ) for parameters as given
in the legend. As appears the approximate results (in solid line) agree asymptotically with numerical simulation (circles) of the original system (3.296) for small
values of |h 1 |, i.e. as the assumptions (3.299) are better fulfilled.
3.9.7 Second-Order Analysis
It is a common experience with nonlinear systems, demonstrated also by the
examples in Sects. 3.9.3–3.9.6, that essential system behavior is revealed by
approximate analysis to lowest (i.e. first) order. However, higher order analysis may
be necessary when better numerical accuracy is required, or if parameters that are
assumed small are not really so, or if certain phenomena are only revealing
themselves at higher order. The general averaging method for vibro-impact analysis
196
3 Nonlinear Vibrations: Classical Local Theory