x instead of u. Hence, for a given e, solutions to the Rayleigh oscillator equation can
be obtained simply by integrating in time the solutions for the van der Pol equation.
This includes the approximate solution for small e (3.247), which for the Rayleigh
oscillator equation becomes:
uðtÞ ¼ À2 cosðt þ u 0 Þ; _
uðtÞ ¼ 2 sinðt þ u 0 Þ; e ( 1;
ð3:249Þ
which just corresponds to a phase shift of p/2 compared to the van der Pol equation.
3.9 Vibro-Impact Analysis Using Discontinuous
Transformations
Here we show how near-elastic vibro-impact problems can be conveniently analyzed by a discontinuity-reducing transformation of variables, combined with an
extended averaging procedure; the presentation is based mainly on Fidlin (2006)
and Thomsen and Fidlin (2008).
Vibro-impact systems are characterized by repeated impacts. Applications
include devices to crush, grind, forge, drill, punch, tamp, pile, and cut a variety of
objects, and vibrating machinery or structures with stops or clearances (Babitsky
1998). Also, vibro-impact is involved in noise- and wear-producing processes, as
with rattling gear boxes and heat exchanger tubes. The analysis of such systems is
often difficult, mainly due to the inherent presence of strong nonlinearity: Even if a
vibro-impact system can be considered linear or weakly nonlinear in-between
impacts, the impacts correspond to a force–displacement relation which cannot be
linearized at the point of impact contact, or has a dominating nonlinear component.
Using kinematic impact formulations, this nonlinear effect is taken into account
though the prescription of a discontinuous change in relative velocity at impact
times. With near-elastic impacts, these velocity discontinuities are “large”, i.e. of
the order of magnitude of the impact velocities themselves. Such problems are not
natural candidates for approximate perturbation analysis. But sometimes they can
be transformed to be so.
In this section we present a general procedure and some application examples,
showing how near-elastic vibro-impact problems, linear or nonlinear in-between
impacts, can be conveniently analyzed by a discontinuity-reducing transformation
of variables, combined with an extended averaging procedure that allows presence
of the resulting small discontinuities of the transformed system. Only first-order
analytical predictions are derived, while the more elaborate extension to second
order is described in Thomsen and Fidlin 2008. We consider only near-elastic
vibro-impact, i.e. with a coefficient of restitution close to unity, though averaging
may be applied also for the inelastic case (Fidlin 2006). A recent application
example of the technique is provided in Rebouças et al. (2019), using it to calculate
the vibro-impact response for a cantilever beam with a one-sided stop, and comparing with laboratory experiments.
3.8 Two More Classical Nonlinear Oscillators
175
be obtained simply by integrating in time the solutions for the van der Pol equation.
This includes the approximate solution for small e (3.247), which for the Rayleigh
oscillator equation becomes:
uðtÞ ¼ À2 cosðt þ u 0 Þ; _
uðtÞ ¼ 2 sinðt þ u 0 Þ; e ( 1;
ð3:249Þ
which just corresponds to a phase shift of p/2 compared to the van der Pol equation.
3.9 Vibro-Impact Analysis Using Discontinuous
Transformations
Here we show how near-elastic vibro-impact problems can be conveniently analyzed by a discontinuity-reducing transformation of variables, combined with an
extended averaging procedure; the presentation is based mainly on Fidlin (2006)
and Thomsen and Fidlin (2008).
Vibro-impact systems are characterized by repeated impacts. Applications
include devices to crush, grind, forge, drill, punch, tamp, pile, and cut a variety of
objects, and vibrating machinery or structures with stops or clearances (Babitsky
1998). Also, vibro-impact is involved in noise- and wear-producing processes, as
with rattling gear boxes and heat exchanger tubes. The analysis of such systems is
often difficult, mainly due to the inherent presence of strong nonlinearity: Even if a
vibro-impact system can be considered linear or weakly nonlinear in-between
impacts, the impacts correspond to a force–displacement relation which cannot be
linearized at the point of impact contact, or has a dominating nonlinear component.
Using kinematic impact formulations, this nonlinear effect is taken into account
though the prescription of a discontinuous change in relative velocity at impact
times. With near-elastic impacts, these velocity discontinuities are “large”, i.e. of
the order of magnitude of the impact velocities themselves. Such problems are not
natural candidates for approximate perturbation analysis. But sometimes they can
be transformed to be so.
In this section we present a general procedure and some application examples,
showing how near-elastic vibro-impact problems, linear or nonlinear in-between
impacts, can be conveniently analyzed by a discontinuity-reducing transformation
of variables, combined with an extended averaging procedure that allows presence
of the resulting small discontinuities of the transformed system. Only first-order
analytical predictions are derived, while the more elaborate extension to second
order is described in Thomsen and Fidlin 2008. We consider only near-elastic
vibro-impact, i.e. with a coefficient of restitution close to unity, though averaging
may be applied also for the inelastic case (Fidlin 2006). A recent application
example of the technique is provided in Rebouças et al. (2019), using it to calculate
the vibro-impact response for a cantilever beam with a one-sided stop, and comparing with laboratory experiments.
3.8 Two More Classical Nonlinear Oscillators
175