chosen for their simplicity; we are not concerned with the accuracy of these models
for describing physical reality, but only with the accuracy of the method we suggest
for analyzing them. No attempt is given to cover abundant literature on the modeling of particular vibro-impact systems and the study of their dynamical behavior.
Sect. 3.9.1 illustrates in a simple setting the basic idea of employing an
unfolding transformation to eliminate discontinuities for purely elastic vibro-impact
systems. With inelastic impacts such transformations will not eliminate the discontinuities, but for near-elastic impacts they will be reduced to a value which is
small as compared to the impact velocities. This motivates the following
Sect. 3.9.2, which shows how to apply asymptotic first-order averaging for general
systems of ordinary differential equations containing small discontinuities. Finally
Sects. 3.9.3–3.9.6 presents four application examples, where discontinuous
unfolding transformation and averaging are combined, resulting in approximate
analytical expressions for key properties such as oscillation amplitudes and frequencies. Problem 3.23 exercises the key techniques in a simple setting.
3.9.1 The Unfolding Discontinuous Transformation:
Basic Idea
We begin by considering a simple vibro-impact system, just to show how an
unfolding transformation can be used to eliminate impact discontinuities. Assuming
purely elastic impacts (coefficient of restitution R = 1), the motions of the harmonic
oscillator in Fig. 3.26(a), bouncing against a rigid stop at s = 0, are governed by:
€ s þ s ¼ 0 for s [ 0;
s þ ¼ s À ; _
s þ ¼ À_ s À for s ¼ 0;
ð3:250Þ
where overdots denote differentiation with respect to time t, and subscripts plus and
minus indicate states immediately before and after impact, respectively. The general
solution is:
sðtÞ ¼ A sin t þ h
ð
Þ
j
j ;
ð3:251Þ
which is shown in Fig. 3.26(b) (solid line) for initial conditions corresponding to
(A, h) = (1, 0); it is nothing but a folded or full-wave rectified sine function. But this
in turn means that the discontinuity in (3.250) can be removed by a discontinuous
transformation defined by:
s ¼ z
j j; z À z þ \0;
ð3:252Þ
3.9 Vibro-Impact Analysis Using Discontinuous Transformations
177
for describing physical reality, but only with the accuracy of the method we suggest
for analyzing them. No attempt is given to cover abundant literature on the modeling of particular vibro-impact systems and the study of their dynamical behavior.
Sect. 3.9.1 illustrates in a simple setting the basic idea of employing an
unfolding transformation to eliminate discontinuities for purely elastic vibro-impact
systems. With inelastic impacts such transformations will not eliminate the discontinuities, but for near-elastic impacts they will be reduced to a value which is
small as compared to the impact velocities. This motivates the following
Sect. 3.9.2, which shows how to apply asymptotic first-order averaging for general
systems of ordinary differential equations containing small discontinuities. Finally
Sects. 3.9.3–3.9.6 presents four application examples, where discontinuous
unfolding transformation and averaging are combined, resulting in approximate
analytical expressions for key properties such as oscillation amplitudes and frequencies. Problem 3.23 exercises the key techniques in a simple setting.
3.9.1 The Unfolding Discontinuous Transformation:
Basic Idea
We begin by considering a simple vibro-impact system, just to show how an
unfolding transformation can be used to eliminate impact discontinuities. Assuming
purely elastic impacts (coefficient of restitution R = 1), the motions of the harmonic
oscillator in Fig. 3.26(a), bouncing against a rigid stop at s = 0, are governed by:
€ s þ s ¼ 0 for s [ 0;
s þ ¼ s À ; _
s þ ¼ À_ s À for s ¼ 0;
ð3:250Þ
where overdots denote differentiation with respect to time t, and subscripts plus and
minus indicate states immediately before and after impact, respectively. The general
solution is:
sðtÞ ¼ A sin t þ h
ð
Þ
j
j ;
ð3:251Þ
which is shown in Fig. 3.26(b) (solid line) for initial conditions corresponding to
(A, h) = (1, 0); it is nothing but a folded or full-wave rectified sine function. But this
in turn means that the discontinuity in (3.250) can be removed by a discontinuous
transformation defined by:
s ¼ z
j j; z À z þ \0;
ð3:252Þ
3.9 Vibro-Impact Analysis Using Discontinuous Transformations
177