The classical approach for analyzing vibro-impact problems is stitching
(Kobrinskii 1969), where the equations of motions are integrated in-between
impacts, kinematic impact conditions are used to switch between time intervals of
solution, and solutions over times involving several impacts are obtained by ‘gluing’
together a suitable number of such partial solutions. Variants of this involve setting
up and analyzing discrete maps between successive impacts (Guckenheimer and
Holmes 1983). For numerical simulation, stitching and its variants can be simple and
effective, with appropriate numerical algorithms (e.g., Wiercigroch 2000). However,
for obtaining purely analytical solutions stitching is elaborate, and any nonlinearity
in-between impacts makes it difficult to apply. Furthermore, for typical applications
it is not necessary to obtain solutions at the level of detail provided by exact methods.
Of more interest may be condensed measures such as oscillation frequencies, stationary amplitudes, and the stability of stationary motions.
Thus, for vibro-impact problems approximate analytical methods are necessary
and useful. Among these are the methods of harmonic linearization (Babitsky
1998), averaging (Fidlin 2006), and direct separation of motions (Blekhman 2000),
each with their particular strengths. In particular the method of harmonic linearization, as introduced by Babitsky (1998) in the 1960s, has been used successfully to solve many vibro-impact problems (e.g. Rebouças et al. 2017). This
approach is convenient in use, but mathematically not well supported; results need
careful validation by e.g. numerical simulation, in particular for systems operating
away from resonance.
The approach presented in this section – discontinuous transformation combined
with extended averaging – is founded on original ideas by in particular Zhuravlev
(1976), Pilipchuk (1988), and Ivanov (1997). In Thomsen and Fidlin (2008) the
approach is mathematically supported by a theorem similar to the standard averaging theorem (Sanders and Verhulst 1985), thus providing estimates of the
accuracy of approximation, and a systematic procedure for increasing the accuracy
to any desired level. By contrast to harmonic linearization, it assumes a kinematic
rather than a kinetic impact formulation; that is: the impact process is described
simply by a coefficient of restitution relating relative velocities just before and after
impact, while the details of impact process itself is not involved. Compared to
classical or semi-analytical stitching, it provides analytical solutions valid at all
times, i.e. free of switching conditions, and also works for systems that are nonlinear in-between impacts; the latter nonlinearities can be weak or strong or even
essential (as e.g. u
3 or sgn(u)), if just the solution of the unperturbed system is
known. Compared to the averaging approach described by Ivanov (1997), the
discontinuous transformations used with the present approach need not to eliminate
the impact discontinuities completely. This is a considerable advantage, since setting up (and physically interpreting) transformations that eliminate discontinuities
completely is far from trivial and requires quite some ingenuity. Still, there is no
general rule for suggesting workable transformations, but the examples provided
may work directly or serve as inspiration for other cases.
The purpose of the presentation is to demonstrate the practical applicability of a
method for analyzing vibro-impact problems. The mathematical models used are
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3 Nonlinear Vibrations: Classical Local Theory
(Kobrinskii 1969), where the equations of motions are integrated in-between
impacts, kinematic impact conditions are used to switch between time intervals of
solution, and solutions over times involving several impacts are obtained by ‘gluing’
together a suitable number of such partial solutions. Variants of this involve setting
up and analyzing discrete maps between successive impacts (Guckenheimer and
Holmes 1983). For numerical simulation, stitching and its variants can be simple and
effective, with appropriate numerical algorithms (e.g., Wiercigroch 2000). However,
for obtaining purely analytical solutions stitching is elaborate, and any nonlinearity
in-between impacts makes it difficult to apply. Furthermore, for typical applications
it is not necessary to obtain solutions at the level of detail provided by exact methods.
Of more interest may be condensed measures such as oscillation frequencies, stationary amplitudes, and the stability of stationary motions.
Thus, for vibro-impact problems approximate analytical methods are necessary
and useful. Among these are the methods of harmonic linearization (Babitsky
1998), averaging (Fidlin 2006), and direct separation of motions (Blekhman 2000),
each with their particular strengths. In particular the method of harmonic linearization, as introduced by Babitsky (1998) in the 1960s, has been used successfully to solve many vibro-impact problems (e.g. Rebouças et al. 2017). This
approach is convenient in use, but mathematically not well supported; results need
careful validation by e.g. numerical simulation, in particular for systems operating
away from resonance.
The approach presented in this section – discontinuous transformation combined
with extended averaging – is founded on original ideas by in particular Zhuravlev
(1976), Pilipchuk (1988), and Ivanov (1997). In Thomsen and Fidlin (2008) the
approach is mathematically supported by a theorem similar to the standard averaging theorem (Sanders and Verhulst 1985), thus providing estimates of the
accuracy of approximation, and a systematic procedure for increasing the accuracy
to any desired level. By contrast to harmonic linearization, it assumes a kinematic
rather than a kinetic impact formulation; that is: the impact process is described
simply by a coefficient of restitution relating relative velocities just before and after
impact, while the details of impact process itself is not involved. Compared to
classical or semi-analytical stitching, it provides analytical solutions valid at all
times, i.e. free of switching conditions, and also works for systems that are nonlinear in-between impacts; the latter nonlinearities can be weak or strong or even
essential (as e.g. u
3 or sgn(u)), if just the solution of the unperturbed system is
known. Compared to the averaging approach described by Ivanov (1997), the
discontinuous transformations used with the present approach need not to eliminate
the impact discontinuities completely. This is a considerable advantage, since setting up (and physically interpreting) transformations that eliminate discontinuities
completely is far from trivial and requires quite some ingenuity. Still, there is no
general rule for suggesting workable transformations, but the examples provided
may work directly or serve as inspiration for other cases.
The purpose of the presentation is to demonstrate the practical applicability of a
method for analyzing vibro-impact problems. The mathematical models used are
176
3 Nonlinear Vibrations: Classical Local Theory