TY - JOUR

T1 - Energy Balance for Random Vibrations of Piecewise-Conservative Systems

AU - Yurchenko, Daniil

AU - Dimentberg, M. F.

PY - 2001/12/13

Y1 - 2001/12/13

N2 - Vibrations of systems with instantaneous or stepwise energy losses,
e.g., due to impacts with imperfect rebounds, dry friction forces(s) (in
which case the losses may be treated as instantaneous ones by
appropriate introduction of the response energy) and/or active feedback
“bang-bang” control of the systems' response are
considered. Response of such (non-linear) systems to a white-noise
random excitation is considered for the case where there are no other
response energy losses. Thus, a simple linear energy growth with time
between “jumps” is observed. Explicit expressions for the
expected response energy are derived by direct application of the
stochastic differential equations calculus, which contains the expected
time interval between two consecutive jumps. The latter may be predicted
as a solution to the relevant first-passage problem. Perturbational
analysis of the relevant PDE for this problem for a certain vibroimpact
system demonstrated the possibility for using the solution to the
corresponding free vibration problem as a zero order approximation. The
method is applied to an s.d.o.f. system with a feedback inertia control,
designed according to a certain previously introduced “generalized
reversed swings law”. Extensive Monte-Carlo simulation results are
presented for this system as well as for several previously analyzed
ones: system with impacts; system with dry friction; system with
stiffness control; pendulum with controlled length. The results are
compared with those due to the asymptotic stochastic averaging approach.
Both methods are shown to provide adequate accuracy far beyond the
expected applicability range of the asymptotic approach (which requires
both excitation intensity and losses to be small), with direct energy
balance being generally superior.

AB - Vibrations of systems with instantaneous or stepwise energy losses,
e.g., due to impacts with imperfect rebounds, dry friction forces(s) (in
which case the losses may be treated as instantaneous ones by
appropriate introduction of the response energy) and/or active feedback
“bang-bang” control of the systems' response are
considered. Response of such (non-linear) systems to a white-noise
random excitation is considered for the case where there are no other
response energy losses. Thus, a simple linear energy growth with time
between “jumps” is observed. Explicit expressions for the
expected response energy are derived by direct application of the
stochastic differential equations calculus, which contains the expected
time interval between two consecutive jumps. The latter may be predicted
as a solution to the relevant first-passage problem. Perturbational
analysis of the relevant PDE for this problem for a certain vibroimpact
system demonstrated the possibility for using the solution to the
corresponding free vibration problem as a zero order approximation. The
method is applied to an s.d.o.f. system with a feedback inertia control,
designed according to a certain previously introduced “generalized
reversed swings law”. Extensive Monte-Carlo simulation results are
presented for this system as well as for several previously analyzed
ones: system with impacts; system with dry friction; system with
stiffness control; pendulum with controlled length. The results are
compared with those due to the asymptotic stochastic averaging approach.
Both methods are shown to provide adequate accuracy far beyond the
expected applicability range of the asymptotic approach (which requires
both excitation intensity and losses to be small), with direct energy
balance being generally superior.

U2 - 10.1006/jsvi.2001.3853

DO - 10.1006/jsvi.2001.3853

M3 - Article

SN - 0022-460X

VL - 248

SP - 913

EP - 923

JO - Journal of Sound and Vibration

JF - Journal of Sound and Vibration

IS - 5

ER -