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A Journal of Russian Academy of Sciences | | Founded
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| L. L. Song and D. V. Iourtchenko, "Analysis of stochastic vibroimpact systems with inelastic impacts," Mech. Solids. 41 (2), 146-155 (2006) |
| Year |
2006 |
Volume |
41 |
Number |
2 |
Pages |
146-155 |
| Title |
Analysis of stochastic vibroimpact systems with inelastic impacts |
| Author(s) |
L. L. Song (Moscow)
D. V. Iourtchenko (Moscow) |
| Abstract |
Vibroimpact systems are of particular interest for studying. This is due to the
fact that even when the behavior of the system between impacts is linear, the
impacts make the system highly nonlinear. All vibroimpact systems can be
divided into the systems with one-sided constraint and the systems with
two-sided constraint. In the case of one-sided constraint, the coordinate of
the limiter, Δ, measured from the equilibrium position of the system can
be negative (interference), zero, or positive (clearance) [1], while in the
case of two-sided constraint, the limiters can be arranged symmetrically
or asymmetrically relative to the equilibrium position.
Vibroimpact systems with deterministic excitation have been well studied and
described in detail in a number of books [1-4]. Systems with elastic impacts
can be studied by means of the method of [5, 6], which transforms the impact
system to a system without impacts. However, this method is effective only in
the cases where the motion between impacts is linear or has a polynomial
nonlinearity. In the case of inelastic impacts, the impact condition is usually
expressed by the Dirac delta function on the right-hand side of the equation of
motion. The coefficient of the delta function expresses the value of the impact
impulse. In this case, the problem can be solved by the method of averaging.
A description of stochastic vibroimpact systems can be found in [7]. Stochastic
systems with linear behavior between impacts and a one-sided limiter located at
the equilibrium position (Δ=0) have been thoroughly studied. The exact
expressions for the probability density and spectral density [8] in such
systems have been obtain by means of the transformation proposed in [5, 6].
In
the other cases, the impact term is introduced into the right hand side of the
equation of motion in the form of the Dirac delta function multiplied by an
appropriate coefficient, as was the case for deterministic systems, and the
system is solved approximately by the quasi-conservative averaging method
of [9].
A detailed description of the statistical linearization method for
vibroimpact stochastic systems is given in [1]. The aforementioned approximate
methods enable one to study vibrations in vibroimpact systems only if the
restitution coefficient is close to unity, i.e., (1−r)«1. These methods
do not enable one to evaluate the response of the vibroimpact system with other
(lower) restitution coefficients.
A new energy balance method has been recently proposed in [10, 11] for systems
with impact-induced dominant losses, i.e., for systems with negligibly low
friction. Such systems were called the piecewise-conservative systems. The
energy loss in such systems occurs at certain discrete time instants. These
instants are unknown beforehand and depend only on the position and/or velocity
of the system. Vibroimpact systems with impact-induced dominant losses are
characteristic representatives of the class of piecewise-conservative systems.
The idea of the method as applied to vibroimpact systems implies the
consideration of the behavior of the energy of the system between impacts and
the balance of the energy before and after an impact. It is important that this
method does not require the change in the system energy per period to be small,
as is the case for the quasi-conservative averaging method. Therefore, one can
anticipate that this method provides more accurate estimate for the average energy of
the system, as compared with the quasi-conservative averaging.
The main purpose of the present study is to obtain analytical estimates for the
average energy of stochastic vibroimpact systems with inelastic impacts. To
that end, we present calculations that enable one to extend the application
area of the energy balance method to stochastic vibroimpact systems with
interference, clearance, and two-sided impact. In addition, the results are
compared with approximate results obtained by means of the quasi-conservative
averaging and the numerical simulation results. The results of the numerical
simulation of a vibroimpact system with two degrees of freedom are given. |
| References |
| 1. | V. I. Babitskii, Theory of Vibroimpact Systems: Approximate Methods [in
Russian], Nauka, Moscow, 1978. |
| 2. | V. I. Babitskii and V. L. Krupenin, Vibrations in Highly Nonlinear Systems [in
Russian], Nauka, Moscow, 1985. |
| 3. | V. I. Babitskii and M. Z. Kolovskii, "To the theory of vibroimpact systems,"
Mashinovedenie, No. 1, pp. 24-30, 1970. |
| 4. | A. E. Kobrinskii and A. A. Kobrinskii,
Vibroimpact Systems [in Russian], Nauka, Moscow, 1973. |
| 5. | V. Ph. Zhuravlev, "Equations of motion of mechanical systems with ideal unilateral constraints,"
PMM [Applied Mathematics and Mechanics], Vol. 42, No. 5, pp. 781-788, 1978. |
| 6. | V. Ph. Zhuravlev and D. M. Klimov,
Applied Methods in Vibration Theory [in Russian], Nauka, Moscow,
1988. |
| 7. | M. F. Dimentberg, Statistical Dynamics of Nonlinear and Time-varying Systems,
Wiley, New York; Research Studies Press, Taunton, 1988. |
| 8. | M. F. Dimentberg, Z. Hou, and M. Noori, "Spectral density of a nonlinear single-degree-of-freedom
system's response to a white-noise random excitation: a unique case of an exact solution,"
Intern. J. of Non-linear Mech., Vol. 30, No. 5, pp. 673-676, 1995. |
| 9. | P. S. Landa and R. L. Stratonovich, "To the theory of fluctuation transitions of various systems
between two stationary positions," Vestnik MGU [Bulletin of the Moscow State University], Ser. 3, No. 1,
pp. 33-45, 1962. |
| 10. | M. F. Dimentberg and D. V. Iourtchenko, "Towards incorporating impact losses into random
vibration analyses," Probabilistic Eng. Mech., Vol. 14, No. 4, pp. 323-328, 1999. |
| 11. | D. V. Iourtchenko and M. F. Dimentberg, "Energy balance for random vibration
of piecewise-conservative systems," J. Sound and Vibrat., Vol. 248, No. 5, pp. 913-923,
2001. |
| 12. | Y. K. Lin and G. Q. Cai, Probabilistic Structural Dynamics: Advanced Theory
and Applications, McGraw Hill, New York, 1995. |
| 13. | R. L. Stratonovich, Selected Issues of
the Theory of Fluctuations in Radio Engineering
[in Russian], Sovetskoe Radio, Moscow, 1961. |
| 14. | V. V. Bolotin, Random Vibration of Elastic Systems [in Russian], Nauka, Moscow, 1979. |
|
| Received |
17 December 2003 |
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