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IssuesArchive of Issues2001-6pp.140-143

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S. E. Zaitsev and O. N. Tushev, "Estimation of the influence of random additive and multiplicative vibrations on the dynamic behavior of a system," Mech. Solids. 36 (6), 140-143 (2001)
Year 2001 Volume 36 Number 6 Pages 140-143
Title Estimation of the influence of random additive and multiplicative vibrations on the dynamic behavior of a system
Author(s) S. E. Zaitsev (Moscow)
O. N. Tushev (Moscow)
Abstract The influence of high-frequency random vibrations of different nature (additive and parametric) on the "slow motion" of a system is analyzed. The system with suspension and other elements can be described by an ordinary vector differential equation. It is assumed that external stationary or nonstationary excitations are defined within the framework of the correlation theory. The solution is sought by representing the vector of phase coordinates of the system as the integro-power series with respect to the matrix containing the random vibrations. Confining ourselves to the quadratic approximation for the multiplicative components, we obtain the convenient explicit function of the entries of the correlation matrix of vibrations, which permits us to analyze the contribution of each vibration component into the dynamic behavior of the system.

The results are illustrated by an example.

The influence of high-frequency additive and parametric vibrations on a dynamic system has been thoroughly studied. A typical example of this effect is the shift of a pendulum as a result of fast sinusoidal vibration of the suspension. To describe this effect, the Mathieu or Hill equations with right-hand side are usually used. Similar results are obtained for random stationary vibrations. Thus, for simple systems and stationary vibrations, this effect is easy to describe qualitatively and quantitatively. Considerable difficulties arise for a system with many degrees of freedom subjected to several random stationary or nonstationary excitations (the last case involves much more difficulties).

The problem considered in the present paper is of particular importance for protecting some mechanical objects, such as measuring devices, from vibrations. Vibrations may not lead to catastrophic results (for example, accelerations and displacements lie within their acceptable limits with large margins of safety), but substantially affect the functional properties of devices. This effect usually manifests itself in the appearance of a spurious signal. The magnitude of this signal essentially depends on the adjustment of the damping system. For the purposes of design, it is important not only to define the spurious signal, but also to evaluate the contribution of each vibration component to this signal.
References
1.  F. G. Gantmakher, Theory of Matrices [in Russian], Nauka, Moscow, 1967.
2.  R. Bellman, Introduction to the Theory of Matrices [Russian translation], Nauka, Moscow, 1969.
3.  V. N. Chelomei (Editor), Vibrations in Engineering [in Russian], Mashinostroenie, Moscow, 1981.
Received 06 October 1999
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