Random Parametric Excitation Research Articles

Non-linear parametric liquid sloshing under wide band random excitation

The stationary response of a liquid free surface, in a partially filled cylindrical container, to a wide band random parametric excitation is investigated. Two analytical approaches are employed. These are the Gaussian closure scheme to truncate the infinite hierarchy moment equations, and the Stratonovich stochastic averaging method. The validity of the two solutions is examined by comparing the two predicted probability densities with the one measured experimentally. The comparison reveals poor agreement with the Gaussian closure results in contrast to those obtained by the stochastic averaging method, which gives a good description for the random surface motion. In general, the results compare well for surface amplitudes greater than the most probable amplitude, but below the most probable amplitude there is a remarkable deviation. This deviation is found to increase as the excitation level decreases. It is concluded that the uncertainty of the experimental measurements and the inaccuracy of the experimental fitting curves are the main source of the resulting difference.

Exact solution of a particular problem of oscillations of a system with random parametric excitation

A second order system with external and parametric perturbations of the white noise type is considered. An exact analytic solution of the steady Fokker—Plank —Kolmogorov equation is obtained for this system at some special relation between the coefficients of modulation intensity with respect to the coordinate and velocity. The solution determines the power relationship for the combined probability density of the coordinate and velocity.

The Stability of a Moving Elastic Strip Subjected to Random Parametric Excitation

The dynamic stability of a thin strip, traveling axially, at a constant speed between two roller supports is investigated for the case of zero mean random in-plane loading. Galerkin’s method is used to reduce the equations of motion to a set of fourth-order stochastic equations. An extention of the method first proposed by Wu and Kozin is developed which allows determination of the sufficiency conditions to guarantee Almost Sure Asymptotic Stability of stochastic systems of order greater than two. Using this method, results in terms of the variance of the random loadings on the moving strip are derived. It is found that the critical noise level to guarantee stability of the strip decreases with increasing mode, approaching asymptotically a level determined solely by the strip stiffness.

Stationary Response of a Randomly Parametric Excited Nonlinear System

This paper presents a review of three truncation schemes for the problem of the infinite hierarchy of moment equations and an investigation of the stationary response of a nonlinear system under a broad band random parametric excitation. The validity of the truncation methods is discussed together with the conditions for preservation of moment properties. One of these schemes is employed to truncate the dynamic moment equations of a nonlinear single-degree-of-freedom system subject to a broad band random parametric excitation. The influence of inertia, stiffness, and damping nonlinearities is discussed and closed-form solutions are obtained for each case. The preservation of the response moment properties is confirmed for certain solutions while it fails for the remaining ones. The invalidity of these solutions is not necessarily attributed to the inaccuracy of the used truncation method as it may be due to the fact that the system may not be able to achieve a stationary response.

Parametric Random Excitation of a Damped Mathieu Oscillator

AbstractThe effect of parametric random excitation on the moment stability of a damped Mathieu oscillator is investigated. Conditions for stability of the first and second moments of the response are obtained when the harmonic excitation lies in the neighbourhood of twice the natural frequency of the oscillator. It is found that the presence of the additional parametric random excitation can cause either a stabilizing or a destabilizing effect depending on the values of certain parameters of the random excitation. In particular, it is shown that parametric excitation by white noise has no effect on the first moments of the response but has a destabilizing effect on the second moments. For the case of exponentially correlated noise, regions of instability are shown in graphical form.

Certain problems of identification of a vibratory system with random parametric excitation

Certain problems of identification of a vibratory system with random parametric excitation

The mean square stability of an inverted pendulum subject to random parametric excitation

This paper describes a theory of the stabilization of an inverted pendulum by means of externally imposed random oscillations of the point of support in a vertical line. Methods previously used in the multiple scattering theory of wave propagation in random media [1] are applied to derive two conditions for stability in the upright position. The first of these is the well-known condition which requires the variance of the support velocity to be large. The second condition has apparently not been explicitly noted hitherto, and relates to the parametric amplification of the pendulum oscillations caused by resonant interactions between the steady pendulum motion at frequency k , say, and those spectral components of the effective excitation at precisely twice that frequency. The theory is compared with recent results of analogue computer simulations and found to be in good qualitative agreement.

The stability of vibratory systems in the case of random parametric excitation

The stability of vibratory systems in the case of random parametric excitation

Probability distributions in simple neutronic systems via the fokker-planck equation

The Fokker-Planck equation for the neutron density probability distribution in a multiplying system, possessing random source and random parametric excitation, is deduced. It is shown that white noise parametric excitation and white noise source excitation both have the same effect on the functional form of the correlation but that the parametric excitation increases its amplitude. A method for obtaining the correlation function directly from the conditional probability distribution is described and applied to a simple point model equation with one group of delayed neutrons.

First order piecewise linear systems with random parametric excitation

The Fokker-Planck equation is used to develop a general method for finding the spectral density and other properties of first order systems governed by stochastic differential equations of the form dx dt + f(x)[1 + m(t)] = n(t) , where f( x) is piecewise linear and m( t) and n( t) represent stationary Gaussian white noise. The method is similar to one used by the authors to deal with the case m( t) = 0, but is complicated by the possible existence of irregular (singular) points of the Fokker-Planck equation. Graphical results for some special cases are presented.

Stability of a linear second order system under random parametric excitation

The stability of a linear second order system with small, random, Gaussian parametric excitation will be investigated. It will be shown that if the spectrum of the parameter variation is continuous near twice the natural frequency of the system, stability in mean square is only dependent upon the value of the spectral density at this part of the spectrum.

On the stability of nonlinear stochastic systems

Stability of systems with random parametric excitation was investigated by several workers. The cases when the stability towards white-noise excitation was investigated were the most successfull ones, since the methods of the theory of Markov processes (see eg. [1 to 4] e.a.) could be utilised. Investigation of stability under nonwhite excitation is much harder, and this is the reason why most authors limited their investigations to either linear [5 to 7] or nonlinear systems of some particular type [7]. In [8] we find the stability criteria for an arbitrary nonlinear system with excitations of any type, but this criterion is effective only in the cases when the solution is a Markov process. Authors of [8 and 9] introduced the use of Liapunov function in problems of stability under random excitation. We shall however utilise that aspect of the Liapunov method, which was first used for similar purposes by the authors of [10 and 7].

Experiments with an Inverted Pendulum Subject to Random Parametric Excitation

Experimental results are presented that verify the theoretical conclusion that an inverted pendulum may be stabilized with second-order stationary, random parametric excitation having a discrete-power spectral density. Experiments also verified that the theoretical condition on frequency spacing is critical.

A Controversy in Problems Involving Random Parametric Excitation

A Controversy in Problems Involving Random Parametric Excitation

Behavior of Linear Systems with Random Parametric Excitation

The behavior of linear systems with parameters that vary as “physical” white noise (very-wide-band Gaussian) is developed. The Fokker-Planck equation is used to develop moment equations, and “equivalent” systems are derived.

A Comment on “The Behavior of Linear Systems with Random Parametric Excitation”

Journal of Mathematics and PhysicsVolume 42, Issue 1-4 p. 336-337 Article A Comment on “The Behavior of Linear Systems with Random Parametric Excitation”† F. Kozin, F. Kozin School of Aeronautical and Engineering Sciences Purdue UniversitySearch for more papers by this authorJ. L. Bogdanoff, J. L. Bogdanoff School of Aeronautical and Engineering Sciences Purdue UniversitySearch for more papers by this author F. Kozin, F. Kozin School of Aeronautical and Engineering Sciences Purdue UniversitySearch for more papers by this authorJ. L. Bogdanoff, J. L. Bogdanoff School of Aeronautical and Engineering Sciences Purdue UniversitySearch for more papers by this author First published: April 1963 https://doi.org/10.1002/sapm1963421336Citations: 2 †Supported by NSF, Contract No. GP-60 AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat No abstract is available for this article.Citing Literature Volume42, Issue1-4April 1963Pages 336-337 RelatedInformation

The Behavior of Linear Systems with Random Parametric Excitation

Journal of Mathematics and PhysicsVolume 41, Issue 1-4 p. 300-318 Article The Behavior of Linear Systems with Random Parametric Excitation T. K. Caughey, T. K. Caughey California Institute of Technology, Pasadena, CaliforniaSearch for more papers by this authorJ. K. Dienes, J. K. Dienes California Institute of Technology, Pasadena, CaliforniaSearch for more papers by this author T. K. Caughey, T. K. Caughey California Institute of Technology, Pasadena, CaliforniaSearch for more papers by this authorJ. K. Dienes, J. K. Dienes California Institute of Technology, Pasadena, CaliforniaSearch for more papers by this author First published: April 1962 https://doi.org/10.1002/sapm1962411300Citations: 57AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Citing Literature Volume41, Issue1-4April 1962Pages 300-318 RelatedInformation