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Flexural instabilities in moving bands under harmonic tension fluctuation are investigated. The discretized equations of flexural motion represent a periodically perturbed linear gyroscopic system. Explicit conditions for the onset of instability are established employing the method of averaging. The effects due to damping, mean band speed, and the band compliance on the band stability are discussed. The present analysis is adequate in the design of moving bands where it is necessary to avoid critical speed ranges.

The tension in power transmitting chains is subject to periodic fluctuation. This is brought about by factors such as polygonal action, eccentricity of sprockets, etc. A study of the dynamic stability of such a chain is presented. Instabilities due to polygonal action are investigated. The analysis is carried out by treating the chain as a travelling uniform heavy string.

Dynamic stability of linear conservative gyroscopic systems under stochastic parametric excitations of small intensity is examined. Conditions for mean square stability of dynamic response are obtained. Results are shown to depend only on those values of the excitation spectral density near twice the natural frequencies and the combination frequencies of the system. These results are applied to the problem of flow induced vibration in a supported pipe conveying fluid with pulsating velocity. The effects of mean flow velocity and virtual mass on the extent of parametric instability regions are then discussed.

For a stochastically perturbed two-dimensional system exhibiting a pitchfork bifurcation, asymptotic expansions for the Lyapunov exponent and the rotation number are obtained in the vicinity of the point of deterministic bifurcation. Using the result for the Lyapunov exponent, the shift in the bifurcation point due to the stochastic perturbation is evaluated

The paper deals with the evaluation of the Lyapunov exponent and the rotation number of two-dimensional linear stochastic systems. Exact analytical results are obtained for these quantities for a two-dimensional nilpotent stochastic system commonly occurring in investigations of the stochastic stability of elastic systems

The chaotic behaviour of a parametrically excited system is studied in some detail. An example of such a system is seen in the transverse vibration of a buckled column under axial excitation. The existence of homoclinic orbits for the parametrically excited system is predicted by Melnikov's method. Lyapunov exponents, Poincaré maps, power spectra and other characteristics of chaotic motions are analysed numerically to confirm the results. An experiment on a buckled column is also carried out to qualitatively verify the theoretical results.

Flexural instabilities in moving bands under random tension fluctuation are investigated. The discretized equations of flexural motion represent a gyroscopic system with randomly varying stiffness. Explicit stability conditions based on the mean square amplitudes are established by employing a stochastic averaging procedure. The effect of mean band speed on the band stability is discussed. Copyright 1993, 1999 Academic Press

The torsional vibration of moving bands subject to harmonic tension fluctuation is investigated. A thin rectangular strip translating longitudinally with a constant speed and simply supported at its end is considered. The linearized equation of motion, when suitably discretized, represents a linear gyroscopic system with periodically varying stiffness. The stability of the trivial solution of this system of equations, for tension fluctuations of small amplitude, is examined using the method of averaging. Analytic conditions for stability of torsional motion are obtained explicitly and shown graphically in the frequency vs excitation parameter space.

The tension in power transmitting chains is subject to periodic fluctuation. This is brought about by factors such as polygonal action, eccentricity of sprockets, etc. A study of the dynamic stability of such a chain is presented. Instabilities due to polygonal action are investigated. The analysis is carried out by treating the chain as a travelling uniform heavy string.

In this paper, the dynamic stability of shallow structures such as arches and curved panels under stochastically fluctuating loads is studied. Sufficient conditions guaranteeing the almost-sure stability in both symmetric modes and unsymmetric modes of deformation are obtained first by Infante's method. Necessary and sufficient conditions are determined by evaluating the largest Lyapunov exponent of the perturbed solution.

Turbulence in wind velocity is shown to change the stability condition for the motion of suspension bridges. When the turbulence is convected at a high velocity, the time variation of the random excitation field becomes fast as viewed by the nonmoving structure. In this case the structural response can be approximated by components of a Markov vector process. Modeling the structure as a linear system and considering only the dominant torsional mode, the stability boundaries for the first and the second stochastic moments are obtained in terms of the mean wind velocity and the spectral level of the turbulence component.

The dynamic stability of nongyroscopic, elastic structures under parametric random loading of small intensity is investigated. Conditions for stability in the second norm of the dynamic response are obtained and shown to depend only on those values of the excitation spectral density near twice the natural frequencies and the combination frequencies of the structure. As an application, the flexural-torsional stability of a simply supported thin elastic beam subjected to stochastically fluctuating end couples is studied.

Vibration mode localization in disordered cyclic structures is studied. The equations of motion of a wrap-rib antenna are formulated. Only a single substructure mode is taken for each rib. The localization factors which characterize the exponential rates at which the vibration amplitudes grow (decay) are determined by two approaches. The first approach uses a transfer matrix formulation and Furstenberg's theorem on asymptotic properties of products of random matrices. The second method employs a Green function formulation.

A method of obtaining a sufficient almost-sure (a.s.) asymptotic stability condition for second-order, linear systems with both ergodic damping and stiffness coefficients is presented. The probabilistic property of the correlation between the damping and stiffness coefficients is taken into account. A sufficient condition for a.s. asymptotic stability is derived and numerical results are presented for the case of Gaussian noise coefficients. Results obtained in some of the previous investigations are included in the present study as special cases.

Moments and mean crossing rates are determined for the response of linear systems subject to polynomial forms in Gauss-Markov processes. Differential equations describing the evolution in time of response moments are developed by the use of Itoˆ’s calculus. Estimators are obtained for mean crossing rates based on higher order moments and approximations of the distribution of the response. The method is applied to the analysis of a simple oscillator subject to the square of an Ornstein-Uhlenbeck process.

The variation of the largest Lyapunov exponent for two-dimensional parametrically excited stochastic systems is studied by a method of linear transformation. The sensitivity to random disturbance of systems undergoing bifurcation is investigated. Two commonly occurring examples in structural dynamics are considered, where the random fluctuation appears in the stiffness term or the damping term. The boundaries of almost-sure stochastic stability are determined by the vanishing of the largest Lyapunov exponent of the linearized system. The validity of the approximate results is checked by numerical simulation.

The almost-sure asymptotic stability of a class of coupled multi-degrees-of-freedom systems subjected to parametric excitation by an ergodic stochastic process of small intensity is studied. Explicit asymptotic expressions for the largest Lyapunov exponent for various values of the system parameters are obtained by using a combination of the method of stochastic averaging and a well-known procedure due to Khas’minskii, from which the asymptotic stability boundaries are determined. As an application, the example of the flexural-torsional instability of a thin elastic beam acted upon by a stochastically fluctuating load at the central cross-section of the beam is investigated.

Vibration mode localization in disordered cyclic periodic structures with multiple substructure modes is studied. The localization factor characterizing the exponential rate of growth or decay of the amplitudes of vibration is related to the smallest positive Lyapunov exponent of the corresponding discrete dynamical system. Themultiplicative ergodic theoremof Oseledec is applied to determine the Lyapunov exponents and the localization factors. As a second approach, the Green function formulation is extended to a block matrix system to determine the localization factors for disordered periodic structures.

The almost-sure stability of linear second-order systems which are parametrically excited by ergodic, “nonwhite,” random processes is studied by an extension of the method of Infante. In this approach, a positive-definite quadratic function of the form V = x′Px is assumed and a family of stability boundaries depending on the elements of the matrix P is obtained. An envelope of these boundaries is then solved for by optimizing the stability boundary with respect to the elements of P. It is found that the optimum matrix P in general depends not only on the system constants but also on the excitation intensities. This approach is, in principle, applicable to study systems involving two or more random processes. The results reported in previous investigations are obtained as special cases of the present study.

A method of obtaining a sufficient almost-sure (a.s.) asymptotic stability condition for second-order, linear systems with ergodic damping coefficient is presented. The probabilistic property of the derivative process of the damping coefficient is taken into account. A sufficient condition for almost-sure asymptotic stability is derived and numerical results are presented for the case of Gaussian noise coefficients. The results are found to be a significant improvement over previously available results for second-order systems with stochastic damping.

Flexural instabilities in moving bands under harmonic tension fluctuation are investigated. The discretized equations of flexural motion represent a periodically perturbed linear gyroscopic system. Explicit conditions for the onset of instability are established employing the method of averaging. The effects due to damping, mean band speed, and the band compliance on the band stability are discussed. The present analysis is adequate in the design of moving bands where it is necessary to avoid critical speed ranges.

In this paper bifurcation behavior of parametrically perturbed nonlinear gyroscopic systems is considered. Parametric excitations are assumed to be periodic and the method of averaging is used to examine bifurcation behavior. Explicit results for bifurcating paths in the presence of both subharmonic and combination resonance are obtained for certain classes of systems that commonly occur in practical situations. As an application, the stability of a light rotating shaft that carries a heavy rotor is discussed.

This paper presents an analytical method to examine the dynamical behavior of periodically perturbed linear conservative gyroscopic systems. Explicit stability conditions for perturbations of small intensity are obtained using the method of averaging. The existence of combination resonance in addition to subharmonic parametric resonance is established. For large periodic excitation, a numerical method based on the Floquet theory is used to extend the stability boundaries. These results are applied to the problem of flow-induced vibration in a supported cylindrical pipe conveying fluid with pulsating velocity. The effects of the mean flow velocity, dissipative forces, boundary conditions, and virtual mass on the extent of the parametric instability regions are then discussed.

This paper is in two parts. The first part treats the optimal control of a linear stochastic system with a. quadratic performance criterion. An explicit solution for the cost functional is given. The second port deals with the optimal filtering of the linear system considered in the first part. Modifications in the filtering equation have been made for the case in which the observation noise is of the truncated white noise type. An explicit solution for the a posteriori density function, defined by the filtering equation, is given. The paper concludes with a numerical solution to both the optimal control and the filtering problem associated with a six-plate chemical absorption tower.

The paper presents a general investigation of the behaviour of randomly excited linear dynamical systems whoso parameters are subject to Gaussian white noise variations. The Fokker-Planek equation governing the first probability density of the response is derived, from which the equations determining the response moments are obtained. General criteria arc derived for the stability of linear stochastic systems mid discussed in relation to certain first and second-order systems. Also, the controversy over the question of stabilizing on inherently unstable system by white noise parameter variations is clarified.

The paper is an extension of two previous papers (Ariaratnam and Graefe 1965 a, b) and presents a method of finding correlation functions and spectral densities of the response of systems governed by n state equations whose coefficients are subjected to Gaussian white noise variations. The method is illustrated by an example of a chemical process characterized by two state equations.

The paper is an extension of a previous one (Ariaratnam and Graefe 1965), and presents the analysis of general linear systems governed by n first-Order state equations whose coefficients are subjected to either of two forms of random variation, namely Gaussian white or incremental Brownian type of process. The differences between the two cases are pointed out. The Fokker-Planck equation is set up for both cases and the equations governing the response moments are obtained. An example is given which incorporates both types of random variation of the coefficients.

A method is given for the determination of the natural frequencies and dynamic response of general plane frameworks. The relevant equations are derived in a form suitable for digital computation.

This paper contains three parts. In (a), the Fokker-Planck equation for a linear stochastic central system with additive disturbances is solved. The performance of the optimal system possesses a term for cost duo to the random disturbances at the input. In (b), the optimal control law for a linear system subject to stochastic fluctuations in its parameters and additive disturbances at the input is derived. Stability conditions of the optimal system are considered. In (c), the optimal control of a linear system subject to parametric fluctuations of the truncated white noise type is considered. The corresponding modified Fokker-Planck equation is given, from which the stability conditions of the optimal system are obtained. A second-order example is given to illustrate the methods.

The first-passage time problem for envelope crossing for a linear oscillator subject to stationary wide-band random excitation is investigated. The amplitude of the response is approximated by a one-dimensional Markov process, and differential equations governing the moments of the first-passage time are derived. Analytical solutions are obtained for the mean and the variance

The condition for bifurcation of equilibrium for an elastic-plastic cylindrical shell under axial compression is established. The shape of the yield surface has been left arbitrary; hence the result is applicable to a wide class of inelastic solids. It is shown that the value of the critical stress is quite sensitive to the variation in the unit normal to the local yield surface.

This paper deals with the vibration of heavy unsymmetrical shafts possessing unequal flexural rigidities in the principal directions and running in symmetrical bearings. The existence of several bands of whirling speeds is shown and the effects of stationary and rotary viscous damping on the free and forced vibrations of the shaft are discussed.

A method of obtaining a sufficient almost-sure asymptotic stability condition for second-order, linear oscillatory systems with an ergodic damping coefficient is presented. In this method, the probabilistic property of the derivative process of the damping coefficient is taken into account. A sufficient condition for almost-sure asymptotic stability is derived and numerical results are presented for the case of a Gaussian noise coefficient. The results are found to be an improvement over previously available results for oscillatory systems with stochastic damping.

The response to white noise excitation of a light elastic string loaded at equal intervals by a number of equal masses is examined using the theory of the Markov random process and the associated Fokker-Planck equation. Taking nonlinear effects due to variation in string tension into account, an exact solution of the Fokker-Planck equation is obtained and used in an approximate analytical evaluation of the mean squared response of the system.

The experimental determination of the critical loads of isolated pin-ended struts by the Southwell plot is well known. In this paper it is shown that the Southwell method may be extended to the prediction of the elastic critical loads of plane frameworks.

A proof of Hoff's convergence criterion for the stability of frameworks when the moment distribution is carried out in stagesis presented. The proof is an extension of that, due to Biezeno and Bottema, of the convergence of the ordinary moment distribution process when stability effects are ignored.