Stephen H. Crandall

Non-gaussian closure for random vibration of non-linear oscillators

Abstract An approximate method for determining estimates of stationary response statistics for a non-linear oscillator driven by wide-band random excitation is described. The differential equation of the oscillator is used to generate relations between unknown response statistics. These relations are then used to fix a corresponding number of unknown parameters in a non-Gaussian probability distribution for the response. The method is illustrated by application to the Duffing oscillator. Elementary methods for evaluating the necessary correlations between excitation and response are discussed in an Appendix.

Non-Gaussianclosure techniques for stationary random vibration

Abstract The classical method of statistical linearization when applied to a non-linear oscillator excited by stationary wide-band random excitation, can be considered as a procedure in which the unknown parameters in a Gaussian distribution are evaluated by means of moment identities derived from the dynamic equation of the oscillator. A systematic extension of this procedure is the method of non-Gaussian closure in which an increasing number of moment identities are used to evaluate additional parameters in a family of non-Gaussian response distributions. The method is described and illustrated by means of examples. Attention is given to the choice of representations of non-Gaussian distributions and to techniques for generating independent moment identities directly from the differential equation of the non-linear oscillator. Some shortcomings of the method are pointed out.

THE EFFECT OF DAMPING ON THE STABILITY OF GYROSCOPIC PENDULUMS

There is a fascinating category of mechanical systems which exhibit the following paradoxical behavior: when modelled as systems without damping they possess stable equilibria or stable steady motions, but when small damping is introduced, those equilibria or steady motions become unstable. The paradox revolves around the fact that energy must be supplied to the growing unstable disturbances, but the damping forces are themselves consumers of energy and are thus incapable of directly supplying energy to the growing disturbances. The explanation of the paradox is that in these systems there are substantial energy sources which remain dormant until the damping forces are introduced. The damping forces act as imperfect transfer agents supplying energy from the previously dormant sources to the growing disturbances while dissipating some additional energy.

Random Vibration of One- and Two-Dimensional Structures

Publisher Summary This chapter focuses on statistical relations between excitation and response processes for a broad range of structural elements including beams, cables, arches, plates, membranes, and shells. The analysis of the dynamic response of continuous structures under random excitation was initiated in connection with Brownian motion (Van Lear and Uhlenbeck, 1931). More recent developments arose out of work on aerospace problems connected with jet-noise excitation. The chapter presents general formulations and a number of solution procedures. It focuses on approximate procedures. The chapter highlights experimental procedures and presents a few comparisons between measurements and analytical predictions. Special consideration is given to stationary wide-band excitation of uniform structures. Under certain circumstances, the random response distributions exhibit patterns that become asymptotically very simple as the number of responding modes increases.

Is stochastic equivalent linearization a subtly flawed procedure

Abstract The standard equivalent linearization procedure for estimating the mean and variance of the response of nonlinear dynamic systems has proved to be an unusually effective technique. For over forty years there has been general agreement about the procedure to be followed. Recently two independent claims have been made that the standard procedure harbors a subtle flaw. In place of the standard procedure, essentially the same alternative procedure was claimed to be the “correct” procedure, even though, in the test cases investigated, the alternative “correct” procedure produced estimates with greater errors than the “incorrect” standard procedure. The present note investigates the claim that the standard procedure is flawed and finds that: (a) there is no subtle flaw in the standard procedure; (b) the proposed alternative procedure differs from the standard procedure in that it employs a different criterion for selecting the optimum linear approximation; (c) there is also no flaw in the proposed alternative procedure; but, (d) there does not seem to be any practical advantage to using the proposed alternative, since the standard procedure is simpler and more accurate.

Numerical Treatment of a Fourth Order Parabolic Partial Differential Equation

which satisfy prescribed initial conditions for ~b a n d ~ / ~ t in 0<x<l at t = 0 and which satisfy prescribed boundary conditions for ~b and ~ / ~ x at x = 0 and x = 1 for subsequent t. This formulation represents (in dimensionless terms) the problem of predicting the transient response of a uniform flexible beam clamped at both ends, whose displacements and velocities are initially known. The particular case, which is later treated numerically, is that occasioned by suddenly removing from a naturally straight beam a continuously distributed load with intensity sin rr x. This problem was first studied by Collatz, i who showed that an apparently straightforward finite difference approach involving an explicit recurrence formula led to completely absurd results. Later, in the light of the stability studies on the • . 2 3 diffusion equation, he showed that his earlier approach would have been successful if his time interval had been taken half as large as it was. In what follows, an implicit recurrence formula is presented and its properties are studied and compared with the explicit formula. In addition to a theoretical analysis, the results of actual computations are compared. 1. Recurrence formulas. The solution domain is covered with a rectangular network with uniform spacings Ax and At. Let

Optimum Recurrence Formulas for a Fourth Order Parabolic Partial Differential Equation

in the unit strip 0 6 x ~ 1, t > 0 represents the transient nmtion of a finite beam according to the Euler-Bernoulli theory. :Approxinmte solutions can be obtained by covering the donmir~ b y a rectangular network with spacing &x and &t and mai-ching out. a finite difference approximation to (1). An explicit recurrence formula for this process was given by Collatz [1]. Implicit formulas were given by Crandall [2], by Royster and Conte [3], and b y Conte [4]. All of these formulas had truncation errors proportional to (&x):. In this note we consider a more general class of recurrence formulas which contain those previously considered but also contain formulas which have truncation errors proportional to (~x) ~ and (~x)6 . The treatment closely para!lels that recently given [5] for tbt:~ linear diffusion equation.

Wide band random vibration of an equilateral triangular plate

Abstract The distribution of mean-square velocity response over a lightly damped uniform plate excited at a point by a wide-band random force tends to be more or less uniform except in certain special zones or lanes of intensified or decreased response. The locations of these zones depend on the symmetry of the plate shape and on the position of the driving-point. In the case of an equilateral triangular plate there is intensified response along a median whenever the driving-point lies on that median. When the driving-point is at the centroid of the triangle there is intensified response along all three medians with a localized concentration at the driving-point. It is shown that the driving point intensification ratio increases slowly, but without limit, as the excitation bandwidth increases due to the increasing multiplicity of the higher-frequency plate resonances. Quantitative estimates for the driving-point response are obtained from a computer evaluation of the multiplicities of the first 32 000 resonances and from an asymptotic approximation based on the theory of numbers.

Implicit vs. Explicit Recurrence Formulas for the Linear Diffusion Equation

in the unit strip 0 K x < 1, t > 0. The c l a s s of problems env i saged are those in which ~b(x,O) = 0 and O~(1, t ) /Ox=O while (_,(O,t) is a prescr ibed function. By using the theory of the superpos i t ion integral1 we find that th is c l a s s of problems reduces to a cons idera t ion of the response to a unit s tep in ~,(O,t). The solut ion domain is covered with a uniform network with spac ing Ax = h = 1/M and At. The ratio A t / ( A x ) 2 is denoted by r. Let g,(jAx, hAt ) be writ ten as ~bi,k. We cons ider t h r ee finite d i f ference recur rences formulas as approximations to (1). They are al l spec ia l c a s e s of the following.

Modal-sum and image-sum procedures for estimating wide-band random response of structures

Abstract The nature of the mean-square velocity response field in one- and two-dimensional structures subjected to point excitation by wide-band random forces is described, with emphasis on the effects of damping and of excitation bandwidth. Two procedures for predicting the response are studied. Both procedures require evaluating an array of integrals. The sizes of the arrays and the relative dominance of the main-diagonal terms in the arrays are shown to depend differently on the damping, excitation and dimensionality of the structure. The modal-sum procedure is advantageous for lightly damped, two-dimensional structures excited by relatively narrow-band processes. The image-sum procedure is advantageous for heavily damped, one-dimensional structures excited by very wide-band processes. The image-sum procedure has the additional feature that it provides a simple means of locating local zones of intensification and reduction of response and estimating the magnitude of the relative intensification or reduction in such zones.