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Elsevier

DISCUSSION

DISCUSSION TO: W.D. IWAN AND Z.K. HOU, EXPLICIT SOLUTIONS FOR THE RESPONSE OF SIMPLE SYSTEMS SUBJECTED TO NONSTATIONARY RANDOM EXCITATION STRUCTURAL SAFETY, 6 (1989) 77-86 J.D. Riera

*

CPGEC, Universidade Federal do Rio Grande do Sul, Porto Alegre, Brasil

C.G. Bucher Institute of Engineering Mechanics, University of Innsbruck, A-6020 Innsbruck, Austria

In their recent paper, Iwan and Hou present explicit solutions for the response of a simple single-degree-of-freedom viscously damped oscillator subjected to an amplitude-modulated white noise excitation. The solution, obtained in terms of convolution integrals derived through a state-vector-formulation in the time domain, is elegant and appropriate to the problem at hand, but appears not to be new. The general concept is given, for instance, by Papoulis [1]. More specific solutions, moreover, are as well available. In fact, Riera [2], after casting the equation for the evolution of the covariance matrix into the form

~-

=

~n

--2~.~°

--~°

0

2o3n

0

Svv

Sxo +

So n~(t)

(1)

Sxx

obtained in 1977 the solution by means of Laplace Transformation (LP). Thus, the LP of the variance of the displacement, for example, is

L(Sxx(t))

s'

2o34S° L(~E(t)) + 6~',o3ns2 + 4 ( 1 + 2~'n) : O3,S ~ + 8~no3~

(2)

in which s is the transformed variable. Inverse transformation of eqn. (2) yields, by resorting to the convolution theorem

So

O32

S ~ = 2 6 7 - 7 f 2) ftexp[--~o3n(t J0 -r)l[1-c°s

2O3d(t - r)] r/2(r) d r

* Present adress: Institute of EngineeringMechanics, University of Innsbruck, Innsbruck, Austria. 0167-4730/90/$03.50 © 1990 - Elsevier Science Publishers B.V.

(3)

74

which is clearly equivalent to eqns. (18) and (29) of the paper by Iwan and Hou. An explicit solution for the step modulating function is also given in Ref. [2]. This explicit equation for the evolution of the variance was later employed by Scherer et al. [3] in the evaluation of filters to characterize the evolutionary spectral density of seismic excitation. In this context the extensive work of Gasparini [4] and Gasparini and DebChaudhury [5] on the state-variable formulation for the analysis of linear multi-degree-of-freedom systems is of utmost interest. In fact, their analytical solution for piecewise linear intensity functions appears to be sufficiently simple for all computational purposes. Additionally, it should be mentioned that a convenient way of further simplifying computations is given by the approximate procedure for multi-degree-of-freedom systems described by Bucher [6].

References 1 A. Papoulis, Probability Random Variables and Stochastic Processes, McGraw-Hill, Singapore, 1984. 2 J.D. Riera, Application of the state-variables method to the analysis of dynamic systems under nonstationary random excitation, in: Anaris Congresso Brasileiro de Engenharia Mecanica, A, Florianopolis, Brasil, Dec. 1977, Vol. A, pp. 255-268 (in Portuguese). 3 R.J. Scherer, J.D. Riera and G.I. Schu6ller, Estimation of the time-dependent frequency content of earthquake accelerations, Nucl. Eng. Des., 71 (1982) 301-310. 4 D.A. Gasparini, Response of MDOF systems to non-stationary random excitation, J. Eng. Mech. Div., ASCE, 105 (1) (1979) 13-27. 5 D.A. Gasparini and A. DebChaudhury, Dynamic response to nonstationary nonwhite excitation, J. Eng. Mech. Div., ASCE, 106 (6) (1980) 1233-1248. 6 C.G. Bucher, Approximate nonstationary random vibration analysis for MDOF systems, J. Appl. Mech., 55 (March 1988) 197-200.