~~ ; .:'All ELSEVIER 0266-8920(94)00003-4 ProbabilisticEngineeringMechanics 10 (1995) 11-21 © 1995 Elsevier Science Limited Printed in Great Britain. All rights reserved 0266-8920/95/$09.50 Structural random vibration analysis by using recursive formulas Dongyao Tan Beijing University of Aeronautics and Astronautics, School of Astronautics, P.O. Box 504, Beijing 100083, China & Qingshan Yang Harbin Architectural and Civil Engineering Institute, Harbin 150006, China (Received October 1992; revised March 1994; accepted April 1994) This paper discusses the application of/3-recursive formula to structural random vibration analysis. This recursive formula is able to calculate the accurate stationary mean and mean square responses of a linear or equivalent linearized nonlinear structure subjected to stationary excitations. To enhance the computational efficiency, the basic formulae for calculating structural mean and mean square responses which are obtained from /3-recursive formula are transformed into other forms which directly use the band-shape and symmetric structural dynamic characteristic matrices in calculation. The detailed formula- tions are given according to different excitation environments, such as white noise excitations, time-domain correlated white noise excitations, independent filtered white noise excitations and spatially correlated filtered white noise excitations. Some numerical examples are given to show the applications of the recursive formula. INTRODUCTION Structural random vibration analysis by using numerical methods became a feature in the field of random vibration theory in the last decade. For examples, Di Paola et al.1 gave an unconditionally stable step-by-step method, based on the eigenvalue analysis, to calculate the mean square response of a multidegree of freedom linear or nonlinear system subjected to nonstationary filtered noises, C. W. S. To 5'6 proposed explicit and implicit direct integrators to calculate the random response of multidegree of freedom systems. Various numerical examples have been given to illustrate the efficiency of these methods. This paper uses/3-recursive formula which was proposed and studied by Dongyao Tan et al. TM to calculate the stationary and non- stationary mean and mean square responses for large linear or equivalent linearized nonlinear structures. /3-recursive formula was obtained by discretizing a general state equation in time domain with the assumption that the state vector changes linearly between two adjacent discrete time instants. 7 It has been proved to be unconditionally stable, and it can 11 carry out the accurate stationary mean and mean square responses for any linear or equivalent linearized non- linear dynamic system. 9'1° It also has second order accuracy in calculating non-stationary mean and mean square responses. 1° This paper discusses its application in structural random vibration analysis. Unlike struc- tural deterministic dynamic analysis, the detailed calculation formulations in structural random vibration analysis depend largely on the types of excitations to which the structures are subjected. Here, the detailed and practical calculation formulations by using /3- recursive formula to calculate structural mean and mean square responses are deduced with respect to four important excitation environments. The first is white noise excitation which roughly describes the engineering background of multi-dimensional earth- quake motions exciting a structure simultaneously. The second is time-domain correlated white noise excitation which describes the situation of a propagat- ing earthquake motion exciting a large span structure from its different supports. The third is independent filtered white noise excitation which describes in some detail the earthquake motion or any other coloured