Equivalent Linearization Analysis of Geometrically Nonlinear Random Vibrations Using Commercial Finite Element Codes

ABSTRACTTwo new equivalent linearization implementations for geometrically nonlinear randomvibrations are presented. Both implementations are based upon a novel approach for evaluatingthe nonlinear stiffness within commercial finite element codes and are suitable for use with anyfinite element code having geometrically nonlinear static analysis capabilities. The formulationincludes a traditional force-error minimization approach and a relatively new version of apotential energy-error minimization approach, which has been generalized for multiple degree-of-freedom systems. Results for a simply supported plate under random acoustic excitation arepresented and comparisons of the displacement root-mean-square values and power spectraldensities are made with results from a nonlinear time domain numerical simulation.1. INTRODUCTIONCurrent efforts to extend the performance and flight envelope of high-speed aerospace vehicleshave resulted in structures which may respond to the imposed dynamic loads in a geometricallynonlinear (large deflection) random fashion. What differentiates the geometrically nonlinearrandom response considered in this paper from a linear response is the presence of bending-membrane coupling, which gives rise to membrane stretching (in-plane stresses) in the former.This coupling has the effect of stiffening the structure and reducing the dynamic responserelative to that of the linear system. Linear analyses do not account for this effect andconsequently may significantly over-predict the response, leading to grossly conservativedesigns. Without practical design tools capable of capturing the nonlinear dynamics, furtherimprovements in vehicle performance and system design will be hampered.Methods currently used to predict geometrically nonlinear random response include perturbation,Fokker-Plank-Kolmogorov (F-P-K), numerical simulation and stochastic linearizationtechniques. All have various limitations. Perturbation techniques are limited to weak geometricnonlinearities. The F-P-K approach [1, 2] yields exact solutions, but can only be applied tosimple mechanical systems. Numerical simulation techniques using numerical integrationprovide time histories of the response from which statistics of the random response may becalculated. This, however, comes at a high computational expense due to the long time recordsor high number of ensemble averages required to get high quality random response statistics.Statistical linearization methods, for example equivalent linearization (EL) [[2-7]), have seen the