Random vibration of a nonlinearly deformed beam by a new stochastic linearization technique

Abstract A new stochastic linearization technique is employed to investigate the large amplitude random vibrations of a simply supported or a clamped beam on elastic foundation under a stochastic loading which is space-wise either (a) white noise or (b) uniformly distributed load and time-wise white noise. The new version of the stochastic linearization method is based on the requirement that the mean square deviation of the strain energy of the nonlinearly deformed beam, and the strain energy of the equivalent beam in a linear state, should be minimal. As a result, the modal mean square displacements are expressed as solutions of a set of nonlinear algebraic equations. Results obtained by the conventional equivalent linearization method and by the new technique are compared with the numerical results obtained from integration of the exact probability density function (when the exact solution is available) or with the result of the Monte Carlo simulations (when the exact solution is unavailable). It is shown that the new stochastic linearization technique yields a much more accurate estimate of the mean square displacement than the classical linearization method, which has attracted the past interest of about 400 investigators in a variety of nonlinear random vibration problems.

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