Contrasting probabilistic and anti-optimization approaches in an applied mechanics problem

Probabilistic and non-probabilistic, anti-optimization analyses of uncertainty are contrasted in this study. Specifically, the comparison of these two competing approaches is conducted for an uniform column, with initial geometric imperfection, subjected to an impact axial load. The reliability of the column is derived for the cases when the initial imperfections posses either (a) uniform probability density, (b) truncated exponential density or (c) generic truncated probability density. The problem is also analyzed in the context of an interval analysis. It is shown that in, the most important near-unity reliability range these two approaches tend to each other. Since the interval analysis…

Stochastic linearization critically re-examined

Abstract The stochastic linearization technique, widely used for the analysis of nonlinear dynamic systems subjected to random excitations, is revisited. It is shown that the standard procedure universally adopted for determining the so-called effective stiffness of the equivalent linear system is erroneous in all previous publications. Two error-free stochastic linearization techniques are elucidated, namely those based on (1) the force linearization and (2) energy linearization.

A subtle error in conventional stochastic linearization techniques

Abstract The stochastic linearization technique as applied to the SDOF system is re-examined. Two standard procedures associated with the stochastic linearization, widely adopted in the literature, are shown to be erroneous. Two new procedures to correct the errors made in previous works are introduced. To gain more insight, the procedures are applied to the quintic oscillator. Comparative numerical analysis is performed.

Accuracy of the finite difference method in stochastic setting.

In this paper we study the accuracy of the finite difference method when the finite difference method is applied to approximately analyze the structure.

A New Look at the Stochastic Linearization Technique for Hyperbolic Tangent Oscillator

Abstract Stochastic linearization technique is reconsidered for oscillator with restoring force in form of hyperbolic tangent. We show that a subtle error was made in the previously known procedure for derivation of the linearized system parameters. Two new error-free procedures, namely, those based on minimization of mean square difference between (a) restoring force or (b) potential energy of the original non-linear system and their linear counterparts, are suggested. The results of numerical analysis are shown.

Convergence of Boobnov-Galerkin Method Exemplified

In this Note, Boobnov–Galerkin’s method is proved to converge to an exact solution for an applied mechanics problem. We address in detail the interrelation of Boobnov–Galerkin method and the exact solution in the beam deflection problems. Namely, we show the coincidence of these two methods for clamped–clamped boundary conditions, using an alternative set of functions proposed by Filonenko-Borodich.12 Received 25 February 2003; accepted for publication 13 March 2004. Copyright c 2004 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. Copies of this paper may be made for personal or internal use, on condition that the copier pay the $10.00 per-copy fee to th…

Booton's problem re-examined

The application of the stochastic linearization technique to the specific problem analyzed by Booton is reexamined. It is shown that Booton has made a subtle error in the procedure for minimization of the mean square force difference between the sharp limiter and its linear equivalent counterpart

Error in the finite difference based probabilistic dynamic analysis: analytical evaluation