Nonlinear Dynamics 17: 95–117, 1998. © 1998 Kluwer Academic Publishers. Printed in the Netherlands. Experimental Validation of Reduction Methods for Nonlinear Vibrations of Distributed-Parameter Systems: Analysis of a Buckled Beam ∗ WALTER LACARBONARA and ALI H. NAYFEH Department of Engineering Science and Mechanics, MC 0219, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, U.S.A. WAYNE KREIDER Dominion Engineering Inc., 6862 Elm Street, McLean, VA 22101, U.S.A. (Received: 4 November 1997; accepted: 20 May 1998) Abstract. An experimental validation of the suitability of reduction methods for studying nonlinear vibrations of distributed-parameter systems is attempted. Nonlinear planar vibrations of a clamped-clamped buckled beam about its first post-buckling configuration are analyzed. The case of primary resonance of the nth mode of the beam, when no internal resonances involving this mode are active, is investigated. Approximate solutions are obtained by applying the method of multiple scales to a single-mode model discretized via the Galerkin procedure and by directly attacking the governing integral-partial-differential equation and boundary conditions with the method of multiple scales. Frequency-response curves for the case of primary resonance of the first mode are generated using both approaches for several buckling levels and are contrasted with experimentally obtained frequency-response curves for two test beams. For high buckling levels above the first crossover point of the beam, the computed frequency-response curves are qualitatively as well as quantitatively different. The experimentally obtained frequency-response curves for the directly excited first mode are in agreement with those obtained with the direct approach and in disagreement with those obtained with the single-mode discretization approach. Keywords: Buckled beam, experiment, Galerkin method, direct approach, method of multiple scales. 1. Introduction Two main approximate analytical approaches are being used to study nonlinear vibrations of distributed-parameter systems: finite-degree-of-freedom discretization and direct approaches. With the discretization approach, one assumes a priori either the spatial or the temporal form of the solution; on the other hand, with the direct approach, one attacks the governing partial-differential equations and boundary conditions by using one of the many available reduction methods, with no a priori assumption of the form of the solution. In the case of space discretization, the Galerkin method is commonly used by taking as trial functions only the linear mode shapes of the directly and indirectly excited modes. The ensuing discretization procedure has two possible shortcomings. First, by minimizing the residuals, one may discard the nonlinear terms that are orthogonal to the eigenmodes retained in the expansion. Second, by fixing the shape of the motion a priori, one may overconstrain the system to behave as a low-dimensional system, there by violating the inherent infinite-dimensional nature of the problem. ∗ Contributed by Professor R. A. Ibrahim.