Nonlinear Vibrations of a Beam With Pinned Ends

Abstract

This paper describes theoretical and experimental investigations of the large-amplitude vibrations of a flexible beam simply supported on a nearly rigid base. The beam is designed to minimize most secondary effects, such as transverse shear flexibility, rotatory inertia, and nonlinearities in curvature and in the stress-strain curve. Detailed attention is given to quantitative verification of the assumptions made in deriving the equation of motion. Three different approaches are used to solve the equation: assumed-space mode, assumed-time mode, and Ritz-Galerkin solutions. In the experiments, the beam was base excited. The resonant frequencies and associated strain distributions, modal shapes, and waveforms were measured for various values of initial tension and amplitude ratios (ratio of maximum amplitude to beam thickness) up to 16. Within experimental accuracy, the experimental results verified all of the assumptions made in the analyses. However, no one method of solution resulted in the best predictions of all of the experimentally observed phenomena.

Sandwich Structures

“…As in the previous research of Yu [144] and Ray and Bert [145], it is assumed that the effect of coupling among individual vibration modes is not quantitatively significant, which has been confirmed by comparing analytical predictions with experimental results.…”

Nonlinear motion of a beam

“…In [1] it was shown that equations (8) are solved by the Jacobian elliptic functions u m = cn(wmt + am, km) (11) in which the natural frequencies (o m and the moduli k m are 2= c~m and k 2= 1-.…”

“…Other reports of investigations were given by Eisley [6], Morris [7], Srinivasan [8], Evenson [9], Bennett and Eisley [10] and Ray and Bert [11]. A succinct summary of developments through 1972 is given by Busby and Weingarten [12], and an encyclopaedic listing of references to work both immediately and closely related is to be found in the research monograph Nonlinear Oscillations, by Ali H. Nayfeh and Dean T. Mook [13].…”

Finite Element Formulation for the Large-Amplitude Vibrations of FG Beams

“…Employing the elliptic integral solution, he investigated the nonlinear vibrations of hinged-hinged beams with axially immovable ends. This issue was afterwards addressed by several authors using perturbation and Ritz-Galerkin methods (Srinivasan, 1965;Evensen, 1968;Ray and Bert, 1969). There can be found some newer analytical studies on the nonlinear vibrations of beam, which seem to be useful, such as (Azrar et al, 1999;Emam, 2009;Emam and Nayfeh, 2009;Pirbodaghi et al, 2009).…”