Chaos in a Harmonically Excited Elastic Beam

Abstract

Numerical integration of the equations for the evolution of amplitudes of a harmonically forced simply supported elastic beam indicates that for certain values of the frequencies of excitation the response of the beam may be chaotically modulated. In an attempt to confirm the presence of chaos, the Lyapunov exponents have been calculated for two cases in which the broadening of the power spectra around the dominant peaks occurs. In both the cases one of the Lyapunov exponents is positive; this provides further evidence of chaos. The Lyapunov dimension of the attractors in these cases is estimated to be 2.09 and 2.11, respectively.

Non-Linear vibrations and chaos in harmonically excited rectangular plates with one-to-one internal resonance

“…The modal amplitudes X 1 and X 2 are functions of time and the nonlinear terms in the system determine their time evolution. Substituting equation (8) into equation (5), the solution for the resulting linear partial differential equation in the stress function F can be written as F(x, y, t) = Fh(x, y, t) + FP(x, y, t) , (9) where F h is the homogeneous solution which includes the effect of the in-plane stretching forces independent of the transverse deflection, and F p is the particular solution that includes the effect of out-of-plane boundary conditions. The particular solution F p can be easily shown to be [13] l(n 2 m 2 )…”

“…modal equations to four first-order differential equations representing the slow-time evolution of amplitudes of harmonic motion of the two modes. These amplitude or averaged equations are a generalization of those that describe the motion of square plates [12] and membranes [10] and, when the additional restriction of circular symmetry is imposed, they have arisen in the study of resonant motions of a spherical pendulum [4], a stretched string [11], and forced response of axisymmetric shells [15] and beams [9]. The amplitude equations for the rectangular plate depend on three nonlinear coefficients, in contrast to the two independent nonlinear coefficients found in the above mentioned studies.…”

A finite element approach to the chaotic motion of geometrically exact rods undergoing in-plane deformations

“…Since the work of Holmes [4], chaotic motion of beams has attracted the interest of many researchers (see, e.g., [5,7,8,10]). In some papers (e.g., [9,[11][12][13]) the arch has been considered too.…”

Nonlinear nonplanar dynamics of a parametrically excited inextensional elastic beam

“…Hyer [9] studied the nonplanar dynamics of a forced inextensional cantilever beam, and Luongo et al [10] investigated the free nonplanar motions of an inextensional elastic beam supported in an arbitrary manner but with no axial restraints. Nonplanar chaotic motions of a harmonically forced and simply supported inextensional elastic beam with an axisymmetric cross section were studied by Maewal [11]. Also, pertaining to the subject of forced nonplanar motions is the recent study of Johnson and Bajaj [12] who showed nonplanar amplitude-modulated and chaotic motions of harmonically forced strings.…”