Observations of nonlinear interactions in a flexible cantilever beam

Abstract

An experimental investigation into the response of a nonlinear continuous system with many natural frequencies in the range of interest is presented. The system is a flexible cantilever beam, whose first four natural frequencies are 0.65 Hz, 5.65 Hz, 16.19 Hz, and 31.91 Hz, respectively. The fourth natural frequency is about fifty times the first natural frequency. Several cases were considered with this beam. For the first case, the beam was excited with periodic base motion along its axis. The excitation frequency fe was near twice the thud natural frequency f3, which for a uniform isotropic beam corresponds to approximately the fourth natural frequency fq. Thus the thud mode was excited by a principal parametric resonance (i.e., fe=2f3) and the fourth mode was excited by an external resonance (i.e., fe=f4) due to a slight curvature in the beam. Modal interactions were observed involving the first, third, and fourth modes. Through a stationary frequency sweep the response transitions from a periodic, to a periodically modulated, to a chaotic motion. For the second case, the beam was excited with band limited random base motion transverse to the axis of the beam. The first and second modes were excited through nonlinear interactions. For the third case, the beam was excited with base excitation along the axis of the beam at 138 Hz. The corresponding response is dominated by the second mode.