| The nonlinear response characteristics of a clamped-clamped beam is investigated analytically, numerically, and experimentally. The beam is under an initial static axial load and subjected to a harmonic excitation of its support. Two ranges of the axial load are considered. These are below (the beam is initially straight) and above Euler buckling load (the beam is initially buckled). Hamilton's principle is used to derive a fourth order partial differential equation of motion which is descritized and reduced to a set of second order ordinary differential equations by applying Galerkin's method. Under certain values of the static load, the normal modes are nonlinearly coupled and this coupling results in a fourth order internal resonance condition between the first three modes when the beam is initially straight. Second and third order internal resonance conditions occur between the first two modes for the case of initially buckled beam. The multiple scales method showed the significant effects of these internal resonance conditions on the system behavior. In the straight beam case, the third mode which is externally excited transfers energy to the first two modes within a small range of internal detuning. Outside this region, the response is governed by a unimodal response of the third mode. In the neighborhood of 1:1 internal resonance, it is found that within the region of two mode interaction, the solution is either stationary or nonstationary depending on the excitation level and system parameters. Saturation and jump phenomena are found to take place in the case of two mode interaction with 2:1 internal resonance. Numerical simulation and experimental testing confirmed these predictions and revealed the occurrence of multifurcation, snap-through (escaping from one well to the other in an irregular manner), and chaotic motion. |
