| The present study deals with the effect of nonlinear modal interaction of a clamped-clamped beam in the neighborhood of internal resonance condition. The beam is subjected to wide band random excitation or filtered white noise excitation in a direction normal to the beam span and a static axial load is initially applied at one end of the beam. Depend on the magnitudes of the axial load, the research is divided into two different cases. The first case deals with the beam dynamic behavior below the first Euler buckling load. The nonlinearity in this case is a geometric type due to stretching of the mid-plane of the beam. The second case considers axial load which exceeds the first Euler buckling load. In this case, in addition to the nonlinearity due to stretching of the mid-plane, the initial curvature of the beam causes other nonlinearity. In each case, the motion of the beam is governed by the nonlinear partial differential equation, which is derived by using Hamilton's principle. The resulted fourth order partial differential equation is reduced to a set of second order ordinary differential equations by applying Galerkin's method. Under certain values of the static axial load, the natural frequencies of normal modes have linear relationships and this linear relationship results in a fourth-order internal resonance condition among the first three modes for the case of initially straight beam and the second-order internal resonance condition between the first two modes for the case of initially buckled beam.; The response characteristics are examined via three different approaches. These are analytical, numerical simulation, and experimental testing of the physical model. The analytical approach is based on Markov vector approach together with the system Fokker-Plank equation. The moment equations of these nonlinear systems are found to form an infinite hierarchy problem. This infinite hierarchy problem is resolved by using Gaussian and non-Gaussian closure schemes. The Monte Carlo simulation method is used for determining the response statistics of the system. Mean square response behavior to wide band random excitation or filtered white noise excitation examined via the analytical and numerical approaches. Both analytical solution and Monte Carlo simulation yield bifurcation boundaries of the non-excited mode in terms of the system parameters, such as, axial load, excitation spectral density level, system damping ratios, and damping ratio of the linear filter equation. The probabilistic descriptions of the response amplitude are established for Monte Carlo simulation and experimental results. Those results are compared qualitatively. |
