Dynamic Stability of a Base-Excited Thin Beam With Top Mass

Abstract

This paper deals with a base-excited clamped-clamped vertical thin beam carrying a top mass. The thin beam is considered to be inextensible and initially not perfectly straight. Based on Taylor series expansions of the inextensibility constraint and the exact curvature, and by using one or more basis functions, a semi-analytical model is derived. This model is numerically validated through a comparison with quasi-static and modal analysis results obtained using finite element modelling. The steady-state nonlinear dynamics of the base-excited beam are investigated using numerical continuation of periodic solutions and bifurcations. Using these numerical tools, the dynamic stability of the beam is investigated for various parameter settings, including the effect of nonlinear damping. The continuation of bifurcations appears to be very suitable to determine whether or not parametric resonance occurs for certain parameter settings.

Nonlinear resonances in an axially excited beam carrying a top mass: simulations and experiments

“…in terms of Q * , see (19), and its first time derivative. By substituting DOFs Q i in (9) and their first time derivatives in the first time derivative of (9), v(t, L/4) andv(t, L/4) are obtained, from which directly the peak-to-peak values can be derived.…”

“…(5)- (6). By considering higher-order expansions of the exact kinematics and a multi-mode discretization, it is shown in [9] that the third-order single-mode semi-analytical model can (to a large extent) accurately describe the first harmonic resonance and the first (large amplitude) 1/2 subharmonic resonance of the base-excited (initially unbuckled) slender beam. It is noted that for accurate steady-state response prediction of an initially buckled beam, in general higher-order approximations of the exact kinematics are required [19].…”

“…As can be noted, the Young modulus E is identified to be a little bit lower than the theoretical value. This may be due to a combination of the following three facts: (1) in the set-up, the clamped-clamped boundary conditions will not be as ideal as considered in the model, (2) for the discretization of v, see (9), not the exact vibrational eigenmodes are used, which will result in a model, which overestimates the resonance frequencies, and (3) in the model, axial inertia and axial strain of the beam are not included, which may also result in a small overestimation of the resonance frequencies of the model. For beam 1, the eigenfrequencies and damping ratios of the resulting 1-MODE model linearized around the static equilibrium position are listed in the first column of Table 4.…”

“…The resulting base acceleration will not be purely harmonic, will not have a constant amplitude but will be influenced by the dynamics of the shaker system carrying the slender beam with top mass. Response results for voltage excitation can thus not directly be compared with results for a prescribed harmonic base acceleration as considered in [9]. To be able to compare the experimental results with the semi-analytical results in a quantitative manner, the equations of motion for the base-excited slender beam with top mass will be coupled with a model of the shaker.…”

“…where DOFs Q i are the N generalized DOFs of the beam, see (9). In the model of the slender beam, the axial motions are defined with respect to an arbitrary base motion U b .…”