Primary Resonance Response of a Beam with a Differential Edge Settlement Attached to an Elastic Foundation

In this work, we study, through one-mode simplification, the influence of frequency curve veering on the primary resonance response of an elastic Euler-Bernoulli attached to a linear Winkler elastic foundation. The beam is a hinged-hinged with one torsional spring at one end, has an initial 1/4 sine deflection shape due to a constant differential edge settlement and is subjected to a uniformly distributed vertical load which is harmonically varying with a time part and a large mean part. A combined numerical-analytical procedure which accounts for the nonlinear interdependence between the vertical deflection and induced axial force due to mid-plane stretching was used to determine the beam static deflection. The assumed single mode approach is used to obtain the reduced nonlinear temporal equation of motion about the static equilibrium deflection which contains quadratic and cubic nonlinear terms. The results of numerical simulation indicate that the coefficients of the quadratic and cubic nonlinear terms in the temporal problem can, depending on the selected range of system parameters, vary widely and take positive and negative values, and thus change the number and stability of equilibrium positions as well as the system behavior which can be a hardening or softening type. The harmonic balance and, for comparison purposes, the method of multiple-scales are used to obtain approximate analytical solution for the primary resonance response and its stability. The obtained primary frequency response results are presented over a selected range of system parameters near and away from veering points which show a significant change in response behavior. The obtained approximate-analytical results were compared with those obtained numerically.