Modeling and numerical results for the argumental transverse vibration of a beam excited through permanent or intermittent elastic contact by a harmonic axial motion

The argumental transverse vibration of a beam excited axially by an harmonic motion transmitted through intermittent or permanent elastic contact is studied. It is shown that this vibration is governed by a nonlinear argumental equation, namely that a vibration in the fundamental transverse mode of the beam can occur when the frequency of the excitation is many times the frequency of the fundamental transverse mode. Two cases are considered: the hinged–(hinged–guided) case and the clamped–(clamped–guided) case. A “natural” model is given and an approached smooth model is derived. The averaging method yields a standard system of differential equations for the smooth model. The stationary condition is represented and studied in the (excitation amplitude, oscillator amplitude)-plane. Stability is studied under symbolic form, and a simple criterion, applicable in said plane, is brought out. The possibility of a stationary condition with permanent contact is discussed. With the help of the Van der Pol representation, numeric simulations allow a comparison between the natural model and the smooth model, leading to clues about the natural model obtained from symbolic relations obtained from the smoothed model.