Non-autonomous Dynamical Systems

Beams on elastic soil, solicited by time-varying forces, inducing external or parametric resonances, are considered. Formally, they are continuous dynamical systems explicitly dependent on time, and therefore said non-autonomous,Non-autonomous systems either in their know term (causing external excitation) or in their coefficients (triggering parametric excitation). Transverse loads cause external excitation. According to the value assumed by their frequency, three possible forms of resonance are examined here, namely: primary, sub-harmonic and super-harmonic. The frequency-response law, linking the amplitude of the periodic motion (limit cycle) to the external frequency, is determined for each of these resonance. As a particular case, undamped free vibrations are studied. The parametric excitation, is related to a pulsating axial load, which makes the stiffness of the beam harmonically depending on time. When the excitation frequency is double of a natural frequency of the beam, the principal parametric resonance phenomenon takes place. For such phenomenon: (i) the instability region in the bifurcation parameter plane is determined, and (ii) the limit cycles bifurcating from its boundary, are evaluated. All the problems dealt with in this chapter are attacked by the Multiple Scale Method, which provides a specific bifurcation equation for each type of resonance, ruling the slow-time evolution of amplitude and phase of the natural mode involved in the motion. The influence of damping, taken as auxiliary parameter, is also investigated. The region of the existence of the limit cycles are discussed, and their stability detected. Multivalued nonlinear responses, and consequent jump phenomena are commented.