RESPONSE OF ELASTIC SHOCK OSCILLATORS AND ELASTO- PLASTIC OSCILLATORS, WHITELY EXCITED, FROM ENERGY BALANCE AND STOCHASTIC ANALYSES

摘要

This paper deals with mechanical nonlinear oscillators, of mass-spring type, subjected to white noise excitation. First, a remarkable property is noted: the energy, per time unit, injected into the oscillator by the external force depends only on the white PSD level (directly), and the oscillator mass (inversely). So, different oscillators, whatever their nonlinear-ities and dissipation mechanisms, receive, and consequently dissipate, the same energy rate from the white noise provided they have the same mass. It is discussed how this simple energy balance, eventually connected to additional conjectures, gives information about the dynamics of oscillators of two classes, (i) the elastic shock oscillators (ESO), and (ii) the elasto-plastic oscillators (EPO), both including different subclasses (for the symmetry of stops or the type of plasticity). But relevant theories are needed to go further. The paper focuses on two items linked to stochastic diffusion processes. Concerning ESO with undamped stops, the theory leads, via the Fokker-Planck equation, to an analytical expression for the probability density function (PDF) of the oscillator in the phase space. Quantities of interest (like Rice frequency, impact frequency or distribution) may then be deduced under closed-form, even for totally asymmetric ESO. Concerning (perfectly) EPO, a completely new approach has appeared for a decade, based on stochastic variational inequality (SVI). The infinitesimal generator proper to the diffusion is degenerated and works in the special phase space defined from the EPO velocity and its elastic displacement. Joint PDF can be numerically determined through heavy computations, as illustrated by an example using finite differences. Finally, the paper comments the “white noise of engineers”, the PSD of which is realizable since truncated in frequency, which justifies some assumptions and simplifies the plastic-elastic transitions for EPO, as well as the stop contact phases for ESO.