Equivalent nonlinear potential systems: Review of previous assumptions

Few exact solutions for the response of stochastic dynamical systems are available. Thus, the methods of the equivalent linearization and of the equivalent nonlinear system are often used. While the former yields a Gaussian response to a Gaussian excitation, the latter gives a non-Gaussian response, which is nearer to the exact unknown response of a nonlinear system. Among the equivalent nonlinear system methods the one based on the replacement of the actual dynamic system by means of a potential system is promising, Cavaleri and Di Paola (2000) [41], as it leads to a fixed procedure differently from other methods. The procedure is developed under the assumption that the ratio of the moments E [ Λ j X ̇ 2 ] and E [ Λ j + 1 ] is a constant α not depending on j , Λ being the mechanical energy of the oscillator and X ̇ its velocity. This relationship is demonstrated only for the so called Robert’s oscillator, that is an oscillator with restoring force g ( X ) = ω 0 sgn ( X ) ⋅ | X | ν , but in the above mentioned reference it is argued that it is valid in general. In this paper, numerical analyses are presented to ascertain its validity in order to apply the method to systems with different restoring forces. It is found that in some cases it is true, some others are doubtful, while there are cases in which α clearly depends on j . The effects of considering α constant when it is not are ascertained for a Duffing oscillator with linear plus cubic damping.