On Restoring Forces Which Admit Forcing Terms of Non-critical Amplitude

In a mechanical system x is the displacement and / is the time. We call g(x) a restoring force and ƒ(/) a forcing term. Throughout this paper f{t) will be periodic and when we speak of a periodic solution of (1) we always mean a solution having the same period as ƒ(/). If g(x) is simply a constant multiplied by x, equation (1) represents a linear oscillator. In this case the amplitude of the forced oscillation is a constant multiple of the amplitude of the forcing function provided this latter is non-resonant. Here, a change in the amplitude of the forcing term merely changes the amplitude of the forced vibration. In the case of a non-linear restoring force, on the other hand, we usually expect [ l ] 1 that changing the amplitude of the forcing term will alter the essential form of the periodic solution. However, as we shall show, there is a class of non-linear restoring forces, for which forcing terms exist with the property that as the amplitude of the forcing function is varied, the periodic solutions resulting all have the same form but vary only in amplitude. A general method is given for determining such a forcing function whenever it exists. As an example, the Duffing equations with g(x) = ±x+bx are discussed, and the forcing terms mentioned are found to exist and are given explicitly in (16).