Forced Oscillations with Nonlinear Restoring Force

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Abstract
Forced oscillations with nonlinear restoring force are studied in transient states as well as in steady states. The original differential equation characterized by a nonlinear term is transformed under certain restrictions to the following differential equation of the first order: dy/dx=Y(x,y)/X(x,y). Following Poincaré and Bendixson, the singularities and the integral curves of the above equation are discussed, the former being correlated with the periodic states of oscillations, the latter with the transient states of oscillations. The stability of the periodic solutions is determined in accordance with that of the singular points, viz., according to the roots of the characteristic equation. The integral curves yield the relationship between the given initial conditions and the periodic solutions. Thus, once the initial conditions are prescribed, we can foresee the final periodic states which are started with those conditions. With this method of investigation, we have first studied the harmonic oscillations presented in Part I and then the subharmonic oscillation of order ⅓ given in Part II. In both cases the theoretical results are compared with the experimental measurements carried out for an electric circuit containing a saturable iron core, and the satisfactory agreement is found between them.

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