Steady Oscillations of Systems With Nonlinear and Unsymmetrical Elasticity

Abstract

Abstract Steady forced oscillations are reduced to free undamped oscillations of the same amplitude. The reduction is accomplished by changing the excitation from a time function F(t) into a space function F(x) through the assumption that x and t are related as in a free motion. By combining F(x) with the elastic restoring force E(x), a new effective E(x) is obtained, to which there corresponds a new “free” motion, which in turn furnishes a second approximation for the relation between x and t, and hence a new function F(x). A new effective E(x) and a new free motion are then found; and the cycle is repeated until the relation between x and t ceases to change. The frequency of the forced motion at the assumed amplitude is then known. In general, the process converges rapidly. The accuracy can be checked at any stage of the work. A general criterion of the stability of the motions is offered.