Transient Response of Nonlinear Systems

The transient response of nonlinear, nonconservative structures, idealized as lumped parameter chains, is treated by a combination graphical-numerical method. The step-by-step solution of the set of ordinary second-order nonlinear differential equations is in the form of trajectory curves in multiple phase-plane space. Examples of one and two-degree-of-freedom systems with cubic hardening springs are treated in which the forcing function is a prescribed foundation velocity. Relative displacement response curves obtained from an analog computer show good agreement with the graphical-numerical results. A scaling law is derived for nonlinear multi-degree-of-freedom systems. Prescribed restrictions are imposed on the parameters and input of the model system and the prototype system. Application of the scaling law is made for a system consisting of a finite number of masses connected by cubic hardening springs and linear dashpots.