Phase Trajectory Construction of High-Order, Nonlinear, Time-Varying, Nonautonomous Systems

Abstract

A graphical procedure for constructing the time response of a system from its phase-plane trajectory is described. From this, a graphical solution of the equation ẍ + θ(ẋ)g(x) + φ(ẋ) + h(x) = F(t) is presented. The construction procedure for the equation ẍ + φ(ẋ) + c(t)X(x) = F(t) is detailed as a special case of the more general form ẍ + a(t)M(x)N(ẋ) + θ(ẋ)g(x) + b(t)Q(ẋ) + φ(ẋ) + c(t)X(x) + h(x) = F(t) which can also be solved in a similar fashion. The extension of these techniques to higher-order systems is presented.