Analysis of Delay Differential Equations via the Lambert Function

by

Professor A. Galip Ulsoy

University of Michigan – Ann Arbor

Department of Mechanical Engineering

Abstract - Time delays are inherent in many physical and engineering systems, e.g., active vibration and noise control, conveyor systems, transmission lines, economic systems, and chatter instability in machining. Delay differential equations (DDEs) are used to model such phenomena, and include a delay operator, which leads to a nonlinear transcendental characteristic equation and an infinite spectrum. Such DDEs are typically solved numerically or graphically, or by using asymptotic expansions, perturbation methods, Pade` approximations, etc. This talk will introduce an analytical solution to DDEs in terms of Lambert functions, first for the scalar first-order case, then for a system of first-order DDEs. The Lambert function W(x), satisfies W(x)e^W(x) = x. The approach based on Lambert functions, analogous to the solution of systems of first-order ordinary differential equations in terms of the matrix exponential function, can be used to study stability, free response and forced response. The new solution method will be validated using several simple examples, and also applied to the engineering problem of chatter stability in machining.

Friday, October 14, 2005

3:30 – 4:30p.m.

Rm. 1500 EECS