The generalized duffing equation with large damping

Abstract The equation x + 2p x + ω 2 0 x + μ n x = 0 where n is an odd integer greater than or equal to 3, x(0) = A0, and x (0) = 0 has received much attention in the literature but always with the restrictions that μ and p are small. It is the purpose of this paper to remove this restriction on p and to only require that p ⩽ω0 which is just the requirement that the linear portion of the equation is underdamped or critically damped. To accomplish this a suitably functional form for the frequency of the solution is first chosen. The principle of harmonic balance is then applied with the modification, however, that the coefficients of the first harmonic terms are minimized rather than set equal to zero. The results obtained, when checked against a computer solution, prove to be satisfactory even when μ is no longer small.