Exact Solutions of the Autonomous Duffing Equation and Their Computation : 3rd Report. Solution of a Characteristic Parameter in Terms of Power Series for the Quasi-linear Vibration

Abstract

A power series expansion of the characteristic parameter is obtained in terms of the natural frequency, which is defined as an implicit function in a transcendental equation. It plays a central role in calculating the Fourier coefficient of exact solutions. A general algorithm is proposed to obtain the solution in the power series x=Σ b_nyⁿ to the equation Σ c_nxⁿ=y. The method is particularly effective when m (the modulus parameter of elliptic functions) approaches zero, which corresponds with the quasi-linear small oscillations in those systems with a hard, a soft or snap-through (half-swing mode) spring. The accuracy of existing (simple) approximate solutions to the autonomous Duffing equation, i.e., the perturbation, the averaging and the (two-term) harmonic balance methods, is examined by comparison with the present exact solutions.