Analytical approaches to oscillators with nonlinear springs in parallel and series connections. (November 2015) - Sanmiguel-Rojas, E.; Hidalgo-Martínez, M.; Jiménez-González, J.I.; Martín-Alcántara, A.
Abstract: We develop analytical approaches and analyze their range of applicability to oscillators with two nonlinear springs in parallel and series connections. Specifically, we focus our study on systems with hardening springs and cubic nonlinearities. In both cases, three dimensionless parameters govern the oscillator namely λ = k 2 / k 1 and ϵ1, 2 = ε 1, 2 A 2 . Here, k 1, 2 and ε 1, 2 are the linear stiffness and the nonlinearity coefficient of both springs, respectively, and A is the amplitude of the position of the mass, in the parallel case, or the deflection of the spring connected to the mass ( k 2, ε 2 ), in the series case. It is found that, in parallel configuration, for λ > 0 and 0 < ϵ1, 2 ≤ 1 the analytical solution gives an excellent approach to the exact solution found numerically. However, in series connection the numerical simulations show that the solution of the oscillator becomes much more complex than in parallel connection, and the analytical approaches work excellently in the ranges 0 < λ ≤ 1, 0 < ϵ1, 2 ≤ 0.1, and, 0 < λ ≤ 0.1, 0 < ϵ1, 2 ≤ 1. Highlights: We study oscillators with nonlinear springs in parallel and series connections. We focus our study on systems with hardening springs and cubic nonlinearities. Three dimensionless parameters govern the nonlinear oscillator. The analytical approaches work excellently in a wide range of parameter values.