ε^2-Order normal form analysis for a two-degree-of-freedom nonlinear coupled oscillator

In this paper, we describe an ε^2-order normal form decomposition for a two-degree-of-freedom oscillator system that has a mass supported with horizontal and vertical support springs. This system has nonlinear terms that are not necessarily ε^1-order small when compared to the linear terms. As a result, analytical approximate methods based on an ε expansion would typically need to include higher-order components in order to capture the nonlinear dynamic behaviour. In this paper we show how this can be achieved using a direct normal form transformation up to order ε^2. However, we will show that the requirement for including ε^2 components is primarily due to the way the direct normal form method deals with quadratic coupling terms rather than the relative size of the coefficients.