Second-order normal form analysis for nonlinear vibration problem

There are several methods of analyzing the vibration response of a weakly nonlinear system. One method is the use of normal forms in which the equations of motion are transformed into a more linear form using a near-identity nonlinear transform. Traditionally this transform has been performed on the first-order derivative representation of the equations of motion. Here we consider a similar approach applied to the second-order derivative representation of the equations of motion. By applying the normal form approach directly to the second-order nonlinear equations of motion, we are able to generate a more compact and algebraically simpler normal form which is more easily comparable to the linear equivalent system, which is normally represented as a set of second-order differential equations. We demonstrate that the method is capable of producing highly accurate predictions of the forced response of a nonlinear system both at the forcing frequency and at the harmonics of the forcing frequency.