Higher order accuracy analysis of the second-order normal form method

The second-order normal form method has shown its intelligence in handling the weak nonlinear vibration problems, especially the lightly damped nonlinearities. The new technology can directly realize near identify transformation to the differential equation, while the first-order method has to change the differential equation to the first-order form at the very beginning. In order to get a more precise result, a lot of effort has been done to realize it through eliminating unnecessary approximations or reconsidering the influence of the nonlinearity in the subsequent processing. It is easy to conduct a simplified nonlinear transformation to get the first-order motion equation. Here in this paper, we focus on the higher-order accurate terms in the dynamic equation. The Taylor series expansion and the Poincaré expansion of the nonlinearity indicate that there are other resonance terms existing in final dynamic equation. A general form of expression for higher-order resonance response function has been derived. The results show that the additional resonance terms cannot obviously increase the accuracy of the second-order normal form method; also, it cannot improve much of the predictions of sub and superharmonic responses.