Using frequency detuning to compare analytical approximations for forced responses

It is possible to use numerical techniques to provide solutions to nonlinear dynamical systems that can be considered exact up to numerical tolerances. However, often, this does not provide the user with sufficient information to fully understand the behaviour of these systems. To address this issue, it is common practice to find an approximate solution using an analytical method, which can be used to develop a more thorough appreciation of how the parameters of a system influence its response. This paper considers three such techniques—the harmonic balance, multiple scales, and direct normal form methods—in their ability to accurately capture the forced response of nonlinear structures. Using frequency detuning as a method of comparison, it is shown that it is possible for all three methods to give identical solutions, should particular conditions be used.