Application of the arc-length method in nonlinear frequency response

Several iterative numerical techniques have been developed to solve nonlinear structural problems and some of these methods are capable to trace complex paths in the space load/displacement. One of those most popular procedures is the arc-length method of Crisfield, which possesses the capability to overcome inflection points, without having the necessity of determining them. A great similarity exists between curves of the nonlinear load/displacement path obtained with the arc-length method, and curves of the frequency response of nonlinear dynamic systems. Both curves present limit points with snap-back and snap-through phenomena. This work consists of the description and the application of the arc-length method to solve a system of nonlinear equations obtaining as a result the nonlinear frequency response. The analysis employs the concept of the describing functions where the fundamental harmonic component is considered the most relevant and for some cases can be considered an approximation to the effect of all harmonics. Some examples involving a cubic stiffness and a gap nonlinearity are employed to illustrate the methodology.