Analytical solution for general nonlinear continuous systems in a complex form

In this paper, the method of multiple scales is used to study free vibrations and primary resonances of geometrically nonlinear spatial continuous systems with general quadratic and cubic nonlinear operators in a complex form. It is found that in the free vibrations of general continuous systems in a complex form, both forward and backward modes are excited. This situation is in contrast to the primary resonances in which only forward modes are excited. Consequently, one may determine the form of solution before applying the multiple scales method to the equation. This analysis is applicable to general continuous systems with gyroscopic and Coriolis effects and includes many nonlinear problems as a special case. As an example of application of this general solution, free vibrations and primary resonances of a simply supported rotating shaft with stretching nonlinearity are considered.