Iterative techniques for analyzing nonlinear vibrating dynamical systems in the frequency domain

The nonlinear response of prototypical structures experiencing harmonic excitation is studied using novel techniques called iterative harmonic analysis (IHA) and iterative modal analysis (IMA). First, a simple damped oscillator with a cubic hardening stiffness nonlinearity is studied, and IHA is used in this single-degree-of-freedom system. In this first section, a high-order harmonic balance is applied, and IHA is employed in order to find the amplitude coefficients for different harmonics and their codependence. Additionally, a set of nested sums are identified that describe the harmonic coupling explicitly. Secondly, a pinned-pinned nonlinear beam of rectangular cross-section is studied, and IMA is applied to find the amplitude coefficients for different modes and their codependence. The nonlinearity is introduced through the membrane effect, where axial strain due to transverse deflection becomes a significant contributor to the system behavior. Typical frequency-domain methods cannot be easily applied to these systems as the solutions of the differential equations lead to intricate coupling between coefficients of the solution, and no analytical expression exists for those coefficients. Based upon these examples, other nonlinear systems may also be considered in future work using either a modal-based or finite element model. Finally, the advantages of the new method (reduced computational cost) as well as the limitations (effectively the same as those of an nth-order harmonic balance) are emphasized.