An analytical method for the optimisation of weakly nonlinear systems

In this paper we discuss how backbone curves can be used to guide the design and optimisation of weakly nonlinear systems with multiple degrees-of-freedom. Aft er decomposing the system using the modes of the equivalent linear system (the linear modes), we show how the backbone curves of the unforced, undamped equivalent system can be calculated. These consist of pure responses in each of the linear modes and, in certain parameter regimes, responses which are a combination of two or more linear modes - a feature which can be linked to internal resonance. Using an example system we will investigate how these backbone curves can be used to describe particular characteristics of the response. An energy balancing technique is also employed to relate the backbone curves to the response of the forced and damped system, and anticipate the conditions for which a particular characteristic will be seen. Finally, we discuss how the analytical nature of these techniques enables us to precisely design and optimise characteristics of such systems and how this can be expanded to systems with a greater number of degrees-of-freedom.