The resonances of structures with quadratic inertial non-linearity under direct and parametric harmonic excitation

The resonant behaviour of general structures under single frequency harmonic excitation is considered. It is shown that the linear parametric terms and inertial non-linearities can dominate the behaviour when there are certain near integer relationships between the excitation frequency and some of the natural frequencies of the structure (external resonance), and/or amongst the natural frequencies themselves (internal resonance). This behaviour is characterized by structural response at frequencies other than the frequency of excitation. A theoretical analysis with retention of both the parametric terms and up to the quadratic inertial non-linearities reveals the conditions required for resonant behaviour, and in the vicinity of each yields an approximate solution. Qualitative experimental evidence demonstrates the behaviour in simple laboratory structures with up to four distinct modal frequencies simultaneously responding to a single frequency of excitation. The theoretical analysis is verified by application to the experimental structures.