One contribution of 11 to a theme issue 'A field guide to nonlinearity in structural dynamics' . Much is known about the nonlinear resonant response of mechanical systems, but methods for the systematic design of structures that optimize aspects of these responses have received little attention. Progress in this area is particularly important in the area of micro-systems, where nonlinear resonant behaviour is being used for a variety of applications in sensing and signal conditioning. In this work, we describe a computational method that provides a systematic means for manipulating and optimizing features of nonlinear resonant responses of mechanical structures that are described by a single vibrating mode, or by a pair of internally resonant modes. The approach combines techniques from nonlinear dynamics, computational mechanics and optimization, and it allows one to relate the geometric and material properties of structural elements to terms in the normal form for a given resonance condition, thereby providing a means for tailoring its nonlinear response. The method is applied to the fundamental nonlinear resonance of a clamped-clamped beam and to the coupled mode response of a frame structure, and the results show that one can modify essential normal form coefficients by an order of magnitude by relatively simple changes in the shape of these elements. We expect the proposed approach, and its extensions, to be useful for the design of systems used for fundamental studies of nonlinear behaviour as well as for the development of commercial devices that exploit nonlinear behaviour.

We present an optimization procedure for tailoring the nonlinear structural resonant response with time-harmonic loads. A nonlinear finite element method is used for modeling beam structures with a geometric nonlinearity and the incremental harmonic balance method is applied for accurate nonlinear vibration analysis. An optimization procedure based on a gradient-based algorithm is developed and we use the adjoint method for efficient computation of design sensitivities. We consider several examples in which we find optimized beam width distributions that minimize or maximize fundamental or super-harmonic resonant responses.

We demonstrate systematic control of mechanical nonlinearities in micro-electromechanical (MEMS) resonators using shape optimization methods. This approach generates beams with nonuniform profiles, which have nonlinearities and frequencies that differ from uniform beams. A set of bridge-type microbeams with selected variable profiles that directly affect the nonlinear characteristics of in-plane vibrations was designed and characterized. Experimental results have demonstrated that these shape changes result in more than a threefold increase and a twofold reduction in the Duffing nonlinearity due to resonator mid-line stretching. The manipulation of this nonlinearity has significant interest in many applications, including precise mass sensing, accurate measurement of angular rates, and timekeeping. Published by AIP Publishing.

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