Structural optimization for nonlinear dynamic response

Abstract

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.

Efficient sensitivity analysis method for chaotic dynamical systems

“…The knowledge of sensitivity for dynamical systems is of considerable interest in structural dynamic reduction [1], optimal control [2], reliability analysis [3], and uncertainty analysis [4]. The sensitivity analysis can be carried out by using different methods such as the finite difference method, the direct differentiation method, and the adjoint variable method [5] [6]. The sensitivity of performance measure computed using the direct and adjoint methods is essentially identical.…”

Nonlinear Resonances in 3D Printed Structures

“…A systematic approach is based on the concepts in "topology optimization" [21]. Applications of topology or shape optimization have now appeared in the literation on nonlinear dynamics as well, with the works in [22][23][24] focusing on general one-dimensional elastic systems whereas the works in [12,13] focusing on plate structures with internal resonances. The overall goal is to tailor the system's dynamic response to some desired form for appropriate external excitations.…”

Mechanism of geometric nonlinearity in a nonprismatic and heterogeneous microbeam resonator

“…In the recent years, substantial efforts have been put forth to tailor the strength of structural nonlinearity [18,28,29]. Saghafi et al [28] provided an analytical scheme based on a continuous system model to predict the onset of nonlinearity in a bilayer clamped-clamped micro-beam to enhance the working dynamic range.…”

“…In order to overcome the limitation of linear MEMS resonators, recent efforts have been devoted to design MEMS devices that implement [18] or harness nonlinearity [19]. Utilization of intentional geometric nonlinearity in MEMS has proved to achieve higher bandwidth resonances, frequency tunability, and instantaneous hysteresis switching that are difficult to attain in a linear setting.…”

“…Saghafi et al [28] provided an analytical scheme based on a continuous system model to predict the onset of nonlinearity in a bilayer clamped-clamped micro-beam to enhance the working dynamic range. Dou et al [18] established a systematic numerical approach combined with the optimization technique to maximize or minimize objective nonlinearity in a clamped-clamped beam by varying its thickness profile. Li at al.…”