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 consider a chain of N nonlinear resonators with natural frequency ratios of approximately 2:1 along the chain and weak nonlinear coupling of a form that allows energy to flow between resonators. Specifically, the coupling is such that the response of one resonator parametrically excites the next resonator in the chain, and also creates a resonant backaction on the previous resonator in the chain. This class of systems, which is being proposed for micro-electro-mechanical frequency dividers, is shown to have rich dynamical behavior. Of particular interest is the case when the high frequency end of the chain is resonantly excited, and coupling results in the potential for a cascade of sub-harmonic bifurcations down the chain. When the entire chain is activated, that is, when all N resonators have non-zero amplitudes, if the input frequency on the first resonator is Ω, then the terminal resonator responds with frequency Ω/2N. The details of the activation depend on the strength and frequency of the input, the level of resonator dissipation, and the mistuning in the chain. In this paper we present analytical results, based on perturbation methods, which provide useful predictions about these responses in terms of system and input parameters. Parameter conditions for activation of the entire chain are derived, along with results about other phenomena, such as bistability and partial activation of the chain. We demonstrate the utility of the predictive results by direct comparison with simulations of the equations of motion, and we also present samples of mechanical and electromechanical systems that realize the desired properties. These results will be useful for the design and operation of mechanical frequency dividers based on subharmonic resonances.
Frequency conversion mechanisms are essential elements in frequency synthesizers, which are used in many applications ranging from microwave and RF transceivers to wireless applications to vibration energy harvesters. In particular, the frequency divider, which is an integral part of the phase-locked loop circuit, is essential in modern day instrumentation and wireless communications. In most systems requiring frequency conversion, electronic frequency converters are used; these components require significant power input and introduce noise into the system. In this dissertation, we introduce a mechanism for eliminating these noisy electronic components by using coupled mechanical elements. This novel mechanism for frequency division using parametric resonance in MEMS relies on finite deformation kinematics and nonlinear coupling between isolated modes in a structure to divide an input signal through multiple stages using purely mechanical coupling. We present the theoretical framework for a generic subharmonic resonance cascade. Design considerations for one specific implementation are discussed, and x a proof-of-concept for low-noise low-power applications is demonstrated. A single input signal is divided through three modal stages, generating output signals at 1 2 , 1 4 , and 1 8 of the input signal. Coupling and boundary conditions are explored, as well as the noise characteristics of this mechanical frequency divider. We show that this type of cascading frequency conversion improves phase noise performance of each individual mode.
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.
We consider a chain of N nonlinear resonators with natural frequency ratios of approximately 2:1 along the chain and weak nonlinear coupling that allows energy to flow between resonators. Specifically, the coupling is such that the response of one resonator parametrically excites the next resonator in the chain, and also creates a resonant back-action on the previous resonator in the chain. This class of systems, which is a generic model for passive frequency dividers, is shown to have rich dynamical behavior. Of particular interest in applications is the case when the high frequency end of the chain is resonantly excited, and coupling results in a cascade of subharmonic bifurcations down the chain. When the entire chain is activated, that is, when all N resonators have nonzero amplitudes, if the input frequency on the first resonator is Ω, the terminal resonator responds with frequency Ω/2N. The details of the activation depend on the strength and frequency of the input, the level of resonator dissipation, and the frequency mistuning in the chain. In this paper we present analytical results, based on perturbation methods, which provide useful predictions about these responses in terms of system and input parameters. Parameter conditions for activation of the entire chain are derived, along with results about other phenomena, such as the period doubling accumulation to full activation, and regions of multistability. We demonstrate the utility of the predictive results by direct comparison with simulations of the equations of motion, and we also present a sample mechanical system that embodies the desired properties. These results are useful for the design and operation of mechanical frequency dividers that are based on subharmonic resonances.
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