Steven W. Shaw

On the dynamic response of a system with dry friction

The response of a single degree of freedom system with dry friction is considered. Den Hartog [1] gave an exact solution for the symmetric steady state motions of such a system in 1930. In this paper these results are extended to include a static coefficient of friction different from the dynamic one. More importantly, the asymptotic stability of the steady state motions and some transient behaviors are also determined. It is shown that for positive viscous damping the non-sticking steady state solutions of the same period as the forcing are nearly always asymptotically stable, but that for negative viscous damping, which may arise from aerodynamic forces [2], such motions can become unstable. By using bifurcation theory it is shown that new behaviors, such as aperiodic motions containing two distinct frequency components, can result from such dynamic instabilities. It is also shown that the symmetric motions with two stops per period can be unstable and that pairs of unsymmetric motions are born at the bifurcation points.

Non-linear normal modes and invariant manifolds

Small-amplitude motions of dynamic systems (structural, fluid, control, etc.) about an equilibrium state are modeled by linear differential equations which have constant coefficients. These are typically obtained by a Taylor series expansion of the forces about the equilibrium point. Under quite general circumstances these equations admit a set of special solutions, called normal mode motions, in which each system component moves with the same frequency and with a fixed ratio amongst the displacements of the components (for a conservative system; for a non-conservative system all displacements and velocities are linearly related to a single displacement/velocity pair).

Nonlinear Dynamics and Its Applications in Micro- and Nanoresonators

This review provides a summary of the work completed to date on the nonlinear dynamics of resonant micro- and nanoelectromechanical systems (MEMS/NEMS). This research area, which has been active for approximately a decade, involves the study of nonlinear behaviors arising in small scale, vibratory, mechanical devices that are typically integrated with electronics for use in signal processing, actuation, and sensing applications. The inherent nature of these devices, which includes low damping, desired resonant operation, and the presence of nonlinear potential fields, sets an ideal stage for the appearance of nonlinear behavior, and this allows engineers to beneficially leverage nonlinear dynamics in the course of device design. This work provides an overview of the fundamental research on nonlinear behaviors arising in micro/nanoresonators, including direct and parametric resonances, parametric amplification, impacts, selfexcited oscillations, and collective behaviors, such as localization and synchronization, which arise in coupled resonator arrays. In addition, the work describes the active exploitation of nonlinear dynamics in the development of resonant mass sensors, inertial sensors, and electromechanical signal processing systems. The paper closes with some brief remarks about important ongoing developments in the field.

The nonlinear response of resonant microbeam systems with purely-parametric electrostatic actuation

Electrostatically-actuated resonant microbeam devices have garnered significant attention due to their geometric simplicity and broad applicability. Recently, some of this focus has turned to comb-driven microresonators with purely-parametric excitation, as such systems not only exhibit the inherent benefits of MEMS devices, but also a general improvement in sensitivity, stopband attenuation and noise rejection. This work attempts to combine each of these areas by proposing a microbeam device which couples the inherent benefits of a resonator with purely-parametric excitation with the simple geometry of a microbeam. Theoretical analysis reveals that the proposed device exhibits desirable response characteristics, but also quite complex dynamics. Of particular note is the fact that the devices nonlinear frequency response is found to be qualitatively dependent on the systems ac excitation amplitude. While this flexibility can be desirable in certain contexts, it introduces additional design and operating limitations. While the principal focus of this work is the proposed systems nonlinear response, the work also contains details pertaining to model development and device design.

Forced vibrations of a beam with one-sided amplitude constraint: Theory and experiment

Abstract An elastic beam with one-sided amplitude constraint subjected to periodic excitation is considered. Experimental results are obtained and compared with results given by a theoretical model based on a single mode analysis of the beam following the work of Moon and Shaw [1]. This model has been studied in some detail by the author in previous works [2–4], it is a single degree of freedom oscillator with periodic excitation and a piecewise linear restoring force. This single mode model is shown to provide good overall qualitative information about the actual physical system. It predicts the multiple subharmonic resonances, period doublings, and some chaotic regimes found experimentally.

The transition to chaos in a simple mechanical system

Abstract A simple mechanical device and its response to periodic excitation is considered. The system consists of an inverted pendulum with rigid barriers which limit the amplitude variation from the unstable upright position. The static stable rest positions correspond to the pendulum leaning against one of the barriers. When subjected to periodic excitation the system response can be quite complicated and may include one or several stable subhannonics and/or chaotic motions. The analysis presented here is based on a piecewise linear model which allows explicit analytic expressions to be determined for many bifurcation conditions including: the appearance of certain types of subharmonics by saddle-node bifurcations, the secondary bifurcations of these subharmonics, and a global bifurcation which results in the creation of horseshoes.

Tunable Microelectromechanical Filters that Exploit Parametric Resonance

Background: This paper describes an analytical study of a bandpass filter that is based on the dynamic response of electrostatically-driven MEMS oscillators. Method of Approach: Unlike most mechanical and electrical filters that rely on direct linear resonance for filtering, the MEM filter presented in this work employs parametric resonance. Results: While the use of parametric resonance improves some filtering characteristics, the

Linear and Nonlinear Tuning of Parametrically Excited MEMS Oscillators

Microelectromechanical oscillators utilizing noninterdigitated combdrive actuators have the ability to be parametrically excited, which leads to distinct advantages over harmonically driven oscillators. Theory predicts that this type of actuator, when dc voltage is applied, can also be used for tuning the effective linear and nonlinear stiffnesses of an oscillator. For instance, the parametric instability region can be rotated by using a previously developed linear tuning scheme. This can be accomplished by implementing two sets of noninterdigitated combdrives, choosing the correct geometry and alignment for each, and applying ac excitation voltages to one set and proportional dc tuning voltages to the other set. Such an oscillator can also be tuned to display a desired nonlinear behavior: softening, hardening, or mixed nonlinearity. Nonlinear tuning is attained by carefully designing combdrive geometry, flexure geometry, and applying the correct dc voltages to the second set of actuators. Here, two oscillators have been designed, fabricated, and tested to prove these tuning concepts experimentally

On the response of the non-linear vibration absorber

Abstract The steady state vibrations of a non-linear dynamic vibration absorber are studied using the method of multiple scales, in conjunction with digital simulations. The main results are concerned with certain dynamic instabilities which can occur if the absorber is designed such that the desired operating frequency is approximately the mean of the two linearized natural frequencies of the system. A combination resonance can occur in this case, resulting in large amplitude almost-periodic vibrations. This motion destroys the effectiveness of the absorber and can coexist with the desired low-amplitude periodic response, which leads to initial condition dependent dynamics.

Large-amplitude non-linear normal modes of piecewise linear systems

A numerical method for constructing non-linear normal modes (NNMs) for piecewise linear autonomous systems is presented. These NNMs are based on the concept of invariant manifolds, and are obtained using a Galerkin-based solution of the invariant manifolds non-linear partial differential equations. The accuracy of the constructed non-linear modes is checked by the comparison of the motion on the invariant manifold to the exact solution, in both time and frequency domains. It is found that this construction approach can accurately capture the NNMs over a wide range of amplitudes, including those with strong non-linear effects. Several interesting dynamic characteristics of the non-linear modal motion are found and compared to those of linear modes. A two-degree-of-freedom example is used to illustrate the technique. The existence, stability and bifurcations of the NNMs for this example are investigated.