Nanomechanical resonators with high aspect ratio, such as nanotubes and nanowires are of interest due to their expected high sensitivity. However, a strongly nonlinear response combined with a high thermomechanical noise level limits the useful linear dynamic range of this type of device. We derive the equations governing this behavior and find a strong dependence [[proportional]dsqrt((d/L)[sup 5])] of the dynamic range on aspect ratio.

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Dynamic range of nanotube- and nanowire-based electromechanical systems

A global nonlinear distributed-parameter model for a piezoelectric energy harvester under para- metric excitation is developed. The harvester consists of a unimorph piezoelectric cantilever beam with a tip mass. The derived model accounts for geomet- ric, inertia, piezoelectric, and fluid drag nonlinearities. A reduced-order model is derived by using the Euler- Lagrange principle and Gauss law and implementing a Galerkin discretization. The method of multiple scales is used to obtain analytical expressions for the tip deflection, output voltage, and harvested power near the first principal parametric resonance. The effects of the nonlinear piezoelectric coefficients, the quadratic damping, and the excitation amplitude on the output voltage and harvested electrical power are quantified. The results show that a one-mode approximation in the Galerkin approach is not sufficient to evaluate the per- formance of the harvester. Furthermore, the nonlinear piezoelectric coefficients have an important influence on the harvester's behavior in terms of softening or hardening. Depending on the excitation frequency, it is determined that, for small values of the quadratic damping, there is an overhang associated with a sub- critical pitchfork bifurcation.

Global nonlinear distributed-parameter model of parametrically excited piezoelectric energy harvesters

This article presents an analytical and experimental investigation of energy harvesting via parametrically excited cantilever beams. To that end, we consider a lumped-parameter non-linear model that describes the first-mode dynamics of a parametrically excited cantilever-type harvester. The model accounts for the beam's geometric and inertia non-linearities as well as non-linearities representing air drag. Using the method of multiple scales, we obtain approximate analytical expressions describing the beam response, voltage drop across a purely resistive load, and output power in the vicinity of the first principle parametric resonance. Using these expressions, we study the effect of the electromechanical coupling and load resistance on the output power. We show that these parameters play an imperative role in determining the magnitude of the output power and characterizing the broad-band properties of the harvester. Specifically, we show that the region of parametric instability wherein energy can be har...

Investigation of Power Harvesting via Parametric Excitations

Nonlinear Vibrations of Cantilever Beams and Plates

This paper studies analytically and numerically the spring softening and hardening phenomena that occur in electrostatically actuated microelectromechanical systems comb drive resonators utilizing folded suspension beams. An analytical expression for the electrostatic force generated between the combs of the rotor and the stator is derived and takes into account both the transverse and longitudinal capacitances present. After formulating the problem, the resulting stiff differential equations are solved analytically using the method of multiple scales, and a closed-form solution is obtained. Furthermore, the nonlinear boundary value problem that describes the dynamics of inextensional spring beams is solved using straightforward perturbation to obtain the linear and nonlinear spring constants of the beam. The analytical solution is verified numerically using a Matlab/Simulink environment, and the results from both analyses exhibit excellent agreement. Stability analysis based on phase plane trajectory is also presented and fully explains previously reported empirical results that lacked sufficient theoretical description. Finally, the proposed solutions are, once again, verified with previously published measurement results. The closed-form solutions provided are easy to apply and enable predicting the actual behavior of resonators and gyroscopes with similar structures.

Nonlinear Dynamics of Spring Softening and Hardening in Folded-MEMS Comb Drive Resonators

An experimental and theoretical investigation into the first- and second-mode responses of a parametrically excited slender cantilever beam is presented. Inclusion of quadratic damping in the analytical model significantly improves the agreement between the experimental and theoretical results. In addition, the experimental results verify that the often ignored nonlinear curvature terms play a dominant role in the response of the first mode and that the nonlinear inertia terms play a dominant role in the response of the second mode.

Experimental Verification of the Importance of The Nonlinear Curvature in the Response of a Cantilever Beam

An experimental investigation into the response of a nonlinear continuous system with many natural frequencies in the range of interest is presented. The system is a flexible cantilever beam, whose first four natural frequencies are 0.65 Hz, 5.65 Hz, 16.19 Hz, and 31.91 Hz, respectively. The fourth natural frequency is about fifty times the first natural frequency. Several cases were considered with this beam. For the first case, the beam was excited with periodic base motion along its axis. The excitation frequency fe was near twice the thud natural frequency f3, which for a uniform isotropic beam corresponds to approximately the fourth natural frequency fq. Thus the thud mode was excited by a principal parametric resonance (i.e., fe=2f3) and the fourth mode was excited by an external resonance (i.e., fe=f4) due to a slight curvature in the beam. Modal interactions were observed involving the first, third, and fourth modes. Through a stationary frequency sweep the response transitions from a periodic, to a periodically modulated, to a chaotic motion. For the second case, the beam was excited with band limited random base motion transverse to the axis of the beam. The first and second modes were excited through nonlinear interactions. For the third case, the beam was excited with base excitation along the axis of the beam at 138 Hz. The corresponding response is dominated by the second mode.

Observations of Nonlinear Interactions in a Flexible Cantilever Beam

The linear modes of vibration of buckled beams are investigated analytically and experimentally. An exact solution is obtained for the linear modes and associated frequencies of initially buckled beams with fixed-fixed, fixed-hinged, and hinged-hinged boundary conditions. The analytical solution is validated experimentally for the fixed-fixed case. The analytically obtained natural frequencies are in excellent agreement with those obtained experimentally. These exact linear modes provide a basis from which to study the nonlinear vibrations of buckled beams.

T. Anderson

Papers

Experimental Verification of the Importance of The Nonlinear Curvature in the Response of a Cantilever Beam

Observations of Nonlinear Interactions in a Flexible Cantilever Beam

An analytical and experimental investigation of the natural frequencies and mode shapes of buckled beams

Experimental verification of the importance of the nonlinear curvature in the response of a cantilever beam