In this paper, the design, modelling and performance characteristics of electrostatically driven vacuum-encapsulated polysilicon resonators are addressed. A one-port configuration is preferably employed for excitation and detection of the vibration. Mechanical instability (pull-in) is discussed on the basis of the energy minimum principle. An expression for the pull-in voltage of a beam is given. The electromechanical behaviour in a limited frequency regime around the fundamental resonance is accurately modelled by an electric circuit consisting of a (static) capacitor shunted by a series (dynamic) RLC branch. The d.c. bias dependence of the circuit components and of the series resonance frequency has been experimentally investigated and is compared with the theory. The large-amplitude behaviour is discussed as well. The plate modulus and residual strain of boron-doped polysilicon are estimated from the resonance frequencies of microbridges of varying lengths. The feasibility of their application as resonant strain gauges is investigated. The 210 m long beams typically have an unloaded fundamental frequency of 324 kHz, a gauge factor of 2400 and an uncompensated temperature coefficient of -135 ppm 0C-1.

/pdf/electrostatically-driven-vacuum-encapsulated-polysilicon-1siqv0cvt3.pdf

Electrostatically driven vacuum-encapsulated polysilicon resonators part II. theory and performance

This paper is divided into two parts. The authors’ purpose in Part I is to formulate a set of mathematically consistent governing differential equations of motion describing the nonplanar, nonlinear dynamics of an inextensional beam. The beam is assumed to undergo flexure about two principal axes and torsion. The equations are developed with the objective of retaining contributions due to nonlinear curvature as well as nonlinear inertia. A priori ordering assumptions are avoided as much as possible in the process. The equations are expanded to contain nonlinearities up to order three to facilitate comparison with analogous equations in the literature, and to render them amenable to the study of moderately large amplitude flexural-torsional oscillations by perturbation techniques. The utilization of the order-three equations in the analysis of nonlinear beam oscillations is the subject of Part II.

Nonlinear Flexural-Flexural-Torsional Dynamics of Inextensional Beams. I. Equations of Motion

Bending, buckling and free vibration of laminated composite and sandwich beams: A critical review of literature

We explore an electrostatic mechanism for tuning the nonlinearity of nanomechanical resonators and increasing their dynamic range for sensor applications. We also demonstrate tuning the resonant frequency of resonators both upward and downward. A theoretical model is developed that qualitatively explains the experimental results and serves as a simple guide for design of tunable nanomechanical devices.

/pdf/tuning-nonlinearity-dynamic-range-and-frequency-of-6banfwja1z.pdf

Tuning nonlinearity, dynamic range, and frequency of nanomechanical resonators

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.

/pdf/dynamic-range-of-nanotube-and-nanowire-based-11cgvd2ax5.pdf

Dynamic range of nanotube- and nanowire-based electromechanical systems

Der Einfluss von Vorspannungen auf die freien und erzwungenen nichtlinearen Schwingungen von Balken und rechteckigen Platten wird mittels einer einfachen Erweiterung der Losungen fur Falle ohne Vorspannung untersucht. Es wird eine einzige Koordinatenfunktion benutzt; es werden einfach aufgelegte und eingespannte Falle betrachtet; und die Diskussion wird auch auf den uberkritischen Bereich ausgedehnt.

/pdf/nonlinear-vibration-of-beams-and-rectangular-plates-2vndkz90av.pdf

Nonlinear vibration of beams and rectangular plates

The steady-state free and forced response and stability for large amplitude motion of a beam with clamped ends is investigated. Elastic restraint of the ends is included in order to relate theory with experiment. A multimode analytical and numerical technique is used to obtain theoretical solutions for both response and stability. Experimental results largely confirm the results of the analysis. It is concluded that, while single mode analyses are adequate in some cases, there are circumstances where a multimode analysis is essential to predict the observed results. Nomenclature Am = amplitude of the rath mode E = Young's modulus F = transverse force FQ = generalized force h = beam thickness I = second moment of area of the cross section Ks = axial spring factor k = axial spring constant ki = rotational spring constant L = beam length PO = initial axial tension Pom = nondimensional amplitude of the generalized harmonic force t = time w = transverse displacement x = axial coordinate Mmkin,Fim, = modal constants GmrsjGmgrs

A Multiple Degree-of-Freedom Approach to Nonlinear Beam Vibrations

It is noted that Eqs. (22, 24, and 25) are essentially those obtained by Beckwith 1 except for the coefficient constants. Finally, for the case of similar flows in the plane of symmetry of an inclined axisymmetric body with zero streamwise pressure gradient and insulated walls, the following conditions prevail: e\ = 1, e» = r(x), KI = 0, ft = 0, and dgr/d^ = 0. In order to transform the resulting equations into a familiar form, we first differentiate Eq. (14) with respect to f *, which is defined as rf * = f. Then, with the aid of the following definitions:

/pdf/large-amplitude-vibration-of-buckled-beams-and-rectangular-3eg5ge7aar.pdf

Large amplitude vibration of buckled beams and rectangular plates

Large amplitude whirling motions of a simply supported beam constrained to have a fixed length are investigated. Equations of motion taking into account bending in two planes and longitudinal deformations are developed. Using the method of harmonic balance, response curves for certain planar and non-planar steady state, forced motions are obtained. Another approximate scheme is used to study the stability of these motions. Stable regions corresponding to non-planar motions are found, thus confirming the existence of whirling motions. Numerical results are presented and discussed for several specific cases.

/pdf/non-planar-non-linear-oscillations-of-a-beam-i-forced-3hhme07hwh.pdf

Non-planar, non-linear oscillations of a beam—I. Forced motions

https://deepblue.lib.umich.edu/bitstream/handle/2027.42/22080/0000504.pdf;sequence=1

Joe G. Eisley

Papers

Nonlinear vibration of beams and rectangular plates

A Multiple Degree-of-Freedom Approach to Nonlinear Beam Vibrations

Large amplitude vibration of buckled beams and rectangular plates

Non-planar, non-linear oscillations of a beam—I. Forced motions

Non-planar, non-linear oscillations of a beam II. Free motions