C.H.J. Fox

A simple theory for the analysis and correction of frequency splitting in slightly imperfect rings

Abstract The flexural radial vibration of circular rings with imperfections represented by small attached masses and springs is considered. The concept of an “equivalent imperfection mass” is introduced. This is used to quantify the frequency splitting which occurs in slightly imperfect axisymmetric bodies. The concept is used to estimate the trimming mass which must be added to or removed from the ring, and at which point, in order to reduce or eliminate the frequency split in a given pair of modes. The method can be applied to any axisymmetric body for which the mode shapes can be measured or computed.

A preliminary investigation of thermo-elastic damping in silicon rings

Zeners model for thermo-elastic loss, when applied to uniform beams undergoing flexural vibrations, gives theoretical predictions of mechanical Q-factor that often agree well with experimental measurements. The use of silicon ring resonators in MEMS devices is now becoming increasingly common. This paper considers the application of Zeners theory to thin, circular rings and presents a simple expression for the Q-factor associated with in-plane flexural modes of vibration. The theoretical predictions are shown to be in good agreement with experimental measurements for a practically relevant range of ring sizes. The relationships between ring dimensions, ambient temperature and Q-factor are explored.

Optimization of a cantilever microswitch with piezoelectric actuation

This paper considers MEMS microswitches with piezoelectric-film actuation. A mathematical actuator model is established that accounts for normal stress at the piezo-substrate interface, in addition to shear stress. This model gives more accurate predictions for relatively thicker piezoelectric layers. Maintaining adequate contact force between switch electrodes is important in ohmiccontact switches, but parallel contact geometry is also important in switches that provide capacitive coupling between the signal lines. Furthermore, ‘off-state’ isolation is governed by electrode gap when open. The presented mathematical model demonstrates the relationships between these factors and the switch geometry. Optimum conditions are derived.

Development of micromachined RF switches with piezofilm actuation

Current developments in RF systems require high-performance switches for applications including signal routing, impedance matching and adjustable gain amplifiers. The use of micro-switches to replace traditional semiconductor components is increasingly common, because of their advantages in terms of electrical isolation and power loss. This paper reports on a research program relating to the development of a silicon micro-machined RF micro-switch that uses thin-film piezoelectric material for actuation. Piezoelectric actuation has potential advantages over electrostatic actuation in terms of achievable forces and simplicity of structural design. This paper gives an overview of the design and analysis of a prototype switch. The design concept, based on a cantilevered silicon beam or plate, is described. A low order mathematical model, incorporating the mechanical and electrical characteristics of the switch and the interaction between the silicon structure and the piezo-drive is summarized. This allows the basic behavior of the switch to be quantified, and provides a useful tool for design and optimization purposes. The outline design and manufacture/processing of a prototype switch is discussed.

In-plane excitation of thin silicon cantilevers using piezoelectric thin films

This paper deals with the actuation of in-plane and out-of-plane motions of silicon cantilevers, using a single thin film of lead zirconate titanate with a divided electrode configuration. In-plane actuation is demonstrated practically, and excellent agreement is obtained between theoretically predicted and experimentally measured resonant amplitudes, for the fundamental out-of-plane and in-plane modes of vibration of the fabricated test cantilevers.

Tapered Timoshenko finite elements for rotor dynamics analysis

Abstract Finite element idealizations of rotor systems, which include shafts with significant taper angles, often model the tapered sections using stepped, uniform cross-section elements. As part of a study of the shock response of rotor-bearing systems, two tapered beam finite elements have been developed. In one a linear approximation is used for the geometrical properties yielding closed form expressions for the element matrices. Scaling factors are incorporated to compensate for errors introduced by the linear approximation. In the other element exact expressions are used with a numerical integration. Included in the analysis are translational and rotational inertia, shear deformation, gyroscopic moment, axial torque, viscous and hysteretic material damping and mass eccentricity.

Review and comparison of different support loss models for micro-electro-mechanical systems resonators undergoing in-plane vibration

Several approaches for calculating support loss in micro-electro-mechanical system resonators undergoing in-plane vibration are reviewed. In each of them, the support is approximated as a semi-infinite domain. The first approach is analytical and models the support as a semi-infinite thin plate. This is compared with two different finite element approaches that introduce artificial boundaries to their finite domain. In order to absorb outgoing waves and model the infinite support, a perfectly matched layer method and the use of infinite elements are considered. Simple test cases are studied and the results for the support losses predicted by the different methods are compared. It is shown that each of the methods yields similar trends. Using the developed analytical model, a parametric study is performed on the support losses of a ring-based resonator. General strategies for improving the quality factor by reducing support losses are provided.

Harmonic response of rotating cylindrical shells

Results derived from Flugge thin shell theory are presented which illustrate the effects due to a small, constant axial rotation upon a thin cylindrical shell with various end conditions. The shell experiences a harmonic, radial excitation acting at a fixed point on the shell. The excitation is set at a frequency equal to that of a natural frequency of the non-rotating shell. It is shown that the rotation produces changes in the nodal pattern which cause a build-up of amplitude, proportional to the magnitude of the rotation, where previously a node existed. In addition, the dependence upon the rotation of other quantities, such as relative phases of the shell displacements, are examined.

Modelling of a tunable capacitor with piezoelectric actuation

A modelling study for a tuneable micro-capacitor actuated by piezoelectric film actuator is presented in this paper. Compared to similar sized electrostatically actuated capacitors, piezoelectrically actuated capacitors have the potential to provide a larger tuning range and provide a more linear relationship between actuation voltage and capacitor gap change. FE and analytical models have been established, including both piezoelectric actuation and the electrostatic force due to bias voltage across the capacitor. The predictions of the two models are in good agreement. The models are used to investigate the displacement of the movable plate and the tuning range of a capacitor.

An Investigation of Residual Stress Effects due to the Anodic Bonding of Glass and Silicon in MEMS Fabrication

Anodic bonding is widely-used in the fabrication of Micro-Electro-Mechanical Systems (MEMS) devices to join silicon and glass components. The process involves the application of temperature, moderate pressure and an electric field. This paper investigates residual stresses arising during anodic bonding, focusing on the resulting induced distortions. Components of a MEMS silicon rate sensor, in which a silicon wafer is anodically bonded to Pyrex™ glass, were used as the vehicle for the investigation. Distortions generated by the anodic bonding process when using two different electrode configurations (point and planar) were measured using a surface optical profiler. These showed a particular pattern across the wafers for both configurations. An efficient FEM study was carried out to model the qualitative effect of the following residual stress sources; thermal stress, glass shrinkage due to structural relaxation and compositional gradients due to ion migration. Importantly, the FE model takes account the actual multi-device wafer-level configuration, as opposed to a single device. The results demonstrate that compositional gradients can make a significant contribution to the observed pattern of distortions.