A Simple Force Feedback Accelerometer Based on a Tuning

Fork Displacement Sensor

by David Stuart-Watson

Thesis Presented for the Degree of

DOCTOR OF PHILOSOPHY in the Department of Electrical Engineering

UNIVERSITY OF CAPE TOWN April 2006

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Declaration

This thesis is being presented for the degree of Doctor of Philosophy in the Department of

Electrical Engineering at the University of Cape Town. It has not been submitted before for any

degree or examination at this or any other university. This author confirms that it is his own

original work. Portions of the work have been published in condensed form in the journal Review

of Scientific Instruments and in the conference proceedings at the First African Control

Conference (2003): the author confirms in accordance with University rule GP7 that he was the

primary researcher in all instances where work described in this thesis was published under joint

authorship.

David Stuart-Watson

3 April 2006.

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Acknowledgments

I would like to thank my supervisor, Prof. J. Tapson, for all his help and support throughout the

project. I would also like to thank B. Prenzlow for all the technical and non-technical discussions

shared in the office. My family, friends and especially Sarah Makin also deserve my thanks, not

so much for the technical stuff, but all the important bits in between.

The author received financial support from the National Research Foundation (NRF) and the

University of Cape Town.

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Abstract

This thesis describes research into the use of a piezoelectric tuning fork as the displacement

sensor in a simple force feedback accelerometer. The research also includes the use of a second

piezoelectric transducer as both the suspension system and the force transducer for the

accelerometer.

A simple inertial accelerometer model, based on a damped mass-spring system, was

developed. This model was used to explore the frequency response of the suspended mass, and its

relative output displacement to an input displacement, velocity or acceleration.

An extended control model for the application of force feedback was discussed. A number of

alternate displacement sensors, and their potential for use in force feedback accelerometer

systems, were investigated.

Each tine of the tuning fork was modelled as a separate vibrating cantilever. This mechanical

model was then combined with an electrical equivalent circuit model. The overall model was then

tested with actual data obtained from a 32.768 kHz piezoelectric tuning fork. The actual data

matched the theoretical response very closely, proving the accuracy of both the mechanical and

electrical model. From a simple noise analysis on the system the fundamental limits of the tuning

forks ability to measure displacement was obtained.

Operating the tuning fork as a displacement sensor required the measurement of its output

magnitude, and the phase measurement between the input and output sinusoidal waveforms.

Digital measurement systems were excluded as they required very high sampling rates to achieve

the required accuracy. Magnitude measurement was done using a simple filtered rectifier. The

importance of isolating the phase measurement from the magnitude measurement led to the

discussion of many different phase detectors. Logic gate phase detectors were, however, the only

simple phase detectors capable of measuring phase without letting changes in magnitude

influence the measurements.

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A shift in displacement was modelled as a change in the forces in the piezoelectric tuning

fork model. This change in force shifts the operating characteristics of the fork, which can then be

modelled as simply a change in operating frequency. For any shift of displacement, modelled as a

change in operating frequency, the output motion of the tuning fork can be divided into two

transient motions and one steady state motion. A new method had to be developed for the

combination of the transient and steady state responses into one total response. This total

response was then used to develop both the control models and the controllers for keeping the

tuning fork operating at a specific point in its resonant band. From the control models it was

found that it is advantageous to use phase rather than magnitude to control the crystal.

For the application of the force feedback response, electro-mechanical models of the

piezoelectric transducers were derived, and the sensitivity of the suspension system was obtained.

Numerous approach tests were also completed to find the most sensitive physical arrangement of

the tuning fork accelerometer. In the application of force feedback, two different control loops

were required. Using phase and resonant frequency as the control variables in these loops proved

to offer a better solution than using magnitude and phase.

A simple tuning fork accelerometer was designed and tested. It was compared to two

conventional devices to establish both the sensitivity and bandwidth. The object of the test was

not to be completely noise free, but rather to test the concept of the tuning fork accelerometer.

The tests gave a bandwidth of DC-25 Hz, with an estimated sensitivity of 13 g, which is close to

the theoretically calculated value. Noise signals produced in the operation and measurement

limited the sensitivity and bandwidth.

This thesis explored the previously unexamined option of using a piezoelectric tuning fork in

conjunction with a piezoelectric transducer to form a simple force balanced accelerometer. The

results obtained go some way in indicating the potential of using this system in future

accelerometer design.