DESIGN OF DUAL ENDED TUNING FORK RESONATORS FOR ROLLER BEARING MICRO-STRAIN MEASUREMENT Robert Azevedo and Julian Lippmann University of California at Berkeley Department of Mechanical Engineering Berkeley, California 94720 Abstract—The strain field within automotive roller bearings is intense but small in area (~150-250 um in length.) Dual Ended Tuning Fork (DEFT) resonators fabricated in a Silicon-on-Insulator process are examined as strain sensors for this application using Euler-Bernoulli beam theory. Frequencies of between 990 r 70and 780 r 40 kHz were found for .25% strained DEFT’s with strain sensitivities greater than 100 MHz. However, the limits of manufacturing suggest that these DEFT resonators may not be feasible for this application. I. INTRODUCTION: During the standard operation of an automobile roller bearing, the small contact areas between each rolling element and the bearing race induce concentrated, high-magnitude strains. Both analytical and Finite Element Method (FEM) analyses have given insight into the magnitude and distribution of these strains [1], but the minute dimensions of the strained area (on the order of 100 microns) has made direct measurements impossible using conventional strain gauges. In addition, the high frequency of strain fluctuations and the harsh environment surrounding an operating automotive bearing further complicate the task of measuring these strains. By virtue of their small scale, MEMS style strain gauges offer the possibility of measuring these “micro-strains” in-situ on an operating bearing. To date, various strain measurement methods have been investigated including using optical [2], piezoelectric [3], and resonant [4] methods. Resonating sensors have been of particular interest due to their simple design and potentially high sensitivity. As a resonating beam undergoes strain it changes length, shifting its natural frequency. By detecting this shift, very minute strains can be monitored. Most recently, work at UC Berkeley [5]-[6], and the University of Southampton [7] has demonstrated that a dual-ended tuning fork (DETF) design fabricated in Silicon on Insulator (SOI) processes offers the frequency response, sensitivity and device-to-device repeatability required to make a practical strain sensor. In this work we intend to investigate the frequency response of an SOI DETF. The focus will be on optimized designs for natural frequency and strain sensitivity. Dimensional uncertainty will also be analyzed. II. DESIGN A. Resonator Layout and Operation The basic layout of a DEFT consists of two parallel, straight tines anchored to each other and the substrate at each end (Figure 1). For our application, external electrodes parallel to the beams electro-statically attract each tine laterally. Varying the frequency of the input sinusoidal voltage controls the frequency of excitation. Resonance is detected by measuring the change in capacitance between each tine and electrode as they vibrate. An electrical schematic is presented in Figure 2. A high frequency signal (50 times larger than the drive frequency) is superimposed on the drive voltage. By then demodulating the output current, the change in capacitance can be detected. A simple digital accumulation of capacitance values can then mark the point of resonance. High Pass Filter Digital Counter Drive Voltage Sense Voltage Figure 2. Electrical Schematic of Actuation and Sense Circuit. Figure 1. DEFT Layout with electrodes Electrode w L wa La Isolation Region Tine AnchorB. Frequency Characteristics A full description of a DEFT frequency response requires analysis of the tines, isolation region and anchors, typically through FEM analysis. However, as long as the DEFT is symmetric, and the mode of vibration is parallel to the anchoring substrate, the frequency analysis can be reduced to an examination of the resonance of the tines [8]. Further, if each tine is slender (L to w ratio greater than 10) their vibrational characteristics can be determined using Bernoulli-Euler beam theory. Bernoulli-Euler beam theory states that a beam undergoing vibration experiences both bending and tensile forces. During its motion, cross-sections through out the beam are assumed not to rotate with respect to each other. These assumptions result in the following differential equation: [9] )(2222222xPtyAxyFxxyEIx=∂∂+∂∂∂∂+∂∂∂∂ρ Where E is the young’s modulus, I is the moment of inertia of the beam, F is the axial force applied to the beam, y is the displacement of the beam in the direction of the applied force and P(x) is the shear force applied to the beam along its length. Applying a clamped-clamped boundary condition and assuming that there is no external shear force (P(x) = 0 for all x) it can be show that the natural frequency, fn, in Hz is: [6] ()3285.46.19821LMEwtLEIfeffnεπ+= where the effective mass, Meff, and I are given by: wtLMeffρ4.= 123twI = Simplifying, fn can be expressed as: ()4224.85.455.1621LELEwfnρεπ+= Differentiating this expression with respect to strain gives: ()nxnfLEf222025.3ρπε=∂∂ the strain sensitivity of the resonator. It is important to notice that both frequency and sensitivity are independent of beam thickness. C. Actuation Each tine will be actuated by electrostatic attraction to external electrodes. The magnitude of the force applied is given by: dAVFelect221ε= Where ε is the permittivity of free space, A is the area of the electrode (t * w), V is the voltage potential and d is the gap distance between the electrode and the tine. Since the gap distance is a function of the force applied, the deflection with respect to charge is very non-linear. Limiting the deflection distance of the beam to small displacements reduces this non-linearity. This is also favorable from an energy perspective since the energy lost to the substrate is reduced with smaller deflection. Thus, in operating the resonator vibration amplitudes are limited to ~.1um. This can be achieved by limiting the gap distance to 2 um and controlling the maximum voltage applied to the electrode appropriately [7]. D. Fabrication The resonator and electrode structures will be fabricated from single crystal silicon using a two mask SOI process with high aspect RIE etch. Mask 1 defines the resonator and electrode structure while Mask 2 defines metal contact layer. Since the frequency and
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