EE C245 - ME C218. Fall 2003

Transcription

1 EE C45 - E C8 Introduction to ES Design Fall Roger Howe and Thara Srinivasan ecture 9 Energy ethods II Today s ecture echanical structures under driven harmonic motion develop analytical techniques for estimating the resonant frequency umped mass-spring case: ADX-5 example Rayleigh-Ritz method and examples: lateral resonator, double-ended tuning for Viscous damping Damped nd order mechanical systems: transfer functions Reading: Senturia, S. D., icrosystem Design, Kluwer Academic Publishers,, Chapter 9, pp

2 Estimating Resonant Frequencies Simple harmonic motion x(t) x o cos(ωt) Potential energy: W ( t) x ( t) xo cos ( ωt) Kinetic energy: K( t) x& ( t) xo ω sin ( ωt) Energy Conservation Ignoring dissipation W max K max W max x o K max ω x o Resonant frequency: ω Applications: lumped masses that dominate the suspension s mass 4

3 Example: ADX-5 Suspension beam: 6 μm, h. μm, W μm Average residual stress through thicness: σ r 5 Pa 5 umped Spring-ass Approximation ass 6 nano-grams, 6% is from the capacitive sense fingers Suspension: four tensioned beams include both bending and stretching terms F/4 Bending compliance b - F/4 Stretching compliance st - A.P. Pisano, BSAC Inertial Sensor Short Courses,

4 ADX-5 Suspension odel Bending contribution: b ( ( / ) / c + / c) E( Wh /) EWh Stretching contribution: st / S σ Wh r.4μm / μn θ 4.μ m / μn S F y S sinθ S(x/) Total spring constant: add bending to stretching 4 ( + ) 4( ) 4.5μ N / μm b st S A.P. Pisano, BSAC Inertial Sensor Short Courses, ADX-5 Resonant Frequency umped mass-spring approximation: 4.48N / m f 6. 5Hz π π 6x g Data sheet: f 4 Hz Note: we have not included the frequency-pulling effect of the DC bias voltage on the sense capacitor plates A.P. Pisano, BSAC Inertial Sensor Short Courses,

5 Distributed echanical Structures Vibrating structure s displacement function can be separated into two parts: y( x, t) yˆ( x) cos( ωt) We can use the static displacement of the structure as a trial function and find the strain energy W max at the point of maximum displacement (t, π/ω, ) v(x,t) -ωy(x)sin(ωt) y(x) 9 aximum Kinetic Energy At times t π/(ω), π/(ω),, then the displacement of the structure is y(x ). The velocity of the structure is maximum and all its energy is inetic (since W ) y(x,t ) v(x,t ) v(x,t ) -ωy(x)sin(ωw ) -ωy(x)

6 Finding the aximum Kinetic Energy A differential length dx along the beam has a inetic energy dk due to its transverse motion: W v(x,t ) - ωy(x) h dx Total maximum inetic energy: dk (/)(dm)v (x,t ) dm ρ(whdx) K Whdxv x t max ρ (, ) ρwhω yˆ ( x) dx The Raleigh-Ritz ethod The maximum potential and maximum inetic energies must be equal K max ρwhω yˆ ( x) dx W The resonant frequency of the beam is therefore: Wmax ω ρwhω yˆ ( x) dx max

7 Example: Folded-Flexure Resonator h W h T umped masses (shuttle, truss) need to be included in the inetic energy expression Kinetic Energy Shuttle mass s, truss mass t K max svs + tvt + ρwhω yˆ ( x) dx Use static deflection as estimate of mode shape yˆ( x) y( x) Y x x agnitude of shuttle velocity is v s ωy o agnitude of truss velocity is v t ωy o / o 4

8 Resonant Frequency Find maximum potential energy from spring constant and shuttle deflection: W max y Y o Rayleigh-Ritz equation: 4EI z y EWh ω s + ( 4) + ( / 5) t y / b both trusses all 8 beams 5 Double-Ended Tuning Fors Trey Roessig, Ph.D., E Dept., UC Bereley, 998 6

9 ode Shapes for Clamped-Clamped Beams φ (ε.5) φ,max φ (ε.5) φ ( ε ) κ[sinh( βε sin( βε) + α(cosh( βε) cos( βε))] ode α β κ [ε x/] First Second Third Applying Rayleigh-Ritz to DETFs Case : σ r Case : σ r Pa model comb drive as a point mass with m pgm Dimensions: μm, h W μm, E 5 GPa, ρ. gmcm - Use first mode shape for both cases: analyze a single tine Trey Roessig, Ph.D., E Dept., UC Bereley, 998 8

10 DETF Results Bending: d φ dε 98.6 dε ω dφ Axial load: dε 4. dε 85 Kinetic energy (distributed along beam): φ ( ε ) dε. 97 z EI d φ dε ρwh σ rwh dε + dφ dε φ dε + mφ (.5) dε eff eff Case : 4 Hz Case : 9 Hz Trey Roessig, Ph.D., E Dept., UC Bereley, Resonant Sensors The double-ended tuning for is an excellent sensor for axial stress, since it converts it into a shift in frequency (or in oscillation period) these are easily and accurately measured. Therefore, we can use the DETF as a building bloc for a variety of resonant sensors, such as accelerometers (T. A. Roessig, Ph.D. E, 998) and gyroscopes (A. A. Seshia, Ph.D. EECS, ) Rayleigh-Ritz ethod yields analytical expressions for frequency shift due to axial stress (residual or applied): 4.85 dω dσ / d eff eff deff r dσ r eff eff dσ r ω Wh dφ dε dε design insight!

11 Finite Element Analysis ar emin, Ph.D., E Dept., UC Bereley, 997 Damping any sources: ignore internal damping and acoustic radiation from the anchors focus on drag from the surrounding gas Gap dimension can be on the order of mean-free path continuum model doesn t apply David Horsley, Ph.D., E Dept., UC Bereley, 998

12 Couette Damping Drag force F d -bv y Damping coefficient: b b μa / y o Effective viscosity at reduced pressure (p << 5 Torr): μ (μ p ) py o (.7 x - ) py o units fixed units of μ p are g-s - -m - Torr - for p in Torr William Clar, Ph.D., EECS Dept., UC Bereley, 997 Squeeze-Film Damping Plates slide in y direction b 7μAz o / y o Note that it s just μ, not μ p 4

13 ass-spring-damper System b F(t) Fcos(ωt) x(t) Xcos(ωt) d x dx + b + x dt dt F( t) Sinusoidal, steady-state response: X(t) X e jωt jωt jωt jωt ω Xe + jωbxe + Xe Fe jωt 5 Second-Order Resonant Transfer Function Solve for the ratio of the phasor displacement X to the phasor driving force F H ( jω) X F ω ω + j ω Qω ω ω b Q Above analysis is for the light damping case (Q >> ) 6

14 ow frequency: ω << ω High frequency: ω >> ω imiting Cases Resonant frequency: ω ω Pea width (- db): Δω ω / Q 7 agnitude Bode Plot 6 H db 4 ω - Δω ω ω + Δω ω - - m/n Q 8

15 Phase Bode Plot Phase(H) -45 ω - Δω ω ω + Δω ω