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EE C245 – ME C218 Fall 2003 Lecture 9
EE C245 - ME C218Introduction to MEMS Design
Fall 2003
Roger Howe and Thara SrinivasanLecture 9
Energy Methods II
2EE C245 – ME C218 Fall 2003 Lecture 9
Today’s Lecture• Mechanical structures under driven harmonic motion
� develop analytical techniques for estimating the resonant frequency
• Lumped mass-spring case: ADXL-50 example• Rayleigh-Ritz method and examples:
lateral resonator, double-ended tuning fork• Viscous damping• Damped 2nd order mechanical systems:
transfer functions
• Reading:Senturia, S. D., Microsystem Design, Kluwer Academic
Publishers, 2001, Chapter 9, pp. 260-264.
3EE C245 – ME C218 Fall 2003 Lecture 9
Estimating Resonant FrequenciesSimple harmonic motion
k Mx(t) = xocos(ωt)
Potential energy:
Kinetic energy:
)(cos21)(
21)( 222 tkxtkxtW o ω==
)(sin21)(
21)( 2222 tMxtxMtK o ωω== &
4EE C245 – ME C218 Fall 2003 Lecture 9
Energy ConservationIgnoring dissipation � Wmax = Kmax
22max
2max 2
121
oo xMKkxW ω===
Resonant frequency:
Mk=ω
Applications: lumped masses that dominate thesuspension’s mass
5EE C245 – ME C218 Fall 2003 Lecture 9
Example: ADXL-50
Suspension beam: L = 260 μm, h = 2.3 μm, W = 2 μmAverage residual stress through thickness: σr = 50 MPa
6EE C245 – ME C218 Fall 2003 Lecture 9
Lumped Spring-Mass Approximation
• Mass = M = 162 nano-grams, 60% is from the capacitive sense fingers
• Suspension: four tensioned beams … include both bending and stretching terms
F/4
F/4Bending compliance kb
-1
Stretching compliance kst-1
A.P. Pisano, BSAC Inertial SensorShort Courses, 1995-1998
7EE C245 – ME C218 Fall 2003 Lecture 9
ADXL-50 Suspension Model• Bending contribution:
• Stretching contribution:
NmEWh
LWhELkkk ccb μμ /2.4
)12/(3)2/(2)/1/1( 3
3
3
31 ==⎥
⎦
⎤⎢⎣
⎡=+=−
NmWhLSLk
rst μμ
σ/14.1/1 ===−
S
Sθ
Fy = S sinθ ≈ S(x/L)• Total spring constant: add bending to stretching
mNkkk stb μμ /5.4)88.024.0(4)(4 =+=+=
A.P. Pisano, BSAC Inertial SensorShort Courses, 1995-1998
8EE C245 – ME C218 Fall 2003 Lecture 9
ADXL-50 Resonant Frequency
kHzkgxmN
Mkf 5.26
10162/48.4
21
21
12 === −ππ
Lumped mass-spring approximation:
Data sheet: f = 24 kHz
Note: we have not included the frequency-pullingeffect of the DC bias voltage on the sense capacitor plates
A.P. Pisano, BSAC Inertial SensorShort Courses, 1995-1998
9EE C245 – ME C218 Fall 2003 Lecture 9
Distributed Mechanical Structures• Vibrating structure’s displacement function can be
separated into two parts:
)cos()(ˆ),( txytxy ω=
• We can use the static displacement of the structure as a trial function and find the strain energy Wmax atthe point of maximum displacement (t = 0, π/ω, …)
y(x)
v(x,t) = -ωy(x)sin(ωt) = 0
10EE C245 – ME C218 Fall 2003 Lecture 9
Maximum Kinetic Energy• At times t¸= π/(2ω), 3π/(2ω), …, then the displacement
of the structure is y(x ) = 0.• The velocity of the structure is maximum and all its
energy is kinetic (since W = 0)
v(x,t)̧
y(x,t)̧ = 0
v(x,t)̧ = -ωy(x)sin(ωW̧) = -ωy(x)
11EE C245 – ME C218 Fall 2003 Lecture 9
Finding the Maximum Kinetic Energy• A differential length dx along the beam has a kinetic
energy dK due to its transverse motion:
dx
Wh
v(x,t)̧ = - ωy(x)dK = (1/2)(dm)v2(x,t´)
dm = ρ(Whdx)
• Total maximum kinetic energy:
∫∫ =′=LL
dxxyWhtxWhdxvK0
22
0
2max )(ˆ
21),(
21 ωρρ
12EE C245 – ME C218 Fall 2003 Lecture 9
The Raleigh-Ritz Method• The maximum potential and maximum kinetic
energies must be equal �
max0
22max )(ˆ
21 WdxxyWhK
L
== ∫ ωρ
• The resonant frequency of the beam is therefore:
∫= L
dxxyWh
W
0
22
max
)(ˆ21 ωρ
ω
13EE C245 – ME C218 Fall 2003 Lecture 9
Example: Folded-Flexure Resonator
Lumped masses (shuttle, truss)need to be included in thekinetic energy expression
h
W
hT
14EE C245 – ME C218 Fall 2003 Lecture 9
Kinetic Energy• Shuttle mass = Ms, truss mass = Mt
∫++=L
ttss dxxyWhvMvMK0
2222max )(ˆ
21
21
21 ωρ
• Use static deflection as estimate of mode shape
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎠⎞
⎜⎝⎛−⎟
⎠⎞
⎜⎝⎛=≅
32
23)()(ˆLx
LxYxyxy o
• Magnitude of shuttle velocity is vs =ωYo
• Magnitude of truss velocity is vt = ωYo/2
15EE C245 – ME C218 Fall 2003 Lecture 9
Resonant Frequency• Find maximum potential energy from spring constant
and shuttle deflection:
2max 2
1oyYkW =
• Rayleigh-Ritz equation:
( ) )35/12(4/ bts
y
MMMk
++=ω
both trusses all 8 beams
3
3
3
224L
EWhLEIk z
y ==
16EE C245 – ME C218 Fall 2003 Lecture 9
Double-Ended Tuning Forks
Trey Roessig, Ph.D., ME Dept., UC Berkeley, 1998
17EE C245 – ME C218 Fall 2003 Lecture 9
Mode Shapes for Clamped-Clamped Beams
( ) ))]cos()(cosh()sin([sinh( βεβεαβεβεκεφ −+−=
Mode α β κ [ε = x/L]First -1.018 4.730 -0.618Second -0.999 -7.853 -0.663Third -1.000 -10.996 0.711
φ1(ε = 0.5) = 1
φ3(ε = 0.5) = 1
φ2,max= 1
18EE C245 – ME C218 Fall 2003 Lecture 9
Applying Rayleigh-Ritz to DETFs
Case 1: σr = 0 Case 2: σr = 30 MPamodel comb drive as a point masswith m = 1 pgm
Dimensions: L = 200 μm, h = W = 2 μm, E = 150 GPa, ρ = 2.3 gmcm-3
Use first mode shape for both cases:analyze a single tine Trey Roessig, Ph.D., ME Dept.,
UC Berkeley, 1998
19EE C245 – ME C218 Fall 2003 Lecture 9
DETF Results
∫ =⎟⎟⎠
⎞⎜⎜⎝
⎛1
0
2
21
2
6.198εεφ d
dd
Bending:
Axial load: ∫ =⎟⎠⎞⎜
⎝⎛1
0
21 85.4ε
εφ d
dd
Kinetic energy (distributed along beam): ∫ =1
0
21 397.0)( εεφ d
eff
eff
rz
Mk
mdWhL
ddd
LWhd
dd
LEI
=+
⎟⎠⎞⎜
⎝⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛
=
∫
∫ ∫1
0
21
21
1
0
1
0
21
2
21
2
32
1
)5.0(φεφρ
εεφσε
εφ
ω
Trey Roessig, Ph.D., ME Dept.,UC Berkeley, 1998
Case 1: 412 kHz
Case 2: 339 kHz
20EE C245 – ME C218 Fall 2003 Lecture 9
Resonant SensorsThe double-ended tuning fork 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 block for a variety ofresonant sensors, such as accelerometers (T. A. Roessig, Ph.D. ME, 1998)and gyroscopes (A. A. Seshia, Ph.D. EECS, 2002)
Rayleigh-Ritz Method yields analytical expressions for frequencyshift due to axial stress (residual or applied):
∫ ⎟⎠⎞⎜
⎝⎛=⎟
⎟⎠
⎞⎜⎜⎝
⎛==
− 0
1
21
1
2/1
1 121
21 ε
εφ
ωσσσω d
dd
LWh
ddk
Mk
Mk
dd
dd
r
eff
eff
eff
eff
eff
rr
4.85
design insight!
21EE C245 – ME C218 Fall 2003 Lecture 9
Finite Element Analysis
Mark Lemkin, Ph.D., ME Dept., UC Berkeley, 1997
22EE C245 – ME C218 Fall 2003 Lecture 9
Damping• Many sources: ignore internal damping and acoustic radiation from
the anchors � focus on drag from the surrounding gas
Gap dimension can beon the order of mean-freepath � continuum modeldoesn’t apply
David Horsley, Ph.D., ME Dept., UC Berkeley, 1998
23EE C245 – ME C218 Fall 2003 Lecture 9
Couette Damping
William Clark, Ph.D., EECS Dept., UC Berkeley, 1997
Damping coefficient: b
b = μA / yo
Drag force = Fd = -bvy
Effective viscosity at reduced pressure (p << 50 Torr):
μ= (μp) pyo = (3.7 x 10-2) pyo
units of μp are kg-s-1-m-2 Torr-1
for p in Torrunits fixed
24EE C245 – ME C218 Fall 2003 Lecture 9
Squeeze-Film Damping
• Plates slide in y direction
Note that it’s just μ, not μp
b ≈ 7μAzo2 / yo
3
25EE C245 – ME C218 Fall 2003 Lecture 9
Mass-Spring-Damper System
k Mx(t) = Xcos(ωt)
b
)(2
2
tFkxdtdxb
dtxdM =++
Sinusoidal, steady-state response: X(t) = X ejωt
F(t) = Fcos(ωt)
tjtjtjtj FekXebXejXeM ωωωω ωω =++− 2
26EE C245 – ME C218 Fall 2003 Lecture 9
Second-Order Resonant Transfer Function
( )
1
2
1
1
1ωω
ωω
ω
Qj
kFXjH
+⎟⎟⎠
⎞⎜⎜⎝
⎛−
==−
Solve for the ratio of the phasor displacement X to the phasor driving force F
Above analysis is for the light damping case (Q >> 1)Mk=1ω b
MQ 1ω=
27EE C245 – ME C218 Fall 2003 Lecture 9
Limiting Cases
Low frequency: 1ωω <<
High frequency: 1ωω >>
Resonant frequency: 1ωω =
Peak width (-3 dB): Q/1ωω =Δ
28EE C245 – ME C218 Fall 2003 Lecture 9
Magnitude Bode Plot
ω1 ω
| H |dB
0
20
40
60
ω1 + Δωω1 - Δω
-20
k-1 = 1 m/NQ = 1000
29EE C245 – ME C218 Fall 2003 Lecture 9
Phase Bode Plot
Phase(H)
-90
-45
ω1 ω0
ω1 + Δωω1 - Δω
-135
-180