N. Mahmoodi1, C. J. Anthony1

1 University of Birmingham, School of Mechanical Engineering, Edgbaston, Birmingham, B15 2TT, UK

Air Damping Simulation of MEMS Torsional Paddle

Excerpt from the Proceedings of the 2014 COMSOL Conference in Cambridge

A MEMS torsional paddle (Figure 1) can be considered as

a potential device for bio/chemical sensing [1]. The Quality

Factor (QF) is inversely proportional to energy loss of the

resonator and can determine the sensitivity.

The effect of geometrical parameters on the behavior of

the MEMS torsional resonator was investigated. It was

shown that by changing these parameters the quality

factor could be enhanced which consequently could have

significant results on the sensitivity of the sensor. The

fabrication of the resonator by Focused ion beam would

be the next step to verify these results.

[1] Boonliang, B. , et al. , "A focused-ion-beam-fabricated micro-paddle

resonator for mass detection." Journal of Micromechanics and

Microengineering 18(1), (2008)

device thickness) on air damping and Quality factor of the

torsional paddle are investigated using COMSOL

Multiphysics® 4.4.

The Fluid-Structure Interaction interface was used.

2-D model was developed. Angular displacement

𝜃(𝑡) = 𝜃0 sin 𝜔𝑡 is applied on opposite sides to

produce the moment (Figure 3).

Fluid is in continuum regime and classified as a laminar

and incompressible fluid.

Time domain analysis was performed.

Figure 3. Applying angular

displacement

Power loss

• 𝑃𝑙𝑜𝑠𝑠 𝑡 = (𝑑𝑎𝑚𝑝𝑖𝑛𝑔 𝑡𝑜𝑟𝑞𝑢𝑒. 𝜔)

Energy loss

• 𝐸𝑙𝑜𝑠𝑠 = 𝑃𝑙𝑜𝑠𝑠(𝑡)𝑇

0

Quality factor

• 𝑄 = 2𝜋𝐸𝑛𝑒𝑟𝑔𝑦 𝑠𝑡𝑜𝑟𝑒𝑑

𝐸𝑛𝑒𝑟𝑔𝑦 𝑙𝑜𝑠𝑠=

𝐼𝜃𝑚𝑎𝑥𝜔𝑛2

𝑇𝑚𝑎𝑥

Figure 2. Steps to calculate the

Quality factor

ρ𝜕𝒖𝑓𝑙𝑢𝑖𝑑𝜕𝑡

=

𝛻. [−𝑝𝑰 + µ 𝛻𝒖𝑓𝑙𝑢𝑖𝑑 + 𝛻𝒖𝑓𝑙𝑢𝑖𝑑𝑇

+ 𝑭

𝜌𝛻. 𝒖𝑓𝑙𝑢𝑖𝑑 = 0

−𝛻. 𝜎 = 𝑭𝑣

Figure 4. Eigenfrequency and air flow distribution around the paddle (length×width

×thickness=10×8×0.2μm3 ),(a,b) anchors at the centre, (c,d) anchors offset from the

centre.

(b)

(d)

Q=127

Q=95 f=1.37 (MHz)

f=2.01 (MHz)

(a)

(c)

Figure 5. (a) phase difference between the

damping torque and angular velocity, (b) sine

and cosine components of damping torque

Figure 6. (a) Damping coefficient

versus angular velocity, (b) damping

torque versus angular velocity

0

5E-16

1E-15

1.5E-15

2E-15

2.5E-15

0 10000 20000 30000 40000

Dam

pin

g d

rag

to

rqu

e

(N.m

)

Angular velocity (rad/s)

2.E-20

3.E-20

4.E-20

5.E-20

6.E-20

0 20000 40000 60000

Dam

pin

g c

oe

ffic

ien

t (N

.m.s

)

Angular velocity (rad/s)

(a)

(b)

L (µm) t (nm) f (MHz) T(max) (N.m) D (N.m.s) QF

1 200 2.86 8.03E-16 4.47E-20 170

2.5 200 2.01 5.31E-16 4.21E-20 127

5 200 1.47 3.48E-16 3.75E-20 105

1 500 6.33 2.36E-15 5.92E-20 712

2.5 500 4.50 1.47E-15 5.19E-20 578

5 500 3.34 9.57E-16 4.56E-20 488

Table 1. Summarized simulation results (L=anchors’ length, t=thickness, f=resonance

frequency, T(max)=damping torque, D=damping ratio, Q=quality factor

L (µm) t (nm) Sensitivity

(Hz/fg)

2.5 (asymmetric) 200 5.58

2.5 200 8.17

2.5 500 10.88 0

1000

2000

3000

4000

5000

6000

0.E+0 2.E+6 4.E+6 6.E+6Maxim

um

dis

pla

cem

en

t (p

m)

Frequency (Hz)

Unloadedresonator

Resonator withadded mass (100pg)

Q=578

Table 2. Summarized sensitivity value for

paddle

Figure 7. Response curve for loaded

and unloaded micro paddle

-4.E-15

-3.E-15

-2.E-15

-1.E-15

0.E+00

1.E-15

2.E-15

3.E-15

4.E-15

2.22E-07 3.22E-07 4.22E-07

Da

mp

ing

to

rqu

e(N

.m)

Time(s)

Cos component

Sin component

Total damping torque

-4E-15

-3E-15

-2E-15

-1E-15

0

1E-15

2E-15

3E-15

4E-15

2.22E-07 5.22E-07 8.22E-07 1.12E-06 Time (s)

Angular velocity*10^-15 (rad/s)

Total damping torque (N.m)(a)

(b)

Prediction of the

QF is important

for optimization

and precise

dynamic performance

analysis of such a

sensor. In this study the

geometrical effect (anchors’

length, position and

Introduction

Figure 1. MEMS Torsional paddle

Results

Computational method

Refrence

Conclusion

𝜃(𝑡)

−𝜃(𝑡)