Non-destructive Damping Measurement for Wafer-level 

Packaged Microelectromechanical System (MEMS)

Acceleration Switches

by Ryan Knight and Evan Cheng

ARL-TR-7094 September 2014

Approved for public release; distribution unlimited. 
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Army Research Laboratory

Adelphi, MD 20783-1138

ARL-TR-7094 September 2014

Non-destructive Damping Measurement for Wafer-level 

Packaged Microelectromechanical System (MEMS) 

Acceleration Switches

Ryan Knight and Evan Cheng

Sensors and Electron Devices Directorate, ARL

Approved for public release; distribution unlimited. 
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Non-destructive Damping Measurement for Wafer-level Packaged 

Microelectromechanical System (MEMS) Acceleration Switches

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6. AUTHOR(S)

Ryan Knight and Evan Cheng

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U.S. Army Research Laboratory

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14. ABSTRACT

Microelectromechanical systems (MEMS) three-axis acceleration threshold sensors have been developed to measure 

acceleration threshold levels using voltage switching when the threshold is reached. Determining damping coefficients is 

important for categorizing how each threshold sensor or switch operates. Switches with different damping coefficients result 

in different mechanical impedances and response times. Analytical and numerical methods to model damping coefficient 

values based on empirical data are needed to characterize three-axis acceleration sensors; traditional methods use the 

displacement of an underdamped system to calculate the damping ratio. Mechanical switches are single output devices that 

distinguish whether closure occurs or not and lack a transduction mechanism to turn acceleration into a readable displacement 

signal. Thus, we have devised a more inventive technique to analyze a closed switch. A shock table and vibration testing 

produces a deterministic acceleration input to close an acceleration switch, which has a defined switch gap distance, and 

mathematical fitting using these deterministic values allows one to determine damping coefficients. By using both an 

analytical equation fit method and a numerical optimization program, the damping coefficients for MEMS three-axis threshold 

acceleration sensors were calculated from the results of the tests and design dimensions of the switches.

15. SUBJECT TERMS

MEMS acceleration impact switch damping coefficient response

16. SECURITY CLASSIFICATION OF:

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 ABSTRACT

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19a. NAME OF RESPONSIBLE PERSON

Ryan Knight

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iii

Contents

List of Figures iv

List of Tables v

1. Introduction/Background 1

1.1 Equations of Motion........................................................................................................2

1.2 Squeeze-Film Damping...................................................................................................4

1.3 Analysis Methods............................................................................................................5

1.3.1 Analysis Methods 1: Acceleration Table Damping Predictor.............................5

1.3.2 Analysis Methods 2: Harmonic Excitation Modal Damping Predictor ..............6

2. Experimental Approach 7

2.1 Linear Shock Machine.....................................................................................................8

2.1.1 System Setup .......................................................................................................9

2.1.2 Sensors...............................................................................................................10

2.1.3 Impact Pads .......................................................................................................10

2.2 Harmonic Excitation Experimental Setup.....................................................................11

2.3 MATLAB Programs......................................................................................................12

2.3.1 MATLAB Programs – Impact Table Damping Model .....................................12

2.3.2 MATLAB Programs – Harmonic Excitation Modal Damping Predictor .........14

3. Results and Discussion 17

3.1 Impact Table Damping Results.....................................................................................17

3.2 Harmonic Excitation Modal Damping Results .............................................................23

4. Conclusions 24

5. References 26

List of Symbols, Abbreviations, and Acronyms 27

Distribution List 29
iv

List of Figures

Fig. 1 Example of the data taken by the data acquisition device...................................................2

Fig. 2 Squeeze-film air damping between a moving plate and a fixed plate .................................4

Fig. 3 MEMS JFTP 60G impact switch layout..............................................................................6

Fig. 4 The 50G MEMS acceleration switch with its lid removed .................................................7

Fig. 5 Open LSM-100 linear shock machine (GHI Systems, Inc).................................................8

Fig. 6 Power supply (left), channel input connections (right), and piston arm control 

(bottom)......................................................................................................................................9

Fig. 7 Three channel (left) and power supply (right) connections...............................................10

Fig. 8 Six 60G JFTP sensors attached to mounting plate ............................................................10

Fig. 9 Two different shock absorber pads: hard plastic (left) and soft rubber (right)..................11

Fig. 10 Numerical method; experimental data (top) and initial calculated model (bottom)........12

Fig. 11 Numerical method; experimental data (top) and initial calculated model (bottom)........13

Fig. 12 Numerical method; experimental data (top) and final calculated model (bottom)..........14

Fig. 13 The –3 dB method to quantify the quality factor and damping of a mechanical 

system. .....................................................................................................................................15

Fig. 14 Switch voltage as a function of frequency (left) and the information retain from an 

amplification ratio standpoint (right).......................................................................................15

Fig. 15 Switch voltage as a function of frequency (top) and the closure frequency window 

plotted on top of a frequency response graph that assume a 0.1 damping ratio (bottom) .......16

Fig. 16 (Top contact) peak acceleration vs. closure time ............................................................18

Fig. 17 (In-plane contact) peak acceleration vs. closure time......................................................18

Fig. 18 (Top contact) damping vs. closure time ..........................................................................19

Fig. 19 (In-plane contact) damping vs. closure time ...................................................................20

Fig. 20 (Top contact) damping vs. peak acceleration ..................................................................21

Fig. 21 (In-plane contact) damping vs. peak acceleration ...........................................................21

Fig. 22 (Top contact) damping vs. maximum velocity................................................................22

Fig. 23 (In-plane contact) damping vs. maximum velocity.........................................................23
v

List of Tables

Table 1 The 50G MEMS acceleration switch – results, 5–10G amplitude excitation.................24

Table 2 The 500G MEMS acceleration switch - results, 5–10G amplitude excitation ...............24
vi

INTENTIONALLY LEFT BLANK.
1

1. Introduction/Background

Mechanical analysis of microelectromechanical system (MEMS) acceleration switches has been 

successfully modeled based off of Newtonian dynamic equations of motion.

1 Damping plays an 

important part in varying the response of MEMS acceleration switches and is an integral part of 

system modeling. In isolated environments filled with a working fluid that dissipates energy, 

damping between parallel and sliding plates that are affected by compression forces and internal 

friction cause energy loss and is often referred to as squeeze-film and Couette damping.

2

Reynolds equations have been applied to model the squeeze-film air damping in MEMS 

switches. The Reynolds number for MEMS is especially small due to extremely small 

geometries and inertial forces are often negligible, so the behavior of fluids is based primarily on 

the viscous effects.

3

Experimental evaluations of MEMS systems required modeling damping by evaluating 

displacement or oscillation using piezoelectric, electromagnetic, or electrostatic (capacitive 

sensing) transduction mechanisms.

4,5 A previous study determined damping in oscillating 

cantilever tuning forks by linearizing the Reynolds equation at a set gap distance.

6 Other studies 

measured the oscillation amplitude at resonant frequencies and estimated the values from the 

logarithmic decrement of those amplitudes.

7,8 Experiments that did not rely on resonant 

frequencies used capacitance measurements or pressure sensors to relate changes in capacitance 

or pressure directly to displacement. The analytical methods and experimental methods for these 

studies all consisted of differentiation of measured displacement to calculate damping forces.

9,10

Previous reported studies were only able to determine damping by measuring displacement. The 

data for isolated MEMS acceleration threshold sensor or switches are limited because switch 

closure only quantifies single instances of voltage change (i.e., switch closure). Two novel 

methods are discussed as non-destructive methods for damping measurement of wafer-level

packaged MEMS contact switches. The first uses empirical acceleration table results and 

modeling a second-order-differential-equation fit to mimic the MEMS acceleration switch 

response. This method iteratively adjusts the damping ratio until the solution converges. 

The second method uses harmonic vibration empirical data to solve the amplification ratio 

equation for both natural frequency and damping. The MEMS acceleration switch is vibrated at

an amplitude where the switch begins to make closure at its natural frequency. The frequency 

span window values where switch closure first occurs and finally opens are the ends points for 

solving a nonlinear amplification ratio equation. This method is constrained to solve only 

underdamped systems; however, it is more advantageous when quantifying single modal 

damping values rather than impact excitation where many modes are excited. 
2

The two damping measurement techniques presented are unique in their application and cover all 

damping ratio values: underdamped and overdamped systems. Damping characterization will 

benefit end users by allowing a framework for modeling acceleration switch response in their 

application and help them correctly choose a closure-acceleration value for the MEMS switch.

1.1 Equations of Motion

A linear shock table was used to test the sensors, allowing the data acquisition device to plot 

acceleration and voltage change over time. An open switch indicated little or no voltage drop,

but at switch closure, the short circuit caused a change in voltage. In Fig. 1, the voltage was 

measured against time, and the change in voltage corresponded to the g-force at the time of 

switch closure. The acceleration and voltage change was analyzed with the system modeled as a 

mass with a spring and damper in a second-order differential equation. 

Fig. 1 Example of the data taken by the data acquisition device

The impact table will produce high amplitude shocks and will terminate the motion with shock 

absorbers. The resulting acceleration forms a half-sine curve, which is applied as a driving 

function in the equation of motion. The governing differential equation for the motion of the 

proof mass is given by Eq. 1

mẍ(t) + cẋ(t) + kx(t) = F sin(ωt) (1)

or Eq. 2 in displacement form

ẍ(t) + 2ζωnẋ(t) + ωn

2x(t) = F

m

sin(ωt) (2)

where m is the mass (kg), c is the damping coefficient, k is the spring constant (N/m), F is the 

driving force (N) (defined by Newton’s law ∑ F = ma), ζ is the damping coefficient 

ζ = c

2√km (3)

and ωn is the natural frequency (rad/s)
3

ωn = �k

m (4)

The general form of the solution to the system, assuming a harmonic driving function and an 

underdamped system, is given by

x(t) = e−ζωnt[c1 cos(ωdt) + c2 sin(ωdt)] + R sin(ωt − φ) (5)

where c1 and c2 are displacement coefficients of the terms of the transient solution obtained from 

the initial conditions (initial displacement and initial velocity is at zero), ω is the harmonic 

frequency (rad/s), ωd is the damping frequency (rad/s)

ωd = ωn�1 − ζ2 (6)

and R is the coefficient of the steady state solution Eq. 2 derived by substituting the steady-state 

component into the equation of motion

R =

F

k�

��1 − � ω

ωn

�

2

�

2

+ �2ζ � ω

ωn

��

2

(7)

and phase angle, φ, is 

φ = tan−1 �

2ζω

ωn

1−� ω

ωn

�

2 � (8)

Solving the equation for c1 and c2 using the initial conditions results in the real part of the

solution

x(t) = e−ζωnt �R sin(φ) cos(ωdt) +R [ζωn sin(φ)−ω cos(φ)]

ωd

sin(ωdt)� + R sin(ωt − φ) (9)

The vibration and impact empirical data discussed in Section 1.3.2 analyzes two forms of Eq. 9,

t ≪ 1 and t ≫ 1.

x(t ≪ 1) = e−ζωnt �R sin(φ) cos(ωdt) +R (ζωn sin(φ)−ω cos(φ))

ωd

sin(ωdt)� + R sin(ωt − φ)

(10)

x(t ≫ 1) =

F

k� sin(ωt − φ)

��1 − � ω

ωn

�

2

�

2

+ �2ζ � ω

ωn

��

2

(11)

Dividing the output x(t) by the input leads to the amplification ratio for a harmonic second-order 

system
4

xo

xi

= 1

��1 − � ω

ωn

�

2

�

2

+ �2ζ � ω

ωn

��

2

(12)

Equation 12 can be solved for ωnand ζ given that xo, xi, and the respective ω’s are known. 

1.2 Squeeze-Film Damping

Squeeze-film air damping, as shown in Fig. 2, is created by the compressing the gas between two 

surfaces moving at a certain decreasing gap distance and creating a resistant force to the 

movement. The damping pressure between two plates consists of a viscous damping force and an 

elastic damping force.

Fig. 2 Squeeze-film air damping between a moving 

plate and a fixed plate

In cases of high frequency oscillations, the elastic force dominates the viscous force and is 

proportional to the displacement of the plate. The coefficient of the elastic damping force ke(σ)

as determined by Bao and Yang3 is described as

ke(σ) = 64σ2PaA

π8ho

� 1

(mn)2 ��m2 + �n�η�

2

�

2

+ σ2

π

� 4� m,n odd

(13)

where Pa is the ambient pressure (Pa), A is the area of the plate (m2), η is the aspect ratio of the 

plate (for a square plate, η = 1), and σ is the squeeze number given by

σ = 12μωl

2

Paho

2 (14)

where μ is the coefficient of viscosity of the gas (Pa-s), and l is the characteristic length of the

plate (m).

3 As shown in Eqs. 13 and 14, greater force between the two plates will be caused by a 
5

higher squeeze number, which can be designedly caused by increasing characteristic length or

decreasing the gap distance. The equation of motion is then be described as

mẍ(t) + cẋ(t) + (k + ke)x(t) = F sin(ωt) (15)

Equation 15 can be solved for the damping coefficient given the input forcing function and 

known input design parameters, m, k + ke, and gap distance when the acceleration switch 

closes.

1.3 Analysis Methods

Two methods were developed that characterize the damping ratio for packaged MEMS 

acceleration switches. MEMS acceleration switches only provide a single point of information in 

an applied acceleration field—when switch closure occurs. Other MEMS devices use transducers

like capacitive, piezoelectric, optical, and pressure to discern a continuous displacement signal. 

Knowing displacement as a function of time allows for damping to be calculated via the classical 

ring-down log decrement technique. Additionally, the damping ratio for underdamped systems 

can be predicted by the frequency response to a harmonic excitation using the –3 dB rule. These 

two methods are frequently used to characterize damping for micro-devices. When displacement 

is no longer continuously known, for example, in MEMS acceleration switches, traditional 

techniques to measure damping can no longer be applied. The two methods discussed below are 

ways to measure damping when switch closure is all that is known.

1.3.1 Analysis Methods 1: Acceleration Table Damping Predictor

The analytic method used the solution for the equations of motion Eq. 9 to compute the damping 

coefficient at the time of specified voltage changes. MEMS acceleration switches developed for 

60G fuzing impact environments were used to evaluate the non-destructive method for damping 

measurement. The Department of Defense’s Joint Fuze Technology Program (JFTP) provided 

the funding behind the production of the 60G MEMS impact switches. The JFTP MEMS impact 

switches are omnidirectional, consisting of two out-plane contact (top and bottom) and one inplane contact for lateral or side acceleration events. Figure 3 illustrate the computer-aided design 

(CAD) layout for the 60G MEMS acceleration switches. For the top and bottom contacts of the 

JFTP 60G sensors, the squeeze-film air damping was taken into account due to the rectangular 

plate design.
6

Fig. 3 MEMS JFTP 60G impact switch layout

The numerical method approximated the damping by first estimating a damping coefficient value 

and then comparing the time value at the determined damping to the actual time of switch 

closure. By adjusting the time value until it was equivalent to the time of switch closure and 

modifying the damping coefficient value accordingly, the damping coefficient value for the 

function was determined at the instance of voltage changes.

1.3.2 Analysis Methods 2: Harmonic Excitation Modal Damping Predictor

To evaluate the harmonic excitation modal damping predictor (HEMDP) method, underdamped 

50–2000G MEMS impact acceleration switches were tested. These switches were chosen 

because they were known to be underdamped, which is needed for the prescribed method. The 

JFTP 60G impact switch was designed to have higher damping and given that it’s a relatively 

new device the damping values were unknown and, hence, not used for HEMDP test setup. 

The 50–2000G MEMS impact switches used in fuzing applications were mounted to an 

inductive shaker and vibrated at constant amplitude for sine sweeps with frequencies ranging 

from 100 Hz–10 kHz. The switches are omnidirectional; therefore, switch closure is dependent 
7

upon switch orientation during vibration. Figure 4 illustrate the 50G omnidirectional acceleration 

switch with its lid removed.

Fig. 4 The 50G MEMS acceleration switch with its lid removed

Modal frequency separation is large enough that modal coupling effects can be ignored. The 

MEMS impact switches were evaluated both in the ±z out-of-plane and in the orthogonal planar 

direction. 

The modal damping values are evaluated by using the harmonic excitation data where the switch 

closes. This frequency is constrained by two points: 1) the frequency where the switch first 

closes and 2) the frequency where the switch opens. Using these two known points and also the 

known designed gap distance the MEMS switch has to travel during closure allows one to then 

solve the nonlinear amplification ratio Eq. 12 for the damping ratio and natural frequency.

2. Experimental Approach

The goal of the experimental approach was to empirically characterize MEMS acceleration 

switches on a linear impact table and an inductive shaker, and use these results to determine 

damping values. During an impact event, typically all modes are excited resulting in a

superposition of modal displacements. During harmonic excitation, a particular mechanical mode 

can be excited generating a very simple motion. Damping values determined from harmonic 

excitation will be very specific to the mode, whereas damping values obtained from impact tests 

will be generalized. 
8

There were many different parts to the experimental system setup used to test the sensors, and 

two MATLAB programs were written to analyze the data. A linear shock table was used to 

produce a high impact shock. Four channels were also used to measure each sensor one at a time. 

Six JFTP sensors were attached to the mounting plate and moves at high speeds into a cushion. 

The harmonic excitation method used an inductive shaker with accelerometer feedback control to 

hold a constant acceleration value through the sine sweeps. Data acquisition records the time 

versus switch voltage data and correlates into frequency versus switch voltage. Switch closure is 

indicated by a non-zero voltage. MATLAB post-processing programs were developed to sort 

through each data set and obtain the relevant information needed to solve for damping and 

natural frequency.

2.1 Linear Shock Machine

The LSM-100 linear shock machine used pressurized air and a piston arm to pull back the plate, 

which compressed spring coils (Fig. 5). The piston arm would hit the trigger and release the 

plate, forcing the plate into a damping pad at high speeds. The trigger could be moved at 

different, measurable distances to vary the acceleration amplitude. Data were collected and 

displayed by a WinCAT Data Acquisition System (GHI Systems, Inc) when the acceleration of 

the plate surpassed a certain g-force threshold specified by the user. 

Fig. 5 Open LSM-100 linear shock machine (GHI Systems, Inc)

DUT

Cushion/pad

Accelerometer

acceleration 

amplitude dial

Piston arm

Trigger
9

2.1.1 System Setup

The system setup and WinCAT software was arranged to analyze one sensor at a time. The 

power supply, channel input connection, and piston arm control are shown in Fig. 6. Four 

channels connected to the data acquisition device were used to collect data. Channel 1 was used 

for measuring acceleration of the table at impact and Channels 2, 3, and 4 (Fig.7) measured the 

voltage changes for each contact (top, in-plane, bottom). For shorter durations with the harder 

cushion, data were taken at a frequency of 125 kHz for a duration of 4.10 ms. For longer 

durations with the softer cushion, 10-bit data were taken at a frequency of 50 kHz for a time 

duration of 10.24 ms. 

Fig. 6 Power supply (left), channel input connections (right), and piston arm control (bottom)
10

Fig. 7 Three channel (left) and power supply (right) connections

2.1.2 Sensors

Six JFTP sensors (Fig 8) were attached to a voltage divider circuit that was connected to the data 

acquisition device. The sensors were attached to a mounting plate in different orientations using 

screws and washers.

Fig. 8 Six 60G JFTP sensors attached to mounting plate

2.1.3 Impact Pads

Different pads (Fig 9) were used to absorb the shock from the impact. Softer cushions were made 

of rubber and the harder cushions were made of a harder plastic.

JFTP sensor

1 in.
11

Fig. 9 Two different shock absorber pads: hard plastic (left) and soft rubber (right)

2.2 Harmonic Excitation Experimental Setup

The harmonic excitation method was conducted at the US Army Research Laboratory’s (ARL)

Weapons and Material Research Directorate, Ballistic Structures and Launch Dynamics 

Laboratory in Adelphi, Maryland. The experimental setup uses an inductive shaker controlled via 

a feedback accelerometer. The harmonic excitation method for modal damping characterization 

requires constant acceleration amplitude. The data acquisition system measures acceleration and 

switch voltage as a function of time and correlates these data to the driving sine sweep frequency 

ranging from 100 Hz–10 kHz. The data acquisition software had to be modified such that switch 

closure remained unfiltered during sine sweeps. Sine sweep software typically has a travelling 

notch filter that removes frequencies outside the sine sweep frequency. 

Figure 10 illustrates the inductive shaker used (left) where the MEMS impact switch are 

mounted on the top of the shaker (right). The printed circuit board along with the MEMS impact 

switches and accelerometer (right) are mounted to the top of the shaker; power, and voltage are 

recorded by the data acquisition system. Figure 8 shows the circuit board developed to power the 

voltage dividing circuitry for each MEMS impact switch. A voltage of zero indicates an open 

switch, whereas a voltage of +5 V indicates a closed switch.
12

Fig. 10 Numerical method; experimental data (top) and initial calculated model (bottom)

2.3 MATLAB Programs

MATLAB programs where developed for each testing method describe in Section 2. The first 

described MATLAB program uses empirical acceleration table results and modeling a secondorder-differential-equation fit to mimic the MEMS acceleration switch response. This method 

iteratively adjusts the damping ratio until the solution converges. 

The second method uses harmonic vibration empirical data to solve the amplification ratio 

equation for both natural frequency and damping. The MEMS acceleration switch is vibrated at 

an amplitude where the switch begins to make closure at its natural frequency. The frequency 

span where switch closure first occurs and finally opens are the ends points for solving a 

nonlinear amplification ratio equation. This method is constrained to solve only underdamped 

systems; however, it is more advantageous when quantifying single modal damping values rather 

than impact excitation where many modes are excited. 

2.3.1 MATLAB Programs – Impact Table Damping Model

MATLAB programs were written for both a numerical and an analytical method. The numerical 

method guessed some damping value and used an ordered differential equation solver (ode45) 

to determine a time of closure at the gap distance between the proof mass and the contact. If the 

calculated closure time did not match the closure time in the data, the program would estimate a 

closer value of damping using the relationship

ζn = ζn−1(1 − Ktn−1) (16)

where k is a refinement factor. The analytical method used the fzero function and applied it to

the real part of the solution of the equation of motion by setting the whole function to zero. The 
13

program calculated the damping using the gap distance between the proof mass and the contact 

and the experimental closure time. 

An example of the convergence method is portrayed in Figs. 11 and 12. In Fig. 11, a graph of 

actual experimental data for a single test is shown as well as the model for the calculated 

acceleration input. The switch closes the gap when the displacement is 30 μm. The incorrect 

damping value in the switch empirical data shows that switch closure happens at a later time, so 

damping is continually increased until the time closure value matches that of the original. In 

Fig. 12, the code converged to a final damping value at 2.041 after 42 iterations, changing the 

damping by a factor of 5.

Fig. 11 Numerical method; experimental data (top) and initial calculated model (bottom)
14

Fig. 12 Numerical method; experimental data (top) and final calculated model (bottom)

2.3.2 MATLAB Programs – Harmonic Excitation Modal Damping Predictor

A well-known method to measure damping for harmonic excitation is by directly measuring 

displacement and correlating that to the input displacement to produce an amplification ratio as a 

function of frequency. If input acceleration is held constant, then the input displacement 

amplitude is proportion to inverse of frequency squared. Therefore at higher frequencies, less 

input displacement is needed to produce a constant acceleration. Figure 13 illustrates that if 

output displacement is known for various excitation frequencies then the quality factor and 

damping of the system can be directly determined. However, in the case of MEMS acceleration 

switches, the peak value of the amplification ratio is unknown because of switch closure. 
15

Fig. 13 The –3 dB method to quantify the quality factor and damping 

of a mechanical system.

Figure 14 illustrates that switch closure only indicates the frequency at which the switch closes 

and then subsequently opens (left). Thus, the only information attained is the two frequency 

values where the switch first closes and then opens and the input displacement, which is input 

acceleration amplitude divided by the frequency squared. In the case of Fig. 14 (right), these 

values are 0.7395 and 1.094 (non-dimensional frequency).

Fig. 14 Switch voltage as a function of frequency (left) and the information retain from an amplification ratio 

standpoint (right)
16

The values shown in Figs. 14–15 are used below as an example to calculate the damping ratio 

and natural frequency of MEMS acceleration switches when all that is known is switch closure 

and open frequencies and input acceleration. Input displacement is determined by dividing the 

constant input acceleration value by the closure and opening frequencies which gives two values 

that Eq. 12 can be solved. MATLAB’s built-in function fsolve can be used to solve a system 

of nonlinear equations (e.g., 17 and 18).

Fig. 15 Switch voltage as a function of frequency (top) and the closure frequency window 

plotted on top of a frequency response graph that assume a 0.1 damping ratio (bottom)

( ) (17)

2 1

1 2.098

2

1

2 2

1

1, 2



































 Ω

+ 





























 Ω −

= =

Ω

n n

o

x t

x

ω

ς

ω

( )

)

 (18)

2 1

1 3.394

2

2

2 2

2

1, 2



































 Ω

+ 





























 Ω −

= =

Ω

n n

o

x t

x

ω

ς

ω

( ) ( )

( ) ( )

Gap distance displacement at closure

 shaker frequency at switch closure

acceleration amplitude (constant) 

sin

sin 

1,2

2 1,2

1,2

=

Ω =

=

Ω

Ω

− =

= Ω

o

o

o

A

t

A

x t



x t A t

( ) ( )

( ) 1,2

Gap distance

sin 2

Ω

=

=

Ω

Ω

− =

x t

x Amp

x

t

A

x t

o

o

o
17

3. Results and Discussion

Data were obtained for MEMS acceleration switches using the two methods defined in the 

Section 2. Results from the linear impact table and harmonic excitations method are organized 

and discussed. The linear impact table method uses acceleration shock profile to induce MEMS 

acceleration switch closure. This method differs from the harmonic excitation method because 

when the MEMS switch is put into a transient shock environment multiple modes are excited. 

The harmonic excitation method uses an inductive shaker to excite single modes of interest. The 

harmonic excitation method only works for an underdamped system, which is a limiting factor 

for this developed testing method. Both testing methods revealed nonlinearity to damping as a 

function of input amplitude. This is due to squeeze-film effects amplifying as a function of 

response velocity and gap closure.

3.1 Impact Table Damping Results

Damping is expected to change in a manner related to the change in velocity of the impact test. 

This is expected due to gap closure of the MEMS acceleration switch. The value of the squeezefilm constant, Eq. 14, increases as gap distance decreases and input frequency increases. Both the 

analytical method and the numerical analysis method were expected to result in similarly 

consistent values for the in-plane contacts. Using both methods, the damping coefficient for top 

contact movement was expected to be consistently higher than in-plane damping.

The analytical method for calculating damping was too sensitive to input variations; the 

acceleration, closure time, and frequency produced impractical damping values when applied to 

the equation. Additionally, manufacturers closure gap tolerances are thought to be inconsistent 

and have a fairly large standard deviation. Static centrifuge testing needs to be conducted to 

determine what the acceleration closure value is for each orientation—determining this will give 

better insight on how the gap distances vary. 

The numerical method proved to be a better way of determining damping because the analytical 

method used many assumptions when solving the characteristic equation. One assumption was 

that the system was an underdamped system, so any cases of overdamped or critically damped 

systems would be inaccurately modeled using the equations of motion. 

In Figs. 16 and 17, there is a general decreasing acceleration pattern over increasing closure time 

for both contacts. There are data points that show constant peak acceleration over different 

closure times.
18

Fig. 16 (Top contact) peak acceleration vs. closure time

Fig. 17 (In-plane contact) peak acceleration vs. closure time

0 0.5 1 1.5 2 2.5 50

100

150

200

250

300

350

Closure Time - [ms]

Peak Acceleration - [G]

[Top Contact] Peak Acceleration versus Closure Time

0.25 0.3 0.35 0.4 0.45 0.5

140

160

180

200

220

240

260

280

300

Closure Time - [ms]

Peak Acceleration - [G]

[In-plane Contact] Peak Acceleration versus Closure Time
19

Two or three outliers above a damping value of three were considered anomalies and were most 

likely due to the proof mass bouncing around or chattering inside the sensor and aliasing due to 

the 125-kHz sampling frequency for data acquisition, seemingly increasing closure time. 

Similar patterns occur with damping as with acceleration. In Figs. 18 and 19, there is a general 

decreasing damping pattern over increasing closure time for both contacts, but there are patterns 

of constant peak acceleration over different closure times. The top contact damping was always 

extremely high and had inflated damping values. There seemed to be cases of overdamping 

because the calculated squeeze-film factor was extremely large. 

Fig. 18 (Top contact) damping vs. closure time

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

Closure Time - [ms]

ζ Damping

[Top Contact] ζ Damping versus Closure Time
20

Fig. 19 (In-plane contact) damping vs. closure time

In Fig. 20, the top contact damping values are inflated due to high squeeze-film damping. For 

both contacts (Figs. 20 and 21), there are clear patterns of increasing damping with increasing 

peak acceleration.

0.25 0.3 0.35 0.4 0.45 0.5 0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

Closure Time - [ms]

ζ Damping

[In-plane Contact] ζ Damping versus Closure Time
21

Fig. 20 (Top contact) damping vs. peak acceleration

Fig. 21 (In-plane contact) damping vs. peak acceleration

100 150 200 250 300 350 0.5

1

1.5

2

2.5

3

Peak Acceleration - [G]

ζ Damping 

[Top Contact] ζ Damping versus Peak Acceleration

150 200 250 300 0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

Peak Acceleration - [G]

ζ Damping

[In-plane Contact] ζ Damping versus Peak Acceleration
22

A relationship of increasing damping with increasing velocity is portrayed in Figs. 22 and 23.

The relationship between damping and velocity were due to the drag force in the system. This 

pattern also accounts for the relationship between damping and acceleration for the same time 

value. The values for damping were inflated or inaccurate due to the sampling frequency and 

large squeeze-film values. These factors severely impacted the calculated damping coefficients, 

but there were clear relationships shown in the results. 

Fig. 22 (Top contact) damping vs. maximum velocity

20 40 60 80 100 120 140

1

1.5

2

2.5

Maximum Velocity - [mm/s]

ζ Damping

[Top Contact] ζ Damping versus Maximum Velocity
23

Fig. 23 (In-plane contact) damping vs. maximum velocity

3.2 Harmonic Excitation Modal Damping Results

The damping results from the harmonic excitation method are shown in Tables 1 and 2 for 50G 

and 500G MEMS acceleration switches. When first comparing the side (in-plane) and out-ofplane (top and bottom) damping values, the results were unexpected. However, upon further 

review of the contact surface area, the side contact had roughly two times surface area as the outof-plane (top and bottom) contacts. Squeeze-film damping would be more pronounced in the side 

contact rather than the top or bottom since it has double the contact surface area. 

Since the top and bottom contacts are inherently identical, the difference in damping values are 

attributed to the manufacturing gap tolerance for the top and bottom contact distances. The static 

acceleration switching threshold may deviate as much as 10% for the top and bottom contacts, 

indicating variation in the gap distances as a result of different acceleration threshold values. 

The data in Tables 1and 2 are for fairly low input acceleration amplitude ranging from 5–10G’s. 

Switch ringing was seen at input acceleration amplitudes greater than 20G’s, where lateral switch 

contact was sensed. This is believed to be from contact happening at such a great velocity the 

switch rings, exciting other modes lateral to the input acceleration direction. Larger input 

acceleration amplitudes often resulted in higher damping values, which are not shown in 

Tables 1 and 2. This is believed to be attributed to velocity dependent damping effects that were 

also seen during the impact table damping characterization tests. 

100 110 120 130 140 150 160 170 180 0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

Maximum Velocity - [mm/s]

ζ Damping

[In-plane Contact] ζ Damping versus Maximum Velocity
24

Table 1 The 50G MEMS acceleration switch – results, 5–10G amplitude excitation

Mode #/Shape FEA modal (Hz) Contact Freq Damping

Mode 2 / Out-plane 680 Top 653.9312 0.004161

Mode 2 / Out-plane 680 Bottom 735.2356 0.005367

Mode 3-4 / In-plane 798 Side 802.6671 0.044097

Table 2 The 500G MEMS acceleration switch - results, 5–10G amplitude excitation

Mode #/Shape FEA modal (Hz) Contact Freq Damping

Mode 2 / Out-plane 2156 Top 2719.009 0.004860

Mode 2 / Out-plane 2156 Bottom 2703.846 0.001689

Mode 3-4 / In-plane 2482 Side 2817.224 0.009231

Mode 3-4 / In-plane 2482 Spin 3216.026 0.000007

4. Conclusions

For MEMS systems and systems characterized by low Reynolds numbers, the damping force of 

bodies moving in a fluid has been found to be linearly proportional to the speed of the mass.

11

This relationship is determined through analyzing the drag force, which is proportional to the 

velocity and damping of the system. In plotting the maximum velocity against the calculated 

damping coefficients, the results support the correlation and show a steady increase in damping 

with increasing velocity. This shows that damping may be quantified as a function of velocity 

ζn(ẋ). The pattern of decreasing damping with increasing closure time can therefore be 

explained with an increasing drag force for larger velocity and damping values. This pattern 

similarly accounts for increasing damping with increasing acceleration; higher peak acceleration 

within the same time duration causes a greater maximum velocity. 

Time-closure values may also have been aliased due to the frequency at which data were taken.

The proof mass bouncing around in the sensor was an unmeasurable and uncontrollable factor in 

the data, so time-closure values may have been incorrectly obtained from the data. The bouncing 

also may have closed the switch with other contacts, causing the MATLAB code to incorrectly 

read the proper closure time for the correct contact. 

The switching threshold was set to 0.79 V, which may have affected the time-closure data. In the 

event of bouncing or almost incomplete closure, the switching threshold may not have been 

exceeded and the data for time of closure may have been missed and resulted in subsequent 

closure being detected thus greatly increasing measured closure time.

In the future, a more accurate convergence algorithm may be required to more efficiently 

converge on the correct damping value. In addition, an increase in sampling frequency would 
25

more accurately detect closure, which would provide more accurate data to be analyzed. The 

inflated values for the top contact could be lessened if gap-dependent squeeze-film value model 

was used. The squeeze-film constant in Bao3 was derived using a small amplitude of vibration 

assumption at a set gap distance and doesn’t take into account a reducing gap that goes to zero. 

Determining the damping values of an impact acceleration switch helps categorize how the 

switch functions. Factors such as mass and spring constants are tailorable and measurable, but 

the majority of the damping is caused by the mechanical switch’s interaction with the N2

packaged gas environment. The novel method developed to measure damping by measuring 

closure time helps model designs to further improve design to more accurately close at the 

expected acceleration. 
26

5. References

1. Muldavin JB, Rebeiz GM. Nonlinear electro-mechanical modeling of MEMS switches.

Microwave Symposium Digest, 2001 IEEE MTT-S International. May 2001;3:2119–2122.

2. Li G, Hughes H. Review of viscous damping in micromachined structures. Proc. SPIE 4176, 

Micromachined Devices and Components VI, August 2000.

3. Bao M, Yang H. Squeeze film air damping in MEMS. Sensors and Actuators A.

2007;136:3–27. 

4. Glynne-Jones PM, Tudor J, Beeby SP, White NM. An electromagnetic, vibration-powered 

generator for intelligent sensor systems. Sensors and Actuators A: Physical. February 

2004;110(1–3):344–349.

5. Williams CB, Yates RB. Analysis of a micro-electric generator for Microsystems. Sensors 

and Actuators A: Physical. April 1996;52(1–3):8–11.

6. Brotz J. Damping in CMOS-MEMS resonators. MS Thesis, ECE, Carnegie Mellon 

University, June 2004.

7. Gorelick S, Dekker JR, Leivo M, Kantojärvi U. Air damping of oscillating MEMS 

structures: modeling and comparison with experiment. COMSOL Conference 2013, VTT 

Technical Research Centre of Finland, 2013.

8. Zhang W. Energy dissipations in MEMS resonators: fluid damping of flexural resonators and 

thermoelastic damping. Doctoral dissertation, University of California, Santa Barbara, UMI

3245943, December 2006. 

9. Steeneken PG, Rijks Th.GSM, van Beek JTM, Ulenaers MJE, De Coster JR. Puers R.

Dynamics and squeeze film gas damping of a capacitive RF MEMS switch. Journal of 

Micromechanics and Microengineering. 2005;15:176–184.

10. Mohite SS, Kesari H, Sonti VR, Pratap R. Analytical solutions for the stiffness and damping 

coefficients of squeeze films in MEMS devices with perforated back plates. Journal of 

Micromechanics and Microengineering. 2005;15:2083–2092.

11. Younis MI. MEMS Linear and Nonlinear Statics and Dynamics. Springer, New York, NY, 

2011.
27

List of Symbols, Abbreviations, and Acronyms

ARL US Army Research Laboratory

CAD computer-aided design 

DUT device under test 

HEMDP harmonic excitation modal damping predictor 

JFTP Joint Fuze Technology Program

MEMS microelectromechanical system

A area of the plate (m2)

F driving force (N)

K refinement factor 

R coefficient of the steady state solution (m)

c damping

c1 , c2 coefficients of the terms of the transient solution

k spring constant (N/m)

ke(σ) coefficient of the elastic damping force 

l characteristic length of the plate (m)

m mass (kg)

Pa ambient pressure (Pa)

t time (s)

ζ damping coefficient

η is the aspect ratio of the plate (for a square plate, η = 1)

μ coefficient of viscosity of the fluid (Pa-s)

σ squeeze number
28

ω harmonic frequency (rad/s)

ωd damping frequency (rad/s)

ωn natural frequency (rad/s)
29

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