EXPERIMENTAL MODAL ANALYSIS FOR MICROSYSTEMS

O. Burak Ozdoganlar†∗, Bruce D. Hansche‡, Thomas G. Carne††Org. 9124 Structural Dynamics Research ‡ Org. 9122 Advanced Diagnostics and Product Testing

Sandia National Laboratories, Albuquerque, New Mexico 87185

1 ABSTRACT

Recent advances in Micro-electromechanical System (MEMS)fabrication and design techniques, including surface micro-machining and LIGA, have made it possible to create complex,multi-task integrated systems. However, current industrial ap-plications of the microsystems are limited to various sensors,actuators, and mirror arrays. Realization of commercially vi-able microsystems products requires thorough understanding,and eventually prediction, of their performance and reliabilitycharacteristics. Structural dynamics behavior of microsystemsthat include moving, overhung, and compliant subcomponentsplays a pivotal role in determining their performance and reli-ability. Traditionally, experimental modal analysis is used tocharacterize dynamic behavior of structures, as well as toderive, validate, update and correct analytical and numericalmodels. Due to their small size, however, conventional modaltesting methods cannot be directly applied to microstructures.This paper provides a brief overview of modal testing tech-niques for microsystems. A particular experimental modalanalysis methodology is then described and evaluated. Themodal tests included base excitation via a piezoelectric shakerand measurement through a laser interferometer. The bound-ary value problem and modal model is derived for the case ofbase excitation of micro-cantilever beams. The effectivenessof the methodology is illustrated through various experimentson polysilicon micro-cantilevers for different geometries andambient pressures. Analysis of the damping data for differentpressures has shown that the structural damping in microsys-tems can be considerably less than damping arising from theforces applied on the structure from the ambient gases.

2 INTRODUCTION

Recent advances in various fabrication techniques have trans-formed miniaturization from being a futuristic dream to reality.Microsystems (also commonly referred to as micro electrome-chanical systems (MEMS)) that perform mechanical function-ality with geometrical features from a few hundred nanome-ters to a few millimeters have been realized. Silicon basedmicrosystems have been manufactured via surface micro-

∗Corresponding author.

machining and bulk micro-machining. Metal (and secondarilyceramic, plastic and polymer) based microsystems have beenproduced using the LIGA (an acronym from the German wordsfor lithography, electroplating, and molding) technique, whichuses X-ray synchrotron radiation to make molds, which arethen filled with the chosen material via electrodeposition. Anumber of novel acceleration, pressure, chemical and biolog-ical sensors, actuators, and various mirror arrays have beenmanufactured using MEMS technology.

As expected from any relatively new technology, the researchin microsystems has been mostly in fabrication techniquesand design[1]. Numerous products have been designed andfabricated within research organizations and universities[2].However, only a few have been realized as commercially vi-able products. In addition to the long-term reliability issues,one of the main reasons impeding commercialization is thelack of tools for understanding, testing, modeling, and per-formance prediction for microsystems, including those for mi-crosystem dynamics[2]. Characterization of mechanical anddynamic properties, testing capabilities, and experimentallyvalidated high-fidelity predictive modeling are required to as-sure that the microsystems reliably perform their intendedfunctionality[3−7]. In addition, validated predictive models formicrosystem dynamics will enable optimization in the designstage[5].

Experimental modal analysis techniques are traditionally usedfor dynamic characterization of “macro” systems for variouspurposes, including (1) validation, correction, and refining ofanalytical and computational models, (2) determination of en-ergy dissipation capacity, i.e., the damping, (3) derivation ofsimplified representative dynamic models of complex systemswithin the frequency range of interest, (4) detection and char-acterization of structural faults and defects, and (5) determi-nation of operational characteristics and fatigue behavior ofstructures[8−10]. The majority of traditional modal testing tech-niques includes excitation of the structure via a function ofknown frequency characteristics (impulse, random, periodic)using, e.g., an impact hammer or a shaker, and measuring theassociated response using force or motion sensors. Gener-ally, a description of system characteristics is obtained by es-tablishing Frequency Response Functions (FRFs) calculatedfrom (auto-) power spectral density and cross spectral den-

sity functions. Frequency response functions commonly pre-sented in the form of motion over force.

Due to size and mass mismatch, the traditional modal testingtechniques are not suitable for microsystems[11−13]. Place-ment of conventional sensors such as accelerometers, if atall possible, can substantially alters the dynamic character-istics of microstructures. Providing transient input with animpulse hammer or directly attaching a shaker to the struc-ture is not possible or acceptable. Ideally, both the excitationand the measurement must be conducted through noncontactinstruments[4, 11, 14]. For these reasons, a number of opticaland electrical techniques, such as video imaging, blur enve-lope (synthesis), and interferometry, to name a few, are em-ployed for modal testing of microstructures.

In addition to dynamic characterization, modeling, and vali-dation, experimental modal analysis of microsystems is alsocritical in other aspects. The minuscule size of microsystemshinders direct measurement of some inherent characteristics.Experimental modal analysis can be used as a tool to in-fer some of these characteristics. For instance, mechanicaland geometric properties, such as Young’s modulus, fatiguecharacteristics, residual stresses, average density, and thick-nesses, can be deduced by dynamically testing various well-characterized structures (e.g., beams or plates) and compar-ing the results with analytical formulas for natural frequenciesand mode shapes[7, 14−16]. Dynamic testing is also used formodeling and validation of coupled-physics phenomena, suchas the effects of electrostatic forces or gas damping. Other im-portant uses of modal testing for microsystems include eval-uating manufacturing processes, assessing changes in fab-rication characteristics, diagnosing structural faults and de-fects before and during operation[4], and in-situ performancemeasurements[5, 17].

This paper, after providing a brief background of modal testingtechniques used for microsystems, presents a methodologyfor experimental modal analysis of microsystems. A piezo-electric shaker is used to base excite the microstructures upto 102.4 kHz frequency. The response is measured using aLaser Doppler Vibrometer coupled with an optical microscope.A vacuum chamber included in the system enabled testing thedynamic response of micro cantilevers of different dimensionsunder various pressures. To show the effectiveness of thetechnique in determining dynamic characteristic of microsys-tems, the damping arising from the gas forces at various pres-sures and gap heights was experimentally investigated formicro-cantilevers of various geometries. The boundary valueproblem and modal model are derived for base-excited can-tilever beams used during experimentation.

3 MODAL TESTING TECHNIQUES FOR MICROSYS-TEMS

Experimental modal analysis involves (1) exciting the structurewith a force (or motion) of known frequency characteristics, (2)

measuring the input and associated response (both amplitudeand phase characteristics), (3) acquiring and processing theinput and output signals to obtain FRFs, and (4) using a curvefitting technique for extracting dynamic parameters based on aparticular modal model. When performing experimental modalanalysis on microsystems, the first two items above pose chal-lenges that cannot be overcome by conventional modal testingtechniques. For this reason, a number of different techniquesare employed for modal testing of microsystems. This sectionprovides an overview of the most popular techniques used formodal testing of microsystems.

3.1 Excitation Methods

Direct excitation methods, such as impulse provided by animpact hammer or input from a shaker that is directly at-tached to the structure, cannot be applied to microsys-tems due to their small size. Therefore, an indirect, non-contact excitation method is required for modal testing ofmicrosystems[4, 10, 11, 14]. In addition, the natural frequenciesseen in microsystems are typically substantially higher thanthose of traditional structures, from a few kilohertz to themegahertz range. The excitation method must be capable ofexciting the required high-frequency range[11].

Electrostatic forces can be applied for excitation of microstruc-tures. At the small size scales of microsystems, significantelectrostatic forces can be induced with the application oflow voltages. Electrostatic excitation can be applied eitherby employing the structure itself (by designing the elementsas electrostatic electrodes, or integrating electrodes into thestructure)[14, 15, 18, 19], or by using a built-in actuator (such asa comb drive in surface-micro machined MEMS)[3, 17, 20]. Al-though fairly simple to apply, this method suffers from the factthat the electrostatic forces themselves are (nonlinear) func-tions of structural motions, and as such, cannot be indepen-dently characterized[1, 18]. In addition, actual input to the struc-ture cannot be measured. Realization of this method as a vi-able modal testing technique for thorough dynamic character-ization of microsystems requires coupling of accurate modelsfor structural dynamics and electrostatic forces. If the excita-tion is conducted by an internal component, although the com-plexity arising from coupling of structure and electrostatics isreduced, structural coupling becomes important, and the re-sults must be closely examined and processed by using sub-structuring techniques.

The excitation can also be established by embedded smartmaterials, such as piezoelectric elements or shape-memoryalloys, integrated into the structure[19] or attached to itsboundaries[21], the latter of which can only be applied forsomewhat larger microstructures as it may require assembly.Either way, this method adds complexities to the fabricationmethods and, in some cases, no fabrication capability of inte-grating such smart materials within the microsystems exists.Dynamic characteristics can be significantly altered by the in-clusion of extra materials[21]. As with the electrostatic meth-

ods, the characteristics of the embedded smart materials arealso nonlinear functions of structural motion.

Magnetic and thermal excitation techniques have also beenused as modal excitation methods[21]. These methods usu-ally require coating the structure with some other material,addition of which can alter the dynamic characteristics of thestructure[7]. Magnetic excitation cannot be used if the struc-ture itself is ferromagnetic. Moreover, magnetic and/or ther-mal forces may adversely impact the structures and othersystem components. Difficulty with measuring the input, andvarying characteristics of the input with structural motion areother drawbacks. Another alternative method is the use ofelectrical discharge pulse[11], which requires very high volt-ages. This method is capable of providing excitation frequen-cies up to the megahertz range.

The problems with the above methods arise from the fact thatnone of them is truly external. In addition, they are not suitablefor generating FRFs, since, in most cases, it is not possible tomeasure the inputs. One of the most promising alternatives isBase excitation using external elements. This is a viable tech-nique that does not alter the structural characteristics[13, 22, 23].In this case, a transducer, e.g., a piezoelectric shaker in theform of a PZT (lead zirconate titanate) film or disk[10, 24, 25],is used to excite the structure[4, 7, 11] (traditional mechani-cal, electromagnetic, and hydraulic shakers cannot be appliedsince they have a limited frequency range). Although the in-put is not directly measured since a sensor, such as a loadcell, cannot be included, it is possible to derive motion-over-motion FRFs taking the base (boundary) motions as a refer-ence. Since the response characteristics can be affected bythe dynamic characteristics of the testing apparatus, includingboth the shaker and the fixture, care must be taken during itsdesign. Once the experimental apparatus is manufactured, athorough dynamic characterization must be performed[10].

3.2 Measurement Methods

Since it is not possible to place sensors on microstructures,a number of alternative detection methods have been devel-oped and/or adapted (mostly from optics). Similar to excitationmethods, measurement methods are also required to employnoncontact devices and to be applicable for the required high-frequency range[4, 10, 11, 14, 26].

Due to the high frequencies involved, the measurement ofthe motion of microstructures poses a critical challenge. Oneof the enabling techniques adopted from optics to effectivelyslow the motion is stroboscopy [27]. Stroboscopy can be usedin conjunction with the measurement methods described be-low to increase the applicable frequency range. In this tech-nique, stroboscopic light is flashed on the moving structure(excited at a prescribed frequency). In effect, the image isfrozen at the time of the flash. Coordinating the stroboscopiclight with the excitation frequency, and changing the time ofthe flash, enables sweeping through the phase and detecting

the image as a function of time. The maximum detectable fre-quency is limited by the pulse time of the stroboscopic light,i.e., the shorter the flash pulse time, the higher the detectablefrequency. Some researchers used a (pulsed) laser diode[26]

instead of the commonly used light emitting diode (LED) asthe stroboscopic light source to decrease the flash pulse timeand increase the modulation bandwidth, thus enabling mea-surement of higher frequency and larger amplitude motions.

Video-imaging techniques can also be applied for measur-ing structural motion[2, 5, 20]. For instance, a CCD (charge-coupled device) camera can be used in conjunction withan optical microscope to record the motion of microstruc-ture, which can then be analyzed using image processingtechniques[26]. Since the motion of microsystems generallyinclude very high frequencies, however, direct observation ofthe motion requires an ultra-high speed camera and associ-ated data acquisition and image-processing system[2], whichmakes this technique costly and complex[17]. Stroboscopy canbe used to “slow” the motion to a range that a regular speedCCD camera (30 frames/second) is capable of recording[17].Advantages of this method includes simplicity of the setup andnot requiring expensive equipment, since only an optical mi-croscope, a regular speed CCD camera, a stroboscopic lightsource, and a device to coordinate the flash times to the exci-tation are required. However, since a large number of imagesmust be collected to describe the motion completely, the mo-tion is required to be either repeatably transient or periodic. Inaddition, image-processing techniques are required to trans-form the data to a time versus motion form. This method ismore suitable for measuring in-plane motions.

Another common technique adapted from optics is interferom-etry [27]. Interferometry measures distance by comparing op-tical path length difference between two beams of monochro-matic light. The comparison is made by re-combining thebeams; if they are in phase, a bright “fringe” is produced,while if they are out of phase, a dark fringe is produced. Thesurface is required to be specular, i.e., mirror-like reflective.For a typical interferometer configuration, such as a Michel-son interferometer, a mechanical motion of half a wavelengthproduces a total optical path change of one wavelength. Vis-ible light has wavelengths of about 0.5 µm, and thus a fringerepresents a distance difference of about 0.25 µm. The res-olution of “fringe counting” is perhaps around 0.1 µm, whilephase stepping techniques can achieve resolution of .002 µmor better. When using interferometry as a measurement tech-nique for microsystems, the structure is observed through amicroscope configured as a classical interferometer[14, 16, 21].The resulting image has a fringe pattern superimposed, wherethe fringes represent a contour map of the microstructure rel-ative to the (usually planar) reference surface. For direct ob-servation, the detectable frequency is limited by the speed ofthe CCD camera, but stroboscopic methods can also be com-bined with classical interferometry[26]. In either case, inter-ferometry is more suitable for detecting out-of-plane motions.One main advantage is that the motion of the whole surface,rather than a single point, is measured at once. This is very

important for establishing mode shapes of the structure. To bepractical, interferometric methods must be coupled with auto-mated image processing techniques. Also, issues related tofringe instability of the interferometry must be considered[14].

Holographic interferometry or ESPI (Electronic Speckle Pat-tern Interferometry)[12, 28] can also be used. These tech-niques measure surface position relative to a reference po-sition, rather than using a flat plane as a reference as do theclassical interferometric techniques. They too can be com-bined with stroboscopic illumination to provide motion analy-sis.

Blur envelope is a very inexpensive technique to measure mo-tion characteristics of microstructures[3, 17]. When the motionof a structure is detected by a camera that integrates a num-ber of cycles onto the image, the high-frequency motion of thestructure is seen as a blur. Sweeping through the excitationfrequency, the amplitude of the blurred image changes, indi-cating modulations of the response amplitude. This method isgenerally applied manually by a user who watches the imageat different excitation frequencies and notes the frequency thatcauses the largest blur envelope. Alternatively, grabbing thecamera images and analyzing the amplitude enables one todescribe the motion more accurately in amplitude-frequencyplane. This method can be used in conjunction with an auto-mated signal generator, a frame grabber, and an image pro-cessing technique to collect complete motion information interms of both phase and amplitude versus frequency[20]. Thistechnique is more suitable for detecting in-plane motions.

Some internal detection methods have also been considered.One such method measures changes in the capacitance aris-ing from the structural vibrations[3, 18−20]. Embedded smartmaterials (especially piezo-sensitive materials) that enable re-lating the changes in electrical properties to structural motionhave also been used[19, 21]. Similar to the internal excitationmethods mentioned above, however, the measured quantitiesthemselves are not independent of structural motion charac-teristics, and the inputs cannot be directly measured[18]. Thus,without thorough knowledge of multi-physics involved, param-eter extraction is not possible. In addition, the signal-to-noiseratios are usually very low. On the positive side, no extraequipment is required to apply these methods.

Use of a Laser Doppler Velocimeter (LDV)1 is another attrac-tive method with noncontact measurement capability and highpositional accuracy[4, 6, 7, 10, 11, 24, 25]. LDVs measure out-of-plane velocities by interferometrically measuring the Dopplershifted wavelength of the light reflected from the surface. Itis generally required that the laser light is perpendicular tothe surface whose motion is being measured. Adequate re-flectivity of the surface is also necessary. In the case of mi-crosystems, the laser light is fed through a microscope to re-duce the focused spot size of the laser light. This is importantsince the velocity is averaged over the spot area and the mo-

1Sometimes referred to as a Laser Doppler Vibrometer.

UndeflectedBeam

DeflectedBeamu(x,t)

y(t)

w(x,t)

x

ExcitedBase

Figure 1: A base-excited cantilever beam.

tion of the small features can only be measured with a smallspot[10]. Single point LDVs require multiple measurements forestablishing the mode shapes. To correctly establish phaseinformation between multiple points measured one at a time,a common, repeatable reference must be used. Since it istypically not possible to measure input forces, one commontechnique is to use the excitation voltage as a reference[10].However, this assumes that the frequency response betweenthe voltage and the structure being tested is essentially flat,and thus ignores the dynamic behavior of the shaker and thefixture. Response measurements can be sequentially taken atseveral points, and then compared to provide mode shapes.Alternatively, a scanning LDV system can be established byincorporating controlled scanning elements, such as posi-tioning stages[4, 10, 25], scanning mirrors[6], or acousto-opticdeflectors[6]. Scanning LDV systems allow automatic mea-surement of many points in a single test. Due to aforemen-tioned advantages and availability, the use of single-point LDVis the measurement technique of choice for the experimentspresented here.

4 MODELING BEAM DYNAMICS

This section presents derivation of the equation of motion ofa cantilever beam undergoing base excitation. This derivationis similar to those in[11, 13, 23], and provided here for the sakeof completeness.

Figure 1 depicts a cantilever beam subjected to base excita-tion. The boundary-value problem that described the motionof such an Euler-Bernoulli beam can be given as[29]

Lw(x, t) + m∂2w(x, t)

∂t2= f(x, t), (1)

where t is time, m is mass per unit length, w(x, t) is the verticaldisplacement at point x, and f(x, t) is the force per unit length.The operator L can be expressed as

L =∂2

∂x2

(EI

∂2

∂x2

),

where E is the Young’s modulus and I is the area moment of

inertia. The boundary conditions for this cantilever beam are

w(x, t)

∣∣∣∣x=0

= y(t),∂

∂xw(x, t)

∣∣∣∣x=0

= 0, (2)

∂2

∂x2w(x, t)

∣∣∣∣x=L

= 0,∂3

∂x3w(x, t)

∣∣∣∣x=L

= 0, (3)

where y(t) is the prescribed base excitation.

One way to solve this equation is to solve the associatedeigenvalue problem and employ the normal mode expansionmethod. However, non-homogeneous boundary conditionspose a formidable challenge for solving the eigenvalue prob-lem. To render the boundary conditions homogeneous, theboundary value problem may be written in terms of relativemotion by applying the transformation u(x, t) = w(x, t)− y(t).Substituting this into Eqs. (1)-(3), the new system of equationsbecomes

Lu + m u = f(x, t) − m y, (4)

subjected to

u = 0, u′ = 0 at x = 0, (5)

u′′ = 0, u′′′ = 0 at x = L, (6)

in which the uniform-homogeneous beam assumptions havebeen applied[29].

To solve the eigenvalue problem associated with this systemof equations with homogeneous boundary conditions, we as-sume synchronous motions, i.e., one in which the whole struc-ture vibrates with the same frequency. Synchronous motionsimply that the solution u(x, t) is separable in its spatial andtemporal components. Solving the separated ordinary differ-ential equations, the eigenvalue equation of the system canbe established as

cos βrL cosh βrL + 1 = 0, (7)

where

β4r =

ω2rm

EI

denotes the rth solution and ωr is the rth natural fre-quency. This transcendental eigenvalue equation can begraphically or numerically solved to yield values of βrL(1.875, 4.694, 7.856, . . .). The associated eigenfunction (modeshape) can be expressed as

Ur(x) = Cr

[(sin βrL − sinh βrL)(sin βrx − sinh βrx)

(cos βrL + cosh βrL)(cos βrx − cosh βrx)]. (8)

where Cr is an arbitrary scaling constant. A means of uniquelyand conveniently expressing Cr is to execute mass normaliza-tion, which uses the orthonormalization conditions∫ L

0

UrLUp dD = ω2pδrp, (9)

∫ L

0

mUrUp dD = δrp, (10)

where Ur is the mass-normalized eigenfunction and δrp is theKronecker’s Delta function. Using these equations, Cr can beanalytically determined from

C2r =

[m

∫ L

0

Ur(x)Ur(x) dx

]−1

. (11)

Since the eigenfunctions of Eq. (8) form a complete set in thedomain of the boundary-value problem (Eqs. (4)-(6)), the so-lution u(x, t) can be expressed as a linear combination of theeigenfunctions as

u(x, t) =

∞∑r=1

ηr(t)Ur(x). (12)

This normal mode expansion procedure inherently assumeslinearity. The upper limit of the summation can be changedto n if the effect of the higher modes are negligible in the fre-quency range of interest.

Substituting this into Eq. (4), multiplying both sides with Ur(x),integrating over the domain, and considering the orthonormal-ity relations, one obtains an uncoupled set of r differentialequations as

ηr + ω2r ηr = Nr, (13)

where

Nr =

∫ L

0

f(x, t) Ur dx − m y(t)

∫ L

0

Ur dx. (14)

The second term of Nr in Eq. (14) is due to the base excita-tion. Clearly, base excitation excites the whole structure withan inertial force scaled by the integral of normalized modeshape[22]. The scaling factor can be analytically evaluated as

Rr =

∫ L

0

Ur dx =Cr

βr(sin βrL − sinh βrL).

The first term in Eq. (14) is the generalized external forcingfunction. For the cases considered here as the applicationof the modal testing methodology, there exist external forcesarising from the effect of ambient gasses. For small motions,the gas forces can be considered as proportional to the veloc-ity, and thus can be represented as a viscous damping force.Setting

−2ζrωr ηr =

∫ L

0

f(x, t) Ur dx,

Equation (13) can be rewritten as

ηr + 2ζrωr ηr + ω2r ηr = −myRr. (15)

In addition to the effect of gas forces, the damping term canalso be considered to account for structural damping.

Solving Eq. (15) for η(t) and substituting into Eq. (12), themotion of the beam in the presence of gas forces and baseexcitation can be completely described.

LDV OpticalMicroscope

BeamPositioning

Mirror

Fixture

Figure 2: Experimental setup.

5 MODAL MODEL

When interpreting the results of modal tests, and transformingthe FRF data to physically meaningful dynamic parameters,an equation that models the dynamic behavior of the structureis employed. This model is usually referred to as the modalmodel. The modal model that will be outlined in this sectionfollows the boundary-value problem described above.

During modal testing with base excitation, the structure is ex-cited by a periodic motion y(t), and both the input (base) andresponse velocities are measured[22]. Assuming that the re-sponse is also periodic, Fourier transform of both sides ofEq. (15) results in

η(jω) =mRrω

2

−w2 + w2r + 2jζrωrω

Y (jω).

Substituting this into the normal mode expansion given inEq. (12), and transforming back to the absolute motion,

1

ω2

(W (jω)

Y (jω)− 1

)=

n∑r=1

Ar

−w2 + w2r + 2jζrωrω

, (16)

where W and Y are the Fourier transformed (absolute) re-sponse and base velocities, respectively, and Ar = RrUr isthe modal constant.

For the case of base excitation, the measured FRF is usu-ally in the form of motion over motion rather than motion overforce[22]. Equation (16) assumes that the FRF is created bymeasuring the absolute response velocity and the velocity ofthe base motion, and algebraic manipulation on the left handside allows representing the FRF in the standard form suitablefor curve fitting procedures.

Figure 3: Fixture/vacuum chamber.

6 EXPERIMENTAL SETUP

Figure 2 shows the experimental setup used during the ex-perimentation. It includes an LDV (Polytech model OFV-302interferometer, with OFV3000/OVD02 velocity demodulator),an optical microscope, a piezoelectric shaker, and a vacuumchamber. During the experimentation, the laser beam createdby the LDV is fed through the camera port of the microscope.In this manner, a 6µm diameter laser spot is established. Thereflecting laser beam allows the LDV to determine the out-of-plane velocity perpendicular to the surface by measuring theDoppler shift. The optical microscope was also used for fo-cusing the laser spot on the desired location on the surface ofthe beams.

The piezoelectric shaker used here was originally designedfor acoustic emission. It is capable of exciting the structurewith up to 104.2 kHz frequency. A high voltage signal (up to100 Volts) from an amplifier is used for driving the shaker tothe required amplitude.

As seen in Fig. 3, the vacuum chamber is a metal cylinder withholes on either side. A continuously running vacuum pump isplaced on one of the sides, whereas a pressure gage is at-tached on the other to measure the pressure. The vacuumchamber also functions as a fixture for the MEMS modulethat includes the test structure, which is attached to the fix-ture by means of superglue. A glass lid is placed on the top ofthe cylinder using removable vacuum grease. This open-looparrangement is sufficient to obtain pressures down to the 30mTorr level.

Figure 4 shows the test structures, which include polysiliconbeams with varying lengths from 100µm to 1 mm, with 20µmwidth fabricated in SUMMiT VTM (Sandia Ultra-planar, Multi-level MEMS Technology) process. Two different nominal gapheights, i.e., the height of the bottom of the beam from thebase surface, were considered: 2µm and 6.3µm. It should benoted that the gap height varies along the length of the beam

Figure 4: Test structures: polysilicon micro-cantilevers.

as a result of the warpage arising from the residual stressesof the fabrication processes. The 2µm-gap beams had a de-sign thickness of 2.5µm, whereas the 6.3µm-gap beams hada design thickness of 2.25µm.

7 EXPERIMENTAL RESULTS

As mentioned in the Introduction section, modal testing hasa wide range of uses for microsystems, including testing andvalidation of coupled physics modeling in microsystems. Asan application of the modal testing setup presented here, theeffect of gas damping has been experimentally investigated.In particular, the goal of the experimentation was to determinethe amount of energy dissipated due to ambient gas forcesas measured by the viscous damping ratios for various beamgeometries, pressures, and gap heights.

Due to the large surface-to-volume ratios seen in microsys-tems, the surface forces play a pivotal role in determining thecharacteristics and performance of microsystems. The gasforces include the air drag and squeeze-film effect, the lat-ter of which is usually dominant for surfaces moving relativeto one another in close proximity. In reality, these forces arenonlinear functions of instantaneous gap height and beam ve-locity.

The aim of the work undertaken here, however, was not tostudy nonlinearities, but rather to establish a nominal quanti-tative measure for the energy dissipation. For this purpose,a true random excitation signal was used during experimen-tation. Used with the Hanning window and spectrum averag-ing, this type of excitation effectively linearizes the measuredFRFs[8, 9, 30]. Considering this fact, the input amplitude wasvaried from test to test to establish the largest signal-to-noiseratio without saturation, while preserving the observed linear-ity of the response. Average input (base) amplitude was lessthan 10 nm, whereas the response amplitude at the tip of thebeams did not exceed 0.5 µm. Care should be taken, how-ever, for the cases where there is very low damping, which

10-10

10-9

10-8

10-7

10-6

0 40 80 102.4

Frequency (kHz)

Ve

loc

ity

PS

D

(in

/s)2

Figure 5: Velocity PSD of the base.

translates to the number of frequency lines included within theband of a resonant peak being very small. In these cases, theHanning window would change the parameters extracted fromthe FRF data, i.e., it would increase the observed damping ra-tio.

Since a single-point LDV was used during the experimenta-tion, it was not possible to measure both the response andreference motions simultaneously to establish the FRFs. Forthis reason, the FRFs were calculated from

H(jω) =Hw

Hy,

where Hw is the frequency response function between the re-sponse velocity (at a particular point) and the supply voltage,and Hy is the FRF between the base velocity and the sup-ply voltage. Each of Hw and Hy was measured separately.Taking the voltage source as the reference, however, enabledcorrectly obtaining both the amplitude ratios and phase infor-mation.

7.1 Base Response

Figure 5 illustrates the velocity power spectrum measured atthe base of the beam. Since the power spectral density (PSD)of the supplied voltage was essentially flat for the whole rangeof excitation frequency, this variation can be attributed to thedynamics of the shaker and the fixture. Several orders of mag-nitude variation in amplitude, associated with a large peak at72 kHz can be seen from this figure. For an ideally linearsystem, such a base response would not be a great concern,since the aforementioned FRF uses this response as a refer-ence, and thus its effect is removed. However, such large vari-ations would lead to unfavorable signal-to-noise ratios even foran ideally linear system. In addition, although the true randomexcitation and spectrum averaging linearizes the FRF, therestill exist nonlinearities that cannot be removed through thisprocess. In other words, the structure will experience differ-

ent base excitation amplitudes at different frequencies and re-spond to those in a nonlinear fashion. Since this input andassociated nonlinearity is systematic, it will not be removedthrough spectrum averaging, and thus will be observed in theFRFs.

7.2 Observations

Frequency response data for 800 µm beams with 6.3 µm nom-inal gap height at different ambient pressures are shown inFig. 6, in which first three bending modes are identifiable (theamplitude scales from (a) through (d) are different). The effectof the non-flat base excitation can be seen in these figures,especially at higher ambient pressures, as a large noise spikeat 72 kHz (see Fig. 6(a)). This spike is due to the peak in thebase input. The noise spike diminishes greatly for the lowerpressure FRFs. However, even for the low pressure caseswhen the amplitude of the resonant peaks become very large,this effect is observable from the phase characteristics (seeFig. 6(d)).

It is interesting to follow the response of the first bending mode(around 4.8 kHz). At the atmospheric pressure (Fig. 6(a)),due to the large damping ratio, this mode cannot be clearlyidentified from the amplitude response, without an examina-tion of the phase. At lower ambient pressures (from Fig. 6(a)-Fig. 6(d)), reduced damping ratio allows identification of theresonant peak from the amplitude response.

The observed natural frequencies, as seen by comparing Fig-ures Fig. 6(a) through Fig. 6(d), do not vary significantly atdifferent pressures. In addition, the responses do not showany indication of nonlinearities other than the effect of thenon-flat base excitation. It can then be argued that the ob-served response is linear, and modelling the effect of the gasforces as viscous damping forces nominally captures the dis-sipative mechanism (stiffness and mass characteristics seemto be pressure invariant).

Figure 7 shows the resonant peak of the second bendingmode for 600 µm beam with 6.3 µm nominal gap height. Sim-ilar to the observations above, the considerable effect of thepressure is observed here. Again, the pressure does not af-fect either of the stiffness or mass characteristics, and gasforces can be considered to influence the structure as a vis-cous damping force (the small change in the damped naturalfrequency is due to the change in viscous damping ratio).

7.3 Analysis

Polynomial curve-fitting methods were applied to determinethe modal parameters of the structures, including the natu-ral frequencies and viscous damping ratios, using the modalmodel derived above. Both partial fraction polynomials andorthogonal Forsythe polynomials were used for curve fit-

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Figure 6: FRFs for 800 µm beams with 6.3 µm gap at (a)610 Torr, (b) 63 Torr, (c) 7 Torr, (d) 500 mTorr pressure.

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Figure 7: The second bending mode of 600 µm beam atvarious pressures.

ting to a single degree of freedom at a time, since thepeaks were separated considerably even when there is largedamping[8, 9, 31]. Whereas the condition numbers of the ma-trices used for partial fraction method were very large (indi-cating near-singularity), the results of this method and the nu-merically stable Forsythe method were identical to the fourthsignificant digit. Synthesized FRFs generated using the es-tablished modal parameters closely matched the experimentalFRFs.

A comparison between the theoretical and experimental nat-ural frequencies is given in Fig. 8, which includes the naturalfrequencies obtained from all tests performed for a range oflengths and boundary conditions. The closer the data is tothe 45o line, the closer the theoretical and experimentally ob-served natural frequencies are. It is interesting to note thatthe difference is very small, even though there are a numberof uncertainties arising from the boundary condition (not ide-ally cantilever), geometry (thickness and length), and materialproperties (Young’s modulus). It can be deduced that, for thisset of test structures, the effect of the aforementioned uncer-tainties effectively cancelled each other.

The measured damping ratios for 700 µm and 800 µm beamswith 6.3 µm nominal gap height are given in Fig. 9. Since thesqueeze-film force is strongly correlated to the vibration am-plitude, larger vibration amplitudes at lower modes result inlarger damping ratios. From the low pressure data, it can alsobe argued that the structural damping also reduces with in-creased frequency. Another observation is that the behaviorof the damping ratio with pressure, as seen by the shape ofthe fitted curves, is similar for different modes and beam ge-ometries.

Figure 10 presents the damping ratios for 300 µm beams fortwo different gap heights at various pressures. The large dif-

45o line

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Figure 8: Comparison between theoretical andexperimental natural frequencies.

ference (about an order of magnitude) between the dampingratios at high pressures corresponds to the squeeze-film ef-fect. At reduced pressures, the damping ratios approach tothe same value, which is less than 0.1 %. This behavior in-dicates that the damping at this frequency range almost com-pletely arises from the effect of gas forces, and the structuraldamping is negligible.

8 SUMMARY AND CONCLUSIONS

This paper presented a methodology for experimental modalanalysis of microsystems. A brief survey of microsystemmodal testing methods is included as a background to thereader. Since the methodology includes base excitation, theboundary value problem and modal model of cantilever beamsunder base excitation were derived. As an application of themethodology, the effect of the gas forces on the energy dis-sipation characteristics of micro cantilevers has been investi-gated for various beam geometries, pressure levels and gapheights, and a number of observations has been made.

The specific conclusions of this work are;

1. The methodology presented in this paper, including baseexcitation via piezoelectric shaker and motion detectionvia laser interferometer, is an effective method for exper-imental modal testing of microsystems.

2. Gas damping, arising from air drag and the squeeze-film effect, was experimentally observed to be the mainmechanism of energy dissipation for microsystems underconsideration (when there exists vertical motion in closeproximity to the “ground”).

3. Deduced from the low pressure data, structural damping

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Figure 9: Measured damping ratios at different pressuresand modes for (a) 700 µm beams, and (b) 800 µm beams.

in surface micro-machined polysilicon-based microsys-tems is much lower than that for most conventional(macro) systems.

4. Due to very low damping ratios seen in microsystems,care must be taken when using particular windows dur-ing data acquisition and signal processing.

9 ACKNOWLEDGEMENTS

The authors would like to thank following individuals from San-dia; David W. Kelton for his help during experimentation; JohnR. Torczynski, Edward S. Piekos and Michail A. Gallis for pro-viding insight about characteristics of gas forces; MichelleA. Duesterhaus, and Darren Hoke for providing test struc-tures and associated photos; and Clark R. Dohrmann, JamesM. Redmond and David O. Smallwood for valuable discus-sions, and David S. Epp and James J. Allen for reviewing themanuscript and providing valuable input.

Sandia is a multiprogram laboratory operated by Sandia Cor-poration, a Lockheed Martin Company, for the United StatesDepartment of Energy under contract DE-AC04-94AL85000.

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Figure 10: The effect of gap height in the damping ratiofor 300 µm beams.

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