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MMACHINERY, ISOLATION

See VIBRATION ISOLATION, APPLICATIONS AND CRITERIA

MAGNETORHEOLOGICAL FLUIDS

See ELECTRORHEOLOGICAL AND MAGNETORHEOLOGICAL FLUIDS

MAGNETOSTRICTIVE MATERIALS

A Flatau, National Science Foundation, Arlington, VA,USA

Copyright # 2001 Academic Press

doi:10.1006/rwvb.2001.0079

The bidirectional coupling between the magnetic andmechanical states of a magnetostrictive material pro-vides a transduction mechanism that is used in bothsensing and actuation. The current interest in thedesign of adaptive smart structures, coupled withthe advent of materials which exhibit high sensorfigures of merit such as Metglas and the so-calledgiant magnetostrictive materials such as Terfenol-Dhas led to a renewed interest in the engineering ofoptimized magnetostrictive transducer designs. Anoverview of magnetostriction and of recent sensingand vibration control applications for magnetostric-tive materials is presented along with a brief discus-sion of some pertinent magnetostrictive device designissues. Schematics of several actuator and sensorconfigurations are included, as are typical experimen-tal results.

Magnetostriction and Magnetism

Magnetostrictive materials convert magnetic energyto mechanical energy and vice versa. As a magneto-strictive material is magnetized, it strains. Conver-

sely, as either a force or torque produces a strain in amagnetostrictive material, the material's magneticstate changes. The application of either an externalmagnetic field or an external stress breaks the materi-al's internal energy balance and leads to both mag-netic responses (domain wall motion, magnetizationrotation, and a change in permeability) and elasticresponses (strain of the material and a change instiffness).

Magnetostriction is an inherent material propertyassociated with electron spin and orbit orientationsand molecular lattice geometries and will not degradewith time or with use. The effect is not present when amagnetostrictive material is heated above its Curietemperature; however, the effect returns fully as thematerial is cooled to below the Curie temperature. Allferromagnetic materials exhibit magnetostriction;however, in many materials the magnitude of thestrain or shape change is too small to be of conse-quence. Nominal longitudinal saturation strains forvarious materials at room temperature are shown inTable 1.

Sensors are commonly based on the reciprocalmagnetostrictive effects, the change in magnetizationand permeability of a sample that accompanies achange in an external stress or torque. The mag-netization change produced due to the applicationof a stress is known as either the Villari effect orthe magnetomechanical effect. The magnetization

change produced due to application of a torque isknown as either the inverse Wiedemann effect or theMatteuci effect.

Actuators are commonly based on the shapechange that accompanies magnetization of a magne-tostrictive material. The most common shape changeis a change in sample length, accompanied by achange in transverse dimensions. This is known asthe Joule effect (after James Prescott Joule (1818±89)for positive identification of the change in length ofan iron sample) or the magnetostrictive effect. Mag-netostriction can also occur as a twisting whichresults from a helical magnetic field, often generatedby passing a current through the magnetostrictivesample. This is known as the Wiedemann effect.

At low applied magnetic field strengths, domainwall motion occurs as a consequence of the growth ofdomains whose magnetization is aligned favorablywith respect to the applied field, at the expense ofdomains with magnetization opposing the field. Atmoderate field strengths, the magnetic momentswithin unfavorably oriented domains overcome theanisotropy energy and suddenly rotate (jump) intoone of the crystallographic easy axes closer to theexternal field direction. This jump coincides with arelatively large change in strain for a small change inapplied field. At moderate to high field strengths, themagnetic moments align along crystallographicallyeasy axes lying closest to the field direction. As thefield is increased further, the magnetic momentsundergo coherent rotation until they reach alignmentwith the applied field resulting in a single-domainmaterial and technical saturation.

Crystalline growth can be tailored or optimized soas to enhance the magnetostrictive effect in a materialof a given stoichiometry. For example, in the highlymagnetostrictive alloy Terfenol-D (TbxDy1ÿxFey,where x � 0:27ÿ0:3 and y � 1:92ÿ2:0) energy

minimization is satisfied with random alignment ofthe magnetic domain orientations along the eightoutward diagonals from the center of the molecularcubic lattice such that the bulk sample has zero (oralmost zero) net magnetization. Commercially avail-able material is grown using directional solidificationtechniques and patented crystallographic growthmethods that lead to crystal growth and magneticmoment orientations such that jumping of magneti-zation between easy axes perpendicular and parallelto the sample growth direction is facilitated.

Exchanges between magnetic and mechanicalenergy are the primary transduction effects employedin magnetostrictive sensing and actuation applica-tions. The two coupled linear equations given ineqns [1] and [2] model these relationships. Theseequations neglect temperature effects and hysteresis,and have been reduced from a three-dimensionalvector form to reflect only axial behavior. Thesemagnetostrictive equations of state are expressed interms of mechanical parameters (strain ", stress s,Young's modulus at constant applied magnetic fieldEH), magnetic parameters (applied magnetic field H,magnetic induction B, permeability at constant stressms), and two piezomagnetic coefficients (these arealso known as magnetomechanical coefficients andas axial strain coefficients: d33 � d"=dHjs, andd�33 � dB=dsjH; where for small strains, these twocoefficients can be shown to be equal):

" � s=EHy � dH �1�

B � d�s� msH �2�

where " and B are dependent on the externally ap-plied quantities s and H. The elastic modulus, per-meability, and piezomagnetic coefficients can varysignificantly with operating conditions. They needto be experimentally determined and are usually pro-vided by the material manufacturer. At present, mostmodeling codes rely on look-up tables to capture thetime-varying and load-dependent nature of thesecoefficients.

These equations capture the low signal, linear,coupled mechanical, and magnetic nature of magne-tostriction. Eqn [1] indicates that the net strain of amagnetostrictive element is the combined response tochanges in stress (Hooke's law behavior, " � s=E)and applied magnetic field (Joule effect magnetostric-tion, l). Eqn [2] indicates the simultaneous change inthe magnetic induction of the element due to changein stress (reciprocal Joule or Villari effect magnetiza-tion) and applied field. This later effect can also be

Table 1 Saturation magnetostrictions for magnetostrictivematerials at 300 8K

Material Magnetostriction�DL=L� 10ÿ6)

Fe 79Ni 735Co 76060% Co±40%Fe 6860% Ni±40%Fe 25TbFe2 1753Terfenol-D 1600SmFe2 71560Metglas 2605SC 40

Data produced from Restorff JB (1994) Encyclopedia of Applied Physics,vol. 9, pp. 229±244.

754 MAGNETOSTRICTIVE MATERIALS

expressed as a change in permeability by writing eqn[2] in a more general form:

B � mH �3�

In eqn [3] the effects of stress are included in thepermeability, m. Permeability can be monitored sinceboth B and H can be related to measurable electricalquantities as described in eqns [4] and [5].

In the early nineteenth century, Oersted discoveredthat a moving charge generated a magnetic field in aplane perpendicular to the direction of chargemotion. Thus, a current in a conductor could beused to produce a magnetic field around the conduc-tor. Ampere's law describes this electromagnetic rela-tionship. For a long, thin solenoid having a number ofturns Nc and a length Lc a simple expression isderived;

H � NcI� �=Lc �4�

Placing a magnetostrictive element inside such anexcitation coil (solenoid) with an impressed currentI provides an efficient means of magnetizing theelement and producing controlled strain and forceoutput.

The law of electromagnetic induction (Faraday±Lenz law) describes how a magnetic flux, j � BAc inarea Ac, induces a potential in an electrical conductorto which it is flux-linked. In its simplest differentialform, the Faraday±Lenz law is given by:

V � ÿNcdjdt� ÿNcAc

dB

dt�5�

where V is the induced voltage in the solenoid ofconstant area Ac. According to this law, a potentialwill be induced in any electrically conducting mate-rial that makes up the magnetic circuit.

The excitation coil described by eqn [4] can be usedto generate a magnetic field in a sample spatiallyseparated from the coil. According to Gauss's lawof magnetism:

r � B � 0 �6�

the divergence of B is zero. This means that themagnetic flux is always conserved. Thus magneticflux lines close, defining a magnetic circuit, and ele-ments of the magnetic circuit through which magneticflux flows are said to be flux-linked. This makes itpossible to magnetize one component of the magneticcircuit by generating a magnetic field in anothercomponent. Based on the principle expressed by eqns

[5] and [6], it is possible to measure the magnetic fluxdensity in a magnetic circuit by the voltage induced ina flux-linked detection coil.

Transduction

Magnetostrictive materials are magnetoelastic in thesense that they do work in the process of convertingbetween magnetic and elastic (or mechanical) energystates. However, magnetostrictive transducers aregenerally classified as electromechanical devices orelectromagnetomechanical because their input andoutput are generally electrical and mechanical innature. The conversion of magnetic energy to andfrom electrical and/or mechanical is transparent tothe device user. A common two-port schematicappropriate for both magnetostrictive sensing andactuation devices is given in Figure 1. The magnetos-trictive driver is represented by the center block, withthe transduction coefficients Tme (mechanical due toelectrical) and Tem (electrical due to mechanical),indicative of both the magnetoelastic attributes char-acterized by eqns [1] and [2] and the electromagneticattributes associated with conversion between elec-trical and magnetic fields characterized by eqns [4]and [5].

The transduction process relating the electrical andmechanical states can be described with two coupledlinear equations. These canonical equations areexpressed in terms of mechanical parameters (forceF, velocity v, mechanical impedance Zm), electricalparameters (applied voltage V, current I, electricalimpedance Ze), and the two transduction coefficients:

V � ZeI � Temv �7�

F � TmeI � zmv �8�

Common magnetostrictive transducer componentsinclude a magnetic circuit, a solenoid for transductionof magnetic-to-electrical energy and vice versa,mechanisms for DC magnetic and mechanical (pres-tress) biasing. A generic magnetostrictive device con-

Figure 1 Two-port transducer schematic.

MAGNETOSTRICTIVE MATERIALS 755

figuration is shown in Figure 2. The permanentmagnet provides a DC magnetic field. The magneticcircuit passes though the magnetostrictive driver, endcaps made of magnetic materials, and the permanentmagnet. Belleville spring washers are used to provide

an initial mechanical prestress. The solenoid can beused to provide both AC and DC magnetic fields foractuation purposes. Alternatively, for sensing appli-cations, the change in voltage induced in the solenoidis to detect a change in strain and/or in the forceapplied to the device.

Figure 3 depicts a major hysteresis loop from anactuator similar in design to that shown in Figure 2.Superimposed on the major hysteresis loop are fiveminor hysteresis loops collected by driving at 0.7 Hzwith an applied of +5 kA m71 as the DC magneticbias was increased from 5 to 45 kA m71 in incrementsof 10 kA m71. AC operation of magnetostrictiveactuators is typically achieved through the use of aDC magnetic field to provide bias operation so that itis centered about the steepest portion of the majorhysteresis loop. This region is called the burst region,and the DC magnetic field amplitude required foroperation about the middle of the burst region iscalled the critical field. All subsequent data are frommagnetically biased transducers.

Figure 4 shows frequency response functions ofacceleration per input current, where the change inthe transducer's axial resonant frequency varies from1350 Hz to over 2000 Hz, reflecting the effect of DCbias on the elastic modulus. The reduction in elasticmodulus below that at magnetic saturation is knownas the delta E effect. Note that in Figure 4 only theaxial resonant frequency of the magnetostrictive dri-ver shifts and that the structural resonance of thedevice housing at 3300 Hz is not affected by thechanging DC field.

Figure 3 Strain vs applied-field major hysteresis loop andminor loops for AC fields of +5 kA m71 at 0.7 Hz at DC biaslevels of 5, 15, 25, 35, and 45 kA m71. Device mechanical loadis a prestress of 6.9 MPa.

Figure 4 Acceleration per input current frequency response functions at DC bias levels of (from left to right): 20, 30, 36, 42, 50,and 60 kA m71.

Figure 2 Components typical of a simple magnetostrictivetransducer.


Figure 5 illustrates the sensitivity of major strain-applied field hysteresis loops to variations in mechan-ical load or prestress. While this attribute allowstailoring of devices preloaded for optimized perfor-mance under a constant load (such as acoustic sourceapplications), this introduces a parameter that mustbe optimized for operation under variable load con-ditions which are typical of vibration control applica-tions. Actuators using tailored magnetostrictivecomposite drivers can minimize device sensitivity tovariations in the external load.

The upper traces in Figures 6±8 are Bode plots ofstrain per applied field, strain per magnetizationand magnetization per applied field, respectively.The Bode plots were obtained using a swept sinu-soidal signal at a relatively low signal excitation ofthe device (e.g., driving the device to produce max-imum strains at resonance that are less than one-third of the device full strain potential, therebyminimizing the presence of undesired harmonics).The lower traces are minor hysteresis loops ofthese quantities recorded at (from left to right)

Figure 5 Strain vs applied-field major hysteresis loop corresponding to device mechanical loads of prestress: 3.5, 5.2, 6.9, 8.6and 10.4 MPa. (± ± ±) 1.5 ksi; (± - - ±) 1.25 ksi; (Ð) 1.0 ksi; (- - - - -) 0.75 ksi; (. . . . . .) 0.5 ksi.

Figure 6 (A) Strain per applied magnetic field Bode plot. (B) Strain vs applied magnetic field minor loops (+400 microstrain vs+10 kA m71). From left to right, hysteresis minor loop data collected at: 10, 100, 500, 800, 1000, 1250, 1500, and 2000 Hz.


Figure 7 (A) Strain per magnetization Bode plot. (B) Strain vs magnetization minor loops (+100 kA m71 vs +10 kA m71). From leftto right, hysteresis minor loop data collected at: 10, 100, 500, 800, 1000, 1250, 1500, and 2000 Hz.

Figure 8 (A) Susceptibility (magnetization per applied field) Bode plot. (B) Susceptibility or magnetization vs applied-field minorloops (+400 microstrain vs +100 kA m71). From left to right, hysteresis minor loop data collected at: 10, 100, 500, 800, 1000, 1250,1500, and 2000 Hz.


frequencies of 10, 100, 500, 800, 1000, 1250,1500, and 2000 Hz.

Information on performance is used to implementcalibration, input and/or output linearization, andcontrol schemes to facilitate the optimized use ofmagnetostrictive devices. For example, informationfrom Figures 3 and 5 might be coupled to optimize theDC magnetic bias in real time to produce specifiedstrains under varying mechanical loads. Figure 4demonstrates the ability to tune a system's resonantfrequency in real time, which is the basis for a patentpending, tunable magnetostrictive vibration absorberdesign. This can be coupled with information onfrequencies that minimize losses by using hysteresisloop data from Figures 6±8 to tailor the frequencyoperation for the most efficient electromechanicalperformance. Additionally, this suggests the abilityto target operating conditions for minimization ofinternal heating under continuous operation, which isof particular concern for ultrasonic operation.

Another significant loss factor that is associatedwith the transduction of electric to magnetic energyunder dynamic operation is eddy currents. Eddy cur-rent power losses increase with approximately thesquare of frequency and thus have a significant impacton the operational bandwidth of devices. Laminationsin the magnetostrictive core help to mitigate theeffects of eddy currents, however, materials such asTerfenol-D are brittle and costly to laminate. Materi-als such as insulated magnetic particles or the siliconsteels in common use in motors and power systems aresuitable for the magnetic circuit components thatmake up the transducer housing, as they simulta-neously support flux conduction and offer high resis-tivities. Magnetostrictive composites that usenonelectrically conducting matrix material yieldreductions in eddy current losses. They have beenproposed for extending device output bandwidth byan order of magnitude, from roughly 10 kHz to closeto 100 kHz. Such composites offer great promise ashigh-frequency magnetostrictive drivers.

Actuation Configurations

Nickel was used in many of the early magnetostrictivesonar devices and is still being used in commerciallyavailable ultrasonic cleaners available from, forexample, Blue Wave Ultrasonics. Other examples ofmagnetostrictive material use in commercial applica-tions include Terfenol-D-based devices such as theunderwater communication systems produced byTrigger Scuba, Inc., the acoustic pressure wave source(P-wave) produced by ETREMA Products, Inc. forenhancing oil well production rates, and precisionmicropositioners produced by Energenic, Inc.

Dynamic strains are of primary importance in low-frequency high-power transducers, namely those forsonar and underwater communications. At and nearmechanical resonance, strains greater than the staticsaturation strain can be obtained. Three of the morecommon devices for sonar applications are the flex-tensional, the piston, and the `ring' types of actuators.Flextensional transducers radiate acoustic energythrough flexing of a shell, usually oval-shaped,caused by the longitudinal extension and contractionof a cylindrical drive motor mounted in compressioninside the shell. The Tonpilz (Tonpilz is German for`sound mushroom') transducer is a common piston-type design. The transducer has a magnetostrictiverod surrounded by a drive coil that provides AC andDC magnetic field excitation, a mechanism for pre-stressing the rod, a front radiating surface (piston),and a countermass. An advantage of piston-typedesigns over conventional flextensional transducersis that they lack parts likely to suffer fatigue-inducedfailure due to bending. A typical ring transducermight consist of four magnetostrictive rods that arearranged to form a square with four curved pistonsthat are the radiating surfaces enclosing the squareand attached to the corners of the square. Monopoleoperation is achieved with the rods acting in unison.Dipole operation is accomplished by switching themagnetic bias on two of the rods and maintaining aconstant direction on the AC magnetic field on allfour rods. The interest in ring transducers duringthose early days was based on their ruggedness andlower cost compared to other available transducertechnologies.

There is a growing interest in use of magnetostric-tive devices as a source for motion and/or force, andin particular for use in conjunction with smart struc-tures for active vibration control. These linear motorsystems fall into the two general categories of pistondevices and inchworm devices. An example of theperformance that can be realized using magnetostric-tive actuators for active vibration control is presentedin Figure 9 in which a simple analog proportioncontrol scheme was applied to achieve significantvibration control (433 dB at both mode 1 and 2).There are several variations on the inchworm motorconcepts that move a shaft relative to the motorhousing, and they all generally rely on two separatecapabilities, a clamping force and a pushing force.Motors with linear shaft output rates of 20 mm s71

have been designed that can develop 1000 N of forcewith a 200 mm stroke.

Although generic ultrasonic devices are similar totheir low-frequency counterparts, e.g., they requirethe same components described in Figure 2, thereare several problems associated with operation at


frequencies above 20 kHz. Hysteresis losses within amagnetostrictive driver will introduce high internalheat dissipation. Eddy currents will introduce bothheat and a skin effect that effectively shields the coreof the driver from an applied magnetic field.Impedance mismatches at ultrasonic frequenciesmake transfer of energy from the driver to the sur-rounding medium difficult. The use of laminated and/or composite magnetostrictive materials reduceslosses associated with eddy currents. Device opera-tion at resonance for enhanced energy output is often

obtained through the use of quarter-wavelength-longdrivers in conjunction with half-wavelength amplify-ing horns having application specific profiles.

Sensing Configurations

Magnetostrictive sensors can be classified as passivesensors, active sensors, and combined or hybrid sen-sors, based on how the magnetomechanical proper-ties of the system components are used to measure theparameters of interest.

Figure 9 Magnetostrictive actuator performance for active vibration control of the transient responses of a 40 � 2 � 1 0.125-thickaluminum `C' channel beam to impact excitation.


Passive sensors rely on magnetomechanical cou-pling to link a measurable change in the magnetos-trictive material to the external property or conditionof interest. For example, according to the Villarieffect, the change in the magnetization of the magne-tostrictive sample is correlated to an externallyimposed change in stress. A coil flux-linked to themagnetostrictive sample can be used to measurechanges in magnetic flux as per eqn [5]. Quantitiessuch as external load, force, pressure, vibration, andflow rates can then be measured.

Active sensors use an internal excitation of themagnetostrictive element to facilitate some measure-ment of the magnetostrictive element that changeswith the external property of interest. For example,an excitation coil could be used to excite the magne-tostrictive sample with a known H as per eqn [4] andthe detection coil used to measure B as per eqn [5].The permeability from eqn [3] can then be monitoredfor changes due to an external condition. Designs thatemploy two coils, one to excite the magnetostrictiveelement and one for measurement, are known astransformer-type sensors. The most common activesensor design mentioned in the literature is the non-contact torque sensor. Many configurations employvariations on a general theme of a magnetostrictivewire, thin film, or ribbon flux-linked to a target shaftsubject to a torque. The change in the magneticinduction or permeability can then be related to thetorque on the specimen.

Finally, combined or hybrid sensors use a magne-tostrictive element actively to excite or changeanother material to allow measurement of the prop-erty of interest. For example, a fiberoptic magneticfield sensor uses the change in length of a magnetos-trictive element in the presence of a magnetic field(Joule effect) to change the optical path length of afiberoptic sensor. Stress can be measured using photo-elastic material, and highly accurate displacementmeasurements can be made with the help of a mag-netostrictive guide.

Magnetostrictive transducers are particularlyattractive for noncontact torque and position sensorapplications. As a magnetostrictive shaft is torqued,stress develops at +458 from the shaft axis, and theVillari effect produces changes in the shaft magneti-zation and permeability. The change in magnetiza-tion can be measured directly with a Hall probe.Alternatively, the change in permeability can be mea-sured using a two-solenoid active technique in whichan excitation solenoid is used to apply a constantmagnetic field to a magnetic circuit flux-linked to theshaft and a detection solenoid monitors the torque-induced changes in the magnetic circuit permeability.These torque-induced changes can also be monitored

by tracking electrical impedance function changes influx-linked excitation and/or detection solenoids, assuggested by the data in Figure 10.

Several approaches exist for extending this technol-ogy to nonferromagnetic materials, by applying mag-netostrictive materials directly to the surface of therotating shafts. One technique relies on using amor-phous magnetostrictive wire or ribbons that are heli-cally wrapped around and bonded to a shaft. Inanother, at a high temperature the wire is explodedinto fine particles which adhere strongly to the shaft.

Similarly, noncontact position sensors and forcedistribution sensors are available that respond tothe influence of reflected signals and gaps on magne-tostrictive components of a magnetic circuit (magne-tostrictive waveguides and magnetostrictive delaylines, respectively). Metglas amorphous ribbons arean attractive strain sensor material because theyexhibit large permeability changes in response tostrain.

Velocity and force can be directly measured in amagnetostrictive transducer, as depicted in Figure 2,where the voltage induced in the solenoid is propor-tional to the velocity of the force applied to the freeend of the magnetostrictive element. Similar transdu-cers have been used as colocated sensor±actuatordevices; e.g., devices that simultaneously actuateand sense. The telephone, scanning sonar, and activevibration control applications have made use of thisdual operation mode. Although this can be accom-plished by separating the voltage induced in theexcitation solenoid via the Faraday±Lenz law fromthe applied (excitation) voltage, the sensed voltages isgenerally much smaller than that needed for actua-tion. Hence, it is common for a dual-mode transducerto use separate excitation and detection solenoids.

Figure 10 Sensitivity of electrical impedance functions tochanges in the stress in the transducer magnetostrictive ele-ment. (Key as in Figure 5.)


Nomenclature

A areaB magnetic inductionF forceH magnetic fieldT currentL lengthN number of termsT transductionv velocityV applied voltageZ impedance" strainl Joule effect magnetostrictions stress

See also: Actuators and smart structures; Electro-rheological and Magnetorheological Fluids; Sensorsand actuators; Transducers for absolute motion;Transducers for relative motion.

Further Reading

Butler JL (1988) Application Manual for the DesignEtrema Terfenol-D. Magnetostrictive Transducers.Ames, IA: Edge Technologies.

Calkins FT (1997) PhD dissertation, Iowa State University.

Cedell T (1995) Magnetostrictive Materials and SelectedApplications. PhD dissertation, Lund University, Swe-den.

Clark A (1980) Magnetostrictive Rare Earth-Fe2 Com-pounds, Ferromagnetic Materials, vol.1. Amsterdam:North-Holland.

Cullity BD (1972) Introduction to Magnetic Materials.Reading, MA: Addison Wesley.

Dapino MJ, Calkins FT and Flatau AB (1999). In WebsterJG (ed.) Wiley Encyclopedia of Electrical and ElectronicsEngineering, vol. 12, pp. 278±305. Chichester, UK:Wiley.

du Tremolet de Lacheisserie E (1993) MagnetostrictionTheory and Applications of Magnetoelasticity. BocaRaton: CRC Press.

Fleming W (1990) Magnetostrictive torque sensors ±comparison of branch, cross, and solenoidal designs.SAE Technical Paper 900264, pp. 51±78.

Hunt FV (1982) Electroacoustics: The Analysis ofTransduction, and its Historical Background. AmericanInstitute of Physics for the Acoustical Society ofAmerica.

Jiles D (1991) Introduction to Magnetism and MagneticMaterials. London: Chapman and Hall.

NDRC (1946) Summary of Technical Report of Division 6.The design and construction of magnetostriction trans-ducers, Vol. 13.

Restorff JB (1994) Encyclopedia of Applied Physics, vol. 9,pp. 229±244.

MATERIALS, DAMPING

See DAMPING MATERIALS, SEISMIC INSTRUMENTS, ENVIRONMENTAL FAC-TORS, LASER BASED MEASUREMENTS

MEASUREMENT

See LASER BASED MEASUREMENTS; SEISMIC INSTRUMENTS, ENVIRONMEN-TAL FACTORS; STANDARDS FOR VIBRATIONS OF MACHINES AND MEASURE-MENT PROCEDURES; TRANSDUCERS FOR ABSOLUTE MOTION;TRANSDUCERS FOR RELATIVE MOTION

MEMBRANES

A W Leissa, Ohio State University, Columbus, OH, USA


doi:10.1006/rwvb.2001.0132

A membrane is a structural element which is verythin in one direction compared with the other two,and is flat. In contrast with a plate, it has no bending

or twisting stiffness. If a membrane is not flat, it iscalled a `membrane shell' (see Shells). The membraneis stretched in its plane by tensile and (perhaps) shearstresses. It is these stresses which provide the restoringforces during transverse free vibrations. It is alsoassumed that these stresses are sufficiently large sothat, if the transverse vibrational displacement (w) iskept small, the stresses will remain constant during

762 MEMBRANES

vibration. Figure 1 depicts a membrane in its staticequilibrium position, stretched over a boundary ofarbitrary curvilinear shape. The stress (force/area)applied externally is shown as a normal stress sn

which may vary along the boundary. In addition, avariable shear stress, which is not shown, may alsoact in the plane (xy) of the membrane, and tangent toits boundary. These boundary stresses cause internalnormal stresses (sx and sy) and shear stress (txy), asshown on an internal element of the membrane inFigure 1, which may vary with the coordinates (x andy), but not with time (t).

The equation of transverse motion of the mem-brane is:

Tx@2w

@x2� 2Txy

@2w

@x @y� Ty

@2w

@y2� q � rh

@2w

@t21� �

where Tx; Txy; Ty are the stress resultants (force/length) obtained by multiplying sx; txy and sy, re-spectively, by the thickness (h). Transverse forcingpressure (force/area) is indicated by q, which could bepresent in a forced vibration situation, and r is themass density/volume. In eqn [1], Tx; Txy; Ty; r and hmay each be functions of x and y, but not t.

Considering free, undamped vibrations �q � 0�,the most widely used form of eqn [1] is the specialcase wherein the in-plane shear stress is zero, and theremaining tensile stress is constant and the same in alldirections �Tx � Ty � T�. In this case eqn [1]becomes:

Tr2w � rh@2w

@t22� �

where r2 � @2=@x2 � @2=@y2 is the Laplacian differ-ential operator.

Rectangular Membranes

Consider first a rectangular membrane with dimen-sions a�b, stretched by uniform tension �T�. Let theorigin of the xy coordinate system be in one corner, sothat the four edges �x � 0; a; y � 0; b� are fixed�w � 0�. An exact solution for the transverse, freevibration displacement is:

w x; y; t� � � sin mpx=a� � sin npy=b� � sin ot � f� �3� �

where m and n are integers, ranging from one toinfinity �m; n � 1; 2 . . .1�, and f is an arbitraryphase angle. It is seen that eqn [3] satisfies the bound-ary condition �w � 0� along all four edges.Substituting it into eqn [2] yields the natural frequen-cies �o�, which are conveniently expressed in terms ofthe nondimensional parameter:

oa��rh=T

p� p m2 � a=b� �2n2

h i1=24� �

Looking at eqn [4] it is clear that the fundamental(i.e., lowest) frequency of a rectangular membrane,regardless of the aspect ratio �a=b� will be for a modeshape having one half-sine wave in each direction�m � n � 1�. Figure 2 shows the first nine nodalpatterns of free vibration for a rectangular membranehaving a=b � 1:5. The dashed lines are node lines�w � 0�. The corresponding nondimensional fre-quencies oa

��rh=T

pare 5.664, 7.854, 9.935,

10.538, 11.327, 13.329, 13.421, 14.482, and15.471, according to eqn [4]. It is seen that somefrequencies may be quite close together.

One may also have two mode shapes having thesame frequency. These are called `degenerate modes'.For example, consider the square membrane �a � b�and its degenerate frequencies o12 � o21. Becausethey both vibrate with the same frequency, super-position of the two mode shapes may be taken as:

W x; y� � � C12 sin px=a� � sin 2py=a� �� C21 sin 2px=a� � sin py=a� � 5� �

where C12 and C21 are arbitrary constants, deter-mined by how the membrane is set into motion (theinitial conditions). Figure 3 shows how the node linevaries with the relative magnitude �C2=C1� of thesuperimposed modes. Figure 3 also shows some

Figure 1 Membrane of arbitrary shape subjected to nonuni-form tensile stress.

MEMBRANES 763

Figure 2 First nine nodal patterns of a rectangular membrane with a=b�1:5.

Figure 3 Some superimposed degenerate nodal patterns for square membranes.

764 MEMBRANES

superimposed degenerate modes for o14 � o41, andfor o15 � o51.

Circular Membranes

For a circular membrane it is convenient to use polarcoordinates, as shown in Figure 4. Thenr2 � @2=@r2 � �1=r�@=@r � �1=r2�@2=@y2. By sepa-ration of variables one finds the transverse displace-ment:

w r; y; t� � � AnJn kr� � � BnYn kr� �� cos ny sin ot � f� �6� �

which satisfies eqn [2] exactly, where Jn and Yn areBessel functions of the first and second kinds, respec-tively, k2�o2rh=T; An and Bn are arbitrary con-stants, and n is an integer, beginning with zero.Because Yn�0� � ÿ1, it is necessary to set Bn � 0to have finite displacement at the center �r � 0�.Setting w � 0 along the boundary �r � a� results inthe frequency equation:

Jn ka� � � 0 7� �

The first five roots ka � oa��rh=T

pof eqn [7] are

given in Table 1 for each n. Nodal patterns for thefirst nine frequencies of a circular membrane areshown in Figure 5, along with the correspondingoa

��rh=T�pIt is seen that modes 1, 4, and 9 are

axisymmetric �n � 0�, having 0, 1, and 2 interiornodal circles, respectively, whereas modes 2, 3, 5,and 7 have 1, 2, 3, and 4 nodal diameters, respec-tively, with no interior nodal circles. Exact shapes ofthe vibration modes along diameters may be seenfrom plots of the Bessel functions Jn�kr�.

An annular membrane is bounded by two con-centric circles, an inner one at r � b, and an outerone at r � a, with both circular boundaries beingfixed and exerting uniform radial tension T (force/length). The solution (eqn [6)] of eqn [3] is again

appropriate. But both An and Bn are now retained,because r � 0 does not apply to the membrane. Set-ting w � 0 at r � a and r � b gives, for a nontrivialsolution, the following frequency equation:

Jn l� �Ynb

al

� �ÿ Jn

b

al

� �Yn l� � � 0 8� �

where l � ka � oa��rh=T

p. For any desired value of

b=a, all frequencies of the annular membrane areobtainable from eqn [8]. The first three frequencies(corresponding to 0, 1, and 2 interior nodal circles)are listed in Table 2 for the n � 0 (axisymmetric) andn � 1 (one nodal diameter) modes of annular mem-branes having various b=a ratios, especially forb=a<0:5. It is interesting to note that, as b=a approa-ches zero, which corresponds to a central point sup-port, the frequency for n � 0 becomes the same as forthe unsupported membrane (Table 1). This is becausemembranes are incapable of transmitting transverselyapplied concentrated forces (reactive in this situation)

Figure 4 Circular membrane and polar coordinates.

Table 1 Frequencies oa��rh=T

pfor circular membranes.

n (nodal diameters)

Numberof roota

0 1 2 3 4 5

1 2.405 3.832 5.136 6.380 7.588 8.7712 5.520 7.016 8.417 9.761 11.065 12.3393 8.654 10.173 11.620 13.015 14.373 15.7004 11.792 13.324 14.796 16.223 17.616 18.9805 14.931 16.471 17.960 19.409 20.827 22.218

a Number of nodal circles plus one.

MEMBRANES 765

Figure 5 First nine nodal patterns of a circular membrane.


pfor annular membranes.

Number of roots

n b=a 1 2 3

0.80 15.698 31.411 47.1210.60 7.828 15.695 23.5530.40 5.183 10.443 15.688

0 0.20 3.816 7.786 11.7320.10 3.314 6.858 10.3770.02 2.884 6.136 9.3760.00 2.405 5.520 8.654

0.80 15.738 31.431 47.1340.60 7.930 15.747 23.5880.40 5.391 10.558 15.766

1 0.20 4.236 8.055 11.9270.10 3.941 7.331 10.7480.02 3.836 7.031 10.2050.00 3.832 7.016 10.173

766 MEMBRANES

without violating the linearizing assumptions made inderiving the theory.

Other Shapes

A sectorial membrane has the shape of a sector of acircle (Figure 6), with a sector angle, a. The transversedisplacement:

w r; y; t� � � AuJu kr� � � BuYu kr� �� sin uy sin ot � f� �9� �

satisfies eqn [2]. Moreover, if noninteger values of uare chosen such that u � np=a�n � 1; 2 . . .1�, thenw � 0 along the radial edges y � 0 and a, as desired.Setting Bu�0 to avoid infinite displacement at r � 0,and w � 0 at r � a yields:

Ju ka� � � 0 10� �

Symmetric modes result from n � 1; 3; . . . ; antisym-metric modes from n � 2; 4; . . . For every u there isan infinite number of nondimensional frequencieska � oa

��rh=T

p. Frequencies arising from half-inte-

ger orders of u are given in Table 3. Results for inter-mediate values of a � 1808; 908; 608; 458, and 368,corresponding to integer u, may be taken from Table 1.The case of u � 1=2, corresponding to a � 3608 inTable 3, yields boundaries at y � 08 and y � 3608,which are the same radial line, and the case of thecomplete circular membrane supported along an ad-ditional single radial line is thereby represented.

Annular sectorial membranes are bounded by thecircle arcs (r � b; r � a) and the two radial lines(y � 0; y � a). Frequencies for arbitrary a and b=amay be found from eqn [8], replacing n by u, whereu � np=a�n � 1; 2 . . .1).

Because node lines have the same zero transversedisplacement as a constrained boundary, consider-able miscellaneous results for other membrane shapes

can be found by studying nodal patterns. For exampleFigure 3 shows that the exact frequency for an iso-sceles right triangle with sides a � a is that of the (1,2) and (2, 1) modes of a square ÿoa

��rh=T

p � ��5pp

.Considering the node lines as boundaries, Figure 3also shows a variety of shapes, each having straightand curvilinear lines as boundaries.

Additional information for straight-sided mem-branes may be obtained from published results forplates having the same configurations and all edgessimply supported, using the correspondence:

o2mrh

T

� �2

� o2prh

D11� �

where the subscripts m and p correspond to mem-brane and plate natural frequencies, respectively (seePlates). Using this one can, for example, obtain theaccurate frequencies for right triangular membranesof various aspect ratios (b=a) presented in Table 4.Corresponding nodal patterns for the second, third,

Figure 6 Sectorial membrane.


pfor sectorial membranes.

Sector angle, a (and value of n)

Number of 3608 1208 728 51.438 408 32.738

rootsa (1/2) (3/2) (5/2) (7/2) (9/2) (11/2)

1 3.142 4.493 5.763 6.988 8.183 9.3562 6.283 7.725 9.095 10.418 11.705 12.9673 9.425 10.904 12.323 13.698 15.040 16.355

a Number of nodal circles plus one.

MEMBRANES 767

2 3 4

1

2

3

1.5

ba

Mode

a

b

Figure 7 Nodal patterns for the second, third, and fourth modes of right triangular membranes.


pfor right triangular membranes

b=a

Modenumber

1 1.25 1.5 2 2.5 3

1 7.025 6.323 5.855 5.269 4.907 4.6792 9.935 8.876 8.099 7.063 6.408 5.9673 11.33 10.25 9.584 8.641 7.781 7.1474 12.95 11.48 10.32 9.018 8.482 8.142

Adapted from more extensive data given in: Gorman DJ (1983) A highly accurate analytical solution forfree vibration analysis of simply supported right triangular plates. Journal of Sound and Vibration 89:107±118.

768 MEMBRANES

and fourth modes are exhibited in Figure 7. (The firstmodes have no interior node lines.)

Some Complicating Effects

Results for frequencies and mode shapes of freevibrations given above have all been for membraneswhich are initially stretched by a uniform tensilestress resultant �T� in all directions. If a rectangularmembrane is subjected to different, but uniform,tensile stresses in its lengthwise and breadthwisedirection �Tx � sx h � constant, Ty � sy h � con-stant), the sine wave mode shapes of eqn [3] remainvalid. Substituting eqn [3] into eqn [1], with q � 0and Txy � 0, yields the exact frequencies:

oa��rh=Tx

p� p m2 � Ty=Tx

ÿ �a=b� �2n2

h i1=212� �

If shear stress is also present then no exact solution ispossible. Let sx; sy, and txy all be constants.Reasonably accurate frequencies for a square mem-brane �a=b � 1�, with sy � sx and varying amountsof additional shear stress �txy=sx�, were obtained bythe Ritz method, and are presented in Table 5. Forthis combination of stresses, the stresses along 458diagonal planes are principal stresses:

s1 � sx � txy; s2 � sx ÿ txy; t12 � 0 13� �

as shown in Figure 8. As txy=sx approaches unity, s2

becomes zero, and s1 becomes 2sx. This is a specialcase of diagonal tension where the membrane isstretched in the direction of one of its diagonals, butis slack in the other diagonal direction. This is alimiting case, for if txy=sx>1; s2 is compressive, and

the membrane will buckle (wrinkle) in its initial state.Contour plots of mode shapes corresponding to someof the frequencies are shown in Figure 9. Fortxy=sx � 1, the node lines are all straight, and parallelto the direction of the applied principal stress �s1�.

Nomenclature

h thicknessT uniform tensionw transverse vibrational displacementa sector angle� arbitrary phase angle

See also: Plates; Shells.

Figure 8 Biaxial diagonal tensile stresses.

Table 5 Nondimensional frequencies oa��rh=sx

pfor a square membrane with biaxial

tension �sy � sx� and additional shear stress �txy�

Mode number

txy=sx 1 2 3 4 5 6 7

0.00 4.443 7.025 7.025 8.886 9.935 9.935 11.3270.10 4.438 6.916 7.118 8.837 9.922 9.962 11.0920.20 4.425 6.790 7.197 8.701 9.886 10.030 10.8010.50 4.322 6.285 7.337 7.934 9.603 9.606 10.6060.75 4.126 5.620 6.864 7.306 8.099 9.062 9.2680.85 3.986 5.211 6.221 7.212 7.221 8.149 8.6780.98 3.591 4.174 4.647 5.110 5.584 6.114 6.8021 3.376 3.677 4.015 4.411 4.889 5.488 6.280

Adapted from more extensive data given in: Leissa AW, Ghamat-Rezaei A (1990). Vibration ofrectangular membranes subjected to shear and nonuniform tensile stresses. Journal of the AcousticalSociety of America 88: 231±238.

MEMBRANES 769

Further Reading

Courant R, Hilbert D (1953) Methods of MathematicalPhysics, vol.1. New York: Wiley Interscience.

Lord Rayleigh (1945) The Theory of Sound, vol. 1. NewYork: Dover Publications (reproduction of 1877 originalpublished by the MacMillan Co.).

Meirovitch L (1967) Analytical Methods in Vibrations.New York: Macmillan.

Timoshenko SP, Young DH, Weaver W (1974) VibrationProblems in Engineering, 4th edn. New York: Wiley.

Figure 9 Mode shape nodal patterns corresponding to some of the frequencies in Table 5.

770 MEMBRANES

MEMS, APPLICATIONS

I Stiharu, Concordia University, Montreal, Quebec,Canada


doi:10.1006/rwvb.2001.0075

Introduction

Microelectromechanical systems (MEMS) applica-tions have been conceived for both substitution andfor new products. The decreased cost associated withvery large production is directing the utilization of themicrosystems into applications that require massproduction. Pressure, temperature, proximity, accel-eration or chemical integrated microsensors for auto-motive application represent a large family ofsubstitution MEMS. Intelligent transportation sys-tems (ITS) technology is another motor of develop-ment for new microsystems. The health care devicesare also regarded as great potential applications foreither substitution or new products.

The primary feature of MEMS applications is thatthey adequately detect and influence phenomenawhere microscale interactions are adjoining to macro-scale effects. The small-size, low-energy consumptionendorsed by high reliability, low cost, and capabilityto process and deliver undistorted measurement sig-nals in a form that the user can easily exploit con-stitute the major strengths of MEMS. Microsensorsusually do not interfere with macrophenomena, whilethey could track microscale events accurately. MEMSare thus capable of perceiving the nature to a muchlower scale size than the normal-sized sensors do.They can be implemented in time-varying spatialmicroscale configurations to survey the borderlinetransition from micro to macro phenomena. Thus,applications of MEMS sensors in resonant structures,buckling, turbulent flow, acoustics, thermo-stressphenomena, and phase change, are befitting. Thelow cost of the microsensors permits allocations ofmuch larger numbers of measurement points in sys-tems that require synchronization of distributedresponse, in dynamic allocation of scarce resourcessystems, or in redundant control and decision man-agement systems.

Although the great majority of the present com-mercially available devices are based on bulk micro-machining, surface micromachined microsystemsrepresent the desired turnout for future MEMS. The

capability to integrate on the same chip when pro-duced during the same process, the integrated circuit(IC) circuitry, and the sensing/actuating device enableapplications that up to now have been unimaginable.The large number of present MEMS-based systemsrepresent just a small fraction of the possible con-ceivable and unique applications that will be avail-able in the very near future. A number of availableapplications involving MEMS are shown in Table 1.The applications listed are at various levels of devel-opment, ranging from laboratory testing to the mar-ket. Most microsystems are still related to themacroscopic perception of the world, whereas thoseconceived to handle the matter at miniature scalehave been implemented more recently. DNA-map-ping kits and bioanalysis systems on a single chip arean application that will probably soon be commer-cially available. However, the endowment of MEMShas become quite common, such that the acknowl-edgment of the microtechnology in newly developedsystems is no longer publicized by the manufacturer.

The related applications of MEMS to the dynamicsof mechanical systems are mostly limited at this timeto those accomplished by the normal-sized sensors.Accelerometers, angular rate sensors, vibrometers,microphones, and pressure sensors, are the mostcommon applications available at this time. Thereare very few MEMS devices in large-scale commercialuse, other than microsensors and microvalves. How-ever, sustained research is being carried out to accom-plish fully integrated systems in one single chip. Thus,single-chip autonomous microrobot or computerhard-drive units on a single chip, implantable drugdelivery systems equipped with sensors, valve, reser-voir, and controller, or ion propulsion miniaturesystems for space application are only a few of themicrosystems which have been thoroughly investi-gated. If the progress in accomplishing fully inte-grated microsystems is considered to be moderate,microelements such as new microsensors and micro-actuator concepts and prototypes are constantlybeing reported.

Mechanical Microsensors

Most MEMS mechanical sensors are alternatives forexisting normal-sized sensors. Regardless of the phy-sical quantity detected, MEMS mechanical sensors

MEMS, APPLICATIONS 771

exhibit a larger measurement range, higher sensitiv-ity, better linearity, and lower hysteresis. Besides, allcommercially available MEMS mechanical sensorseasily comply with the rule of 20% for a new product:newly developed MEMS-based products are at least20% less expensive to build and/or perform 20%better. Many MEMS devices are developed at a costwhich is 10 times less expensive and perform 10 times

better than normal-sized sensors. The most signifi-cant impact of MEMS, however, arises from theirapplications, which in most cases cannot be realizedwith normal-sized systems. A multiactive linkingcatheter, arrayed mirror high-resolution display, ora swimming microrobot represent only a few of theresearch topics which are most likely to grow intocommercial products which will have a large impacton our life.

The automotive industry has had most benefit fromMEMS so far. The mass production of cars requiresmass production of sensors, which can be made moreeconomical and rugged through micromachining.Research reports predict a growth of the MEMSdevice industry to about $35 billion annually by theyear 2002: more than half of this is attributed to theautomotive industry. Figure 1 illustrates the distribu-tion of MEMS applications on the end-user industry.

Pressure and acceleration sensors have alreadybeen successfully implemented as typical MEMScommercial products. The excellent features ofmicromachined accelerometers enabled a tremendousimpact in the improvement of car dynamics. The lowmass and reduced power consumption associatedwith a reduced cost in microaccelerometer havefacilitated the implementation of the MEMS struc-ture in various vehicle applications, ranging from airbag deployment system to the active suspension.

Microaccelerometers

Microaccelerometers are miniature inertial masseselastically suspended on inertial frames that arerigidly connected to the body on which the accelera-tion should be measured. The inertial mass moves inproportion to the acceleration amplitude of the mov-ing body. The amplitude of motion is detected andconverted into an electric signal. The lower the massof the accelerometer, the higher the natural frequencyof the microaccelerometer is. Thus, the main require-ment for an accelerometer could be accomplishedthrough miniaturization. The natural frequency ofmicroaccelerometers is often beyond 10 kHz, whiletheir mass can be as low as 50 mg.

Such a microsystem is illustrated in Figure 2, as anADXL50 micromachined accelerometer. This is acomplete one-axis acceleration measurement systemon a single monolithic IC. Detection of acceleration isbased on inertial displacement under external accel-eration of an elastically suspended mass. The fixedarmatures detect both amplitude and direction ofthe displacement through capacitive measurementwith respect to the fixed electrodes. Along with thesensing structure, the chip comprises an oscillator,reference signal circuit, preamplifier, demodulator,

Table 1 Potential applications of MEMS

Area of application Application

Mechanicalmicrosensors

MicroaccelerometersMicrogyroscopesRate, speed, and position linear and

angular sensorsVibration measurement systemsPressure, temperature, flow, gas

composition sensorsFluid control Microvalves, smart valves

MicropumpsGas/liquid chromatographyLogic fluidic elementsMultisensing arrays

Biomedical Multiple microchemical test kitsMultisensing and surgery cathetersDrug deliveryCell handlingCell fusionBiomolecular handlingBlood test integrated cellMultiple virus integrated test kits

Human sensingstimulation

Neural activity detection andmeasurement

Tactile stimulationHearing and visual aidSmell sensorsDistributed arrays of multiple sensing

Microoptics Fiberoptic alignmentScanningModulationInterferometerOptical headOptomechanical integrated circuitsArrays of mirrors and varying-focal

mirrorsMicroprobing and

microtestingAtomic force microscopy probesScanning tunneling microscope probesNear-field microscopy probesTunneling probe arraysMicrobalances

Computer/peripheralcomponents

Magnetic, printer, compact disk headLaser scanMicromechanical memory

Integrated circuittechnology

Micromanipulators, microprobesMicropositioners

Robotics MicrorobotsMicroteleoperatorsMobile sensors

772 MEMS, APPLICATIONS

bias compensation circuit, and self-test circuit.Traditionally, such microaccelerometers have beenused in air bag deployment systems. However, otherapplications in vehicle and seat active suspensionsmake use of such microaccelerometers. The low costand high reliability of MEMS represent an importantdriving factor in the progress of implementation ofthe intelligent vehicle system (IVS). This conceptcomprises implementation of sensors and marks,such that roads and vehicles would be able to interact

and improving the traffic on congested roads. Proxi-mity sensors, impact avoidance sensors, position andguidance sensors, and remote communication devicesare essential elements in the implementation of effi-cient adaptive traffic schemes on highways. In thevehicle industry, many other solid-state microsys-tems, such as thermal, optical, magnetic, or chemicalsensors, have been implemented and are intensivelyused. Engine management, wipers, mirrors and lightsand air quality control in the car are among the best

Figure 1 Applications of MEMS and end-user industries.

Figure 2 Schematics of the microaccelerometer ADXL50. (A) With no excitation, the proof mass rests in a balanced position. (B)Under the acceleration, the proof mass moves along with the fingers. The relative position of the fingers with respect to the fixedelectrodes is related to the magnitude of the acceleration. (C) Schematic view of the entire microsystem.


known applications of optical and chemical sensorsin vehicles. However, such sensors are semiconduc-tor-based devices and do not exhibit any detectablemechanical dynamic outcome. The distinctionbetween solid-state sensors and MEMS is less percei-vable from the point of view of function, since mostMEMS utilize one or more measurement principlesused by solid-state microsensors, such as piezoresis-tive, magnetoresistive, piezoelectric, capacitive, andion-sensitive field effect.

One of the most critical problems faced by MEMSis packaging. The diversity of measurands makespackage standardization unfeasible for all possibletypes of microsensors and microsystems. Figure 3illustrates a standard packaging configuration thatis presently used in IC. Some microsystems requiresealed chips, whereas others call for access of thesurrounding environment to the sensing area. Incertain cases, simple sealed or perforated lids mayaccomplish specific requirements for specific sensors.In more complex situations, packaging issues aredealt with individually, from case to case. Thus,microactuators may require access to the plant anddirect physical contact with the environment. More-over, rotating parts need protection and lubrication.MEMS devices must also meet the milieu conditionsof the application. To illustrate, Table 2 presents theenvironmental conditions that a sensor used in theautomotive industry must face, as prescribed by SAEJ1221 and SAE J5756.

A larger potential of MEMS applications withmechanical microsensors resides in new integratedsystems for innovative and emerging applications.As an example, MEMS employed in global position-ing systems (GPS) create navigation systems that canprovide accurate positioning on a geographic areawhich are sufficiently accurate to make them appro-priate for the ground vehicle traffic managementapplications. The low cost of MEMS sensors and

GPS enables the current implementation of suchdevices on standard consumer vehicles.

Since the 1980s the research on MEMS hasmatured tremendously. The microsystems are perma-nently heading towards smaller features and integra-tion of mechanical, electrical, and control subsystemson a single chip.

Acoustic Microsensors

The acoustic sensor class exhibits considerable poten-tial for mechanical and chemical sensing. The planargeometry configuration enables compliant fabrica-tion of sensors using standard IC technology. Theprinciple of detection is based on the high sensitivityof the elastic wave velocity, traveling through aresonant media while crossing an active material,which is subjected to excitation from the physical

Figure 3 Standard packaging of an integrated circuit.

Table 2 Basic requirements for automotive sensors (SAE J1221and SAE J5756)

Exposure to: Range

Temperature 7408C to 858C inside the vehicle7408C to 1258C under the hood7408C to 1508C on the engine7408C to 200±6008C in the exhaust and

combustion areas

Mechanical shock 3000 g during assembly50±500 g in service

Mechanical vibration 15 g, 100 Hz to 2 kHz

Chemical exposure to: FuelOilFreonHumiditySalted waterExhaust gasesEthylene glycol


quantity to be measured. The acoustic wave-basedsensors, such as the surface acoustic waves (SAW)and the flexural plate waves (FPW), are based on thepiezoelectric properties of classes of monocrystalsand ceramics. Figure 4 illustrates the schematic con-figuration and functioning of an SAW sensor. One ofthe SAW interdigital transducers is used to excite thepiezoelectric substrate of the sensor. When an elec-trical charge is applied to a piezoelectric structure, thestructure tends to vibrate near its natural frequency orone of its harmonics. The vibration wave propagatesthrough the substrate and travels throughout thecoating, which is the active material sensitive to thequantity to be measured. The change in the activematerial will modify the speed of the crossing wave.The second SAW transducer detects the phase shift oramplitude modification due to the structural changesin the bulk of the sensing material. The acousticsensors have been known since the early 1960s fromtheir applications with large monocrystalline quartz.The piezoelectric effect can be successfully used in theminiature devices, microsensors, and microactuators.If the structure is a thin plate of mass m and thicknesst, the resonance shear-mode wavelength l (thicknessis chosen such that l � t) will be modified due to themass change of the plate by the linear relationship:

Dm

m� Dl

l

Modifications to the structure of the delay line will bereflected in the signal perceived at the receiver. Thisinformation can always be related to the input exci-

tation, which is directly related to the physical quan-tity to be measured. The interdigital transducerconfiguration is very appropriate for MEMS imple-mentation. IC-like technologies have been used tocomplete monolithic piezoelectric accelerometersalong the full scale ranging from 1 to 100 g. Thepiezoelectric substrate deposition requires some spe-cial processing which is not specific to IC processes.Among the thin film piezoelectric materials that canbe implemented with IC technology are ZnO, AlN,Pb(Zr,Ti)O3 (PZT), and polyvinylidene fluoride(PVDF).

Microactuators

Microactuators are conceived to transform variousforms of energy, mostly electrical, into mechanicalrotation or translation on a miniature scale.

The first electrostatic microactuators were createdmaking use of modified surface micromachining tech-nology in which thick polysilicon rotors actuated byside-driver electrodes were held at the center by ahollow hub. Early designs of micromotors were con-ceived as small replicas of normal-sized electricmotors. At 500 rpm, the life of the motor would notexceed 1 h due to the substantial friction and wear onthe sliding surface. Eccentrically rotating motors(wobble motors) were then created to overcome thesliding friction which would be reduced to a rollfriction. Moreover, elastically suspended movingelements driven by electrostatic or magnetic fieldsor actuated by fluid or thermal power have beenproposed. Such designs fully eliminate the contactfriction.

Some other designs use the friction force betweenthe moving and fixed parts as driving force. Such anexample is represented by ciliary motion systems.Miniature actuators making use of piezoelectric,magnetostrictive, or photostrictive or phase transfor-mation effects or thermal effects could generate largerpower per mass unit efficiency factors. More creativeconcepts have been proposed and tremendousprogress was achieved in increasing the life of themicromotor and in improving the force/torque. Inter-digitated comb-like resonant structures, linear ther-mal actuators, shape-memory elements, or capillaryforce-actuated elements are among the manyproposed solutions to raise the actuation force ofthe micromotors. In addition to modified surfacemachining, high-aspect ratio machining (lithografiegalvanik abeforming or LIGA, and deep X-ray litho-graphy) was used to create microactuators. Figure 5illustrates the concepts of micromachined actuators.Forces in the range of mN can be accomplished bysingle microactuators. Optical applications are most

Figure 4 Schematic of a surface acoustic sensor concept. Theoscillation generated by the source is changed when passingthrough the coating which represent the active material. Thechange is perceived by the surface acoustic wave (SAW) trans-ducer, which detects the change due to the modification in thestate of the active (sensing) materials.


suitable with the force range generated by single unitmicromotors. Table 3 illustrates the work per unit ofvolume and the frequency range of various classes ofmicroactuators.

Certain macroscopic tasks can be performed byarrays of microactuators. Larger displacementscould be achieved by multiple microactuators withsynchronized motion, as illustrated in Figure 5F, inthe case of a ciliary motion conveyer where the

actuators are connected in series. Larger loads couldalso be supported by distributed synchronized micro-actuators connected in parallel. Arrays of microsen-sors and microactuators connected conveniently inseries and parallel can produce higher forces andlarger displacements and perform more complexfunctions than the unit microactuator. Although theenergy loss in direct-driven microactuators isminimal, the undesired friction limits the type of

Figure 5 Microactuators. (A) Side-drive electrostatic micromotor. (B) Wobble electrostatic micromotor. (C) Comb-drive microac-tuator. (D) Thermal actuator (bimorphous cantilever beam). (E) Electrostatic linear microactuator. (F) Ciliary motion conveyer.


transmission elements to the supported beams aslinkages. Joints and gears cannot be used in micro-systems due to the microscale high friction level thatreduces the efficiency of the transmission to valuesclose to zero.

Remarkable applications of microactuators tomacroscopic systems have been demonstrated in thecontrol of turbulent flow on surfaces. Augmentationof a disturbance created by a small motion of amicroactuator on the surface enable formation of avortex, which upholds the capability to control theaerodynamic macrophenomena.

Microvalves and Micropumps

MEMS applications in this group are related to fluidhandling, chemical analysis, microbiology, aerody-namics, and controls. Early investigations on fluidicmicrodevices were directed towards the accomplish-ment of micropumps, microvalves, and shaped flowchannels. Most devices have been accomplishedthrough hybrid bulk and special micromachiningprocedures. Silicon membranes actuated by electro-magnetic, electrostatic, or thermal expansion forceshave been used as valves, as shown in Figure 6. Themetallic conductor deposited on the thin membrane

will move the flexing element when a current ispassed through the coil. The motion can be conveni-ently used to obstruct one of the valve orifices. Similarstructures can be used as a pump if the openings of thetwo orifices are synchronized. The variation ofvolume inside the cavity due to the motion of themembrane will induce flow from the input to theoutput orifices.

Fluidic elements require special packaging suchthat the moving element is enclosed within a specificstructure. Wafer-to-wafer or glass on silicon anodicbonding is usually used to seal the fluidic elements.

Fluidic logic elements have been developed as analternate solution to the electronic logic circuits.MEMS enable downsizing of these circuits such thatfull logic controllers can be built on a few squaremillimeters of wafer. Figure 7 illustrates such a cir-cuit. The flow stream switches the flow channel if thecorresponding control signal is applied through theappropriate control channel.

Chemical analysis integrated systems comprisingpumps, valves, sensors, and controllers are underintensive investigation.

Other Applications

Optical microsystems are among the most commonapplications of MEMS. Here, the mechanical loadcarried by the actuator is reduced to the intrinsicmass of the moving part. Besides, very performantoptical cells can be built in IC technology. Microma-chined high-resolution displays comprising arrays ofmicromirrors, each of 16mm size, which can be tiltedsuch that the reflected beam creating the pixel imagebecomes visible or invisible, integrated on metal oxidecomplementary circuits have been accomplished.Micromechanical optical switches and resonators,optical modulators, tunable filters, or deformableoptical wave guides are also available. Applications

Table 3 Work per unit of volume and frequency range forselected types of microactuators

No. Microactuator type Work/volume(J mÿ3)

Frequency(Hz)

1 Shape memory alloy(Ni-Ti)

6� 106 ± 25� 107 102

2 Solid±liquid phasechange

4:7� 106 1

3 Thermal expansion 4:6� 105 102

4 Electromagnetic 1:6� 103 ± 4� 105 102

5 Electrostatic 7� 102 ± 1:8�105 104

6 Piezoelectric 1:8� 102 ± 1:2� 105 107

Figure 6 Scheme of a cross-section of the microvalve (actuation chip plus encapsulation and permanent magnet with iron pot).


to communications and data storage systems as wellas in very high-accuracy measurement devices of theoptical MEMS are quite common.

Another type of MEMS application features theinvestigative systems to nanoscale level of the matter.Thus, scanning tunneling or lateral force microscopyprobes and arrays of probes accomplished in silicondramatically improved the performances of themicroscopes, at the same time reducing their cost.The mass of microobjects can be accurately mea-sured, making use of simple principles such as thebending of the free end of a cantilever beam under agravity force. Figure 8 illustrates such a measurementsystem made of an array of beams conceived tomeasure a larger range of masses.

Biotechnology and medicine also benefit fromMEMS. Catheters equipped with micropressure sen-sors can detect the pressure variation on a portion ofan artery, which indicates the existence of clots.Research on similar catheters bearing together withpressure sensors and biochemical sensors that detectenzymes composition and concentration has beencarried out. Less invasive probes for tests have beenmanufactured in MEMS technology and the resultsobtained claim more effort in developing such sys-tems. Neural probes used to detect the electric poten-tial generated by nerve activity have been built using ahybrid surface and bulk postprocessing of silicon.Such a microelectrode is shown in Figure 9. Implan-table artificial organs, a microautonomous tool forminimally invasive surgery or drug delivery systems,are among the most desired MEMS due to the enor-mous positive impact they will have on humans' life.

The ultimate goal of the research effort in MEMSis to create useful micromachines as full systems.Successful applications make use of standard IC

processes such that the microsystems could benefitfrom the coupling of the micromechanical compo-nents with the conditioning circuitry and controlleron the same miniature unit. Besides, many originalsystems have been accomplished through bulkmicromachining comprising anisotropic Si etching,wafer bonding, and advanced packaging. Since bulk

Figure 7 Schematics of a fluidic logic element.

Figure 8 Microcantilever beam used to scale a small mass.The intrinsic interlayer stress will bend the beam. A micromassis used to bend back the beam. The new value of the deflectionis directly related to the mass of the microparticle.


micromachining could not enable the fabrication ofthe electronics on the same chip with the microstruc-ture, the circuit is built separately and packaged onthe same unit. The variance in sensitivity of themechanical structure may range within +20%. Com-pensation is achieved through tuning of the electroniccircuitry. Laser trimming as well as look-up compen-sation tables inscribed in EPROM circuits are com-pensation methods which are currently used. The lowcost and high reliability of microsystems make themserious competitions to the present sensors and con-trollers. The capability of MEMS to investigatefurther into the matter of phenomena means thatmicrotechnology has a strong future. Breakthroughsin the following five general areas are predicted to beaccomplished by MEMS and MEMS technology:

1. implementation of hierarchic intelligent systems inmachines which can analyze, and conclude whatthe decision is to be taken

2. downsizing and redundancy of systems and deeperinvestigation of the matter

3. biomimetics: the development of flexible, intelli-gent, and sensitive autonomous micromachines

4. informatics: the acquisition of information on thestate of systems equipped with sensors; exchangeof real-time data among interacting systems, deci-sion taking towards the optimization of multiplesystem functioning

5. environment preservation: measurement, detec-tion, cleaning

MEMS have had, and will have, a profound impacton future technology and society.

See also: Actuators and smart structures; MEMS,dynamic response; MEMS, general properties; Trans-ducers for absolute motion.

Further Reading

Bhushan B and Gupta BK (1997) Handbook of Tribology:Materials, Coatings, and Surface Treatments. New York:McGraw-Hill.

Campbel SA and Lewerenz HJ (eds) (1998) SemiconductorMicromachining. Chichester: Wiley.

Fraden J (1993) AIP Handbook of Modern Sensors;Physics, Design and Applications. New York: AmericanInstitute of Physics.

Gardner JW (1994) Microsensors; Principles and Applica-tions. New York: John Wiley.

Kovacs TA (1998) Micromachined Transducers Source-book. Boston: McGraw-Hill.

Lee HH (1990) Fundamentals of Microelectronic Proces-sing. New York: McGraw-Hill.

Madou M (1997) Fundamentals of Microfabrication. NewYork: CRC Publishing.

Neamen DA (1992) Semiconductor Physics and Devices;Basic Principles. Homewood, IL: Irwin.

Ohba R (ed.) (1992) Intelligent Sensor Technology.Chichester: John Wiley.

Sze SM (ed.) (1994) Semiconductor Sensors. New York:John Wiley.

Tiller WA (1990) The Science of Crystallization: Micro-scopic Interfacial Phenomena. Cambridge, UK: Cam-bridge University Press.

Weste NHE and Eshraghian K (1994) Principles of CMOSVLSI Design. New York: Addison Wesley.

MEMS, DYNAMIC RESPONSE



doi:10.1006/rwvb.2001.0074

Introduction

A microstructure subjected to an input signal wouldrespond with the same oscillation pattern of ampli-

tude that correlates with the characteristics of thestructure. Microstructures are particularly associatedwith high resonant frequencies. This phenomenon ismainly due to the extremely small oscillating massesof the microstructures, but it is also due to the notableinfluence of the electrostatic and electromagneticfields, and the built-in interlaminar stress in thestructural layers. Detecting the natural resonancefrequency of a microstructure in conjunction withthe appropriate operating frequency should be con-sidered during the design process.

Figure 9 Neural probe.

MEMS, DYNAMIC RESPONSE 779

The dynamic response of microelectromechanicalsystems (MEMS) involves multiphysics interactions.The association between various physical quantitiesmust be accurately established to evaluate the perfor-mance of a microsystem. Geometric downscaling by afactor does not necessarily mean downscaling withinthe same range for various physical quantities inter-acting at the microstructure level. For a given one-dimension downscaling factor, the mass and themoment of inertia decrease by the power of 3, theforce decreases by the power of 2 and the accelerationgrows by the power of 1, while the speed remainsunchanged. Such dependence will definitely influencethe dynamic response of a miniature structure.

From a dynamic point of view, most existingmicrosensors (optical, chemical, electromagnetic,temperature, and radiation) are static devices. Theyusually comprise both the transducer and the circui-try, which are built on the chip or packaged on thesame circuit board. Under vibrations, the mechanicalstructure moves together with the circuitry and thechip. The stiffness of the connections is very highwhile the free pendent mass of the sensor is verysmall. Such devices can thus be considered as simplemasses. Thus, for most microsystems mechanicalvibration has no detectable influence on their func-tioning. However, the mechanical vibrations couldinduce quite destructive effects to the package andwire connections on the chip.

The influence of MEMS on the dynamics of theparts supporting the microsystem, on the other hand,is quite negligible. The mass of a standard 363 mmsilicon chip does not exceed 10 mg, while the mass of afully packaged chip does not exceed a few grams.Thus, rigid mounting of MEMS to normal-sizemechanical systems would not interfere or yield adetectable influence on the overall dynamic responseof the mechanical structure under most circumstances.

MEMS comprising flexing or free elements,which are free-standing structures released form thesilicon chip by postprocessing, are clamped, sup-ported, or free but constrained in their motion. Suchstructures are strongly affected by the oscillatorymotion induced to their support or foundation. Thedynamic response can be roughly modeled as a lumpmass±spring±damper system fixed on a vibratingfoundation, as shown in Figure 1. The motion alongthe free direction x of the swing mass of the micro-system is described by the force equilibrium equation:

m�x� c _xÿ _x0� � � k xÿ x0� � � X0 sin ot� � �1�

The deflection x of the free-standing microstruc-ture follows, after a lag, the pulse of the excitation:

x � A sin ot ÿ f� �

where the amplitude A and phase shift f are given by:

A � X0

��k2 � co� �2

kÿmo2� �2� co� �2

s�2�

and:

f � tanÿ1 mco3

k kÿmo2� � � co� �2 !

�3�

The main feature related to the dynamics of MEMS isthe high value of the resonant frequency, which ismainly due to the extremely low mass of the micro-structure. This attribute makes them valuable mea-surement tools for high-frequency bandwidthvibrations in mechanical systems.

In the case of microaccelerometers, microvibrom-eters, atomic force probes, and microphones, themotion of the free-standing element is beneficialand regarded as the measurable input signal, so thestructure must always behave in the predicted way.The motion of the free-standing element is furthertransformed into electric effectual signal. Thedynamic behavior of the microstructure must bepredicted through modeling, before such a microsys-tem can be accomplished. The modeling of themechanical structure often involves fluid, thermal,and electromagnetic interactions. The electrical cir-cuitry requires further attention, since the resonantfrequency of the mechanical microstructures is ingeneral very high. The shift in critical naturalfrequency makes the microsystem exhibit strongercoupling with the electronic circuitry than the

Figure 1 Lump model of damped mass elastically supportedon a harmonic vibratory base.

780 MEMS, DYNAMIC RESPONSE

normal-size electromechanical systems. AlthoughMEMS are subjected to multiple physical interactionswith very nonlinear effects, the lump mass model isalways conceived to estimate the range of thedynamic linear response of the system.

When modeling multiple interactions among thelinear physical system, it is perhaps more convenientto convert all the components to one common type ofanalogous system. Traditionally, all the mechanical,fluid, or thermal elements are converted to standardelectrical components. The network models and theanalysis methods for the electrical circuits are verypopular and intuitive, such that the analysis of theanalog systems is notably simplified. Furthermore,software packages can be conveniently used for theequivalent electrical network analysis.

For the normal-size world, Table 1 gives theequivalent circuit components and signal variablesamong electrical, mechanical, fluid, and thermal phy-sical quantities. The same analogy can be extended tothe microworld, with some reservations due to themajor influence of certain phenomena that are sig-nificant at a microscale level. However, attention isrequired when the figure number for the damping orstiffness or even mass is fed into the equation of themodel. Moreover, it should be emphasized thatmicrominiature systems usually exhibit noticeablenonlinearities, which should be taken into accountwhen more accurate modeling is carried out.

Dynamic Response of MEMS

MEMS which are structured in micromechanicalconfigurations comprising masses elastically sus-pended exhibit specific dynamic behavioral patterns

similar to those of the normal-sized mechanical sys-tems. The small masses associated with relativelylarge stiffness bring the resonant frequency of thestructure in the range of kHz. However, other forces,which are associated with the processes or with thefunctioning principles of MEMS, may considerablyinfluence their dynamic behavior. Two of the mostsignificant influences, the electrostatic field and inter-laminar stress, will be further discussed. A briefsummary that helps to resolve the figure number ofmass, stiffness, and damping for the lump model ofsimple MEMS structures is also given below.

Mass

Mass (the equivalent capacitance in electrical circui-try) can be evaluated assuming the geometry of thefeature. However, due to the selectivity of the etch-ant, the initial shape of the feature may be distorted,as shown in Figure 2. The etchant removes thesacrificial film at a high rate but it will also etchfrom the structural layer to a lower extent. The innerwalls of the structural film are sloped after the etchingis carried out because the inner walls are graduallyexposed to the etchant, whereas the external surfacesare permanently in contact with the etchant and thusuniformly etched. In certain conditions, the density ofmatter of the thin films is slightly smaller than that ofthe bulk, so the density figure number should beregarded with some caution. The equivalent massmethod stands, when concentrated masses are con-sidered along with the distributed masses.

Table 2 shows the standard formulation for theequivalent mass of cantilever and double-supportedbeams loaded with concentrated weight. The formu-lation presented in Table 2 covers in the first iteration

Table 1 Analogous elements and variables

Electrical Mechanical Fluid Thermal

Current ( j ) Force (F) Volume flow (q) Heat flow (H)

Potential (V ) Speed (v) Pressure (p) Temperature (y)

Resistance

R � DV

i

Mechanical resistance (damping)

1

b� Dv

f

Fluid resistance

r � Dp

q

Thermal resistance

Ry � DyH

Inductance

L � DV

di=dt

Mechanical resistance (spring)

1

k� Dv

df=dt

Fluid inertance (inertia)

L � Dp

dq=dt

Thermal inertance Ð(no significance)

Capacitance

1

C� DVR

i dt

Mechanical capacitance (mass)

1

m� DvR

f dt

Fluid capacitance

1

C� DpR

q dt

Thermal capacitance (thermalmass)

1

C� DyR

H dt


most of the MEMS structures, which are conceived asbeams simple or double supported.

Stiffness

Stiffness (the equivalent inductance in the electricalcircuit) is an essential parameter in the evaluation ofthe dynamic properties of microstructures. Assumingthat the structures have uniform geometric features,the analogy with the helical spring can only beapplied for concentrated loading. The stiffness can

thus be expressed as the ratio between load anddeflection:

k � dF

ddor; for single concentrated torque;

ky � dM

dy

�4�

When the loading is uniformly distributed on a sec-tion of the structure, the equivalent loading method

Figure 2 Cross-section through the geometry of a surface micromachined structure transformed due to the limited selectivity ofthe etchant.

Table 2 The equivalent mass of beams loaded with concentrated mass

Cantilever beam of mass mcarrying an end mass M

meq � M� 0:23m

Cantilever beam of mass mcarrying an end mass M

meq � m1 � l2l1

� �2

m2� l3l1

� �2

m3

Simply supported beam of massm carrying a mass M at themiddle

meq�M�0:5m


may be employed to determine the equivalent stiff-ness. Table 3 shows the equivalent stiffness for se-lected types of beams loaded with concentrated anddistributed loads.

When the structure is subjected to multiple con-centrated loads, the energy method is instrumental inevaluating the equivalent stiffness of the system. Thestrain energy in the structure is equal to the equivalentenergy of a theoretical spring:

X Zl

M2bdx

2EI

24 35 � keqd2

2�5�

for bending and:

X Zl

M2t dx

2GIp

24 35 � kyeqy2

2�6�

for torsion where Mb is the equivalent bending mo-ment in each element of structure, Mt is the equiva-lent torsion moment in each element of structure, E isthe Young's modulus of elasticity, G is the shearmodulus of elasticity, I is the moment of inertia ofthe section of the structure, Ip is the polar moment ofinertia of the section of the structure, Keq is theequivalent bending stiffness of the structure, Kyeq isthe equivalent torsion stiffness of the structure, d isequivalent deflection of the structure, and y is theequivalent torsion angle of the structure.

Damping

Damping evaluation requires the most laborious ana-lytical effort. The small geometric size of the flexuralelements and the reduced gap between the mobile andfixed part of the structure promote squeeze damping,which can reach remarkably large values of dampingcoefficients. Damping is critical to the performance ofthe microsystem throughout the frequency band-width of the system. Large values of squeeze filmdamping, which are likely to occur in MEMS, canreduce the dynamic bandwidth of the system to anunacceptable level. Damping is determined by thegeometry of the gap and of the elements, as well asby the viscosity or the pressure of the surroundingfluid. Damping is of great concern for microaccele-rometers and miniature microphones. The frequencyresponse of such systems can be improved by redu-cing damping by configuring flow channels throughthe flexing or fixed elements or through low-pressureencapsulation. For laterally oscillating microstruc-tures, like comb-like resonant structures, the domi-nant nature of the damping is due to Couette-type

Table 3 The equivalent stiffness of beams loaded with concen-trated and distributed load

k � 6EI

2l3 ÿ 3l2aÿ a3

k � 3EI l � 2a� �22 l ÿ a� �2a3

k � 48EI

a� 1

3l2 ÿ 4a2

� �

k � 384EI

5l3

k � 8EI

l3


flow. Table 4 gives the equivalent damping values forsome energy dissipation arrangements. Reduceddamping requires reduced pressure of the immersingfluid/gas.

Certain classes of application require very lowdamping. The resonant structures (piezoelectricactuators and acoustic sensors) function based onthe oscillatory properties of specific active materials(quartz, LiNbO3, ZnO, AlN, PZT, etc.). Low intrin-sic damping properties of the material ensure that thesystem is more efficient through reduced energy dis-sipation/loss.

When damping is low, the dynamic over the staticamplitude ratio at resonance, the quality factor Q,

yields considerable information about the dynamicresponse of the system. The bandwidth of the systemrepresents the range of frequencies for which theamplitude ratio falls beyond Q=2. The Q factor of astructure is associated with the ratio of the strainenergy stored in the system to the energy dissipatedthrough damping.

When the natural frequency of a more complexsystem is evaluated, it is more appropriate to use oneof the energy methods. One of the remarkable advan-tages of such a method is that one can incorporatevarious effects of the electric and magnetic field, andcapillary forces, which would otherwise be exces-sively difficult to incorporate in the model. Equating

Table 4 The equivalent damping for various mechanical structures

Relative motion betweenparallel surfaces

ceq � ZA

h

Vibrating mass in rigidcontact with a fixedsurface (Coulombfriction)

ceq � 4mN

poX

Squeeze film damping c�eq �4Zl3

h3

*per unit of length

Couette-type damping c�eq �pu0Zod


the maximum potential energy and the kinetic energy,one can assess the influence of the external forces andfields to the natural frequency of the structure.

The Influence of the Electrostatic Field

Assume two parallel plates separated by a small gap.One plate is fixed while the other moves along thedirection z normal to the two plates. The mobile plateis subjected to elastic restoring forces. The motion ofthe plate in the fluid filling the gap between the twoplates yields a dissipative force which is proportionalto the speed of the plate. The distance between thetwo plates is usually measured through the value ofthe impedance of the variable capacitor formed by thetwo plates. The small size of the micromechanicalstructures of MEMS enables the electric field which isalways present to strongly influence the dynamicresponse of a microstructure. The energy due to theelectrostatic field can be expressed by:

ÿ @U

@z� V2

2

@C

@z�7�

where U represents the stored electrostatic energy, Vis the voltage differential across the capacitor plates,C is the capacitance of the capacitor, and z representsthe deflection of the flexible elastic armature of thecapacitor.

Eqn [7] shows that enhanced displacement of themobile armature tends to augment the intensity of theelectrostatic field, reducing the gap and increasingcapacitance, and this field will incline to pull the

flexing element even more. The armature under theelectrostatic field thus renders a larger displacementthan that experienced in the absence of the electro-static field, which could be comprehended as negativestiffening of the structure due to the existence of theelectrostatic field. It must be noted that the electro-static field, regardless the potential of the armatures,produces a weakening effect on the stiffness of themobile plate. However, the detectable phenomenon isthe attraction between the two armatures.

It has also been found that each structure exhibits aspecific threshold value of static deflection due to theelectrostatic force, and, if this is exceeded, the flexiblearmature becomes dynamical unstable. The mobilearmature is perceived to be collapsing on the fixedback-plate. From the point of view of the electricalfield, a threshold voltage is associated with eachstructure, at which pull-in and ultimately snappingoccur. Usually, this condition is associated withmechanical failure of the structure, so it must becarefully avoided. The natural frequency of a platesubjected to electrostatic field will drop due to thenegative stiffness effect induced by the electrostaticfield. When threshold displacement is reached, thenatural frequency approaches zero; this is when thebalance between positive and negative stiffness isreached. The trend of the natural frequency of adouble-supported plate, which is subjected to apotential difference with respect to the fixed arma-ture, is illustrated in Figure 3. The natural frequencyand the first few resonance frequency and modeshapes for simple stuctures are given in Table 5.

Figure 3 (A) Natural frequency of a microbeam 10006 10006 2 mm which is double-supported and subject to a variable differ-ence of potential with respect to the fixed armature, parallel and spaced at 2mm about the beam. The natural frequency decayssuch that, for voltage beyond 15 V the natural frequency of the plate is close to zero. The plate exhibit dynamic instability at evenlower difference of potential, as shown in (B). (B) Phase state plot of the microbeam described in (A), subject to sinusoidal excitationand under an electric field created by 12 V potential difference between the microbeam and the fixed armature.


Table 5 The first resonance frequencies and mode shapes for simple structures.

No. Beam configuration Natural frequencies

1

fi � ki

2p

��EIg

wl4

r

2

fi � ki

2p

��EIg

wl4

r

3

fi � ki

2p

��EIg

wl4

r

4

fi � ki

2p

��EIg

wl4

r

5

fi � ki

2p

��6ki

j � �js=3�� l

s

lj

Js

l

l

l

l


In carrying out the design of capacitive, piezoresis-tive, or acoustic sensors and actuators, the influenceof the electrostatic field on the natural frequencydecay of the structure must be carefully considered.Such analysis can be carried out using one of theenergy methods (Rayleigh for natural frequency orRayleigh±Ritz for the first few resonant frequencies).By equating the strain energy with the kinetic energyplus the electrostatic energy, the natural frequency orfirst few resonant frequencies can be calculated.Finite element analysis (FEA) represents anotheroption in identifying the first few natural frequenciesof a complex-shaped structure. Finite element model-ing permits complex analysis of multilayered complexshape structures, including stress±strain, dynamic,

thermal, or electro-magnetic loading, and even com-binations of the above.

The Influence of the Interlaminar Stress

Besides the influence of the potential fields, fabrica-tion of thin films may be responsible for the peculiardynamic behavior of the microstructures. Thin depos-ited films induce either compressive or tensile stress,which is mainly due to the thermal mismatchingbetween the substrate and the coating encounteredat the low ambient temperatures after high-tempera-ture deposition process. The residual stress may pro-duce unwanted bending, warping, and fracture of thestructures. However, the existence of the stress in

Table 5 continued.

No. Beam configuration Natural frequencies

6

fi � ki

2p

��Tg

wl2

r

7

f � k

2p

��Et3g

12wa4 1ÿ v2� �

s

8

f � ki

2p

��Et3g

12wr4 1ÿ v2� �

s

j, concentrated end-mass moment of inertia; js, distributed mass moment of inertia; G, shear modulus of elasticity; T, tension in string; i, the rank of thefrequency; fi, natural frequencies (Hz); ki, constant; r, radius of the circular path; t, thickness of the plate; l, length of the beam; w, load per unit length(including beam weight); I, area moment of inertia; g, gravitational acceleration; E, Young's modulus of elasticity.


structure produces changes in the dynamic behaviorof the structure. The amount of change in thedynamic properties depends mostly on the quantityof stress built into the structure. The amount of stressis strongly dependent on the parameters of the deposi-tion process, and the values of such stress can rangefrom a few MPa to a few GPa. The low-stressed filmsare almost always present when accomplishing sur-face micromachined structures. Certain sequences ofprocesses and thermal treatments could lower theamount of residual stress to close to zero. The mea-surement of the residual stress can be carried outusing experimental techniques that have been createdfor various stress ranges and specimen size. For fullwafer measurement and stress exceeding 10 MPa, thevariation of curvature yields the value of the built-instress, as given by the relationship (Stoney):

sr � Et2s

6 1ÿ v� �t1

Riÿ 1

Rf

� ��8�

where ts is the thickness of the substrate, t is thethickness of the thin film, Ri is the curvature of thewafer before deposition (expected Ri � 1), and Rf isthe curvature of the wafer after deposition.The mea-surement of the large curvatures is performed withoptical stress gauge machines, which can detect largeradii of curvature of the substrate before and afterthin film deposition.

For stress lower than 1 MPa, buckling of a subset ofdouble-supported witness beams is used to range the

compressive stress value in a narrow bandwidth. Forthe tensile stress, special sets of structures consistingof a ring (Guckel rings) supported by a diameter beamare used to evaluate the built-in stress between thesubstrate and the film. The witness test structures aremade from the thin film deposited on the substrate.The two structures described above are shown inFigure 4. The array of structures can be containedwithin a certain design and the measurement consistsof confronting the observation through a microscopewith a table illustrating the critical stress in bucklingversus the geometric dimensions of the beams. Theresidual stress can be evaluated out of the criticalstress at buckling:

sr � ÿ p2h2E

3L2for double-supported beams �9�

sr � ÿ p2h2E

12z R� �R2for Guckel rings �10�

where h is thickness, L is length, R is radius of theGuckel ring, and x�R� is the function of the inner andouter diameters.

The residual stress strongly influences the vibrationcharacteristics of the free-standing structures, whichare part of MEMS. Flexing multilayered structuressuch as microbeams and microplates, as shown inFigure 5, tune their dynamic response according tothe built-in residual stress. The general equation(Lagrange) describing the plate motion is:

Figure 4 In situ structure for low residual stress measurement in thin films. (A) Double-supported beam buckles under compres-sive stress. (B) Gluckel ring configuration; the diameter beam buckles under the tensile stress in the ring.


Dr4z� r@2z

@t2� 0 �11�

with:

D � Eh3

12�1ÿ v2� flexural rigidity of the plate

r4 @4

@x4� 2

@4

@2x@2y� @4

@y4

r mass by surface unit

If the plate is subjected to stress, which is equiva-lent to the stress yielded by the contour lineal forcesNx, Ny, and Nxy the previous equation changes to:

Dr4z� r@2z

@t2� Nx

@2z

@x2� 2Nxy

@z

@x@y�Ny

@2z

@y2

�12�

The eigenfrequencies fi of the plate not subjected toany stress are given by:

fi � 1

2pli

a2

��D

r

s�13�

The eigenvalues of the plate subjected to decoupledlineal forces Nx and Ny (Nxy � 0) are given in theform:

l2si � l2

i �a2p2

DNym

2 �Nxn2 a

b

� �2� �

�14�

From the above last two equations, one can identifythe set of lineal forces Nx and Ny which satisfy therelationships:

4ra2 f 2i ÿ fsi

ÿ � � Nym2 �Nxn2 a

b

� �2�15�

where fi is eigenfrequency of the unstressed plate, fsi iseigenfrequency of the stresses plate, a, b is dimensionsof the sides of the plate, h is thickness of the plate, andm, n are mode numbers.

The last relationship can serve to solve the inverseproblems, which are of practical interest. When thestress at the interface of the substrate with the thinlayer is known, the shift in the vibration frequenciesof a given structure can easily be predicted.

Microplates subjected to forced vibration will thusbehave similarly, with the exception that their naturalfrequency is much higher than that of the normal-sized plates.

Annealing performed on the structures at 800±10008C in inert gases for 2±4 h can substantiallyreduce residual stress. However, such treatment istoo severe for the electronic circuit components and itcan only be applied in the very early stages of astandard integrated circuit process, before the defini-tion of the semiconductor junctions.

Signal Conditioning

Microsensors like any other sensors, must convert themeasured physical quantity into meaningful signals.The function of the signal conditioning circuitry con-sists of converting the sensor output into convenientelectrical signals that can be further streamed to thecontroller or to the data acquisition system. Theoutput signal of the sensor should be a voltage wave-form, which can be easily processed by the dataacquisition circuit. The process of converting theoutput signal from the sensor to a voltage can beused to reduce the noise or partially to compensate

Figure 5 Mismatching stress at the thin film interface. When the substrate or the sacrificial film is etched out to reveal themicrostructure, the contour stress or the built-in stress at the interface of the bimorph film will modify the static and dynamicresponse of the structure.


for the objective errors collected by the signal fromthe transducer to the conditioning circuitry.

The new generation of sensors is conceived tocomprise on the same chip the signal-processingcircuit, the analog to digital converter (A/D), anddedicated controllers. The resolution of the A/D canbe maximized by matching the dynamic range of theinput signal to the dynamic range of the data acquisi-tion system. The source impedance of the input signalmust be low, such that changes in the impedance ofthe data acquisition system or controller do not affectthe signal. The sampling rate of the A/D must be atleast twice as fast as the bandwidth of the outputsignal of the sensor (practically, one order of magni-tude higher yields a reasonably accurate conversion).

The output signal of the sensor may include noisethat must be removed before it is fed to the A/D. Thenoise is often associated with stray capacitance of theconnection line of the circuit. The high level ofintegration in MEMS presents the advantage oflow-level noise in the signal. However, the electro-static of the electromagnetic fields may negativelyinfluence the performance of the conditioning circui-try. This problem can be addressed through appro-priate packaging of the integrated microsensor.

The output signal of the microsensor may notalways match the required input signal of the con-ditioning circuitry. Thus, appropriate conversion ofthe signal may be required before being fed to the A/D.The conversion is carried out from voltage, current,resistance, and capacitance, to voltage. Most commonconversion schemes are briefly presented below.

Voltage-to-voltage Conversion

Voltage-to-voltage conversion is required whendynamic range modifications (bandwidth reductionor impedance alteration) are necessary, when the out-put signal from the sensor is a voltage. The above-mentioned conversions are carried out by standardcircuits. The amplifiers are used to match the outputlevel of the sensor to the input of the data acquisitionsystem. Some of the standard circuits, their scheme,the function carried out and their main properties areshown in Table 6. The output signals from the sensorcan collect noise. If the frequency of the noise is off thefrequency of the significant signal, the unwantedsignal can be filtered out using low/high-pass filters.The filtering is usually applied before the signal isamplified. The DC component of the signal, if any,can be removed with an offset circuit.

Current-to-voltage Conversion

Current-to-voltage conversion is usually performedwith an operating amplifier in an inverting config-

uration, since noninverting amplifiers draw very lowcurrent. This configuration scheme is most suitablewith current-generating sensors. When the currentoutput increases, the output potential increasesproportionally.

Resistance-to-voltage Conversion

Resistance-to-voltage conversion is used when thesensor detects the electric resistance variation due tothe input signal. Piezoresistive accelerometers, pres-sure sensors, or microphones are examples of suchtypes of sensors. The resistance variation needs to betranslated into standard voltage output to feed the A/DC, the data acquisition circuit, or the controller.The two basic methods to convert resistance variationto voltage output are presented below. The simplestway consists of applying a voltage to a resistor dividernetwork made of the variable resistor RS (the trans-ducer) and a reference fixed resistor RR, as shown inFigure 6. The voltage detected across the sensor isamplified before being converted to digital signal.The drawback of this method is that the amplifierboosts the entire voltage across the sensor, whichtranslates through a high DC offset. Thus, the secondmethod of converting the resistance to voltageaddresses this disadvantage. The resistance bridgeconfiguration, as shown in Figure 7, only amplifiesthe change in voltage due to the shift in the resistanceof the sensor. The resistance bridge presents theadvantage of zero offset, high sensitivity, and thermalcompensation.

Capacitance-to-voltage Conversion

Capacitance-to-voltage conversion is used when theoutput from the transducer is a variation of capaci-tance. The capacitance-based sensing presents multi-ple advantages. The capacitance varies in proportionto the distance between the armatures of the capaci-tor. The capacitors are sensitive to the material thatresides between their electrodes. The most popularmethods used to convert capacitance to voltage aresimilar to those used to measure resistance. However,besides these methods, more innovative principlesand circuits have been created. For example, thecharge amplifier circuit is used to detect small capa-citance variations in the fF range.

The voltage divider uses the variable capacitor anda reference capacitor. The circuit configuration is thesame as the resistance voltage divider: the only dif-ference is that, instead of the resistors RS and RR,capacitors are used. Similarly, the capacitor bridgecircuit configuration looks like the resistance bridgein which resistors are replaced by capacitors.


Table 6 Standard circuits

Circuit Scheme Input±output relationship Characteristics

Amplifier inverterVout � ÿR2

R1� Vin

Input Impedance � R1

Output Impedance � 0

Amplifier noninverterVout � 1� R1

R2

� �� Vin

Input Impedance � 1Output Impedance � 0

SummingVout � V1

R3

R1� V2

R3

R2

Combines the output of arrays tosensors

ME

MS

,D

YN

AM

ICR

ES

PO

NS

E791

Table 6 Continued

Circuit Scheme Input±output relationship Characteristics

DifferenceVout � V2 ÿ V1� � � R2

R1

Removes unwanted DC offset

Single-polelow-pass filter A � ÿR2

R1

1

1� j�f=fo�

Single-polehigh-pass filter A � ÿR2

R1

j�f=fo�1� j�f=fo�

792M

EM

S,

DY

NA

MIC

RE

SP

ON

SE

The reference signal is sinusoidal, since the capaci-tance obstructs DC. The impedance of the capacitor isinverse in proportion to frequency. An appropriatereference frequency yields a voltage across the capa-citor that is suitable for amplification. When thereference frequency is too low, the high impedanceof the sensor creates high levels of noise. When thereference frequency is too high, the low impedance ofthe sensor allows the sensor to saturate the circuit.

The main feature of MEMS containing free-stand-ing elements is the high natural frequency exhibitedby the mechanical structure that may be stronglycoupled with the electrical circuit system. MEMSperform in a very nonlinear way in dynamic charac-teristics. However, in a first approximation, lumpmass linear models of the miniature microsystemsare satisfactory. Simple formulations of the equiva-lent mass, stiffness, and damping are analogous withthe formulations used in normal-sized systems. For amore accurate analysis of either a simple flexingstructure or complex microactuators, the nonlinearphenomena must be considered and examined. Theyinclude, but are not limited to, electrostatic andelectromagnetic fields, capillary forces, and interlayermismatch stress. Analysis of the influences of non-linear physics on the dynamics of the microstructuremay be carried out using standard energy methods.

Nomenclature

A amplitudeC capacitanceE Young's modulus of elasticityg gravitational accelerationG shear modulus of elasticityh thicknessI inertiaL length

Figure 7 Resistance bridge measurement scheme.

Figure 6 Resistance voltage divider measurement scheme.


Q quality factorR radiusRR reference fixed resistorRS variable resistorT tensionU electrostatic energyV voltage� phase shiftr mass by surface unit

See also: Damping materials; Dynamic stability; MEMSapplications; MEMS, general properties; Signal inte-gration and differentiation.

Further Reading

Bhushan B and Gupta BK (1997) Handbook of Tribology:Materials, Coatings, and Surface Treatments. New York:McGraw-Hill.

Campbel SA and Lewerenz HJ (eds) (1998) SemiconductorMicromachining. Chichester: Wiley.

Fraden J (1993) AIP Handbook of Modern Sensors;Physics, Design and Applications. New York: AmericanInstitute of Physics.

Gardner JW (1994) Microsensors; Principles and Applica-tions. New York: John Wiley.

Kovacs TA (1998) Micromachined Transducers Source-book. Boston: McGraw-Hill.

Lee HH (1990) Fundamentals of Microelectronic Proces-sing. New York: McGraw-Hill.

Madou M (1997) Fundamentals of Microfabrication. NewYork: CRC Publishing.

Neamen DA (1992) Semiconductor Physics and Devices:Basic Principles. Homewood, IL: Irwin.

Ohba R (ed.) (1992) Intelligent Sensor Technology. NewYork: John Wiley.

Sze SM (ed.) (1994) Semiconductor Sensors. New York:John Wiley.

Tiller WA (1990) The Science of Crystallization: Micro-scopic Interfacial Phenomena. Cambridge, UK: Cam-bridge University Press.

Weste NHE and Eshraghian K (1994) Principles of CMOSVLSI Design. New York: Addison Wesley.

MEMS, GENERAL PROPERTIES



doi:10.1006/rwvb.2001.0073

Introduction

MEMS, the acronym for Micro-Electro MechanicalSystems, represents an emerging technology throughwhich miniature mechanical systems are built makinguse of the standard Integrated Circuits technologieson the same chip as the electronic circuitry. Micro-mechanical structures along with afferent electroniccircuitry often perform functions with high accuracy,which cannot be accomplished by so called `normal'sized structures. The strength of MEMS is providedthrough the high effectiveness of the fabrication pro-cess, which yields uniform performances for thestructures at a relatively low cost, in conjunctionwith the low power consumption. Moreover, micro-structures may probe into phenomena deeper thanactual electromechanical systems do.

Microelectromechanical systems (MEMS) are min-iature mechanical devices built from silicon usingsolid-state technology; they contain the necessaryelectronic circuitry built on the same chip with the

mechanical structure. These devices are also knownas microsystems or micromechatronic devices.Monocrystalline silicon is a very attractive mechan-ical material which, in addition to its enabling elec-trical properties, exhibits remarkable mechanicalproperties. The elastic modulus is close to that ofsteel while the specific mass is about three timeslower. Silicon enables the parallel batch fabricationof microsystems, which yields higher reliability of thedevices together with a dramatic drop in the fabrica-tion cost. The concept of MEMS emerged from theneed for miniaturization of the mechanical sensingelements, enabled by the anisotropic crystallographicetching properties of silicon and driven by the enor-mous progress made in integrated circuits (IC) tech-nology over the last two decades. The field of MEMShas been acknowledged at the end of the 1980s,although the roots of silicon-based sensors and actua-tors go back to the 1970s.

MEMS are classified into two large functionalclasses: microsensors and microactuators. Microsen-sors are conceived as fully signal-conditioned trans-ducers that provide the same function as the classicsensors along with their conditioning circuit board.Since the main functions of sensors are detection andmeasurement, they should interfere as little as poss-ible with the environment, except for measured

794 MEMS, GENERAL PROPERTIES

physical quantity. Thus, the energy exchange betweenthe environment and the microsensor should be mini-mized. Miniature transducers easily comply withthese requirements.

Microactuators are conceived to convey energy totheir surrounding environment. Their size limits tre-mendously their performance and thus, the area ofapplication is mainly reduced due to their limitedenergy storage capability. Arrays of microsensorsand microactuators together with implemented con-trols enable the creation of `smart matter', whichrepresents a natural step in the development of tech-nology.

The first silicon-based miniature devices were pro-duced in the early 1970s through bulk micromachin-ing. This fabrication principle takes full advantage ofbatch fabrication but the minimum feature size islimited to fractions of millimeters. Besides, bulkmicromachining cannot enable full implementationof the electronic circuitry with the mechanical com-ponent on the same silicon chip, using a uniqueprocessing flow. The earlier devices thus involvedconsiderable efforts in assembling and packagingthe mechanical components and the electronic con-ditioning circuitry. Further, the research focus hasbeen directed towards developing a comprehensivetechnology to realize both the reliable electronic

circuits and robust mechanical structures integratedwithin a single design. Attempts to fabricate mechan-ical devices using modified standard IC processeswere successful, so that the conditioning circuitrycould be included on the same chip and fabricatedduring the same process flow. The surface microma-chining processes yield a minimum feature size in therange of micrometers. Here, superimposing severalpatterned layers of conducting, insulating, and tran-sistor-forming materials to build the structural andthe sacrificial configurations along with the condi-tioning circuitry enables the realization of MEMSstructures. After the completion of both mechanicalstructure and electronic circuitry, a sacrificial etchingis performed to release the free-standing structurewithout affecting the function of the electronic cir-cuitry. This step, called postprocessing, requireshighly selective etchants and a series of precautionsin order to retain the integrity of the microstructure.

The feasibility of the microactuators and micro-mechanisms has been established through a hybridtechnology that enables surface micromachinedMEMS based upon thicker polysilicon as the struc-ture and SiO2 as sacrificial films (MUMPS).

Figure 1 illustrates the difference between bulk andsurface micromachining. The bulk-micromachinedstructures are obtained by removing the excess from

Figure 1 (A) Bulk-micromachined plate in section. The structural element is built from silicon substrate by deep crystallographicanisotropic etching. (B) Surface micromachined step-up free end beam in section. The structural element is built in the top of thesacrificial film. Postprocessing will remove the sacrificial film and release the free-standing structure. (C) Surface-micromachinedmicrobeams released by postprocessing. Mismatching between the materials constituting the polymorphous beams induces highstress which has evolved into spectacular bending of the beam.

MEMS, GENERAL PROPERTIES 795

the substrate material, such as that shown inFigure 1A, through isotropic or anisotropic etchingof the material. Surface micromachining yields micro-systems enabled by the structural layers deposited andpatterned in a specific sequence with the sacrificiallayers. The sacrificial layers are removed during post-processing and the microstructure is released as illu-strated in Figure 1B. Figure 1C shows a magnifiedpicture of a set of micromachined cantilever beams,which are released through postprocessing etching.Thermal mismatching in thin films of various materi-als, which constitute the beam as a composite mate-rial, induces high stresses, resulting in a dramaticbending of the free end of the beam.

Sensors and microsensors are conceived to detectand measure electrical, chemical, magnetic, mechan-ical, radiant, and thermal measurands. The mostcommon physical quantities and measurands aregiven in Table 1. However, mechanical microsensorsare the most interesting from the point of view ofvibration. They modify mechanical measurands inelectrical signals, using resistive, capacitive, induc-tive, piezoelectric, or other physical conversion prin-ciples.

The primary sensing element is usually a simpleelastic structure such as a cantilever beam, whichdeflects under input measurand. This deflection,although small, is related to the input measurand,which is detected and translated into an electric signalby the built-in subminiature secondary sensingdevices, such as piezoresistive, piezoelectric or capa-citive elements. The rendered electric signal must bewell shielded from the noise of the environment forfurther conversion into a meaningful indication bythe conditioning circuitry. The sensitivity of the sen-sing element to the measured signal must be high,while sensitivity to irrelevant excitations should be aslow as possible. Limitations in sensitivity are relatedto the measured range and the maximum permittedoverloading.

The force/position feedback servosensors structureis more complex, since they comprise both micro-sensors and microactuators. When distorted by theinput signal, the sensing element is returned to itsstatic position by actuators, which are fed by thefeedback signal. The signal used to restore the posi-tion of the element is meanwhile used for measure-ment, as shown in Figure 2. Such a design concept,although more complex, addresses issues related tooverloading and hysteresis.

Implementation of the IC technology for microsys-tem fabrication represents a very different concept inthe sensors technology. Resolutions of 0:001�A=Hz1=2

or sensitivities in the order 10710 N are usual featuresof such microsensors. The actual achieved precision

and the accuracy of MEMS sensors can improve up toone order of magnitude the performances of theclassic sensors. The detected input signal can befurther fed into the electronic circuitry, conveniently

Table 1 The most common measurands for microsensors

Physical quantity/measurementprinciples

Measurand

Mechanical Acceleration, velocity, positionElastoelectric Force, momentElastomagnetic Stress, pressureThermoelastic StrainThermoelectric Mass, densityPhotoelectric Flow, flow speedPhotoelastic Stiffness

ViscosityEnergy

Acoustic Wave amplitude, wave velocity, wave phaseElastoelectric PolarizationElastomagnetic Frequency spectrumPhotoelastic IntensityMagnetoelectric Energy

Thermal TemperatureThermoelectric Thermal fluxThermomagnetic Specific heatThermoelastic Thermal conductivityThermooptic Thermal energy, thermal power

Energy

Electrical Charge, currentMagnetoelectric PotentialThermomagnetic Electric fieldThermoelectric ConductivityPhotoelectric PermittivityElastoelectric Energy

Magnetic Magnetic fieldMagnetoelectric Magnetic fluxElastomagnetic PermeabilityThermomagnetic Energy

Optical Wave amplitude, wave velocity, wave phasePhotoelectric PolarizationPhotomagnetic Frequency spectrumPhotoelastic Energy

Radiation TypeThermoelectric IntensityPhotoelectric EnergyThermooptic

Biochemical CompositionPhysical

transformationConcentration

Chemicaltransformation

State

ElectrochemicalSpectroscopy


located next to the transducer, which modifies theinput into a signal which matches the data acquisitionsystem. This is usually equivalent to changing theoutput to a voltage. The function of the electroniccircuitry extends to modifying the sensors' dynamicrange to maximize accuracy, filtering the noise, andlimiting the measurement spectrum.

Micromachined microactuators are conceived toconvert electrostatic, electromagnetic, thermal, orfluid flow energy into mechanical rotation, transla-tion, or oscillation. Although presently the microac-tuators are created for applications in which therequired forces do not exceed mN range, tremendousapplications in microbiology have been forecasted.Both linear and rotary microactuators have been builtusing bulk and surface micromachining technology,although the three-dimensional attributes of theactuators have required special technologies forhigh-quality vertical pattern fidelity. One main pro-blem of the microactuators relates to the friction andwear of moving parts. The small sized pivots, thecomplexities associated with total control of clear-ances between the pivot and the bearing, and the lackof appropriate lubrication methods dramaticallyreduce the life of the electrostatic motor. However,IC technology has not been modified to accomplishreadily assembled devices, such as hinges and articu-lations. Thus, the alternative technologies permit tocompensate for such limitation. Elastic suspensionsor levitation principles have been implemented toovercome the friction and wear in rubbing parts.Compliant mechanisms represent another option forMEMS. The compliant mechanisms allow motionthrough elastic deformations of one or more ele-ments. Motion of compliant segments generallyinvolves large-magnitude nonlinear deflections,which are difficult to predict using the linear beamtheory. More accurate prediction of microstructurebehavior will require multiphysics nonlinear analyses

of the miniature structures. Modeling such devices isthus quite challenging. Application of the surface-micromachined compliant multistable mechanismson microvalves and microswitches is presentlyunder investigation.

MEMS Materials

Microsystems manufacturing (micromachining)makes use of a large variety of materials. However,only a few are used to build the structural elements.The mechanical properties of these materials are thusof prime interest for the mechanical designer ofMEMS. Although the mechanical properties of bulkmaterials are comprehensively known, the propertiesof the thin deposited films are strongly dependent onthe deposition process and the deposition parameters.Table 2 provides a list of the most frequently usedmaterials for MEMS structures together with theiravailable relevant properties, as given in the availableliterature. The physical values range between broadlimits, which may be attributed to excessive varia-tions in the measurement methods employed and tothe large variance in the properties of the materials.Besides, the thinly deposited films exhibit high inter-nal compressive or tensile stresses, which may to acertain extent influence the mechanical properties ofthe thin structures. It has been acknowledged that thematerials in thin films exhibit higher mechanicalstrength than the same materials in bulk state.

Apart from the mechanical properties, some mate-rials require other special properties and conditionsfor the material processing, as they may not with-stand subsequent processing. The active materials arespecifically used to build the transducer on the mea-surement element, which is mostly associated withthe flexing element in the mechanical microsensors.The deposition and patterning sequences rigorously

Figure 2 General configuration of a sensor. The dashed line with the feedback actuator shows the configuration of a servo sensor.


require certain standing limits for the deposited mate-rial. One should bear in mind that the same materialsare used to build the IC, and thus, material selectionmust avoid any incompatibility problems that couldjeopardize the IC performances due to the fabrica-tion.

MEMS Fabrication Technology

MEMS fabrication mostly makes use of establishedIC processes. Apart from the rigid processingsequence, some new techniques have been developedand implemented to substitute for the lack of flex-ibility of the standard IC. They appropriately use theunconventional machining together with the standardmachining processes. Thus, LIGA (deep-etch litho-graphy, electroforming, and plastic molding), laser-assisted chemical vapor deposition (CVD), electro-plating, chemical plating, and even conventionalmachining form the most common manufacturingtechniques.

In lay terms, accomplishment of MEMS throughsurface micromachining is done through depositionof specific thin films (required by the IC process),patterning and etchings of the deposited material onundesired areas, followed by other successive deposi-tions, patternings, and etchings arranged in stacks.The desired structure is thus fully completed andembedded into the consecutive layers of depositions.Postprocessing is performed to remove the sacrificialfilms and release the free-standing structure.

Table 3 schematically illustrates the core of the IC-for-MEMS fabrication processes. The table presents

the three main processes: deposition, patterning, andthe removal of sacrificial layers. These processes aresubject to considerable variations in the process para-meters depending upon the technology implemented.The deposition techniques have reached performancelevels that enable the deposition of films with ade-quate mechanical and electrical properties.

Further, Table 4 gives a list of most other commonprocesses used to realize MEMS. As can be seen,standard machining processes are included in thelist, since hybrid accomplishment of MEMS throughIC and standard processes is an alternative that is stillunder appraisal. Some of the standard machiningprocesses are most instrumental for packaging.Most of the standard machining processes, however,require individual manipulation of parts, whichrepresents a bottleneck for a batch fabricationstream.

The etching process marks the postprocessing ofthe microstructure. Release of the free-standing struc-ture represents a challenge, since the electronic cir-cuitry, the contact pads, and the structure itself haveto withstand the etching. The costs associated withpostprocessing and packaging of MEMS often con-tribute to a significant increase in the cost of theMEMS. The low-cost postprocessing techniques,invariably performed outside the silicon foundry,are thus extremely desirable.

The selectivity of the etchant with respect to thepassivation and structural films is of most interest insurface micromachining. Table 5 presents the mostpopular etching recipes for Si, glass, Si3N4, Al, andAu along with the corresponding etch rates.

Table 2 Mechanical properties of selected materials used in MEMS processing

Material Thermalexpansioncoefficient(61076 8C71)

Strain limit(610-6)

Yield strength(MPa)

EYoungmodulus(GPa)

Density(kg m73)

HardnessKnoops(GPa)

Poisson ratio Thermalconductivity(W m7168C)

Al: film 23 2 0.124 47.2±74 2700±3500 1.3 236Si: bulk 2.3±4.3 ± 7 62±202 2330 5.3±13.1 0.22±0.278 157Polysilicon:

film3±7 ± 120±200 ±

SiO2: film 0.41±31.9 112 8.4 46±95 2200 8.2±18 0.17 1.10±1.7Si3N4: film 0.8±3.9 37 1.4 100±380 3200 34.8 0.22±0.27 9.2±30SiC: film 4.7±5.8 ± 21 302±450 3200 24.8±29.6 0.183±0.192 20.8±120PVDF: film 6±250 ± ± 2 1780Polymide: film 3±140 ± ± 7.5±15 0.1±0.45Pt: film 8.9±10.2 1 ± 170 21 440 73Monosilicate

glass t059:film

± ± 40±60 2250 5±7

Diamond-likecarbon(DLC)

± ± 150±800

Au: gold 14.1±16.5 ± ± 80 19 280 315


Table 3 Description of fabrication processes for MEMS

Process Type and method Sketch

Thin filmdeposition

Uniform depositionof selectedmaterials on theentire surface ofthe wafer

Chemical vapordeposition (CVD)

Physical vapordeposition(sputtering,vaporization)

Lithography

Batch printing onsubstrate 2-D and3-D features onphoto-sensitivepolymer

UV photolithography


Table 3 Continued

Process Type and method Sketch

E-beam lithography

Etching Deep etching(crystallographic)

Selective removal ofundecidedmaterial fromdesignated areasexposed to theetchant

Anisotropic ± (100)silicon wafer

Isotropic: Si wafer

Thin film etching

Sacrificial: anymaterial etching


The selectivity of the anisotropic etchant versuscertain crystallographic planes is responsible for thesculpturing into silicon of V or U trenches that areconveniently used to create bulk micromachinedstructures. Table 6 presents the most used siliconanisotropic etchants, typical etching temperature,etch rate, and selectivity.

The essential electrochemical etch reactions involveoxidation±reduction and dissolution of the oxidationproducts by complex etching agents. Both electrons

and hole carriers exist at the silicon surface. Oxida-tion of silicon atoms takes place at the anodic sites,while the oxidant is reduced at the cathodic sites. Theetching mechanism can be summarized as follows:

. Injection of holes into the semi-conductor Si � 2�� ! Si2

. Attachment of hydroxyl group OHÿ to Si2�,e.g. Si2� � 2�OHÿ� ! Si�OH�2

. The dissociation of water H2O$ OHÿ �H�

. Reaction of the hydrated Si with complexing agentin the solution

. Dissolution of the reacted product into etchantsolution

These etching mechanisms, however, do not fullydescribe the crystallographic anisotropic etching ofsilicon, and the temperature and stress dependencesor the selectivity.

More accurate etching can be carried out in reac-tive gases. The anisotropic etching of silicon in gas isno longer driven by the crystallographic structure.Thus, the etchant no longer follows the crystallo-graphic direction (111), like the wet anisotropic etch-ants, but the cut strictly follows a perpendiculardirection to the masked surface. Figure 3 illustratesisotropic and anisotropic etching. Table 7 gives thegas etchants for materials most commonly used inMEMS fabrication, together with specific etch rateand selectivity versus various masking materials.

MEMS Packaging

The basic considerations for MEMS packaging are:

. Connections to the chip for signal and power leads

. Means of cooling of the chip if necessary

. Mechanical structure to support and protect thechip

. Controlled-pressure environment to reduce the vis-cous damping of the flexing structure and provideaccess for the measurand to the sensing area

Satisfying all the requirements for a robust packagehas proven to be a very difficult task. The selectedmaterials must possess compatible thermal coeffi-cients of expansion, or the structure must comprisethe means to release the resulting mechanical stress,and excess of heat.

Temperature changes may arise either from thechip operation or changes in the operating environ-ment. The connections and the package should enablelow noise transmission and a high-fidelity path forsignals to and from the chip, and robustness ofconnections. Applications with specific considera-tions include, but are not limited to, operations inhostile environments, hybrid multichip packages,

Table 4 Alternate processes used in MEMS manufacturing

Process Alternatives and issues

Electroforming Electrochemical deposition (plating):contamination

Anodizing: contaminationPolishing: not very productive

Surface processing Ion implantationDiffusion: high temperaturesAnnealing: high temperatures

Coating Dipping: all sides are coveredSpinning: high process variabilitySpraying: uniformity issuesSerigraphy: alignment

Forming Hydro: contamination, accuracyDrawing: high temperature or high built-

in stressExtrusion Direct/indirect: accuracyMolding High-pressure injection

Vacuum: accuracyDipping: fillingCompression

Machining Turning: fixtureMilling: high forcesDrilling: chip removalSawingElectrochemical discharge

Finishing machining Grinding: high forcesPolishing: low efficiencyLapping: low efficiency

Microcutting Laser: low efficiencyElectron beamElectrochemical discharge:

contamination, minimum sizeJoining Riveting: size (microsize riveting under

investigation)Screwing: sizeSoldering: strengthBrazing: high temperatureElastic

Bonding Anodic: high temperature and voltageThermal: high temperatureAdhesives: limited life

Welding LaserElectron beamElectric spot: sizeElectrochemical

Wire bonding UltrasonicThermal: high temperature


Table 5 Wet etching solutions and etch rates for MEMS materials

Material Etchant solution Etch rate

Polysilicon 6 ml HF, 100 ml HNO3, 40 ml H2O 800 nm min71, smooth edges1 ml HF, 26 ml HNO3, 33 ml 150 nm min71

CH3COOH

SiO2 Buffered HF (BHF) (28 ml HF, 170 ml H2O, 113 g NH4F) 100±250 nm min71 at 258C1 ml BHF, 7 ml H2O 80 nm min71

Borosilicate glass (BSG) 1 ml HF, 100 ml HNO3, 100 ml H2O 30 nm min71 for 9 mol%B2O3, 5 nm min71 for SiO2

4.4 ml HF, 100 ml HNO3, 100 ml H2O 75 nm min71 for 9 mol% B2O3,13.5 nm min71 for SiO2

Phosphosilicate glass (PSG) Buffered HF (BHF) 550 nm min71 for 8 mol% P2O5

1 ml BHF, 7 ml H2O 80 nm min71

Si3N4 HF 14 nm min71, CVD at 11008C75 nm min71, CVD at 9008C100 nm min71, CVD at 8008C

BHF 0.5±10 nm min71

H3PO4 10 nm min71 at 1808C

Al 4 ml H3PO4, 1 ml HNO3, 4 ml CH3COOH, 1 ml H2O 35 nm min71, high resolution16±19 ml H3PO4, 1 ml HNO3, 0±4 ml H2O 150±240 nm min71, coarse0.1 mol K2Br4O7, 0.51 mol KOH, 0.6 mol K3Fe(CN)6 1 mm min71, coarse

Au 3 ml HCl, 1 ml HNO3: aqua regia 25±50 mm min71

4 g KI, 1 g I2, 40 ml H2O 0.5±1mm min71, resist mask

Table 6 Moncrystalline silicon etchants, etching rates and selectivity

Etchant Typicalcomposition

Temperature(8C)

Etch rate(mm min-1)

Anisotropy(100)/(111)etch rate ratio

Dopantdependence

Masking films(etch rateof mask)

Ethylene diaminepyrocatechol(EDP) in water

750 ml 115 0.75 35:1 761019 cm-3 boron reducesetch rate by about 50

SiO2

(0.2 nm min71)120 g100 ml Si3N4

(0.01 nm min71)750 ml 115 1.25 35:1120 g240 ml

KOH in water 44 g100 ml

85 1.4 400:1 1020 cm73 boron reducesetch rate by about 20

Si3N4, SiO2

(1.4 nm min71)KOH in isopropyl

alcohol50 g 50 1.0 400:1

100 ml

Tetramethylammoniumhydroxide inwater (TMAH)

25 g 85 0.6 200:1 � 2�1020 cm73 boronreducesetch rate about 25 times

Si:SiO2

(1000:1)75 ml15 g 85 0.5 200:185 ml

Isotropic etchant 10 ml 22 0.7±3 Reduce rate by 150 forphosphorous and boron

SiO2

(30 nm min71)HF 80 ml30 mlHNO3 1017.cm73

CH3COOH 25 ml 22 4 Si3N4

50 ml SiO2

(70 nm min71)25 ml


vibration control, and direct exposure of portions ofthe chip to outside stimuli (i.e., light, gas, pressure,and temperature).

MEMS Design

The observations on the dynamic behavior of MEMSindicate that they do not follow the pattern of similargeometric macrostructures. The design of MEMSrequires a perfect knowledge of the capabilities anddrawbacks of the manufacturing processes. Both bulkand surface micromachining impose specific con-straints on the design that are often reflected in thegeometry of the features. Both bulk and surfaceMEMS could be better designed using a computer-aided design (CAD) tool which are built to include theconstraints of the manufacturing process. The layoutdesign level is usually carried out on IC-compatibledesign tools, which enable the design of the electroniccircuitry. Simulations of the dynamics of both elec-tronic circuitry and the mechanical structure modeledas an electronic circuit help to improve the perfor-mance of the coupled electromechanical system. Thesimulation uses models which are created based onanalogous electrical circuit elements. A comparisonof the mechanical, fluid, and thermal elements withthe electrical element is given in Table 8. It is impor-tant to note that other first-order analogs areaccepted and used in simulations. The selection ofanalogs, however, may depend on the likeness ofcurrent and voltage used in describing the mechanicalphysical quantities.

Figure 3 Section through anisotropic and isotropic etched sili-con features. (A) Wet etching of (100) Si orientation; (B) wetetching of (110) Si orientation; (C) dry isotropic etching of Si; (D)isotropic etching of Si in XeF2.

Table 7 The gas etchants for materials used in MEMS, etch rate, and selectivity versus Si, SiO2, and photoresist

Material Etching gas Etch rate(nm min-1)

Selectivity

vs silicon vs SiO2 vs photoresist

Si SF6 + Cl2 100±450 80 5Polysilicon Cl2 50±80 30 5SiO2 CF4 + H2 50 20 5PSG CF4 + H2 80 32 8Al BCl3 + Cl2 50 4 22 7

Table 8 Comparison of physical elements

Dissipative Current storage Potential storage

Electric Resistance Inductance CapacitanceMechanical Damper Spring MassFluid Fluid resistance Fluid inertance Fluid capacitanceThermal Thermal resistance Ð Thermal capacitance


One of the challenges a designer has to face inmodeling is to establish the figure number of thesqueeze damping and the elastic constant for thecontinuum beam or plate element. Attempts toexpress analytically the equivalent stiffness of canti-lever and double-supported micromachined beamshave been done, but the simple analytical expressionsdo not always match the experimental available data.More accurate solutions are accomplished when thefinite element analysis (FEA) is used in conjunctionwith the appropriate material data. Nonlinear char-acteristics of the material or nonlinear behavior of thestructure can be simulated through appropriatedescription of the material and elements. Sufficientaccuracy can be achieved by conveniently refining themesh. A parametric design used in conjunction withmultiple variable optimization algorithms canenhance the performance of the structures. Stress±strain, modal, dynamic, thermal, electromagnetic orCFD analyses can be run once a suitable model hasbeen conceived. The output can provide essentialinformation that may lead to improved performanceof the microstructure. Some commercial codes canperform multivariable optimization, which wouldease the task of the designer.

However, an overall strategy of the entire manu-facturing flow, including packaging, must be plannedbefore the analysis is carried out. Important pieces ofinformation such as damping of the structure reside inthe environment in which the structure will work andthe operating temperature. Thus, the dynamic prop-erties will strongly be influenced by the surroundinggas/fluid in which the flexing structure performs.

Before the layout design is finalized, simulations ofselected deposition and etching processes may be runto insure that critical features will be accomplishedwithin the permitted tolerances.

Figure 4 illustrates the graphic output of a stressanalysis performed on the section of a plate subjectedto uniform pressure loading. Appropriate meshingyields the stress distribution across the thickness ofthe plate, which can further be used by the designer toestablish the most appropriate location of the piezo-resistive gauges.

After prototyping, manufacturing, and testing, themodels are adjusted based on the empirical dataobtained from measurements. Presently, the inte-grated modeling of MEMS is still in its infancy,although efforts have been made to understand andmodel the individual manufacturing processes, suchas etching, crystal growth and physical deposition.Although not explicitly stated, it is apparent thatanalytical models of microstructures are also usedand continuously improved, as test data becomeavailable. Modeling of electromechanical±fluid±ther-mal systems is thus of much interest to the designer.One of the main needs at this time is to build a goodmicroscale data bank of parameters and materialproperties that can be used in the modeling anddesign of microsystems.

Miniaturization of mechanical systems establishesa unique momentum in both classic and emergingtechnologies. MEMS enables a new approach inmaking, running, and controlling systems and pro-cesses. Micromechanical devices and systems areessentially smaller, lighter, faster, more precise, and

Figure 4 (see Plate 42). The output graphic file of an analysis of a silicon plate subjected to uniform distributed pressure loading.


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