Mechanics of Microelectromechanical Systems - N.Lobontiu and E.Garcia Part 10 potx

The staticequilibrium equations are used for either translatory or rotary motion in order to qualify the performance of various classes of MEMS, starting from thesimplest designs with on

The bloc force of the two-beam thermal actuator is up to 80 times largerthan the bloc force of a bent beam actuator, for very small lengths of theshort leg of the two-beam design

Problem 4.5

A transverse (plate-type) electrostatic device is used as a capacitivesensor Design the sensor such that when the mobile plate travels between thelimit positions and the capacitance variation is noless than a minimum value

Answer:

Problem 4.8

is used as an electrostatic actuator Find the gap between themicrocantilever and its mating pad (assuming both have the same length) thatwill produce a tip deflection of under application of a 80 V voltage.The material properties are E = 120 GPa and

Answer:

the area Can be chosen arbitrarily;

n = 2

Figure 4.62 Magnetic-electromagnetic interaction via flexible microbridge

Answer:

Problem 4.11

A microcantilever is magnetized as indicated in Fig 4.39 (c) Determinethe actuation effects at the free end when the external filed is defined by thelaw: and the magnetic field source is located underneath the free

end at a distance h Known are the cross-sectional dimensions w and t, andthe magnetization m of the magnet.

Answer:

Problem 4.12

A thin sheet is constructed of piezoelectric material of cross-sectionaldimensions w and t, and dielectric constant The piece is polarized overits thickness direction 3 and is subject to bending moments M that areapplied at its ends A total strain is measured on one of the sides of thepiezoelectric sheet Calculate the corresponding voltage that is generated.Answer:

Problem 4.13

Find the bloc force that has to be applied to a piezomagnetic bloc ofcross-sectional area A and Young’s modulus E The piece is magnetizedabout its height direction by means of a magnetic field H Also determine thecoupling coefficient Known are the magnetic charge constant themagnetic permeability and the magnetic compliance

Answer:

Problem 4.14

A clamped square membrane constructed of SMA with side l andthickness t is deformed while in martensitic state by application of a uniformpressure p A temperature variation, which brings the martensite intoaustenite phase, is applied subsequent to removing the initial pressure.Determine the variation in the maximum deflection Known are Young’smodulii in martensitic and austenitic states, and as well as Poisson’sratio (Hint: use Eq (1.233) with the first two terms of the infinite seriesexpansion.)

Answer:

Problem 4.15

A bimorph is constructed of two different materials, having

Find the

produced by a temperature increase of

Answer:

Problem 4.16

A bimorph is formed of two layers, the active one being piezomagneticand the substrate being polysilicon Find the length of the bimorph when a tipbloc force of needs to be produced with an induced strain

Answer:

Problem 4.19

A three-layer multimorph is constructed to function as a thermal actuator.Determine the tip deflection that this actuator is capable of producing under atemperature increase of It is known that

Problem 4.20

Positive and negative magnetostrictive layers, of identical thickness, areattached on both sides of a polysilicon layer Determine the tip slope when anexternal magnetic field induces opposite strains in the twopiezomagnetic layers Known are:

Answer:

References

1 L Que, J.-S Park, Y.B Gianchandani, Bent beam electrothermal actuators – Part I: single

beam and cascaded devices, Journal of Microelectromechanical Systems, 10 (2), pp 247-254,

2001.

2 Y.B Gianchandani, K Najafi, Bent beam strain sensors, Journal of

Microelectromechanical Systems, 5 ( 1 ) , pp 52-58, 1996.

3 G.T.A Kovacs, Micromachined Transducers Sourcebook, McGraw-Hill, Boston, 1998.

4 M.N.O Sadiku, Elements of Electromagnetics, Third Edition, Oxford University Press,

7 M McCraig, A.G Clegg, Permanent Magnets in Theory and Practice, Second Edition,

John Wiley & Sons, New York, 1989.

8 S Seely, A.D Poularikos, Electromagnetics – Classical and Modern Theory and

Applications, Marcel Dekker, New York, 1979.

9 A Kruusing, V Mikli, Flow sensing and pumping using flexible magnet beams, Sensors

12 E Garcia, N Lobontiu, Induced-strain multimorphs for microscale sensory actuation

design, Smart Structures and Materials, 13, pp 725-732, 2004.

13 R.H Liu, Q Yu, D.J Beebe, Fabrication and characterization of hydrogel-based

microvalves, Journal of Microelectromechanical Systems, 11 (1), pp 45-53, 2002.

1 INTRODUCTION

This chapter studies the static response of microsystems by modeling thecombined effects of actuation, sensing and elastic suspension The number ofmicrodevices that can be custom-built by integrating spring designs such asthose presented in Chapters 2 and 3 with rigid parts and transductionprinciples, as the ones analyzed in Chapter 4, is vast, and the present chaptercontains just a sample of the extended pool of MEMS applications The staticequilibrium equations are used for either translatory or rotary motion in order

to qualify the performance of various classes of MEMS, starting from thesimplest designs (with one suspension unit and one transduction unit) tomore complex ones (comprising several spring microsuspensions togetherwith either actuation or sensing units or with both actuation and sensingcapabilities) The large deformations of mechanical microsuspensions areanalyzed in MEMS applications that deform either axially or throughbending The buckling phenomenon, as applied to straight and curvedmicrocomponents, is also addressed together with the post-buckling andaccompanying large-deformation phenomena Later, the stresses and yieldcriteria for combined stresses are presented for several MEMS applications.Fully-solved examples supplement the text in order to better explain thevarious topics of this chapter, and a set of proposed problems completes thepresentation

2 SINGLE-SPRING MEMS

One of the simplest MEMS configurations comprises onemicrosuspension (spring) and the actuation/sensing component Theequilibrium in such situations is produced when the actuation force/momentand the opposing elastic force/moment are equal Several practicalapplications will be analyzed next, including microdevices that are designedfor linear or rotary (mainly electrostatic) transduction and flexure microhingeMEMS

STATIC RESPONSE OF MEMS

2.1 Transverse Electrostatic Actuation with

Microsuspension

By coupling the transverse (plate-type) electrostatic transduction that hasbeen introduced in Chapter 4 to one of the microsuspensions presented inChapter 3 leads to the model shown in Fig 5.1

Figure 5.1 Model of transverse electrostatic actuation and microsuspension

The maximum gap between the fixed and the mobile plates, occursinitially for y = 0 The static equilibrium sets in when the two opposingforces, the electrostatic and the spring force, are equal:

The force produced through transverse electrostatic actuation was given inChapter 4 and is rewritten here as:

whereas the elastic force is:

Figure 5.2 shows the force-displacement plots of these two forces As Fig.5.2 indicates, there are two points of equilibrium, and where the twoforces are equal for specified spring and electrostatic actuation properties.However, only the first equilibrium point, is stable because of the factthat the slope of the electrostatic force is smaller than the (constant) slope ofthe elastic force, whereas at point the slope of the electrostatic force islarger than the one of the elastic force

Figure 5.2 Electrostatic and spring forces versus displacement

It is known that an equilibrium point, one for which the total force,defined as:

is zero, has stable properties when the force derivative is negative, namely:

The limit point separating the stable region from the unstable one can befound by solving the equation system:

where the total force F is determined by means of Eqs (5.2), (5.3) and (5.4)

By solving the equation system (5.6) in terms of position and correspondingvoltage, the following solution is obtained:

The force corresponding to this point can be found by substituting of Eq.(5.7) into either Eq (5.2) or Eq (5.3), and its expression is:

The values of and define the point P of Fig 5.2, which characterizes the

phenomenon known as pull-in For forces less than of Eq (5.8), the slope

of the electrostatic force is smaller than the one of the spring force (which isequivalent to saying that the slope of the total force F is less than zero) andthe system is stable When the forces are larger than the situation reversesand the slope of F is greater than zero, which means that the system becomesunstable As a consequence, for displacements that are larger than one-third

of the initial gap the mobile plate collapses (it is pulled-in) against the

fixed one, irrespective of the microspring design This also explains thereason why the equilibrium point is stable (it is positioned to the left ofand the other equilibrium point is unstable

The particular situation where Eqs (5.7) and (5.8) are valid is pictured inFig 5.3 Compared to the generic case of Fig 5.2, the actuation voltage Uneeds to be increased or the spring has to adequately be redesigned, in orderfor the spring force characteristic to be tangent to the electrostatic forcecharacteristic, as shown in Fig 5.3 By increasing the voltage for instance,the force-displacement curve representing the electrostatic actuation willtranslate upward until it becomes tangent to the spring characteristic

Figure 5.3 Single-point equilibrium in transverse electrostatic actuation and microspring

Example 5.1

A transverse electrostatic actuator is serially coupled to a spring ofstiffness Find the actuation voltage that will result in the stableequilibrium position being related to the pull-in position as: Knownare the following amounts:

As previously shown, the conditions for stable static equilibrium are:

The electrostatic force and the spring force are given in Eqs (5.2) and(5.3), respectively The value of the pull-in displacement is also given inthe second Eq (5.7) By combining these equations with the relationshipbetween and it is found that the voltage is U = 86.6 V

2.2 Flexure-Spring Microdevices

Flexure-spring microdevices are used as acceleration sensors inautomotive control systems of airbags, chassis or navigation monitoring Thesimplest microaccelerometer consists of a flexure hinge and a tip mass, aspictured in Fig 5.4 (a)

Figure 5.4 Flexure-hinge microaccelerometer: (a) side view with schematic configuration;

(b) detail with displaced proof-mass

By assuming that the acceleration of the monitored system actsperpendicularly to the flexure-hinge microaccelerometer, it is possible toevaluate this acceleration by means of a measured amount, such as thedeflection or the slope of the deformed flexure as shown next It can beconsidered that the inertia force acts at point 2, the center of the proof mass –Fig 5.4 (a), which means the flexure hinge is loaded at its tip 3 by the inertiaforce and the moment The slope and deflection at point 3 can be found

by using the compliance formulation of Chapter 2 as:

where the compliances above define any of the flexure microhinges that havebeen analyzed in Chapter 2 The inertia force and moment are:

The unknown acceleration a can be determined when either the slope or thedeflection of Eqs (5.10) can be measured directly (experimentally), namely:

or:

Example 5.2

Determine the external acceleration by means of a flexure-hingemicroaccelerometer (as the one sketched in Fig 5.4 (a)) whose gap ismeasured electrostatically

Solution:

An elementary electrostatic force can be formulated that corresponds

to a length dx (not shown in Fig 5.4 (b)) and to the gap g(x) This force is:

The variable gap g(x), as shown in Fig 5.4 (b), is:

The total electrostatic force can be calculated by integrating Eq (5.14) overthe length and by using Eqs (5.10) and (5.11), namely:

This electrostatic force is equal to the inertia force that is generated by theexternal acceleration and the mass of the proof mass, as given in the first Eq.(5.11) The acceleration a can be determined by solving the third-degreealgebraic equation that results by equating Eq (5.16) and the first Eq (5.11)

2.3 Rotary Microdevices

Rotary actuation and sensing, together with appropriate suspensions, areused in microgyroscopes for instance that are utilized to detecting changes inthe direction of rotation of navigational systems such as those implemented

in cars The rotary portion of a microgyroscope consists at its minimum ofactuation, sensing and suspension An example, similar to the one analyzed

by Geiger et al [1], is sketched in Fig 5.5, where two pairs of rotaryelectrostatic actuators (connected in parallel) are disposed symmetrically toensure balancing of the microdevice Two similar sensing pairs (also inparallel connection) are disposed 90° with respect to the actuator units Theouter mobile hub and the inner fixed post are connected by means of a spiralspring

Figure 5.5 Rotary electrostatic transduction with spiral-spring microsuspension

Other suspension solutions, with several microsprings, such as those of Figs.5.6 (a) and (b), are also possible.

Figure 5.6 Rotary electrostatic transduction with: (a) straight-beam microsuspensions; (b)

curved-beam microsuspensions

Application of a voltage differential between the fixed and mobileelectrodes of the two actuation units generates a couple that will rotate themobile hub The maximum rotation angle under static actuation is given bythe equilibrium between the actuation torque and the elastic restoring coupleproduced by the spiral spring The two sensing units will detect the rotationangle as a change in capacitance, as shown in Chapter 4 Comparing theangle predicted by capacitance reading to the angle that results from thetorque balance equation can give an insight on the actuation losses, asdetailed in the following example

Example 5.3

The microdevice pictured in Fig 5.5 operates in an environment with

and is actuated electrostatically by a voltage U = 80 V Thereadout units indicate a capacitance variation of

Consider that each transduction unit is formed of n = 10 gaps and that

and – see Fig 4.29 The spiral

Fig 3.43, (the cross-sectional dimensions) and Young’smodulus is 160 GPa Find the relative error in the rotation angle between themodel-predicted value and the actual value read by the sensing units

Solution:

The torque equilibrium in the position of static balance requires that theelectrostatic actuation torque be equal to the restoring torque produced by thespiral spring, namely:

where the actuation torque is twice the value determined in Eq (4.47) ofChapter 4 (there are two parallel actuators in Fig 5.5) The elastic torque can

be expressed as:

where is the rotation angle of the mobile hub and is the rotationcompliance of the spiral spring, which has been defined in Eq (3.137) ofChapter 3 for thin spiral springs Equation (5.18) uses the simplifyingassumption that the rotation compliance is simply the inverse of thecorresponding stiffness

By combining Eqs (5.17), (5.18), (4.47) and (3.137) together with thenumerical data of this example, the predicted value of the rotation angle isfound to be The capacitance change, as provided by the twosensing units, relates to the actual rotation angle according to Eq (4.49) ofChapter 4 (the total capacitance variation is twice the value given by Eq.(4.49) because there are two sensing units in parallel) and a value ofresults from the measurement The relative (percentage) error betweenthe model and actual rotation angles is therefore equal to

3 TWO-SPRING MEMS

Two springs can be coupled either in series or in parallel and theresulting stiffness is found as a combination of the individual springs’stiffnesses Figure 5.7 illustrates the models that give the equivalent stiffnessfor spring parallel/serial connection

The equivalent parallel and series stiffnesses, as well-known fromelementary mechanics, are calculated as:

Equations (5.19) are specified for linear springs, but they are also valid forrotary springs, which can similarly be coupled either in series or parallel Thesame equations can be extrapolated in design cases where more than twomicrosprings are connected in parallel/series

Figure 5.7 Two-spring connection: (a) Parallel; (b) Serial

3.1 Flexure-Spring Microdevices

The model of a MEMS comprising a mass physically supported by twosuspensions that enable linear motion about an out-of-the-plane axis z, issketched in Fig 5.8 (a)

Figure 5.8 MEMS with two flexure springs in parallel: (a) Configuration; (b)

Equivalent spring model

The device of Fig 5.8 (a) can be used as an electrical switch or amicroaccelerometer for instance The middle link (shuttle mass formicroaccelerometers) will displace about the z-direction through either