Similarity theory based method for MEMS dynamics analysis

Journal of Harbin Institute of Technology (New Series), Vol. 15, No.3, 2008Similarity theory based method for MEMS dynamics analysisu Gui-xian' , PENG Yun-feng' , ZHANG Xin2李瑰贤,彭云峰,张欣(1. School of Mechanical and Electronic Engineering, Harbin Instiute of Technology, Harbin 150001，China;2. Dept. of Computer Science and Engineering, Arizona State University, Arizona, A285287 ,USA)Abstract: A new method for MEMS dynamics analysis is presented, ased on the similarity theory. With thismethod, two systems ' similarities can be captured in terms of physics quantities/ govemed-equations amongstdifferent energsy fields , and then the unknown dynamic characteristis of one of the systems can be analyzed ac-cording to the similar ones of the other system. The probability to establish a pair of similar gystems amongMEMS and other energy systems is also discussed based on the equivalent between mechanics and electrics, andthen the feasibility of applying this method is proven by an example, in which the squeezed damping force inMEMS and the curent of its equivalent cireuit established by this method are compared.Key words: similarity theory; MEMS dynamics; equivalent between mechanics and electics; equivalent cir-cuitCLC number: TH164Document code: AArticle ID: 1005-9113(2008 )03-0312-05Micro-electro-mechanical systems (MEMS) has ex- tem(2]. The similar theory has broad application. Forperienced great developrnent in the past decades. Byexample, the theory can be used to guide the experi-now, some basic problems in MEMS have been solved ,ment and dala processes in heat transmission', andsuch as the micro-scaled fabrication of the micro-partsalso can be used to simulate the rillery system.and the function development. These issues ( the lawsBased on the similarity theory, a similar mappingthat govem the micro machine movement, the physicsmethod for MEMS’dynamic analysis is presented andproperty of micro parts and their dynamics川) are allits description can be described as follow.related to the dynamic analysis of micro-parts. Due toThere exist two energy systems 14 and I, whosethe small size scale of MEMS, there exist micro-scalephyical quantties are I, = {x,*n"xm} and I。effect and strong coupled energy-fields , which cause that= {y2***y.} (m and n are constants , which denotethe traditional methods can be no longer suit for the anumber of physical quantities). The characteristicsnalysis of the dynamics characteritics of MEMS. There-and analyzing method of I2 are assumed to known, whilefore, there need a new method to analyze the dynamics.In this paper, a similar mapping method is presented,those of 4 not. The governed equations of two systermsaref = f(x;,x,..xm) andf2 = f(yr2*yn). Ifwhich base on the introduction of similarity theory.The paper is organized as fllows: in the firstthere existi(i ≤m,n) similar govemed equation be-chapter , the theory of similar mapping method is given ,tweenI and l2, then a similar systems can be estab-the probability to establish similar system betweenlished and similar mapping relationship can be definedMEMS and other energy systems and the feasibility ofbetween1 and l2:the application of this method is discussed in the sec-h:4→l2ond and third chapter.and also the mapping relationship between correspond-ing physical quantities:Heory Backgroundπ:x→y k = 1,2...iSo a numerical operator ψ can be defined to relateAccording to the similaritytheory, systemsthe corresponding physical quantities between the simi-have similar features can constitute a pair of similarlar systems, which issystems'“. So, by the similar relationship between dif-x =ψ(r,) k = 1,2,iferent energy systems or different physical quantities ，similar system can be established and the unknowncharacteristics in one of the systems can be analyzedthe中国煤化工路rlinihe anderistics of physicalmutually by the known ones of the other similar sys-quan:HTHCN M H G the correspondingReceived 2005 -12 -09.●312●Joumal of Harbin Insitue of Technology (New Series)， Vol. 15, No. 3, 2008similar ones in 1| after it have been analyzed with ma-characteristics of a circuit are govemed by Kirihhofanture method. The similar reverse mapping relationshiplaws, which are Kirhhofan current laws ( KCL) andand numerical operator areKirihhofan voltage laws ( KVL).π":I2(x)→1(yn),k = 1,2,..iye =ψ~'(x}),k = 1,2,..i|iIn a word, the similar mapping method can beconcluded as: Based on the similar relationship be-4 ].itween physical quantities/ governed equations of differ-ent energy systems, a similar system can be established(a)Scheme of forces ( torques)(b)Scheme of eurenunents troughand then similar mapping relationship and numericaloperator between different physical quantities can beFig. 1 Similarity of forces ( torques) and currentdefined. Then the unknown characteristics of one ofthe similar systems can be achieved by the correspond-Based on the Kirihhofan laws, the similar systeming similar quantities in the other system.of MEMS and an electric circuit will to be establishedin two ways, of which one is based on the KCL, and2 The Probability to Establish a Similar Systemthe other is based on KVL.Between Mems and Other Energy Systems2. I. I Similarity based on classical Newton mechanicslaws and Kirihhofan curent lawsMEMS is a complicated system, which involvesThe characteristics of nod current in a circuit ( seemulti-energy fields, such as mechanical field , electricFig. 1(b)) is governed by Kiribhofan current lawsfield, fluid field and magnetic field. When analyzing(KCL), that the sum of curents in and out a nod e~the dynamic characteristics of micro-parts, it usuallyquals zeroinvolves solving multi physics problems. When beingEi=0(2)act as an actuator or sensor, MEMS is often integratedin an electric circuit system. So, some scholars like towhich i denotes the jth branch current.establish its equivalent circuit model and get the equiv-Obviously, there exists similar relationship be-alent voltage and current, which then will be insertedtween the physical quantities in Eqs. (1) and (2),into a system-level simulator such as SABER to simu-which relales forces ( torques) in a mechanical systemlate the MEMS dynamicss.In this paper a eircuitand current in a circuit. Then it is easily to create thesystem is chosen to verify the establishment probabilitysimilar mapping relationship π of the physical quanti-of similar system between MEMS and other systems.ties between the mechanical system and the nod systemHere, the problem will be discussed in two facets.in a circuit. When the number of forces ( torques) e-2.1 Similarity Based on Classical Newton Me-quals that of branch current in and out the nod ( that ischanics Laws and Kirihhofan Lawsh =jorl = j), the similar system is a complete similarThe basic description physical quantities in a me-system2). Then a similar mapping relationship can bechanical system are forces ( torques) and displace-constructed and the similar mapping operator ψ be-ment.Without losing its generality， the forcestween current and forces( torques ) can be defined( torques) are chosen as the description physical quan-i; =ψ(F) orij = ψ(T;)tity in mechanical system here. Current and voltage areFrom above, when the physical quantities ’char-chosen as the representative quantities of an electricacteristics of one system are known, it is undoubtlycircuit system.that those of the corresponding quantities of the otherIn a mechanical system, the structure is oftensimilar system are also known.simplified as a mass point m, on which is appliedThe similar mapping relationship above can be ex-forces and torques ( as shown in Fig. 1 (a)). Thetended to other relationship. For a mechanical system offorces and torques must satisfy the classical Newton 1 DOF, the solid mechanical structure can be simplifiedmechanics laws to maintain the equilibrium of the sys-as a spring mass- damping system( as shown in Fig. 2).tem, which iswhich F: is the hth forces and T, is the lth torques ap-ZF. =0and 2T=0(1)?Trplied on the mass m.中国煤化工An electric cireuit is often modeled as a network,whose branch corresponding to an electric component.MHCNMHGFigs“oeeme uspHIou uyuarnic model ofand a nod to a connection joint in a circuit. In the net-mechanical systemwork, one nod connects at less three branches. The，313●Joumnal o Harbin Insinte o[ Technology (Nevw Series), Vol. 15, No.3, 2008The forces apalled on a mass point must satisfied2.1.2 Similarity sysems based on Kirihhofan tolagelaws (KVL)8R.=0(3)The characteristics of voltage in a closed eircuitwhichF: are forces applied on the mass caused by(as shown in Fig. 4(a)) are govermed by KVL, thatdamping, spring and extemal forces separalely.the sum of the voltage-drop across all components in aBy the natural characteristics of each component,closed-circuit is zero.Eq. (3) can be rewitten as2u=0(7)md'x(C)+γdx()+kx(1) = F(I) (4)which uy is voltage drop across the jh component.d2which x(t) is the displacement in the direction of load;Eqs. (1); and (7) resemble on their correspond-m is the flective mass of the mechanical system; k ising forces ( torques) and voltage, and then the similarthe siffness of spring; y is the damping cofficient andmapping relationship can defined between the mechani-F(t) denotes the load.cal system and the cireuit, and also the numerical op-There are many energy systems whose characteris-erator based on the similarity + of.volage and forcestics of physical quanties are similar with that described(torque).Similarly, by the natural characeristics of theirby Eq. (4), and the RLC circuit is the one, whichcomprise of resistance, capacitance and inductance.constitutive component, Eq. (7) can be rewilten asFrom the similar relationship between forces and currentL出+Ri+亡fid=u()given above, an equivalent circuit is given, shown inFig.3, whose characteristics of corresponding quanitiesBy derivation, it becomesare similar with that of the mechanical system.ld+rd+ti= d()(8)Obviously, Eqs. (4) 'and .(8) constitute a pair ofsimilar systems, which based on the similarity of forceand voltage. The corresponding similar physical quan-tities between the similar systems are shown in Tab.2.Tab.2 The similar mapping relationship of correspondingFig.3 Equivalent dircuit based on the sinilarity of forcephysical quantitiesand currentMEMSMas mcolfroieply coefficient kDampingBy Kirihhofan current laws (KCL) , the govermedCireuitInductangg Conduetance CopacityCurrentequation of current out the source in the circuit issystemI/Ccu() + +气Ju()d=() (s) 2.2 Simiarity betwee Veloity Impedance otMechanical System and Circuit Systemwhich C is the capaciance , R is the resistance andL isthe inductance; i is the current out the elctricalVelocity impedances of a simple spring, dampingsource; U is the voltage of the source.and mass component areBy derivation, Eq. (5) becomesZ=气di(1)C最+RH+u=(6)z, =γObviously, Eqs. (4) and (6) resembles eachzm = jomother, by which the systems described constitute a pairwhich w is the round frequency; j is the plurality.of similar system. The corresponding similar physicalThe velocity impedance of the 1DOF mechanicalquantities between the similar systems are shown insystem shown in Fig.2 isTab.1.Z. =一=jwm+γ+Tab.1 The similar mapping relationship of correspondingwhich x denotes the velocity of the mass.The impedances in a RLC eireuit component areSiffness Displacement中国煤化工CircuitCaparcity cConduciance. InduetionVolage uYHCNMH G .ystem1/R1/L‘Gl = JwFrom above, there exists similar mapping relation-●314.Journal of Harbin Institue of Technology (New Series)， Vol. 15, No. 3, 2008ship between mechanical components.A simple RLC circuit of the 1 DOF mechanical3 The Application Feasibility of the Method insystem can be constructed based on the similar map-MEMSping relationship of impedance( as shown in Fig. 4(b)). In the RLC circuit, the electric impedanceIn this paper, an example was presented to dem-across the rouse of the circuit network isonstrate the application feasibility of this method inMEMS. By the similar mapping method, the equivalent2.=一=jwL+R+;circuit of the squeezed damping forcies' between twoThen the similarity relationship π3 between twoperpendicular moving microstructures, which is a gen-systems can be defined aseral part in microsensors, was constructed ( as shownin Fig. 5), and the damping forces and current in theequivalent circuit are compared.The corresponding relationships between physicalThe efeet of the damping forces between the mi-quantities of a simple mechanical system and circuitcroplates can be modeled as a lumped parameter mod-system based on the impedance similarity are shown inel, which constitute a series of damping and springTab. 3.components in parallel-connected( as shown in Fig.5).The spring and damping in the branch are'年h.，年k，年k(a) Voltage drmp across component in a cireuitFig.5 Lumped parameter model of squeezeddamping forceR6lwP.(mn)°π(g +z)768(lw)'ηo(b) Impedance of component in a cireuitC.s= (mn)2(mP +nw2 )π(g+z) .which n,m are odds; I,1 are the width and length ofFig.4 Scheme of a simple RLC circuitthe plate respectively; P。is the atmosphere pressure ;Tab.3 The corresponding mapping relationship based onηei is the effective viscosity; g is the height and z is theimpedances of mechanical system and electric sys-vibration amplitude of plate.In the lumped parameter model, the velocity im-Veloeity impedanee Velocity impedance Velocity impecdancepedance of the m ,nth branch isZm. =while theof spring Zg(k)of damperZ,(r) of mass Z[(m)Cireuilmpedance ofImpedance ofimpedance of a circuit branch is Y.nObvious-sytem capacity Z(1/C) resistance Zx(R) inductance z(L)ly, the velocity impedance of the spring and dampingThe similar mapping operator ψ3 can be defined asin series-connected and the admittance of the circuit(Zc,Zp,Z) = 4s(Z,Z,,Zm)take on similarity( as shown in Fig. 6).Then the similar relationship between a mechani-cal system and a circuit system can be establishedbased on the impedance similanity and the similar sys-tem between MEMS and circuit system can be con-一却structed.Also, the similar system between MEMS and aFig. 6 Similar relationship between mechanicalcircuit system can be constructed based on the similari-中国煤化工ty between the admittance of a mechanical componentand that of a circuit component.YHCNMHGybetweentheim-In a word, the similar relationship between MEMSpedance in series-connected spring and damper and theand other energy systems can be established.admittance in parallel-connected capacitance and re-●315●Journal of Harbin Institute of Technology (New Series)，Vol. 15，No.3, 2008sistance, the equivalent circuit can be constructed( asApparently, once the dynamics of the current inshownin Fig. 7), which has the same number of paral-the circuit through the rouse in Fig.7 is known, that oflel-connected capacitance and resistance components asthe squeezed force on the microplates is also known bythat of the series-connected spring and damper compo-the similar mapping method. Therefore, the methodnents in the lumped parameter model.presented in this paper can be used in analyzing theThe impedance in series- connected mechanicalMEMS' dynamics.branchis 7 = 2一while the admitance in paral-IConclusionlel-connected of the circuit branchisY= E Y, So theIn this paper, a new method was presented to an-impedance of the mechanical branch isalyze the MEMS’dynamics, which is based on thesimilarity theory. The possibility and feasibility of thisZm. =R2 -(k.. - jacm.n)method for MEMS were venfied. Due to the method isThe admittance of the circuit branch isfar from mature and 80 can't be applied efficiently.The discussion here focuses on the theory. So furtherY.. =一+jwCm.work should be emphasized on its improvement and theFrom above, the similar mapping relationship πapplication.between two systems can be defined asReferences:[1] Wen Shizhu, Ding Jianning. Study on the fundamental de~sign issues of micro elecchanical systems ( MEMS).Chinese Joumal of Mechanical Engineering, 2000,6(7):and the similar mapping operator ψm.n of the physical39 -42 (in Chinese).[2] Zhou Meili. Similarity Theory. Beijing: China Machinequantities in m ,nth branch isPress, 1998 ( in Chinese).(-Co)= 4..[3] Yang Shiming, Tao Wenquan. Themal Conduetivity. Bei-jing: Higher Education Press, 1998 ( in Chinese).The total force applied on the micro-plate is F =[4] Feng Jinfu, Cao Shuzi. The similarity theory for gun sys-2 Fm. by the lumped parameter model, and the tolaltem. Jounal of Nanjing University of Science and Technolo-gy, 1996, 20(5) :405 - 408( in Chine).current out the soure in Fig.7isI= E lm., then the[5 ] Timo Vejola, Heikki Kuisma. Equivalent-circuit modal ofthe squeezed gas film in a silicon accelerometer. Sensorsdamping forces act on the plate can be thought to equaland Actuator, 1995,A 48: 239 -248.with the current I.[6] Vemuri s, Fedder G K, Mukherjee T. Low order squeefilm model for simulation of MEMS devices. Proe 3rd Int'lConf on Modeling and Simulation of Microsystems ( MSM2000. San Diego: CA, 2000.0 205 -209.000~Fig.7 Equivalent circuit of damping force中国煤化工MYHCNMHG.316●.