EE 245: Introduction to MEMSModule 9: Energy MethodsCTN 10/19/091Copyright © 2009 Regents of the University of CaliforniaEE C245 – ME C218Introduction to MEMS DesignF ll 2009Fall 2009Prof. Clark T.-C. NguyenDept. of Electrical Engineering & Computer SciencesUniversity of California at BerkeleyEE C245: Introduction to MEMS Design LecM 9 C. Nguyen 9/28/07 1University of California at BerkeleyBerkeley, CA 94720Lecture Module 9: Energy MethodsLecture Outline• Reading: Senturia, Chpt. 10• Lecture Topics:ªEnergy MethodsªEnergy Methods( Virtual Work( Energy Formulations( Tapered Beam Example( Estimating Resonance FrequencyEE C245: Introduction to MEMS Design LecM 9 C. Nguyen 9/28/07 2EE 245: Introduction to MEMSModule 9: Energy MethodsCTN 10/19/092Copyright © 2009 Regents of the University of CaliforniaEnergy MethodsEE C245: Introduction to MEMS Design LecM 9 C. Nguyen 9/28/07 3More General Geometries• Euler-Bernoulli beam theory works well for simple geometries• But how can we handle more complicated ones?• Example: tapered cantilever beam•Objective: Find an expression for displacement as a function •Objective: Find an expression for displacement as a function of location x under a point load F applied at the tip of the free end of a cantilever with tapered width W(x)50% taperWEE C245: Introduction to MEMS Design LecM 9 C. Nguyen 9/28/07 4xyEE 245: Introduction to MEMSModule 9: Energy MethodsCTN 10/19/093Copyright © 2009 Regents of the University of CaliforniaSolution: Use Principle of Virtual Work• In an energy-conserving system (i.e., elastic materials), the energy stored in a body due to the quasi-static (i.e., slow) action of surface and body forces is equal to the work done by these forces …y• Implication: if we can formulate stored energy stored energy as a function of the deformation of a mechanical object, then we can determine how an object responds to a force by determining the shape the object must take in order to minimizeminimize the differencedifference UU between the stored energy and the work done by the forces:EE C245: Introduction to MEMS Design LecM 9 C. Nguyen 9/28/07 5U = Stored Energy - Work Done• Key idea: we don’t have to reach U = 0 to produce a very useful, approximate analytical result for load-deflectionMore Visual Description …EE C245: Introduction to MEMS Design LecM 9 C. Nguyen 9/28/07 6EE 245: Introduction to MEMSModule 9: Energy MethodsCTN 10/19/094Copyright © 2009 Regents of the University of CaliforniaFundamentals: Energy Density• Strain energy density: [J/m3]ª To find work done in straining material• Total strain energy [J]:ªIntegrate over all strains (normal and shear)EE C245: Introduction to MEMS Design LecM 9 C. Nguyen 9/28/07 7ªIntegrate over all strains (normal and shear)Bending Energy Densityy(x) = transverse displacementof neutral axisxNeutral Axis• First, find the bending energy dWbendin an infinitesimal length dx:yEE C245: Introduction to MEMS Design LecM 9 C. Nguyen 9/28/07 8EE 245: Introduction to MEMSModule 9: Energy MethodsCTN 10/19/095Copyright © 2009 Regents of the University of CaliforniaEnergy Due to Axial Loadx• Strain due to axial load S contributes an energy dWstretchin length dx, since lengthening of the different element dx (to ds) results in a strain εxyEE C245: Introduction to MEMS Design LecM 9 C. Nguyen 9/28/07 9Shear Strain Energy• See W.C. Albert, “Vibrating Quartz Crystal Beam Shear ModulusEE C245: Introduction to MEMS Design LecM 9 C. Nguyen 9/28/07 10,gQyAccelerometer,” Proc. ISA Int. Instrumentation Symp., May 1982, pp. 33-44EE 245: Introduction to MEMSModule 9: Energy MethodsCTN 10/19/096Copyright © 2009 Regents of the University of CaliforniaApplying the Principle of Virtual Work• Basic Procedure:ª Guess the form of the beam deflection under the applied loadsª Vary the parameters in the beam deflection function in order to minimize:iiijjuFWU∑∑−=Sum strain energiesAssumes point loadEE C245: Introduction to MEMS Design LecM 9 C. Nguyen 9/28/07 11ª Find minima by simply setting derivatives to zero• See Senturia, pg. 244, for a general expression with distrubuted surface loads and body forcesDisplacement at point loadExample: Tapered Cantilever Beam• Objective: Find an expression for displacement as a function of location x under a point load F applied at the tip of the free end of a cantilever with tapered width W(x)Adjustable parameters: minimize U50% taperxyWEE C245: Introduction to MEMS Design LecM 9 C. Nguyen 9/28/07 12• Start by guessing the solutionª It should satisfy the boundary conditionsª The strain energy integrals shouldn’t be too tedious( This might not matter much these days, though, since one could just use matlab or mathematica3322)( xcxcxy +=EE 245: Introduction to MEMSModule 9: Energy MethodsCTN 10/19/097Copyright © 2009 Regents of the University of CaliforniaStrain Energy And Work By F(Bending Energy)(Using our guess)EE C245: Introduction to MEMS Design LecM 9 C. Nguyen 9/28/07 13Tip DeflectionEWh3Find c2and c3That Minimize U• Minimize U → basically, find the c2and c3that brings U closest to zero (which is what it would be if we had guessed correctly)•The c2and c3that minimize U are the ones for which the The c2and c3that minimize U are the ones for which the partial derivatives of U with respective to them are zero:•Proceed:EE C245: Introduction to MEMS Design LecM 9 C. Nguyen 9/28/07 14Proceed:ª First, evaluate the integral to get an expression for U:EE 245: Introduction to MEMSModule 9: Energy MethodsCTN 10/19/098Copyright © 2009 Regents of the University of CaliforniaMinimize U (cont)• Evaluate the derivatives and set to zero:• Solve the simultaneous equations to get c2and c3:EE C245: Introduction to MEMS Design LecM 9 C. Nguyen 9/28/07 15The Virtual Work-Derived Solution• And the solution:• Solve for tip deflection and obtain the spring constant:EE C245: Introduction to MEMS Design LecM 9 C. Nguyen 9/28/07 16• Compare with previous solution for constant-width cantilever beam (using Euler theory):13% smaller than tapered-width caseEE 245: Introduction to MEMSModule 9: Energy MethodsCTN 10/19/099Copyright © 2009 Regents of the University of CaliforniaComparison With Finite Element Simulation• Below: ANSYS finite element model withL = 500 μmWbase= 20 μm E = 170 GPah = 2 μmWtip= 10 μm• Result: (from static analysis)ª k = 0.471 μN/m• This matches the result from energy
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