Progress ReportOptimization of an inertialgrade micro-accelerometerMarch 29, 2001Sung-Joon HwangAlexis Perez-DuarteBrian Trease12ContentsIntroduction 51 Suspension mechanism 101.1 Nomenclature ........................... 131.2 Model ............................... 141.3 Constraints ............................ 151.4 Initial Results ........................... 192 Force multiplier 222.1 Topology Optimization ...................... 232.1.1 Principles ......................... 232.1.2 Application ........................ 252.2 Analytical Investigation ..................... 262.2.1 Analytical Model ..................... 262.3 FEA ................................ 303 Tuning forks 303.1 Nomenclature ........................... 323.2 Model ............................... 343.3 Results ............................... 383.3.1 Simplified model ..................... 403.3.2 More sophisticated model ................ 42References 423AbstractMicro-accelerometers are MEMS-scale devices that are able to detect accel-erationsby allowing a proof mass (PM) free to move in one direction to actupon a force sensor. Based on the measured force and the mass of the PM,acceleration is determined. This project is suggested by and based on a pre-vious ME555 project, Optimal Design of a Force-Multiplication Device forResonant Accelerometers, by Michael Farina and Brian Jensen. Their workspecifically addressed a force-multiplier component, while we seek to optimizethe entire system. The goal of their project was to maximize force magnifi-cation, which contributes to the overall system goal of increased resolution.However, their work concentrated on a single component, and did notconsider the effect that the mutual interaction between the different partsmight have on the global result. Their conclusion evokes the possible trade-offs when their component is integrated in the whole mechanism. This iswhat leads us to believe that the optimization of the system as a whole canbring completely new results, and yield a resolution previously unseen inmicro-accelerometers.This high degree of resolution should make possible an inertial grademicro-accelerometer. The output signal over time should be accurate enoughto integrate twice, giving precise position information. Combined with twoother orthogonal accelerometers, any arbitrary path of the object they arefixed to can be precisely determined. The small size would allow the device4to be integrated with mobile GPS systems, increasing their functionalityand reliability. Previous attempts at increased resolution include the originaldouble-edged tuning fork design which provided precision to 11 mg (g =acceleration due to gravity). An improved levered design increased this to 89µg. Jensen and Farina’s force-multiplier theoretically increases this furtherby a factor of about 160 times.IntroductionSeveral types of micro-accelerometers exist and we are investigating the typebased on a double-edged tuning fork (DETF). There are three major com-ponents to this system. The proof mass is attached to a suspension, whichallows it to move back and forth in only one direction. As the PM moves dueto outside accelerations on the system, it pushes on the ends of the DETF,which are two parallel suspended beams that are attached to the ground ontheir other end. Two electrodes on either side of the beam make-up a closedloop circuit that charge the forks and cause them to vibrate. A simplifiedsystem without the force-multiplier is shown in the figure 1.The frequency of the beams change as the PM pushes on them. Oneelectrode senses the vibration and the control system seeks to keep the DETFvibrating at its natural frequency. The changes in frequency are recorded anddirectly correlate to acceleration.In addition to the peculiarities of the micro scale, the accelerometer in-5Figure 1: Double Ended Tuning Fork (DETF).6volves technologies from different fields, including mechanics and electronics.This will translate into tradeoffs when put together. One of the most basicones is the MEMS-scale condition, since —as can be expected— the perfor-mance of a mechanical system increases as its area gets bigger (which wasproved by Jensen and Farina). Another important issue is the tradeoff be-tween resolution and linearity of the response. It has been noticed that thegain in sensitivity is often echoed by a loss of linearity in the response, whichin turn reduces the bandwidth of the device. This tradeoff will repeatedly bea source of concern in the different components :• in the force multiplier as shown by Jensen and Farina,• in the suspension system, we can expect nonlinearities in the event oflarge amplitude deformations, which are likely to occur if we are lookingfor high force output/input ratios.• in the tuning forks, where we will have mechanical nonlinearities forlarge amplitude oscillations, and electrical nonlinearities coming fromthe electrostatic actuation.We can now describe more qualitatively the subsystems and the linksbetween them.Suspension1The suspension would be a spring system, indicated in figure 2, shownbelow. This is one possible topology; much research has been done in1part done by Brian7Figure 2: Suspension mechanism.compliant micro-suspensions suggesting other possibilities. The lengths,cross-sectional areas, and ground connections of the beams are designvariables. The objective would be maximum flexibility (or minimumstiffness) in the desired direction and maximum stiffness in all otherdirections. Notice that the proof mass is strongly related to the de-sign of the suspension system, where it will be used as either a designvariable or a parameter. The limiting factors are not only the MEMSrules of thumb but also the fact that the whole proof mass is by thinbeams, and must not touch the base. Lateral stiffness (which we needto minimize) will add to the stiffness of the force multiplier and thatof the tines.8Force Multiplier2The solution by Jensen and Farina was based on a pseudo-rigid bodycompliant mechanism. It is interesting to reconsider the topology theyused, since their design space was fixed and ours is a system variable,linked to the optimization of the suspension system. However, we haveused the topology synthesis homogenization method to generate theoutline of a new design. Redefining this topology in terms of beams,we have parametrized it and will optimize using beam theory or possi-bly finite element methods in conjunction with optimization software.The