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major failure mechanism for microscale devices is not fatigue failure but wear between contacting surfaces A detailed study conducted by Sandia National Laboratories on polycrystalline silicon micromechanical transmissions concluded that the major failure mechanism for operating MEMS was the wear of the rubbing surfaces Friction and wear can be most effectively minimized or avoided if elastically exible structures are used This is probably the reason for the widespread use of simple elastic structures in microtransducers However, intermittent surface contacts are unavoidable when applications such as the ones described in the previous section involve sophisticated mechanical movement Cams and gears suffer from the problem of friction and wear but provide the designer with numerous possibilities As the tribological issues at the microand nanoscale are better understood, cams are likely to play an increasing role in MEMS devices Some past and current studies on friction and wear pertinent to the successful operation of cams are described next Two early studies on in situ friction measurement on electrostatically actuated micromotors reported friction coef cients in the range of 021 to 038 for polysilicon on silicon nitride and 025 to 035 for polysilicon on silicon In the latter study, the micromotors stopped running after 075 to 1 million revolutions and this was attributed to wear Another study used a polysilicon wobble micromotor in which the wear between the rotor and the central hub changes the gear ratio of the motor This was used to quantify wear A very detailed three-year study (Sandia 2000) considered the effects of humidity, temperature, shock, vibration, and storage The device on which this study focused was called a microengine, which as described earlier has many polysilicon parts that rubbed against each other This study outlined the observed failure modes in operating and nonoperating conditions and recommended some design rules to avoid such failure Even though the study was based entirely on a particular type of polysilicon microdevice, their ndings could be generalized to any MEMS device with rubbing surfaces or surfaces in close contact Wear was observed to be the major failure mechanism The debris and the asperities caused by wear lead to momentary and intermittent sticking and eventually permanent adhesion and seizure The three-body wear in which debris gets caught between the rubbing surfaces was found to be a major contributor to wear It was also found that humidity helps to mitigate wear by acting as a lubricant Therefore, relative humidity of 30 percent to 60 percent is recommended for operation at room temperature Reduction of three-body wear by removing the initial debris is recommend but it may not always be practical Minimizing rubbing surfaces by design and minimizing the impact forces at the rubbing surfaces by controlled actuation were recommended to reduce wear All of these are relevant when cams are used in MEMS because rubbing surfaces are inevitable in cams Several studies have been conducted to study friction at the microscale In one such study (Bhushan, 1999), a silicon nitride (Si3N4) probe whose tip had a radius of 50 nm (50 10-9 m) was scanned over a 1 1 mm sample area at a scanning speed of 5 mm/s in the load range of 10 to 150 nN Several samples of single-crystalline, polycrystalline, and oxide-coated silicon were used It was found that the coef cient of friction was between 002 and 004 for all the samples This is considerably smaller than the coef cient of friction at the macroscale, which was found to be about 018 for the same samples but using a Si3N4 ball of radius 3 mm Two reasons are cited for this First, the indentation hardness and elastic modulus are higher at the microscale and this reduces wear Second, there is a smaller apparent area of contact, which contributes less to plowing Even though the coef cient of friction is an order of magnitude smaller than the macroscale value, it should be remembered that friction forces at the microscale are still dominant because of the scaling effect mentioned earlier.

lim 5x = 5 [ lim x] = 5 3 = 15

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This chapter began by stating that the scope of cams has somewhat expanded to include gears and ratchets and other types of devices that involve intermittent and sustained contact between rigid bodies The main reason for this is the limited applications where sophisticated mechanical motions are required This has to do with the evolution of the MEMS eld, where the emphasis has been on microtransducers (sensors and actuators) The slower development of MEMS applications involving mechanical transmission may also be due to the limited capabilities of micromachining techniques compared to the macroscale machining techniques As the need for mechanical transmission at the microscale increases, linkages, gears, and cams will appear as much as beams and membranes do at present It is also possible that future developments involving mechanical manipulation might require compliant mechanisms where there is no relative rigid-body motion but complex motions are not precluded At this juncture, it is important to note that nature seems to avoid sliding contacts at the microscale and prefers compliant designs where there are only elastic deformations However, engineered designs can overcome the limitations and make sliding contacts more viable even at the microscale Whether rigidbody motions are good or elastic deformations are good de nitely depends on a number of factors These factors include materials, manufacturing processes, and an understanding of friction and wear and how to overcome them at the microscale

Allen, JJ, and Schriner, HK, Micromachine Wedge Stepping Motor, Micro-Electro-Mechanical Systems Symposium at the International Mechanical Engineering Congress and Exhibition, DSCVol 66, ASME Press, New York, 1998 Bhushan, B, ed, Handbook of Micro/Nano Tribology, Boca Raton, Fla, CRC Press, 1999 Kovacs, GTA, Micromachined Transducers Sourcebook, WCB McGraw-Hill, Boston, 1998 Madou, M, Fundamentals of Microfabrication, CRC Press, Boca Raton, Fla, 1998 National Research Council, Microelectromechanical Systems: Advanced Materials and Fabrication Methods, National Research Council Report, National Academy Press, Washington, DC, 1997 Petersen, K, Silicon as a Mechanical Material, Proceedings of the IEEE, 70 (2): 420 57, 1982 Sacks, E, and Barnes, SM, Computer-aided Kinematic Design of a Torsional Ratcheting Actuator, Proceedings of the Fourth International Conference on Modeling and Simulation of Microsystems, Computational Publications, Boston, 2001 Tanner et al, MEMS Reliability: Infrastructure, Test, Structures, Experiments, and Failure Modes, SAND2000-0091, Sandia National Laboratories report, Albuquerque, NM, 2000 Trimmer, WS, ed, Micromechanics and MEMS: Classic and Seminal Papers to 1990, IEEE Press, New York, 1990

Based on experience with y = 1/ x, this function has the same shape, it is just displaced (or translated) 1 unit to the right In y = 1/ x, x = 0 is the asymptote line, but in y = 1 / (x - I ) , x = 1 is the asymptote line Follow the logic of the limit calculations and verifL the graph as shown in Fig 2-3 Graph y = 1/ x2 using limit concepts and notation

Of course limx 3 9 = 9 Putting all this information into equation ( ) gives lim 4x 3 7x 2 + 5x 9 = 108 63 + 15 9 = 51

161 BACKGROUND 529 162 INTRODUCTION 531 163 CAMSHAFT MATERIALS 531 1631 Sliding Follower Camshafts 532 16311 Cast Iron 532 16312 MIiscellaneous Materials 534 1632 Roller-Follower Camshafts 534 16321 Steel 534 16322 Ductile Iron 535 16323 Miscellaneous Materials 16324 Roller Follower 535 164 CAMSHAFT CURVES 535 165 CAMSHAFT PROFILE GEOMETRY 166 HARMONIC ANALYSIS 537 167 ENGINE PERFORMANCE 541 168 SUMMARY 541

Thus we may apply the Pinching Theorem Obviously lim g( x) = lim h( x) = 0

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