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drive is rotated by 25 counterclockwise, the pawl engages the inner teeth of the ring gear and rotates it counterclockwise When the comb drive is deactivated, its torsional spring attached to the comb drive restores it to its start orientation, which disengages the pawl The antireverse prevents the gear from rotating clockwise The outer involute teeth of a ring gear drive a load A SEM image of the prototype is shown in Fig 157b Although this design intended that the gear advance by one tooth for one cycle of the oscillating angular comb drive, the fabricated microprototype exhibited some overrun caused by the impact between the pawl and the ring gear The impact made the gear jump by two or more teeth Therefore, an antioverrun and a riser were added in the modi ed design

Fig 2-2 This relationship xy = 1 or y = l / x is best understood in the context of its graph Numbers can be assigned to x, and y values calculated but note how the concept and language of limits make graphing so much easier

Let S be the set of all people and T be the set of all people Let f be the rule that assigns to each person his or her brother Is f a function

FIGURE 157 Torsional ratcheting actuator (a) schematic (courtesy, Elisha Sacks, Purdue University) (b) microfabricated prototype (courtesy, Sandia National Labs MEMS, S&T Department, wwwmemssandiagov)

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In this case f is not a function For many people have no brother (so the rule makes no sense for them) and many people have several brothers (so the rule is ambiguous for them)

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In the modi ed design, a riser is attached to the xed frame and an antioverrun is attached to the comb drive with a revolute joint A sequence of images shown in Fig 158 illustrates how the modi ed design works The pawl rotates with the comb drive, engages the inner teeth of the gear (Fig 158b), and rotates it counterclockwise (Fig 158c) The antioverrun rotates with the driver, rides over the riser (Fig 158b), engages the gear (Fig 158c), and prevents further gear motion When the comb drive is deactivated, torsional springs restore it to its start position, which disengages the pawl and the antioverrun (Fig 158d and 158e) As before, the antireverse prevents the gear from rotating clockwise This design modi cation was analyzed and veri ed using a kinematic analysis and tolerencing software (Sacks and Barnes, 2001) 1563 Mechanical Lock I Spiral Groove Cam A mechanical locking device has been developed by Sandia researchers for applications where access to or activation of some crucial components should not occur until a certain sequence of events takes place In the device shown in Fig 159, a pin runs through a maze in a speci c way to unlock another component As shown in the gure, the large gear has a spiral groove that has radial and circumferential segments The gear is driven by Sandia s electrostatic microengine (described in Sec 1573) through a pinion that can be seen in Fig 159 The pin is attached to a linear electrostatic comb drive to allow radial movements The rotation of the large gear and the radial movement of the pin are to be coordinated in a speci c manner to let the pin run through the spiral groove from the beginning to the end Only then can the gear rotate by some angle to release a latch to activate.

(d )

Let S be the set of all people and T be the set of all strings of letters not exceeding 1500 characters (including blank spaces) Let f be the rule that assigns to each person his or her legal name (Some people have rather long names; according to the Guinness Book of World Records, the longest has 1063 letters) Determine whether f : S T is a function

Refer to Fig 2-2 during this discussion First consider positive values The point x = 1, y = 1 is so easy to calculate it should not be ignored The curve goes through this point Now as x is made a larger and larger positive number, y approaches zero, but remains positive This can be expressed in a simple sentence,

FIGURE 158 Schematic illustration of the modi ed design of Fig 157a with an antioverrun (courtesy, Elisha Sacks, Purdue University)

This f is a function because every person has one and only one legal name Notice that several people may have the same name (such as Jack Armstrong), but that is allowed in the definition of function

FIGURE 159 A micromechanical lock with a spiral groove cam (courtesy, Sandia National Labs MEMS, S&T Department, wwwmemssandiagov)

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