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Lecture #21 SPUR GEARS Course Name : DESIGN OF MACHINE ELEMENTS Course Number: MET 214

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To develop the relationships that characterize spur gears, consider a pair of wheels transmitting torque from one shaft to another via friction existing at a point of contact. As friction wheel #1 rotates, wheel #2 rotates due to friction between wheel #1 and wheel #2. The common point of contact has a velocity that can be expressed in terms of either friction wheel. where radius of friction wheel #1 angular speed of wheel #1 rads/ sec radius of friction wheel #2 angular speed of wheel #2 rads/ sec

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Friction wheels are prone to slippage. When transferring large amounts of power, slippage can cause shock and must be avoided. To avoid slippage, friction wheels can be modified by forming a periodic arrangement of protrusions and furrows around the periphery of each friction wheel to interlock (mesh) one wheel to another. The protrusions on one friction wheel mate with the furrows on the adjacent wheel to provide a mechanical means for interlocking the wheels and avoiding slippage. The protrusions are referred to as teeth, and the modified friction wheels are termed gears. When forming a gear, it is desirable to shape the teeth and/or form the teeth so that a pair of meshing gears retain the same speed and/or torque ratios that would be associated with the friction wheel implementation from which the gears are formed. The figure on the next slide shows how the friction wheels may be modified to produce a pair of mating gears. The friction wheels implementations for the gears shown in the figure are represented by circles referred to as pitch circles. The teeth of each gear extends above and below the corresponding pitch circle of the gear as shown in the diagram.

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The pitch circle of each gear contact each other at one point referred to as the pitch point which is designated as point P in the figure above. The pitch point P lies on the line connecting the centers of each gear. The radius of the pitch circle enables the rotational rate of the friction wheel implementation to be related to the tangential linear velocity at the pitch point.

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The following figure identifies additional gear teeth characteristics. A detailed description of the gear teeth will be presented in a subsequent lecture.

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When two gears mesh, the smaller gear is called the pinion and the larger the gear. Pitch diameter of pinion Pitch diameter of gear

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N = number of teeth uniformly distributed around the periphery of a gear. P = circular pitch : the distance from a point on a tooth of a gear to the corresponding point on the next adjacent tooth measured along the pitch circle. Accordingly, each of the expression above that involve P can be rearranged and equated to develop relationships between gears in mesh. eq page 363 in book by Mott

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Since the teeth and/or gears are designed to have the pitch circles contact at a single point in order to retain the desirable properties of friction wheels, the following relationships also hold for meshing gears. Equating the above expression to the previous expression involving teeth numbers results in the following relationships The above expressions can be utilized to develop alternative expressions involving torque transmitted between gear pairs as shown below.

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Teeth size as a function of diametral pitch are shown in the figure below.

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Several pairs of gears may be cascaded to form a gear train. A gear train is one or more pairs of gears operating together to transmit power. Frequently, as power is transmitted from one shaft to another, the particular speed/torque combination associated with the power needs to be adjusted. The speed/torque ratios existing from shaft to shaft may be adjusted by meshing gears with the appropriate ratio of teeth. As demonstrated by the example provided below, there are some techniques that can be used to simplify the determination of the speed of the output shaft of a gear train given the speed of the input shaft. For the example shown below, determine the speed of output shaft #3 if the input shaft has a speed of 1750 rpm clockwise as shown below.

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For the gear train shown below, compute TV and compute the speed of the shaft carrying gear E if the shaft carrying gear A rotates at 1750 rpm.

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