434 DESIGN OF MACHINERY CHAPTER 9 Pitch point Gear Sheave Sheave Gear 3 Belt (a)Gear train (b)Belt train FIGURE 9-3 Gear and belt trains each have an equivalent fourbar linkage for any Instantaneous position The principal drawbacks to the rolling cylinder drive (or smooth belt)mechanism are its relatively low torque capability and the possibility of slip.Some drives require absolute phasing of the input and output shafts for timing purposes.A common example is the valve train drive in an automobile engine.The valve cams must be kept in phase with the piston motion or the engine will not run properly.A smooth belt or rolling cyl- inder drive from crankshaft to camshaft would not guarantee correct phasing.In this case some means of preventing slip is needed. This usually means adding some meshing teeth to the rolling cylinders.They then become gears as shown in Figure 9-4 and are together called a gearset.When two gears are placed in mesh to form a gearset such as this one,it is conventional to refer to the Pinion smaller of the two gears as the pinion and to the other as the gear. 9.2 THE FUNDAMENTAL LAW OF GEARING Conceptually,teeth of any shape will prevent gross slip.Old water-powered mills and windmills used wooden gears whose teeth were merely round wooden pegs stuck into the rims of the cylinders.Even ignoring the crudity of construction of these early examples of gearsets.there was no possibility of smooth velocity transmission because the geome- try of the tooth"pegs"violated the fundamental law of gearing which,if followed,pro- vides that the angular velocity ratio between the gears of a gearset remains constant throughout the mesh.A more complete and formal definition of this law is given on p. 436.The angular velocity ratio (my)referred to in this law is the same one that we de- rived for the fourbar linkage in Section 6.4 and equation 6.10(p.257).It is equal to the ratio of the radius of the input gear to that of the output gear. Gear mv=oau=±n=土dm (9.1a) in dowt FIGURE 9-4 功T= 0n=tot=士 out (9.1b) An external gearset out m din