Among other things, it is necessary thatyou actually be able to draw the teeth on apair of meshing gears .You should understand, however, that youare not doing this for manufacturing orshop purposes.Rather, we make drawings of gear teeth toobtain an understanding of the problemsinvolved in the meshing of the matingteeth.
1st it is necessary to learn how toconstruct an involutes curve. Divide thebase circle into a number of equal parts, andconstruct radial lines OA0, OA1, OA2, etc.Beginning at A1, construct perpendicularsA1B1, A2B2, A3B3,etc. Then along A1B1 layoff the distance A1A0, along ay off twice thedistance A1A0, etc., producing pointsthrough which the involute curve can beconstructed.
When two gears are in mesh, their pitchcircles roll on one another without slip-ping.Designate the pitch radii as r1andr2andthe angular velocities asω1andω2,respectively.Then the pitch-line velocity isV=|r1ω1| =|r2ω2|Thus the relation between the radii on theangular velocities is ω1ω2=r2r1
VELOCITY RATIO OF GEAR DRIVE 2 N2 d1 velocity ratio (n) = 1 N1 d2 d= Diamter of the wheel N =Speed of the wheel ω= Angular speed
Suppose now we wish to design a speedreducer such that the input speed is1800rev/min and the output speed is 1200rev/min.This is a ratio of 3:2; the gear pitch diameterswould be in the same ratio,For example, a 4-in pinion driving a 6-in gear.The various dimensions found in gearing arealways based on the pitch circles.
Suppose we specify that an 18-toothpinion is to mesh with a 30-tooth gearand that the diametral pitch of the gear-set is to be 2 teeth per inch. The pitchdiameters of the pinion and gearare, respectively,d1 =N1P=182=9 ind2 =N2P=302=15 in
The first step in drawing teeth on a pair ofmating gears the center distance is the sumof the pitch radii, in this case 12 in. So locatethe pinion and gearcentersO1andO2, 12 inapart . Then construct the pitch circles ofradii r1andr2.These are tangent at P, thepitch point. Next draw line ab, the commontangent, through the pitch point. We nowdesignate gear 1 as the driver, and sinceit is rotating counter-clockwise, we draw aline cd through point P at an angle φ to thecommon tangent ab.
The line cd has three names, all of whichare in general use. It is called the pressureline.The generating line , and the line of action.It represents the direction in which theresultant force acts between the gears.The angle φ is called the pressureangle,and it usually has values of 20 or25◦, though 14*12 was once used.
Next, on each gear draw a circle tangent tothe pressure line. These circles are thebase circles. Since they are tangent to thepressure line, the pressure angle determinestheir size.The radius of the base circle isrb =rcosφwherer is the pitch radius.The addendum and dedendum distances forstandard interchangeable teeth are, aswe shall learn later, 1/P and 1.25/Prespectively. Therefore, for the pair of gearswe are constructing,
a=1p =>12=>.500inb=1.25p =>1.252 =>0.625Next, using heavy drawing paper, orpreferably, a sheet of 0.015- to 0.020-in clearplastic, cut a template for each involute, beingcareful to locate the gear centers prop-erlywith respect to each involute. A reproductionof the template used to create some of theillustrations for this book. Note that only oneside of the tooth pro-file is formed on thetemplate. To get the other side, turn thetemplate over. For some problems you mightwish to construct a template for the entiretooth.
To draw a tooth, we must know the tooththickness and the circular pitchp=πPp=π2=1.57 int =p2t=1.572=0.785 in The portion of the tooth between theclearance circle and the dedendum circleincludes the fillet. In this instance the clearanceis c=b−a =0.625−0.500 =0.125 in
Referring again to Fig. the pinion withcenter at O1 is the driver andturnscounterclockwise. The pressure, orgenerating, line is the same as the cordused in to generate the involute, andcontact occurs along this line. The initialcon-tact will take place when the flank ofthe driver comes into contact with the tip ofthedriven tooth. This occurs at point wherethe addendum circle of the dri-ven gearcrosses the pressure line. If we nowconstruct tooth profiles through point aand draw radial lines from the intersectionsof these profiles with the pitch circles to the
As the teeth go into mesh, the point ofcontact will slide up the side of the drivingtooth so that the tip of the driver will be incontact just before contact ends. The finalpoint of contact will therefore be wherethe addendum circle of the driver crossesthepressure line. This is point bin Fig. 13–12. By drawing another set of toothprofilesthroughb, we obtain the angle of recess.
The sum of the angle of approach and theangle of recess for either gear is calledthe angle of action.The line ab is called the line of action.
The base pitch is related to the circu-lar pitchby the equationpb =pccosφwhere pb is the base pitch.Figure shows a pinion in mesh with aninternal,orring, gear.Note that bothof the gears now have their centers ofrotation on the same side of the pitch point.Thus the positions of the addendum anddedendum circles with respect to the pitchcircle are reversed;
the addendum circle of the internal gear liesinside the pitch circle. Note, too,from Fig.that the base circle of the internal gear liesinside the pitch circle near the addendumcircle.
Thus the pitch circles of gears really do notcome into existence until a pair of gears arebrought into mesh.Changing the center distance has no effect onthe base circles, because these wereused to generate the tooth profiles. Thus thebase circle is basic to a gear. Increasing thecenter distance increases the pressureangle and decreases the length of the lineof action, but the teeth are still conjugate, therequirement for uniform motion transmis-sion is still satisfied, and the angular-velocityratio has not changed.