940

THEORY AND PRACTICE OF ENGINEERING.

Book II.

angular velocities of these two points will be in the constant given ratio (—-^) of the radii Bd^

A B, whatever their absolute velocity may be.

A machine composed only of the two circles A and rf, which should revolve and touch atthe point B would only transmit the motion of the first circle to the second by simplefriction at the point of contact, and as soon as the contact should cease, which would happen,in a short time, the transmission of motion would no longer take place.

Of the Plane Epicycloid .—"When two circles which touch are in the same plane, and onerolls on the other without ceasing to touch, a certain point in the movable circle describesa curve called a plane epicycloid. The movable circle may roll either on the exterior orinterior of the fixed one; in the latter case, if the movable circle has a radius of the fixedcircle for its diameter the epicycloid would become a right line ; this right line is the radiusof the fixed circle which passes through, and in which it is touched by the movable circle,considered in its first position.

Let B be the point in which the movable circle in its first position A II K B touches thefixed circle BEFG, in another position ACD; it touchesthe fixed circle in D, and taking the arc D C of the samelength as B D, the point C, the extremity of the arc D C, willbe one of the points of the line described through the movablepoint B of the circle A II K B ; further, this point will be onthe radius A B. Let us suppose that it is at C ; beyond thisradius the angle B A D has for its measure either the entire arcB D, or half the arc C D. This arc C 1) is only half the radiusof B D ; then in order that it should have twice the number ofdegrees of the arc B I), it is necessary that the two arcs shouldbe of the same length as D B. Then these two arcs whichbelong to the same circle are equal, and the points C and C'are the same ; whence it follows that the epicycloid describedthrough the point B of the circle A II K B is reduced to a right line AB.

When the movable circle is of infinite radius, it becomes a right line; and the planeepicycloid a development of the fixed circle; if the radius of the fixed circle is infinite,the epicycloid becomes a cycloid.

Roemer , a Danish mathematician, was the first to adopt the epicycloidal curve to theteeth of wheels, in 1674, about which time De Lahire published a work upon thesame subject, and demonstrated that if a tooth of either a wheel or pinion be formed by aportion of an exterior epicycloid, described by a generating circle of any dimensions what-ever, the tooth of its follower will be properly formed by a portion of an interior epicycloiddescribed by the same generating circle. The object here was to obtain a uniform pressureand velocity throughout the machine, so that the wheels, whatever might be their position,should act uniformly and equally, and that the surfaces of the teeth touching at a pointshould roll over each other and avoid all friction.

The cycloid and epicycloid have been treated on by numerous writers, and Camus inparticular has applied their principles to the forms of the surfaces of the teeth of wheels andpinions acting against each other under different circumstances, as the wheel and trundle,the wheel and pinion, the rack and pinion, conductor or conducted, spur or bevel gear, andothers. De Lahire thought the involute of a circle the best of the exterior epicycloids,and considered the generating straight line as a curve of infinite radius, which it is whenapplied to a pinion acting on a rack.

M. Ilachette investigated this question with great ability, and decided the form of curvesmost applicable to the teeth of wheels: and that the mechanical effect w’hich one wheelproduces upon another should be the same in all positions, for which it is necessary thatthe line perpendicular to the surfaces of the teeth at the point of contact should intersectthe line joining the centres at a fixed point that divides the line into two parts, the ratio ofwhich is the mechanical power.

The teeth ought to work a little before and after the line of centres, and some mathe-maticians have preferred a union of the epicycloid with the hypercloid to effect this moreperfectly, as their line of action is nearly perpendicular to the radius, which causes lessfriction as well as strain upon the axles.

The teeth of wheels have been made so small since the introduction of iron fortheir manufacture, that their form is not so important; such excellence of workmanship isarrived at, that they are now set out with the template and dividers only. The practicalrule made use of by millwrights for finding the depth of the teeth from the bottom to thepitch line, and from thence to the top of the tooth, is simply to multiply the pitch by 5,and divide by 9, and vice versa.

Sometimes the curves of the exterior and interior surfaces of the teeth are set out by atracer, fixed in the radius line of one of two pieces of board, the edges of which are madeto resemble the primitive circles of the wheels: then by fixing a template to one edge, and

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Fig. 1235.