8-7 



WHEEL-CUTTING. 



WHEEL-CUTTING. 



693 



therefore, again by mechanics, 



: : p + w : pressure on 



p + w 



p.cc / +m(r + w)\ 

 m + n I* 



n(P+w)-p.cc' 

 and, m like manner, expresses the pressure on M. 



Consequently the whole pressure on H is 



) p.cc' 



JB + 



m + n 



and on N, is 



m + n 





If the wheel and cylinder are in a state of motion about their mathe- 

 matical axis, the pressure on the supports will evidently be diminished 

 by the force with which the common centre of gravity of the weights 

 p and w tends to descend ; the value of this force is investigated in 

 treatises on dynamics. 



If two wheels and cylinders are connected together by a string 

 badf (in Jif/. 1), or by teeth in the circumferences, as in most forms 

 of rack-work, the ratio between the power I 1 and the resistance w, in 

 the case of equilibrium, may be determined by the same rule as would 

 be employed if those weights were at the opposite extremities of a 

 double lever of the first or second kind. For the power p may be 

 conceived to be applied at A perpendicularly to the semidiameter CA, 

 and it will be in equilibrio with a resistance at a, perpendicular to 



ca, which may be expressed by P . : let this be represented by p. 



Now this force at a may, in consequence of the string passing round 

 the axle c B and the circumference of the wheel n s, be conceived to be 

 a moving power applied at A perpendicularly to E& ; and this will be in 

 equilibrio with a resistance w at r, acting perpendicularly to F E, which 



Kb 

 may be expressed by p ; therefore, substituting in it the above 



CA F.'< 



value of p, we have w = p. . . And in like manner may the 



relation between the power and resistance be found, in the case of 

 equilibrium, whatever be the number of wheels and axles. 



It is to be understood, in the above description, that the axles of 

 the two wheels MN and us are supposed to lie parallel to one another 

 and to the horizon ; and that the parts of the string li a df are in a 

 vertical plane perpendicular to those axes, in order to avoid the 

 reductions which would be necessary on account of a loss of power 

 resulting from an oblique action of the forces at a and b. The forces 

 acting in AP and BW, or FW, are also supposed to be exactly or very 

 nearly in one vertical plane, in order to avoid the strain on the axle 

 which would otherwise take place. [MATERIALS, STRENGTH OF.] 



If the string passing over the circumference of the wheel H s and 

 the axle CB were to cross itself, as represented by the lines ltdaf, the 

 relation between the powers would be the same as before, but the 

 weight ttf" would be raised in the direction w'r' instead of F. 



It is easy to perceive that (as in the lever and other mechanical 

 powers) the spaces described by the weights B and w, in a given time, 

 when in motion, are to one another in the inverse ratio of those 

 weights ; for the spaces described are respectively equal to the lengths 

 of the strings which pass over the circumferences of the wheel-and-axle 

 in the given time ; and these lengths are proportional to the circum- 

 ferences, or radii, that is, inversely as the weights acting at the 

 circumferences. 



Hence the advantage in the wheel-and-axle may be increased either 

 by increasing the radius of the wheel, or by diminishing that of the 

 axle. In the latter case, of course, the axle would soon become too 

 weak to sustain the weight. This is beautifully avoided by the use of 

 innnd axle, one part of which is of smaller radius than the other. 

 One end of the cord carrying w is wound round the thicker, and the 

 other, in a contrary direction, round the thinner part. As P descends, 

 some of the cord unwinds from the thinner axle, while another part is 

 wound up round the thicker ; but as the latter part, of course, exceeds 

 the former in length, the weight is raised in this proportion. Thus 

 we may have an axle of virtually vanishing radius, and may, con- 

 sequently, almost indefinitely increase the power, but, of course, only 

 at the expense of time. 



Taking the measurements as in fy. 1, and representing the radius of 

 the l/iirter axle by BC, and that of the thinner by a quantity o, less 

 than BC; since the whole weight w is supported by the two parts of 

 the cord, the tension of the cord = Jw. 



Hence, by mechanics, taking moments about c, we get, 



p . CA+ {wo=Jw . CB 

 .'. P.CA = $W(CB a) 

 .'. P: w : : 4(CB a) : CA. 



WHEEL-CUTTING, a term applied to a particular branch of prac- 

 tical mechanics, which comprehends the modes of cutting the teeth in 

 the wheels used by watch and clock makers and for other mechanical 



ARTS A!CD SCI. DIV. VOL. VIII. 



purposes. The engines used for this purpose vary in their construc- 

 tion according to the wants or caprice of the artists who use them. 

 We shall content ourselves with giving a description of the engine 

 commonly employed, with a few remarks on the kind of tools used 

 for cutting the spaces between the teeth, which operation is usually 

 termed cutting the teeth of a wheel, although in reality the teeth are 

 those portions of the metal which are left standing. We shall, however, 

 employ the common phrase, as it will perhaps be best understood by 

 all who feel an interest in the art. 



Description of the engine commonly used : AAAA a strong frame 



of cast-iron consisting of two parallel plates, the stouter the better, 

 firmly connected together, but so that the plates are from 3 to 5 inches 

 apart, to allow the dividing-plate PP to revolve between them. The 

 plate PP is fixed firmly to the axis c (about 8 or 9 inches long), which 

 works at its upper end in a collar d, in the upper plate, and its lower 

 end in the centre of a screw, R : this axis c has a hole down from its 

 upper end, about three-fourths of its length, to receive the smaller 

 axes, arbors, or pinions of the wheels which are to be cut. E is a hori- 

 zontal slide, of which the vertical part E' is formed into a dove-tail, on 

 which slides the vertical slide F, to which is securely attached a frame 

 o, having two projecting sides through which pass two screws, one of 

 which is seen at H. These screws have female centres to receive the 

 ends of the arbor which carries the cutter I : _;', a pulley on the cutter- 

 arbor which receives the band by which motion is communicated 

 to the cutter I ; I, the handle of a lever, whose centre of motion is at 

 f on a piece projecting from the back of the fixed dovetail F.', to which 

 is attached the connecting-rod m, for depressing the slide F, and 

 thereby passing the cutter through the wheel ; o, a piece attached to 

 the back of the dovetail E', for the purpose of fixing the spring n, one 

 end of which is attached to the slide F, and operates to bring up the 

 slide after the cutter has passed through the wheel. The slide F. is 

 for the purpose of bringing the cutter to the requisite distance from 

 the centre of the wheel to be cut, and has a screw, not seen in the 

 drawing, for the purpose of setting it fast when brought by the screw 

 Q to its proper place. The dividing- plate pp has on its surface a 

 number of concentric circles, which occupy that portion of the plato 

 nearest its circumference : these circles are each accurately divided 

 into such a number of equal parts as are likely to be suitable for the 

 wheels required to be cut ; the outer circles, being the largest, gene- 

 rally contain high numbers, such as 400, 360, 192, 168, 160, 150, 140, 

 136, 130, Ac., and with these almost any common number of teeth can 

 be cut. Firmly fixed on a moveable centre or joint attached to the frame 

 of the engine is an index , capable of a motion on its joint parallel to 

 the plate PP, and having at its end a pin t, with a rather long conical 

 point. On the plate pp, at the intersection of each division with its 

 corresponding circle, is drilled a hole ; and if th&se holes are drilled 

 quite through the plate all the better. The pin k is attached to the 

 index M by a moveable piece which is acted upon by the screw s, and 

 serves the purpose of shifting the plate p p any small quantity less than 

 the distance of a single division on the plate ; and 10 is a nut to set 

 the pin k fast in any required position. 



The index, being placed with its conical point It in one of the holes 

 in any circle by means of slit v (say that of 360 divisions), is screwed 

 fast by the screw t, and the elasticity of the index keeps it sxifficiently 

 tight in the hole to prevent the plate and arbor from moving round : 

 if the end of the spring or index be now lifted up by the hand, and the 

 plato be moved round till the next division or hole in the same circle 



3 M 



