of a wheel intercepted between similar points of consecutive teeth (the chords of which arcs are called the pitches of the teeth), it is evident that all these arcs must be equal, since the teeth are all equal and similarly placed; so that each tooth of either wheel, as it passes through its contact with a corresponding tooth of the other, carries its pitch line through the same space CD, over the point of contact C of the pitch lines. Since, therefore, the pitch line of the one wheel is carried over a space equal to CD, and that of the other over a space equal to cd by the contact of any two of their teeth, and since the wheels revolve by the contact of their teeth as they would by the contact of their pitch circles at C, it follows that the arcs CD and cd are equal. Now let r, and r, represent the radii of the pitch circles of the two wheels, then will 2, and 27r, represent the circumferences of their pitch circles; and if n, and n, represent the numbers of teeth cut on them respectively, then CD= and cd= 2πη Therefore the radii of the pitch circles of the two wheels must be to one another as the numbers of teeth to be cut upon them respectively. Again, let m, represent the number of revolutions made by the first wheel, whilst m, revolutions are made by the second; then will 2nr,m, represent the space described by the circumference of the pitch circle of the first wheel while these revolutions are made, and 2r,m, that described by the circumference of the pitch circle of the second; but the wheels revolve as though their pitch circles were in contact, therefore the circumferences of these circles revolve through equal spaces, therefore 2r,m,= 2πr,m,; The radii of the puch circles of the wheels are therefore inversely as the numbers of revolutions made in the same time by them. Equating the second members of equations (227) and (228) The numbers of revolutions made by the wheels in the same time are therefore to one another inversely as the numbers of teeth. 205. In a train of wheels, to determine how many revolutions the last wheel makes whilst the first is making any given number of revolutions. When a wheel, driven by another, carries its axis round with it, on which axis a third wheel is fixed, engaging with and giving motion to a fourth, which, in like manner, is fixed upon its axis, and carries round with it a fifth wheel fixed upon the same axis, which fifth wheel engages with a sixth upon another axis, and so on as shown in the above figure, the combination forms a train of wheels. Let n,, ng, ng,... nap represent the numbers of teeth in the successive wheels forming such a train of p pairs of wheels; and whilst the first wheel is making m revolutions, let the second and third (which revolve together, being fixed on the same axis) make in, revolutions; the fourth and fifth (which, in like manner, revolve together) m, revolutions, the sixth and seventh m,, and so on; and le the last or 2pth wheel thus be made to revolve mp times whilst the first revolves m times. Then, since the first wheel which has n, teeth gives motion to the second which has n, teeth, and that whilst the former makes m revolutions the latter = makes m, revolutions, therefore (equation 229), and since, while the third wheel (which revolves with the second, makes m, revolutions, the fourth makes m, revolu tions; therefore, = Similarly, since while the fifth wheel, which has n, teeth, makes m, revolutions (revolving with the fourth), the sixth, which has n, teeth, makes m, revo lutions; therefore Mp n2p-1 mp-1 n2p M/3 Multiplying these equations together, and striking out factors common to the numerator and denominator of the first member of the equation which results from their multiplication, we obtain The factors in the numerator of this fraction represent the numbers of teeth in all the driving wheels of this train, and those in the denominator the numbers of teeth in the driven wheels, or followers as they are more commonly called. If the numbers of teeth in the former be all equal and represented by n,, and the numbers of teeth in the latter also equal and represented by n,, then Having determined what should be the number of teeth in each of the wheels which enter into any mechanical combination, with a reference to that particular modification of the velocity of the revolving parts of the machine which is to be produced by that wheel, it remains next to consider, what must be the dimensions of each tooth of the wheel, so *The reader is referred for a more complete discussion of this subject (which belongs more particularly to descriptive mechanics) to Professor Willis's Prin ciples of Mechanism, chap. vii., or to Camus on the Teeth of Wheels, by Haw kins, p. 90. that it may be of sufficient strength to transmit the work which is destined to pass through it, under that velocity, or to bear the pressure which accompanies the transmission of that work at that particular velocity; and it remains further to determine, what must be the dimensions of the wheel itself consequent upon these dimensions of each tooth, and this given number of its teeth. 206. To determine the pitch of the teeth of a wheel, knowing the work to be transmitted by the wheel. Let U represent the number of units of work to be transmitted by the wheel per minute, m the number of revolutions to be made by it per minute, n the number of the teeth to be cut in it, T the pitch of each tooth in feet, P the pressure upon each tooth in pounds. U Therefore nT represents the circumference of the pitch circle of the wheel, and mnT represents the space in feet described by it per minute. Now U represents the work transmitted by it through this space per minute, thereforemnT represents the mean pressure under which this work is transmitted (Art. 50.); The pitch T of the teeth would evidently equal twice the breadth of each tooth, if the spaces between the teeth were equal in width to the teeth. In order that the teeth of wheels which act together may engage with one another and extricate themselves, with facility, it is however necessary that the pitch should exceed twice the breadth of the tooth by a quantity which varies according to the accuracy of the construction of the wheel from th to th of the breadth.* Since the pitch T of the tooth is dependant upon its breadth, and that the breadth of the tooth is dependant, by the theory of the strength of materials, upon the pressure P which it sustains, it is evident that the quantity P in the above equation is a function of T. This functiont may be assumed of the form *For a full discussion of this subject see Professor Willis's Principles of Mechanism, Arts. 107-112. See Appendix, on the dimensions of wheels. where c is a constant dependant for its amount upon the nature of the material out of which the tooth is formed. Eliminating P between this equation and the last, and solving in respect to T, The number of units of work transmitted by any machine per minute is usually represented in horses' power, one horse's power being estimated at 33,000 units, so that the number of horses' power transmitted by the machine means the number of times 33,000 units of work are transmitted by it every minute, or the number of times 33,000 must be taken to equal the number of units of work transmitted by it every minute. If therefore II represent the number of horses' power transmitted by the wheel, then U=33,000H. Substituting this value in the preceding equation, and representing the constant 33,000c by C3, we have The values of the constant C for teeth of different materials are given in the Appendix. 207. To determine the radius of the pitch circle of a wheel which shall contain n teeth of a given pitch. Let AB represent the pitch T of a tooth, and let it be supposed to coincide with its chord AMB. Let R represent the radius AC of the pitch circle, and n the number of teeth to be cut upon the wheel. Now there are as many pitches in the circumference as teeth, therefore the angle ACE subtended by each pitch is represented by 2 Also T=2AM 2AC sin. ACB=2R sin. |