\\HKEL-AND-AXI.i:. 



WIIKKI. AM) AXI.K. 



tnrmrilly tn tlio province of applied mechanics. Before closing thi 

 article, h.-urvrr. it if ovMhtiAl to allude cunorily to some of the 

 mechanical condition* involved in the application of wheels. 



The aimplmt manner in which those contrivances for the train- 

 minion of power are used, i the one known by the UTIII of the wheel 

 anfl axle. [WiiEKL AKD-AxLE.] In the mndJau the power is applied 

 to the axle by means of a cranked lever revolving in ;i circular path, 

 instead of by a wheel ; but the principle of the action of this form of 

 lerer U precisely the same u that of the wheel, with only the difference 

 that the moments of inertia of the moving machinery are slightly 

 changed. [WINDLASS.] In trtad-vKerlt the power is obtained by 

 causing the men or animals who act upon the machinery to exercise 

 tlieir i-tHi-t, l.y tin- application of their weight, u]x>n the periphery of 

 the wheel. When equable motion u required to be produced by a 

 wheel animated by a variable power, it in obtained by the interposition 

 of a cone, or of some such contrivance for increasing the leverage of 

 the power in proportion as the power itself diminishes ; as, for instance, 

 in the spiral springs and fusees of watch-work. 



The power of a combination of two cog-wheels is ascertained by 

 multiplying tin: distance at which the power is applied from the centre 

 of the first wheel, by the radius of the second wheel ; and dividing 

 that sum by the sum of the distance at which the resistance acts from 

 the centre of the second wheel, multiplied by the radius of the first : the 



rlient will represent the ratio of the power to the resistance it is 

 to overcome. In a combination of any number of teethed wheels 

 the power of the system may be ascertained by taking the radii of the 

 wheels as the even terms of a series, and the distances at which the 

 power and resistance act from the centres of their respective wheels as 

 the odd terms (or the intermediate ones) of the series ; then the pro- 

 duct of the odd terms, divided by the product of the even terms, will 

 represent the ratio of the power to the resistance. The even terms 

 will in this cose represent the flyers or drin-n, and the odd ones the 

 foilomrt, and the product of the former will give the velocity of the 

 power, whilst that of the latter will give the velocity of the weight 

 or resistance. 



Very good practical rules, and examples for their application, of the 

 relations of the various parts of a system of wheel- work are to be 

 found in the Memorandum-book of Mr. Telford, inserted in his 

 ' Biography,' and reprinted in the ' Engineer's Pocket Book ; ' but 

 none of these easy practical solutions of the mechanical problems 

 involved in this branch of applied mechanics, can dispense the engineer 

 from the study of their principles. These are discussed at considerable 

 length in such works as those previously mentioned, and in Moscley's 

 ' Mechanics applied to the Arts ; ' his work on ' Engineering and 

 Architecture ; ' in Worr's ' Dynamics ; ' Weisbach's ' Mechanics of 

 Machinery ; ' BorgnU, ' Traitc complet de Meconique appliquce aux 

 Arts;' Dupin's ' Geometric et Mccanique des Arts;' Lanz et Bdtan- 

 court, ' Essoi sur la Composition des Machines,' &c. ; and to them the 

 student is earnestly referred. 



WHEEL-AND-AXLE, is a machine consisting usually of a cylinder 

 to which a wheel is firmly united, so that the mathematical axes 

 of both are coincident. The wheel and cylinder are of wood or 

 metal, and the diameter of the former is greater than that of the 

 Utter. 



A cylinder on the circumference of which are fixed exteriorly boards 

 whose planes, if produced, would pass through the axis, and which 

 (being turned by the force of running water, or by the weight of meu 

 in the act of stepping from one board to the next above it) is employed 

 to raise a heavy body by means of a rope passing over a smaller 

 cylinder on the some axis, as in the treadmill, is a simple machine of 

 this kind : the same may be said of a hollow cylinder which, with its 

 axle, is made to revolve by men or animals walking in the direction of 

 its circumference, in its interior surface. The capstan, the windlass, 

 and the helm-wheel of a ship are only so many different forms of the 

 same class of machines. Frequently also the axle is made to carry a 

 wheel with teeth on its circumference, in order that, by revolving, 

 motion may be communicated to machinery : such ore the wind and 

 water mills which are employed for grinding corn. 



When it is required to exhibit the mechanical properties of the 



Fig. 1. 



wheel-and-axle, a weight representing the moving power in applied at 

 OLC extremity of a string which at the other extremity is attached to 



and pane* round the circumference of the wheel ; and a weight, repre- 

 senting the resistance to be overcome, is applied in like manner at one 

 end of a string which puses round the axle or cylinder. 1 

 (in /'/. 1) be a section passing through the wheel and cylinder perpen- 

 dicularly to their common axis, and let CA, or CA', and CB be the 

 semi-diameters of the circles in that section : let r represent the 

 moving power and w a weight to be raised, or held in equilibrio ; A P 

 or A' i 1 ', and BW, being the directions of the strings to which those 

 weights are attached; and for simplicity, let these lines be in .-tie 

 plane and coincident with tAugenU to the circles at A. or A', and at n. 

 Here it iseviileiitthatthemechiuiic.il power of the wheol-ond-axle U 

 the same as that of a lever of the first kind ; for (the thickness of the 

 ropes and the weight and inertia of the materials being disregarded) 

 the forces P and w acting perpendicularly to the arms i A and . 

 effect is the same as if those forces were applied immediately at the 

 extremities of the straight line AH, or of the I nut line A'CII, and c 

 being the fulcrum or )xiint of support, we have, by the nature of thu 

 lever, in the case of equilibrium, 



r : w : : BC : AC (=A'c),orr=w. . 



The wheel and axle has manifestly however a great advantage over 

 the simple lever, since the weight w may bo raised to any i 

 which is consistent with the lengths of the ropes, by winding the ropo 

 round the axle. 



If the power r or r' do not act in the direction of a tangent to 

 the circle, but in some other, as AT"; then letting fall CD IT]>CII 

 dicularly on I'"A, produced if necessary, we have, by the lever, 



p" : w : : BC : CD. 



If the ropes to which the weights are attached have sensible thick- 

 nesses, and it is thought proper to take those thicknesses int > 

 consideration, the ropes may be conceived to be reduced to their 

 mathematical axes, and these to pass over the circumferences of the 

 wheel anil cylinder at distances equal to the semidiauieters : thus, if 

 r and R be the semidiameters of the ropes passing over those circum- 

 ferences, respectively, we obtain, in the case first supposed, 



p : w : : BC + B : AC*r. 



If it be required to determine the pressures on the supports of a 

 wheel-and-axle when the weights applied to it arc in equilibrio, and 

 the whole machine is at rest, the investigation may be conducted in 

 the following manner : Let the weight of the wheel be represented 

 by A and that of the cylinder by B ; also let H and N (in Jig. 2) be the 



Fig. 2. 



C.P 



on which the two pivots rost ; then 4 is evidently the pressure 

 supported on each of the point* M and N, arising from the weight of 

 the cylinder alimt. Let the weight A be supposed to act at c, the 

 centre of the wheel, and let CM = m, cn = ; then, by mechanics. 



m + : m : : A : pressure at N, 



tn 



m + n l 



in like manner, 



m + n 



A expresses the pressure at H; each of these 



pressures arising from the weight of the wheel 



In order to find the pressures arising from the weights P and w, the 

 sum of those weights must bo considered as applied at a point >. in 

 the axis of the machine, where that axis would bo cut by a vertical 

 plane passing through the common centre of gravity of the two 

 weights : let c and c' be points in which the axis is cut by vertical 

 planes pawing through the respective centres of gravity of P and w ; 

 then, in order to find u, we have, by mechanics, 



p + w 



/ IM,'\ 



:cc ::P:CO(= r + w ); 



p. co' 

 hence +in, or 



p.cc' 

 and n T ... , or 



p + w 



p.cc' 

 r-rw ' 



= MO, 



-ay. 



