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To calculate the bending moment which each of the arms of a wheel has to resist, let M be the greatest moment of the effort transmitted by the wheel; n, the number of arms; r, the geometrical radius of the wheel, from the axis to the pitch-line; x, the length of an arm, from the boss to the rim; M', the bending moment on each arm; then two cases may be distinguished.

I. If the arms and rim are made in one piece, either by casting or by welding;

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and this formula is applicable also to wheels like the paddle wheels of a steamer, in which wrought-iron arms are rigidly bolted or rivetted both to the boss and to the rim.

II. If the arms are cast along with segments of the rim, and fastened into sockets in the boss;

M' =

Mx nr

This second formula is based on the where an arm is inserted into the boss bear any part of the bending moment. but it is an error on the safe side.

.(2.)

supposition that the joint cannot safely be trusted to This is not strictly correct,

The transverse section of the arms is to be adapted to bear safely the working moment thus found, by the aid of the rules of Article 437, pages 514 to 516.

In Case I, the greatest bending moment is exerted on each arm at two points, close to the rim and close to the boss respectively; in Case II, the greatest bending moment is exerted close to the rim.

Another way of adjusting the strength of the arms to the moment exerted through them is as follows:-Having fixed the figure and dimensions of an arm according to convenience, calculate the working moment to which it is adapted; let this be denoted by M'; then the number of arms required is given by the following formulæ :

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The real working modulus of stress for cast iron in these calculations should not exceed 4,500 lbs. on the square inch, or 3.2 kilogrammes on the square millimètre; and for wrought iron,

9,000 lbs. on the square inch, or 63 kilogrammes on the square millimètre.

472. Centrifugal Tension in Wheels and Pulleys.—The rim of a wheel, moving with the velocity v, is subjected to a centrifugal v 2 tension whose amount is equal to the weight of a length of that

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rim (including teeth, if it is a toothed wheel). This is resisted by the tenacity of the rim at its smallest cross-section (or by the fastenings of the rim, if it is made in segments), partly assisted by the tenacity of the arms. Each of the arms has to bear its own centrifugal tension, which, at a point close to the boss, is equal to the weight of a length of the arm itself expressed

v2

by (1 − 2-3); r being the radius of the wheel, and that of the

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boss; and on the whole, it is an error on the safe side to make the rim strong enough to bear its own centrifugal tension without aid from the arins. This fixes a limit of safety as to speed, for a rim of a given material and construction. Let m be the ratio in which the mean sectional area is greater than the effective sectional area, f the greatest working tensile stress, and let w be the heaviness of the material; then the greatest proper velocity, being that which produces the stress, is given by the formula :

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The modulus, for cast iron may be taken at 800 feet, or 244

mètres; so that when m = 1, as in a pulley, or a fly-wheel without teeth, we have v = 160 feet, or 49 mètres, per second nearly. Let a cast-iron spur fly-wheel be so designed that m = 2; then v = 113 feet, or 34 mètres, per second nearly.

w'

The modulus, for wrought-iron wheel-tyres that are not welded, but rolled out of perferated discs, may be taken at 2,400 feet, or 730 mètres. (See page 585.)

473. Tension-Arms of Vertical Water-Wheels.-The weight of a great vertical water-wheel, of the construction introduced by Hewes, is hung from a cast-iron boss by means of wrought-iron tension-rods. The load is distributed amongst the rods which, at a given instant, point obliquely or vertically downwards from the boss; and the amount of the tension on each rod is proportional nearly to the square of the cosine of its inclination to the vertical.

1

The mean value of that square is nearly; and, at any instant, half the total number of rods point downwards. Hence, let ƒ be

the intensity of the tension on the rods which point vertically downwards; S, the sectional area of a rod; n, the number of rods; W, the load; then

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For the working value of f, we may take from 9,000 to 10,000 lbs. on the square inch; or from 63 to 7 kilogrammes on the square millimètre. (See A Manual of the Steam-Engine and other Prime Movers, Article 155, pages 181, 182.)

474. Braced Wheels. Instead of transmitting power between the boss and the rim of a wheel by means of the resistance of the arms to bending, the arms may be so placed as to transmit power by their direct tension and thrust; and for that purpose they must not be radial, but must lie in the direction of tangents to a circle of a radius somewhat smaller than that of the boss. Let r" denote the radius of this circle; n, the number of arms; M, the greatest moment transmitted; then the amount of the greatest stress along an arm is given by the following expression :

M nn"

-

This is tension for one half of the arms, and thrust for the other half; and their dimensions are to be determined by the rules of Article 459, pages 537 to 539.

475. Levers, Beams, and Cranks have usually one or two arms, as the case may be; and each arm is in the condition of a bracket; the greatest bending moment being exerted at that cross-section which traverses the fulcrum, or axis of motion. In the crank of a steam engine, the greatest bending moment is identical with the greatest twisting moment exerted on the shaft to which the crank is fixed.

In ordinary cases it is unnecessary to add anything to the rules which have already been given in Article 434 to 438, pages 504 to 517, for determining bending moments, and the transverse dimensions required in order to resist those moments. Cranks are usually rectangular in section; levers and walking beams are sometimes rectangular and sometimes I-shaped. The bending moment is in most cases exerted in contrary directions alternately, so that the cross-section must be made symmetrical about the neutral axis; and for the modulus of stress must be taken a safe working value of that kind of stress against which the material is

weakest; tension for cast iron, thrust for wrought iron; for example:

Lbs. on the square inch,

Kilogrammes on the square millimètre,...

3000
2.I

Cast Iron. Wrought Iron. 6000

4.2

Holes made in a lever for the purpose of inserting pins should be as near as possible to the neutral layer; that being the position in which the removal of a given area of material weakens the lever least.

The following rules for the proportionate dimensions of a steamengine crank, made to be keyed on the end of a shaft, are those deduced by Mr. Bourne from the practice of Messrs. Boulton and Watt:

Diameter of Shaft-Journal multiplied by

Crank-web; thickness produced to centre of shaft,... 075 breadth produced to centre of shaft,..

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Large Eye; breadth,.

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I'50

175

0'45

Crank-web; thickness produced to centre of pin,...........
breadth produced to centre of pin,............

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Small Eye; breadth,

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Diameter of

Piston-Rod

multiplied by

I'IO

1.60

1.87

0.63

I'40

1.60

Trussed or framed levers are sometimes used; as in the walking beams of American river steamers. A beam of that sort consists mainly of a cast-iron cross, having the ends of its arms tied together by four wrought-iron tie-rods, forming a lozenge-shaped figure. The long arms are from twice to three times the length of the short arms. The long arms are always in a state of thrust; the upper and lower tie-rods alternately are subjected to tension; and the upper and lower short arms of the cross alternately are subjected to a thrust equal to twice the load. The load, the thrust along a long arm, and the tension on a tie-rod, are to each other nearly in the proportions of the length of a short arm, the length of a long arm, and the length of a tie-rod.

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ON THE PRINCIPLES OF THE ACTION OF CUTTING TOOLS.

476. General Explanations.—In making the bearing and working surfaces of the parts of a machine, it is only a rough approximation to the required figure that can be obtained by casting, by forging, or by pressure. The precision of form which is essential to smooth motion and efficient working is given by means of cutting tools. The object of the present chapter is to give a brief statement of the principles upon which the action of such tools depends. For detailed information respecting them, reference may be made to the second volume of Holtzapffel's Treatise on Mechanical Manipulation, extending from page 457 to page 1025, and to Mr. Northcott's Treatise on Lathes and Turning; and for a very clear summary account of their nature and use, to an Essay by Mr. James Nasmyth, published at the end of the later editions of Buchanan's Treatise on Millwork.

The appendix to Holtzapffel's volume contains two essays of much value on the general principles of cutting tools-one by Mr. Babbage, and the other by Professor Willis.

477. Characteristics of Cutting Tools in General.-The usual material for cutting tools is steel, of a degree of hardness suited to that of the material to be cut. Every cutting tool has at least one cutting edge; and sometimes three or more edges meet and form a point, two or more of those edges being cutting edges, so that the form of the cutting part of a tool is that of a wedge, or of a pyramid, as the case may be. A cutting edge is formed by the meeting of two surfaces, generally plane, and sometimes curved. When a surface forming a cutting edge is oblique to the original surfaces of the bar out of which the tool is made, that surface is called a chamfer or bevel. The angle at which those surfaces meet may be called the cutting angle. It ranges from about 15° to 135°, according to the nature of the material to be cut, and the way in which the tool is to act upon it. Examples of cutting angles of tools for different purposes will be mentioned further on. A narrow cutting edge at the end of a bar-shaped tool is often called the point of the tool; the body of the tool is called the shaft or the blade; the term shaft being usually applied to tools with a cutting point or narrow edge at one end, and blade to those which have a longitudinal cutting edge; but blade is often applied to

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