there is no use in entering upon detailed explanations of what a learner has before him. shafts are seen wherever there is machinery; it is easy to see the extent to which they are employed to transmit power, and the usual manner of arranging them. various text-books afford data for determining the amount of torsional strain that shafts of a given diameter will bear; explain that their capacity to resist torsional strain is as the cube of the diameter, and that the deflection from transverse strains is so many degrees; with many other matters that are highly useful and proper to know. i will therefore not devote any space to these things here, but notice some of the more obscure conditions that pertain to shafts, such as are demonstrated by practical experience rather than deduced from mathematical data. what is said will apply especially to what is called line-shafting for conveying and distributing power in machine-shops and other manufacturing establishments. the following propositions in reference to shafts will assist in understanding what is to follow:—

1. the strength of shafts is governed by their size and the arrangement of their supports.

2. the capacity of shafts is governed by their strength and the speed at which they run taken together.

3. the strains to which shafts are subjected are the torsional strain of transmission, transverse strain from belts and wheels, and strains from accidents, such as the winding of belts.

4. the speed at which shafts should run is governed by their size, the nature of the machinery to be driven, and the kind of bearings in which they are supported.

5. as the strength of shafts is determined by their size, and their size fixed by the strains to which they are subjected, [45] strains are first to be considered.

there were three kinds of strain mentioned—torsional, deflective, and accidental. to meet these several strains the same means have to be provided, which is a sufficient size and strength to resist them; hence it is useless to consider each of these different strains separately. if we know which of the three is greatest, and provide for that, the rest, of course, may be disregarded. this, in practice, is found to be accidental strains to which shafts are in ordinary use subjected, and they are usually made, in point of strength, far in excess of any standard that would be fixed by either torsional or transverse strain due to the regular duty performed.

this brings us back to the old proposition, that for structures which do not involve motion, mathematical data will furnish dimensions; but the same rule will not apply in machinery. to follow the proportions for shafts that would be furnished by pure mathematical data would in nearly all cases lead to error. experience has demonstrated that for ordinary cases, where power is transmitted and applied with tolerable regularity, a shaft three inches in diameter, making one hundred and fifty revolutions a minute, its bearings three to four diameters in length, and placed ten feet apart, will safely transmit fifty horse-power.

by assuming this or any other well-proved example, and estimating larger or smaller shafts by keeping their diameters as the cube root of the power to be transmitted, the distance between bearings as the diameter, and the speed inverse as the diameter, the reader will find his calculations to agree approximately with the modern practice of our best engineers. this is not mentioned to give proportions for shafts, so much as to call attention to accidental strains, such as winding belts, and to call attention to a marked discrepancy between actual practice and such proportions as would be given by what has been called the measured or determinable strains to which shafts are subjected.

as a means for transmitting power, shafts afford the very important advantage that power can be easily taken off at any point throughout their length, by means of pulleys or gearing, also in forming a positive connection between the motive-power and machines, or between the different parts of machines. the capacity of shafts in resisting torsional strain is as [46] the cube of their diameter, and the amount of torsional deflection in shafts is as their length. the torsional capacity being based upon the diameter, often leads to the construction of what may be termed diminishing shafts, lines in which the diameter of the several sections are diminished as the distance from the driving power increases, and as the duty to be performed becomes less. this plan of arranging line shafting has been and is yet quite common, but certainly was never arrived at by careful observation. almost every plan of construction has both advantages and disadvantages, and the best means of determining the excess of either, in any case, is to first arrive at all the conditions as near as possible, then form a "trial balance," putting the advantages on one side and the disadvantages on the other, and footing up the sums for comparison. dealing with this matter of shafts of uniform diameter and shafts of varying diameter in this way, there may be found in favour of the latter plan a little saving of material and a slight reduction of friction as advantages. the saving of material relates only to first cost, because the expense of fitting is greater in constructing shafts when the diameters of the different pieces vary; the friction, considering that the same velocity throughout must be assumed, is scarcely worth estimating.

for disadvantages there is, on the other hand, a want of uniformity in fittings that prevents their interchange from one part of a line shaft to the other—a matter of great importance, as such exchanges are frequently required. a line shaft, when constructed with pieces of varying diameter, is special machinery, adapted to some particular place or duty, and not a standard product that can be regularly manufactured as a staple article by machinists, and thus afforded at a low price. pulleys, wheels, bearings, and couplings have all to be specially prepared; and in case of a change, or the extension of lines of shafting, cause annoyance, and frequently no little expense, which may all be avoided by having shafts of uniform diameter. the bearings, besides being of varied strength and proportions, are generally in such cases placed at irregular intervals, and the lengths of the different sections of the shaft are sometimes varied to suit their diameter. with line shafts of uniform diameter, everything pertaining to the shaft—such as hangers, couplings, pulleys, and bearings—is interchangeable; the pulleys, wheels, bearings, or hangers can be placed at pleasure, or changed from one part of the shaft to another, or from [47] one part of the works to another, as occasion may require. the first cost of a line of shafting of uniform diameter, strong enough for a particular duty, is generally less than that of a shaft consisting of sections varying in size. this may at first seem strange, but a computation of the number of supports required, with the expense of special fitting, will in nearly all cases show a saving.

attention has been called to this case as one wherein the conditions of operation obviously furnish true data to govern the arrangement of machinery, instead of the determinable strains to which the parts are subjected, and as a good example of the importance of studying mechanical conditions from a practical and experimental point of view. if the general diameter of a shaft is based upon the exact amount of power to be transmitted, or if the diameter of a shaft at various parts is based upon the torsional stress that would be sustained at these points, such a shaft would not only fail to meet the conditions of practical use, but would cost more by attempting such an adaptation. the regular working strain to which shafts are subjected is inversely as the speed at which they run. this becomes a strong reason in favour of arranging shafts to run at a maximum speed, provided there was nothing more than first cost to consider; but there are other and more important conditions to be taken into account, principal among which are the required rate of movement where power is taken off to machines, and the endurance of bearings.

in the case of line shafting for manufactories, if the speed varies so much from that of the first movers on machines as to require one or more intermediate or countershafts, the expense would be very great; on the contrary, if countershafts can be avoided, there is a great saving of belts, bearings, machinery, and obstruction. the practical limit of speed for line shafts is in a great measure dependent upon the nature of the bearings, a subject that will be treated of in another place.

(1.) what kind of strains are shafts subjected to?—(2.) what determines the strength of shafts in resisting transverse strain?—(3.) why are shafts often more convenient than belts for transmitting power?—(4.) what is the difference between the strains to which shafts and belts are subjected?—(5.) what is gained by constructing a line shaft of sections diminishing in size from the first mover?—(6.) what is gained by constructing line shafts of uniform diameter?