RULE VI. — (Figs. 5 and 6.) To find the length of the radius bar; P Q, and the horizontal distance of the centre H of the radius bar from the main centre being given.
To the given horizontal distance, add half the versed sine[1] (D N) of the arc described by the extremity (D) of the side lever. Square this sum and multiply the square by the length of the side rod (P D). Call this product A. Take the same horizontal distance as before added to the same half versed sine (D N), and multiply the sum by the length of the side rod (P D): to the product add the product of the length of the side lever C D into the length of Q D, and divide A by the sum. The quotient will be the length required.
When the centre H of the radius has its position determined, rules 4 and 6 will always give the length of the radius bar F H. To get the length of C G, it will only be necessary to draw through the point F a line parallel to the side rod D P, and the point where that line cuts D C will be the position of the pin G.
In using these formulas and rules, the dimensions must all be taken in the same measure; that is, either all in feet, or all in inches; and when great accuracy is required the corrections given in Table (A) must be added to or subtracted from the calculated length of the radius bar, according as it is less or greater than the length of C G, the part of the beam that works it.
Examples.
1. Rule 4.—Let the Horizontal distance (M C) of the centre (H) of the radius bar from the main centre be equal to 51 inches ; the half versed sine D N=3 inches, and DC = 126 inches. Then by the rule we will have
EQUATION = 16.2 inches
which is the required length of the radius bar (F H).
2. Rule 5.— The following dimensions are those of the Red Rover Steamer: CG=32 DP=94 QD=74 CD=65 PQ=20
By the rule we have, A = equation = 96256 and
EQUATION = 53.4,
which is the required length of the radius bar.
3. Rule 6.—Take the same data as in the last example, only supposing that C G is not given, and that the centre H is fixed at a horizontal distance from the main centre, equal to 83.5 inches. Then the half versed sine of the arc D’ D D" will be about 2 inches, and we will have by the rule
A= EQUATION =54.8 inches,
the required length of the radius bar in this case.
In both of the last two examples , CG/HF = 0.6 nearly. The correction found by Table (A), therefore, would be 54 x 0.027 = 1.458 inches, which must be subtracted from the lengths already found for the radius bar, because it is longer than C G. The corrected lengths will therefore be
in example 2 ... F H = 51.94 inches.
in example 3 ... F H = 53.34 inches.
PROPORTIONS OF THE DIFFERENT PARTS OF A STEAM ENGINE. In constructing the different parts of a steam engine, and indeed of any other machine, it is desirable that they should be as light as is consistent with a due provision for safety, and with a due provision against the risk of acddents. The heaviness of the different parts brings along with it a correspondtng amount of friction, and a corresponding amount of inertia, both of which must be overcome by the motive power before the machinery can be set in motion, and consequently before any useful effect can be derived from it. It is obvious, therefore, that the weight of the different parts of the engine is, to a certain extent, an evil, which can only be tolerated because it is the necessary concomitant of their strength, and consequently of a due provision against the risk of accidents. It is therefore desirable to reduce this evil as far as is consistent with that due provision which we have mentioned. We would not be understood to infer from these remarks, that the different parts of the steam engine should be constructed just of sufficient strength to resist the strains to which they are subjected. If this were to be the constant aim of an engineer who is conversant with the laws which govern the resistances of beams, we hare no doubt but that the sarcastic remark, "the stability of the engine is inversely proportional to the theoretical skill of the engineer," would often be made. What we would be understood to infer is, that the strength of the different parts of the engine ought to be proportioned to the strains to which they are subjected. If one part of the engine be occasionally exposed to a strain double that of some other part, then it ought to have double the strength; if triple, triple ; if quadruple, quadruple; &c. If we suppose the former part to have only 3/2 the strength of the latter, then, on the supposition that it is subjected to a double force, 1/4 of the strength of the latter is useless, at least in so far as regards exposure to accidents; on the supposition that it is subjected to a triple force, 1/2 of the strength of the latter is useless; on the supposition that it is subjected to a quadruple force, 5/8 of the strength of the latter is useless; and generally, on the supposition that it is subjected to a force m times as great, (2m-3)/2m of the strength of tbe latter is useless.
These are obvious; for, on the first supposition, the former would break when only 3/4 of the strength of the latter was required, leaving the remaining 1/4th useless; on the second supposition, the former would break when only 3/6 or 1/2 of the strength was exerted, leaving 1 - 1/2 or 1/2 useless; on the third supposition, the former would break when only 3/8 of the strength of the latter was exerted, leaving 1 — 3/8 or 5/8 useless ; and on the general supposition, the former would break when only 3/2m of the strength of the latter was exerted; leaving 1 - 3/2m or (2m-3)/2m which would never be called into exercise. This superabundant strength is not merely useless; it is positively prejudicial, inasmuch as it brings along with it additional expense, and, what is more serious, as unnecessary and prejudicial weight. We therefore proceed to investigate the proper strength for some of the principal parts of the engine. The investigation will depend upon those laws which regulate the resistances of beams to fracture, to compression, to tension, to torsion, &c. These laws, we may observe, are derived from theory, and abundantly confirmed by experiment, so that we have no doubt with respect to them. In their application, however, a difficulty occurs in determining the proper values of the constants, which vary for different materials or even for different qualities of the same material, and which can only be determined by numerous experiments. It is obvious, also, that the proper strength of the different parts depends principally upon the pressure exerted in the cylinder upon the piston ; and this depends upon the diameter of the cylinder and the pressure of the steam in the boiler, or rather the greatest power which it may acquire before escaping at the safety valve. The proper strength of some of the parts also depends upon the length of the stroke; as, for example, in a marine engine, the diameter of the paddle shaft journal, and the exterior diameter of the large eye of the crank, &c. We may remark, that the parts of marine engines ought in general to possess a greater strength, since accidents with them are more serious and more difficult to repair.
Before proceeding farther, it may be necessary to give a few general propositions upon the resistances of solid materials to different sorts of strains.
A piece of solid matter may be exposed to four different kinds of strains, which are different in the manner of their operation.
I. It may be acted upon by a tensile force, which tends to pull it asunder; as, for example, in the case of ropes, stretchers, king-posts, tie-beams. &c.; or, in a steam engine, the piston rod of a single-acting engine, the connecting rod, &c.
II. It may be acted upon by a force which tends to break it across; as in the case of a joist or lever of any kind; and, in a steam engine, the crank, the main beam, &c.
III. It may be acted upon by a force which lends to wrench or twist it; as in the case of the axis of some wheels. &c.; or, in a land engine, the shaft of the fly wheel ; or. in a marine engine, the paddle shaft.
IV. It may be crushed, as in the case of pillars, posts, and truss beams; and, in a steam engine, the piston rod when it is a double-acting engine, parallel motion rods, air pump and force pump rods, and the like.
Each of these forces is resisted by the cohesion of the material, differently modified for the particular species of force. When they are increased to a certain extent the force of cohesion is so far overcome as to allow the material to take a new shape; for example, when a beam is subjected to a considerable transverse force it is often observed to take a curvilinear form. If the applied force be not increased above a certain limit, then, immediately upon its removal, the beam resumes its former shape ; but if the applied force be greater than a certain limit, then, although it be removed, the beam does not resume its former shape, but takes permanently a shape different from its original one. The beam is then said to have taken a "set," and it is remarkable that after a beam has once taken a set it cannot resist so great a force as previously without undergoing permanent derangement. These remarks are not confined to those materials of which beams are usually constructed — they are applicable to all sorts of material. If we take a quantity of clay, and form it into a rod, and apply to the rod a tensile force, then the tensile force will stretch the rod; but if it be not greater than a certain limit, upon its removal the rod will resume its former length ; if, however, it be greater than a certain limit, the rod will take a length different from its original one. The same process may be renewed, and the rod will take up a series of sets, each set requiring a smaller force to cause permanent change than the previous one. The force of cohesion varies for different materials, and is constant for the same material; or, we should rather say, it is constant for the same quality of the same material. Its efficacy in resisting an applied
- ↑ If great accuracy is required, in place of D N in this rule take D N multiplied by D Q and divided by P Q.
