§ 2.Support of Structures.—Every structure, as a whole, is maintained in equilibrium by the joint action of its ownweight, of theexternal loador pressure applied to it from without and tending to displace it, and of theresistanceof the material which supports it. A structure is supported either by resting on the solid crust of the earth, as buildings do, or by floating in a fluid, as ships do in water and balloons in air. The principles of the support of a floating structure form an important part of Hydromechanics (q.v.). The principles of the support, as a whole, of a structure resting on the land, are so far identical with those which regulate the equilibrium and stability of the several parts of that structure that the only principle which seems to require special mention here is one which comprehends in one statement the power both of liquids and of loose earth to support structures. This was first demonstrated in a paper “On the Stability of Loose Earth,” read to the Royal Society on the 19th of June 1856 (Phil.Trans.1856), as follows:—Let E represent the weight of the portion of a horizontal stratum of earth which is displaced by the foundation of a structure, S the utmost weight of that structure consistently with the power of the earth to resist displacement, φ the angle of repose of the earth; thenS=(1 + sin φ)2.E1 − sin φTo apply this to liquids φ must be made zero, and then S/E = 1, as is well known. For a proof of this expression see Rankine’sApplied Mechanics, 17th ed., p. 219.§ 3.Composition of a Structure, and Connexion of its Pieces.—A structure is composed ofpieces,—such as the stones of a building in masonry, the beams of a timber framework, the bars, plates and bolts of an iron bridge. Those pieces are connected at their joints or surfaces of mutual contact, either by simple pressure and friction (as in masonry with moist mortar or without mortar), by pressure and adhesion (as in masonry with cement or with hardened mortar, and timber with glue), or by the resistance offasteningsof different kinds, whether made by means of the form of the joint (as dovetails, notches, mortices and tenons) or by separate fastening pieces (as trenails, pins, spikes, nails, holdfasts, screws, bolts, rivets, hoops, straps and sockets.)§ 4.Stability, Stiffness and Strength.—A structure may be damaged or destroyed in three ways:—first, by displacement of its pieces from their proper positions relatively to each other or to the earth; secondly by disfigurement of one or more of those pieces, owing to their being unable to preserve their proper shapes under the pressures to which they are subjected; thirdly, bybreakingof one or more of those pieces. The power of resisting displacement constitutes stability, the power of each piece to resist disfigurement is itsstiffness; and its power to resist breaking, itsstrength.§ 5.Conditions of Stability.—The principles of the stability of a structure can be to a certain extent investigated independently of the stiffness and strength, by assuming, in the first instance, that each piece has strength sufficient to be safe against being broken, and stiffness sufficient to prevent its being disfigured to an extent inconsistent with the purposes of the structure, by the greatest forces which are to be applied to it. The condition that each piece of the structure is to be maintained in equilibrium by having its gross load, consisting of its own weight and of the external pressure applied to it, balanced by theresistancesor pressures exerted between it and the contiguous pieces, furnishes the means of determining the magnitude, position and direction of the resistances required at each joint in order to produce equilibrium; and theconditions of stabilityare, first, that theposition, and, secondly, that thedirection, of the resistance required at each joint shall, under all the variations to which the load is subject, be such as the joint is capable of exerting—conditions which are fulfilled by suitably adjusting the figures and positions of the joints, and theratiosof the gross loads of the pieces. As for themagnitudeof the resistance, it is limited by conditions, not of stability, but of strength and stiffness.§ 6.Principle of Least Resistance.—Where more than one system of resistances are alike capable of balancing the same system of loads applied to a given structure, thesmallestof those alternative systems, as was demonstrated by the Rev. Henry Moseley in hisMechanics of Engineering and Architecture, is that which will actually be exerted—becausethe resistances to displacement are the effect of a strained state of the pieces, which strained state is the effect of the load, and when the load is applied the strained state and the resistances produced by it increase until the resistances acquire just those magnitudes which are sufficient to balance the load, after which they increase no further.This principle of least resistance renders determinate many problems in the statics of structures which were formerly considered indeterminate.§ 7.Relations between Polygons of Loads and of Resistances.—In a structure in which each piece is supported at two joints only, the well-known laws of statics show that the directions of the gross load on each piece and of the two resistances by which it is supported must lie in one plane, must either be parallel or meet in one point, and must bear to each other, if not parallel, the proportions of the sides of a triangle respectively parallel to their directions, and, if parallel, such proportions that each of the three forces shall be proportional to the distance between the other two,—all the three distances being measured along one direction.Fig. 86.Considering, in the first place, the case in which the load and the two resistances by which each piece is balanced meet in one point, which may be called thecentre of load, there will be as many such points of intersection, or centres of load, as there are pieces in the structure; and the directions and positions of the resistances or mutual pressures exerted between the pieces will be represented by the sides of a polygon joining those points, as in fig. 86 where P1, P2, P3, P4represent the centres of load in a structure of four pieces, and the sides of thepolygon of resistancesP1P2P3P4represent respectively the directions and positions of the resistances exerted at the joints. Further, at any one of the centres of load let PL represent the magnitude and direction of the gross load, and Pa, Pb the two resistances by which the piece to which that load is applied is supported; then will those three lines be respectively the diagonal and sides of a parallelogram; or, what is the same thing, they will be equal to the three sides of a triangle; and they must be in the same plane, although the sides of the polygon of resistances may be in different planes.Fig. 87.According to a well-known principle of statics, because the loads or external pressures P1L1, &c., balance each other, they must be proportional to the sides of a closed polygon drawn respectively parallel to their directions. In fig. 87 construct such apolygon of loadsby drawing the lines L1, &c., parallel and proportional to, and joined end to end in the order of, the gross loads on the pieces of the structure. Then from the proportionality and parallelism of the load and the two resistances applied to each piece of the structure to the three sides of a triangle, there results the following theorem (originally due to Rankine):—If from the angles of the polygon of loads there be drawn lines(R1, R2, &c.),each of which is parallel to the resistance(asP1P2, &c.)exerted at the joint between the pieces to which the two loads represented by the contiguous sides of the polygon of loads(such asL1, L2, &c.)are applied; then will all those lines meet in one point(O),and their lengths, measured from that point to the angles of the polygon, will represent the magnitudes of the resistances to which they are respectively parallel.When the load on one of the pieces is parallel to the resistances which balance it, the polygon of resistances ceases to be closed, two of the sides becoming parallel to each other and to the load in question, and extending indefinitely. In the polygon of loads the direction of a load sustained by parallel resistances traverses the point O.2§ 8.How the Earth’s Resistance is to be treated.... When the pressure exerted by a structure on the earth (to which the earth’s resistance is equal and opposite) consists either of one pressure, which is necessarily the resultant of the weight of the structure and of all the other forces applied to it, or of two or more parallel vertical forces, whose amount can be determined at the outset of the investigation, the resistance of the earth can be treated as one or more upward loads applied to the structure. But in other cases the earth is to be treated asone of the pieces of the structure, loaded with a force equal and opposite in direction and position to the resultant of the weight of the structure and of the other pressures applied to it.§ 9.Partial Polygons of Resistance.—In a structure in which there are pieces supported at more than two joints, let a polygon be constructed of lines connecting the centres of load of any continuous series of pieces. This may be called apartial polygon of resistances. In considering its properties, the load at each centre of load is to be held toincludethe resistances of those joints which are not comprehended in the partial polygon of resistances, to which the theorem of § 7 will then apply in every respect. By constructing several partial polygons, and computing the relations between the loads and resistances which are determined by the application of that theorem to each of them, with the aid, if necessary, of Moseley’s principle of the least resistance, the whole of the relations amongst the loads and resistances may be found.§ 10.Line of Pressures—Centres and Line of Resistance.—The line of pressures is a line to which the directions of all the resistances in one polygon are tangents. Thecentre of resistanceat any joint is the point where the line representing the total resistance exerted at that joint intersects the joint. Theline of resistanceis a line traversing all the centres of resistance of a series of joints,—its form, in the positions intermediate between the actual joints of the structure, being determined by supposing the pieces and their loads to be subdivided by the introduction of intermediate jointsad infinitum, and finding the continuous line, curved or straight, in which the intermediate centres of resistance are all situated, however great their number. The difference between the line of resistance and the line of pressures was first pointed out by Moseley.Fig. 88.§ 11.* The principles of the two preceding sections may be illustrated by the consideration of a particular case of a buttress of blocks forming a continuous series of pieces (fig. 88), where aa, bb, cc, dd represent plane joints. Let the centre of pressure C at the first joint aa be known, and also the pressure P acting at C in direction and magnitude. Find R1the resultant of this pressure, the weight of the block aabb acting through its centre of gravity, and any other external force which may be acting on the block, and produce its line of action to cut the joint bb in C1. C1is then the centre of pressure for the joint bb, and R1is the total force acting there. Repeating this process for each block in succession there will be found the centres of pressure C2, C3, &c., and also the resultant pressures R2, R3, &c., acting at these respective centres. The centres of pressure at the joints are also calledcentres of resistance, and the curve passing through these points is called aline of resistance. Let all the resultants acting at the several centres of resistance be produced until they cut one another in a series of points so as to form an unclosed polygon. This polygon is thepartial polygon of resistance. A curve tangential to all the sides of the polygon is theline of pressures.§ 12.Stability of Position, and Stability of Friction.—The resistances at the several joints having been determined by the principles set forth in §§ 6, 7, 8, 9 and 10, not only under the ordinary load of the structure, but under all the variations to which the load is subject as to amount and distribution, the joints are now to be placed and shaped so that the pieces shall not suffer relative displacement under any of those loads. The relative displacement of the two pieces which abut against each other at a joint may take place eitherby turning or by sliding. Safety against displacement by turning is calledstability of position; safety against displacement by sliding,stability of friction.§ 13.Condition of Stability of Position.—If the materials of a structure were infinitely stiff and strong, stability of position at any joint would be insured simply by making the centre of resistance fall within the joint under all possible variations of load. In order to allow for the finite stiffness and strength of materials, the least distance of the centre of resistance inward from the nearest edge of the joint is made to bear a definite proportion to the depth of the joint measured in the same direction, which proportion is fixed, sometimes empirically, sometimes by theoretical deduction from the laws of the strength of materials. That least distance is called by Moseley themodulus of stability. The following are some of the ratios of the modulus of stability to the depth of the joint which occur in practice:—Retaining walls, as designed by British engineers1 : 8Retaining walls, as designed by French engineers1 : 5Rectangular piers of bridges and other buildings, and arch-stones1 : 3Rectangular foundations, firm ground1 : 3Rectangular foundations, very soft ground1 : 2Rectangular foundations, intermediate kinds of ground1 : 3 to 1 : 2Thin, hollow towers (such as furnace chimneys exposed to high winds), square1 : 6Thin, hollow towers, circular1 : 4Frames of timber or metal, under their ordinary or average distribution of load1 : 3Frames of timber or metal, under the greatest irregularities of load1 : 3In the case of the towers, thedepth of the jointis to be understood to mean thediameter of the tower.Fig. 89.§ 14.Condition of Stability of Friction.—If the resistance to be exerted at a joint is always perpendicular to the surfaces which abut at and form that joint, there is no tendency of the pieces to be displaced by sliding. If the resistance be oblique, let JK (fig. 89) be the joint, C its centre of resistance, CR a line representing the resistance, CN a perpendicular to the joint at the centre of resistance. The angle NCR is theobliquityof the resistance. From R draw RP parallel and RQ perpendicular to the joint; then, by the principles of statics, the component of the resistancenormalto the joint is—CP = CR · cos PCR;and the componenttangentialto the joint is—CQ = CR · sin PCR = CP · tan PCR.If the joint be provided either with projections and recesses, such as mortises and tenons, or with fastenings, such as pins or bolts, so as to resist displacement by sliding, the question of the utmost amount of the tangential resistance CQ which it is capable of exerting depends on thestrengthof such projections, recesses, or fastenings; and belongs to the subject of strength, and not to that of stability. In other cases the safety of the joint against displacement by sliding depends on its power of exerting friction, and that power depends on the law, known by experiment, that the friction between two surfaces bears a constant ratio, depending on the nature of the surfaces, to the force by which they are pressed together. In order that the surfaces which abut at the joint JK may be pressed together, the resistance required by the conditions of equilibrium CR, must be athrustand not apull; and in that case the force by which the surfaces are pressed together is equal and opposite to the normal component CP of the resistance. The condition of stability of friction is that the tangential component CQ of the resistance required shall not exceed the friction due to the normal component; that is, thatCQ ≯ ƒ · CP,where ƒ denotes thecoefficient of frictionfor the surfaces in question. The angle whose tangent is the coefficient of friction is calledthe angle of repose, and is expressed symbolically by—φ = tan−1ƒ.Now CQ = CP · tan PCR;consequently the condition of stability of friction is fulfilled if the angle PCR is not greater than φ; that is to say, ifthe obliquity of the resistance required at the joint does not exceed the angle of repose; and this condition ought to be fulfilled under all possible variations of the load.It is chiefly in masonry and earthwork that stability of friction is relied on.§ 15.Stability of Friction in Earth.—The grains of a mass of loose earth are to be regarded as so many separate pieces abutting against each other at joints in all possible positions, and depending for their stability on friction. To determine whether a mass of earth is stable at a given point, conceive that point to be traversed by planes in all possible positions, and determine which position gives the greatest obliquity to the total pressure exerted between the portions of the mass which abut against each other at the plane. The condition of stability is that this obliquity shall not exceed the angle of repose of the earth. The consequences of this principle are developed in a paper, “On the Stability of Loose Earth,” already cited in § 2.§ 16.Parallel Projections of Figures.—If any figure be referred to a system of co-ordinates, rectangular or oblique, and if a second figure be constructed by means of a second system of co-ordinates, rectangular or oblique, and either agreeing with or differing from the first system in rectangularity or obliquity, but so related to the co-ordinates of the first figure that for each point in the first figure there shall be a corresponding point in the second figure, the lengths of whose co-ordinates shall bear respectively to the three corresponding co-ordinates of the corresponding point in the first figure three ratios which are the same for every pair of corresponding points in the two figures, these corresponding figures are calledparallel projectionsof each other. The properties of parallel projections of most importance to the subject of the present article are the following:—(1) A parallel projection of a straight line is a straight line.(2) A parallel projection of a plane is a plane.(3) A parallel projection of a straight line or a plane surface divided in a given ratio is a straight line or a plane surface divided in the same ratio.(4) A parallel projection of a pair of equal and parallel straight lines, or plain surfaces, is a pair of equal and parallel straight lines, or plane surfaces; whence it follows(5) That a parallel projection of a parallelogram is a parallelogram, and(6) That a parallel projection of a parallelepiped is a parallelepiped.(7) A parallel projection of a pair of solids having a given ratio is a pair of solids having the same ratio.Though not essential for the purposes of the present article, the following consequence will serve to illustrate the principle of parallel projections:—(8) A parallel projection of a curve, or of a surface of a given algebraical order, is a curve or a surface of the same order.For example, all ellipsoids referred to co-ordinates parallel to any three conjugate diameters are parallel projections of each other and of a sphere referred to rectangular co-ordinates.§ 17.Parallel Projections of Systems of Forces.—If a balanced system of forces be represented by a system of lines, then will every parallel projection of that system of lines represent a balanced system of forces.For the condition of equilibrium of forces not parallel is that they shall be represented in direction and magnitude by the sides and diagonals of certain parallelograms, and of parallel forces that they shall divide certain straight lines in certain ratios; and the parallel projection of a parallelogram is a parallelogram, and that of a straight line divided in a given ratio is a straight line divided in the same ratio.The resultant of a parallel projection of any system of forces is the projection of their resultant; and the centre of gravity of a parallel projection of a solid is the projection of the centre of gravity of the first solid.§ 18.Principle of the Transformation of Structures.—Here we have the following theorem: If a structure of a given figure have stability of position under a system of forces represented by a given system of lines, then will any structure whose figure is a parallel projection of that of the first structure have stability of position under a system of forces represented by the corresponding projection of the first system of lines.For in the second structure the weights, external pressures, and resistances will balance each other as in the first structure; the weights of the pieces and all other parallel systems of forces will have the same ratios as in the first structure; and the several centres of resistance will divide the depths of the joints in the same proportions as in the first structure.If the first structure have stability of friction, the second structure will have stability of friction also, so long as the effect of the projection is not to increase the obliquity of the resistance at any joint beyond the angle of repose.The lines representing the forces in the second figure show theirrelativedirections and magnitudes. To find theirabsolutedirections and magnitudes, a vertical line is to be drawn in the first figure, of such a length as to represent the weight of a particular portion of the structure. Then will the projection of that line in the projected figure indicate the vertical direction, and represent the weight of the part of the second structure corresponding to the before-mentioned portion of the first structure.The foregoing “principle of the transformation of structures” was first announced, though in a somewhat less comprehensive form, to the Royal Society on the 6th of March 1856. It is useful in practice, by enabling the engineer easily to deduce the conditions of equilibrium and stability of structures of complex and unsymmetrical figures from those of structures of simple and symmetrical figures. By its aid, for example, the whole of the properties ofelliptical arches, whether square or skew, whether level or sloping in their span, are at once deduced by projection from those of symmetrical circular arches, and the properties of ellipsoidal and elliptic-conoidal domes from those of hemispherical and circular-conoidal domes; and the figures of arches fitted to resist the thrust of earth, which is less horizontally than vertically in a certain given ratio, can be deduced by a projection from those of arches fitted to resist the thrust of a liquid, which is of equal intensity, horizontally and vertically.§ 19.Conditions of Stiffness and Strength.—After the arrangement of the pieces of a structure and the size and figure of their joints or surfaces of contact have been determined so as to fulfil the conditions ofstability,—conditions which depend mainly on the position and direction of theresultantortotalload on each piece, and therelativemagnitude of the loads on the different pieces—the dimensions of each piece singly have to be adjusted so as to fulfil the conditions ofstiffnessandstrength—conditions which depend not only on theabsolutemagnitude of the load on each piece, and of the resistances by which it is balanced, but also on themode of distributionof the load over the piece, and of the resistances over the joints.The effect of the pressures applied to a piece, consisting of the load and the supporting resistances, is to force the piece into a state ofstrainor disfigurement, which increases until the elasticity, or resistance to strain, of the material causes it to exert astress, or effort to recover its figure, equal and opposite to the system of applied pressures. The condition ofstiffnessis that the strain or disfigurement shall not be greater than is consistent with the purposes of the structure; and the condition ofstrengthis that the stress shall be within the limits of that which the material can bear with safety against breaking. The ratio in which the utmost stress before breaking exceeds the safe working stress is called thefactor of safety, and is determined empirically. It varies from three to twelve for various materials and structures. (SeeStrength of Materials.)PART II. THEORY OF MACHINES§ 20.Parts of a Machine: Frame and Mechanism.—The parts of a machine may be distinguished into two principal divisions,—the frame, or fixed parts, and themechanism, or moving parts. The frame is a structure which supports the pieces of the mechanism, and to a certain extent determines the nature of their motions.The form and arrangement of the pieces of the frame depend upon the arrangement and the motions of the mechanism; the dimensions of the pieces of the frame required in order to give it stability and strength are determined from the pressures applied to it by means of the mechanism. It appears therefore that in general the mechanism is to be designed first and the frame afterwards, and that the designing of the frame is regulated by the principles of the stability of structures and of the strength and stiffness of materials,—care being taken to adapt the frame to the most severe load which can be thrown upon it at any period of the action of the mechanism.Each independent piece of the mechanism also is a structure, and its dimensions are to be adapted, according to the principles of the strength and stiffness of materials, to the most severe load to which it can be subjected during the action of the machine.§ 21.Definition and Division of the Theory of Machines.—From what has been said in the last section it appears that the department of the art of designing machines which has reference to the stability of the frame and to the stiffness and strength of the frame and mechanism is a branch of the art of construction. It is therefore to be separated from thetheory of machines, properly speaking, which has reference to the action of machines considered as moving. In the action of a machine the following three things take place:—Firstly, Some natural source of energy communicates motion and force to a piece or pieces of the mechanism, called thereceiver of powerorprime mover.Secondly, The motion and force are transmitted from the prime mover through thetrain of mechanismto theworking pieceorpieces, and during that transmission the motion and force are modified in amount and direction, so as to be rendered suitable for the purpose to which they are to be applied.Thirdly, The working piece or pieces by their motion, or by their motion and force combined, produce some useful effect.Such are the phenomena of the action of a machine, arranged in the order ofcausation. But in studying or treating of the theory of machines, the order ofsimplicityis the best; and in this order the first branch of the subject is the modification of motion and force by the train of mechanism; the next is the effect or purpose of the machine; and the last, or most complex, is the action of the prime mover.The modification of motion and the modification of force take place together, and are connected by certain laws; but in the study of the theory of machines, as well as in that of pure mechanics, much advantage has been gained in point of clearness and simplicity by first considering alone the principles of the modification of motion, which are founded upon what is now known as Kinematics, and afterwards considering the principles of the combined modification of motion and force, which are founded both on geometry and on the laws of dynamics. The separation of kinematics from dynamics is due mainly to G. Monge, Ampère and R. Willis.The theory of machines in the present article will be considered under the following heads:—I.Pure Mechanism, orApplied Kinematics; being the theory of machines considered simply as modifying motion.II.Applied Dynamics; being the theory of machines considered as modifying both motion and force.Chap. I. On Pure Mechanism§ 22.Division of the Subject.—Proceeding in the order of simplicity, the subject of Pure Mechanism, or Applied Kinematics, may be thus divided:—Division 1.—Motion of a point.Division 2.—Motion of the surface of a fluid.Division 3.—Motion of a rigid solid.Division 4.—Motions of a pair of connected pieces, or of an “elementary combination” in mechanism.Division 5.—Motions of trains of pieces of mechanism.Division 6.—Motions of sets of more than two connected pieces, or of “aggregate combinations.”A point is the boundary of a line, which is the boundary of a surface, which is the boundary of a volume. Points, lines and surfaces have no independent existence, and consequently those divisions of this chapter which relate to their motions are only preliminary to the subsequent divisions, which relate to the motions of bodies.Division 1. Motion of a Point.§ 23.Comparative Motion.—The comparative motion of two points is the relation which exists between their motions, without having regard to their absolute amounts. It consists of two elements,—thevelocity ratio, which is the ratio of any two magnitudes bearing to each other the proportions of the respective velocities of the two points at a given instant, and thedirectional relation, which is the relation borne to each other by the respective directions of the motions of the two points at the same given instant.It is obvious that the motions of a pair of points may be varied in any manner, whether by direct or by lateral deviation, and yet that theircomparative motionmay remain constant, in consequence of the deviations taking place in the same proportions, in the same directions and at the same instants for both points.Robert Willis (1800-1875) has the merit of having been the first to simplify considerably the theory of pure mechanism, by pointing out that that branch of mechanics relates wholly to comparative motions.The comparative motion of two points at a given instant is capable of being completely expressed by one of Sir William Hamilton’s Quaternions,—the “tensor” expressing the velocity ratio, and the “versor” the directional relation.Graphical methods of analysis founded on this way of representing velocity and acceleration were developed by R. H. Smith in a paper communicated to the Royal Society of Edinburgh in 1885, and illustrations of the method will be found below.Division 2. Motion of the Surface of a Fluid Mass.§ 24.General Principle.—A mass of fluid is used in mechanism to transmit motion and force between two or more movable portions (calledpistonsorplungers) of the solid envelope or vessel in which the fluid is contained; and, when such transmission is the sole action, or the only appreciable action of the fluid mass, its volume is either absolutely constant, by reason of its temperature and pressure being maintained constant, or not sensibly varied.Let a represent the area of the section of a piston made by a plane perpendicular to its direction of motion, and v its velocity, which is to be considered as positive when outward, and negative when inward. Then the variation of the cubic contents of the vessel in a unit of time by reason of the motion of one piston is va. The condition that the volume of the fluid mass shall remain unchanged requires that there shall be more than one piston, and that the velocities and areas of the pistons shall be connected by the equation—Σ · va = 0.(1)§ 25.Comparative Motion of Two Pistons.—If there be but two pistons, whose areas are a1and a2, and their velocities v1and v2, their comparative motion is expressed by the equation—v2/v1= −a1/a2;(2)that is to say, their velocities are opposite as to inwardness and outwardness and inversely proportional to their areas.§ 26.Applications: Hydraulic Press: Pneumatic Power-Transmitter.—In the hydraulic press the vessel consists of two cylinders, viz. the pump-barrel and the press-barrel, each having its piston, and of a passage connecting them having a valve opening towards the press-barrel. The action of the enclosed water in transmitting motion takes place during the inward stroke of the pump-plunger, when the above-mentioned valve is open; and at that time the press-plunger moves outwards with a velocity which is less than the inward velocity of the pump-plunger, in the same ratio that the area of the pump-plunger is less than the area of the press-plunger. (SeeHydraulics.)In the pneumatic power-transmitter the motion of one piston istransmitted to another at a distance by means of a mass of air contained in two cylinders and an intervening tube. When the pressure and temperature of the air can be maintained constant, this machine fulfils equation (2), like the hydraulic press. The amount and effect of the variations of pressure and temperature undergone by the air depend on the principles of the mechanical action of heat, orThermodynamics(q.v.), and are foreign to the subject of pure mechanism.Division 3. Motion of a Rigid Solid.§ 27.Motions Classed.—In problems of mechanism, each solid piece of the machine is supposed to be so stiff and strong as not to undergo any sensible change of figure or dimensions by the forces applied to it—a supposition which is realized in practice if the machine is skilfully designed.This being the case, the various possible motions of a rigid solid body may all be classed under the following heads: (1)Shifting or Translation; (2)Turning or Rotation; (3)Motions compounded of Shifting and Turning.The most common forms for the paths of the points of a piece of mechanism, whose motion is simple shifting, are the straight line and the circle.Shifting in a straight line is regulated either by straight fixed guides, in contact with which the moving piece slides, or by combinations of link-work, calledparallel motions, which will be described in the sequel. Shifting in a straight line is usuallyreciprocating; that is to say, the piece, after shifting through a certain distance, returns to its original position by reversing its motion.Circular shifting is regulated by attaching two or more points of the shifting piece to ends of equal and parallel rotating cranks, or by combinations of wheel-work to be afterwards described. As an example of circular shifting may be cited the motion of the coupling rod, by which the parallel and equal cranks upon two or more axles of a locomotive engine are connected and made to rotate simultaneously. The coupling rod remains always parallel to itself, and all its points describe equal and similar circles relatively to the frame of the engine, and move in parallel directions with equal velocities at the same instant.§ 28.Rotation about a Fixed Axis: Lever, Wheel and Axle.—The fixed axis of a turning body is a line fixed relatively to the body and relatively to the fixed space in which the body turns. In mechanism it is usually the central line either of a rotating shaft or axle having journals, gudgeons, or pivots turning in fixed bearings, or of a fixed spindle or dead centre round which a rotating bush turns; but it may sometimes be entirely beyond the limits of the turning body. For example, if a sliding piece moves in circular fixed guides, that piece rotates about an ideal fixed axis traversing the centre of those guides.Let the angular velocity of the rotation be denoted by α = dθ/dt, then the linear velocity of any point A at the distance r from the axis is αr; and the path of that point is a circle of the radius r described about the axis.This is the principle of the modification of motion by the lever, which consists of a rigid body turning about a fixed axis called a fulcrum, and having two points at the same or different distances from that axis, and in the same or different directions, one of which receives motion and the other transmits motion, modified in direction and velocity according to the above law.In the wheel and axle, motion is received and transmitted by two cylindrical surfaces of different radii described about their common fixed axis of turning, their velocity-ratio being that of their radii.Fig. 90.§ 29.Velocity Ratio of Components of Motion.—As the distance between any two points in a rigid body is invariable, the projections of their velocities upon the line joining them must be equal. Hence it follows that, if A in fig. 90 be a point in a rigid body CD, rotating round the fixed axis F, the component of the velocity of A in any direction AP parallel to the plane of rotation is equal to the total velocity of the point m, found by letting fall Fm perpendicular to AP; that is to say, is equal toα · Fm.Hence also the ratio of the components of the velocities of two points A and B in the directions AP and BW respectively, both in the plane of rotation, is equal to the ratio of the perpendiculars Fm and Fn.§ 30.Instantaneous Axis of a Cylinder rolling on a Cylinder.—Let a cylinder bbb, whose axis of figure is B and angular velocity γ, roll on a fixed cylinder ααα, whose axis of figure is A, either outside (as in fig. 91), when the rolling will be towards the same hand as the rotation, or inside (as in fig. 92), when the rolling will be towards the opposite hand; and at a given instant let T be the line of contact of the two cylindrical surfaces, which is at their common intersection with the plane AB traversing the two axes of figure.The line T on the surface bbb has for the instant no velocity in a direction perpendicular to AB; because for the instant it touches, without sliding, the line T on the fixed surface aaa.The line T on the surface bbb has also for the instant no velocity in the plane AB; for it has just ceased to move towards the fixed surface aaa, and is just about to begin to move away from that surface.The line of contact T, therefore, on the surface of the cylinder bbb, isfor the instantat rest, and is the “instantaneous axis” about which the cylinder bbb turns, together with any body rigidly attached to that cylinder.Fig. 91.Fig. 92.To find, then, the direction and velocity at the given instant of any point P, either in or rigidly attached to the rolling cylinder T, draw the plane PT; the direction of motion of P will be perpendicular to that plane, and towards the right or left hand according to the direction of the rotation of bbb; and the velocity of P will bevP= γ·PT,(3)PT denoting the perpendicular distance of P from T. The path of P is a curve of the kind calledepitrochoids. If P is in the circumference of bbb, that path becomes anepicycloid.The velocity of any point in the axis of figure B isvB= γ·TB;(4)and the path of such a point is a circle described about A with the radius AB, being for outside rolling the sum, and for inside rolling the difference, of the radii of the cylinders.Let α denote the angular velocity with which theplane of axesAB rotates about the fixed axis A. Then it is evident thatvB= α·AB,(5)and consequently thatα = γ·TB/AB.
§ 2.Support of Structures.—Every structure, as a whole, is maintained in equilibrium by the joint action of its ownweight, of theexternal loador pressure applied to it from without and tending to displace it, and of theresistanceof the material which supports it. A structure is supported either by resting on the solid crust of the earth, as buildings do, or by floating in a fluid, as ships do in water and balloons in air. The principles of the support of a floating structure form an important part of Hydromechanics (q.v.). The principles of the support, as a whole, of a structure resting on the land, are so far identical with those which regulate the equilibrium and stability of the several parts of that structure that the only principle which seems to require special mention here is one which comprehends in one statement the power both of liquids and of loose earth to support structures. This was first demonstrated in a paper “On the Stability of Loose Earth,” read to the Royal Society on the 19th of June 1856 (Phil.Trans.1856), as follows:—
Let E represent the weight of the portion of a horizontal stratum of earth which is displaced by the foundation of a structure, S the utmost weight of that structure consistently with the power of the earth to resist displacement, φ the angle of repose of the earth; then
To apply this to liquids φ must be made zero, and then S/E = 1, as is well known. For a proof of this expression see Rankine’sApplied Mechanics, 17th ed., p. 219.
§ 3.Composition of a Structure, and Connexion of its Pieces.—A structure is composed ofpieces,—such as the stones of a building in masonry, the beams of a timber framework, the bars, plates and bolts of an iron bridge. Those pieces are connected at their joints or surfaces of mutual contact, either by simple pressure and friction (as in masonry with moist mortar or without mortar), by pressure and adhesion (as in masonry with cement or with hardened mortar, and timber with glue), or by the resistance offasteningsof different kinds, whether made by means of the form of the joint (as dovetails, notches, mortices and tenons) or by separate fastening pieces (as trenails, pins, spikes, nails, holdfasts, screws, bolts, rivets, hoops, straps and sockets.)
§ 4.Stability, Stiffness and Strength.—A structure may be damaged or destroyed in three ways:—first, by displacement of its pieces from their proper positions relatively to each other or to the earth; secondly by disfigurement of one or more of those pieces, owing to their being unable to preserve their proper shapes under the pressures to which they are subjected; thirdly, bybreakingof one or more of those pieces. The power of resisting displacement constitutes stability, the power of each piece to resist disfigurement is itsstiffness; and its power to resist breaking, itsstrength.
§ 5.Conditions of Stability.—The principles of the stability of a structure can be to a certain extent investigated independently of the stiffness and strength, by assuming, in the first instance, that each piece has strength sufficient to be safe against being broken, and stiffness sufficient to prevent its being disfigured to an extent inconsistent with the purposes of the structure, by the greatest forces which are to be applied to it. The condition that each piece of the structure is to be maintained in equilibrium by having its gross load, consisting of its own weight and of the external pressure applied to it, balanced by theresistancesor pressures exerted between it and the contiguous pieces, furnishes the means of determining the magnitude, position and direction of the resistances required at each joint in order to produce equilibrium; and theconditions of stabilityare, first, that theposition, and, secondly, that thedirection, of the resistance required at each joint shall, under all the variations to which the load is subject, be such as the joint is capable of exerting—conditions which are fulfilled by suitably adjusting the figures and positions of the joints, and theratiosof the gross loads of the pieces. As for themagnitudeof the resistance, it is limited by conditions, not of stability, but of strength and stiffness.
§ 6.Principle of Least Resistance.—Where more than one system of resistances are alike capable of balancing the same system of loads applied to a given structure, thesmallestof those alternative systems, as was demonstrated by the Rev. Henry Moseley in hisMechanics of Engineering and Architecture, is that which will actually be exerted—becausethe resistances to displacement are the effect of a strained state of the pieces, which strained state is the effect of the load, and when the load is applied the strained state and the resistances produced by it increase until the resistances acquire just those magnitudes which are sufficient to balance the load, after which they increase no further.
This principle of least resistance renders determinate many problems in the statics of structures which were formerly considered indeterminate.
§ 7.Relations between Polygons of Loads and of Resistances.—In a structure in which each piece is supported at two joints only, the well-known laws of statics show that the directions of the gross load on each piece and of the two resistances by which it is supported must lie in one plane, must either be parallel or meet in one point, and must bear to each other, if not parallel, the proportions of the sides of a triangle respectively parallel to their directions, and, if parallel, such proportions that each of the three forces shall be proportional to the distance between the other two,—all the three distances being measured along one direction.
Considering, in the first place, the case in which the load and the two resistances by which each piece is balanced meet in one point, which may be called thecentre of load, there will be as many such points of intersection, or centres of load, as there are pieces in the structure; and the directions and positions of the resistances or mutual pressures exerted between the pieces will be represented by the sides of a polygon joining those points, as in fig. 86 where P1, P2, P3, P4represent the centres of load in a structure of four pieces, and the sides of thepolygon of resistancesP1P2P3P4represent respectively the directions and positions of the resistances exerted at the joints. Further, at any one of the centres of load let PL represent the magnitude and direction of the gross load, and Pa, Pb the two resistances by which the piece to which that load is applied is supported; then will those three lines be respectively the diagonal and sides of a parallelogram; or, what is the same thing, they will be equal to the three sides of a triangle; and they must be in the same plane, although the sides of the polygon of resistances may be in different planes.
According to a well-known principle of statics, because the loads or external pressures P1L1, &c., balance each other, they must be proportional to the sides of a closed polygon drawn respectively parallel to their directions. In fig. 87 construct such apolygon of loadsby drawing the lines L1, &c., parallel and proportional to, and joined end to end in the order of, the gross loads on the pieces of the structure. Then from the proportionality and parallelism of the load and the two resistances applied to each piece of the structure to the three sides of a triangle, there results the following theorem (originally due to Rankine):—
If from the angles of the polygon of loads there be drawn lines(R1, R2, &c.),each of which is parallel to the resistance(asP1P2, &c.)exerted at the joint between the pieces to which the two loads represented by the contiguous sides of the polygon of loads(such asL1, L2, &c.)are applied; then will all those lines meet in one point(O),and their lengths, measured from that point to the angles of the polygon, will represent the magnitudes of the resistances to which they are respectively parallel.
When the load on one of the pieces is parallel to the resistances which balance it, the polygon of resistances ceases to be closed, two of the sides becoming parallel to each other and to the load in question, and extending indefinitely. In the polygon of loads the direction of a load sustained by parallel resistances traverses the point O.2
§ 8.How the Earth’s Resistance is to be treated.... When the pressure exerted by a structure on the earth (to which the earth’s resistance is equal and opposite) consists either of one pressure, which is necessarily the resultant of the weight of the structure and of all the other forces applied to it, or of two or more parallel vertical forces, whose amount can be determined at the outset of the investigation, the resistance of the earth can be treated as one or more upward loads applied to the structure. But in other cases the earth is to be treated asone of the pieces of the structure, loaded with a force equal and opposite in direction and position to the resultant of the weight of the structure and of the other pressures applied to it.
§ 9.Partial Polygons of Resistance.—In a structure in which there are pieces supported at more than two joints, let a polygon be constructed of lines connecting the centres of load of any continuous series of pieces. This may be called apartial polygon of resistances. In considering its properties, the load at each centre of load is to be held toincludethe resistances of those joints which are not comprehended in the partial polygon of resistances, to which the theorem of § 7 will then apply in every respect. By constructing several partial polygons, and computing the relations between the loads and resistances which are determined by the application of that theorem to each of them, with the aid, if necessary, of Moseley’s principle of the least resistance, the whole of the relations amongst the loads and resistances may be found.
§ 10.Line of Pressures—Centres and Line of Resistance.—The line of pressures is a line to which the directions of all the resistances in one polygon are tangents. Thecentre of resistanceat any joint is the point where the line representing the total resistance exerted at that joint intersects the joint. Theline of resistanceis a line traversing all the centres of resistance of a series of joints,—its form, in the positions intermediate between the actual joints of the structure, being determined by supposing the pieces and their loads to be subdivided by the introduction of intermediate jointsad infinitum, and finding the continuous line, curved or straight, in which the intermediate centres of resistance are all situated, however great their number. The difference between the line of resistance and the line of pressures was first pointed out by Moseley.
§ 11.* The principles of the two preceding sections may be illustrated by the consideration of a particular case of a buttress of blocks forming a continuous series of pieces (fig. 88), where aa, bb, cc, dd represent plane joints. Let the centre of pressure C at the first joint aa be known, and also the pressure P acting at C in direction and magnitude. Find R1the resultant of this pressure, the weight of the block aabb acting through its centre of gravity, and any other external force which may be acting on the block, and produce its line of action to cut the joint bb in C1. C1is then the centre of pressure for the joint bb, and R1is the total force acting there. Repeating this process for each block in succession there will be found the centres of pressure C2, C3, &c., and also the resultant pressures R2, R3, &c., acting at these respective centres. The centres of pressure at the joints are also calledcentres of resistance, and the curve passing through these points is called aline of resistance. Let all the resultants acting at the several centres of resistance be produced until they cut one another in a series of points so as to form an unclosed polygon. This polygon is thepartial polygon of resistance. A curve tangential to all the sides of the polygon is theline of pressures.
§ 12.Stability of Position, and Stability of Friction.—The resistances at the several joints having been determined by the principles set forth in §§ 6, 7, 8, 9 and 10, not only under the ordinary load of the structure, but under all the variations to which the load is subject as to amount and distribution, the joints are now to be placed and shaped so that the pieces shall not suffer relative displacement under any of those loads. The relative displacement of the two pieces which abut against each other at a joint may take place eitherby turning or by sliding. Safety against displacement by turning is calledstability of position; safety against displacement by sliding,stability of friction.
§ 13.Condition of Stability of Position.—If the materials of a structure were infinitely stiff and strong, stability of position at any joint would be insured simply by making the centre of resistance fall within the joint under all possible variations of load. In order to allow for the finite stiffness and strength of materials, the least distance of the centre of resistance inward from the nearest edge of the joint is made to bear a definite proportion to the depth of the joint measured in the same direction, which proportion is fixed, sometimes empirically, sometimes by theoretical deduction from the laws of the strength of materials. That least distance is called by Moseley themodulus of stability. The following are some of the ratios of the modulus of stability to the depth of the joint which occur in practice:—
In the case of the towers, thedepth of the jointis to be understood to mean thediameter of the tower.
§ 14.Condition of Stability of Friction.—If the resistance to be exerted at a joint is always perpendicular to the surfaces which abut at and form that joint, there is no tendency of the pieces to be displaced by sliding. If the resistance be oblique, let JK (fig. 89) be the joint, C its centre of resistance, CR a line representing the resistance, CN a perpendicular to the joint at the centre of resistance. The angle NCR is theobliquityof the resistance. From R draw RP parallel and RQ perpendicular to the joint; then, by the principles of statics, the component of the resistancenormalto the joint is—
CP = CR · cos PCR;
and the componenttangentialto the joint is—
CQ = CR · sin PCR = CP · tan PCR.
If the joint be provided either with projections and recesses, such as mortises and tenons, or with fastenings, such as pins or bolts, so as to resist displacement by sliding, the question of the utmost amount of the tangential resistance CQ which it is capable of exerting depends on thestrengthof such projections, recesses, or fastenings; and belongs to the subject of strength, and not to that of stability. In other cases the safety of the joint against displacement by sliding depends on its power of exerting friction, and that power depends on the law, known by experiment, that the friction between two surfaces bears a constant ratio, depending on the nature of the surfaces, to the force by which they are pressed together. In order that the surfaces which abut at the joint JK may be pressed together, the resistance required by the conditions of equilibrium CR, must be athrustand not apull; and in that case the force by which the surfaces are pressed together is equal and opposite to the normal component CP of the resistance. The condition of stability of friction is that the tangential component CQ of the resistance required shall not exceed the friction due to the normal component; that is, that
CQ ≯ ƒ · CP,
where ƒ denotes thecoefficient of frictionfor the surfaces in question. The angle whose tangent is the coefficient of friction is calledthe angle of repose, and is expressed symbolically by—
φ = tan−1ƒ.
Now CQ = CP · tan PCR;
consequently the condition of stability of friction is fulfilled if the angle PCR is not greater than φ; that is to say, ifthe obliquity of the resistance required at the joint does not exceed the angle of repose; and this condition ought to be fulfilled under all possible variations of the load.
It is chiefly in masonry and earthwork that stability of friction is relied on.
§ 15.Stability of Friction in Earth.—The grains of a mass of loose earth are to be regarded as so many separate pieces abutting against each other at joints in all possible positions, and depending for their stability on friction. To determine whether a mass of earth is stable at a given point, conceive that point to be traversed by planes in all possible positions, and determine which position gives the greatest obliquity to the total pressure exerted between the portions of the mass which abut against each other at the plane. The condition of stability is that this obliquity shall not exceed the angle of repose of the earth. The consequences of this principle are developed in a paper, “On the Stability of Loose Earth,” already cited in § 2.
§ 16.Parallel Projections of Figures.—If any figure be referred to a system of co-ordinates, rectangular or oblique, and if a second figure be constructed by means of a second system of co-ordinates, rectangular or oblique, and either agreeing with or differing from the first system in rectangularity or obliquity, but so related to the co-ordinates of the first figure that for each point in the first figure there shall be a corresponding point in the second figure, the lengths of whose co-ordinates shall bear respectively to the three corresponding co-ordinates of the corresponding point in the first figure three ratios which are the same for every pair of corresponding points in the two figures, these corresponding figures are calledparallel projectionsof each other. The properties of parallel projections of most importance to the subject of the present article are the following:—
(1) A parallel projection of a straight line is a straight line.
(2) A parallel projection of a plane is a plane.
(3) A parallel projection of a straight line or a plane surface divided in a given ratio is a straight line or a plane surface divided in the same ratio.
(4) A parallel projection of a pair of equal and parallel straight lines, or plain surfaces, is a pair of equal and parallel straight lines, or plane surfaces; whence it follows
(5) That a parallel projection of a parallelogram is a parallelogram, and
(6) That a parallel projection of a parallelepiped is a parallelepiped.
(7) A parallel projection of a pair of solids having a given ratio is a pair of solids having the same ratio.
Though not essential for the purposes of the present article, the following consequence will serve to illustrate the principle of parallel projections:—
(8) A parallel projection of a curve, or of a surface of a given algebraical order, is a curve or a surface of the same order.
For example, all ellipsoids referred to co-ordinates parallel to any three conjugate diameters are parallel projections of each other and of a sphere referred to rectangular co-ordinates.
§ 17.Parallel Projections of Systems of Forces.—If a balanced system of forces be represented by a system of lines, then will every parallel projection of that system of lines represent a balanced system of forces.
For the condition of equilibrium of forces not parallel is that they shall be represented in direction and magnitude by the sides and diagonals of certain parallelograms, and of parallel forces that they shall divide certain straight lines in certain ratios; and the parallel projection of a parallelogram is a parallelogram, and that of a straight line divided in a given ratio is a straight line divided in the same ratio.
The resultant of a parallel projection of any system of forces is the projection of their resultant; and the centre of gravity of a parallel projection of a solid is the projection of the centre of gravity of the first solid.
§ 18.Principle of the Transformation of Structures.—Here we have the following theorem: If a structure of a given figure have stability of position under a system of forces represented by a given system of lines, then will any structure whose figure is a parallel projection of that of the first structure have stability of position under a system of forces represented by the corresponding projection of the first system of lines.
For in the second structure the weights, external pressures, and resistances will balance each other as in the first structure; the weights of the pieces and all other parallel systems of forces will have the same ratios as in the first structure; and the several centres of resistance will divide the depths of the joints in the same proportions as in the first structure.
If the first structure have stability of friction, the second structure will have stability of friction also, so long as the effect of the projection is not to increase the obliquity of the resistance at any joint beyond the angle of repose.
The lines representing the forces in the second figure show theirrelativedirections and magnitudes. To find theirabsolutedirections and magnitudes, a vertical line is to be drawn in the first figure, of such a length as to represent the weight of a particular portion of the structure. Then will the projection of that line in the projected figure indicate the vertical direction, and represent the weight of the part of the second structure corresponding to the before-mentioned portion of the first structure.
The foregoing “principle of the transformation of structures” was first announced, though in a somewhat less comprehensive form, to the Royal Society on the 6th of March 1856. It is useful in practice, by enabling the engineer easily to deduce the conditions of equilibrium and stability of structures of complex and unsymmetrical figures from those of structures of simple and symmetrical figures. By its aid, for example, the whole of the properties ofelliptical arches, whether square or skew, whether level or sloping in their span, are at once deduced by projection from those of symmetrical circular arches, and the properties of ellipsoidal and elliptic-conoidal domes from those of hemispherical and circular-conoidal domes; and the figures of arches fitted to resist the thrust of earth, which is less horizontally than vertically in a certain given ratio, can be deduced by a projection from those of arches fitted to resist the thrust of a liquid, which is of equal intensity, horizontally and vertically.
§ 19.Conditions of Stiffness and Strength.—After the arrangement of the pieces of a structure and the size and figure of their joints or surfaces of contact have been determined so as to fulfil the conditions ofstability,—conditions which depend mainly on the position and direction of theresultantortotalload on each piece, and therelativemagnitude of the loads on the different pieces—the dimensions of each piece singly have to be adjusted so as to fulfil the conditions ofstiffnessandstrength—conditions which depend not only on theabsolutemagnitude of the load on each piece, and of the resistances by which it is balanced, but also on themode of distributionof the load over the piece, and of the resistances over the joints.
The effect of the pressures applied to a piece, consisting of the load and the supporting resistances, is to force the piece into a state ofstrainor disfigurement, which increases until the elasticity, or resistance to strain, of the material causes it to exert astress, or effort to recover its figure, equal and opposite to the system of applied pressures. The condition ofstiffnessis that the strain or disfigurement shall not be greater than is consistent with the purposes of the structure; and the condition ofstrengthis that the stress shall be within the limits of that which the material can bear with safety against breaking. The ratio in which the utmost stress before breaking exceeds the safe working stress is called thefactor of safety, and is determined empirically. It varies from three to twelve for various materials and structures. (SeeStrength of Materials.)
PART II. THEORY OF MACHINES
§ 20.Parts of a Machine: Frame and Mechanism.—The parts of a machine may be distinguished into two principal divisions,—the frame, or fixed parts, and themechanism, or moving parts. The frame is a structure which supports the pieces of the mechanism, and to a certain extent determines the nature of their motions.
The form and arrangement of the pieces of the frame depend upon the arrangement and the motions of the mechanism; the dimensions of the pieces of the frame required in order to give it stability and strength are determined from the pressures applied to it by means of the mechanism. It appears therefore that in general the mechanism is to be designed first and the frame afterwards, and that the designing of the frame is regulated by the principles of the stability of structures and of the strength and stiffness of materials,—care being taken to adapt the frame to the most severe load which can be thrown upon it at any period of the action of the mechanism.
Each independent piece of the mechanism also is a structure, and its dimensions are to be adapted, according to the principles of the strength and stiffness of materials, to the most severe load to which it can be subjected during the action of the machine.
§ 21.Definition and Division of the Theory of Machines.—From what has been said in the last section it appears that the department of the art of designing machines which has reference to the stability of the frame and to the stiffness and strength of the frame and mechanism is a branch of the art of construction. It is therefore to be separated from thetheory of machines, properly speaking, which has reference to the action of machines considered as moving. In the action of a machine the following three things take place:—
Firstly, Some natural source of energy communicates motion and force to a piece or pieces of the mechanism, called thereceiver of powerorprime mover.
Secondly, The motion and force are transmitted from the prime mover through thetrain of mechanismto theworking pieceorpieces, and during that transmission the motion and force are modified in amount and direction, so as to be rendered suitable for the purpose to which they are to be applied.
Thirdly, The working piece or pieces by their motion, or by their motion and force combined, produce some useful effect.
Such are the phenomena of the action of a machine, arranged in the order ofcausation. But in studying or treating of the theory of machines, the order ofsimplicityis the best; and in this order the first branch of the subject is the modification of motion and force by the train of mechanism; the next is the effect or purpose of the machine; and the last, or most complex, is the action of the prime mover.
The modification of motion and the modification of force take place together, and are connected by certain laws; but in the study of the theory of machines, as well as in that of pure mechanics, much advantage has been gained in point of clearness and simplicity by first considering alone the principles of the modification of motion, which are founded upon what is now known as Kinematics, and afterwards considering the principles of the combined modification of motion and force, which are founded both on geometry and on the laws of dynamics. The separation of kinematics from dynamics is due mainly to G. Monge, Ampère and R. Willis.
The theory of machines in the present article will be considered under the following heads:—
I.Pure Mechanism, orApplied Kinematics; being the theory of machines considered simply as modifying motion.II.Applied Dynamics; being the theory of machines considered as modifying both motion and force.
I.Pure Mechanism, orApplied Kinematics; being the theory of machines considered simply as modifying motion.
II.Applied Dynamics; being the theory of machines considered as modifying both motion and force.
Chap. I. On Pure Mechanism
§ 22.Division of the Subject.—Proceeding in the order of simplicity, the subject of Pure Mechanism, or Applied Kinematics, may be thus divided:—
Division 1.—Motion of a point.Division 2.—Motion of the surface of a fluid.Division 3.—Motion of a rigid solid.Division 4.—Motions of a pair of connected pieces, or of an “elementary combination” in mechanism.Division 5.—Motions of trains of pieces of mechanism.Division 6.—Motions of sets of more than two connected pieces, or of “aggregate combinations.”
Division 1.—Motion of a point.
Division 2.—Motion of the surface of a fluid.
Division 3.—Motion of a rigid solid.
Division 4.—Motions of a pair of connected pieces, or of an “elementary combination” in mechanism.
Division 5.—Motions of trains of pieces of mechanism.
Division 6.—Motions of sets of more than two connected pieces, or of “aggregate combinations.”
A point is the boundary of a line, which is the boundary of a surface, which is the boundary of a volume. Points, lines and surfaces have no independent existence, and consequently those divisions of this chapter which relate to their motions are only preliminary to the subsequent divisions, which relate to the motions of bodies.
Division 1. Motion of a Point.
§ 23.Comparative Motion.—The comparative motion of two points is the relation which exists between their motions, without having regard to their absolute amounts. It consists of two elements,—thevelocity ratio, which is the ratio of any two magnitudes bearing to each other the proportions of the respective velocities of the two points at a given instant, and thedirectional relation, which is the relation borne to each other by the respective directions of the motions of the two points at the same given instant.
It is obvious that the motions of a pair of points may be varied in any manner, whether by direct or by lateral deviation, and yet that theircomparative motionmay remain constant, in consequence of the deviations taking place in the same proportions, in the same directions and at the same instants for both points.
Robert Willis (1800-1875) has the merit of having been the first to simplify considerably the theory of pure mechanism, by pointing out that that branch of mechanics relates wholly to comparative motions.
The comparative motion of two points at a given instant is capable of being completely expressed by one of Sir William Hamilton’s Quaternions,—the “tensor” expressing the velocity ratio, and the “versor” the directional relation.
Graphical methods of analysis founded on this way of representing velocity and acceleration were developed by R. H. Smith in a paper communicated to the Royal Society of Edinburgh in 1885, and illustrations of the method will be found below.
Division 2. Motion of the Surface of a Fluid Mass.
§ 24.General Principle.—A mass of fluid is used in mechanism to transmit motion and force between two or more movable portions (calledpistonsorplungers) of the solid envelope or vessel in which the fluid is contained; and, when such transmission is the sole action, or the only appreciable action of the fluid mass, its volume is either absolutely constant, by reason of its temperature and pressure being maintained constant, or not sensibly varied.
Let a represent the area of the section of a piston made by a plane perpendicular to its direction of motion, and v its velocity, which is to be considered as positive when outward, and negative when inward. Then the variation of the cubic contents of the vessel in a unit of time by reason of the motion of one piston is va. The condition that the volume of the fluid mass shall remain unchanged requires that there shall be more than one piston, and that the velocities and areas of the pistons shall be connected by the equation—
Σ · va = 0.
(1)
§ 25.Comparative Motion of Two Pistons.—If there be but two pistons, whose areas are a1and a2, and their velocities v1and v2, their comparative motion is expressed by the equation—
v2/v1= −a1/a2;
(2)
that is to say, their velocities are opposite as to inwardness and outwardness and inversely proportional to their areas.
§ 26.Applications: Hydraulic Press: Pneumatic Power-Transmitter.—In the hydraulic press the vessel consists of two cylinders, viz. the pump-barrel and the press-barrel, each having its piston, and of a passage connecting them having a valve opening towards the press-barrel. The action of the enclosed water in transmitting motion takes place during the inward stroke of the pump-plunger, when the above-mentioned valve is open; and at that time the press-plunger moves outwards with a velocity which is less than the inward velocity of the pump-plunger, in the same ratio that the area of the pump-plunger is less than the area of the press-plunger. (SeeHydraulics.)
In the pneumatic power-transmitter the motion of one piston istransmitted to another at a distance by means of a mass of air contained in two cylinders and an intervening tube. When the pressure and temperature of the air can be maintained constant, this machine fulfils equation (2), like the hydraulic press. The amount and effect of the variations of pressure and temperature undergone by the air depend on the principles of the mechanical action of heat, orThermodynamics(q.v.), and are foreign to the subject of pure mechanism.
Division 3. Motion of a Rigid Solid.
§ 27.Motions Classed.—In problems of mechanism, each solid piece of the machine is supposed to be so stiff and strong as not to undergo any sensible change of figure or dimensions by the forces applied to it—a supposition which is realized in practice if the machine is skilfully designed.
This being the case, the various possible motions of a rigid solid body may all be classed under the following heads: (1)Shifting or Translation; (2)Turning or Rotation; (3)Motions compounded of Shifting and Turning.
The most common forms for the paths of the points of a piece of mechanism, whose motion is simple shifting, are the straight line and the circle.
Shifting in a straight line is regulated either by straight fixed guides, in contact with which the moving piece slides, or by combinations of link-work, calledparallel motions, which will be described in the sequel. Shifting in a straight line is usuallyreciprocating; that is to say, the piece, after shifting through a certain distance, returns to its original position by reversing its motion.
Circular shifting is regulated by attaching two or more points of the shifting piece to ends of equal and parallel rotating cranks, or by combinations of wheel-work to be afterwards described. As an example of circular shifting may be cited the motion of the coupling rod, by which the parallel and equal cranks upon two or more axles of a locomotive engine are connected and made to rotate simultaneously. The coupling rod remains always parallel to itself, and all its points describe equal and similar circles relatively to the frame of the engine, and move in parallel directions with equal velocities at the same instant.
§ 28.Rotation about a Fixed Axis: Lever, Wheel and Axle.—The fixed axis of a turning body is a line fixed relatively to the body and relatively to the fixed space in which the body turns. In mechanism it is usually the central line either of a rotating shaft or axle having journals, gudgeons, or pivots turning in fixed bearings, or of a fixed spindle or dead centre round which a rotating bush turns; but it may sometimes be entirely beyond the limits of the turning body. For example, if a sliding piece moves in circular fixed guides, that piece rotates about an ideal fixed axis traversing the centre of those guides.
Let the angular velocity of the rotation be denoted by α = dθ/dt, then the linear velocity of any point A at the distance r from the axis is αr; and the path of that point is a circle of the radius r described about the axis.
This is the principle of the modification of motion by the lever, which consists of a rigid body turning about a fixed axis called a fulcrum, and having two points at the same or different distances from that axis, and in the same or different directions, one of which receives motion and the other transmits motion, modified in direction and velocity according to the above law.
In the wheel and axle, motion is received and transmitted by two cylindrical surfaces of different radii described about their common fixed axis of turning, their velocity-ratio being that of their radii.
§ 29.Velocity Ratio of Components of Motion.—As the distance between any two points in a rigid body is invariable, the projections of their velocities upon the line joining them must be equal. Hence it follows that, if A in fig. 90 be a point in a rigid body CD, rotating round the fixed axis F, the component of the velocity of A in any direction AP parallel to the plane of rotation is equal to the total velocity of the point m, found by letting fall Fm perpendicular to AP; that is to say, is equal to
α · Fm.
Hence also the ratio of the components of the velocities of two points A and B in the directions AP and BW respectively, both in the plane of rotation, is equal to the ratio of the perpendiculars Fm and Fn.
§ 30.Instantaneous Axis of a Cylinder rolling on a Cylinder.—Let a cylinder bbb, whose axis of figure is B and angular velocity γ, roll on a fixed cylinder ααα, whose axis of figure is A, either outside (as in fig. 91), when the rolling will be towards the same hand as the rotation, or inside (as in fig. 92), when the rolling will be towards the opposite hand; and at a given instant let T be the line of contact of the two cylindrical surfaces, which is at their common intersection with the plane AB traversing the two axes of figure.
The line T on the surface bbb has for the instant no velocity in a direction perpendicular to AB; because for the instant it touches, without sliding, the line T on the fixed surface aaa.
The line T on the surface bbb has also for the instant no velocity in the plane AB; for it has just ceased to move towards the fixed surface aaa, and is just about to begin to move away from that surface.
The line of contact T, therefore, on the surface of the cylinder bbb, isfor the instantat rest, and is the “instantaneous axis” about which the cylinder bbb turns, together with any body rigidly attached to that cylinder.
To find, then, the direction and velocity at the given instant of any point P, either in or rigidly attached to the rolling cylinder T, draw the plane PT; the direction of motion of P will be perpendicular to that plane, and towards the right or left hand according to the direction of the rotation of bbb; and the velocity of P will be
vP= γ·PT,
(3)
PT denoting the perpendicular distance of P from T. The path of P is a curve of the kind calledepitrochoids. If P is in the circumference of bbb, that path becomes anepicycloid.
The velocity of any point in the axis of figure B is
vB= γ·TB;
(4)
and the path of such a point is a circle described about A with the radius AB, being for outside rolling the sum, and for inside rolling the difference, of the radii of the cylinders.
Let α denote the angular velocity with which theplane of axesAB rotates about the fixed axis A. Then it is evident that
vB= α·AB,
(5)
and consequently that
α = γ·TB/AB.