Page:Encyclopædia Britannica, Ninth Edition, v. 15.djvu/785

APPLIED MECHANICS.] MECHANICS 753 26. Lateral Deviation or Deflexion Angular Velocity of Deria- timt Revolution. See pp. 681, 682, 31-38. 27. 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, the velocity ratio, which is the ratio of any two magnitudes bear ing to each other the proportions of the respective velocities of the two points at a given instant, and the directional 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 their comparative motion may 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. Willis has the merit of having been the first to simplify con siderably 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. 28. Resolution and Composition of Motion. See p. 681, 30, 31. 29. Rectangular Projection, Resolution, and Composition. Seep. 681, 31. 30. Resolution and Composition of Deviations. See p. 681, 31. Division 2. Motion of the Surface of a Fluid Mass. 31. General Principle. A mass of fluid is used in mechanism to transmit motion and force between two or more movable portions (called pistons or plungers) 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 2. va = Q ........ (1). 32. Comparative Motion of two Pistons. If there be but two pistons, whose areas are a 1 and a a , and their velocities v i and v.^ their comparative motion is expressed by the equation (2); that is to say, their velocities are opposite as to inwardness and out wardness, and inversely proportional to their areas. 33. Applications Hydraulic Press Pneumatic Power-Trans mitter. 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. (See HYDROMECHANICS.) In the pneumatic power-transmitter the motion of one piston is transmitted to another at a distance by means of a mass of air con tained in two cylinders and an intervening tube. When the pres sure 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, or THERMODYNAMICS (g.v.), and are foreign to the subject of pure mechanism. Division 3. Motion of a Rigid Solid. 34. 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 com binations of link-work, called parallel motions, which will be described in the sequel. Shifting in a straight line is usually reciprocating; 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. 35. 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 bear ings, 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 a = -^ , dt then the linear velocity of any point A at the distance r from the axis is ar ; 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. 36. Telocity Ratio of Components of Motion. As tlie distance between any two points in a rigid body is invariable, the projections of their velocities upon the line join ing them must be equal. Hence it follows that, if A in fig. 4 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 in, found by letting fall Fm perpendicular to AP j that is to say, is equal to 4. Hence also the ratio of the com ponents 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. 37. Instantaneous Axis of a Cylinder rolling on a Cylinder. Let a cylinder bbb, whose axis of figure is B and angular velocity y, roll on a fixed cylinder aaa. whose axis of figure is A, either outside (as in fig. 5), when the rolling will be towards the same hand as the Fijr. 5. Fig. 6. rotation, or inside (as in fig. 6), when the rolling will be towards the opposite hand ; and at a given instant let T be the line of con tact of the two cylindrical surfaces, which is at their common inter section with the plane AB traversing the two axes of figure. The line T on the surface bbb has for the instant no velocity in

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