and E receive motion from four number of component motions. For example, in fig. 181, A, B, D, different drivers: C has a motion whose components depend 239. Harmonic Motion in Aggregate Linkwork.-By harmonic motion is to be understood the motion of a point which moves to and fro in a straight line in such a manner that its velocity at every instant is equal to the component, parallel to that straight line, of another point which revolves uniformly in a circle. The length of the straight line is called the travel of the reciprocating point, and is equal to the diameter of the circle. (As to the component velocities of a revolving point, see Article 55, pages 34, 35.) Harmonic motion is exactly realized by any point in a slotheaded sliding rod, driven by an uniformly rotating crank, as explained in Article 159, page 169. The angle which the crank makes with its dead points is called, in mathematical language, the phase of the motion. The velocity of the reciprocating point varies proportionally to the sine of the phase; and the distance of that point from its middle position varies as the cosine of the phase. Harmonic motion is approximately realized by any point in a piece, such as a piston, which is driven by means of a connectingrod and an uniformly rotating crank. The extent of error in that approximation may be expressed either in the form of greatest error in position or of greatest error in velocity. The greatest error in position is the distance of the reciprocating point from the middle of its travel, when the crank is midway between its dead points; and when the line of stroke passes through the axis of the crank, its value may be found either by constructing a figure, or by the following formula: 1- √ 12 – c2; in which denotes the length of the line of connection, and c that of the crank-arm. The comparative error in position is the ratio of this error to the half-travel c; that is to say, which, when is many times greater than c, is nearly equal to C 21 The greatest error in velocity is the proportionate excess of the greatest velocity of the reciprocating piece above that of the crank-pin, as found by the rules of Article 188, pages 199 to 201. When is not less than 2 c, the value of the error in velocity is given approximately by the expression When the line of stroke does not pass through the axis of rotation of the crank, there are other errors arising from the two dead points not being diametrically opposite. Those errors may be found by applying the rules of Article 196, page 198. The present and the following Article relate to cases in which two points in a bar receive given transverse movements, which are either exactly harmonic, or so nearly so that they may be treated as harmonic for practical purposes, and are also of equal period, and have a given constant difference of phase; and it is required to find the extent of travel and the relative phase of the motion of a third point, situated either exactly or nearly in one straight line with the first two. The following is the general rule for the solution of all such cases. Some of its applications will be given in the next Article: RULE. In fig. 182 draw the straight line A B to represent the bar in question, and let A and B represent the points whose motions are given, and C the point whose motion is to be found. Perpendicular to A B, draw A a to represent the half-travel of A, and Bb to represent the half-travel of B. These distances may be laid off in both directions, so that a a shall represent the whole travel of A, and b b that of B. The difference of phase of A and B is supposed to be given; that is to say, A moves as if driven by a crank A' A" (= A a), and B as if driven by a crank B' B" (= B b), which cranks rotate with the same angular velocity, and make a given constant angle with each other. = At A and B lay off the angles B A D A B D, each equal to half the difference of phase; and about the triangle A D B describe a circle. Join a b, a b, and through the point of intersection, E, draw the straight line D E, cutting the circle in F. Join F A, F B; then the angle A F B will be equal to the given difference of phase. Lay off Fa A a, and Fb' Bb; then Fa' and Fb' will represent the two cranks which actually or virtually drive A and B, in their angular position relatively to each other. Join a b'; this will be parallel to A B (because it can be shown by plane = = geometry that FAB and Fa'b' are similar triangles). Finally, draw the straight line F C, cutting a b'in d'; then the point C will move almost exactly as if it were driven by a crank-arm, C' C', equal in length to F c', and having the angular position relatively to the cranks that drive A and B which F c' has relatively to Fa' and F b'; that is to say, being in advance of the crank which drives A by the angle a' F c', and behind the crank which drives B by the angle b' F c'. The travel of C may be represented in the figure by drawing, perpendicular to A B, the straight line c c = 2 Cc 2 F c. When the extent of travel of A and B is the same, part of the trouble of the construction is saved; for the point F is found simply by laying off the angles BAF = A BF, each equal to half the supplement of the difference of phase. The construction which has been described solves the problem by drawing alone. Sometimes it may be convenient to use calculation combined with drawing; and then the whole process consists in drawing the triangle F a b' in any convenient position, with its legs, F a' and F b', equal to the half-travel of the points A and B respectively, and its angle, a' F b', equal to the difference of phase of their motions, and dividing, by calculation, the base a' b'at c' in the same proportion in which A B is divided at C. 240. Link-Motions for Slide-Valves belong to the kind of combinations mentioned in the preceding Article. The bar which receives harmonic motion is called the link; it is in general slightly curved, and only sometimes straight. Two points in it, marked A and B in figs. 183 to 186, receive approximately-harmonic motions from two eccentrics, E and F, on the engine-shaft, O, called respectively the forward and the backward eccentrics. The link carries a slider, C. That slider is attached to the head of the slidevalve spindle either directly (as shown at C in figs. 183, 184, and 185), or by means of an intermediate rod, C X (as in figs. 186, 187). The slider is capable of being adjusted to different positions in the link, either by shifting the link (as in figs. 183, 184, and 185, which represent Stephenson's link-motion) or by shifting the slider (as in fig. 186, which represents Gooch's link-motion), or by shifting the link and the slider at the same time in opposite directions (as in Allan's link-motion, represented in fig. 187). In Stephenson's link-motion the form of the link is an arc of a circle, concave towards the shaft, and of a radius equal to the length of the eccentric rods E A, F B. In Gooch's link-motion the figure of the link is an arc of a circle described about the head, X, of the valve-spindle. In Allan's link-motion the link is straight, and the adjustment of the proportions of the mechanism for shifting it will be described presently. In each case the object is, that the shifting of the position of the slider, C, relatively to the link, A B, shall not cause any sensible alteration of the middle position of the slide-valve. In each of the figures, O D represents the crank of the engine to which the link-motion belongs, the positions of the parts being those which they take when that crank is at a deadpoint. In each of the figures, also, the eccentrics are represented simply by points, E, F, which mark the centres of the eccentric discs. It has already been explained, in Article 195, page 197, that an |