APl LIED MECHANICS.] MECHANICS 755 42. To find the Motion of a Rigid Body from the Motions of Three Points in it. See p. 690, 71, and p. 692, 78. Division 4. Elementary Combinations in Mechanism. 43. Definitions. An elementary combination in mechanism con sists of tvo pieces whose kinds of motion are determined by their connexion with the frame, and their comparative motion by their connexion with each other, that connexion being effected either by direct contact of the pieces, or by a connecting piece, which is not connected with the frame, and whose motion depends entirely on the motions of the pieces which it connects. The piece whose motion is the cause is called the driver ; the piece whose motion is the effect, the follower. The connexion of each of those two pieces with the frame is in general such as to determine the path of every point in it. In the investigation, therefore, of the comparative motion of the driver and follower, in an elementary combination, it is unnecessary to consider relations of angular direction, which are already fixed by the connexion of each piece with the frame ; so that the inquiry is confined to the determination of the velocity ratio, and of the directional relation, so far only as it expresses the connexion between forward and backward movements of the driver and follower. When a continuous motion of the driver produces a con tinuous motion of the follower, forward or backward, and a recipro cating motion a motion reciprocating at the same instant, the directional relation is said to be constant. When a continuous motion produces a reciprocating motion, or vice versa, or when a reciprocating motion produces a motion not reciprocating at the same instant, the directional relation is said to be variable. The line of action or of connexion of the driver and follower is a line traversing a pair of points in the driver and follower respectively, which are so connected that the component of their velocity re latively to each other, resolved along the line of connexion, is null. There may be several or an indefinite number of lines of connexion, or there may be but one ; and a line of connexion may connect either the same pair of points or a succession of different pairs. 44. General Principle. From the definition of a line of connexion it follows that the components of the velocities of a pair of connected points along their line of connexion are equal. And from this, and from the property of a rigid body, already stated in sect. 36, it follows, that the components along a line of connexion of all the points traversed by that line, whether in tJie driver or in the follower, arc equal ; and consequently, that the velocities of any pair of points traversed by a line of connexion are to each other inversely as the cosines, or directly as the secants, of tJie angles made by the paths of those points with the line of connexion. The general principle stated above in different forms serves to solve every problem in which the mode of connexion of a pair of pieces being given it is required to find their comparative motion at a given instant, or vice versa. 45. Application to a Pair of Shifting Pieces. In fig. 9, let PiPg be the line of connexion of a pair of pieces, each of which has a motion of translation or shift ing. Through any point T in that line draw TV lf TV,, re spectively parallel to the simul- taneous direction of motion of the pieces ; through any other point A in the line of connexion draw a plane perpendicular to that line", cutting TV 1; TV 2 in Vj, V,. ; then, velocity of piece 1 : velocity of piece 2 : : TVj : TV 2 . Also TA represents the equal components of the velocities of the pieces parallel to their line of connexion, -and the line V X V 2 represents their velocity relatively to each other. 46. Application to a Pair of Turning Pieces. Let o 1; a,, be the angular velocities of a pair of turning pieces ; 6 lt 6. 2 the angles which their line of connexion makes with their respective planes of rotation ; r 1( r n the common perpendiculars let fall from the line of connexion upon the respective axes of rotation of the pieces. Then the equal components, along the line of connexion, of the velocities of the points where those perpendiculars meet that line direction of motion of the shifting piece, v. 2 the linear velocity of that piece. Then Fig. 9. consequently, the comparative motion of the pieces is given by the equation 47. Application to a Shifting Piece and a Turning Piece. Let a shifting piece be connected with a turning piece, and at a given instant let ai be the angular velocity of the turning peice, r x the common perpendicular of its axis of rotation and the line of con nexion, 0j the angle made by the line of connexion with the plane of rotation, 2 the angle made by the line of connexion with the which equation expresses the comparative motion of the two pieces. 48. Classification of Elementary Combinations in Mechanism. The first systematic classification of elementary combinations in mechanism was that founded by Monge, and fully developed by Lanz and Betancourt, which has been generally received, a.nd has been adopted in most treatises on applied mechanics. But that classification is founded on the absolute instead of the comparative motions of the pieces, and is, for that reason, defective, as Willis has pointed out in his admirable treatise On the Principles of Mechanism. Willis s classification is founded, in the first place, on comparative motion, as expressed by velocity ratio and directional relation, and in the second place, on the mode of connexion of the driver and follower. He divides the elementary combinations in mechanism into three classes, of which the characters are as follows : Class A: Directional relation constant; velocity ratio constant. Class B : Directional relation constant ; velocity ratio varying. Class C : Directional relation changing periodically ; velocity ratio constant or varying. Each of those classes is subdivided by Willis into five divisions, of which the characters are as follows: Division A : Connexion by rolling contact. ,, B: ,, ,, sliding contact. ,, C: ,, ,, wrapping connectors. ,, D: ,, link-work. ,, E: ,, ,, reduplication. In the present article the principle of Willis s classification is followed ; but the aVrangement is modified by taking the mode of connexion as the basis of the primary classification, and by removing the subject of connexion by reduplication to the section of aggregate combinations. This modified arrangement is adopted as being better suited than the original arrangement to the limits of an article in an encyclopedia ; but it is not disputed that the original arrangement may be the best for a separate treatise. 49. Rolling Contact Smooth Wheels and Racks.- -In order that two pieces may move in rolling contact, it is necessary that each pair of points in the two pieces which touch each other should at the instant of contact be moving in the same direction with the same velocity. In the case of two shifting pieces this would involve equal and parallel velocities for all the points of each piece, so that there could be no rolling, and, in fact, the two pieces would move like one ; hence, in the case of rolling contact, either one or both of the pieces must rotate. The direction of motion of a point in a turning piece being per pendicular to a plane passing through its axis, the condition that each pair of points in contact with each other must move in the same direction leads to the following confluences : I. That, when both pieces rotate, their axes, and all their points of contact, lie in the same plane. II. That, when one piece rotates and the other shifts, the axis of the rotating piece, and all the points of contact, lie in a plane per- pendicular to the direction of motion of the shifting piece. The condition that the velocity of each pair of points of contact must be equal leads to the following consequences : III. That the angular velocities of a pair of turning pieces rn rolling contact must be inversely as the perpendicular distances of any pair of points of contact from the respective axes. IV. That the linear velocity of a shifting piece in rolling contact with a turning piece is equal to the product of the angular velocity of the turning piece by the perpendicular distance from its axis to a pair of points of contact. The line of contact is that line in which the points of contact are all situated. Respecting this line, the above principles III. and IV. lead to the following conclusions : V. That for a pair of turning pieces with parallel axes, and for a turning piece and a shifting piece, the line of contact is straight, and parallel to the axes or axis ; and hence that the rolling sur faces are either plane or cylindrical (the term "cylindrical " includ ing all surfaces generated by the motion of a straight line parallel to its elf). VI. That for a pair of turning pieces with intersecting axes the line of contact is also straight, and traverses the point of intersec tion of the axes ; and hence that the rolling surfaces are conical, with a common apex (the term " conical " including all surfaces gene rated by the motion of a straight line which traverses a fixed point). Turning pieces in rolling contact are called smooth or toothless wheels. Shifting pieces in rolling contact with turning pieces may be called smooth or toothless racks. VII. In a pair of pieces in rolling contact every straight line traversing the line of contact is a line of connexion. 50. Cylindrical Wheels and Smooth Racks. In designing cylm- drical wheels and smooth racks, and determining their comparative motion, it is sufficient to consider a section of the pair of piec
made by a plane perpendicular to the axis or axes.