that is to say, the angular velocity about each axis is proportional to the sine of the angle between the other two.
Demonstration.—From C draw CF perpendicular to OA, and CG perpendicular to OE
| Then CF = 2 × | area EC | , |
| CE |
| and CG = 2 × | area ECO | ; |
| OE |
Let vc denote the linear velocity of the point C. Then
∴ γ : α :: CF : CG :: OE : OD,
which is one part of the solution above stated. From E draw EH perpendicular to OB, and EK to OA. Then it can be shown as before that
Let vE be the linear velocity of the point E fixed in the plane of axes AOB. Then
Now, as the line of contact OT is for the instant at rest on the rolling cone as well as on the fixed cone, the linear velocity of the point E fixed to the plane AOB relatively to the rolling cone is the same with its velocity relatively to the fixed cone. That is to say,
therefore
which is the remainder of the solution.
The path of a point P in or attached to the rolling cone is a spherical epitrochoid traced on the surface of a sphere of the radius OP. From P draw PQ perpendicular to the instantaneous axis. Then the motion of P is perpendicular to the plane OPQ, and its velocity is
The whole of the foregoing reasonings are applicable, not merely when A and B are actual regular cones, but also when they are the osculating regular cones of a pair of irregular conical surfaces, having a common apex at O.
§ 32. Screw-like or Helical Motion.—Since any displacement in a plane can be represented in general by a rotation, it follows that the only combination of translation and rotation, in which a complex movement which is not a mere rotation is produced, occurs when there is a translation perpendicular to the plane and parallel to the axis of rotation.
 |
| Fig. 94. |
Such a complex motion is called screw-like or helical motion; for each point in the body describes a helix or screw round the axis of rotation, fixed or instantaneous as the case may be. To cause a body to move in this manner it is usually made of a helical or screw-like figure, and moves in a guide of a corresponding figure. Helical motion and screws adapted to it are said to be right- or left-handed according to the appearance presented by the rotation to an observer looking towards the direction of the translation. Thus the screw G in fig. 94 is right-handed.
The translation of a body in helical motion is called its advance. Let vx denote the velocity of advance at a given instant, which of course is common to all the particles of the body; α the angular velocity of the rotation at the same instant; 2π = 6.2832 nearly, the circumference of a circle of the radius unity. Then
is the time of one turn at the rate α; and
is the pitch or advance per turn—a length which expresses the comparative motion of the translation and the rotation.
The pitch of a screw is the distance, measured parallel to its axis, between two successive turns of the same thread or helical projection.
Let r denote the perpendicular distance of a point in a body moving helically from the axis. Then
is the component of the velocity of that point in a plane perpendicular to the axis, and its total velocity is
The ratio of the two components of that velocity is
where θ denotes the angle made by the helical path of the point with a plane perpendicular to the axis.
Division 4. Elementary Combinations in Mechanism
§ 33. Definitions.—An elementary combination in mechanism consists of two 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 continuous motion of the follower, forward or backward, and a reciprocating 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 relatively 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.
§ 34. 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 § 29, it follows, that the components along a line of connexion of all the points traversed by that line, whether in the driver or in the follower, are 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 the 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.
 |
| Fig. 95. |
§ 35. Application to a Pair of Shifting Pieces.—In fig. 95, let P1P2 be the line of connexion of a pair of pieces, each of which has a motion of translation or shifting. Through any point T in that line draw TV1, TV2, respectively parallel to the simultaneous direction of motion of the pieces; through any other point A in the line of connexion draw a plane perpendicular to that line, cutting TV1, TV2 in V1, V2; then, velocity of piece 1 : velocity of piece 2 :: TV1 : TV2. Also TA represents the equal components of the velocities of the pieces parallel to their line of connexion, and the line V1V2 represents their velocity relatively to each other.
§ 36. Application to a Pair of Turning Pieces.—Let α1, α2 be the angular velocities of a pair of turning pieces; θ1, θ2 the angles which their line of connexion makes with their respective planes of rotation; r1, r2 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 are—
consequently, the comparative motion of the pieces is given by the equation
| α2 | = | r1 cos θ1 | . |
| α1 | r2 cos θ2 |
§ 37. 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 α1 be the angular velocity of the turning piece, r1 the
common perpendicular of its axis of rotation and the line of connexion,
θ1 the angle made by the line of connexion with the plane
of rotation, θ2 the angle made by the line of connexion with the
direction of motion of the shifting piece, v2 the linear velocity of
that piece. Then
which equation expresses the comparative motion of the two pieces.
§ 38. 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 Bétancourt, which has been generally received, and has been adopted in most treatises on applied mechanics. But that classification is founded on the absolute instead of the comparative
