If, as in Fig. 7, the sleeve is fixed at the end of a rod B, any point P on A will have two motions: s along A, and u normal to OP, their resultant being v. The reader should ob serve that Figs. 6 and 7 were solved by using the principle of superposition.
V. Sliding pieces. These occur in two forms. In an iris diaphragm, a photographic shutter, or a pair of shears, two blades slide over each other, it being required to find the velocity of the point common to the two overlapping edges; in cams and gear teeth two tangential surfaces slide and roll on each other, the prob lem being to find the velocity ratio of the two pieces.
In Fig. 8 i, A is fixed and B rotates. The velocity v of P is the resultant of u, due to rotation, and s due to sliding along B; u is tangent to A and P. In the toe and lift mech anism, shown schematically in Fig. 8 ii, v is the resultant of u, due to rotation, and s due to sliding, along the common tangent. If both pieces in Fig. 8 i turn use the method of super position : hold A and B fixed in succession. Fig. 9 shows two cams or parts of two gear • teeth turning about fixed centres C. and CI; it is required to find the angular velocity ratio t1/4/64. The point of contact on A has a veloc ity tr, normal to C,P and on B the velocity v, is normal to GP; these are the actual velocities of points fixed on A and /3 hut coincident at the point of tangency. If the two surfaces are not to separate, v, and v, must have equal projec tions on the. common normal. If the normal components are equal the tangential components cannot be, for otherwise vt and v, would be along the same line; hence sliding takes place and the motion is not pure rolling. Now sup pose the contours to touch at some other point C. The velocities there can be collinear only if C lies on if they are equal there will be pure rolling without sliding. In this case the relative motions of A and B are rotations about C, hence as the relative velocities at P are along the tangent and are due to relative rotations about C, C must lie on the normal at P. C is the pitch point and C,C and the radii of the pitch circles. The angular ve locity ratio is found from if it is constant, as in gear wheels, C must remain fixed, whence the normals at successive points of contact of the teeth must pass through the pitch point. This is the basic law of gear teeth
design.
VI. Pure Rolling. The simplest case of pure rolling is that of a gear or wheel train; Fig. 10 shows the pitch circles, all centres being fixed in position. By equating rim velocities, /tut =re.% Rsui--- rola, Rws= nu, the product of which gives itit.R.41/4=rvir4414 in which capitals refer to drivers and lower case letters to followers. Sometimes the shafts of all the wheels are mounted in a rigid frame. Let the frame turn ± n times around the shaft of the second wheel while this wheel is pre vented from turning. The number of turns made by the other wheels is found by the method of superposition as follows. First let R2 (and of course r,) turn with the frame; then R,, R, and r. make + is turns, there being how ever no relative motion of the wheels. Now hold the frame fixed and give — n turns to 1?“ then 12i, rs and .1, turn respectively —n and — is times. Superpose the rs fan two motions or imagine them to occur simul taneously. Similar considerations lead to the occasionally surprising result that if a 25-cent piece is rolled once around the circumference of a fixed 25-cent piece it will made two turns around its own centre.
Quadric Chain; Inversion; ,Theorem of Three The simplest mechanism con sists of four or hinged links, Fig 11, called by Reuleaux the quadric it may be regarded as the kinematic unit, of. mechanisms having surface contact. A con siderable number of different mechanisms may be derived from it by inversion; this consists in fixing any two points on any one link. The idea, due to Reuleaux, is useful in giving an insight into the relationships in families of mechanisms. (Consult Durley, 'Kinematics of Machines,' 1903). Inversion does not change relative motions. The points (12), (23), (34) and (14) are the relative instantaneous centres of the links whose numbers they bear. By holding 4 or 2 fixed we find that (24) is the I. C. (instantaneous centre) of 2 and 4 for motion relative to each other; likewise (13) is the relative I. C. of 1 and 3. Observe that (24), (23) and (34), and (12), (23) and (13) are collinear triplets of the form (xy), (yz), (se). This is a special case of the theorem of three centres now to be proved. Consider Fig. 12, representing three bodies having relative plane motion.