Toothed gear therefore so that a tab b1 tab also measures the ratio of the angular velocity of wheel A to that of wheel B. The condition that the angular velocity ratio should remain constant may thus also be expressed by the condition that the line a b in the velocity diagram, representing the velocity of sliding of tooth over tooth, should be divided in a constant ratio by the fixed point tab. [This point tab is only fixed if the angular velocities themselves, as well as their ratio, remain constant, the magnitudes of these angular velocities being proportional to p tab.] Whether this proposition can be utilised in simplifying the practical drawing out of the teeth-profiles, so as to secure a constant velocity ratio, the author has not yet investigated. This kinematic method has been applied to many actual mechanisms. The diagrams when fully drawn are to a certain degree complicated, although not difficult to produce; and in order that a large number of points should be distinctly marked and lettered on each of the various curves, the diagrams should be drawn to a moderately large scale. This fact prevents any of these diagrams being usefully reproduced in the atlas to this book; but the student should not fail to work out for himself a sufficient number of practical examples to familiarise him with the method. CHAPTER X. STATIC STRUCTURES, FRAMES, OR LINKAGES. of balance 1. WE have already described the construction of the 'Moment Diagram' for parallel forces, and in Chapter VIII., Fig. 57, this construction is extended to non-parallel co-planar forces. The construction consists in drawing a connected series of lines across the spaces lying between the force-lines, the joint-points of the lines lying on the joint-lines of the spaces, and the lines being drawn parallel to the radii drawn from a 'pole' to the joints of the vector-addition diagram. This series of lines across these spaces forms a closed polygon if the group of forces balance' both with regard to vector sum and with regard to moment. This state of balance' involves two distinct conditions: firstly, that the vector sum Conditions is zero, the graphic test of which is that the vector-addition diagram forms a closed polygon, and the physical meaning of which is that this group of forces when applied to any portion of material has no influence in changing the velocity of the centre of inertia of this material; secondly, that the sum of the moments of the forces is zero round any and every axis, the graphic test of which is that the moment-diagram forms a closed polygon, and the physical meaning of which is that the application of this group of forces to any mass does not result in any change of the angular velocity of this mass round any axis whatever. Ill-taught students are apt to think that when the vector sum is zero the moment must necessarily also be zero round every axis, because a resultant of zero magnitude can have, it is supposed, no moment. of balance In doing so, they forget the case of force-couples, which have a moment although the vector resultant has zero magnitude. Conditions The locor resultant in this case is not zero; it is two equal and opposite forces. If the vector resultant is zero, and the moment diagram does not close, then the locor resultant is a couple. This is indicated by the first and last lines of the moment diagram being parallel. Momentdiagram Closed chain 2. Confusion also sometimes arises from the fact that a group of forces which balances and has no moment round any axis still has a 'moment-diagram' from which one measures bending moments produced by this group of forces. It is to be remembered that these bending moments are due to partial sets only of the whole balancing group. Thus the height of the diagram at any point measures the moment round an axis lying in the line on which the height is measured, of all those forces of the group that lie on one side, either right or left, of that line. When the forces are not parallel, it was explained that this polygon was inconvenient when used directly as a 'moment-diagram,' although it always remains of the greatest utility in finding the moments because it supplies the resultants of the various partial groups of forces. 3. We must now look upon this closed polygon in another light, namely, as a skeleton drawing of a jointed linkage which would keep its shape and position-i.e. be generally in equilibrium, under the action of the forces. Looked on in this light, it is sometimes called a chain,' sometimes a 'linkwork,' or a 'linkage.' The latter term is preferred by the author. 4. A stiff 'link' may for our purpose be defined as an approximately rigid piece of material forming one part of a Definition structure or machine and jointed to the other parts in such a way that the forces exerted between it and those other parts act in lines passing through definite and easily ascertainable points in the link, which points are called the 'joints' of the of link link. The word 'point' must not here be understood in its mathematical sense. No forces act through mathematical points; they are all more or less distributed through certain volumes or masses and act always through sectional areas of finite magnitude. When one speaks of a force acting through Definition a point, one either refers to the point through which passes the centre-line, or line of the resultant (as previously defined), of a group of distributed forces; or else one means that the distributed force is concentrated through a very small mass and acts on a very small sectional area. 5. If two links be connected by a cylindrical pin-and-eye joint in which the pin fits the eye loosely, each link acts on the pin over a very small portion of its circumference; and if there be very little friction between the pin-and-eye surfaces, then the force, being nearly exactly normal to the surface, passes very nearly through the centre of the pin. Whether the pin be loose or tight, provided there be very little friction, the pressure must be almost symmetrically distributed on either side of a line passing through the pin-centre, and, therefore, the centre-line of the pressure passes approximately through the pin-centre. of link joint Thus a rigid mass connected to other pieces of material Flexible only by pin-and-eye joints is a 'link' according to the above definition, because it is definitely known that it is acted on only by forces whose lines pass through the centres of the pins; pass through those centres, at any rate, with a degree of approximation quite equal to that of the other data assumed in the course of engineering calculations. For example, our knowledge of the loads on the structure or machine is always far from accurate, as is also our knowledge of the strength and the modulus of elasticity of the materials out of which the structure or machine is built. We will need afterwards to admit into our investigations flexible, extensible, and compressible links; but in the meantime, and except when specially mentioned as flexible, &c., a 'link' will here be understood ot mean an approximately rigid' link. Stiff joints 6. As applied to actual structures, the accuracy of this idea is interfered with in the first place by the friction between pin and eye. In the case of machine-joints the coefficient of friction is always small and may usually be known with considerable exactitude. In this case it is not difficult to take into account the effect of friction in the graphic calculations of the forces; the method of doing so will be explained in a subsequent chapter devoted to the dynamics of machines. In the pin-joints of roofs, bridges, and similar static structures, the coefficient of friction may often be very large, and the friction, moreover, may be increased very much beyond the product of the effective force through the joint and the coefficient of friction in consequence of the pin being driven in tight. Also, since there is no relative motion between the links, it is impossible to say in which direction the friction at each joint is active. By laborious calculation of the stretching and contraction of the various links under stress, it would be possible to discover in which direction the friction at each joint acts during the process of erection of the structure, that is, when the stresses are first produced. But the friction during the permanent life of the structure is not necessarily the same as that during erection, and it may evidently be reversed at various joints many times in consequence of to-and-fro straining due to rise and fall of temperature and variation of load. Thus it becomes practically impossible to take account of joint friction in these structures; it may be theoretically possible to do so, but, as a matter of fact, it is never done, and the labour of the operation would be out of proportion to the utility of the result. It is useful to remember, nevertheless, that, however great the friction may be, it can never shift the centre-line of the force through the joint away from the centre of the pin so far as the radius of the pin, except in the case of a pin driven in extremely tight. Probably in actual structures the deviation seldom exceeds two-thirds the radius of the pin. 7. Secondly, it must be noticed that the forces applied to |