A short course in graphic statics

The purpose of this book is to provide students of mechanical engineering with a brief course in Graphic Statics which will serve when the time to be devoted to this subject is short. Owing to the necessary limitations as to size, the treatment has been restricted mainly to the properties and general uses of the force and equilibrium polygons, these polygons being sufficient for the solution of most of the problems met in practice by mechanical engineers. While the design of trusses is, in general, the duty of the civil engineer, some attention has been given this subject, since such constructions as the Warren girder for an overhead crane, the walking beam of an engine, etc., fall under this classification. Examples, in full detail, have been included, so far as space would admit, since one good example is often of more service in instruction than many pages of theoretical investigation.

The discussion of principles, as given in this book, is largely a summary of similar portions of the authors' treatise, "The Elements of Graphic Statics," although there have been some minor additions. With the latter and the examples, the new material comprises nearly two-thirds of the text.

The resistance due to friction is both helpful and hurtful in the performance of mechanical work. It is friction which gives the driving wheels of a locomotive the grip on the rails which enables them to move the train; and, again, it is the friction of the car wheels on the track which, disregarding the resistance of the air, forms the total work of pulling the train and develops the total stress in the draw-bar. The analysis of the action of friction is, in some cases, complex, but it is often simplified materially by the use of graphic methods.
 
30. Friction. The sliding friction of solids which only will be treated herein is the resistance to relative motion of surfaces in contact and under pressure. This resistance is caused by the interlocking of the minute projections and indentations of these surfaces. If the latter were absolutely smooth and perfectly hard, there would be no projections to disengage and override, no frictional resistance would occur, and hence no mechanical work would be required to produce relative motion. In practice, the action of pure friction, as above, is complicated by adhesion, abrasion, the viscosity of lubricants, etc.
 
Sliding friction is the friction of plane surfaces; the friction of warped surfaces, such as screw threads, and of cylindrical surfaces, such as journals, is a modified form of this action. Rolling friction is the friction of a curved body, as a cylinder or sphere, when moving over a plane surface or one of greater curvature. In this case, contact occurs theoretically on a line or point only; but, as all materials are more or less elastic, there is actually a surface of contact, and therefore rolling friction is identical in cause with sliding friction. The resistance or force of friction acts in the plane of contact of plane surfaces, and, with curved surfaces, along their common tangent. In most mechanisms, the friction of motion only is encountered; with belting, the friction of rest occurs. The latter is greater theoretically than the former since, during rest, the harder body has better opportunity to indent and engage the softer surface with which it is in contact. The slightest jar, however, nullifies this action.
 
The force of sliding friction is:   F = fN
 
in which F is the total resistance or force which opposes the relative motion of two surfaces in contact, N is the total pressure normal to those surfaces, and f is the factor or coefficient of friction. This formula, which expresses the far from well-established laws of sliding friction assumes that the total force of friction is independent of the area of the surfaces in contact, of their relative velocity, and of the intensity of pressure, i.e., the pressure per unit of surface. According to this expression, F is equal simply to the product of the total normal pressure by the factor/ If the normal pressure and velocity be low, and the surfaces dry or but slightly lubricated, f may be considered as having a constant value for the same materials and state of surfaces; but, in the wide range of conditions met in practice, it has been found that the value of f is affected also by the velocity, the intensity of pressure, the temperature of the surfaces, and the viscosity of the lubricant, so that the formula, as above, is incomplete and approximate. If, however, these effects be all considered in the value assigned to the factor f, and this value be determined independently for each case, the formula will hold.
 
32. Friction of Screw Threads. The screw thread is essentially but an inclined plane wrapped around a cylinder. In a square thread, the radial elements of the thread surface are perpendicular to the axis of the cylinder; in a triangular thread, these elements are inclined to that axis. The axial load W is borne usually by the bolt, whose thread thus corresponds with the contact-surface of the body D in Fig. 43; the nut thread is then similar in its action to the surface AB in that figure, the nut being supported by a bearing surface.
 
The pressure on these threads is assumed to be concentrated on the mean helix, or the circumference of the mean thread-diameter d, of pitch angle a, as in Fig. 44. Each element of the thread surface is regarded as sustaining an equal elementary portion of the total load or stress W on the bolt, and each element has therefore a frictional resistance of the same magnitude. Since the conditions for all elements are thus identical, the total thread resistance, the axial load, and the external turning forces on the nut may be assumed to be each equally divided and concentrated at two points, 1 80 degrees apart, on the circumference of diameter d. The forces P, for lifting the load, thus form a couple whose arm is d, and similarly the forces P' t for lowering, have the same arm and points of application. In Fig. 44, these points are F and K.

  34. Journal Friction; Friction Circle. (a) Journal friction is complex in analysis, since the fit is, in most cases, more or less free, and pure sliding friction be- tween cylindrical surfaces rarely occurs in ordinary machinery, although in the mechanism of precision it is the rule. The usual shaft journal rests in its bearing somewhat loosely, giving theoretically a line bearing between cylindrical surfaces of different radii, which is rolling friction. When the shaft revolves, friction causes the journal to roll up the side of the bearing, like a car-wheel moving up an inclined track. The journal ascends until the weight makes it slide backward, and this rise and fall continue with every variation in the coefficient of friction. The method of analysis which follows is that first established by Rankine and later developed by Hermann and others. It applies to close fitting bearings only.
 
35. Link Connections: Friction Axis. A link is a straight, rigid machine member employed to transmit power between two rotating or oscillating members, or one rotating and one sliding member, as is the case with the connecting rod. Journal friction therefore exists at each end of the link ; the principles governing its action are those established in Art. 34. If there were no friction, the radius of the friction circle would be zero and the force would be transmitted from one member to the other along the line joining the centres of the journals or bearings of the link. With friction the resultant of the transmitted force and the force of friction acts along the friction axis, which is a line tangent to the friction circles of the two journals.
 
There are four fundamental cases of this action, two of which are represented in Figs. 49 and 50 and the others deduced therefrom. In each of these mechanisms, A is a link and B and C are hinged levers, P is a force acting upward on the lower or driving lever, and the dotted lines show the path in ascending. It will be observed that:


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