The graphic statics of mechanism

Since the appearance of Culmann's work, which marked an epoch in the history of graphical statics, the graphical method has attained pretty general dissemination in engineering circles; its advantages over the analytical have been recognized more and more, and its further development kept constantly in view. It is universally applied in the designing of stationary structures, - such as bridges, - for determining the requirements of the individual parts. In machine design, also, the graphical method gives valuable aid in finding the moments to which the machine parts are subjected, and in determining dimensions. Accordingly courses in graphical statics have been introduced in all technical schools.

In all these determinations, however, friction and the special hurtful resistances to motion have not been taken into account. Heretofore all attempts to ascertain these hurtful resistances in machines, and to determine the efficiency which is dependent upon them, and is of such great importance in practice, have been confined to the analytical method, which is often awkward and at times utterly inapplicable. No method is as yet known to me by which the frictional resistances and efficiency of any desired mechanism can be graphically determined. In my lectures on machinery in the polytechnic schools of this place I have endeavored to show the relations existing between the forces in mechanism in a simpler form than that offered by the analytical method. Out of that endeavor has grown the following treatise, which in reality amounts to nothing more than a wider application of the long known but little used angle of friction.

My object in the present treatise was' principally to facilitate study for the students of the technical schools, upon whose time and industry increasing demands are made from day to day; perhaps the work may also be of interest and value to those more advanced.

Although mechanisms, by their very nature, can only effect their object while in motion, or by virtue of the same, yet, for the ascertaining of the relation existing between the various forces, we may always assume as a basis that condition of equilibrium which corresponds to the limit where the slightest increase of the driving-force would produce a motion in the "sense," or direction, of that force. In what follows P will again represent the driving-force and Q the useful resistance. Neglecting for the present any acceleration of the masses, we will suppose a uniform motion in which, at each instant, the work of the force during a small portion of time is just sufficient to overcome the useful resistance Q after the hurtful resistances W have been disposed of. It will then easily appear in what way the accelerating force working upon the mass M in the case of variable motion can be ascertained.

The exterior forces P and Q working upon any mechanism, call forth certain internal forces, or reactions R, between the members of the machine wherever two parts come in contact. These re-actions are to be regarded as two equal and opposing forces occurring at every surface of contact. Every pair of forces thus arising at the same point is, therefore, in equilibrium. We must imagine such re-actions wherever two bodies come in contact, whether the bodies move relatively to each other or not. We can, therefore, in every case neglect the bodies in contact and think only of the reactions offered by those bodies. Under this supposition, any member of a machine which is acted upon by certain exterior forces P and Q and which is supported at certain points by neighboring bodies, must be under the influence of the exterior forces P and Q and of the reactions R, which are sufficient to replace the imagined supports, in order to be in the supposed limiting condition of equilibrium. The conditions of equilibrium furnish us, in general, with a means by which from the known elements, - direction and magnitude of individual forces, - we may ascertain the unknown. In the majority of cases the intensity of the re-actions of the supports is unknown; of the exterior forces, there is, as a rule, one element - the direction, or intensity, of one force - unknown at first. As regards the direction of the re-action replacing a support, it is determined empirically by the condition that it shall be inclined to the supporting surface at a certain determinate angle whose magnitude depends upon the nature of the two bodies in contact, as to smoothness, hardness, etc. The hurtful resistances to motion, W, which, as previously remarked, arise only at the point of contact between two bodies (i.e., at the supporting surfaces), depend on the nature of the material, and of the sur faces constituting the supports. The size of the angle at which the surfaces of contact will be cut by the direction of the re-action existing between them depends closely, as will be shown in what follows, upon the amount of hurtful resistance generated between the surfaces.

If we suppose, in the next place, that no hurtful resistance W exists, - a condition of affairs which, of course, never occurs in practice, - the angle formed by the direction of reaction and the supporting surface would be a right angle; in other words, when there are no hurtful resistances a supporting surface can only react, at each of its points, in the direction of a normal. With the assumption which thus entirely ignores all hurtful resistances, it is a simple matter to determine the proportion of power and load at every point in a machine; and, for this purpose, a graphical method can be used to good advantage. A few examples will serve to illustrate this procedure.