It must be remembered in this connection that useful energy can be stored in the machine only in the plane of horizontal motion and gravity; in other words, vertically and horizontally. Any elasticity at an angle to this plane can only be of use in reducing the concussion upon the rider in a lateral direction; and since, upon a single-track machine, but little if any shock can occur in such direction, it should be seen to that no undue side motion is permitted.
In order to fully comprehend the loss of power that it is possible to save by proper springs, observe as a particular case the annexed diagram showing two thirty-inch wheels arranged substantially as in the present rear-driving Safety.
Let c be the centre of gravity, and let the line c o, drawn to the obstacle, pass through the centre of the front wheel and make an angle of forty-five degrees with the horizontal.
Rover momentum.
The momentum c l is split up into two equal components, one acting in the direction c o, and the other in the direction c k perpendicular to c o, tending to turn the system about o as a centre. The numerical value of the c k component, calling m the momentum, is m√2, and its value in the forward direction c o is m√2 cos 45° = m√2 1√2 = m2, which is the forward momentum retained, showing that in this case one-half of the forward momentum is saved and the other half lost.
It is scarcely necessary to say that the use of an imaginary four-inch obstruction, in our study of momentum and concussion, is entirely arbitrary. Of course obstructions of all heights will evolve proportional results. This proportion would not, however, be linear; the nearest we can come to it is to say that the annoyance begins with an obstruction of zero height, and increases about as a trigonometrical sine increases when the angle grows larger.
It is evident that all this theory applied to one obstruction is simply repeated in a number of them, and a number of them make up a rough road, bearing in mind that a rut is but one form of an obstacle.
Some makers of late seem to realize the importance of springs which will allow of a horizontal as well as a vertical motion, and have in them not only provided against the loss of momentum in the man, but also in the entire machine exclusive of the front wheel. This has apparently been done with another object in view,—i.e., that of relieving the annoyance to the hands and arms by reducing the vibration in the handle-bar. This object, though worthy, is far short of the ideal. Such springs might properly be called storage springs or power economizers; they are, however, generally nominated Anti-Vibrators and Spring Forks.