CHAPTER IX.
SPRINGS IN RELATION TO THE CURVES OF TRANSLATION, MOMENTUM, AND CONCUSSION.
It was a pet scheme of the writer’s to treat of the matter of the annoyance to the rider resulting from a shock or jolt and change in momentum in the various styles of bicycles in a purely mathematical form, and to some extent it can be done; but it is found that so many considerations enter that the question becomes almost interminable. The aim was to find a formula with the sizes of wheels, distances between centres, and position of saddle as variables, which would, when applied, give us a result representing the sum total of annoyance felt by the rider in passing over an obstacle or any depression, rut, or ditch of given height or depth on any combination of wheels likely to be used in one machine. The difficulty in the question is in determining just what that annoyance results from or consists in; no doubt the initial impact, change of direction, and sudden reduction of momentum, and also the duration of the shock, all enter into the grand total.
From a theoretical stand-point there need be no loss of power and consequently no annoyance in running over an obstacle, since all the momentum lost in a forward direction ought to be transmitted vertically in mounting the obstacle, thereby establishing a potential energy which would again be transformed into momentum forward as the wheel rolls down from the elevation. Neither should a rut have to be avoided, since by running into it we gain a momentum that should carry us out; hence, as per theory, the cycler should not worry about riding over rough roads, for in mounting each obstaclehe only loans a bit of power in going up, which will be returned to him in going down, and in running down into a rut momentum will be loaned to him sufficient to bring him out. But, alas! he does not fancy the thing; somehow he has a like prejudice against rough roads that he has to hills, and as this prejudice cannot arise from purely theoretical considerations, we must look for some violation of nature’s laws, or some cause why such laws are not directly applicable. In my judgment there is a reasonably definite connection between the annoyance felt by the cycler in riding over a rough road and the actual loss of energy, though not a similar one in all respects to that which applies in regard to hills. A shock produced by a sudden check or deviation of the momentum is not only hurtful in causing a direct loss of kinetic energy, which the rider has stored up and to regain which he must afterwards do work, but also in contusing and jarring the muscular system, which makes him less able to do the work. In so far as the machine is concerned, the loss of energy goes into vibration and into extra friction of the machine; we cannot see any other means by which it can escape; but as to the rider, while energy is of course similarly lost, the motive power is also interfered with. Now, the application I wish to make of this fact,i.e., that the annoyance or shock felt by the rider in wheeling over rough roads is comparable to an actual loss of kinetic energy, as well as in addition thereto, is that the nearer we can approach to an even rolling motion affecting the rider least disastrously, the nearer we will come to a perfect road bicycle without loss of momentum. In other words, the dynamical and physiological considerations lead us to the same end,—to relieve the annoyance by means of proper springs, and to so distribute the inequalities of the momentum and modify the change in direction thereof as to minimize the loss of energy. From experiments tried with properly-constructed springs, I findthat momentum can be diverted in striking the obstacle into its required new course, upward and forward, with very slight loss indeed, and that much waste of power in rolling off the obstacle can also be saved, the desired conditions and effect being as follows:
The wheel strikes the obstacle, springs back a little, and begins to rise upon it; at the same time an upward thrust is given, additionally compressing the vertical components of the springs, the man going on forward at the usual pace of momentum and being gradually raised. When the top is reached and the wheel starts down, the weight of man and machine causes the wheel to spring forward a little at first, and then, when the weight would drop too slowly and the momentum would otherwise pull the wheel bodily off, the vertical spring, being compressed, will, by its quick action, together with the pressure backward of the horizontal spring against the obstruction, hold the wheel in contact and make it roll off. This action is reversed in the case of a rut, and is quite similar in either fore or hind wheel.
The principle is to avoid a too sudden attack upon the inertia, to change the course of momentum gradually, and to avoid concussion against inelastic parts.
The direct vertical amplitude in the springs of a cycle is of most benefit in regard to momentum in giving the vertical power time to act; that is, if the wheels are raised quickly the momentum is transmitted to and stored up in the springs and allowed to act gradually in raising all the parts without violent concussion or vibration and consequent loss of power. When the machine drops suddenly in descending from an obstacle the springs will act more quickly than gravity can overcome the inertia of the system, and the wheel will then remain in contact with the obstacle; that is to say, sufficient spring acting horizontally in the direction of the acquired momentum, together with the necessary amount of vertical spring, will store theenergy otherwise lost in riding suddenly upon an obstacle; said energy will then be given time to act and be utilized in raising the rider and such parts of the system which the springs control to a certain height, establishing a potential, which will be given out in increasing the forward momentum as the wheel rolls down to the common level.
Springs having a horizontal movement relieving only the saddle can prevent loss of momentum in the man, but cannot prevent the weight of the machine from being thrown dead against the obstacle. This can only be remedied by elastic connections of a kind that prevent the shock from ever reaching the heavier parts, which condition would save almost the entire work lost against the obstacle.
We see, then, that the subject of springs comprehends not only the question of comfort in regard to the shock sustained by the body, but also the most serious and interesting factor in relation to the economy of power; nor is this a theme at all confined to cycles; it has been egregiously overlooked by makers and riders of many other vehicles. No better illustration can be had of man’s selfishness, as against the brute creation, than the fact that now, in machines in which we have to pull our own load, we are just beginning to contrive and apply all possible means to prevent a loss of momentum, whereas in all our carriages drawn by horses we looked only to the ease and comfort of our bodies, and provided good springs with a vertical give for that especial purpose, having little care for any loss of power, to avoid which loss we should also use horizontal springs so placed as to relieve the entire weight of the heavy running gear, as well as that of the man, from forward concussion. I know full well, even then, that a horizontal spring has still some little to do with the ease of riding, but with a heavy conveyance the advantage to the rider is slight as compared with the advantage that it would be to the horse which furnishesthe power. The time will come when the evil will be remedied in general carriages, if only for the gain it will be to the comfort of the man. There would be little hope, indeed, if the poor horse were the only party interested, but when man is directly concerned we can expect more rapid development.
When we start our machines for a run it is considerable work to get up an initial velocity or momentum; however, after that there should be only the friction of the machine within itself and upon the road to be overcome, together with the friction against the air; that is to say, if inequalities in the road could be run over without a loss of momentum being caused thereby, there would not be nearly so much work in travelling upon the cycle as is now necessarily required.
The principal parts of the cycle should be as rigid and firm as possible, so as not to respond at random in vibration to every little shock they should chance to receive, for the spring or elasticity wants to be such as can be controlled,—that is, made to store energy in the right way and give it out at the proper time with a desired effect upon the momentum.
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.
Letcbe the centre of gravity, and let the linec o, drawn to the obstacle, pass through the centre of thefront wheel and make an angle of forty-five degrees with the horizontal.
Rover momentum.
Rover momentum.
The momentumc lis split up into two equal components, one acting in the directionc o, and the other in the directionc kperpendicular toc o, tending to turn the system aboutoas a centre. The numerical value of thec kcomponent, callingmthe momentum, ism√2, and its value in the forward directionc oism√2cos 45° =m√21√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, belinear; 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.