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 obstacle he 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 find that 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 the energy
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
furnishes the 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.
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.
[Illustration: 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,
_m_
is ———, and its value in the forward direction _c o_ is
√2
_m_ _m_ 1 _m_
——— cos 45°= ——— ——— = ———, which is the forward momentum retained,
√2 √2 √2 2
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.