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ARTICLE XV.
OF VIBRATING MOTIONS OF MACHINERY.
FEW mechanicians have considered what power is expended in giving quick vibrations to heavy parts of machinery; such as saw gates, engine beams, &c. Writers on the principles of mechanics have generally agreed in laying down as an axiom that the weight of a body in motion multiplied into its velocity is a true measure of its momentum: but few have informed us that the weight of a body multiplied into the square of its velocity is the true measure of the effects it will produce; which is the truth.* We are thus frequently led into great errors, and to suppose that a double impulse will give double velocity to a body; whereas a quadruple impulse is required to give double velocity; and if so, a quadruple resistance is required to check a double velocity,** consequently the power required to produce
*See the Millwright's Guide, art. 6. **Ibid.
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vibrating motions, is, as the squares of their velocities
multiplied into the weight of the bodies moved. Aware of those principles,
I have guarded against the use of the heavy lever beam in the construction
of my steam engines; as by an injudicious arrangement nearly the whole power
of the engine may be expended in giving a quick motion to a heavy beam.
The natural vibrations of a beam are regulated by its length as much as
those of a pendulum; and if we attempt to vary this motion to a quicker
one we expend much of the power of the engine to do it. I know no better
way of explaining this than by the laws of spouting fluids.* Suppose water
to issue from under a head of 1 foot; it moves with that power 8.1 feet
per second: here, a power equal to 1/8 of the weight of the body moved is
expended to give that velocity. Suppose water to issue from under a head
of 4 feet; it moves with that power 16.2 feet per second: in this case,
a power equal to 1/4 the weight of the body moved is expended to give it
velocity. If it issues from under a head of 16 feet it moves with velocity
34.4 feet per second, and a power equal to 1/2 the weight of the body is
expended to give it velocity. If it issues from under a head of 64 feet,
its velocity is 64 feet per second: here the power expended to produce the
motion is equal to the weight of the body moved. But as an equal power is
required to check the motion, therefore to give a body a vibrating motion
equal to 32.4 feet per second, requires a power equal to the weight of that
body, and 64.8 feet per second requires a power equal to double the weight
of that body.
*See the Millwright's Guide, art.45.
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Great as this evil may appear, yet in most cases it almost entirely vanishes, where the vibrations are produced by the revolutions of that simple instrument, the crank, attached to the axis of a wheel, to which the
power is applied, as in the constrtiction of saw-mills; where the power is applied immediately to move the vibrating body attached by a connecting rod to a crank on the axis of which is put a heavy fly wheel, as in steam engines without lever beams. In both cases, the line of vibration continued, should pass through the centre of the circle described by the crank, coinciding with the diameter of the circle. In the case of saw-mills where the power is applied to the wheel and the crank moves the saw, while the crank is receding from the vibrating line, it moves the saw with a very gradual accelerated motion, and as it approaches the line of vibration it
retards the motion again as gradually. If the saw is not
applied to do work, the momentum communicated to it and its frame by the
crank in giving the motion is recommunicated or returned to the crank during
the retarded motion; therefore very little power will be required to keep
up this vibrating motion. While the saw is cutting, the momentum is expended
performing the work. In the ease of steam engines, the reverse takes place,
the power being immediately applied to produce the vibrating motion in the
piston, which communicates momentum to the fly wheel, while the crank recedes
from the vibrating line; and the momentum communicated to the piston and
all attached to it is communicated to the fly wheel, while the crank approaches
that line, and very little power is required to keep up this vibrating
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motion, without producing any other effect. When the engine is at work, the power is expended to produce the effects, and the momentum of the fly wheel regulates the motion and carries the crank past the vibrating line, (where it would stop) and brings it to a position to receive the power from the vibrating motion to keep up the circu1ar motion of the fly.
The late ingenious Robert Leslie of Philadelphia, to whose
memory and judgment great deference ought to be paid, was of opinion that
the case is widely different when moving a heavy lever beam past its natural
vibrating velocity, although it be attached to a crank.* We know that if
the beam be nicely poised but little power is required to cause it to vibrate
with its natural motion, which is as exactly governed by fixed principles
as the vibrations of a pendulum; but what power is required to give it any
greater number of vibrations per minute proportionate to its length and
weight, I have not known to be ascertained; nor can I say whether or not
the momentum received from the crank, while it recedes from the vibrating
line, will be returned to it, with the same exactness, while approaching
that line as has already been stated, but I am inclined to believe it will
not. My ideas are not mature on the subject, not having given it a full
investigation, although 1 think it important.
*I was well acquainted with Mr.Leslie. He was generally correct in his ideas of the principles of mechanics, and made many useful discoveries and improvements.