First
of all, if you are able to create a force or combination of forces into a
single force which always
points
to a fixed direction in any 3-dimensional space, you have found the solution to
create a
unidirectional
linear force.
The
approach here is to convert 2 or more centrifugal forces into a unidirectional
linear force.
Basically,
there are 2 ways to create a centrifugal force:
(1)
A mass or pendulum rotates on a pivot,
(2)
A mass or pendulum swings back and forth on a pivot.
A pendulum is defined as a mass or a heavy
object which has its own centre of gravity a distance (r)
away
from its own pivot, figure (1).
A heavy mass at a far end of a very
light arm moves the centre of gravity further away from the pivot.
The
total mass of the heavy mass and arm themselves determine the actual centre of
gavity of the
pendulum.
A pendulum in full rotation
on a pivot
When a pendulum rotates on a pivot at an
angular speed (w), centrifugal force (F) will be generated.
A
drive shaft supplies the torque, thus the input force (Fn), to rotate the
pendulum. An equal amount
of
reaction force will be exerted on the pivot.
The
centrifugal force along the pendulum pulls the pivot outward into all
directions as it completes a
circle,
figure (1a).
The
kinectic energy from the drive shaft is being converted into potential energy
to throw the pendulum
outward.
Without
a grip, the pendulum will be thrown outward.
A
centrifugal pump is an example in which the fluid is being thrown outward by
the rotating blades,
as
the centrifugal force is being exerted on it but there is nothing to hold the
fluid in place.
Centrifugal force, F = mw²r (Newton)
wherein
m= mass (kg)
w = angular velocity (rad/s)
r = distance between centre
gravity of mass and pivot (m)
As
the pendulum rotates at a constant speed, a constant amount of centrifugal
force will be generated,
figure
(1b).
The amount of centrifugal force is dependent on
the value of m, w and r, based on the formula above.
A pendulum with a small value of m, w or r
generally creates
a small amount of centrifugal force, and
vice
versa.
A pendulum in a back and forth swing
A
drive shaft with cam is mounted on the arm of the pendulum, and is used to
swing it back & forth
within
a small angle, figure (2a).
As
the cam rotates, it forces the sliding block to travel inside the slot of the
swing arm. At the same
time,
it exerts a input force (Fd) in the -x-direction to push the pendulum to swing
a small angle on the
pivot. The same amount of reaction forces (red
arrow) are reflected to the pivoting shaft, which must
be
born by the whole casing of the device.
The cam transfers the rotary
motion of the drive shaft into linear force in x-direction and pushes the
arm
of the pendulum, of course at a slightly varying point of contact along the
slot.
The
position of a cam on the arm of the pendulum also acts to stop the heavy
pendulum at the 2
designed
end points of the back and forth swing.
The
rotating cam exerts an input force (Fd) in x-direction to swing the pendulum. An
equal amount of
reaction
force will be exerted on the pivoting shaft, thus the whole body of the device.
Centrifugal
force being generated is confined to the space within the angle Ɵ1
and Ɵ2 only, figure (2c).
The
kinetic energy from the rotating drive shaft is now being transferred to the
potential energy on the
swinging
pendulum, tending to throw it outward.
This
one-sided centrifugal force is the first but important property in our quest to
convert centrifugal
force
into unidirectional force.
As
the pendulum swings back and forth from point A to point B then back to point
A, angular velocity
(w)
increases from zero to the maximum ( at y-direction) then decreases again to
zero as it decreases
again
to a stop at the opposite side of the swing.
In
other word, there is an acceleration and deceleration of speed from point A to
point B, and vice
versa.
The centrifugal force being generated will thus increases from zero to the
maximum value, then
decreases
again back to zero.
There
is no force acting on the Z-direction, as shown in the top view and right side
view in figure (2b).
What
we can see from the top view is the driving force (Fd) pushes the drive shaft
C-D to the right, as
the
pendulum is being pushed to swing from point A to point B.
The
creation of centrifugal force is in close correlation to the input force. A
larger input force (Fd)
create
a bigger centrifugal force, an vice versa. The energy to swing the pendulum is
now being
transferred
to the potential energy to throw it outward.
Subsequently,
as the pendulum swings from B back to A, the similar form of centrifugal force
will be
generated, figure (2e).
Now,
the shaft C-D will be pushed to the left.
Figure (2f) Centrifugal force on a playground swing
Playground swing is a typical
example of centrifugal force being generated on the back and forth
swing
of a mass, figure (2f).
Apply
a push ( input force ), the swing will swing forward to the maximum potential
height, stops and
then
reverses course. The
generated centrifugal force will pull the overhead hanging bar to sag in
the
direction of the swing.
Next,
when the gravity pulls it to swing back to the right, it will pull the pivot to
the right.
The
repetitive back and forth swing forces the structure to wobble from the left to
the right, then
back
to the left.
Gradually
the centrifugal force fades out as the input energy is being depleted, and the
swing will
come
to a stop.
The
whole structure of the playground swing must be able to withstand the force and
thus the
moment
of the swing.
Try
to imagine what forces will be exerted to the hanging bar of the if both loaded swings are
a).
making a harmonic swing against each other, or
b).
making a synchronic swing.
Two pendulums swing against each
other
This
section shows a way to convert 2 centrifugal forces into a unidirectional
linear force (Fn), with only a small deviation from the y-direction.
The solution is to put 2 identical pendulums
closely on the same pivot, and to
make a synchronic swing against each other.
(i) 2 pendulums swing toward
each other
Figure (3a) shows the 3 planes of
the 2-pendulum device. Our goal is to create a force which will always
point to one fixed direction in
within any 3-dimensional space in the the device.
In figure (3a), 2 identical pendulums m1 & m2
are aligned closely side by side on the same pivot.
The
2 offset cams on the same rotating drive shaft swing both pendulum m1 and
pendulum m2 from their respective starting point A and point B toward the
y-axis.
As
both pendulums accelerate from their respective stops (point A and point B),
they will create their own centrifugal forces (F) with increasing values.
Both
horizontal input forces (Fd) are very close to each other, and their respective
reaction forces always pointing away from each other on the pivoting shaft.
They are cancelling each other in the x-y plane, as shown on figure (3a1). This
means there is no apparent net force in the x-direction.
Instead,
the 2 input forces create a moment along the pivoting shaft , as shown in the
top view. The pivoting shaft C-D will be twisted in anti clock-wise direction.
The
moments will be much smaller if both pendulums are closely mounted side by
side.
There
is no force in either the z-direction along the pivoting shaft.
In
the front view of figure (3a1), we have 2 centrifugal forces at the positive
side of the y-axis; never at the negative side of it.
The
2 centrifugal forces are now being
merged to form a net resultant force (Fn) on the pivot.
Figure
(3a2) shows the vector of the net resultant force (Fn), which ideally only
points to the y-axis.
The
value (Fn) increases from zero to the maximum as both pendulums swing toward
each other.
The
net resultant force (Fn) on the pivot increases toward the mid-point of the
swing, i.e. y-axis.
On the x-axis, both 2 opposing input
forces cancelling each other. Instead, they create the moment
to twist the pivoting shaft counter
clock-wise, as shown in the top view of figure (3a).
(ii) 2 pendulums swing away
from each other
After
stage (i), the 2 pendulums swing pass each other, figure (3b1).
Both
horizontal input forces (Fd) are always pointing away from each other on the
pivoting shaft, thus they are cancelling each other on the x-direction.
As
both pendulums decelerate from y-axis toward point B and point A respectively,
the 2 pendulums will have their own centrifugal forces decrease.
Thus
the net resultant force (Fn) on the pivot decreases toward the end of the
swing, i.e. points A and point B.
Figure
(3b2) shows the vector of the net resultant force (Fn), which also ideally only
points into the +y-direction.
Figure
(3c) shows the reversing swings of the pendulums. The only change is the
reversal of the
moment
on the pivoting shaft C-D.
The
continuous and repetitive back and forth swing of the 2 pendulums from (i) to
(ii) create wavy net
resultant
force (Fn), as shown in figure 3(d) below.
With
this simple 2 counteracting pendulums working in tandem, we are able to convert
2 centrifugal
forces
created by the pendulums into one unidirectional linear force ( Fn) which
always pull the pivot
into
the +y-direction; this is what we are searching for!
3-pendulum configuration reduces vibration
Figure (3e) A 3-pendulum
configuration eliminates the moment along
the pivoting shaft
There
is way to reduce the moment exerted on the pivoting shaft. The solution is to
have 3 closely
mounted
pendulums instead of 2 pendulums, figure
(3e).
The
2 conditions are:
(a).
Both pendulum 1 and pendulum 3 are identical, while pendulum 2 is 2 X the size
of pendulum 1.
(b).
Both pendulum 1 and pendulum 3 swing synchronously in the same direction while
pendulum 2
swings
in opposite direction.
The
2 opposing moments exerted on the pivoting shaft is confined to the span
between pendulum 1
and
pendulum 3, and they cancels out each other.
The
2 moments will reverse directions when the pendulums reverse their respective
swings.
Nevertheless,
the net moment in the pivoting shaft C-D is almost zero at all time.
By minimizing the oscillating forces on the
bearings of the pivoting shaft, this
configuration reduces
vibration
on the pivoting shaft.
What
being left is the net resultant force which pulls the pivoting shaft into
+y-direction.
Summary
One
pendulum swinging back and forth on a pivot will generate centrifugal force but
pulls and wobbles the pivot along the direction of the swing.
2
pendulums swinging against each other on the same pivot will combine 2
centrifugal forces into a singular linear force on the pivoting shaft. The
resultant force is always at the fixed +y-direction.
A
3-pendulum configuration is better than a 2-pendulum configuration in order to
minimize moment
being
generated in the z-direction of the pivot.
The
counteracting pendulum system is able to convert centrifugal forces into a
singular linear force.
Application Of Linear Force
Irrespective of how complex the
machine is, the main purpose of a jet engine or propeller is to
produce a linear force.
Linear force is used to move a
vehicle or an object, eg. car, truck, train, ship, submarine, aircraft,
helicopter, drone, rocket, missile,
space-ship, lift, personal flyer, ... etc.
The
advantage of the pendulum system is it does not rely on the ejection of
material, such as hot jet
and
accelerated ions, in order to create the linear force, as what happens in a conventional jet engine,
propeller and ion propulsion.
The
pendulum system converts kinetic energy of a drive shaft into potential energy
which always pushes to one direction only.
The source of kinetic can be a rotating
electric motor or an engine.
Next,
I will show you an idea of a device called Force Engine, which utilizes this
theory.
I have posted the idea on the Google+ (
ceased in
April 2019) & Facebook back in year
2012.
The
main error on my previous posts was in the force analysis of the system, in
which I did not
mention
the driving force and reaction force in the x-direction. Anyway, there is no
net force in the
x-direction, but instead a moment being generated in the z-direction.
Nevertheless,
the fundamental idea to convert 2 centrifugal forces into one unidirectional
force
remains
intact.
Here is my video clip on YouTube about this subject :
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