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Top Dead Center and Singularity

The Pedal 30 degs off T.D.C.

There is a mathematics term thrown around a lot by the “free energy” people and after the last post about the do-nothing machine, this is probably the best time to deal with it. The term is singularity or as used by them “the singularity point”.  According to Wikipedia the term singularity means a discontinuity in a function.  Most often in my experience it is the point where you try to divide by zero and the equation “blows up” or goes to + or – infinity.  Believe it or not, we have all experienced that if we ever have ridden a bicycle or tricycle.  There really is nothing magic about it. In the last post, I described where the do-nothing machine experienced that.

In both the first, and the header picture I show a pedal attached to a crank.  For now imagine that is connected directly to a wheel as on a tricycle.  Most bicycles have mechanisms preventing the pedals from making the wheel go in reverse.  Most tricycles do not have that mechanism, so the easy to understand example for now is the tricycle.  Also we will not consider the pedal on the opposite side that is 180 degrees opposite of this one.

The effective “lever arm” of the pedal with the crank at 30 degrees from vertical.

For the rest of this discussion we will consider 0 degrees as the point where the crank is vertical and upward and when the crank is moved to the right from there (clockwise) will be called positive degrees.  This would be the view from the right side of the tricycle.  At 30 degrees we could diagram the twisting force, torque, on the crank as shown in the picture to the right of this.  With a downward force the effective “lever arm” is the horizontal component of the crank.   If the crank arm is assumed to be 1 foot, the lever arm is the sine(30 degrees). This length would also in feet.

Imagine now that we stopped the tricycle with the pedal at various angles and applied a 50 pound force directly downward on the pedal.  If we had the pedal at 90 degrees (horizontal) we would get a lot of twisting force (torque) on the wheel. At 60 degrees we would get less and at 30 degrees even less.  At Top Dead Center (T.D.C. or 0 degrees) no matter how hard we pushed we would get no torque.   Since our example is a tricycle, at a negative angle we would get negative torque and the tricycle would go in reverse.   This is shown in the first graph.

The force required to create 10 ft-lbs of torque at various crank angles.

Now, as we all know, we are not going to push any harder than necessary to get the tricycle moving.  So our real question is how much force is necessary to get the a given amount of torque at the various starting angles.  Torque = Force X Lever Arm distance,   So Force = Torque / Lever Arm Distance.  This is graphed in the final graph where the value of required torque was set to be 10 ft-lbs.   On the negative degrees values we have a negative force.  That would mean we would be pulling up, but actual real life we would be pressing on the other pedal.   As the negative angle approaches zero the lever arm becomes very very small and the required force becomes larger (more negative) and as it get very close to zero this “blows up” and would become “negative infinity”.  Approaching zero degrees from the positive side exactly in inverse would happen and it would become “positive infinity”.  At exactly zero it is undefined and that is the singularity point.
All of us that have ever rode a bicycle have eventually ran into the T.D.C. problem and have learned to work around the problem.   I understand that when steam engines were first being developed for rail roads they also had this problem and coped with it by mounting two cylinders driving the wheels at 90 degrees apart.  (Steam engines produced power on each 1/2 cycle of a revolution of the wheels, both on the extension and retraction of the cylinder.)

I chose the example to be the simplest one I could think of.  The actual motion on a bicycle or tricycle is more complicated because we often move our foot at the ankle joint.   If I had chosen a typical train steam engine, it is complicated by the the extra linkage.  In other words, this model is the first approximation and is not really valid, but it was a good introduction to both the T.D.C. problem and the term singularity.

In mathematics and models of physical processes and devices the divide by zero problem is often present.   In controls classes I had in school we had to make maps of equations and worried about the zero terms in the numerator and in the denominator. A zero in the numerator meant that nothing would happen at that input, and a zero in the denominator meant that we had a “singularity”. it was really undefined but more than likely it would reach a resonance place and things would go badly out of control.

In most of the free energy articles I have read, singularity is held up as some magic point where abundant and free energy exists.  It hides things in mystery and makes it sound as if someone really knows something the rest of us don’t.   Mankind has been surprised by things before.  For example Einstein’s famous equation E-mc^2 where energy and mass are found to be equivalent. However, most of the free energy stuff I have read tends to want to throw big words around to mesmerise people yet does not really show me something. While I will not say something is impossible, I tend to not believe most of those claims.  (Actually I have yet to find one I do believe, but I am still open to be surprised.)

Gary

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