Newsgroups: sci.physics
Subject: Re: Davis Mechanics
From: jim@netlink.cts.com (Jim Bowery)
Date: Fri, 04 Sep 92 08:40:22 PDT
Organization: NetLink Online Communications, San Diego CA

(continuation of "The Fourth Law of Motion" article -- part 2 of 2)

 
The class of anomalous behavior which we wish to study 
involves in every case the presence of surge as we have just 
defined it.  Under conditions of constant or zero acceleration 
mechanical bodies or systems of bodies obey Newton's Laws 
reasonably well.  It is under conditions of changing acceleration 
that difficulties arise.
 
The key word in our analysis of dynamic systems will be 
"simultaneity."  The Laws of Motion presuppose exact 
simultaneity of action and reaction.  In other words, if the force 
exerted by Mass #1 on Mass #2 is suddenly changed, the force 
exerted by Mass #2 on Mass #1 must change at the same instant to 
satisfy Newton's image of the universe.  Einstein recognized that 
this condition would not be met in the case of bodies separated by 
astronomical distances since the change in gravitational field 
would have to be propagated at some finite velocity, presumably 
that of light, but he did not carry his investigations into the simpler 
field of everyday mechanics.
 
If it is true that two stars cannot act as a Newtonian system in 
less time than it takes gravity to propagate from one to the other 
and back, it is equally true that any real body or system of bodies 
suffers to a greater or lesser extent from the same problem.  
Newton's Laws, strictly speaking, apply only to mathematically 
infinitesimal particles or perfectly rigid bodies, neither of which 
exists in the real world.
 
Consider, for example, a simple steel rod one meter long which 
I wish to move by applying a force to one end.  The instant I start 
to apply the force a message leaves the end of the rod as a plastic 
or elastic compressive wave which travels at a speed of 
approximately, 5,000 meters/second.  The compressive wave 
travels to the far end of the rod where it is reflected as a rarefaction 
and returns to the point of application of force at the same speed.
Until the wave returns, 4/10,000 of a second later the rod as a 
whole cannot move according to the Second Law!  No matter how 
much force is applied, the center of gravity of the rod cannot obey 
F=Ma  in less than this time.  It would be oversimplifying to say 
that the rod acts as though it had infinite mass during this time, 
since the center of gravity will be moved somewhat by 
compression, but for all practical purposes, the rod acts as though 
it had a much larger mass than it actually has.
 
This phenomenon is observed with all real mechanical systems 
to a greater or lesser extent and is normally described in the 
engineering textbooks as the "starting transient."  Although 
engineers must concern themselves with starting transients in such 
matters as determining peak power requirements for motors, 
science has devoted relatively little attention to study of this 
phenomena, preferring, in general, to consider steady-state 
conditions where the Laws of Motion and the Laws of 
Thermodynamics may be more comfortably applied.  Therefore, it 
is precisely with the starting transient that we will begin our 
investigation of the dynamic behavior of real bodies.
 
We have concluded that all real bodies or systems of bodies 
experience a starting transient when a force is applied.  The 
particular behavior of a given system will depend on: (a) how 
rapidly the force is applied and, (b) the built-in delay time, or 
"critical action time" of the system.  More exactly, we have found 
that the behavior depends upon how rapidly one attempts to 
change the acceleration applied to the body.  The ultimate 
acceleration of the body, the "a" of F=Ma, is not what is critical; it 
is the rate of onset or "surge" of the acceleration which is vital.
Colonel John Paul Stapp, the United States Air Force Flight 
Surgeon who subjected himself to crushing accelerations to 
investigate the hazards of seat ejection and airplane crashes, 
observed that the extent of damage to personnel and equipment 
depended in an important way on surge as well as the acceleration 
itself.  In fact, today the Air Force specifies limiting rates of onset 
as well as limiting accelerations for men and equipment.  On one 
case, Colonel Stapp estimated that the effect of a given 
acceleration on his body was over twice as great if applied at very 
high rate, than would be predicted by F=Ma!
 
Thus, it is not wholly illogical to attempt to solve the puzzle of 
high rate-of-onset anomalies by postulating that there is a force 
proportional to the rate of change of acceleration as well as 
Newton's force proportional to acceleration itself.  There is no 
conflict here with Newton, for Newton considered only systems 
where either velocity or acceleration were constant.  Since his data 
inputs were from astronomy and since he had no instruments 
capable of investigating effects of changing acceleration  *effects 
that may occupy milliseconds or less * this is not particularly 
surprising.
 
What form should the Equation of Motion now take if we 
assume a force proportional to the surge as well as a force 
proportional to acceleration?  The simplest assumption, and one 
which seems to be supported by preliminary data, is that the new 
force is additive, in the same way that forces due to viscous drag 
and displacement are additive.  In other words, we now write the 
Equation of Motion:
        
        F=Ma+Aa'                (5)
 
where A is a new term which we have labeled "intractance" and 
which has units of mass-seconds.  Because the solutions of the 
equations in some cases yield the ratio A/M as the critical action 
time of the system, we have in those cases assumed that the 
intractance is the product of the mass and the critical action time:
        
        A=DM                    (6)
Let us now see how this equation of motion can be used to 
analyze and predict the anomalous behavior of a simple system.  
Starting transients normally are considered only in connection 
with the beginning or end of a motion and hence are accorded no 
particular attention.  However, there are certain types of cyclic 
motion where the transient behavior is continuous, or repetitive 
and we will see later that even certain single transients may have 
critical importance in understanding natural events.
First consider the simple case of a real body subjected to 
harmonic oscillation.  To make matters easier, we will assume that 
there is no restoring force proportional to displacement present and 
no viscous damping in the system.  If applied force is simple 
harmonic in nature it may be expressed in the form:
        
        F=F0*cos(2*pi*f*t)              (7)
 
where f is the frequency of the vibration in cycles per second, 
and t is the time in seconds after the beginning of the motion.  The 
Newtonian equation for vibration in one dimension would be:
 
    F0*cos(2*pi*f*t)=Ma=M(d^2)x/(dt^2)          (8)
 
and the displacement of the driven body would be:
 
     x=x0*cos(2*pi*f*t)=cos(2*pi*f*t)*-F0/(4(pi^2)(f^2)M)       (9)
 
In other words, the displacement will obey Newton's Second 
Law and will be exactly opposite in phase to the driving force, 
thus also satisfying the Third Law.
 
However, if the body to be oscillated is real, it will not respond 
instantaneously, so that the rate of onset must be considered.  The 
equation of motion then assumes the form:
 
    F0*cos(2*pi*f*t)=(Ma+DM)da/dt=M(d^2)x/(dt^2)+DM(d^3)x/(dt^3)  
 
    (10)
 
where D is the delay time, or "critical action time" of the body and 
da/dt is the surge.  D is assumed to be constant.
 
This equation has the solution:
 
        X=X0*cos(2*pi*f*t-phase)        (11)
 
where tan(phase)=2*pi*f*D               (12)
 
and X0=-F0/(4*(pi^2)(f^2)MZ)            (13)
 
where Z is a sort of complex impedance and
 
Z=sqrt(1+4(pi^2)(f^2)(D^2))             (14)
 
The important feature to note in this solution is that the 
amplitude of the oscillation is less than predicted by the 
Newtonian equation and even more important, there is a phase 
angle between the driving force and the displacement, even in the 
absence of viscous damping.  This solution demonstrates the most 
significant characteristic of real bodies, to wit:  not only is 
displacement somewhat less than Newton would predict for a 
given force, leading to an increased apparent mass, but reaction 
is no longer exactly opposite to the applied force:  there is a phase 
angle which will be larger the longer the critical action time of the 
system!  Action and reaction are not simultaneous!
 
A technical paper will be published in the near future which 
will present a more general discussion of solutions to the equations 
of motion.  The solution presented here will serve only to show the 
effect of critical action time in a system under the simplest 
situation.  The disastrous effect on the theory of mechanics of the 
existence of critical action time in real bodies may be realized at 
once when we consider a system so arranged that the critical 
action time is not the same in all directions.
 
If, for example, we build a mechanical oscillator such that the 
critical action time is short compared to the period of oscillation in 
one direction, the mass will appear to be approximately Newtonian 
and the phase angle of reaction will be negligible when the 
oscillator is moving in that direction.  If we now change the system 
to make the critical action time much longer during the time the 
oscillator is moving in the opposite direction, the mass will appear 
to be greater and the phase angle larger, even though the total 
applied force is exactly the same in the two directions.  Thus there 
will be a net unidirectional acceleration of the driven mass in the 
direction of least apparent mass even though the applied force is 
balanced! 
 
This conclusion is merely a logical consequence of the 
existence of a force proportional to the third derivative.  We have 
not yet, of course, considered how such a behavior could be 
possible in terms of physical reality.
 
In recent years a number of "reactionless" drives have been 
proposed and some demonstrated, including the controversial 
Dean Drive.  To the extent that any of these devices have produced 
useful thrust, they presumably have embodied this principle.  It is 
not the purpose of this article to attempt to explain the operation of 
these systems as such, nor to claim that any of them are practical 
devices.  
 

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