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Q: Physics and electromagnetics ( Answered 5 out of 5 stars,   0 Comments )
Subject: Physics and electromagnetics
Category: Science > Physics
Asked by: drbaker-ga
List Price: $40.00
Posted: 19 Feb 2004 15:30 PST
Expires: 20 Mar 2004 15:30 PST
Question ID: 308586
Consider a 2.5 cm diameter, 27.3 cm long rod composed of five sections
with its ends fixed. The middle section is an electromagnet (iron rod
core) that is 25 cm long, has a mass of 4.7 kg, and produces a
magnetic flux of B = 0.82 T when a current energizes it. Glued to each
end of this middle section is a 0.2 cm thick Plexiglas (or Lucite)
separator disk (also 2.5 cm in diameter) that exhibits a modulus of
elasticity of 50,000 psi, tensile strength of 7,000 psi, compression
strength of 12,000 psi,  elongation of 8%, Brinell Hardness number of
19, and a mass of 12 milligrams. Glued flush against these separation
disks at both ends of the rod are two 0.95 cm thick fixed
permanent-magnet disks (also 2.5 cm in diameter) each having a mass of
36 milligrams. When a half cycle of AC current passes through the
electromagnet?s coils the center section or electromagnet core is
urged to the right with the sequence of polarities left-to-right: N
(permanent magnet) to N (electromagnet) and S (electromagnet) to N
(permanent magnet). When the next half cycle of AC current passes
through the electromagnet?s coils the electromagnet?s core is urged to
the left with the sequence of polarities left-to-right: N to S and N
to N. The questions are: (1) When a 60 cycle AC current is applied to
the electromagnet?s coils, what is the force (in Newtons) that the
core rod produces against the fixed permanent-magnet ends, what is the
acceleration of the core rod in g?s, and what are the equations
utilized to obtain these values? (2) What is the power (watts)
consumed each cycle (in addition to power lost due to the resistance
of the coils, which happen to amount to about 5 watts)? (3) If the AC
current has a frequency of one kHz does the answer to (1) and to (2)
change? (4) If the AC current is one GHz does the answer to (1) and to
(2) change and if so, then what are the new results?
Subject: Re: Physics and electromagnetics
Answered By: hedgie-ga on 28 Feb 2004 05:11 PST
Rated:5 out of 5 stars
Good day, Dr. Baker

      This time you have come up with an interesting
electro-mechanical system, a system which can be modeled by a set of
ordinary differential equations.
I cannot provide a numerical answer, and I therefore considered
posting this as a comment.
However, considering GA pricing guidelines and fact that I can provide
relevant information which may help you to refine the question and
supply missing data , I have decided to post this as an answer. I hope
this decision will meet with your approval.

The essential information missing in your quesdtion is a description of the
"rheological properties of the polymer element (Lucite or Plexiglas)"
in your linear stack.
 These properties, the rheological or viscoelastic properties of your
materials, are key to answering your main questions:
 (3) If the AC current has a frequency of one kHz, does the answer to
(1) and to (2)
 (4) If the AC current is one GHz, does the answer to (1) and to
(2) change and if so, what are the new results? 

The answer is: Yes, the answer will change dramaticaly.

The parameters you have given:  modulus of elasticity of 50,000 psi, 
tensile strength of 7,000 psi, compression strength of 12,000 psi ...

are sufficient for many standard engineering  problems, but insufficient
for dynamical systems such as yours.  

 I should add that the material properties, too, not just the
geometry, should be given in SI units.
(Mixing systems can lead to ugly numerical errors, as fate of the
Adriane rocket illustrates).
So, you need to describe materials by complex modulus of elasticity,a
function of frequency, as explained here

The first SEARCH TERM after that is :
 Dynamic response of materials, of polymers, linear viscoelasticity,
rheological equation of state,..

 From many links which come up as a reponse to that, I am picking this one:

A Worked Example of Time Temperature Superposition Data.   

 because it shows an example of actual data and illustrates a very
important organizing principle - namely, the   "Time/Temperature

 For you, since you are interested in high frequency response, this
plus the operating temperature of your device are important. Very
simply: If you cool your stack, it behaves as if you move to higher
frequencies.   The relation is quantitative over many decades of

 This principle, is part of a field of  science (rheology) described e.g. here:   

Even though this science began with the characterization of mechanical
properties of materials, it now applies to all material properties.
For your  iron element, both mechanical and ferromagnetic properties
would have to be characterized by the spectrum of relaxation and
retardation times (by rheological equation of state).
(Actually, hysteresis of ferromagnetic materials was an early example
of behavior, which when manifested in mechanical properties is called
"creep."  In hysteresis, as in creep, response depends on  history
(thermal, magnetic field B, deformation,..) not just on  the current
value and rate of change of a generalized coordinate).
Mechanical, optical, and electrical/magnetic properties usually
exhibit parallel properties, since the same molecular micro-structures
contribute to the viscoelastic spectra of all properties.
The classic book in the field of polymer rheology  is "Ferry":  

The math needed to solve your system is well described in a slim volume:  

  Gross: Mathematical Theories of Linear Viscoelasticity, 

  which Amazon does not list, but which is available in most technical libraries.

 So,  in summary:  

  Once your materials are properly characterized,
  in the range of temperatures and frequencies  of interest,
  the solution of your stack is relatively  easy. Equations,
 differential or integral, are linear, since the amplitudes of
vibration will be small.

 Large impedance, and therefore low amplitudes behaviour is more pronounced at
 low-temperatures/high frequencies, and may  become a problem.
 Qualitatively, you need stiff components with low inertia to get a measurable 
 mechanical effect at  very high frequencies. For this reason, people tend
to use piezoelectric, rather then electromagnetic actuators, to
explore the high frequency response of materials and properties of
such electro-mechanical devices.
	An important and perhaps obvious point:  the mechanical properties of
your components will be manifested in electrical properties (impedance
of your coil). That is common to all electro-mechanical devices of
this type and quite useful when characterising dynamical response of
	Of course, at very low temperatures, let's say a few K's,
all material becomes stiff and you may be entering  poorly-explored,
and therefore interesting territory.

A final point, possibly useful, possibly just a factoid: At very high
pressures, all materials approach the same equation of state;
individual differences become less pronounced. Something like that may
happen at very low temepratures. So, deciding on a range of
temperatures may be the first engineering parameter to select and
specify for your device.Low temperature range may be necessary to
achieve a stiffness necessary to a high frequency motion of measurable

Hope this helps.


Request for Answer Clarification by drbaker-ga on 01 Mar 2004 13:34 PST
Hi: I am posting this as a request for an answer clarification, but
since it is more than that I will also ask another question for you to
respond to and be compensated.

First, I discovered the answer to my question (1) by visiting , clicking on calculations and inserting the
appropriate dimensions for the magnetic material Neo48.

Second, I see no need to include the Plexiglas spacers, so I have
removed them from the question.

Third, your remark concerning the temperature is very helpful. Since
liquid Nitrogen is said to be ?... cheaper than beer...? I can easily
keep the entire apparatus at the Nitrogen boiling point temperature.

Forth, the question of coil impedance in the low-temperature and
high-frequency regime is important, but I have no idea how to get a
handle on it.

Fifth, and most important, you suggest ?... stiff components with low
inertia ...? Here is the crux of the problem: At the most fundamental
level what I require is a very large FORCE delivered at a high
frequency (the size of the mass being acted upon by the force and its
displacement are unimportant except as to a tolerable acceleration). I
have looked at piezoelectric vibrators; specifically, a Film Bulk
Acoustic Resonator (FBAR). The problem seems to be that for any
appreciable force the vibrating element of the FBAR or other
piezoelectric devices would have to endure an acceleration of about a
billion g?s or so and that is unacceptable for a number of reasons.


Clarification of Answer by hedgie-ga on 02 Mar 2004 05:30 PST
Indeed.  If you remove spacers you simplify the problem significantly.

 First, you remove polymer material, which contribute viscoelastic behaviour
 second, you now have an isolated coil, which is described by a
standard           engineering formula.

But you also lost something, namely chance to operate at resonant frequency,
which is one way how to lower the impedance and so increase the amplitude.

Mathematicaly, and I suspect physicaly as well, interesting case is a
composite stack with rigid boundary you had before.

 There are many ways how to model such stack. I have found easiest the
method which describes each element by complex transfer matrix
(sometime called twoports by EEs)  as described e.g. here

 Transfer Function and Transfer Matrix
 ..Determine the transfer functions and transfer matrix. Solution
 The system under consideration has one input, f(t), and two ...

There are software system which handle such algebra.
French Scilab is free and pretty good.

For active element (magnet, pzt ..) you use 4 x 4 matrix, which connects
electrical and mechanical properties. That will provide the 'missing link',
between all mechanical elements and electrical impedance, once the stack is solved.

So, I will look for you next question, even though, as I said elsewehere
I still believe that this service is not best medium for software consulting.

drbaker-ga rated this answer:5 out of 5 stars

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