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.