View Question
 Question
 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?```
 ```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) change? (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 http://www.vilastic.com/tech3.htm 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. http://web.umr.edu/~wlf/Mechanical/timetemp.html because it shows an example of actual data and illustrates a very important organizing principle - namely, the "Time/Temperature Superposition". 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 frequencies. This principle, is part of a field of science (rheology) described e.g. here: http://dmoz.org/Science/Physics/Rheology/ 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. References: The classic book in the field of polymer rheology is "Ferry": http://www.amazon.com/exec/obidos/tg/detail/-/0471048941/ 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 materials. 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 amplitude. Hope this helps. hedgie``` 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 www.magnetsales.com , 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. RB``` 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 ... www.csulb.edu/~hvu/control_systems/pp36-43.pdf There are software system which handle such algebra. French Scilab is free and pretty good. http://scilabsoft.inria.fr/doc/intro/node24.html 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 http://answers.google.com/answers/threadview?id=269933 I still believe that this service is not best medium for software consulting. hedgie```