How to Calculate Mechanical and Electromagnetic Damping?

[henlus]

I will appreciate any contribtion to the question below. > You have a system that consists of a coil and magnet. The magnet, which is resting on a spring, is free to move up and down in the coil. This system can be modeled as a simple mass-spring-damper system with a damping coefficient that consists of the mechanical and electromagnetic damping. - the question here is the formula i can use to calculate the mechanical and electromagnetic damping. Thanks as you contribute.

 

[jsgruszynski]

I will appreciate any contribtion to the question below. > You have a system that consists of a coil and magnet. The magnet, which is resting on a spring, is free to move up and down in the coil. This system can be modeled as a simple mass-spring-damper system with a damping coefficient that consists of the mechanical and electromagnetic damping. - the question here is the formula i can use to calculate the mechanical and electromagnetic damping. Thanks as you contribute.

The general way to solve it systematically:

1. Write lumped-element circuit equations for the electrical portion
2. Write lumped-element circuit equations for the mechanical portion (mass-spring-damper are analogous to inductor-capacitor-resistor, respectively, only the KVL/KCL of electrical circuits don't apply but instead you use mechanical rules)
3. Determine an electromechanical transducer 2-port network between the two circuits that defines the conversion and impedances - just like h- or y- or z- parameters

Doing this you get most of the solution using simple circuit analysis and linear algebra.

 

[henlus]

The general way to solve it systematically:

Write lumped-element circuit equations for the electrical portion...

Thank you for your contribution sgruszynski, i will put up an attempt at solution soon.

 

[wil3]

Hello. If the magnet is entirely within the cylinder of the coil and the coil is large enough, we can approximate all flux lines as contained within the coil (except those that are parallel to the axis). If this is the case, then no current will flow in the coil, because the magnet has two poles that are each inducing equal but opposite currents:
$$\dfrac{\delta {\bf B_{top}}}{\delta t} = \dfrac{\delta {\bf B_{bottom}}}{\delta t}$$
In this case all damping would be mechanical.

I'm guessing that in your problem the above condition is not the case, which suggests that
$$\dfrac{\delta \Phi}{\delta t}$$
will be a more complicated expression. Would you mind providing some additional information? The resistance of the coil may need to be known.

Best of luck

 

[henlus]

Sorry for my long delay. Preparation for a seminar defence has been tying me down. @jsgruszynski > the lumped circuit equation for the electrical portion is Lq''+Rq'+kq=e(t). That of mechanical portion is mx''+Cx'+kx=f(t). I don't know how to get an expression 4 the damping coefficient, C from here. I know that for a mass-spring-damper system whereby the mass is a magnet that moves in a coil of wire, the damping consists of mechanical and electrical damping coeff. Also, how can I get an expression 4 the power output of such a setup, assumming f(t) to be sinusoidal (eg 5sint) @wil3 > the magnet is not entirely within the cylinder of the coil and the magnet is just slightly smaller than the coil. This system is just like a simple linear generator. >The coil resistance is not actually known but I know it can be calculated from the resistivity & length of the wire. What i actually want to know is an expression for the 2 damping coefficients and the power output when the magnet is moved with a sinusoidal forcing function for example.

 

[jsgruszynski]

There is some content in these documents that should help. Basically your electromechanical transform matrix this damping component (it's both mechanical and electrical).

http://www-lar.deis.unibo.it/euron-geoplex-sumsch/files/lectures_2/lectures-batlle.pdf

http://www-ma4.upc.edu/~carles/Modeling of electromechanical systems.pdf

The "elementary electromagnet" shows the mechanical and electrical transform. This also addresses some of the "not entirely in" aspects.

 

[henlus]

There is some content in these documents that should help. Basically your electromechanical transform matrix this damping component (it's both mechanical and electrical).

http://www-lar.deis.unibo.it/euron-geoplex-sumsch/files/lectures_2/lectures-batlle.pdf

http://www-ma4.upc.edu/~carles/Modeling of electromechanical systems.pdf

The "elementary electromagnet" shows the mechanical and electrical transform. This also addresses some of the "not entirely in" aspects.

I have downloaded the two ebooks and looked into them, but I couldn't read much meaning into them. If you can help me with a step by step solution I will appreciate it. However, let me make the following attempt at solution:


I wish to calculate the power output of the system when it is vibrating. The induced voltage is given by:
E=N*B*v(t)
N=number of turns of the coil; B=magnetic flux density; v(t)= velocity of the magnet wrt time.
N & B can easily be known. To calculate v(t), I intend differentiating the displacement formula for an overdamp mass-spring-damper system.

my"+cy'+ky=F(t) ……..mechanical equation

where m=mass of the magnet
c=damping coefficient
k=spring constant
y=displacement of the magnet
y’=velocity of the magnet
y’’=acceleration of the magnet

With v(t) known, I can calculate power output from:
Power=(E)^2/R

I found the electrical damping coefficient to be:
Ce=(NBl)^2/(Ri+RL+jwL)

Where Ce=electrical damping
l=length of wire
Ri=wire resistance
RL=coil resistance
L=inductance of the coil
w=angular frequency
j=complex number
Is my approach correct?

 

[jsgruszynski]

Fig 4 and sect 2.5 describe the relevant model for what you're talking about.

 

[henlus]

Fig 4 and sect 2.5 describe the relevant model for what you're talking about.

I have gone through Fig 4 and section 2.5. Actually I have come across that configuration in one of my ebooks, but the configuration is very different from the one I'm working on. Mine is just a magnet moving in a coil of wire as shown in the diagrams.

 

[jsgruszynski]

I have gone through Fig 4 and section 2.5. Actually I have come across that configuration in one of my ebooks, but the configuration is very different from the one I'm working on. Mine is just a magnet moving in a coil of wire as shown in the diagrams.

Think of the magnetic flux path as a circuit. Broken into pieces, it's equivalent. The bits in the coil are the same as the model's coiled section. The bits outside of the coil are equivalent to the moving iron. There is no actual gap but there is a variance in the partition between $\phi$_l and $\phi$_m based on the offset of the slug.

The magnet's polarization is just an offset to be added in compared to just iron which is a zero polarization offset.

 

[henlus]

Hi jsgruszynski, thanks for coming back.

Actually, I don't really understand most of what you said. Can you please go through the attempt at solution and tell me if I'm correct. I just want an expression for the power output of the arrangement.