Matlab simulation of motor torque used to control damping 1

[sundep]

I am trying to simulate a spring-mass-damper system which uses an electric generator connected to a resistive load as a variable damper. The damper is varied by controlling the current to the resistive load.

I seem to have a mental deficiency pertaining to the correlation between the damping and power.

The system is forced to oscillate by a sine wave with amplitude = h and period = T. The force from damping is usually reperesented as Force = damping * velocity or F=c*v.

How can I express the power consumed by the damping action in terms of h, T, c and v?

Please assume everything is in phase just to keep this discussion simple.

 

[sundep]

Corrected Free Body Diagram

and

Equivalent Free Body Diagram

 

[electricpete]

I would put the damping element in parallel with the mass element as I had drawn on 26 Aug 09 22:46

Outside of ground and the applied displacement / velocity, there is only one nodal position (displacement or velocity) that matters - it is the node between the spring and where the damper / mass connect. The extra node you created between mass and damper has no meaning and would cause an error. The force on the mass is 2nd derivative of position - no reference other than ground and position of the mass is required. The force on the damper is 1st derivative of position (or it's rotational analogue = angle). I imagine you have a rotating magnet and stationary coil which feeds resistor. The stationary coil forms the reference on which the motion is based. Therefore once again the only reference points for damping are the point of connection to the system and ground (ground = stationary reference frame).

Here is the derivation associated with the 6 rules of the analogy above (I should have posted this before):

Derivation of the impedances and their electrical equivalencies is as follows:
Spring:
f = k x
f = k (v/jw)
Zk = v/f = j*w / k
By comparison to ZL = j*w*L, we find L = 1/k

Damping
f = c v
Zc = v/f = 1/C
By comparison to ZR = R, we find R = 1/c

Mass
f = m a (where a is referenced to ground)
f = m* (jw * v) (where v is referenced to ground)
Zm = v/f = 1/ (jw*m)
By comparison to ZC = 1/(j*w*C), we find C = m
(where C is connected between the location of the mass and ground).

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[sundep]

electricpete said... "It leads to an important question - what are we really trying to achieve....what is the purpose of varying resonant frequency? Are we trying to minimize movement of the mass inertia elements? "

I would like to extract the maximum power out of the system by way of the generator/damper for a displacement wave input on the spring in the range of 3 to 5 second periods.

I think this can be done by keeping the force vector and the velocity vector in the same direction. In other word, make the angle between force and velocity close to zero. In other words, make the damped resonant frequency equal to the frequency of the input dispacement wave.

It is not hard to imagine that if the mass was not oscillating at the same frequency as the displacement, that there would be moments in time where the movement of the mass and damper would diminish to zero and the power extraction would also diminish to zero.

At the damper... Power = Force * Velocity

 

[sundep]

It has been a busy couple of days for me, so I have only been able to think about this for about a minute and a quarter. There is a lot of info here for me to digest so I have some catching up to do so we are on the same page.

Thanks again for all of your input.

 

[sundep]

Okay. I understand the point that you are making about creating an extra node by drawing the damper below the mass. I agree that the mass and damper should be in parallel to create an electrical equivalent.

 

[sundep]

electricpete said "We don't exactly have a simple series or parallel RLC system (we have L in series with parallel R/C) so it will not behave exactly that way same and I do believe R does not have any influence on wd for this particular circuit."

The bode plot of the transfer function of the spring/mass/damper definitely shows how changing the damping effects the resonant peak of the system. I don't know how to explain this in terms of the electrical equivalent circuit.

Here is a m-file that varies Ce from zero damping to over damping.

 

[electricpete]

I can see why you drew the FBD the way you did - in the mechanical world, it is understood a lumped mass has only one position. The position of the "input" connection to the mass is same as the position of the "output" connection to the mass. Only in the electrical analogy when we substitute a capacitor do we find two distinct terminals with different voltages (related to velocity and displacement) and there would be a need to draw the Cap (representing mass) in parallel with the R (represienting damping). So the choice of how to connect that element depends on whether you are thinking electrical or mechanical. To use the electrical analogy it needs to be drawn as I described. If you are viewing it in the mechanical world as a mass, your FBD was fine. So I withdraw my comment.

I agree also for this system changing damping changes the damped resonant frequencies. I think I was previously working with a slightly different transfer function before I completely understood what are the inputs and ouptuts of your physical problem.

Attached I have done an analysis of the systems using mechanical relationships (part 2) and using the electrical analogy (part 3). I was glad to see the results (transfer function H) are the same with both methods. In part 4 I used the electrical analogy system to determine the "Rmax" which is the value of electrica-analogue parameter R (related to mechanical c) that will maximize power transfer to the generator. In part 5, I show that setting R to maximize power transfer (R=Rmax) does not appear to be equivalent to adjusting R toward resonance. You are welcome to double-check my logic... maybe I missed something.

Also I wonder if the wave excitation is truly independent of the energy extracted... or does the system interact with the waves to change them.

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[electricpete]

Above I had a polarity error inside the sqrt in the expression for wde (wd from electrical analogy).

I have corrected this in the attached. The solution Rmax makes sense only for w below the undamped natural freq of the system. I am not sure what value to choose if w is above that undamped natural freq... would require some more thought and possibly analysis. Probably not today.

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[electricpete]

My solution for Rmax was
Rmax = w L/ (1 ? w2 L C)
and it applies far below w=1/sqrt(LC), not sure about above yet.

One please reality check is to set w very small w<1/sqrt(LC). The the denominator is approx 1 and the numerator approx w*L. For this condition the parallel capacitive impedance is aprox infinite and makes no affect on the circuit. The optimum R is that which matches the supply impedance w*L. That's a good sign that the solution is correct.

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[electricpete]

It could have been solved much easier I think.
Find the Thevinin equivalent of the circuit supplying R. That is the impedance looking at the rest of the circuit with the voltage source shorted - it is ZC in parallel with ZL.

Zth = ZL*ZC/(ZL+ZC) = j*w*L/(1-w^2*L*C)
The R to maxmimize power transfer has a value equal to the magnitude of Zth
Rmax = w*L/(1-w^2*L*C)

This confirms the analysis for w below undamped resonant frequency which gives postive R. I am not sure what to make of the above-undamped-resonant-frequency case where the denominator turns negative resulting in negative R.

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[electricpete]

Duh, sorry, the magnitude of Z never turns negative. I think we should just use the magnitude of thevinin impedance which is always positive:
Rmax = |w*L/(1-w^2*L*C)|

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[electricpete]

To summarize:

We started with mechanical system:
Ground===dapplied==k===meq====ceq====Ground
where dapplied=applied displacement, k=spring, meq=mass, ceq=damping

We converted the input by known relationship d->v:
Ground===vapplied==k===meq====ceq====Ground
where vapplied= applied velocity

We converted to electrical analogy using 6 "rules" above:
Ground===Vapplied ===L ======R === Ground
|
===C=====Ground
where L = 1/k, R = 1/c, C = m, Vapplied = applied voltage

We converted to Thevinin equivalent:
Ground===Vth ===Zth ======R === Ground
Where Zth = j*w*L/(1-w^2*L*C) is impedance from the middle node to ground with Vapplied shorted.
Vth – can be determined by voltage divider to find voltage betwen L and C with R removed, but doesn't matter for purposes of finding the optimum R.

Maximim power transfer occurs when R="Rmax" = |Zth|= |w*L/(1-w^2*L*C)|
This corresponds to c="cmax" = |w/(k-w^2*m)|


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[electricpete]

One last point I should have included in my summary:
adjusting damping (R or c) to create max power transfer for a given applied frequency does not lead to the same solution (of R or c) as adjusting damping (R or c) to change the system natural frequency to be equal to the applied frequency.

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[electricpete]

.... the last point said another way:
Choosing R to maximize I (=current thru resistor) or to maximize V (voltage accross the resistor)
is not the same as
Choosing R to maximize I^2*R


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[sundep]

electricpete said, "Also I wonder if the wave excitation is truly independent of the energy extracted... or does the system interact with the waves to change them."

If the energy in a wave is to be completely extracted, then the wave is completely destroyed. To destroy a wave is to make a wave which is equal and opposite.

Falnes does a good job explaining the idea in the attached paper. See Section 3 on page 190.

 

[sundep]

electricpete,

This electrical equivalent stuff seems to be quite useful.

What is your source of information for this technique?

Can you suggest a good book or a few good papers?

Thanks

Paul

 

[electricpete]

Thanks for the paper.

My original reference for the vibration / electrical analogy was Harris’ Shock and Vib Handbook chapter on “Mechanical Impedance Analysis”. He works several problems. The bizarre thing is that he never once mentions anything electrical (not voltage, current, resistance, inductance, or capacitance), even though he uses terms like Thevinin equivalent, Kirchoff’s laws etc. Must have been written by a mechanical type ;-)

From studying Harris, I wrote down the analogy in terms of the 6 rules (posted 25 Aug 09 11:32) and the derivation (posted 28 Aug 09 17:48). I keep those in a word file for easy reference – for me that captures everything I need to know about it.

I googled “Mechanical Impedance Analysis” and came up with a better reference than Harris (see particularly Table 4.4 which summarizes the analogy):

http://books.google.com/books?id=Yy...epage&q=mechanical impedance analysis&f=false

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[electricpete]

I should have mentioned that the analogy I used is referred to in Table 4.4 of above link as “Force Current Analog”. Not to be confused with “Force Voltage Analog”.

For me the Force Current Analog has good intuitive advantage over the Force Voltage Analogue. In the force current analog, forces flow through branches like current… and relative velocity is a difference between nodes like voltage. The only downside is that the thing called "mechanical impedance” transforms to the inverse of electrical impedance under the Force Current Analog. I just tend to use Z to represent electrical impedance and ignore the conflict with the term “mechanical impedance” (I don’t’ use that term).

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[sundep]

electricpete,

Thanks for the Maple calculations. Very nice.

I almost finished going throught the calcuations, when my lovely Jeannabelle's water broke and we had to leave for the birthing center. I may have one question about the solution using the equivalent circuit. I will re-examine the calculations after the three of us are back home.

Thanks again for walking me through the electrical equivalent.

Paul