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.

 

[electricpete]

I imagine you are talking about a linear generator although the answer should apply to any linear damping element.
Assume the velocity V(t) appears across the damper
V(t) = V0*sin(w*t)
The force is
F = C*V = C*V0*sin(w*t)
Power = F(t)*V(t) = C*V0*sin(w*t) * V0*sin(w*t)
Power(t) = C*V0^2*sin^2(w*t)
<Power> =0.5* C*V0^2


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[electricpete]

Whoops. Needs a correction. Let me prepare that...

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[electricpete]

Nope, it was right as written. It is a perfect illustration of the electrical analogy for mechanical vibration. The mechanical damper C acts like a electircal resistor r=1/C. Mechanical velocity V plays the role of electrical voltage v, mechanical force F plays the role of current i.
<Power> = 0.5*C*V^2 (mechanical) = 0.5*(1/r)*v^2 = vrms^2/r (electrical)
We could also express it as
<Power> = 0.5*F^2/C (mechanical) = 0.5*r*i^2 = irms^2*r (electrical)

Here is the electrical analogy for mechanical vibration, according to electricpete:

You need only 6 simple rules in order to solve any sinusoidal steady state lumped linear m/k/c mechanical vibration problem using electrical circuit impedance (R/L/C) analysis techniques:

1 - mechanical force f plays the role of electrical current I

2 - mechanical velocity difference v plays the role of electr voltage difference V

3 - mechanical mobility v/f plays the role of electrical impedance Z = V/I
(use the symbol Z throughout. We use the term impedance to represent either electrical impedance or mechanical mobility .. avoid the term mechanical impedance which is the reciprocal of mobility).

4 - A spring (k) acts like an electrical inductor of inductance L=1/k
The impedance is Zk = j*w/*k

5 - A damper (c) acts like an electrical resistor of resistance R = 1/c
The impedance is Zc = 1/c

6 - A mass (m) acts like an electrical capacitor of capacitance C =m **
The impedance is Zm = 1/(j*w*m)
** one terminal of this mass/capacitance is connected to ground.

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[electricpete]

My answer is based on the presumption that a generator can be used as a linear damping element characterized by it’s damping constant C. That may or may not be the case. I tend to think it is a close approximation if your coil inductive reactance is much less than total circuit resistance (w*L << R).

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[sundep]

Yes, Power = Force * Velocity. Thanks, electricpete.

 

[sundep]

The electrical analogy to mechanical vibration is an interesting approach.

The next step in this problem is to choose m, k and c to maintain resonance over a range of wave periods by varying the power that is extracted power from the damper. The electrical analogy might help.

I will try to upload a free body diagram for discussion.

 

[sundep]

electricpete said... "My answer is based on the presumption that a generator can be used as a linear damping element characterized by it's damping constant C."

This is a valid presumption because the time constant of the generator and circuit is much, much smaller that the wave period. In this case, tenths of second versus 3 to 5 seconds.

 

[sundep]

Here is the FBD for this project.

The equivalent inertia of the motor, gearbox and sheave is lumped together and called Js.

I will attempt to convert this mechanical system into an electrical equivalent.

Please post an electrical equivalent if anyone is comfortable with this method of analysis.

Thank you in advance.

 

[electricpete]

It is a little more of a challenge to apply and communicate the analogy for this system due to a couple of wrinkes: 1 – combining linear inertia and rotary inertia; 2 – speed transformation ratio; 3 – mechanical damper has physical electrical parameters (load resistance etc)– likely to get mixed up with the analogue electrical parameters.

I am almost inclined to say we should avoid the analogy in the face of these difficulties due to increased potential for confusing what the variables represent. Nevertheless, it can be done.
1 – convert all inertia's to equivalent linear inertia
2 – convert everything to left side of the gears

Here is the electrical analogue circuit I would use:

Ground===Vapplied ===Zk =====Zi === Ground
|
===Zd=====Ground

Where
Vapplied = d/dt(x) = w*H*cos(w*t) / 2
Zk = j*w / k
Zi (inertia) = m + r^2*(Js + n^2*Jm)
Zd (damping) = 1/Ceq
r = radius of shaft upon which you've wound that string
Ceq = is equivalent damping factor seen on left side of gears


I don't know what your bm is supposed to mean but we need a model of your generator.
Let's say your generator voltage is Eg = K' * w' where w' is speed on left side of the gears (w' = w*n if w is speed of generator on right side of gears)

Expressing in terms of velocity on left side v' = w'/r...
Eg = K' * v' * r

Let's say this voltage is applied to resistance R
P = Eg^2 / R = K'^2 * v'^2 * r^2
Ceq = P/v'^2 = K'^2 * r^2

Once you solve the system, the current flowing from Vapplied is the spring force. All other mechanical parameters can also \be solved from the elctrical solution as well although not as easily... not sure which mechanical parameters you are interested in.

I have assumed the "string" that you showed does not stretch. I also assumed the angular velocity of the shaft is low enough that the string always remains tight between shaft and mass (i.e. the downward acceleration of the mass below always remains less than 1 g). Under this assumption we have ideally a linear system with two forcing functions (displacement on top and gravity below) and we divide the solutions by superposition and discard the uninteresting static gravity response.

By the way, what do you mean by "wave period"....what wave?


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[electricpete]

Correction

Zi (inertia) = m + r^2*(Js + n^2*Jm)

should have been:

Zi (inertia) = 1/(j*w*m_equivalent)
m_equivalent = m + r^2*(Js + n^2*Jm)

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[electricpete]

I think I have inverted the n ratio in at least one place.

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[electricpete]

Maybe just one place. Should be:

Zi (inertia) = 1/(j*w*m_equivalent)
m_equivalent = m + r^2*(Js + (1/n^2)*Jm)

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[electricpete]

I have assumed the "string" that you showed does not stretch. I also assumed the angular velocity of the shaft is low enough that the string always remains tight between shaft and mass (i.e. the downward acceleration of the mass below always remains less than 1 g). Under this assumption we have ideally a linear system ....

If that assumption is not met, then it is a non-linear system and there is no linear circuit analogy. I would venture to say that a numerical solution would be the best in that case. (You don't need numerical solution if the simple linear model applies).

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[sundep]

It is a little more of a challenge...

... but after all of the variables of the mechanical parameters are transfered to the left of the gearbox and combined then the model becomes more straight forward.

The variable 'bm' was supposed to be the motor damping. I should have wrote 'cm'

The string does not stretch and the string always remains tight.

The gear ratio of 1/n is correct. The standard seems to be that a speed reducer has a gear ratio greater that 1 and a speed increaser has a gear ratio less than one.

The wave period that I mentioned refers to the sine wave displacement applied to the free end of the spring.

 

[sundep]

The damping transfers throught the gearbox just like the inertia does.

Ce = c + r^2*(cs + n^2*cm)

In this case I choose to ignore the damping in the rope, rope sheave, and gear box so c = 0 and cs = 0 and then,

Ce = r^2*n^2*cm

 

[sundep]

CORRECTION

The damping transfers throught the gearbox just like the inertia does.

Ce = c + r^2*(cs + 1/n^2*cm)

In this case I choose to ignore the damping in the rope, rope sheave, and gear box so c = 0 and cs = 0 and then,

Ce = r^2*1/n^2*cm

 

[sundep]

Which mechanical parameters am I interested in?

The original goal of this experiment was to maintain resonance by varying Ce. In other words, vary Ce so that the resonant frequency of the mechanical system is the same as the frequency of the displacement that is driving the free end of the spring.

Ideally, the power absorbed by Ce would be 1000W, but a range from 500W to 3000W is acceptable. There are other physical constraints that limit k to a maximum of 1000 kg/m. The equivalent mass, Me, can be any value.

 

[ijl]

In other words, vary Ce so that the resonant frequency of the mechanical system is the same as the frequency of the displacement that is driving the free end of the spring.

If I recall it right, the resonant frequency does not depend on the damping.

 

[electricpete]

Yes, good point.

As we know, in a simple series or parallel RLC system, while the undamped natural frequency w0=1/sqrt(LC) does not depend on damping R, the actual damped natural frequency wd does depend on R. You can decrease wd from w0 all the way down to 0 (critical damping) by increasing damping (resistance) for the simple system.

We don't exactly have a simple series or parallel RLC system (we have L in series with paralle R/C) so it will not behave exactly that way same and I do believe R does not hae any influence on wd for this particular circuit. But it can certainly have a big impact on the magnitude of the impedance at a particular frequency.

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?

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