Dynamic Force at resonance 3

[FeX32]

Hello all,

I have a little delima in the with a colleague in the work place.
We are determining the dynamic force acting on an object by subsequently exciting the object with an unknown force at different known frequencies. Say, Fo*cos(w*t).
Lumping the system to a 1DOF model can result in a well known relation m*xdd+c*xd+k*x=Fo*cos(wt).
If I know m, c, k and x, xd, xdd (using sensors), it is possible to determine Fo.
However, I believe that the accuracy of the testing is dependent on the frequency 'w' that we subject the system to. At resonance I expect the determined Fo's accuracy to be significantly degraded.
However, a colleague does not believe so at all.

Does anyone have experience in this respect. I cannot find proof of my "gut feeling". Any input is appreciated.
Thanks


Fe

 

[FeX32]

I just realized. Maybe you misunderstood my logic for solving the linear system using ode45. It was just to check m*xdd+c*xd+k*x=Fo. such that I should get the same Fo out using
F0_s=(m*xdd+c*x(1:length(xdd),2)+k*x(1:length(xdd),1))
as I put in.
And this is not the case.
Ok. The reason I was doing this is because I wanted to use a nonlinear system to characterize the actuator. So, I though I could use a known nonlinear stiffness with the DE:
m*xdd+c*xd+k*x+ks*x^3=Fo (Exactly the duffings oscillator.) and collect the data for all the left hand side to obtain the right. I now see that it may not that simple.

So right now I have this:

1) calculate the dynamic forces by using a linear system by the relation: F(j*w) = X(j*w) * [-w^2*m + j*w*c + k]

or

2) if I use our nonlinear system (that it will eventually be used on) I would have to use nonlinear optimization coupled with a numerical solution to the DE: m*xdd+c*xd+k*x+ks*x^3=Fo
to figure out the dynamic force.
Or I could use the inverse vibration problem: {There is a paper I found in this respect.(attached if anyone is interested)}


The first is more reasonable for now. I will eventually perform both however.
Oh, and if anyone is wondering why I am doing this force characterization dynamically, it is because there is a lot of aluminum close to the proximity of where the actuators will be (and also PM's). This is well known to effect the forces dude to eddy currents which are only present in a dynamic environment.
Interestingly, the AL even heats up quite a bit.





Fe

 

[FeX32]

The attachment.


Fe

 

[MikeyP]

Here's an article which explains force drop-out

http://www.sandv.com/downloads/0210varo.pdf

Note that many folk assume that using a current amplifier instead of a voltage amplifier eliminates the problem. This is not true. There is always a mechanical interaction between structure and shaker aside from the issues of the amplifier.

M

--
Dr Michael F Platten

 

[electricpete]

Maybe you misunderstood my logic for solving the linear system using ode45. It was just to check m*xdd+c*xd+k*x=Fo. such that I should get the same Fo out using
F0_s=(m*xdd+c*x(1:length(xdd),2)+k*x(1:length(xdd),1))
as I put in.
And this is not the case.

I understand you want to use ode because you have a non-linear system.
As I understand it, before you did the non-linear system you did some analysis on pure linear system.
The results of your pure linear system were (?) that ode45 does not match the steady state response predicted by the transfer function. If so that indicates some kind of error.

One thing I noticed:
F0_s=(m*xdd+c*x(1:length(xdd),2)+k*x(1:length(xdd),1))
Should be
F0_s=(m*xdd+c*xd (1:length(xdd),2)+k*x(1:length(xdd),1))

Sorry if I am in left field – but if you solve the same linear model for sinusoidal steady state two different ways (ode45 and transfer function) you should get the same results.


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[FeX32]

Thanks Mike for the paper. I was unaware of that. I am rather new compared to most in the field. If anyone is curious it's my first year.

electricpete,
Thanks for your comments. Actually, xd=x,2) when solving using ode45. Thanks for checking.

I will mention on more thing.
I agree with your statement completely:
"but if you solve the same linear model for sinusoidal steady state two different ways (ode45 and transfer function) you should get the same results" And I do if we are talking about displacement x here. Its the force as a function of 'w' where its fishy.

This is actually the original inquiry.

If I simplify the problem so that we are only talking about a linear system in which we are interested in the steady state force.
Ok think of the problem as simply this:
Lets say I have x, xd, xdd and m, c, k. (in time domain)

If I use: Fo=m*xdd+c*xd+k*x
and compare it with: |F(j*w)| = |X(j*w)| * SRSS(k-w^2*m, w*c)
They differ at different 'w'. However originally I thought they would not. I know it has to do with 'w' as in the first relation Fo is not a function of 'w' but in the TF it is.
Does anyone know how to rewrite the first relation to take into account the effect of the forcing frequency 'w'?





Fe

 

[FeX32]

I should have said:
"Their magnitudes differ at different 'w'."


Fe

 

[electricpete]

If I simplify the problem so that we are only talking about a linear system in which we are interested in the steady state force.
Ok think of the problem as simply this:
Lets say I have x, xd, xdd and m, c, k. (in time domain)

If I use: Fo=m*xdd+c*xd+k*x [EQ 1]
and compare it with: |F(j*w)| = |X(j*w)| * SRSS(k-w^2*m, w*c) [EQ 2]
They differ at different 'w'. However originally I thought they would not.

Your original thought is correct imo. If you are considering only sinusoidal steady state conditions, then EQ1 and EQ2 should give the same result at any frequency w (with the obvious difference that EQ1 gives time domain representation of the waveform and EQ2 gives frequency domain representation of the waveform). If they are not, then there must be some kind of logical or numerical error.


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[FeX32]

Ok.
But for some reason I don't see what the error could be.
It is not a complicated code that I use to simulate x, xd and xdd.


Fe

 

[electricpete]

I wonder about this statement:
xdd=diff(x,2));

I think it is incorrrect. Especially considering the time spacing between points of x may be uneven.

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[FeX32]

Thank you very much electricpete! I can't believe I missed that. That is not usual of me.
A quick modification to the time step and using xdd=w*x(t+T/2), or xdd=w^2*xd(t+T/4) yields very good results to calibrate the dynamic force.
I would give you another star if I could.




Fe

 

[FeX32]

Correction:
xdd=w^2*x(t+T/2), or xdd=w*xd(t+T/4)


Fe

 

[electricpete]

Glad I could help.

Your method to calculate xdd from phase-shifted xd or x will work for a pure sinusoidal signal, but could introduce error when you apply it to non-linear systems with signals that aren’t perfectly sinusdoidal.

For equally-spaced samples at interval dt, you could estimate Ak = (Vk+1 - Vk-1) / (2*dt)

For unequally spaced samples as output from ode45, you could calculate quantities on left side of the point (“-“) and right side of the point (+) as follows:

Ak+ = (Vk+1 - Vk) / dt+
Ak- = (Vk - Vk-1) / dt-
Where
dt+ = tk+1 - tk
dt- = tk - tk-1

Then estimate the acceleration at time tk by linear interpolation between Ak- and Ak+
Something like:
Ak = Ak- + (Ak + -Ak-) * dt-/(dt- + dt+)
(double check that interpolation)

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[FeX32]

Thanks again.
One can fix the spacing between the time samples in ode45 by using something like: tspan=0:dt:Tfinal; instead of tspan=[0 Tfinal].
Then I could use the built in diff fcn. ie. xdd=diff(xd)/dt;
This yields almost identical results compared to using the phase shifted estimation (on the linear system).
Thanks for the tips about the nonlinear system, much appreciated. If it were not possible to fix the step, interpolation is the way to go.


Fe