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Dynamic Force at resonance
3

Dynamic Force at resonance

Dynamic Force at resonance

(OP)
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

peace
Fe

RE: Dynamic Force at resonance

You are correct. B&K and MSC discuss this in their literature. If you are especially concerned about the response at resonance then it may be appropriate to use one of 4 different methods (H0 thru H3) for the transfer function. Each is differently sensitive to noise in the force and response channels.

the specific issue is that at resonance the force signal drops away so any noise becomes more important.

There are also the usual effects to with coherence.




 

Cheers

Greg Locock

SIG:Please see FAQ731-376: Eng-Tips.com Forum Policies for tips on how to make the best use of Eng-Tips.

RE: Dynamic Force at resonance

(OP)
Thanks Greg. Much appreciated.
I will see what I can do about getting some literature on the matter.
Initially, I would be concerned about how the systems dynamic force changes at resonance since I suspect that the actuators force changes slightly with the frequency of the applied voltage. (they are inductors)
However, I am not sure exactly how far a frequency away from resonance for the relation in the fist post to apply. Possibly, much lower or much higher than resonance?

peace
Fe

RE: Dynamic Force at resonance

Quote:

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.

I am trying to sort out the question, and I don't really understand it.

Here are some thoughts, probably obvious, just my way of thinking through to figure out what is the question.

IF the system is linear, coefficients constant,  and the measurements are accurate, then the equation m*xdd+c*xd+k*x=f(t) applies for ANY frequency w.

The solution neglecting initial conditions comes in the Laplace domain:
F(s) = X(s) * [s^2*m + s*c + k]

Letting s = j*w for sinusoidal steady state solution:
F(j*w) = X(j*w) * [-w^2*m + j*w*c + k]
|F(j*w)| = |X(j*w)| * SRSS(k-w^2*m, w*c)
where SRSS is square root of sum of squares of the two arguments

The angle can also be calculated if needed.

So, which non-ideal effects are you concerned about?

You were concerned about change in magnitude of force affecting the accuracy of the measurment?  I don't get that.   The above solution calculates the force.  You can also calculate ratio |F|/|X|.  Again with linear system the ratio isn't going to change based on change in magnitude of the excitation.

Greg mentioned concern about measurement noise.  You are measuring X where  |F| is approx constant and I ASSUME measurement noise of X is roughly constant.  The highest magnitude of X occurs near resonance so any small constant noise should have least effect near resonance.   So I must be missing something.

** Now if there undertainty in knowledge of the parameter c as is often the case, then I can very well see the ability to estimate F from X is most questionable at resonance when the ratio depends heavily on c.

In case it is not obvious that the ration F/X depends on c at resonance more than far away from resonance, here is a proof of this fact:
F(j*w) = X(j*w) * [-w^2*m + j*w*c + k]

Divide each side by X(j*w) * m and call the left side H(j*w)
H(j*w) = F(j*w) / X(j*w) = -w^2 + j*w*c/m + k/m]

Define w0 = sqrt(k/m)
H(j*w) = w0^ -w^2 + j*w*c/m
|H(j*w)| = SRSS(w0^ -w^2, w*c/m).

For lightly damped system with w far away  from w0, the first term in the SRSS dominates, and the ratio H is very insensitive to errors in c.

For w very near w0, the 2nd term in SRSS dominates and the ratio H is very sensitive to errors in c.

[end of proof]

I will also mention there is some redundancy built into the measured data.  If you measure displacement only, you have enough to solve the force (if parameters m, k, c are known).   With v and a also available these are extra info that could assist to evaluate the accuracy of the model and measurements.

=====================================
Eng-tips forums: The best place on the web for engineering discussions.

RE: Dynamic Force at resonance

Correction in bold:
H(j*w) = F(j*w) / [X(j*w) *m]

=====================================
Eng-tips forums: The best place on the web for engineering discussions.

RE: Dynamic Force at resonance

Quote:

Initially, I would be concerned about how the systems dynamic force changes at resonance since I suspect that the actuators force changes slightly with the frequency of the applied voltage. (they are inductors)
Is the inductor included within the F or included within the k?  If it is included within the F, then change in F shouldn't matter if the remainder of the system remains linear (as described above) and if the force remains sinusoidal.  If the inductor affects the stiffness K (i.e. K represents electromagnetic attraction), then the system is non-linear and it will be important to account for the change in K.

=====================================
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RE: Dynamic Force at resonance

(OP)
Hello electricpete. Thank you for your interest.
My first thought was exactly yours. Lets assume the systems parameters are linear and that the inductors do not effect the stiffness of the system.

Thus far my understanding is that the relation H(j*w) = F(j*w) / [X(j*w) *m] or m*xdd+c*xd+k*x=Fo*cos(wt) holds for certain frequencies (to determine the magnitude of F0). I am in the precess of deriving why, due to the fact of a lack of references in this nature. However, I know they much exist to a certain extent.

Physically, I think of it in this manner:

Take the well known relation: m*xdd+c*xd+k*x=Fo*cos(wt).
Let Fo be 1 and vary 'w' through resonance.
If we do this we observe xdd, xd, and x all drastically increase with other parameters staying the same. Now, assume we don't know Fo and we do the same thing. If we use the above differential relation to model 'Fo' then for the same 'real' Fo we will obtain different magnitudes of 'Fo' at different frequencies, depending on the frequency response (or force transmissibility). (largest error at resonance).
I have attached 2 matlab codes I just wrote that use Runge-Kutta to solve this  system and then very simply estimate Fo based on the above equation.
2 interesting point are seen:
1) the error of the estimation is almost zero for very low frequencies
2) at about 2*wn the error is also very low

Everywhere else it is unacceptable.
There should be a way to derive this. Does this make sense to you?


 

peace
Fe

RE: Dynamic Force at resonance

(OP)
Please excuse my grammatical errors in the posts smile

peace
Fe

RE: Dynamic Force at resonance

(OP)
Damn those grammatical errors get to me. Especially the very
"VARY" annoying smile.  

peace
Fe

RE: Dynamic Force at resonance

Fundamentally from a practical point of view the force signal drops away because the actuator has a finite output impedance, so as the system goes into resonance there is a change in the impedance mismatch.

The simple way to check that is to look at the input force spectrum.

Cheers

Greg Locock

SIG:Please see FAQ731-376: Eng-Tips.com Forum Policies for tips on how to make the best use of Eng-Tips.

RE: Dynamic Force at resonance

(OP)
Thanks again Greg. Although, I am not sure I understand your analogy.  

peace
Fe

RE: Dynamic Force at resonance

(OP)
Interestingly, I get much much better results using F(j*w) = X(j*w) * [-w^2*m + j*w*c + k] as opposed to the differential equation. ( I am not so sure as to why). This posses a little problem, as in reality my stiffness k is nonlinear. dazed. I was hopping to use the DE on my real data.

peace
Fe

RE: Dynamic Force at resonance

(OP)
But, thanks to you both thus far.  

peace
Fe

RE: Dynamic Force at resonance

F(j*w) = X(j*w) * [-w^2*m + j*w*c + k] won't help much with non linear k, unless k(w), whereas I suspect you have k(x).

If the latter is the case then you can either linearise k for the x you are using, which will lead to tears, or resort to a time based simulation, which is not elegant but can be made to cope with any degree of non linearity you can think of.

Needless to say a bit of a hint as to what you are really trying to do would be appreciated.

If k(x) then the concept of a natural frequency goes out the window, as the frequency of maximum response/input will vary depending on the magnitude of the input.


 

Cheers

Greg Locock

SIG:Please see FAQ731-376: Eng-Tips.com Forum Policies for tips on how to make the best use of Eng-Tips.

RE: Dynamic Force at resonance

(OP)
Greg, I agree.
What I am doing is an experimental analysis of a certain actuator. This actuator came from an unknown place and thus I have no idea what precise force they would handle dynamically. I need a dynamic environment because the end result is the implementation dynamically. It is very important that I precisely determine the dynamic forces using an indirect method. That is why I started with the differential equation. If push comes to shove I can use a linear system to characterize the actuation force.  

peace
Fe

RE: Dynamic Force at resonance

OK. From a practical pespective do a series of sine sweeps, driving a mass. Change both the mass and the amplitude of excitation.

I don't know how you'll characterise the response if/when you get a response at frequencies other than the driving frequency.

The alternative is to use white or pink noise excitation, but that will disguise your non linearities as noise.

 

Cheers

Greg Locock

SIG:Please see FAQ731-376: Eng-Tips.com Forum Policies for tips on how to make the best use of Eng-Tips.

RE: Dynamic Force at resonance

(OP)
Thanks for the help. Much appreciated.  

peace
Fe

RE: Dynamic Force at resonance

If I understand correctly, you built a linear model of the system for ode45 based on m/k/c and it gave different sinusoidal steady state response  than predicted by the steady state transfer function H(j*w) = F(j*w) / X(j*w) = -w^2 + j*w*c/m + k/m] ?

That would not be logical.  Both models assume a linear system with the same parameters m k c and both should give same steady state results.   Here is one aspect to be careful of: it takes a lot longer for the system to reach steady state under resonance conditions than it does under off-resonant conditions... I think maybe you just need to run the simulation longer.

=====================================
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RE: Dynamic Force at resonance

(OP)
Exactly. I just increased the damping coefficient.
Interestingly, the relation:
F0_s=(m*xdd+c*x(1:length(xdd),2)+k*x(1:length(xdd),1))
only gives correct results when the transmissibility ratio is 1. Everywhere else it is not correct.
Thus, if incidentally I have a nonlinear model that method wont work. I would have to use nonlinear optimization to fit the model.  

peace
Fe

RE: Dynamic Force at resonance

(OP)
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.


 

peace
Fe

RE: Dynamic Force at resonance

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

RE: Dynamic Force at resonance

Quote:

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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RE: Dynamic Force at resonance

(OP)
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'?


 

peace
Fe

RE: Dynamic Force at resonance

(OP)
I should have said:
"Their magnitudes differ at different 'w'." smile

peace
Fe

RE: Dynamic Force at resonance

Quote:

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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RE: Dynamic Force at resonance

(OP)
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.  

peace
Fe

RE: Dynamic Force at resonance

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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RE: Dynamic Force at resonance

(OP)
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. smile

 

peace
Fe

RE: Dynamic Force at resonance

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

peace
Fe

RE: Dynamic Force at resonance

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)

=====================================
Eng-tips forums: The best place on the web for engineering discussions.

RE: Dynamic Force at resonance

(OP)
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.  

peace
Fe

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