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

 

[GregLocock]

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

SIGlease see FAQ731-376 for tips on how to make the best use of Eng-Tips.

 

[FeX32]

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?


Fe

 

[electricpete]

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.

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[electricpete]

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

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[electricpete]

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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[FeX32]

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?





Fe

 

[FeX32]

Here is the other file.

You can very the frequency 'w' in the first to obtain a feel.


Fe

 

[FeX32]

Please excuse my grammatical errors in the posts


Fe

 

[FeX32]

Damn those grammatical errors get to me. Especially the very
"VARY" annoying .


Fe

 

[GregLocock]

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

SIGlease see FAQ731-376 for tips on how to make the best use of Eng-Tips.

 

[FeX32]

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


Fe

 

[FeX32]

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. . I was hopping to use the DE on my real data.


Fe

 

[FeX32]

But, thanks to you both thus far.


Fe

 

[GregLocock]

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

SIGlease see FAQ731-376 for tips on how to make the best use of Eng-Tips.

 

[FeX32]

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.


Fe

 

[GregLocock]

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

SIGlease see FAQ731-376 for tips on how to make the best use of Eng-Tips.

 

[FeX32]

Thanks for the help. Much appreciated.


Fe

 

[electricpete]

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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[FeX32]

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.


Fe