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
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
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Fe





RE: Dynamic Force at resonance
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
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
RE: Dynamic Force at resonance
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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RE: Dynamic Force at resonance
H(j*w) = F(j*w) / [X(j*w) *m]
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RE: Dynamic Force at resonance
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RE: Dynamic Force at resonance
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
RE: Dynamic Force at resonance
You can very the frequency 'w' in the first to obtain a feel.
Fe
RE: Dynamic Force at resonance
Fe
RE: Dynamic Force at resonance
"VARY" annoying
Fe
RE: Dynamic Force at resonance
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
Fe
RE: Dynamic Force at resonance
Fe
RE: Dynamic Force at resonance
Fe
RE: Dynamic Force at resonance
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
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
RE: Dynamic Force at resonance
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
Fe
RE: Dynamic Force at resonance
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
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
RE: Dynamic Force at resonance
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
RE: Dynamic Force at resonance
Fe
RE: Dynamic Force at resonance
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
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
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
RE: Dynamic Force at resonance
"Their magnitudes differ at different 'w'."
Fe
RE: Dynamic Force at resonance
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RE: Dynamic Force at resonance
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
RE: Dynamic Force at resonance
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
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
RE: Dynamic Force at resonance
xdd=w^2*x(t+T/2), or xdd=w*xd(t+T/4)
Fe
RE: Dynamic Force at resonance
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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RE: Dynamic Force at resonance
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