## measured PSD: why peaks are multiples of a frequency?

Hi all,

I have a suspension system that is mounted in a vehicle and I must evaluate its behaviour. So I have measured the acceleration at the suspended mass and at the base. Since I haven't a bench I was obbliged to test it during vehicle driving. All signal are sampled at 500Hz. In the next picture you can see the PSD of: the excitation signal (blue line), suspended mass acceleration signal (black line) and its fitting (red line).

http://img717.imageshack.us/f/fitting.png/

As you can see from the picture the excitation PSD has 5 peaks at 1.7Hz,2.25Hz, 3.4Hz 4.5Hz and 5.1Hz.

As you can understand 3.4, 5.1 are multiple of 1.7, instead 4.5 is a multiple of 2.25. Is it usual to have peaks that are multiples from another? Should it be related to a fundamental frequencies of a specific mode or to a system non linearities? I need to understand this, for improving my mathematical model. I hope to have correctly explained my question, if not please ask me.
Any suggestions are really appreciated.
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 Recognitions: Science Advisor Most likely, you have a nonlinear system. For example the force-displacement curve of a wheel+tire against the road is nonlinear because the contact area of the tire depends on the downwards force on the wheel. In the extreme case when the wheel loses contact with the road completely, the contact force can't reverse its direction. Therefore the response to a sine wave component of force will not be a sine wave, and it will appear as a series of harmonics. Even if you don't have a bench, you may be able to test the system by applying impulse loads at different amplitudes to the vehicle without driving it. Just apply a "step" force by pushing down suddenly on the vehicle, holding the force roughly constant for a few seconds till the transient response dies away, then release suddenly. You may be able to measure if small responses are more linear than large ones.

 Quote by AlephZero Most likely, you have a nonlinear system. For example the force-displacement curve of a wheel+tire against the road is nonlinear because the contact area of the tire depends on the downwards force on the wheel. In the extreme case when the wheel loses contact with the road completely, the contact force can't reverse its direction. Therefore the response to a sine wave component of force will not be a sine wave, and it will appear as a series of harmonics. Even if you don't have a bench, you may be able to test the system by applying impulse loads at different amplitudes to the vehicle without driving it. Just apply a "step" force by pushing down suddenly on the vehicle, holding the force roughly constant for a few seconds till the transient response dies away, then release suddenly. You may be able to measure if small responses are more linear than large ones.
So peaks at 1.7, 3.4 and 5.1 are related to a non linearity at a specific mode (i.e. roll), instead peaks at 2.25 and 4.5 are related to a non linearity at another mode (i.e. heave), isn't it?

## measured PSD: why peaks are multiples of a frequency?

Is there a way for understanding the non linearity structure from a measured psd?

Recognitions:
 Quote by serbring So peaks at 1.7, 3.4 and 5.1 are related to a non linearity at a specific mode (i.e. roll), instead peaks at 2.25 and 4.5 are related to a non linearity at another mode (i.e. heave), isn't it?
Yes, that's the general idea.

Of course there may be a "genuine" mode at say 3.35 which you can't distinguish from the 2nd harmonic of 1.7, or whatever.

Recognitions:
 Quote by serbring Is there a way for understanding the non linearity structure from a measured psd?
Probably not from your test data, because your excitation forces will be vary in amplitude at different times and therefore create different amounts of nonlinearity, but that is all mixed up together in your PSD.

A simple way to measure the nonlinearity is use a sine-wave force input where you can control both the frequency and the amplitude. Even if you don't have access to proper vibration testing equipment, you might be able to improvise something using a low frequency signal generator, an amplifier, and a solenoid.

 Quote by serbring Is there a way for understanding the non linearity structure from a measured psd?
The way you understand it is by looking at the suspension system and modeling it with a differential equation. That differential equation will have nonlinearities, which you can examine using various perturbation methods such as the Method of Multiple Scales. It is definitely a nontrivial problem and there are plenty of people that make a living on things like this.

 Quote by AlephZero Yes, that's the general idea. Of course there may be a "genuine" mode at say 3.35 which you can't distinguish from the 2nd harmonic of 1.7, or whatever.
Ok..but do non linearities introduce peaks just at multiples frequencies?

 Quote by AlephZero Probably not from your test data, because your excitation forces will be vary in amplitude at different times and therefore create different amounts of nonlinearity, but that is all mixed up together in your PSD. A simple way to measure the nonlinearity is use a sine-wave force input where you can control both the frequency and the amplitude. Even if you don't have access to proper vibration testing equipment, you might be able to improvise something using a low frequency signal generator, an amplifier, and a solenoid.
ok...the situation is really more complex than I could expected

 Quote by boneh3ad The way you understand it is by looking at the suspension system and modeling it with a differential equation. That differential equation will have nonlinearities, which you can examine using various perturbation methods such as the Method of Multiple Scales. It is definitely a nontrivial problem and there are plenty of people that make a living on things like this.
I have just taken a look on the suspension system, but I don't understand, what it is missing in my model..

 Do you have a differential equation that describes (or approximately describes) your model?

Recognitions:
 Quote by serbring Ok..but do non linearities introduce peaks just at multiples frequencies?
Not necessarily, it depends on the system.

They can also introduce frequencies at 1/2, 1/3, etc of the forcing frequency.

They can also introduce completely new frequencies that are not simply related to anything else. For exmple "oil whip" in lubricated bearings often produces a response between about 0.8 to 0.9 times the forcing frequency.

Usually, you can't make much progress with any nonlinear situation unless you measure what is actually happening and use that to help you create a useful model. Of course in many common situations other people have already done the measurements, created the models, and written textbooks or papers about them.

Common causes of nonlinearity in something like a suspension system would be large geometric rotations (e.g. the approximation that sin theta = theta is not accurate enough), individual components that have a nonlinear response, for example springs or dashpots moving with large amplitudes or velocities, or "contact" situations where the force between two components can only act in one direction (e.g. if a wheel leaves the ground).

 Quote by AlephZero Not necessarily, it depends on the system. They can also introduce frequencies at 1/2, 1/3, etc of the forcing frequency. They can also introduce completely new frequencies that are not simply related to anything else. For exmple "oil whip" in lubricated bearings often produces a response between about 0.8 to 0.9 times the forcing frequency. Usually, you can't make much progress with any nonlinear situation unless you measure what is actually happening and use that to help you create a useful model. Of course in many common situations other people have already done the measurements, created the models, and written textbooks or papers about them. Common causes of nonlinearity in something like a suspension system would be large geometric rotations (e.g. the approximation that sin theta = theta is not accurate enough), individual components that have a nonlinear response, for example springs or dashpots moving with large amplitudes or velocities, or "contact" situations where the force between two components can only act in one direction (e.g. if a wheel leaves the ground).
On reality I would like to identify the system parameters: I just need a semplified model, that can fit peaks in frequencies lower than 5Hz.

 Quote by boneh3ad Do you have a differential equation that describes (or approximately describes) your model?
Yes I have. I try to understand more the behaviour of the system.
 Since I don't need to predict the real suspension behaviour, for my topic I need only a way for simulating the system response from a special input. Whereas system output PSD has 5 peaks, in your opinion is it a good idea to model my system with 5 masses for getting a good fit?
 Recognitions: Science Advisor The best way to answer that question would be try it and see if it works. But here's a story from my own work experience. 20 or 30 years ago, we used to attempt to model the dynamics of some rotating machinery, using models with about 100 degrees of freedom. The results were OK when the response was small but poor when it was large. The general accepted explanation back then was that 100 DOF was much too few to model the system accurately, but bigger models took too long to create and run so that was the best that could be done in practice. As time went by and computers got more powerful, we increase the model size from 100 up to about 3000 DOF, but the end result was not much different. Eventually, a small group of the people doing this (including me!) decided that the poor model results were caused by ignoring nonlinearity, not because the model was "not big enough". After several years of work (and a lot of effort convincing senior engineers that we were right) we can now get very good results, not from 100 DOF, not from 3000 DOF, but from about 10 or 12. The take-home message is: if you want to make a model work properly, the most important thing is that it contains the right basic physics, not that the model parameters are all accurate to 6 decimal places.

 Quote by AlephZero The best way to answer that question would be try it and see if it works. But here's a story from my own work experience. 20 or 30 years ago, we used to attempt to model the dynamics of some rotating machinery, using models with about 100 degrees of freedom. The results were OK when the response was small but poor when it was large. The general accepted explanation back then was that 100 DOF was much too few to model the system accurately, but bigger models took too long to create and run so that was the best that could be done in practice. As time went by and computers got more powerful, we increase the model size from 100 up to about 3000 DOF, but the end result was not much different. Eventually, a small group of the people doing this (including me!) decided that the poor model results were caused by ignoring nonlinearity, not because the model was "not big enough". After several years of work (and a lot of effort convincing senior engineers that we were right) we can now get very good results, not from 100 DOF, not from 3000 DOF, but from about 10 or 12. The take-home message is: if you want to make a model work properly, the most important thing is that it contains the right basic physics, not that the model parameters are all accurate to 6 decimal places.
Thanks, I have understood, I'll try since it is quite simple to verify the goodness of the method. Actually for identifying the model parameters I'm using the output-error method from measured inputs, however I have a big doubt regarding the mass of a system since I convert a non linear single degree of freedom system in a multi degree of freedom system. In particular the suspendend mass weighs 700kg and as we know, many linear systems with different masses can have the same transfer function. For my topic is important to calculate the right masses, so in your opinion, how can I calcolate the system masses? Should the sums of the system be equal to 700kg? I hope to have stated properly my question
 up! :)