static stiffness from dynamic stiffness? 2

[harryhaller]

Hi, I'm an amateur vibration engineer. My real work is building bowed string instruments. But I have used modal analysis to observe the dynamic properties of the instruments and help me make descisions on how I could improve them. My question is it possible to calculate the static stiffness and static mass distribution of the curved wooden backs and fronts of the instruments from the modal data? is it perhaps calculated from the 0Hz data?, but, I don't know how to calculate?

Maps showing the static stiffness and mass distribution would be very helpful. It seems to me extremely tricky to calculate the static stiffness any other way, the wood being non homogenous and the thickness of the shells being of varying thickness, also the wood fibres are parallel to the curve of the shell in some places and in other places the fibres are cut through.

any ideas are gratefully received.

 

[GregLocock]

Yes, the assymptotic trend of the compliance line towards zero Hz IS the static stiffness.

You will have to be careful how you mount the block of wood - a vice on a workbench won't be good enough, you'll need to bolt it to a large concrete wall.

So, if you have an FRF in m/s^2/N, you'll need to double integrate in the frequency domain (multiply each line by (1/(2*pi*f))^2), that will put the y axis units in to m/N, which is compliance, the inverse of stiffness.

This is not especially accurate, but, in general measuring the static stiffness of real world objects is quite difficult, that's why stiffness measuring machines are made of huge lumps of cast iron and kept very clean.



Cheers

Greg Locock

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

 

[harryhaller]

Thank you Mr. Locock,

I am interested in the stiffness of the finished musical instrument in playing condition, so bolting it to a large block of concrete is out of the question. Until now I have been measuring the frf's in free/free condition. Is it possible to calculate the static stiffness from free/free measurements? the other possible problem is that my accelerometer only measures down to 1Hz.

 

[MikeyP]

harryhaller,

For an unconstrained structure the static stiffness is undefined (there is nothing for any applied force to react against). For structure suspended on elastic to represent a free-free condition, the low frequency behaviour will be dominated by the 6 rigid body modes who's stiffness will be defined by the elastic, not by the structure.

Is free-free really a representitative boundary condition for a bowed musical instrument? Certainly not for double bass or cello, probably not for a violin or viola.

The 1 Hz limit of the accelerometer is probably not the lower limiting frequency, noise and other parts of the measurement chain will probably raise that frequency to maybe 5 or 6 Hz.

Greg,
I remember you mentioning obtaining static stiffness from a modal analysis in a thread a couple of weeks ago. Have you ever tried it? or is it one of those things that should work in principle but the practicalities make it nearly impossible? I note that Ewins mentions it but I was just wondering if anyone had actually done it successfully.

M

--
Dr Michael F Platten

 

[harryhaller]

Thankyou Dr. Platten,

It's true that the player does have an influence, although involving a player while taking measurements introduces too many factors that are not reproducible and therefore comparisons between instruments are made impossible. When making measurements, I usually support the instruments at the point of player contact.

so to take a much simpler example, suppose I took a simple metal rectangular plate of even thickness, and did a modal test of it in free/free condition, wouldn't it be possible to deduce the static stiffness of the plate from the modal results? though I know that in this case the plate has a static stiffness that would be straight forward to calculate.

Thank you again

 

[MikeyP]

No you can't get static stiffness from a free-free test. Think of the ideal free-free test (eg a plate hanging in zero gravity in outer-space). static stiffness is a measure of how much an object deflects when a constant force is applied to it. If we apply a static force to our plate in outer space (by strapping a rocket to it, say). The plate will fly off to inifinity. Now, stiffness k = F/x, but x = infinity, therefore k is undefined.

Now think about what we do in practice when we perform a free-free test. We usually suspend the item using the lowest possible stiffness (using elastic bands, bungeees, air bags etc.). This means that the stiffness of the suspension system is much lower than the stiffness in the structure. Now when we apply a static force, the plate doesn't fly off to infinity as it is constrained by the suspension system, but the displacement caused by the static force is due almost entirely to the stiffness of the suspension.

What this illustrates is that 1) A free-free structure has an undefined static stiffness. and 2) The static stiffness is very heavily dependent on the boundary conditions.

M

--
Dr Michael F Platten

 

[cincibcats]

Mikey P,

I've read some conference proceedings that show a modal analysis being done on a bridge. The modal analysis is used to form a modal model of the structure and then the modal model is interpolated back to zero frequency to obtain the static deformation of the bridge due to certain loading conditions.

 

[harryhaller]

Hmm, I think I'm confused. Dr. Platten, what you explain makes sense, but are not the modes present in the metal plate suspended in free/free condition entirely due to the stiffness and mass distribution and damping coefficient of the said plate? Maybe my terminology is wrong, I am interested in the mass and stiffness distribution of the plate.

Thanks again

 

[pja]

Your best bet would be to use a known forcing with a measured response and then do some sort of system identification to back out approximate stiffness and mass.
There are plenty of methods out there for this.

 

[electricpete]

I think the big picture of what you want to do is a modal analysis to determine the frequency-depedendent response or resonant frequencies, right?

Then you are going to build perhaps a lumped model and you need to know what m's and c's and k's to use in the M, K, C matrices. The k's might be referred to as static stiffness but in general we just call them stiffnesses.

If you clarify your objective I think these folks can help you with modal analysis if that's what you want


=====================================
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[MikeyP]

harryhaller,

The static stiffness of any structure depends on how and where it is supported. i.e. it is a property of both the struture AND its constraints. Put two pencils on a table 29 cm apart and parallel to each other and balance a 30 cm plastic ruler on them

________________
o              o

Now press on the centre of the ruler with your finger. You can deform the ruler easily. The centre of the ruler has a LOW STATIC STIFFNESS.

Now, move the pencils half way to the centre of the ruler.

________________
    o      o

Now when you press on the centre of the ruler with the same force it doesn't move as much as before. Its static stiffness has increased.

Finally, bring the pencils to the centre

________________
       oo

Now when you press on the centre of the ruler it doesn't really move at all because you are effectively pressing directly on the pencils. The static stiffness of the centre of the ruler is VERY HIGH

The ruler is the same in each case. Type and mechanism of support is the same in each case. Only the position of the supports (constraints) has changed.

Conclusion 1: The static stiffness of a structure depends on the position of its constraints.

It doesn't matter how you measure the static stiffness. You can put weights on it and measure the deflection with a dial gauge; you can extrapolate to zero frequency from a modal analysis. The conclusion still holds.

A second thought experiment
Hang the ruler by its ends using something very stiff (thick wire for example).

 |            | 
 |            |
_|____________|_

When you press on the centre of the ruler and measure its deflection you find that the static stiffness is quite high and also the ruler bends quite a lot. Now replace the wire with something less stiff, e.g. bungee cords. Press on the centre with the same force as before and you will find the displacement is much greater AND the ruler doesn't bend as much as it does before. Now replace the bungee cords with thin elastic bands. Now when you press on the centre, it deflects quite a long way BUT THE RULER HARDLY BENDS AT ALL.
Finally take this to the extreme and suspend the ruler on imaginay elastic bands with zero stiffness (idealised free-free boundary conditions). Now when you apply force to the centre of the ruler it displaces an infinite amount and what is more, the ruler doesn't bend at all. It moves as a RIGID BODY.

Conclusion 2: The static stiffness of a structure depends on the stiffness of its supports. When the stiffness of the supports is very low any static stiffness you measure will effectivly be the static stiffness of the supports not the test structure.
Conclusion 3: The static stiffness of an unconstrained (free-free) structure is undefined.
Conclusion 4: As the supports tend to the free-free condition, the relative static displacements between various parts of the structure become smaller and smaller and so any information about the stiffness distribution becomes harder and harder to measure.

Back to the stringed instrument.
You can't measure the static stiffness with ideal free-free boundary conditions because the static stiffness does not exist under those conditions. You can't measure the static stiffness using pseudo free-free boundary conditions (eg hang it on elastic bands) because what you will measure will be mostly the static stiffness of the elastic bands. The only way to ensure the static stiffness that you measure is due to the structure and not the supports is to make the supports very stiff indeed (like Greg says, "bolt it to a concrete wall"). Unfortunately you probably don't want to do that to your instrument and also those boundary conditions are not really representitative of what happens when you actually play the instrument.

Oh, by the way the behaviour at zero Hz depends on stiffness only. The mass has no effect so you don't get any information about the mass.

Sorry to be so gloomy.

M



--
Dr Michael F Platten

 

[GregLocock]

Mikey - yes, I have. A good example is the longitudinal (recession) stiffness of the suspension. You get a 2 lb hammer fitted with a force gauge, and an accelerometer on tha wheel centre. You then give the tire a mighty thump on the front, and from that you can estimate the stiffness of the suspension bush, typically 500 N/mm, and the unsprung mass of the suspension, typically 40 kg.

We also use it for estimating the static stiffness of mounting points on the car.

It is rough and ready, but it correlates reasonably well with FEA etc. Solves many arguments in 5 minutes.

Cheers

Greg Locock

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

 

[harryhaller]

Thank you very much Dr. Platten, that is very helpful indeed, I understand now from your excellent explaination, the importance of constraint.

If I could bother you with one more question, If we did a modal analysis on the ruler in your example in free/free condition and determined the resonant frequencies, damping cooeficients, and mode shapes, would it be possible to determine anything about the relative stiffness of the ruler from one measurement point to the next?

For example, if I made the centre section of the ruler twice as thick, it would effect the modal frequencies and mode shapes, the center section having greater mass, and being a lot stiffer than the unmodified ruler, this would be apparent in the modal test, even in perfect free/free conditions. would it be possible to determine from the modal ananlysis that the centre section is relatively stiffer in the middle than the ends and has greater mass than the unmodified ruler?
As I understood it, the mode shapes and frequencies and damping of an object are due to a unique distribution of mass and stiffness inherent in the object. In other words it's not possible to obtain the same mode frequencies, damping and mode shapes from two objects with similar external sizes but with different stiffness and mass distribution? Or am I still confused?

Thank you again.

 

[GregLocock]

I don't think you are confused at all. That is a series of interesting and hard questions.

If I measured the frequency alone I could not tell whether the ruler had a thick middle, or was uniform, from the first flexural mode. I could probably tell by looking at the relationships between the frequencies of successive natural frequencies, and could certainly pick up big clues from the mode shapes. So yes, by doing sufficient analysis I could figure out the mass/thickness distribution in the ruler.

" In other words it's not possible to obtain the same mode frequencies, damping and mode shapes from two objects with similar external sizes but with different stiffness and mass distribution?"

So long as I have enough variables to play with I can make any arbitrary system match the frequency response of another system, for a given number of modes. That is in theory you can create an organ that sounds like a violin, up to a certain frequency, for a given fundamental.

BUT you quickly run into practical limitations.










Cheers

Greg Locock

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