How to calculate natural freq. with a spring/mass w/constant preload? 8

[stillfeelme]

Hello,

I hope I am asking this in the right location. I have a fairly simple question that I have not been able to determine. I want to know how to calculate the natural frequency of a spring mass system that undergoes a constant preload in the spring. This would be for a simple compression spring. I am trying to compare FEA results to hand calculations. I have looked in a couple different textbooks and online and I can't find anything that has this. I did a search here and I couldn't find anything as well.

 

[FeX32]

All I am trying to figure is what is the natural frequency in this example. In my analysis I fixed the displacement and then calculated what the natural frequency is. So I have one spring a mass and preload disacement

There doesn't exist a simple formula for 99% of engineering problems. Thus, don't be disappointed if the engineers here can't give you one. However, there is a theoretical solution (approximation or otherwise) for almost all problems. (I hope you understand what I mean here) I suspect that this problem of yours does have one. It is just hard to understand how you are setting up the system.
The above quote reinforces what I said before. Likely, you are fixing the displacement which must fix the added mass thus rendering it pointless.
A sketch would say a 1000 words


Fe

 

[stillfeelme]

All,

1. The spring is preloaded and has a mass attached
2. The spring is preloaded by about 25% and is always under preload.
3. The mass has more freedom to move in the positive direction but and a shorter stroke limit in the negative direction.

Hopefully the attachment helps but I tried to keep the problem simple asking on here what is the correlation between the preload deflection and the natural frequency? In my case there is always preload.

 

[electricpete]

3. The mass has more freedom to move in the positive direction but and a shorter stroke limit in the negative direction.

I don't know exactly what that means, but it's clearly non-linear. I'm not sure how you decided to withhold that until now. As stated above, there are some smart people willing to help you, but no mindreaders here.

=====================================
(2B)+(2B)' ?

 

[stillfeelme]

Electricpete,

The spring has displacement as I said but I am interested in knowing the correlation between preload and natural frequency. This is why I left out some of the details because I was trying to see what is the correlation between the preload and the natural frequency before I attempted to tackle anything else. If I can't prove the effects of preload, I will have little chance proving anything else. That is why I didn't ask anyone to solve the problem I just wanted to know is there a correlation between spring preload and natural frequency.

I hope I am asking this in the right location. I have a fairly simple question that I have not been able to determine. I want to know how to calculate the natural frequency of a spring mass system that undergoes a constant preload in the spring. This would be for a simple compression spring. I am trying to compare FEA results to hand calculations. I have looked in a couple different textbooks and online and I can't find anything that has this. I did a search here and I couldn't find anything as well.

 

[desertfox]

http://www.roymech.co.uk/Useful_Tables/Springs/Springs_Surge.html#Natural

 

[drawoh]

stillfeelme,

I still do not understand this fixed preload. Is the diagram I have attached a better description of what you are doing?

Assume the spring is nominally compressed.

JHG

 

[stillfeelme]

Drawoh,

Yes that is it. The spring is always under precompression and is never at a free length. The spring load is suppose to keep the mass to the right just as your picture.

 

[stillfeelme]

Desertfox,

I have looked at the surge frequency but that is not the first mode. The first mode is much lower than the surge frequency.

 

[Tmoose]

"The first mode is much lower than the surge frequency."

Maybe some lateral bending frequency? Like a buckling column?

 

[desertfox]

Hi stillfeelme

I'm having trouble understanding what the preload as to do with the natural frequency, on that site I gave you it states "Natural frequency of a loaded spring system" I assume the word loaded spring system means the spring is preload, so I cannot see any difference between that and your sketch.

desertfox

 

[stillfeelme]

Tmoose,

First mode is not a buckling or lateral frequency. It acts in the same direction as with no preload

 

[izax1]

desertfox

The preloaded spring will change the natural frequency only if the preload will shift the spring stiffness. And for most mechanical springs that is the case. f=k/m and if k changes, f will change.

The system illustrated by drawoh is a non-linear system (in the direction of the spring), and will not have a natural frequency. (Assuming the illustration is showing the initial position) The system need to be able to move in a sinusoidal pattern to have a natural frequency.

 

[btrueblood]

A coil spring won't change its stiffness much for a 25% of free length deflection.

The answer is that the frequency for the system that causes the mass to move now depends on the amplitude of the forcing function as a percentage of the spring preload. The lowest frequency of vibration is still sqrt(k/m), with the caveat that the amplitude of the forced motion has to be high enough to overcome the preload and cause the mass to move. The spring might move at higher frequencies, but lower amplitudes too. The impact boundary condition imparts the significant nonlinearity that epete points out. A part of the problem solution is to determine the coefficient of restitution for the impact that occurs when the mass impacts the fixed boundary, in order to find its rebound velocity. I am not at all certain that an FEA will give you anything close to a correct eigenvalue solution for the problem you have described, and would ignore its output.

See how much a simple drawing can do?

 

[drawoh]

stillfeelme,

In the model I have shown, the mass remains forced against the side wall unless the inertia force exceeds the spring preload. There is no mass spring interaction that would make vibration analysis relevant.

The compressed spring will have internal resonant frequencies due to its own distributed mass.

If the maximum acceleration due to vibration exceeds the spring force, the mass will disengage the wall and move. There will be a series of impacts. I do not think an equation will help you here. You will have to solve this numerically.

This is how a lot of optical alignment fixtures work. You apply a preload. The preload and the resulting friction force grossly exceed any force your fixture is likely to see, and everything remains aligned without you having to lock it.

JHG

 

[stillfeelme]

All,

Thanks for the help. How can this be solved numerically? I guess my model is useless back to the drawing board

 

[drawoh]

stillfeelme,

If you can write out the acceleration of your base as a function of time, you can work out the force on your mass. The function probably is a sine wave of some kind.

You can then use numerical integration to work out how your mass moves with respect to your base. Note how, much of the time, it won't.

JHG

 

[PNachtwey]

The formula for the calculating the natural frequency of a mass on a spring was given long ago. However, there is a difference between the natural frequency and the damped frequency of oscillation.

I wonder if stillfeelme is asking the right question because I thought the original question was answered. It isn't clear to me if there is more.

If stillfeelme knows the mass, damping, spring constant and forcing function then the solution should be easy. I can do it.




Peter Nachtwey
Delta Computer Systems

http://www.deltamotion.com

 

[electricpete]

PNachtwey - you might want to read the recent posts starting 19 Jul 11 20:12 to understand the system being discussed.

=====================================
(2B)+(2B)' ?

 

[flash3780]

In the situation that drawoh posted, the spring displacement is a step function, and hence non-linear. If the system is non-linear, the stiffness is a function of the displacement (f = k * dx ---> k(x) = df/dx).
The natural frequencies of the system becomes a bit more difficult to determine.
[tt]
| k1 |-----| k2 |
|^^^^| m |^^^^|
| |_____| |
|
|->x(t), f(t)
[/tt]
You know that k1 is the spring stiffness, but k2 is the stiffness of the wall. It goes from zero to infinity at x=0.

Of course, the fundamental equation of motion still applies, but the math gets a lot uglier.
[tt]a*d2x/dt2 + c*dx/dt + k*x = f(t)[/tt]
Or, in the undamped case:
[tt]a*d2x/dt2 + k*x = f(t)[/tt]
[tt]k1(x) = const[/tt]
[tt]k2(x) = inf (x> 0)[/tt]
[tt] = 0 (x<=0)[/tt]
Hence,
[tt]ktot(x) = inf (x> 0)[/tt]
[tt] = k1 (x<=0)[/tt]

The way to solve this problem is to solve the differential equation (unless you can find it worked out somewhere already). The tricky thing is, most nonlinear problems don't have closed-form solutions, so a numerical solution might be all that's available.

Perhaps a continuous approximation of the stiffness function might work (a heavyside step function or the like with a very high amplitude).

I'd look into Duffing's equation, which represents an un-damped, harmonically excited, single degree of freedom system with a nonlinear spring. It will require an iterative solution.

 

[desertfox]

Hi stillfeelme

Attached are two links the second one is about the natural frequency of a preloaded spring in a mechanism.
I would like to ask what vibrations if any, the device you have will under go and the reason I ask is the first natural frequency is the one you have a formula for, so if your vibrations are lower than this value there shouldn't be a problem because if preloading the spring increases the natural frequency your moving it further away from danger.

http://www.efunda.com/designstandards/springs/spring_frequency.cfm

http://docs.google.com/viewer?a=v&q=cache:S1XHFDFldDUJ:

http://www.dtic.mil/cgi-bin/GetTRDo...eP7nVi&sig=AHIEtbRxBYgsNNkL0YdseTJDf-AmzWidGA