Mass spring damper problem.

[RoarkS]

I would like a link to a solved real world solution of a simple mass spring damper with an impulse force input.

All I want is to see a completed example of a critically or under dampened solution to an impulse force input with real numbers...

(not in reference to the imaginary solution component, im talking about start with x force for y seconds on a mass lbs spring k lbf/ft and damper c lbs*s/ft)

Is that to much to as for? It seems like it so that's why I am asking here. I'm not much of a high level math guy... I just need a tool to work with.

-Thanks

 

[chicopee]

About showing a sketch of the mass, spring and damper and the apllication of the impulsive force as there could be several examples already available in vibration textbooks

 

[RoarkS]

At least I hope I know what I am asking for.
Please see the quick drawing I attached.

If someone could tell me how to solve a problem like this...
I would be infinitely greatful.

 

[RoarkS]

Okay... So I changed the label on the drawing before I posted it...

The force gauge mount=ridid wall.

I basically need to develop a system that will result in a reasonable decrease in felt force on the wall.

At current the system is too fragile and will suffer fatigue failure in a fairly short amount of time.

 

[electricpete]

I am not sure why you are asking for "real world numbers". The problem you have posted can be solved by elementary textboook methods.

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[electricpete]

I apologize for the word "elementary". I really meant to say "straightforward".

And I have to admit that what is textbook to one person in one field can be not so straightforward to another person in another field. I have you not studied vibration I can see this would be more of a challenge.

No hard feelings I hope.

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[RoarkS]

I was asking for real world numbers because it seems like when textbooks and such get to talking about this type of thing after all the calculus substitutions and Euler stuff they never solve a real world problem. They just talk about the concept which I am man enough to say is beyond my math skills to figure out on my own... but if I actually saw one worked out I might be able to make sense of it.

So far I have asked a few of my old prof's from school... but still I keep getting answers like its "Straightforward" yet no one has been able to share this knowledge. so IMHO it's not straightforward.

No worries... no hard feelings... It's just I wish someone would be willing to share.

 

[Compositepro]

I don't care much for complex math myself. Let's look at basic principles. Since damping is asymptotic, the term "coming to rest" makes your problem unsolvable.

The lowest force that can be transferred to the wall is an opposing force applied for 0.042 seconds that balances the impulses of 670 lbf for 0.001 seconds. That would be 670/42=16.

Damping will only increase this force.

The lower your spring constant is, the lower the variation in force on the wall, but the greater the compression of spring. As a practical matter you may want to have stops on the mass so the spring has a preload.

The lowest force will also be at a resonant frequency that matches your impulse frequency so that the mass is moving counter to your impulse at the moment of impulse - not with the mass at rest.

 

[Compositepro]

It would also seem that in the real world you would use a shock absorber which would only damp on the return stroke so that the mass would come to close to zero velocity before coming back to the initial stops.

 

[electricpete]

Attached is a numerical solution. The "model" is listed in model tab (just sdof translated to state space form with state variables x and v).

You input the variables in green within tab Main.

Execute the simulation using grey buttons (2nd button to clear previous results, 3rd button to execute simulation). The first thing you should do after you download the sheet is hit the 3rd button to execute (I removed all data to make it small enough to upload... don’t know why it’s still bloated without data).

Results appear numerically lower down in tab main. Results can be viewed in chart tab (note that there are 2 axes).

Note that the state variables x and v are plotted, along with applied force. Also the base force is plotted (Fbase = k*x + c*v)

You can change the input parameters (green) and re-run as desired. Note input and output parameters are all mks units. You may have to play with the plot to get it to look the way you want.

Also in the model tab is a a pdf file with an attempted analytical solution assuming impulse-like input of duration << natural frequency of the system. It simplifies to solving SDOF with initial condition V0 = Tpulse*Fpulse/M. The solution is presented in correct form for over-damped system. Needs some work for underdamped system.


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(2B)+(2B)' ?

 

[electricpete]

Also in the model tab is a a pdf file with an attempted analytical solution assuming impulse-like input of duration << natural frequency of the system. It simplifies to solving SDOF with initial condition V0 = Tpulse*Fpulse/M. The solution is presented in correct form for over-damped system. Needs some work for underdamped system.

Attached is the similar solution applying to underdamped system. It simply assumes (approximates) that the pulse is so short that initial displacement is 0, initial velocity is Tpulse*Fpulse/M. Initial form of displacement is assumed x(t) = Xmax*sin(wd*t) where wd=sqrt(w0^2-sigma^2), w0=sqrt(k/m), sigma = c/(2*m). Note that the phase angle is assumed 0 as required to establish x(0)=0 by assumption. Xmax is solved by setting derivative of x(t) equal to the initial velocity Tpulse*Fpulse/M. As before Fbase = k*x(t)+c*v(t)


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[electricpete]

Attached is a better more sensible version of my previous attachment 19 Sep 11 10:59
(previous version is corrrect, but includes solution to Xmax before it has been solved).

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[electricpete]

I will say, my comment about this being a “textbook” problem applied to original post – determining the behavior of a SDOF system.

The followup question of what K and C to select for your application are more complicated. I’m not sure it has completely been defined what is the objective function to be minimized (maybe you just want to minimize the max value of base force, with knowledge that another impulse may appear at 0.042 sec or anytime thereafter?) , but even if it becomes precisely defined it is still more than typical vib textbooks.

With the spreadsheet, you have the ability to run simulations of single impact starting from rest. You’d have to sweep through a 2-D grid of K and C to get a feel for those solutions. And then it does not account for the behavior if the system is not approximately at rest for the next pulse. Further if pulses are exactly periodic, there can be a resonance concern (for example damped resonant frequency is a multiple of pulse repetition rate).

It strikes me this falls under the category of problems encountered in the field of“vibration isolation” (preventing vibration from being transmitted to the base) and this problem or at least pieces of it have undoubtedly been solved before. That’s not something I know much about... you might find something useful in the “Vibration Isolation” chapter Chapter (28) of Harris’ Shock and Vib Handbook 6th ed.


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[RoarkS]

I Just got in this morning and just started looking it over and will probably take me a while to wrap my brain around it.
Thank you! I owe ya a beer

As far as what you said.... I'm sure it has been solved before... at least I hope it has. It seems like most things in my industry seems to be a tinker art more than a science.

The device driving the the impulse also relies upon that impulse (recoil) to function and yes poorly "designed" systems create a destructive resonance frequency causing the system to shut down within a few cycles (I have two prototypes sitting in storage that are great for shutting it down within a few seconds). My reason for stating the .042s is actually half of the time until the next impulse. I did this in order to attempt to stay away from any type of resonance.

I really do have to say thank you on this, I honestly don't know if I am even asking the right questions yet but it seems like I am closing in on it finally.

 

[RoarkS]

Oops... to answer...

yeah the objective is to minimize the max value of the base force, with knowledge that another impulse Will occur repeatedly at about 700hz and resonance will kill/cause problems with the system.

what that means to me in common sense is that I need a pretty stiff spring to keep the displacement (reset time) to a minimum... and in my case I'm including the dampening for a friction of the system.

I will look into Vibration Isolation... sounds like where I need to be looking.

 

[RoarkS]

Whoops. ~12hz not 700.

 

[RoarkS]

Okay looking it over... I think what I would be looking for is a critically damped solution.

Yup... found this on Wiki

http://en.wikipedia.org/wiki/Damping

When ? = 1, there is a double root ? (defined above), which is real. The system is said to be critically damped. A critically damped system converges to zero as fast as possible without oscillating. An example of critical damping is the door closer seen on many hinged doors in public buildings. The recoil mechanisms in most guns are also critically damped so that they return to their original position, after the recoil due to firing, in the least possible time.

So in order to get this thing so that it doesn't oscillate I just need to play with the K and C values correct?

 

[electricpete]

So in order to get this thing so that it doesn't oscillate I just need to play with the K and C values correct?

That’s what I understand from your attachment – the only parameters that are variable in the sketch are K and C (although maybe that an artificial box inside which to think).

I envisioned a systematic 2-D sweep through K and C to get a rough idea of the behavior. I tend to think it is not an extreme value (highest K, highest C, lowest K, lowest C) that is optimum. Critical damping may well be the answer. Ideally I guess you’re searching among solutions that virtually decay away at the end of the period for the one that gives the lowest peak force. If I get a chance I may put some work into that tonight or tomorrow night.

A quick scan of “vibration isolation” literature indicates they focus on frequency domain method. That does not seem immediately relevant when the task is to minimize the instantaneous peak vibration, which may have contributions from many different frequencies. Perhaps the more relevant field is “shock isolation”


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[RoarkS]

Sweet knowing that Critically dampened is ?=1=c/(2*(m*k)^(1/2))
Based upon my mass I worked that equation to give me properly matched K and C values to get critically dampened solutions. Now all I have to do is find solution set that works within my time frame then figure out how to make something as close as possible to it!
Will keep ya updated

 

[MechEngNCPE]

Do your machine dynamics homework somewhere else.