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Modeling a Mechanical System

Modeling a Mechanical System

Modeling a Mechanical System

(OP)
I have rotating equipment (pump) problem I need some help with.  We have a close coupled vertically mounted volute pump with a confirmed structural resonance problem.(In one plane the natural frequency is about 11.1 Hz.  In the other plane it is 15Hz.  During field testing, weight [a 50# sandbag!] was added to the system, and the critical peaks shifted down about 1.5-1.7Hz.  This is a classic structural ressonant condition)  

The plan to fix the issue is to add weight to the system (likely a thick plate between motor and motor support) to move the critical frequencies down.  We will also modify the pump so that it runs at a faster RPM.  This will result in moving our operating speed, and natural frequencies away from each other....pretty simple, really.

My question is with regards to creating a "simple" analytical model for this system.  We have determined that 90% of the flexibility in this structure is in the pump, backhead and base.  The motor support and motor are relatively rigid.  As such I want to model the entire base, pump and backhead as a cantaleverd beam, and model the motor and motor base as a rigid lumped mass hung on the end of the beam with the center of mass some distance from the beam end.  The motor and motor support mass and dimensions are known in detail.  As such, I can tweak the model of the "beam" (pump, base, backhead structure) to agree with the field data  obtained.  After getting a model that is close, I can then vary the mass of the lumped mass on the end of the beam to estimate how much mass addition is necessary to move the critical.  

(I could of course just guess, based on the field data measured when we added 50# to the top of the motor and the wing it....but that could waste a lot of time and money.  i would at least like to make an educated guess as to how much mass we add to the system to get the desired effect of shifting the peaks down by 1.7 Hz/100 RPM))   

This model does not have to be perfect.  Even a 20% error in critical frequency change will only cause an error of 20 RPM.

One approach is:

1.  Consider the pump/base/back head beam to be a perfect beam with no mass.  There are two loading schemes we must consider:

- Force acting at end of beam:  For this case deflection is given by PL^3/3EI   

- Moment acting at end of beam:  For this case deflection is given by MoL^2/2EI

So my question is how do I take this system and come up with a solution (frequency)?  I am sure this will end up as some sort of  crazy Partial Diff Eq problem...it's been way too long since I did that stuff.  Anyone know where to go from here.  If need be I will put a drawing together and email it to illustrate my approach.

Thanks for any help.

RE: Modeling a Mechanical System

You could use component modal analysis...finding the
modal frequencies of the beam is easy..from there you could
add masses,springs etc and with the proper constraint equations you could get a very good estimate of the system
frequencies. I am not sure how familiar you are with the method..most good vibrations books go over it. Meirovitch has a good explanation of it. If you need more help let me know.

RE: Modeling a Mechanical System

Perhaps I don’t fully understand your question, but it sounds to me like you are making it more complicated than it needs to be.  You have done the experimental work necessary to find the modal mass and stiffness, so now it is a simple matter of manipulating the equation for the resonance frequency w=sqrt( k/m). Or to put it in terms of Hz for the frequency and pounds weight  w= 1/2pisqrt(k*g/W).

w= natural frequency in Hz
pi= 3.1415
k= modal stiffness in lb/in
g= 386.4 in/s^2
W= modal weight in lbs
sqrt= square root

If I am interpreting the given information correctly, and looking at only one plane, the original natural frequency was 15 Hz. You then added a 50 lb weight, and did your test again and got 15- 1.6 or 13.4 Hz.  So you have two equations and two unknowns:

15 Hz= 1/2pi sqrt(k*g/W),     and
13.4 Hz=  1/2pi sqrt (k*g/ W+50 Lbs)

solving the first equation for k gives  k= 23*W

plugging this into the second gives  
13.4= 1/2pi sqrt (23*W*g/W+50)

solving for W you get  W= 196.85 lbs  so then k= 23*196.85, or k=4527.6 lb/in.

You now have modal weight and modal stiffness, set the natural frequency where you want and solve for the modal weight required, then subtract the existing modal weight and you know exactly how much weight to add. Lets say you want a natural frequency of 10 Hz.

10Hz= 1/2pi sqrt( 4527.6*g/Wnew)    Wnew= 442.85 lbs

subtract the original modal weight, and the weight to be added is

Wadd= Wnew-W=  442.85-196.85    Wadd=246 lbs

I hope this is what you were looking for. You will have to do the calculation for both planes to make sure that you are ok in both directions.

–Steve Schultheis

RE: Modeling a Mechanical System

Steve - Good answer. I vote you a star.

RE: Modeling a Mechanical System

(OP)
sms,

Thanks for the response.  I wish there were some way to attach a drawing with a post.  Sometimes a picture is worth a thousand words.  I think you are on the right track.  There is some additional information I am still gathering which I did not provide in my original post.  

For instance:
- Weight of motor
- Approx weight of motor base
- CG of motor above bottom of motor
- CG of motor base
- Height of motor base

I really wish there we some way to attach PDF's to posts....


RE: Modeling a Mechanical System

Slagathor - Do you see any problem with Steve's approach?

I think that it will work as long as:
#1 - You will be adding weight at the same location and attached with the same rigidity as the weight added during the test run.
#2 - The system is acting equivalently to a single mass-spring damped system.

If those assumptions are met, the actual mass and geometry are not all that important.... Steve has computed an equivalent mass and stiffness which should work for purposes of predicting resonant frequency.  

RE: Modeling a Mechanical System

One more item - I can post a file for you on my web page and add a link on this forum if you'd like.  Just e-mail me.

To get my email address, see my profile, cut-paste my home-page into your browser and go to the bottom of my homepage for my e-mail link. (I would post it here but them web spiders would snatch it up and give me more spam mail!).

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