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# A harmless looking SDOF vibration problem but...3

## A harmless looking SDOF vibration problem but...

(OP)
It took me a while to dig up the solution to this one. It's a fairly realistic SDOF dynamic model of a suspension system subjected to a half-sine pulse of duration T. Since I don't know how to get math into this little forum box, the problem is attached as a pdf. All those who believe they are mathematically-inclined, I challenge you to solve and verify the solution by hand.

H. Bruce Jackson
ElectroMechanical Product Development
UMD 1984
UCF 1993

### RE: A harmless looking SDOF vibration problem but...

Dunno about the analytical solution but it is a terrible model of a car suspension. The tires/unsprung mass are crucially important. also single wheel bumps rather ignore the effect of the other axle. I'd always use a 4dof model for bumps.

Cheers

Greg Locock

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### RE: A harmless looking SDOF vibration problem but...

(OP)

#### Quote (GregLocock)

Yes you're right of course. As I said, it is a "fairly realistic" rendition. The exercise is more math than anything and I would not recommend anyone using the solution for design purposes.

H. Bruce Jackson
ElectroMechanical Product Development
UMD 1984
UCF 1993

### RE: A harmless looking SDOF vibration problem but...

There is an analytical solution to this. I derived one many years back. I can check my notes. This is a traditional base excitation problem. Solution will be in obtained by taking the Laplace and solving for abs magnitude of |Z/Y|.
Quick google search turn up similar problems: https://www.tcsme.org/Papers/Vol34/Vol34No1Paper8....
https://www.researchgate.net/profile/Nguyen_Nguyen...

### RE: A harmless looking SDOF vibration problem but...

And it seems - Chapter 2.4 of the text "Engineering Vibration" by Daniel J Inman has the time domain solution derived.

### RE: A harmless looking SDOF vibration problem but...

standard 1st year undergrad stuff surely unless I missed something !
theory of vibration with application, Chapter 4, pg 102-103

### RE: A harmless looking SDOF vibration problem but...

(OP)

#### Quote (JXB0809)

standard 1st year undergrad stuff surely unless I missed something !
theory of vibration with application, Chapter 4, pg 102-103
Surely you are joshing that your book has the full derivation. It takes, like, ten pages of algebra to get it into simplified form!

H. Bruce Jackson
ElectroMechanical Product Development
UMD 1984
UCF 1993

### RE: A harmless looking SDOF vibration problem but...

I thought you were after the solution only - that's the bit I missed I guess
do a search for vibrationdata dot com + sdof response due to half sine pulse. The full derivation is there Laplace transform , etc - all the steps

### RE: A harmless looking SDOF vibration problem but...

(OP)
Yes JXB0809, that's where I found the solution.

ElectroMechanical Product Development
(aka Electronic Packaging)
UMD 1984
UCF 1993

### RE: A harmless looking SDOF vibration problem but...

http://www.vibrationdata.com/sbase.pdf

Seems like you have been interested in this for awhile. That's ok... I agree it's a challenging one. Without a computer to do the thinking for me, I wouldn't dare try to solve it.

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

### RE: A harmless looking SDOF vibration problem but...

(OP)

#### Quote (electricpete)

Seems like you have been interested in this for awhile. That's ok... I agree it's a challenging one. Without a computer to do the thinking for me, I wouldn't dare try to solve it.

Indeed I finally did find the derivation and simplified solution with Tom Irvine. Tom is a wealth of knowledge on vibration and shock and I have been a member of his vibrationdata.com site for longer than I care to admit.

Yeah, I was trying to 'stir the pot' with a seemingly impossible math exercise. As you use Laplace transforms to solve it, the derivation just grows and grows. And I thought I was good at math. The algebra just spun out of control!

ElectroMechanical Product Development
(aka Electronic Packaging)
UMD 1984
UCF 1993

### RE: A harmless looking SDOF vibration problem but...

Indeed the solution in Inman's text doesn't show the whole derivation in the manner shown in that link. It does refer to the method of undetermined coefficients- which is just an algebraic solution method.
I took a few pics since it seems it may be of interest: https://www.dropbox.com/sh/z4bt03c01tj8b06/AADbi9o...

### RE: A harmless looking SDOF vibration problem but...

(OP)

#### Quote (electricpete)

Without a computer to do the thinking for me, I wouldn't dare try to solve it.

Does that mean that programs such as MathCAD, Matlab, etc. will provide a closed form solution? That would be amazing! Whoever writes the code for these programs to do this kind of thing must be utter geniuses!

ElectroMechanical Product Development
(aka Electronic Packaging)
UMD 1984
UCF 1993

### RE: A harmless looking SDOF vibration problem but...

Not geniuses (persons of genius), just people coding rules. The programs themselves are good at following simple rules for analytical derivatives and keeping a track of the explosive details, no matter how big they get.

Steve

### RE: A harmless looking SDOF vibration problem but...

(OP)

#### Quote (Somptinguy)

Not geniuses (persons of genius), just people coding rules. The programs themselves are good at following simple rules for analytical derivatives and keeping a track of the explosive details, no matter how big they get.

Steve will your math program provide the closed-form solution? Inquiring minds want to know!

ElectroMechanical Product Development
(aka Electronic Packaging)
UMD 1984
UCF 1993

### RE: A harmless looking SDOF vibration problem but...

Tunalover: I don't have a math program (have never tried to write one either). I'm not promoting a product here.

I have witnessed a colleague using one to provide all the terms of a big jacobian matrix with analytical expressions before (a chemical kinetics problem with dozens of species and a non-sparse jacobian). The thought of trying to do all that by hand made the task seem like madness, I would have just used perturbation to approximate them. As it happened, he gave up on the analytical derivatives anyway, since they became unwieldy - hundreds of terms to transcribe from the math program into actual executable code.

The math program just churned through the known rules without making mistakes, going deeper and deeper into chain, product and quotient rule complexities. It wasn't asked to solve anything. Try writing down something seemingly simple, like piston jerk (third time derivative of displacement of a piston in a crank-slider mechanism) without making mistakes.

Steve

### RE: A harmless looking SDOF vibration problem but...

(OP)

#### Quote (Somptinguy)

The math program just churned through the known rules without making mistakes, going deeper and deeper into chain, product and quotient rule complexities. It wasn't asked to solve anything. Try writing down something seemingly simple, like piston jerk (third time derivative of displacement of a piston in a crank-slider mechanism) without making mistakes.

First question:
Am I right to say that maybe the math program provided the right solution to the chemistry problem but the guy was unable to use the data because there was too much to deal with?

Second question:
If it wasn't asked to solve anything, then what was the program doing?

Third question:
What's so special about a piston jerk problem?

ElectroMechanical Product Development
(aka Electronic Packaging)
UMD 1984
UCF 1993

### RE: A harmless looking SDOF vibration problem but...

Tunalover,

These algebraic manipulation programs(software) exist. My favorite a few years ago was Maple (https://www.maplesoft.com/products/Maple/academic/). It is still the best IMO.
Here are 2 screen shots of one of my worksheets.
It can solve (closed form) in a few seconds what would take you days to do and many times not even realize.
This could for sure be used as a means to an end of your original problem in this thread.

There is also http://www.wolframalpha.com/ - it is not too bad sometimes but not that great for anything complicated I found.

Personally, I don't really use this kind of software much anymore - the same end can be obtained numerically with some code much faster and most times there is no closed form solution to real-world type problems.
Cheers,

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