Single DOF spring-mass-dashpot system excited by half-sine wave shock pulse 1

[Tunalover]

Does anyone have closed form solutions for the differential equation providing the transient responses x(t)? I pulled out my old differential equations book and used Laplace transforms. The algebra got out of hand! I DO THINK IT'S SOLVABLE!




Tunalover

 

[FeX32]

Take it as a response to an arbitrary input.
Or you could take the general solution to a unit impulse and simply discritise the "half-sine" with a number or impulses starting with a very small one, ending with the magnitude of the half sine.


"Simplicity is the ultimate sophistication." L. da Vinci
- Gian

 

[IRstuff]

Or, you could possibly treat the half sine as the product of a full sine multiplied by the Rect function, which would be a convolution in frequency domain.

TTFN
faq731-376

Need help writing a question or understanding a reply? forum1529

 

[electricpete]

Here is one I had done on Maple in 2006 (based on file date).
Using Laplace transforms, but on undamped system. The sinusoidal pulse is built the way IRstuff described.
I suppressed the general algebraic output because it was messy I suppose. I did include algebraic output in the example values part.

I'm sure it can be done for damped system. But I myself wouldn't do it without computer algebra system. Do you have one of those? If there's value for you (let me know), I can do it on Maple and post the results, but I think the algebraic solution would be too long to be useful to you.

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

 

[Tunalover]

electricPete:
Thanks for that! I don't have a math program. I tried it the old fashioned way. There is certainly value in the solution as this physical problem is very real with transit drop events. I just don't have the money to poney up for MathCAD (I tried unsuccessfully to install a free trial version of MathCAD Express from PTC).

Tunalover

 

[FeX32]

You can try WolframAlpha online.


"Simplicity is the ultimate sophistication." L. da Vinci
- Gian

 

[Tunalover]

I finally successfully download MathCAD Express so I can try that now. Thanks guys!


Tunalover

 

[IRstuff]

FYI, Most of the advanced functions are disabled in Expres.

TTFN
faq731-376

Need help writing a question or understanding a reply? forum1529

 

[boyze]

This on line calculator maybe of help.

Shock Response Calculator

Have Fun!

James A. Pike

http://www.xl4sim.com

http://www.erieztechnologies.com

 

[spongebob007]

The solution for an undamped SDOF system response to a half sine pulse can be found in any textbook on vibrations. The Harris Shock and Vibration Handbook discusses the effect of damping on the response of a system to a pulse. This is a pretty well known and easy to solve problem. You should be able to find the solution to the problem on the internet.

 

[Tunalover]

Spongebob07-
Yes I am aware that the undamped solution is published but my OP was seeking the damped solution. See the OP.

Tunalover

 

[spongebob007]

What kind of damping values are you looking at? For lightly damped systems, say less than 10%, damping has a very small effect on the maximum shock response. In the Harris book as well as "Vibration Analysis for Electronic Equipment" by Dave Steinberg, the shock response spectra for a single DOF system with damping are plotted. For an undamped system, the peak amplification is 1.76. At 10% damping, the peak amplification ratio is right around 1.5. Going from 1% damping to 10% damping is a ten fold increase, but the peak shock response only decreases about 10%. This is different than the response to steady state vibration, where a ten fold increase in damping will have a similar effect on decreasing the response at resonance. In the isloation region, defined as the region of the spectra where the ratio of the system frequency to the pulse frequency is 0.5 or less, up to 10% damping, there is no difference in the response from the undamped model.

You should be able to find the shock response spectra for a half sine pulse for various damping ratios in shock and vibration literature. Just use these spectra to determine the response of your system.

Do you have access to FEA software? You can model a spring mass system and get the response to a half sine pulse.

 

[boyze]

Here's an extension (Drop Shock Calculator) of the Shock Response Online Calculator cited above. Had to do this for another project so might as well share it.

Have Fun!

James A. Pike

http://www.xl4sim.com

http://www.erieztechnologies.com

 

[Tunalover]

Thanks James! I saved the Shock Response Online Calculator as a favorite. I'll do the same with the Drop Shock Calculator. Of course I'll have to validate them both with handbook calculations before I can use them.
Regards,
Howard Bruce Jackson


Tunalover

 

[Tunalover]

James: I just sent you an invitation to connect on LinkedIn. Please accept.
Regards,
Howard Bruce Jackson
aka Tunalover


Tunalover