Response of single DOF spring-mass-dashpot model to half-sine pulse 4

[Tunalover]

Folks-
The problem is to find the solution to mx''+cx'+kx=f where f is a half-sine pulse. The initial condition are x(0)=a positive constant, say B and x'(0)=0. Years ago I attempted (over many hours!) a closed form solution to this problem because it is a practical problem in electronics packaging design (e.g. MIL-STD-810 Functional Shock). The forcing function is easy to construct with unit step functions and a sine function. I tried several methods concluding with attempt with Laplace transorms. Each solution got so long and convoluted and lengthy that I tried a new approach. Now I'm asking you folks. BTW I've no luck finding a solution in the literature.



Tunalover

 

[GregLocock]

So the forcing function is

F(t)

=(t<0)* (B*k)
+( t>=0 * t=<.5/f)*(sin(2*pi*f*t))
+(t>.5/f)*????

is that right? what is the forcing function after the end of the pulse?

I think Laplace is the right approach, but I bet a crafty bugger could use a superimposition of two systems with a time delay to do it classically.


Cheers

Greg Locock

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[Tunalover]

At the end of the pulse the forcing function must be zero. As for your suggestions, I tried them all.



Tunalover

 

[GregLocock]

not B*k?

That's a bit hard to understand physically.

Cheers

Greg Locock

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[btrueblood]

See:

Cyril M. Harris, Shock and Vibration Handbook, McGraw-Hill, Chapter 8. My copy is the 3rd Edition, 1988. Amazon says the 5th ed. is now out; you should be able to find a copy in a good uni. library.

While the book doesn't give an explicit closed-form solution, it does present a plot of the response function versus reduced frequency. Also, this chapter provides two references for that problem (good luck finding them, you'll have to probably dig in the university library archives):

Criner, H. E., G. D. McCann, and C. E. Warren: Journal of Applied Mechanics, 12:135 (1945).

and

Mindlin, R. D., F. W. Stubner, and H. L. Cooper: Proceedings of the Society of Experimental Stress Analysis, 5:2, 69 (1948)

 

[CESSNA1]

TUNALOVER: This is a problem that is covered by using a metnod known as SRS (Shock Response Spectrum). It is not covered very well in the "Shock and Vibration Handbook" by Harris. You need a practical, not theoretical application of this method. This method is included in one of the vibration courses offered by

http://www.tti4edu.com,

and possibly elsewhere.

Regards
Dave

 

[spciesla]

Look at Mechanical Vibrations (3rd Edition) by Singiresu S. Rao, Example 4.8 (page 283-285).

He solves this problem for initial conditions of x(0) = 0 and x'(0) = 0. You should be able to use the same methodology to solve for x(0) = constant.

 

[Tunalover]

GregLocock-
The waveform replicates a drop shock. It's really not that unreasonable. The duration is typically 10-12ms.


Tunalover

 

[GregLocock]

If it is a drop shock then x is zero initially (system is in free-fall), then a half sine pulse (as it strikes), then a static load (lying shattered on the ground).

Cheers

Greg Locock

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[electricpete]

I looked at Rao. He solved by breaking the solution into two regions, one while the forcing funciton is present and one when it is removed. The final value of the first solution provides the initial condition for the second. That looks like the easiest exact solution method.

Your forcing function can easily be represented in laplace transforms (produce of sin and window function which can be created as combination of step functions or "heaviside" functions).

When you transform the differential equation into Laplace domain, it becomes an algebraic equation. Solve for X(s) and then inverse transform to find x(t).

Attached is solution for the complete problem including initial conditions using Maple to help with the algebra. Without Maple it would have been ugly (especially the partial fraction expansion to find the inverse laplace transform), but do-able (if you had several hours).

http://home.houston.rr.com/electricpete/SinPulseMassSpring.pdf

If for some reason that doens't work, go here:

http://home.houston.rr.com/electricpete/link3.htm

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[electricpete]

If I made B smaller, you would have been able to see the effects of the pulse better.

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[Tunalover]

bukac1-
I'm looking for the closed form solution but thanks anyway.

electricpete-
Thanks for the input! As I said earlier, I tried the Laplace transform solution but that led to excessive algebra. I looked at your solution though and it looks like you managed it. What software generated the formulae?


Tunalover

 

[electricpete]

That software is Maple.

http://www.maplesoft.com/

If you qualify for student version, it's a helluva lot cheaper

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[electricpete]

Would it do you any good to see the partial fraction expansion of X(s). I can have the program do that and post it if you'd like. Would make it easier if you are trying to recreate the result using hand calculations.

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[Tunalover]

electripete-
I have no problem with partial fraction expansions. What I have to do is find a quiet place for two hours to try to come up with your result. I'm one of those geeks who has to do the math in order to believe it!


Tunalover

 

[solider]

A general rule of Thumb:
If the duration of the forcing pulse is less than 1/3 of the period of the natural frequency of the single DOF system, then the response will behave as if the pulse is 1/3 duration. IE if the pulse duration gets shorter, then the systems just behaves the same.

As I write this, I have a little gremlin going off that says it might be less than 1/10 of the natural period.

Does this help?

Cheers
Solider

 

[Tunalover]

solider-
I also learned a rule of thumb that, for manually portable (you don't need a truck, forklift, or dolly to move it) electronic equipment, if the pulse duration is greater than 13ms then the shock pulse can be fairly replaced with a static load on the CG with magnitude AW where A is the amplitude of the pulse (in g's) and W is the weight of the assembly. Has anyone else heard this one?


Tunalover

 

[tomirvine]

Here is the derivation for a single-degree-of-freedom system subjected to a half-sine pulse base excitation.

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

Tom Irvine

http://www.vibrationdata.com

 

[Tunalover]

Thank you Mr. Irvine!


Tunalover

 

[GregLocock]

Well I'm surprised everyone didn't know how to do that.

Star for you Tom, amazing.

Cheers

Greg Locock

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