Shock Isolation 3

[blacktalon]

Hey guys...

Im working on a new project at work that basically requires that an Electrical Control Box be isolated from a 50g saw-tooth shock pulse as per mil-std....

I will be adding more info to this topic tomorrow as I post data etc... but I have some questions...


I have set-up and solved the differential equation of motion for the system numerically by maple and set-up plots of the displacement, velocity, and acceleration.

Now, I keep running into conflicting results with other applied theory....

The Eqn of motion I solved is:


x"(t) + 2*z*w*x'(t) + w^2*x(t) = -h(t)

where:

x"(t) = absolute accleration of the module
x'(t) = relative velocity of the module
x(t) = relative displacement of module with respect
to the structure it is embeded in.
(distance between module and structure wall
which represent spring deflection)
z = damping ratio
w = natural frequency of the isolator(s) chosen
h(t)= acceleration of the surrounding structure which
I defined to be a unit step function
(heaviside function) for 3 pulses.

**This is a base excitation problem


Now the base excitation a sawtooth/50g/0.006s duration pulse 3 times.

The manufacturer of the module states the module cannot withstand more than 10g of shock.

Now I have solved the differential equation and the results look correct. (I will post later)

I am probably going to use a rubber bushings (damp ratio = .05, nat freq's 5-30Hz)


Now.... questions

1) Technically isnt the forcing frequency 1/.006 and not 1/(2*.006)? I mean thats where the same ref point on the pulse re-occurs... why do some texts do that?

For the time being I assume my forcing freq = 1/.006 = 167Hz

2) Despite the space contraints, I cant seem to be able to get the shock down below 10g's, even if I alter the nat freq of the isolators or the damp ratio to rediculous values.... but by the theory of transmissibility....

damp ratio = .05
forcing freq = 167Hz
isolator nat freq = 68Hz or lower

should provide me with a shock isolation 0f T=0.2....

50g *0.2 = 10g

but the differential equation solution doesnt do this... and im 99% sure its right....

Even if I change the nat freq input in the diff eqn model closer to 167Hz... the equation simulates resonance.. (as expected)

***And yes I am converting the nat frqu to angular frequency properlly***


I guess my questions is... is it even possible to reduce a 50g shock to below 10g? Cause I cant seem to simulate it...

Ideas? I will be posting better info tomorrow...

 

[electricpete]

If "shock isolation" means that the acceleration is applied to the mass (rather than base), I would agree with the differential equation in the intitial post. However the following statements lead me to believe h(t) was the acceleration of the base, not the mass:

h(t)= acceleration of the surrounding structure which
I defined to be a unit step function
(heaviside function) for 3 pulses.

**This is a base excitation problem


Now the base excitation a sawtooth/50g/0.006s duration pulse 3 times.

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[blacktalon]

yes... I am referring to h(t) being the acceleration of the surrounding structure, not the mass... I have almost finished typing up my derivations etc and will deifnatley post later tonight. You should find it quite interesting.

 

[rybose]

I've been working on the same problem recently. The differential equation for a base excited damped SDOF system is:

mz'' + cz' + kz = -my''

x - abs. mass displacement
y - abs. base displacement
z = x - y, relative displacement of mass (disp. of mass wrt to base)

z(t) can be solved for any acceleration time history y''(t) using Duhamel's integral aka the convolution integral aka filtering in the time domain.

z(t) = integral{F(tau)*h(tau) dtau}
- Where h(t) is the relative displacement impulse response function of a SDOF system. (Similar to the free vibration solution.)

h(t) = exp(-zeta*wn*t)*(1/(m*wd))*sin(wd*t)
F(t) = -m*y''
- Note: As you noted earlier, the masses cancel.

In MATLAB the convolution integral can be solved easily:
"z = conv(F,h)*dt"

Differentiate z(t) twice to get the relative acceleration. Use x'' = z'' + y'' to get the absolute acceleration.

Check the accuracy of whatever solution method you use by numerically deriving a Shock Response Spectrum for a common input acceleration shape (half-sine pulse).

see "Mechanical Vibrations" Rao pg. 348, "Harris Shock and Vibration Handbook" pg. 23.14 for more details.

I ran your input with my MATLAB program. My stuff seems to agree with your assessement that it might be impossible to get under 10g and 6mm relative displacement with a linear system.

 

[FeX32]

Nothing is impossible.

Fe

 

[electricpete]

OK, I see the "trick" is that x is defined not as position but as mass position minus base position. That makes the solutions equivalent to mine.

The word relative was used in the original post but it didn't catch my attention. The symbols x, y, z made it plain enough for me to understand.

Thanks for explaining that.

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[blacktalon]

I have read in great detail Duhamel's Integral, but did not like having to superimpose the initial conditions on the response. I decided to simply solve the equation of motion.

Interesting enough, when i tried both method's, I did not really get the same results... its was very frustrating.

I have attached what I have done so far as terms as a derivation of the ewuation of motion and a proper laplace transform....

RYBOSE... did you get the same response curve by solving for the differential equation of motion and also duhamel's integral. You should try it and see ...

 

[blacktalon]

why didnt the file show up???

 

[blacktalon]

The difference with mine Rybose is that I have a shock input function to the base.

 

[blacktalon]

I also have the exact solution for Duhamel's Integral to solve for the the acceleration response. I pulled it from a book somewhere... I will post it later tonight.

 

[electricpete]

why didnt the file show up???

There are a few steps you have to follow.
Pres "Upload your file..."
Press Browse... select file...Upload your file
"Click Here To Insert Your File's Link Into Your Post & Return To Eng-Tips Forums"

The last step is the one that I am prone to forgetting.

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[blacktalon]

Alright... here is what I have typed up so far regarding documentation. What I am struggling with right now is getting the final acceleration plot to be in absolute acceleration of the mass.... so I am reversing the substituiton for z"= x"-y"... but im kinda gettingmessed up results and the thing is just because the relative velocity is in one direction, doesnmt mean the absolute velocity isnt in the other... the accelerations are still coming out way to high and there is a mistake somewhere I know it...

HELP ME TRACK IT DOWN!! lol...

 

[blacktalon]

NOTE: You must have equation editor in Microsoft Word Installed to read it properly...

 

[rybose]

You can't use the classic transmissibility curves to predict shock response. These curves plot the steady state reponse to a SDOF system forced at a single frequency.

In shock loading, where the duration of the pulse is usually much less than the natural period of the system, the situation is more like a free vibration problem with a non-zero initial velocity due to the rapid base motion. This siutation where the pulse is short relative to the natural period of the system is referred to as "velocity shock."

For undamped systems in velocity shock, there are closed form solutions for max acceleration and displacement. Check out:

http://www.barrycontrols.com/defenseandindustrial/isolatorselectionguide/iso_select.pdf

http://www.bdproduct.ca/barry pdfs/Passive shock isolation.pdf

For shock loading, Shock Response Spectrums are used in place of the classic transmissiblity curves.

Before you tackle the entire 3 pulse load with damping, try the undamped system excited by a single pulse and verify your results with hand calculations described in the above references.

Attached (hopefully) are some plots of my MATLAB program for a single sawtooth load. Hopefully they will help you debug your Maple stuff.

Enjoy

 

[blacktalon]

Thank you so much for your response Rybose.... taht clears up a lot. I will go over your attachemnt. THANKS!!!

more comments welcome.

 

[blacktalon]

RYBOSE.. that is a kickass GUI yo got goin on there... wish i had access to that! Did you make it yourself? Must have taken sometime if you did.

 

[blacktalon]

RYBOSE... I dont wanna bug you too much, you have already been so helpful... but if you want, could you run the same simulation with more damping.

Realistaclly, each rubber/neprne bushing has a damping ratio of 0.05 and when you have more than one inline, the damping ratios are added. So realistically, I will be using 4-8 bushings so the damping ratio would be .2-.4.

I guess you dont need to adjust the natural frequency because of the spectrum plot, but could you fire in 25 or 30Hz?

It would confirm my findings for sure.

In addition, my relative acceleration plot is exactly the same as you absolute acceleration plot for 68Hz .05 damping! Weird!

 

[blacktalon]

RYBOSE.. is that the GUI that cost $40 to use the matlab scripts?

If so, im will subscribe. Its worth it to prove my model.

 

[blacktalon]

MY DISPLACEMENT PLOT IS THE SAME TOO... BANG ON

 

[blacktalon]

NEVER MIND THE DISPLACEMENT PLOT ISNT EXACTLY THE AME AS THERE IS ONLY ONE SAWTOOTH PULSE CONSIDERED IN THE THE MATLAB MODEL.

 

[spongebob007]

Your displacement results may seem crazy, but they are probably correct. We routinely isolate down to 15 G's, and at an isolated frequency of ~7Hz, we experience up to 4 inches of deflection.

One other thing is that you may be neglecting the non-linear load deflection characteristics of the isolator. When I first tried the approach of solving the equations in MATHCAD, I came up with some crazy deflections. When I switched to a model that used a lookup table for the spring constant, my results were more in line with what I expected.


The deflection under a shock load is given by Acc/(2*pi*f)^2, so for a 10G shock at a 7 Hz isolated system, you will need ~2 to attenuate the shock.