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...

 

[rybose]

If your relative displacement plot is the exact same as mine, you should be pretty close.

It sounds like you might be making a simple sign error.

z = x - y ...... x'' = z'' + y''

 

[blacktalon]

Thats what I thought too. Yes, I am using the same identity as

x" = z" + y"

So here is my derivation:

Solve for Relative Acceleration:

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

where:
h(t) = the unit step function for the sawtooth (one pulse
in this case to match yours)
z" = relative acceleration
z' = relative velocity
z = displacement
c = damping ratio
w = natural angular frequency

now technically h(t) = y"

so

x" = z" + y" = z" + h(t)

so then it just goes back to:

x" = -2*c*w*z'(t)-w^2*z(t) (the h(t)'s cancel)

what am i doing wrong here! arghhh

 

[rybose]

Those equations look fine. What's the point of deriving x'' as a function of z' and z??

I figure once you solve for z just double differentiate and add y''.

 

[blacktalon]

Basically Rybose, my relative acceleration plot is very similar to your absolute acceleration plot. In fact, after the first peak at 48.9 g's... it is EXACTLY the same. This of course makes sense as the base is considered stationary after that point.

I have attached a file comaring the plots. Mine is relative acceleration and yours is absolute.

When i apply the identity though... the acceleration spikes to 60g.

 

[rybose]

It looks like you're almost there! You appear to have a glitch in your code. Check your signs and make sure you adding things on the same time scale.

The only difference between relative and absolute acceleration is only during the 6 ms impulse. It should be pretty clear what's going wrong.

 

[blacktalon]

Got it to work Rybose.... I just differentiated twice like you suggested. The only reason I didnt before was because it was a numerical solution I obtained and you cant differentiate that.

Your method made my results come out bang on.

Thanks for keeping in touch.