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...

 

[blacktalon]

I should add the space constraints are tight... only 6 mm allowable deflection...

also... I have played with damping ratios from 0.05 all the way to 2.0 with nat freq's from 1Hz to 250Hz... NO LUCK in the differential equ model.

I can only get down to 35g realistically

and when I put a damp ratio of 2 with 68 Hz nat freq ... i got 25g

 

[Bribyk]

Your deflection may be the limiting case and could make it impossible with your current design. With your initial velocity and allowable deflection figure out the lowest possible deceleration and see if it "can" be done.

 

[blacktalon]

I totally agree with the possiblility of the space constraints copuld make it impossible.

My greater concern is aside form space constraints, I cant seem to get below 10g's even if I allow a deflection of say 20mm and more..

Ill post data from the solution curves later....

Im I wrong to conclude its impossible to isolate a 50g shock down to 10g?

 

[Bribyk]

Okay, I pulled out the old dynamics textbook. If you can figure out the initial velocity of your module from the test program and module mass, required minimum deflection to bring it to rest with 10g's constant deceleration is going to be:

req'd deflection = (initial velocity [m/s])^2/(2*10*9.81 [m/s])

This is going to be your lower bound for minimum deflection. Depending on your system's response and the forcing frequency you may (will likely) require more than this and possibly even a little more to give you a safety factor to account for deviation/changes to your spring/damping rates.

 

[blacktalon]

Actually I cant seem to get below 35g's realistically... again ill post graphs later.

 

[Bribyk]

Can you add mass to the module?

 

[blacktalon]

But my initial velocity before the shock pulses is 0 m/s... the system is at rest.

 

[blacktalon]

Adding mass to the module to provide further isolation only works in 2 degree of freedom systems.... I did consider that.

 

[blacktalon]

Ironically the differential equation is indpendant of the masses... it only tracks transmitted g's which can be converted to force later...

 

[FeX32]

Theoretically, you should easily prove a zero shock by tuning the absorber frequency to that of the forcing function frequency. However, if in your case there exist a range of frequencies that you wish to isolate then I suggest active rather then passive control.
This is a huge topic. To put it simply, you can design your own by creating a mechanism that can change stiffness in real time. Usually, this is done by means of electromagnetic springs.

I noticed the words 'space constraints'.
This can be fixed with what I mentioned above with the addition of another theory. This is the implementation of a 'high static-low dynamic stiffness absorber' (HSLDS).




Fe

 

[GregLocock]

"ero shock by tuning the absorber frequency to that of the forcing function frequency"

No you can't since a shock doesn't have a single forcing frequency

Cheers

Greg Locock

SIGlease see FAQ731-376 for tips on how to make the best use of Eng-Tips.

 

[FeX32]

Greg,
Yes, I agree with you. There would exist a number of excited modes in a shock that is dependent on the energy present.
However, would the system not vibrate at its natural frequencies when subjected to a shock. Then we should be able to alter these accordingly.
I will be honest. My knowledge of vibration absorption is mainly focused on known forcing functions or forcing functions that exist withing a range of freq's.
I have never actually designed an absorber for a system subjected to shock. Intuitively, I would try increasing the damping as much as possible.
I'm sure someone with more knowledge has a good answer. This has actually intrigued my interest.







Fe

 

[FeX32]

I forgot to mention. Also, please disregards what I mentioned about the 'zero shock' (as Greg said) I previously misinterpreted the problem with that of a simpler one.


Fe

 

[electricpete]

I haven't read the thread carefully other than initial post (my apologies in advance if what I say has become irrelevant based on subsequent discussion).

This equation does not look correct to me:

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

Here is my attempt at derivation... comes up with something very different looking:

Let x(t) = position of mass, X(s) = laplace transform of x(t)
Let x0(t) = position of the "base" *, X0(s) = laplace transform of x0(t)
* the base is what accelerates according to h(t)

For simplicity assume initial conditions are 0.

Now write the force equation on mass 1 in Laplace domain sing X0(s) = H(s)/s^2:
s^2 * m X(s) + s*c*[X(s) – X0(s)] + k * [X(s) – X0(s)] = 0

s^2 * m X(s) + s*c*X(s) + k * X(s) = [s*c + k] *X0(s)

Now substitute H(s) = s^2*X0(s)
s^2 * m X(s) + s*c*X(s) + k * X(s) = [s*c + k] H(s)/s^2

Divide by m
s^2 * X(s) + s*c/m*X(s) + k/m * X(s) = [s*c/m + k/m] *H(s)/s^2

Substitute c/m = 2*zeta*w0 and w0^2 = k/m

s^2 * X(s) + s*2*zeta*w0*X(s) + w0^2 * X(s) = [s*2*zeta*w0 + w0^2] *H(s)/s^2

Take inverse laplace transform
x''+2*zeta*w0*x'+w0^2*x =2*zeta*w0* h~+w0^2 * h~~
where ~ is integration and ~~ is double integration

My expression looks very much different than yours. Maybe I have made a big error or gone off in left field somehow. Maybe you can point out my error or provide your derivation or tell me if I'm in left field


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[blacktalon]

I see where you are coming from, but you are ignoring the acceleration input of the shockpulse defined by h(t)

I have done a successful derivation of the laplace transform of the unit step function for shock pulse and have set-up the equation previously to solve the differential equation in term of laplace transforms. The inverse laplace transform is very messy in Maple and it was much easier to solve numerically.

I am in the process of documenting the calculation in a word file which I will post.

I will post a link to the laplace transform of the saw-tooth function tomorrow hopefully... its taking a while to type up.

 

[electricpete]

Thanks - I will wait and see what you come up with. I have included h(t) in the model, just not in the first few equations. I start with X0(s) and then substitute
H(s) = s^2 * X0(s).

My model is as follows:
Base == Parallel{k/c} === Mass
where the position of mass is given by x(t) and the position of base is given by x0(t) and the acceleration of the base is given by x0'' = h(t)


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[electricpete]

Assuming the model I described above is correct, we can apply a test case to check the validity of the solution.

A good test case would be a very stiff spring and very small mass and slowly varying h(t). In this case we know the acceleration seen at the mass is the same as the acceleration seen at the base.

Let's see how the two solutions act when w gets very high
My solution:
x''+2*zeta*w0*x'+w0^2*x =2*zeta*w0* h~+w0^2 * h~~
Everything without w0 becomes vanishingly small as we increase w0 very high:
w0^2*x =2*w0^2 * h~~
x = h~~ as expected (*)

Your solution:
x"(t) + 2*z*w0*x'(t) + w0^2*x(t) = -h(t)
Everything without w0 becomes vanishingly small as we increase w0 very high:
w0^2*x(t) = 0 (*)

(note this is an approximation rather than equality, but I avoided using the symbol ~ for obvious reasons)

Based on the above, I still think your expression is incorrect. Unless the problem is different than what I posted 29 Mar 09 21:41.

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[electricpete]

Correction in bold:

Let's see how the two solutions act when w0 gets very high

(I prefer to use w0 for sqrt(k/m) since w typically refers to a frequency as an independent variable).


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[FeX32]

I thought this was a 'shock' isolation problem. Not a textbook base excitation problem. Please correct me if I am wrong.


Fe