Shock Isolation
Shock Isolation
(OP)
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...
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...





RE: Shock Isolation
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
RE: Shock Isolation
RE: Shock Isolation
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?
RE: Shock Isolation
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.
RE: Shock Isolation
RE: Shock Isolation
RE: Shock Isolation
RE: Shock Isolation
RE: Shock Isolation
RE: Shock Isolation
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
RE: Shock Isolation
No you can't since a shock doesn't have a single forcing frequency
Cheers
Greg Locock
SIG:Please see FAQ731-376: Eng-Tips.com Forum Policies for tips on how to make the best use of Eng-Tips.
RE: Shock Isolation
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
RE: Shock Isolation
Fe
RE: Shock Isolation
This equation does not look correct to me:
Here is my attempt at derivation... comes up with something very different looking:
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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RE: Shock Isolation
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.
RE: Shock Isolation
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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RE: Shock Isolation
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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RE: Shock Isolation
(I prefer to use w0 for sqrt(k/m) since w typically refers to a frequency as an independent variable).
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RE: Shock Isolation
Fe
RE: Shock Isolation
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RE: Shock Isolation
RE: Shock Isolation
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.
RE: Shock Isolation
Fe
RE: Shock Isolation
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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RE: Shock Isolation
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 ...
RE: Shock Isolation
RE: Shock Isolation
RE: Shock Isolation
RE: Shock Isolation
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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RE: Shock Isolation
HELP ME TRACK IT DOWN!! lol...
RE: Shock Isolation
RE: Shock Isolation
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://
http://w
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
RE: Shock Isolation
more comments welcome.
RE: Shock Isolation
RE: Shock Isolation
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!
RE: Shock Isolation
If so, im will subscribe. Its worth it to prove my model.
RE: Shock Isolation
RE: Shock Isolation
RE: Shock Isolation
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.
RE: Shock Isolation
RE: Shock Isolation
RE: Shock Isolation
RE: Shock Isolation
Damping ratio effects - Look at Figure 9 in the Passive Shock Isolation paper. You can see that the optimal damping ratio for SDOF velocity shock is 0.26. Unfortunately the isolation difference between 0.05 and 0.26 isn't too dramatic.
Relative vs. Absolute Acceleration - During the residual response (after the load is applied) the relative and absolute acceleration curves are the same (base acceleration = 0). During the primary response (when the load is being applied) they should be different.
Damage thresholds are in terms of absolute accleration so in order to make your code as general as possible I'd recommend gettin things ITO abs. acceleration. (Try making your pulse width really big and see what happens to the relative acceleration curve)
My sweet MATLAB GUI - It's actually much easier than it looks. You just need to get the hang of passing the variables through the "handles" structure. I learned how to do it in a couple of days using these great tutorials:
http://blinkdagger.com/category/matlab/gui-matlab
RE: Shock Isolation
I have attempted to attach a plot of the pulse
RE: Shock Isolation
I think everyone here is making too much out of this problem. Assuming a linear spring as a first attempt, here is how I design an isolated system.
1) Generate SRS of shock pulse. Make sure you use the correct damping ratio for the isolated system when generating the response spectrum.
2) Select the frequency that corresponds to the desired attenuation level.
3) Determine the displacement for the isolated frequency.
4) Using the isolated frequency and isolated mass, back out a spring constant.
5) Select an isolator that has the approriate stiffness, static load rating, and desired stroke.
RE: Shock Isolation
Fe
RE: Shock Isolation
RE: Shock Isolation
But 0.4" to attenuate a 50g shock is very unlikely. how did you arrive at that?
I got 35mm displacmenet with those numbers which only isolates to 40g and dies out after that...
RE: Shock Isolation
You might not be familiar with a pseudo velocity plot. The diagonal lines that rise from left to right represent relative displacement. The diagonal lines that fall from left to right represent acceleration. The axes are logarathmic. Start by following the diagonal line for 10G's. The point where the 10G line intersects the red SRS curve, is your ideal operating point for the isolated system.
RE: Shock Isolation
RE: Shock Isolation
RE: Shock Isolation
RE: Shock Isolation
I took a look at your pseudo velocity plot. Is the horizontal axis the frequency of the forcing function or the tuning frequency?
Fe
RE: Shock Isolation
RE: Shock Isolation
The software I use to generate the SRS has a lower frequency bound of 1/T. For this time history that gives 1/.018=55.5 Hz. You can request it to go to a lower bound, but as I found out, the accuracy of the result below 1/T are questionable. So what I did is augment the original pulse with a "tail" that has an acceleration of 0 out to 2 sec. Now I was able to generate an SRS that went down to 1Hz.
The bad news is, that with 5% damping, it is going to take almost 7 inches to attenuate the shock to 10G's @ 3.8Hz. I ran a SDOF model in NASTRAN to verify the results, and sure enough the results agreed.
Like I said earlier, attenuating COTS equipment to 10-20 G's is going to take on the oredr of 3-4 inches. This has been my experience, and we use isolators with nonlinear stiffness characteristics and damping on the order of 15-20% of critical.
I have attached a zip file with the new SRS and response time history plots.
RE: Shock Isolation
Another book I would highly suggest is "Vibration Analysis for Electronic Equipment" by David Steinberg.
RE: Shock Isolation
It says... there are 3 ways to analyse shock isolation... it covers the first too with some equations and at the end it says if the isolators are non-linear like rubber etc... you might as well toss them out th window. Then when it speaks of other methods... its sys the math is real complex and not to be covered in the textbook.
RE: Shock Isolation
http://w
I think Tom Irvine may have some papers also.
Jim Kinney
Kennedy Space Center, FL
RE: Shock Isolation
With the substition of z=x+y for the accleration... I solve for relative acceleration.
When i try to us this z=x+y relationship to bring out the absolute acceleration of the mass.... its adds up to 60g over the 50g!
See the derivation on the file I attached earlier? Any idea what im missing because your acceleration plot for the parameters I gave you seem more realistic. My displacement and velocity plots are exactly the same as yours though... its just bringing out the absolute acceleration.
RE: Shock Isolation
It sounds like you might be making a simple sign error.
z = x - y ...... x'' = z'' + y''
RE: Shock Isolation
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
RE: Shock Isolation
I figure once you solve for z just double differentiate and add y''.
RE: Shock Isolation
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.
RE: Shock Isolation
The only difference between relative and absolute acceleration is only during the 6 ms impulse. It should be pretty clear what's going wrong.
RE: Shock Isolation
Your method made my results come out bang on.
Thanks for keeping in touch.