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Shock Isolation
3

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...  


 

RE: Shock Isolation

(OP)
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

RE: Shock Isolation

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.

RE: Shock Isolation

(OP)
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?

RE: Shock Isolation

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.

RE: Shock Isolation

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

RE: Shock Isolation

Can you add mass to the module?

RE: Shock Isolation

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

RE: Shock Isolation

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

RE: Shock Isolation

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

RE: Shock Isolation

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).


peace

Fe

RE: Shock Isolation

"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

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

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.   


peace


 

Fe

RE: Shock Isolation

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. upsidedown
 

Fe

RE: Shock Isolation

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:

Quote:

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:

Quote (ElectricpeteDerivation):


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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RE: Shock Isolation

(OP)
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.  

RE: Shock Isolation

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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RE: Shock Isolation

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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RE: Shock Isolation

Correction in bold:

Quote (electricpete):

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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RE: Shock Isolation

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

Fe

RE: Shock Isolation

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:

Quote:

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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RE: Shock Isolation

(OP)
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.

RE: Shock Isolation

2
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.

 

RE: Shock Isolation

Nothing is impossible. pipe

Fe

RE: Shock Isolation

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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RE: Shock Isolation

(OP)
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 ...   

 

RE: Shock Isolation

(OP)
why didnt the file show up???   

RE: Shock Isolation

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

 

RE: Shock Isolation

(OP)
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.  

RE: Shock Isolation

Quote:

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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RE: Shock Isolation

(OP)
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...   

RE: Shock Isolation

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

RE: Shock Isolation

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%20pdfs/Passive%20shock%20isolation.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

RE: Shock Isolation

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

more comments welcome.  

RE: Shock Isolation

(OP)
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.

RE: Shock Isolation

(OP)
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!

RE: Shock Isolation

(OP)
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.  

RE: Shock Isolation

(OP)
MY DISPLACEMENT PLOT IS THE SAME TOO...   BANG ON

RE: Shock Isolation

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

RE: Shock Isolation

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.
 

RE: Shock Isolation

(OP)
MY DISPLACEMENT PLOT IS THE SAME HOWEVER, WHEN I SIMPLY PUT IN ONE SHOCK PULSE INSTEAD OF 3

RE: Shock Isolation

(OP)
Thanksm for the input spongebob...    that gives me more confidence in my results.  

RE: Shock Isolation

I made a mistake in my original post, our system isolated frequencies are around 5Hz with a displacement of about 4".   

 

RE: Shock Isolation

Some comments:

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

Here is the pseudo-velocity plot (SRS) for the pulse I posted above.   

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

That doesn't seem to solve the problem as the deflection is still high.

peace

Fe

RE: Shock Isolation

(OP)
SPONGEBOB what was your total damping ratio for the system for that set-up with a 4" deflection?

RE: Shock Isolation

(OP)
SPONGEBOB.....    sorry I idnt read enough...    got it at 0.05.

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

My results were for a single saw pulse.  I created a pulse train of 3 saw pulses and for 5% damping my results are f=8.8Hz and the displacement is about 1.3 inches for 10G's.

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

I was actually going to run an FEA model of this to confirm my results, but apparently NASTRAN has a bug in the source code and it won't run until further notice.  I kid you not.  If you use NASTRAN, you are in for a surprise.

RE: Shock Isolation

(OP)
NASTRAN...   surprise? how so...   

RE: Shock Isolation

There is an error in the source code that returns a divide by zero on any machine where the date is past March 31, 2009.  The program will crash a few seconds after you submit a job.  As far as I know it applies to all flavors of Nastran.

RE: Shock Isolation

spongebob007,

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

Fe, the frequency axis is the natural frequency of the isolated system.

RE: Shock Isolation

Okay, I made an error with the SRS plots I generated.  I thought something seemed funny, so I decided to validate my model with FEA.   For the work I do, our input acclerations are low frequency and second in duration.

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

If you want a better explanation of pseudo velocity plots, you can find it in the "Shock And Vibration Handbook" by Cyril M. Harris.   

Another book I would highly suggest is "Vibration Analysis for Electronic Equipment" by David Steinberg.

RE: Shock Isolation

(OP)
I have read throught that Steinburg book and when it comes to shock anaylsis ...   it is very basci and it even says so in the book.

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

(OP)
Rybose...     if you read this...    I cant seem to get the absolute acceleration to come out right.

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

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''

RE: Shock Isolation

(OP)
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

RE: Shock Isolation

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''.  

RE: Shock Isolation

(OP)
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.

 

RE: Shock Isolation

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.

RE: Shock Isolation

(OP)
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.  

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