Response to impact pulse 3

[davidmandis]

My problem involves a spring mass system with a triangular impulse acceleration as the input on impact. I have simplified the force-deflection response as two linear behaviors as shown in attached figure.
I will need to model two separate responses for the input as two ramp loads, and using the results of the first one as the initial conditions to the second. The two ramp inputs will be,

f(t)=2P(t/tp) 0<t<tp/2
f(t)=2P(1-(t/tp)) tp/2<t<tp

I also will have to calculate two responses for the two different spring behaviors,
F=kx x<3750
F=k x>3750

Now, I cannot seem to figure out how to superimpose all the results. Should I just add all the responses (by response I mean the output displacements) or is there some other way to superimpose the results?

Thanks,
David

 

[GregLocock]

I think that is a mistake. You'll have to work through it.

Incidentally it is probably worth drawing a diagram of the system you think you are analysing, that pdf is contradictory

Cheers

Greg Locock

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

 

[davidmandis]

My system is similar to the one shown on page 5 of the paper attached except that it does not contain a damper, the spring is non-linear and the G forces are higher.
Greg what are you referring to when you say it is contradictory?

 

[GregLocock]

Some of the numbers didn't make sense.



Cheers

Greg Locock

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

 

[electricpete]

A small beginning: The triangle pulse in acceleration is integrated twice to give an expression for displacement:

http://home.comcast.net/~electricpete/eng-tips/trianglepulse/TrianglePulseTwiceIntegrated.pdf

The displacement corresopnding to your acceleration curve (assuming 0 initial velcoity and displacement) is:
([1/3*Gm/tp*t^3, t < 1/2*tp]
[Gm*t^2-1/3*Gm/tp*t^3-1/2*Gm*tp*t+1/12*Gm*tp^2, t < tp],[1/2*Gm*tp*t-1/4*Gm*tp^2, otherwise])

It is not required if you do it numerically, but it's done anyway.

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[electricpete]

Attached is the problem solved using the vba program that I mentioned.

http://home.comcast.net/~electricpete/eng-tips/trianglepulse/TriPulseRKF45.xls

You can see the graphical results. If you'd like to change the parameters and re-run the simulation, I think the controls in the "main" tab are self-explanatory. The model parameters are input in green. The three buttons give you control over clearing the workspace, running, running the program, and generating a chart. The chart is not great, you'll probably need to work to adjust the scales to your liking since the various parameters have much differnet scales.

The main things to pay attention to are the input parameters (do you agree?) and the model. The model is as I described above:
State variables:
dm = displacement of mass (man)
vm = velocity of mass (man)

Derivatives of state variables
d/dt(dm) = vm
d/dt(vm) = (1/M) * f(dg - dm)
where f is the nonlinear function giving force depending on difference in displacment between ground (airplane) and mass (man).

By this weekend, I will post a simpler version that does the simplest possible integration (Euler method)
xk = xk-1 + deltaT * dx/dt (xk-1)

x represents either (both) x1 = dm and x2=vm

k refers to the time step. step k is time deltaT later than k-1.

I will also provide a matlab solution. That is unless someone beats me to it (you guys are welcome to post your own for comparison).

What do you guys think? Is the model (derivative of the two state variables) correct?


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[GregLocock]

I was specifically inclined not to trust an analytical model, because plastic deformation of crash structures, which is what that k graph seems to represent, are rather hard to analyse. Simple question - if the wheel starts to move back down, does the force go to zero? negative? The graph implies that the same force is still exerted in the same direction. A good trick, if you can pull it off. Usually that requires a fundamental force.

ep's ss gives reasonable answers in agreement with a crude handcalc.







Cheers

Greg Locock

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[electricpete]

I'm sure Greg is right. Building a good model is the key.
I defer to his comments on that.

Sticking with the original model, here is an Euler simulation which I made a simple as humanly possible. I think will be much much easier to follow for both the calculations and the graphs.
(by the way it's 6 MB)

http://home.comcast.net/~electricpete/eng-tips/trianglepulse/TriPulseEuler.xls

The input parameters in green are the same.
Colmun C is time and the other columns stretch to the right.

Row 12 are the starting conditions at t=0.

At row 13 we look a small time (h=0.000005 seconds later).
Work through the columns of row 13 from left to right:

Column D: a_plane is acceleration of the ground=plane - that's out triangle pulse.

Column E: v_plane is velocity of the ground=plane. For that we need to integrate a. For integral we take the previous value (from row 13) and add to the time step (h) multipled by the slope. The slope is a_plane. I used the average of a across that interval. So the formula is
= v_plane(0) + [a_plane(0)+a_plane(h)]/2 * h
=+E13+(D14+D13)/2*h
That's all there is to the integration. We'll use that same trick over and over.

Column F: d_plane is displacement of the ground=plane. We do integration in the identical manner as we did in column E. In fact all I did was copy the forumla and let the relative references update.

Column G: delta_spring is the compression of the spring
delta_spring = d_plane - d_man

Column H: This is the spring force. Implemented by an if/then statement which determines which side of the cutoff we're one and applies the correct forumula

Column I: a_man - This is acceleration of the man. It is given by spring force over man's mass.

Column J: v_man - velocity of the man is integral of acceleration, same as before.

Column K: d_man - displacement of the man is integral of velocity, same as before.

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[davidmandis]

wow thank you so much!! I really appreciate your help! Your logic seems to be right and excel calculations also look good. I will go through it again and try to validate it........

Thanks,
David

 

[electricpete]

You're welcome. It's a good chance for me to try out some things.

Below is program and output for simulation of the same system using Matlab's ODE45

http://home.comcast.net/~electricpete/eng-tips/trianglepulse/TriPulseMatlab1.doc

The results appear very close to before. I went ahead and did numerical integration of the plane acceleration (rather than using the algebraic solution for plane displacement) so I could have access to plane velocity for plotting.

Very similar to the simpler euler excel file output, there is a separate plot for acceleration and for velocities and for displacement. That works out to be the easiest way to display the plots based on the relative magnitudes (if plotted a and d on same scale, you couldn't see a).

There is a glitch at the peak of the triangle of the acceleration curve which I generated after the simulation using the t_ode vector of time values representing the times established by the variable step algorithm in ode45. Looking at the numeric output, the program uses a pretty big step size in stepping through that area (0.002 seconds), which is why when the plot connects those points on each side of the peak with a straight line, a big chunk of the peak gets cut off. My variable step size vba program used a lot finer step size through that region. I'm not sure why that is, since I tried to set them up with the same tolerance as an input to the algorithm. I think maybe I failed in that effort and misinterpretted the matlab tolerance argument.

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[electricpete]

If the link gets hung up when you open it, alt-tab and find the pane where comcast is asking for a password. Hit cancel and you should be able to see the file

(does it ask you guys for password when you try to access it?)

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[IRstuff]

nope... downloads fine.

TTFN

FAQ731-376

 

[electricpete]

ok, that's good to know. There must be a cookie on my computer that tells them I'm the webpage owner or something.

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[electricpete]

I am not sure what were the important output parameters of this simulation. But in retrospect, we know we could determine the max force and max accel without doing any simulation at all:

Since the spring is the only force acting on the man:

Fmax = 3000 pounds.
Accelmax = 3,000 lbf / 170 lbm = 17.7 g's
That agrees with the plot output which shows somewhere around 170 m/sec^2 (conversion is 9.8)

I'm going to assume that the velocity of the man and displacement of the man were important also. Otherwise, I feel kind of silly for spending all that time on the solution ;-)

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[davidmandis]

Pete thank you again for your valuable help!! the matlab code runs fine and the next step for me would be to add some damping to the system to represent the crash of the aircraft floor.
The force on the man will depend on the spring force which is an independent function of time. So even though we know the acceleration with time , we do not know how much spring force with respect to time. So assuming here a maximum spring force worked out but it might not always be the case. The max spring spring force depends on the relative displacement of the man and the plane which might or might not always be the maximum spring force in the graph. But, the displacements and velocities of the man are also important parameters for me. So in short your effort was definitely not in vain.

Thanks,
David

 

[electricpete]

That makes sense. I feel better now.

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[Tunalover]

Why solve a problem that has already been solved? This problem is so common that it's probably available from any good vibration handbook (e.g. Cyril Harris' book). Also Tom Irvine (

http://www.vibrationdata.com)

must certainly have a downloadable program to solve the problem (for a $40 one-time fee).

I attempted to solve the SDOF spring-mass-dashpot problem with a half-sine impulse at the base. I found that the solution is best done with Laplace transforms but I gave up because the algebra got out of hand. That is likely to happen with this problem as well because the forcing function has to be written with Heavyside (or Dirac delta) functions before the problem can be solved. Of course MathCAD, MatLab, or other software is good because they can provide a closed-form solution.

I eventually found the solution to my problem in Harris' book.




Tunalover

 

[electricpete]

it's probably available from any good vibration handbook (e.g. Cyril Harris' book).

I have Harris' Shock and Vib Handbook 5th edition and as far as I can tell it doesn't provide any closed form solution of this problem (Note the problem includes triangle acceleration pattern and non-linear spring = different spring constant in two regions). Could you give us a specific paragraph? Or a paragraph some some other reference that addresses this specific problem (including triangle accleration input and non-linear spring)?

I attempted to solve the SDOF spring-mass-dashpot problem with a half-sine impulse at the base. I found that the solution is best done with Laplace transforms but I gave up because the algebra got out of hand. That is likely to happen with this problem as well because the forcing function has to be written with Heavyside (or Dirac delta) functions before the problem can be solved. Of course MathCAD, MatLab, or other software is good because they can provide a closed-form solution.

I eventually found the solution to my problem in Harris' book.

Yes, I remember that: thread384-157462

Your problem was a linear system. And the only abrupt change in behavior of the forcing function (other than t=0) was at the end of the pulse.

This problem has two abrupt changes in behavior of the forcing function (one at the peak of the triangle and one at the end of the pulse) AND another abrupt change in spring constant (non-linear). And we don't initially know at what point in time that change in spring constant occurs, or even whether it's before or after the change in behavior of the forcing function (peak of the acceleration triangle). So while your problem could be broken into two time regions (separate closed-form solution in each region) whose time boundaries were known before the problem started, this problem is broken into three regions and the time of one of the boundaries (where the spring changes behavior) is not known until we solve at least the first region using the specific numerical values for this problem. In short, this problem (triangle pulse on nonlinear system) certainly can be solved analytically (vs numerically), but it is a lot more challenging than it is for your problem of the half-sin pulse on the linear system.

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[electricpete]

One more thing to mention about the difference in solutions of these two problems.

In the case of the half-sin problem, one solution method was Laplace transforms. The change in forcing function at the end of the sin pulse was taken care of by representing it in Laplace transform domain. So once transformed the system and the input into Laplace transform domain, we have one Laplace transform solution for displacement and we inverse Laplace transform to find displacement as a function of time. Note there was no need to consider two different time regions in this process... the Laplace transform of the input funciton took care of it.

For the case of the triangle-input non-linear spring system, we can again represent the input function in Laplace transforms, but it doesn't do us much good. We can't apply that to the system accross the entire time range because it's anon-linear system. We would still have to break it into multiple time ranges even if we used a Laplace trasnform solution.

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[Tunalover]

electricpete-
My mistake. I missed the nonlinear part. But I still recommend using a math program that can provide a symbolic solution. Of course I wouldn't want to miss out on all the studly mathematics in this forum!


Tunalover

 

[davidmandis]

Hi everyone, I seem to be having a little conceptual problem. The spring-mass model here was for a helicopter seat (where mass represents the mass of man and seat) and the acceleration pulse was taken from a military standard for dynamic testing of a helicopter seat.
Now, I cant understand if applying an acceleration pulse to the fixed support (the fixed support represents the helicopter sub-floor) of the mass-spring system will be the same as applying an initial velocity to the mass? The acceleration pulse is given for a time period of 0.054 sec but applying an initial velocity will not need a transient analysis and this has got me confused..........can someone pls help in figuring this out.

Thanks,
David