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Response to impact pulse
3

Response to impact pulse

Response to impact pulse

(OP)
   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

RE: Response to impact pulse

2
IF the acceleration represents ths input to the system AND you had a linear system, you could decompose the input acceleration as you have done and add the "responses" to the acceleration.  But that does not apply for a non-linear system which you have.

It also seems unusual to me to have acceleration an an input.  Is there a mass in the system?  If not, displacement is second integral of acceleration and force is known function of displacement.. seems like a pretty trivial problem. More details of the total problem to be solved would help.

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RE: Response to impact pulse

(OP)
    The system actually represents a landing gear with a composite crash tube to absorb the energy of impact. Now the input would be a force since I know the rate at which the system (system includes mass and spring) deccelerates, so the mass times the g force would give the force applied which would be a function of time and will have the same time step as the acceleration.
    I have converted the non-linear spring behavior into a linear response. But I am not sure how to combine the results of the steps as shown in first post.


RE: Response to impact pulse

Just trying to follow the logic.  

If everything you said is true ( you know the acceleration and therefore the spring must absorb force F(t)=m*a(t) where a(t) is as given in the jpg file above), then you can trace F(t) and determine a displacement.  But displacement of what?  You told me you already know acceleraiton.

How can you know a(t) ahead of time if you haven't factored the non-linear spring into the problem yet?  Doesn't it affect the acceleration?

I would not consider that representing the non-linear curves as two straight lines makes it linear.  A linear system has output proportional to input (and that type of linearity is a prerequisite to using superposition).  It would be represented by a straight line that goes through 0.  

I think if you can break the problem into separate time intervals, then it could be linear within each time interval.  In the time when the spring is in the linear region, using your linear spring constant F = k1.  In the time when the spring is in the flat region, model it as a static applied force F = k2.

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RE: Response to impact pulse

Correction:
"...then you can trace F(t) and determine a displacement"
Should have been:
"...then you can double integrate F(t)/m and determine a displacement"

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RE: Response to impact pulse

(OP)
  You are right about the non-linearity...the spring is indeed non-linear and i meant it is linear if we consider it in two different sections but I dont think that is the right approach.
  What I mean by acceleration as input is that the whole system including the mass and the spring is deccelerating at the rate shown in the figure but I do not know the acceleration of the mass in the system itself. In other words the input to the system is acceleration which can be converted into a force. I have shown the spring mass system in the figure attached.
  A transient analysis would have worked out if the spring was liner but I don't know how to include the non-linearity of the spring in the system.

David

RE: Response to impact pulse

I strongly recommend that you build a time based model that just builds up the displacements from the velocities from the accelerations from the forces.

Cheers

Greg Locock

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RE: Response to impact pulse

Yes.  Break it up into regions.  The breakpoints between the regions are tp/2, tp,  and anytime the spring changes between it's two regions.

If you have numerical values AND units, it might be a little easier to see where the breakpoint might fall.

In your file impact_dynamics.jpg, the horizontal axis is labeled displacement in one figure and load in the other. Supposed to be displacement on horizontal axis in both?

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RE: Response to impact pulse

I meant cruder than that. Every 0.01 seconds at a rough guess. It takes a while to write your first spreadsheet like that but it is a very useful technique , and can be extended to including multidof systems.

Cheers

Greg Locock

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RE: Response to impact pulse

Yes, a numerical integration of the initial value problem would be easier and save a lot of algebra.  

for Dave's benefit, here's one way to set it up:

State variables:
dm = displacement of mass
vm = velocity of mass

Derivatives of state variables
d/dt(dm) = vm
d/dt(vm) = (1/M) * f(dg - dm)

where f is the nonelinear spring function which gives force as function of displacement
dg is displacement of the ground node which is a known function of time that we can determine ahead of time from two integrations of the function a(t) plotted above.   (actually of course ground doesn't move but we can consider the initial inertial frame of the mass where the ground does move).

The simplest numerical integration is Euler's method
xk = xk-1 + deltaT * dx/dt (xk-1)

I have a spreadsheet at home where I programmed in vba a variable step size Runge-Kutta Feldberg 4th order algorithm with 5th order correction of step size.... seems to work just as good as Matlab ODE45.  If you provide the numerical values and units I will set it up and post it.

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RE: Response to impact pulse

By the way, it seems like one assumption needs checking. Is the spring force really a function of displacement only with no dependence on time history... or does it act differently if you apply the force slowly or suddenly (i.e. some damping)?

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RE: Response to impact pulse

If you throw out the 5th order variable step size correction and stick with 4th order fixed step size, the fourth order Runge Kutta (RK4) is very very easy.  It is a helluva lot easier than RKF45 and a helluva lot more accurate than Euler... RK4 is O(dt^4) while  Euler is O(dt^1) where dt is step size.

RK4 is as follows:

Let  f(x,t) be the computed derivative

f1 = f(x,t)
f2 = f(x+ f1*dt/2, t+dt/2)
f3 = f(x+ f2*dt/2, t+dt/2)
f4 = f(x+ f3*dt, t+dt)

estimate of x(t+dt) is
x(t) + dt*(1/6) (f1+2*f2+2*f3+f4)

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RE: Response to impact pulse

(OP)
Thank you Pete and Greg for your replies, I really appreciate it. I think I will solve it first using Greg's method and use the results as validation for the MATLAB results.
Pete, I am trying to figure from your directions how to solve my problem in MATLAB using ode45. Do you know of any reference I could use and would love to see how you solve it using a spreadsheet. I have attached the problem statement with all the units and other details I could gather. Let me know if I missed out any details.

Thanks again,
David  

RE: Response to impact pulse

What is the value and units of M?

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RE: Response to impact pulse

Also can you respond to this query:

Quote:

In your file impact_dynamics.jpg, the horizontal axis is labeled displacement in one figure and load in the other. Supposed to be displacement on horizontal axis in both?

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RE: Response to impact pulse

Sorry, I didn't see your last attachment. Got it now. Thanks.

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RE: Response to impact pulse

You show a single degree of freedom system but list two masses.

Is the spring located between the aircraft and the occupant and the aircraft plays the role shown in the SODF diagram as  ground (except that it's accelerating with the curve given)?

Or are the aircraft and occupant masses lumped together as the mass in the SDOF and the ground accelerating with respect to their initial inertial reference frame?

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RE: Response to impact pulse

(OP)
yes the spring is located in between the aircraft floor and the occupant and the aircraft floor is accelerating according to the given curve. So when the aircraft crashes, the ground imparts a force of 48 G's to the aircraft floor. one thing I missed out is that the force should be multiplied by 42 to get the max force Fm. So that will make,
Fm = 4738230 N = 1075197 lbs-force

David

RE: Response to impact pulse

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: Eng-Tips.com Forum Policies for tips on how to make the best use of Eng-Tips.

RE: Response to impact pulse

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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RE: Response to impact pulse

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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RE: Response to impact pulse

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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RE: Response to impact pulse

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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RE: Response to impact pulse

(OP)
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

RE: Response to impact pulse

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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RE: Response to impact pulse

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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RE: Response to impact pulse

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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RE: Response to impact pulse

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 winky smile

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RE: Response to impact pulse

(OP)
  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

RE: Response to impact pulse

That makes sense. I feel better now.

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RE: Response to impact pulse

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 (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

RE: Response to impact pulse

Quote:

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)?

Quote:

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: Response of single DOF spring-mass-dashpot model to half-sine pulse

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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RE: Response to impact pulse

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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RE: Response to impact pulse

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

RE: Response to impact pulse

(OP)
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

RE: Response to impact pulse

(OP)
Ok I take that back.........applying an initial velocity will also require a transient analysis since the spring is nonlinear........but will applying an intial velocity to the mass be same as applying the acceleration crash pulse to the fixed support?

David

RE: Response to impact pulse

I'm not familiar with how these things are specified, but I can offer some thoughts.

There are two inertial reference frames you could solve the problem in. One where the plane and passenger are initially at rest (as we did) and one where the real outside ground is at rest (the normal reference frame).  

Force and acceleration and spring compressions are all the same in either frame.  The difference in frames is the initial velocity of the plane with respect to the real outside ground is at rest.

The acceleration curve seems intended to replicate the acceleration experienced during a real life situation. For the acceleration curve to be an accurate replication of a crash starting at initial velocity v0, the integral under the acceleration curve must be equal to the initial relative velocity between the two frames.  There is also the shape of the curve to consider... don't know where that came from but I'm sure some thought was put into it.

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RE: Response to impact pulse

To make it a little more concrete, the integral under your  acceleration curve is Gm* tp/2.

It could correspond to a crash starting from initial velocity Gm*tp/2  (assuming the shape has been properly defined).  It could not correspond to a crash starting from any other initial velocity.

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RE: Response to impact pulse

(OP)
I just noticed a change in velocity, deltaV = 42 ft/sec....which is equal to ((48G's*32.2*0.054sec)/2) corresponding to the triangular acceleration crash pulse also given in the specifications, but will this change in velocity be the same as initial velocity?
I am trying to verify my answer using a FEA software (ANSYS). The FEA model consists of a mass and a nonlinear spring with load-deflection characteristics defined. Now since I cannot apply an acceleration pulse to the base, I wanted to apply an initial velocity to the mass. Do you think I should apply the change of velocity as the initial velocity or would this approach be wrong?

Thanks,
David   

RE: Response to impact pulse

I don't have a clube about Ansys so maybe someone else can answer better.

But my thoughts:
There are three nodes of interest:
Xm = the man
Xp = the plane
(spring attached between man and plane)
Xg = the ground

If you want an initial velocity, it is not between Xm and Xp, it would be between Xm = Xp (initially moving together initially at same velocity) and Xg.  Then you have to accomplish deceleration of Xp according to an inverted (negative) triangle deceleration curve.  The acceration, velocity and displacement of Xp are all known by starting with initial conditions (initial velocity V0) and applying the deceleration curve (integrate to give velocity and integrate again to get displacement).  So if you have capability to pre-program a profile of acceleration or velocity or displacement of the plane position Xp, you should be able to do it.

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RE: Response to impact pulse

Did I say clube?  Oh what I wouldn't give for an edit button.  I guess that would be the contraction of "clue" and "'bout". As in: I don't have a clube out nottin.

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RE: Response to impact pulse

(OP)
hmmmm that is what i am trying to figure out....i.e. if I can pre-program an acceleration, disp, vel profile in ANSYS. will let you know as soon i figure it out.

Thanks,
David

RE: Response to impact pulse

(OP)
Hi,
   Sorry for the late reply. I forgot to mention that I got the same results from ANSYS for this simulation. So Pete's approach and code is right!

Thanks,
David

RE: Response to impact pulse

Hi Pete,
I was looking at your MATLAB file for this problem because I am trying to solve a similar problem with an acceleration pulse, and found it difficult to visualize the flow of your code.
First the input to the function x_dot.m would be time t, and state x
These are the state variables,
% state variables:
  % x1 is displacement of man
  % x2 is velocity of man
  % x3 is displacement of the plane
  % x4 is velocity of the plane.
Next you calculate time derivatives of state variables in which,
% dx1/dt = x2
% dx2/dt = (1/Mman) * Fspring = (1/Mman) * f(delta)
  % f is nonlinear spring function
  % f(delta1) = k1*x1 for delta<k_cutoff, k2 for delta1>k_cutoff
  % delta = x3-x1 is the compression of the spring
  % dx3/dt = x4
  % dx4/dt = acceleration of plane = triangle pulse = given function of time

Now, when you calculate the acceleration of man i.e. dx2/dt, why is the input acceleration not being considered? I mean when we write the equations of motion for the two parts of the acceleration pulse we include the input acceleration on the right side of the equation right?
Also why have you calculated acceleration of man again in the main function (main.m)? Is it different than the one calculated in the x_dot.m function?

I have attached my equations of motion for this problem. Can you please tell me if they look right, and where I am going wrong in my reasoning?

Regards,
Mike

RE: Response to impact pulse

Hi Mike. vt = Virginia Tech?

Quote:

Now, when you calculate the acceleration of man i.e. dx2/dt, why is the input acceleration not being considered?
The force acting on the man is the non-linear spring.  The "input" acceleration is applied at the position of the plane (corresopnding to your location z(t)), not the position of the man (corresponding to your u(t)). Draw a free body diagram.

Quote:

Also why have you calculated acceleration of man again in the main function (main.m)? Is it different than the one calculated in the x_dot.m function?
xdot computes displacements and velocities.  Once those are solved the problem is known.  I certainly could have computed the accelerations from the output of x_dot.m using finite difference equations (Ak = (Vk+1 - Vk)/dt)
I chose an alternate method to compute them.


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RE: Response to impact pulse

I am actually from Texas A&M University

I apologize for my ignorance in advance. This is my second year of engineering & dynamics is still a new topic for me.

You said that x_dot.m calculates displacements and velocities. But it is mentioned in your code that x2 is the velocity of man, so won't that make dx2/dt (i.e. state_dot(2)) the acceleration of man?

RE: Response to impact pulse

Quote:

You said that x_dot.m calculates displacements and velocities. But it is mentioned in your code that x2 is the velocity of man, so won't that make dx2/dt (i.e. state_dot(2)) the acceleration of man?
Yes.  But the ode45 routine does not return the derivative of the state variables ... it returns the state variables themselves. So the output of ode45 is
x1(t), x2(t), x3(t), x4(t) etc
(in this case x1(t),v1(t),x2(t),v2(t))

It may seem a little strange ....after all, we fed matlab an function for the derivatives of the state variables.   Those derivatives are in fact computed by matlab and used in solving the initial value problem, but the derivatives themselves are not stored/saved.  All that is stored/returned is the values of the state variables as a function of time.

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