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

 

[davidmandis]

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

 

[electricpete]

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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[electricpete]

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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[davidmandis]

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

 

[davidmandis]

I have attached the reference used to get the acceleration crash pulse and time period data. Fig. 1 (a) on page 3 shows the crash pulse and also the change in velocity to be 42 ft/sec.

 

[electricpete]

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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[electricpete]

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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[davidmandis]

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

 

[davidmandis]

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

 

[vtmike]

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

 

[electricpete]

Hi Mike. vt = Virginia Tech?

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.

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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[vtmike]

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?

 

[electricpete]

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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