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
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
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.
=====================================
Eng-tips forums: The best place on the web for engineering discussions.
RE: Response to impact pulse
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
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.
=====================================
Eng-tips forums: The best place on the web for engineering discussions.
RE: Response to impact pulse
"...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"
=====================================
Eng-tips forums: The best place on the web for engineering discussions.
RE: Response to impact pulse
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
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
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?
=====================================
Eng-tips forums: The best place on the web for engineering discussions.
RE: Response to impact pulse
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
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.
=====================================
Eng-tips forums: The best place on the web for engineering discussions.
RE: Response to impact pulse
=====================================
Eng-tips forums: The best place on the web for engineering discussions.
RE: Response to impact pulse
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)
=====================================
Eng-tips forums: The best place on the web for engineering discussions.
RE: Response to impact pulse
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
RE: Response to impact pulse
=====================================
Eng-tips forums: The best place on the web for engineering discussions.
RE: Response to impact pulse
=====================================
Eng-tips forums: The best place on the web for engineering discussions.
RE: Response to impact pulse
RE: Response to impact pulse
=====================================
Eng-tips forums: The best place on the web for engineering discussions.
RE: Response to impact pulse
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?
=====================================
Eng-tips forums: The best place on the web for engineering discussions.
RE: Response to impact pulse
Fm = 4738230 N = 1075197 lbs-force
David
RE: Response to impact pulse
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
Greg what are you referring to when you say it is contradictory?
RE: Response to impact pulse
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
htt
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.
=====================================
Eng-tips forums: The best place on the web for engineering discussions.
RE: Response to impact pulse
http://h
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?
=====================================
Eng-tips forums: The best place on the web for engineering discussions.
RE: Response to impact pulse
ep's ss gives reasonable answers in agreement with a crude handcalc.
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
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://h
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.
=====================================
Eng-tips forums: The best place on the web for engineering discussions.
RE: Response to impact pulse
Thanks,
David
RE: Response to impact pulse
Below is program and output for simulation of the same system using Matlab's ODE45
http://hom
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.
=====================================
Eng-tips forums: The best place on the web for engineering discussions.
RE: Response to impact pulse
(does it ask you guys for password when you try to access it?)
=====================================
Eng-tips forums: The best place on the web for engineering discussions.
RE: Response to impact pulse
TTFN
FAQ731-376: Eng-Tips.com Forum Policies
RE: Response to impact pulse
=====================================
Eng-tips forums: The best place on the web for engineering discussions.
RE: Response to impact pulse
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
=====================================
Eng-tips forums: The best place on the web for engineering discussions.
RE: Response to impact pulse
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
=====================================
Eng-tips forums: The best place on the web for engineering discussions.
RE: Response to impact pulse
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
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.
=====================================
Eng-tips forums: The best place on the web for engineering discussions.
RE: Response to impact pulse
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.
=====================================
Eng-tips forums: The best place on the web for engineering discussions.
RE: Response to impact pulse
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
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
David
RE: Response to impact pulse
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.
=====================================
Eng-tips forums: The best place on the web for engineering discussions.
RE: Response to impact pulse
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.
=====================================
Eng-tips forums: The best place on the web for engineering discussions.
RE: Response to impact pulse
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
RE: Response to impact pulse
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.
=====================================
Eng-tips forums: The best place on the web for engineering discussions.
RE: Response to impact pulse
=====================================
Eng-tips forums: The best place on the web for engineering discussions.
RE: Response to impact pulse
Thanks,
David
RE: Response to impact pulse
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
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
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.
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.
=====================================
Eng-tips forums: The best place on the web for engineering discussions.
RE: Response to impact pulse
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
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.
=====================================
Eng-tips forums: The best place on the web for engineering discussions.