What is the explanation for the torpedo's behavior during its free fall?

[edmundfo]

In an attempt to teach myself rigid body kinematics I'm trying to model the trajectory of a torpedo falling through air using Matlab. I encountered somewhat strange results from a full 6-DOF parametrization, and consequently reduced the problem to a 3-DOF parametrization using only horizontal position x, vertical position z and pitch angle theta.

I assume quadratic damping. Furthermore I assume that this quadratic damping is substantially smaller along its body-centric x-axis than along its body-centric z-axis. The modelling can be found in the two attached Matlab files. It is (with gross simplifications and some changes of convention) based on Fossen & Fjellstad: Position and Attitude tracking of AUV's: A Quaternion Feedback Approach, IEEE-JOE vol 19. No. 4 Oct. 1994. The chosen values for damping parameters (xUU, zWW and mQQ) are based on Timothy Prestero's modelling of an AUV in water (master's thesis at WHOI), but I cannot see any reason why his approach should not be valid for an object moving in air as well. All units are standard SI units (kg, m, s).

The weird result is that the torpedo after something like 60 seconds of free fall reaches a vertical velocity close to zero. This only happens when x-damping (xUU) is much smaller than z-damping (zWW). If you run the script with xUU closer to zWW, you will see the torpedo reach a terminal velocity around 50 m/s downwards, which I find much more reasonable.

So my questions would be:

1. Is it plausible that a torpedo falling in air can behave like this?
2. If not, then why?

a) Any bugs in the code that I have overlooked?
b) Because I have (deliberately) ignored lift-forces or any other forces?
c) Because quadratic damping isn't applicable for high velocities?
d) Because damping coefficients for water cannot be transferred to damping coefficients for air just like that?
e) Because of numerical inaccuracies due to transformations between body and world frame etc, possibly related to the Coriolis terms which I find a bit confusing?
f) Any other reason?

Any thoughts on this would be greatly appreciated.

 

[SteamKing]

edmundfo:

I can't say why your simulation is giving you these strange results. You didn't link to your MATLAB files.

You say that after 60 s of free fall, the velocity of the torpedo is zero. Is it possible that instead of the velocity being zero at this time, the change in velocity with respect to time is zero? In other words, the torpedo has reached what is known as terminal velocity.

You realize that if falling bodies reached zero velocity before returning to earth, there would be multitudes of objects suspended over ground, incapable of further movement.

 

[edmundfo]

I tried to attach the Matlab files. For some reason they were not displayed. Hopefully you will find them here View attachment script_test_3dof.m and here View attachment rotmat2d.m.

I did not say that the norm of the velocity reaches zero. Rather, the torpedo attains a rather high horizontal velocity when the vertical velocity approaches zero. Then as the horizontal velocity is being damped, the vertical velocity increases again. So the rod is moving all the time. Its behavior does not seem impossible to me, only implausible.

What actually seems weird to me is that a vertically falling object should attain such a high horizontal velocity after a while, irrespectively on what its initial horizontal velocity is. I would think that this happens because I implement the dynamic equations in body frame, and Coriolis force is not taken properly into account. However, if it was so, I would also expect the degree of weirdnesss to disappear with a sufficiently small stepsize in the Runge-Kutta method, which is not the case.