Katyusha

[Yekaterina]

Let's model the trajectory of a self-propelled rocket! I have made a start here, now it's up to you diligent people to solve the equations numerically

 

[Yekaterina]

Let's model the trajectory of a self-propelled rocket! I have made a start here, now it's up to you diligent people to solve the equations numerically

 

[Yekaterina]

The assumptions of the models above are:

1. Gravitaitonal acceleration is constant. 

2. The rocket experiences an air resistance force which acts antiparallel to the direction of its velocity and is dependent on the square of its velocity (Newton Drag), which dominates only if the Reynold's Number is large. 

3. Fuel is ejected at a constant rate and the exhaust velocity is always constant. 

4. The direction of thrust force is always parallel to the velocity. 

 

If 3 and 4 hold, then the rocket may eventually reach a point where the vertical component of thrust is no longer able to counter-balance its weight (because the direction is too horizontal) and the rocket will accelerate downwards, into a dive of no return. This is disastrous and we don't want it. 

 

[Yekaterina]

katyusha.py

My first attempt. Python screws up your calculation with inaccurate decimals noises. 

A Katyusha without any gyroscope will lead to disaster for sure...

[Loki1950]

You might find solving it numerically with a the fourth order Runge-Kutta method more accurate it converges quickly too.We use it in Vegastrike for each physics frame.This is to calculate the position of about several hundred ships each frame as well as the planets in that solar system.The problem you describe reminds me of an article from early 70's in the "the British Interplanetary Society" journal.

Enjoy the Choice     

[Yekaterina]

katyusha.py

Updated version: corrected some mistakes and added a graph plotting functionality

 

[Yekaterina]

data.txtdata.txt 

katyusha_acceleration.py katyusha.py

Please ignore the first 4 files!

Edited August 22, 2022 by Yekaterina

[Yekaterina]

Air resistance forces acting on the rocket should be a mixture of linear terms (due to friction between the side of the rocket casing and the air) and quadratic terms (due to displacing air in front of it). Also there is the Coanda effect.

 

[Yekaterina]

4 hours ago, Helicity said:

linear terms

Yes, viscous force. There is also sheer pressure. 

However, I believe that drag forces should be significantly smaller than the thrust forces. In order to not over-complicate the differential equations, I only kept the dominating terms. 

On second thought, the rocket can exhibit a translational motion as well. It should be placed in a launch tube or a rack before firing, using the rack as a guide to gain some initial velocity. The momentum built up inside the rack should be able to prevent it rotating. 

The thrust would not always be constant; for solid fuels, the thrust is a programmable function of time as you can change the shape of the fuel surface. Thrush is directly proportional to surface area of fuel exposed to air. 

[Yekaterina]

 

In this footage, the missile was aimed almost vertically but quickly gained a horizontal component of velocity, causing its final direction to be somewhat parallel to the ground. This was because of short launch rack and the slight deviation off the vertical gave it a significant horizontal component of speed. Its direction of thrust is parallel to its main axis. 

These Russian missiles are following the trajectory I described:

 

However, the traditional Katyusha has gained a significant initial velocity and momentum in its launch direction, so the acceleration due to gravity or thrust in the vertical direction is no longer significant.