Pitch Analysis of Rigid Body Rocket (Part 1: Equations of Motion)

Review of Dynamic Modeling and Ascent Flight Control of the ARES-I Crew Launch Vehicle Section 3.2

Soooo as my first real post I will be going over the basic equations of a rigid rocket during the ascent phase. In particular I will be looking at the following thesis available online(Dynamic Modeling and Ascent Flight Control of the ARES-I Crew Launch Vehicle) and under section 3.2 of the paper. The author of the paper, Wei Du, does a good job explaining the basics of this analysis and I will be recreating this in here.

Like I have said before there is no real progression on topics with this blog. So there is some assumption that the reader will know certain topics. I believe this is a good starting point to hash out the basics of rocket dynamics. Now bear in mind that this is not going to be a full examination of the dynamics of the rocket. We will be simplifying the analysis to only include a 2-D analysis, and will involve only involve one rotational degree of freedom and a single translational degree of freedom.

I have copied the drawing from the thesis that shows all the terms in use.

Rocket_2D
Figure 1. 2D Rocket

The equations of motion for the rocket are shown below.

Equations_2D_Rocket
Figure 2. Equations of Motion for 2D Rocket

Rotational Equation of Motion

Before going too deep into the simplified equations, lets derive how we get them from the full Euler Rotational and translational equations of motion. The full equations of rotational motion are shown below.

Euler Equtions of MotionSince we are only talking about a 2D rocket, two of the equations vanish. For example let the y and z equations vanish. Since there is no rotation about the y and z axis, we are left with:

EOM_2D_Euler

Translational Equations of Motion

So what about the translational equations!!!

Well not to worry. We use Newtons 3rd law F=ma to figure out the drift in the Z direction.

Translational Equtions of Motion

And once again since we are only talking about 2D motion, two of the equations disappear.

Now that we have figured out the equations of motion, we should focus on the forces being felt by the rocket.

Thrust Force

The thrust force is generated by the actual propulsion system of the rocket. In the ARES-I, the propulsion was performed by a solid fuel rocket. I believe that the rocket is the same as the space shuttle solid rocket boosters. A neat thing about this rocket (and it is the same with many others) is that it has the ability to direct the thrust by a certain amount δ. In future posts we may cover the mechanics of thrust vector control, but for now lets just say that the thrust vector system is able to direct the thrust instantaneously by a certain angle. I believe that in the ARES-I the maximum displacement is about ±10°.

Aerodynamic Forces

A rocket that is travelling through the atmosphere will have aerodynamic forces applied to it. This is just like an airplane. The forces can be simplified  by assuming that the aerodynamic forces will be applied to a location on the rocket called the center of pressure. The amount of force or torque applied to the rocket will be a function of mach or velocity and the angle of attack. Just in case you are wondering what is mach, mach is the velocity of the rocket (or any object travelling through a fluid) divided by the speed of sound in that fluid. Fortunately aerodynamicists are very smart people and have come up with tables that describe the expected force of moment at different flight conditions.

So the forces affecting the rocket are Mα*α  Nα*α. Shown below are the descriptions of the aero coefficients. Notice that V is the velocity of the rocket, S is the reference area of the rocket and Cnα is a table of values that a smart aero guy has provided us with.

aeroforces_2D_rocket

State Space Model

So now that we have defined what our forces are, it is time to combine everything into state space format.

state_space_2D_rocket.png

Now you may be thinking why do we want the equations in state space format? Well once we have this format, we can find the eigenvalues of the system and see if it is stable or not. Turns out that since we have no controller, the system is unstable. We can plug it into matlab or some other software and easily run a simulation with it. But what I really like about this format, is that we can figure out a controller that will control the rocket. That is going to be part II of this series.

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