Dynamics Project #1
2011.12. 23
Team1
Dong-Kyeong Lee
Eunjeong Hyeon
Uihwan Choi
Jeongyeon Lee
1. Problem definition
At t=0, the thrust, T(t), begins. The launching is at angle 𝜃0 to the ground, which is the
surface of the Earth. At time t = 𝑡0 , the fuel runs out, so the rocket falls back down to the
ground. During this, all the reaction forces acting upon the rocket are the same, but there is no
longer a thrust force. Find the exact coordinate of the rocket landing. The initial mass of the
rocket (including the fuel) is 𝑀0 and the total mass of the fuel is 𝑚0 . Although there may
need to be several assumptions made to simplify and solve this problem, use as few
(significant) assumptions as possible.
2. Method approach
(1) Assume that the surface of the Earth is perfectly round.
(2) Use the drag coefficient of cylinder.
(3) Assume that the Earth does not undergo rotation.
(4) Assume that there is no centripetal force.
(5) The initial launch velocity is 𝑣0 , initial launch angle is 𝜃0
(6) The density is changed with altitude. (We used U.S. Standard Atmosphere).
(7) There is no lift and angle of attack on the rocket. (That is, the moving direction of the
rocket is parallel to wind (drag) direction)
(8) The rocket does not escape the atmosphere of Earth. Therefore the time of fuel
consumption does not exceed the minimum period needed for the rocket to go out into space
and the rocket does not have enough speed to perform satellite motion.
(9) Our rocket model is KSLV-1. It is simple (or streamlined) cylinder with maximum
diameter 2.9m.
** <Step 1>~<Step 7> contains history of our projects. These are in the previous reports of
our team.
<Step 8> Considering mass consumption, thrust, drag, curved ground (and
gravity) together – Summation of works of <Step 5>, <Step 6>, <Step 7> and
considering more conditions
Works of <Step 5>, <Step 6> and <Step 7> are individual solution for differential equation
of certain condition. In each step above, we were able to find real mathematical solution.
However we should sum all of these works to get better model for real rocket trajectory. So in
this step, we want to make perfect differential equation model that includes above 3 steps
together.
Before we start, it is good to review the basic differential forms for polar coordinate in
<Step 2>.
Then, at first, we assume that the mass of rocket linearly decreases until the constant thrust
exists. From the <Step 6> discussion and aerospace text book, we know the formula for the
thrust:
𝑇 = 𝑚̇ × 𝑉𝑒
We can consider the direction of the thrust is the same with rocket’s velocity vector, that is,
⃗⃗⃗𝑇
= ⃗⃗⃗
(treated as unit vector)
|𝑇|
|𝑣|
𝑣
So, we can get the vector form of thrust like:
⃗⃗⃗⃗⃗
𝑣
|𝑣|
In this case, we should treat 𝑣 as polar coordinate vector like:
𝑑𝑟
𝑑𝑟
𝑑𝜃
𝑣 =
= 𝑟̂ + 𝑟 𝜃̂
⃗ = 𝑚̇ × 𝑉𝑒
𝑇
𝑑𝑡
𝑣 = √(
𝑑𝑡
𝑑𝑡
𝑑𝑟 2
𝑑𝜃 2
) + (𝑟 )
𝑑𝑡
𝑑𝑡
From this, we can write the thrust as
𝑚̇ × 𝑉𝑒
𝑑𝑟
𝑑𝜃
⃗ =
𝑇
( 𝑟̂ + 𝑟
𝜃̂)
2
2 𝑑𝑡
𝑑𝑡
√(𝑑𝑟) + (𝑟 𝑑𝜃)
𝑑𝑡
𝑑𝑡
− − − − (1)
Second, we assume that the earth is (2 dimensional) circle, and also the direction of gravity
is toward that center of circle. From this, we can write the gravitational force in polar
coordinate like below (gravitational acceleration is not constant):
𝐺𝑀𝑚
⃗⃗⃗⃗
𝐹𝐺 = − 2 𝑟̂ − − − − (2)
𝑟
Third, the drag force is proportional to the square of velocity and the direction of it is
opposite to velocity vector. From the flight dynamics, we know drag force is Fd =
1
𝜌𝑣 2 𝑆 𝐶𝐷 . That is,
2
𝑑𝑟 2
𝑑𝜃 2 𝑑𝑟
𝑑𝜃
⃗⃗⃗⃗
𝐹𝐷 = −𝑘𝑑 𝑣 𝑣 = −𝑘𝑑 √( ) + (𝑟 ) ( 𝑟̂ + 𝑟
𝜃̂) − − − − (3)
𝑑𝑡
𝑑𝑡
𝑑𝑡
𝑑𝑡
1
, where k d = 2 𝜌 𝑆 𝐶𝐷 . At this point, we should consider that the air density 𝜌 is not
constant with altitude. To consider this fact, we used U.S. Standard Atmosphere data to
interpolate air density value at specific altitude. ( http://www.engineeringtoolbox.com/standardatmosphere-d_604.html)
We know the Newton’s 2nd law in this case:
𝑑2 𝑟
⃗ + ⃗⃗⃗⃗
𝑇
𝐹𝐺 + ⃗⃗⃗⃗
𝐹𝐷 = 𝑚𝑎, where 𝑎 = 2
𝑑𝑡
2
2
𝑑𝑟
𝑑𝜃
Then, we put (1) ~ (3) to this Newton’s 2nd law. That is, (v = √( 𝑑𝑡 ) + (𝑟 𝑑𝑡 ) )
𝑚̇𝑉𝑒 𝑑𝑟
𝑑𝜃
𝐺𝑀𝑚
𝑑𝑟
𝑑𝜃
( 𝑟̂ + 𝑟
𝜃̂) − 2 𝑟̂ − 𝑘𝑑 𝑣 ( 𝑟̂ + 𝑟
𝜃̂)
𝑣 𝑑𝑡
𝑑𝑡
𝑟
𝑑𝑡
𝑑𝑡
𝑑2𝑟
𝑑𝜃 2
𝑑2 𝜃
𝑑𝑟 𝑑𝜃
= 𝑚 (( 2 − 𝑟 ( ) ) 𝑟̂ + (𝑟 2 + 2
) 𝜃̂) − − − − (4)
𝑑𝑡
𝑑𝑡
𝑑𝑡
𝑑𝑡 𝑑𝑡
Also, we should consider mass ‘decreasing’. So we consider m as 𝑚 = 𝑚0 − 𝑚̇𝑡, a
function of time.
Solving for (4) in terms of 𝑟, 𝜃 will show us the trajectory of the rocket in polar coordinate.
But it is very complicated for us to solve directly. So, to solve it numerically, we will use the
Runge-Kutta 4th order method. <Step 9, 10> will cover this.
<Step 9> Transformation of equitation (4) to 1st order differential form to use
Runge-Kutta 4th order method
Equation (4) is composed of 3 variables (𝑟, 𝜃, 𝑡). However, in view of programming, this
equation is composed of below:
𝑑𝑟 𝑑𝜃 𝑑 2 𝑟 𝑑 2 𝜃
𝑟, 𝜃, ,
,
,
,𝑡
𝑑𝑡 𝑑𝑡 𝑑𝑡 2 𝑑𝑡 2
t is independent variable, and others are dependent variable to t.
We should eliminate 2nd order differential part because RK4 method need 1st order form.
Therefore we define 4 variables(x1, x2, x3, x4) that are correspondent like:
x1 = 𝑟
x2 = 𝑑𝑟/𝑑𝑡
{
x3 = 𝜃
x4 = 𝑑𝜃/𝑑𝑡
Substituting these to (4) and sorting (4) with 𝑟̂ , 𝜃̂ terms lead us to:
𝑑𝑥1
= 𝑥2
𝑑𝑡
𝑑𝑥2 1
𝑥2 𝐺𝑀𝑚
= (𝑘𝑚 − 2 − 𝑘𝑑 𝑣𝑥2 ) + 𝑥1 𝑥42
𝑑𝑡
𝑚
𝑣
𝑥1
− − − −(5)
𝑑𝑥3
= 𝑥4
𝑑𝑡
𝑑𝑥4
1
𝑥1 𝑥4
2𝑥2 𝑥4
=
(𝑘𝑚
− 𝑘𝑑 𝑣𝑥1 𝑥4 ) −
{ 𝑑𝑡
𝑚𝑥1
𝑣
𝑥1
1
where 𝑣 = √𝑥2 2 + (𝑥1 𝑥4 )2 , 𝑘𝑚 = 𝑚̇𝑉𝑒 , 𝑘𝑑 = 2 𝜌 𝑆 𝐶𝐷 , 𝑚 = 𝑚0 − 𝑚̇𝑡
(5) is a kind of
d𝐗
dt
= 𝑭(𝑿) form. That is, we are ready to operate RK4 method using (5).
3. Results
Solving Equitation (5) through using Runge-Kutta 4th order method
We already wrote the Matlab code for solving equation (5) with Runge-Kutta method. Our
remaining task is to consider the reasonable constants. For doing this, we consider the model
of Naro rocket.
The reference is Dynamics lecture,
http://www.astronautix.com/lvs/kslvi.htm,
http://en.wikipedia.org/wiki/Thrust_specific_fuel_consumption,
http://en.wikipedia.org/wiki/Drag_coefficient,
http://www.engineeringtoolbox.com/standard-atmosphere-d_604.html, etc.
m0 (rocket total mass) = 140 t, Fuel mass = 130t
𝑚̇ =384.6kg/s (treating it as a constant)
𝑤𝑓𝑢𝑒𝑙 ×101972
(𝑚̇ =
, where 𝐼𝑠𝑝 = 338𝑠𝑒𝑐, wfuel = 130(× 103 𝑘𝑔) × 9.8, )
𝐼
𝑠𝑝
Ve = 3315m/s (fuel exit velocity relative to rocket)
Cd = 0.04(streamlined body), 0.82(long cylinder)
2.9 2
S = π ( 2 ) = 6.6𝑚2 (reference area of rocket)
G = 6.6742 × 10−11 (Gravitational constant)
M = 5.9742 × 1024 (Earth mass)
We take the density data from U.S. Standard Atmosphere. To get density value at specific
altitude, we used Cubic Spline method (Matlab built-in function: interp1(…,’spline’)).
Our rocket is launched at the North Pole. Also, we choose some value for initial velocity and
launch angle. That is,
v0 = 1000 𝑚/𝑠
θ0 = 𝜋/2.5 (𝑟𝑎𝑑)
Then we can get the result like below:
If we change the launch angle more small, that is,
θ0 = 𝜋/2.7 (𝑟𝑎𝑑)
We get
4. Analysis
Our rocket model considers many realistic things. Round Earth, change of gravity and
density with altitude, fuel consumption, thrust, aerodynamic drag force, etc. However our
model has also limits.
First one is initial velocity and initial angle. In real case, initial velocity is always zero.
But our model starts with 1000m/s velocity – about Mach 3. Also, the initial angle is flexible
in real case, but to get better result using our code, we took just some specific angle value.
Second one is about constant thrust. In real case, thrust value is changing. But our model
uses constant mass consumption and constant fuel exit velocity – it means constant thrust.
Other limits are about external wind, aerodynamic force (wave drag, induced drag, lift, etc.),
etc. These are also to be considered.
Now, our team has some question about this project.
1. In plotting the rocket trajectory using Matlab, there are many cases that shows the
results about ‘lnf’, ‘NaN’ especially when we use a little bigger values of initial
condition, drag coefficient. For example, if we choose initial velocity as 2000m/s,
there are many ‘NaN’ results in Workspace and graph shows strange trajectory of
rocket(changing its direction to 180 degree at short time, exceeding 10^100 m/s, etc.).
We wanted to find any error in our code or our deriving, but we cannot find. We now
think it is a little bit related to the fact that the values we are using in this model are
very big (Earth radius, Earth mass, etc.). We want to know the way to avoid this kind
of situation.
2. We want to know the analysis method to treat external wind, aerodynamic force,
fluctuation thrust, etc.