ROCKET EQUATIONS (V)
(Numerical Approximation for the Rocket velocity)
Because of the complex nature of these equations, we fail to find an analytic solution, so we opt for the numerical solution to these problems.
The numerical solution of rocket equations is an important tool in the design and analysis of rocket missions, as it allows engineers to predict the performance of a rocket and make necessary adjustments to ensure a successful mission.
Katherine Johnson (1918–2020) used Euler’s Method at NASA in 1961 to perform the trajectory analysis that enabled astronaut Alan Shepard to make the first human space flight. (As told in the book “The Hidden Figures”)
A BRIEF IDEA OF THE EULER METHOD
The Euler method is a numerical integration technique that can be used to solve the equations of motion for a rocket. The basic idea is to approximate the derivatives of the state variables (such as velocity, position, and mass) using finite differences and then use these approximations to update the state variables at each time step.
Methodology:
Here is a general outline of how to use the Euler method to solve the rocket equation:
1.Define the state variables: Start by defining the state variables that you want to solve for, such as velocity, position, mass, and thrust.
2. Write the equations of motion for the rocket.
3. Discretize the time: Divide the time interval over which you want to solve the equations into a series of smaller time steps.
4. Approximate the derivatives: Use the Euler method to approximate the derivatives of the state variables at each time step.
5. Update the state variables: Use the approximated derivatives to update the state variables at each time step.
6. Repeat steps 4th and 5th steps: Repeating steps 4th and 5th is essential for each time step until you have reached the end of the time interval.
7.Analyze the results: Analyze the results of the Euler method to see if the rocket is performing as expected. If necessary, adjust the time step or the initial conditions and repeat the calculation.
Representation of Euler’s method
The Euler’s Method can be represented as:
It is the general form of Euler’s method.
Here velocity is denoted by “v” and time by “t”.
$\Delta t $ is the time step.
v_{n+1} denotes the current velocity
v_{n} denotes the previous velocity.
Using model values:
Using the data for a model rocket:
{M_{R}}(t)=(0.05–0.002t)kg
The rocket is assumed to start at 50 grams.
Mass is variable so the expression becomes (M-\lambda*t)\\
$\lambda=\frac{dM_{R}}{dt}$=$-0.002$kg/sec\\
$M_{R}(1)=0.048kgs$ i.e 48 grams
R=15 Newton
k=0.02 which is the drag coefficient.
Plugging the values in (2) we get (Check ROCKET EQUATIONS IV for clarity)
$(0.05–0.002t)\frac{dv}{dt}+v(-0.02v)=14.51+0.196t-0.002v²$\\
$\frac{dv}{dt}=\frac{14.51+0.196t-0.02v²+0.002v}{0.05–0.002t}$\\
We set up the initial conditions:
$v(t_{0})=v_{0}$
$v(0)=0$
\Delta t =0.01\\;
Manual Implementation of Euler method in Rocket equation:
Isolating $\frac{dv}{dt}$, we solve the differential equation with changing time.
We get the following graph showing how the change in velocity over time becomes constant.
This shows the stability of the solution thus obtained using the Euler method. In the next blog, I would try to show its stability and mathematical software. Till then, Take Care❤
Following are my references:
“Improved Accuracy in Numerical Integration of the Rocket Equation using the Euler Method” by J.A.G. Ortega and W.C. Rheinboldt (Journal of Spacecraft and Rockets, 2003)
“Numerical Solution of the Rocket Equation using the Euler Method” by J.M. Anderson (Journal of Spacecraft and Rockets, 1979)
https://www.coursera.org/learn/computers-waves-simulations/home/welcome
https://www.coursera.org/learn/newspace-access-to-space/home/week/1
