AAE 450 Project Bellerophon

A.6.2.2.1 Balloon and Ground Launchp.g.1

A.6.2.2.1 Balloon and Ground Launch

The steering law, which controls the direction of the velocity vector of the launch vehicle, is one of the most crucial challenges in our project. A slight change in the steering law affectsthe ∆Vdrag, ∆Vgravityandthe eccentricity of the orbit; therefore, the rocket will not obtain an acceptable orbit without a good sub-optimal steering.

Although we eventually incorporate the spherical Earth model, the aerodynamic drag due to the atmosphere, and the gravity field as a function of altitude, the starting point of the construction of the steering law was to consider the flat Earth problem without drag.This simplified problem is a well-defined two point boundary value problem, which is analytically solvable by forming Hamiltonian and applying Euler-Lagrange equations, Transversality condition and Weierstrass condition. The optimal solution obtained is the Linear Tangent Steering Law1:

(A.6.2.2.1.1)

where is the steering angle, is time, and and are the coefficients.

The Linear Tangent Steering Law is the optimal steering law for the flat Earth when there is no atmosphere,which is analogous to “the flat Moon”. As the steering laws that private companies and the governmental space agencies actually use are notavailable to the public, we decide to apply the Linear Tangent Steering Law for our ground and balloon launches. We recognize that it is not the optimal steering law any more when we apply it to the spherical Earth model with atmosphere; however, we also assume that the difference is small enough to treat the Linear Tangent Steering Law as a good sub-optimal steering law.

We implement the steering law in our ordinary-differential-equations-solvers and numerically integrate our equations of motion for each stage. Also, our rocket flies vertically without any steering for the first ten seconds of the first stage, so we do not need to implement the steering law for the very first part of the flight.

Figure A.6.2.2.1.1 shows how we measure the steering angles and depicts the final steering angles of each stage, which numerically define our steering law. Figure A.6.2.2.1.2 shows the steering law versus time when the final steering angles at first, second and third stages are 40°, -20° and -50° respectively. We should note that the initial steering angle is 88° rather than 90° since the tangent function is undefined at 90°.

Fig. A.6.2.2.1.1: Schematic of the steering angle at the end of each stage.

(Amanda Briden)

Fig. A.6.2.2.1.2: Sample of the plot of steering law versus time.

(Amanda Briden)

By changing the final steering angles at each stage degree by degree, we are able to find the sub-optimal steering law that makes it possible to attaina nominal orbit with the eccentricity of as small as 0.0055.In Section A.6.2.3, we will discuss how we actually deal with the computationally expensive process of obtaining the sub-optimal steering law, which requires running the entire trajectory code once for each set of the final steering angles,by using a normal PC in the year 2008 rather than an expensive super computer.

In the process of choosing the launch type, ground launch or balloon launch, we needed to compare the corresponding ∆Vdrag and ∆Vtotal. We had not hadthe final structural configurations yet when we did the analysis on the week 5 of the project, and we did not have a sub-optimized trajectory for each case either. However, the following results are still valid since the trend never changes for our launch vehicles regardless of the modifications since the week 5.

Table A.6.2.2.1.1 Delta V Comparison
Payload [kg] / Launch type / ∆Vdrag / ∆Vtotal / Units
0.2 / Balloon / 21 / 10,027 / m/s
1.0 / Balloon / 21 / 10,011 / m/s
5.0 / Balloon / 20 / 9,932 / m/s
0.2 / Ground / 2,904 / 14,033 / m/s
1.0 / Ground / 2,899 / 13,978 / m/s
5.0 / Ground / 2,875 / 13,711 / m/s

Table A.6.2.2.1.1 shows ∆Vdrag of the ground launch is bigger than that of the balloon launch by the factor of 150, and ∆Vtotal of the ground launch is significantly bigger since ∆Vdragis the major source of ∆Vtotal. Consequently, it is obvious that the balloon launch has a very big advantage in reducing ∆Vdrag, although the accuracy in the numerical values is not perfect since we use the preliminary analysis on the week 5.

Our final configurations of small, medium and big launch vehicle, whose payloads are 0.2 [kg], 1 [kg] and 5 [kg], for the balloon launch have∆Vdragof 6 [m/s], 6 [m/s] and 4 [m/s] and require ∆Vtotalof 9,313 [m/s], 9,379 [m/s] and 9,354 [m/s] respectively.These results validate that our preliminary analysis on the week 5 are numerically close to our final results; therefore, we confidently conclude that the balloon launch is better than the ground launch in terms of ∆Vtotal, which exponentially affects the total cost of our launch vehicles.

References:

1Longuski, J.M. “AAE 508 Optimization in Aerospace Engineering Lectures,”PurdueUniversity, West Lafayette, IN, January 2008.

Author: Junichi Kanehara