Motion of the rocket is simulated using two numerical analysis methods. From the simulation different parameters such as altitude, velocity, acceleration and range for initial fuel flows were calculated. Two numerical methods, Euler’s integration and 4th order Runge-Kutta integration are used for calculating different parameters for the vertically launched rocket. The efficiency and the accuracy of the methods were compared. It was found out that the 4th order Runge-Kutta is more efficient than Euler’s integration method for the given time step. Also for the rocket given the optimal initial fuel mass flow rate for attaining the highest altitude is found to be 35.5 Kg/S which gives an altitude of 1362594 m.
1.0 INTRODUCTION
Rockets are important part of space travelling. But rockets are also in many other important applications. The basic understanding of the physics behind rocket motion is easier to understand as it obeys Newton’s laws of motion. But this understanding is not enough to design and test a rocket as there are other critical parameters that must be taken into account. It is critical to know the trends in the rocket parameters such as its velocity, distance travelled and acceleration in order to design rocket for its appropriate application. For this simulating the motion of the rocket and analysing the data measured is one of the efficient ways. In this report two numerical methods, Euler’s integration and 4th order Runge-Kutta integration are used for calculating different parameters for a vertically launched rocket. This report also discusses the trends in behaviour of some of the parameters measured and in particular the efficiency and accuracy of the both methods.
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2.0 THEORY
Figure x: Forces acting on Rocket
From Figure X and using Newton’s Laws of Motion the Net Force acting on the Rocket is given by the equation: Fnet = Thrust- Drag- Fgravity(1)
Where,
Fnet = Total Force on rocket
Fgravity = Force exerted on rocket by Earth or Weight of rocket(as indicated in Figure 1)
Thrust = Force from burning rocket fuel
Drag = Air resistance Force on rocket
But,
F=Ma(2)
Therefore,
a=FnetMrocket(3)
So,
Acceleration of Rocket = Force on rocket / Mass of Rocket
But the mass of the rocket changes with fuel flow rate. i.e. mass changes with time. So acceleration can only calculated by predicting the altitude (distance rocket travelled from launch to return) and velocity with respect to time. The density of atmosphere changes as the altitude increases. This means that the drag acting on the rocket is not constant and changes with the altitude as is the function of density and velocity. See equation (5) and (6) Density ρ, =1.2* e-y7000(5)
Drag Fd, =0.1*ρ*v*v(6)
3.0 PROCEDURE
For calculating the both methods Visual Basic Application (VBA) with Microsoft Excel was used. Codes for implementing Euler’s method and 4th order Runge-Kutta method was done in VBA and results were tabulated and graphs were plotted in Excel. See Appendix x for the Codes for Euler’ integration and 4th order Runge-Kutta. The initial values of the parameters were set to initialise the calculations; When time t =0,
Altitude y, = 0
Velocity v, = 0
Initial fuel flow rate Q0, = 20 Kg/S
time step h, = 1 (this was changed depending on the method used)
4.0 RESULTS
4.1 Euler Method
The results below are obtained by using Euler’s integration method for the motion of rocket. Figure 2 shows the graph of maximum altitude against a range of time step, 0.01 to 1. From this graph the time step that is needed for the error in the maximum altitude to be approximately 0.1% was found to be 0.7, which is indicated by the straight horizontal line drawn in the graph. As the time step increases the altitude decreases which is an indication of divergence from the actual result.
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Figure 2: Maximum Altitude vs. Time Step for 0.1% error in altitude
Figure 3: Altitude vs. Time
Figure 3 shows graph of altitude y, measured over the time t, using time step h of 0.7.
Figure 4: Velocity vs. Time
The figure 4 shows the graph of velocity v for time step of 0.7, measured against time t.
Figure 5: Acceleration vs. Time
The figure 5 shows the graph of velocity v for time step 0.7, measured against time.
Figure 6: Maximum Altitude vs. Fuel flow rate
Figure 6 shows the graph of maximum altitude y, measured against different values of fuel flow rate Q, at time step of 0.7. The maximum altitude was achieved at 35.5 Kg/S flow rate and the altitude achieved at this flow rate was 1362594 m. See section 5.2 for more detailed discussion about the trend in figure
5. 4.2 Runge-Kutta Method
The results below are obtained by using Runge-Kutta’s integration method for the motion of rocket. Figure 7 shows the graph of maximum altitude against a range of time step, 1 to 15. This range was chosen as it showed the divergence from the actual result is negligible for very small time steps. From this graph the time step that is needed for the error in the maximum altitude to be approximately 0.1% was found to be 14, which is indicated by the straight horizontal line drawn in the graph.
Figure 7: Maximum Altitude vs. Time Step for 0.1% error in altitude
Figure 8: Altitude vs. Time
Figure 8 shows graph of altitude y, measured over the time t, using time step h of 14.
Figure 9: Velocity vs. Time
Figure 9 shows graph of Velocity v, measured over the time t, using time step h of 14.
Figure 10: Acceleration vs. Time
Figure 10 shows graph of acceleration a, measured over the time t, using time step h of 14.
5.0 DISCUSSIONS
5.1 Velocity & Acceleration
Velocity vs. Time graphs for both Euler (Figure 4) and 4th order Runge-Kutta (Figure 9) clearly shows the expected trend. Both graphs are similar for both methods. The velocity increases linearly at launch of the rocket as a result of the huge thrust. During this linear transition the thrust force is higher than the combined drag and gravitational force. This transition is reflected in the acceleration graphs Figure 5 and Figure 10 for Euler and 4th order Runge-Kutta respectively. The constant linear increase in velocity results in the increase acceleration, which results in the small oscillation in the acceleration vs. time graph. As the velocity increases the air resistance increases which in turn increases the drag force on the rocket.
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The combined drag force and gravitational force is higher than thrust which results in the deceleration of the rocket. This could be clearly seen from both graphs as the velocity decreases constantly till the rocket reaches maximum attitude. At this stage all the fuel in the rocket is used and the velocity becomes zero. After the rocket reaches the maximum altitude the rockets start to return back to earth, i.e. fall to earth. At this period of time no thrust force is produced by the rocket. The only forces acting on the rocket is drag and gravitational force.
During this period of time the velocity increases constantly. The negative increase in velocity shows that the rocket is travelling in the opposite direction. The point from the rocket reaching maximum altitude till the rocket reaches the surface of earth the change in velocity is constant. Since, a = dv/dt acceleration is zero at this transition period in both graphs. At the end of the motion of the rocket there is sudden increase in acceleration. This is due to the sudden drop in linear velocity on the returning to the earth, i.e. reaching the lowest point for altitude. This sudden and huge liner velocity drop results huge oscillation in acceleration vs. time graph.
5.2 Fuel Flow Rate
The trend in figure 6 is due to the fact that the atmosphere air density changes with the altitude. The range taken was from the 20 Kg/S to 50 Kg/S. As mentioned in section 4.1 the initial fuel flow mass rate which attained the highest altitude if 35.5. The air density nearer to the earth’s surface is larger than the air density above a certain altitude. So the thrust produced by the rocket is proportional to the air density. The higher the air density the more the thrust produced. To attain the maximum altitude for the given quantity of fuel the rocket must gain enough momentum from the thrust at higher air density area and also have fuel to move forward when enters the lesser air density area. Increasing the initial fuel flow rate more than 35.5 Kg/S decreases the fuel available to move forward at the lesser air density area. Also decreasing the initial fuel flow rate decreases the momentum gained which takes covering the higher air density area over a long period of time
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Efficiency OF EULER and 4th Order Runge-Kutta Method
By comparing the figure 2 and figure 7 it could be clearly understood that the 4th order Runge-Kutta method is more efficient and accurate for large time steps compared to Euler’ integration method. The Euler’s integration tends to diverge from the actual result as the time step increases. But the Runge-Kutta only start to diverge from the actual result for very large time step when compared to Euler’s integration.
6.0 CONCLUSIONS
The aim of the simulation was achieved. For calculating the different parameters more efficient way is to use 4th Order Runge-Kutta as it is stable and the divergence from the actual value is negligible for same time step used in Euler’s integration. Since 4th order Runge-Kutta is more stable and accurate for larger time steps this method is more ideal to use for analysis. Also the air density could be modelled more accurately in order to achieve more accurate data.
