Vega is an expendable launch system in use by Arianespace. Its jointly developed by the Italian Space Agency and the European Space Agency. First time it was launched from Guiana Space Center on 13 February 2012. In this example we show how to simulate rocket launch trajectory. VEGA rocket data from VEGA Users Manual/ The full sub-orbital ascent profile simulation coming soon !!

clc; clear all; global m0 g0 T A Cd rh0 H0 Re hgr_turn tf md % Launch Site: Guiana Space Center Alt = 1; %[m] Alt above sea level % VEGA Rocket m_stage_gross = [95796, 25751,10948];% 1st, 2nd,3d % First stage(Solid Fuel) m_prop = 88365; % [kg] Propellant mass Isp = 280 ; % [s] Specific impulse d = 3; % [m] Diameter g0 = 9.81; % [m/s^2] Constant at its sea-level value m0 = 137000; % [kg] Initial mass A = pi*d^2/4; % [m^2]Frontal area Cd = 0.5 ; % Drag coefficient,assumed to have the constant value rh0 = 1.225; % [kg/m^3] H0 = 7500; % [m] Density scale height Re = 6378e3; % [m] Earth's radius hgr_turn = 200; % [m] Rocket starts the gravity turn when h = hgr_turn tburn = 106.8; % [s] Fuell burn time, first stage md = (m_prop)/tburn; % [kg/s]Propellant mass flow rate T = md*(Isp*g0); % [N] Thrust (mean) mf = m0 - m_prop; % [kg] Final mass of the rocket(first stage is empty) t0 = 0; % Rocket launch time tf = t0 + tburn; % The time when propellant is completely burned %and the thrust goes to zero t_range = [t0,tf]; % Integration interval % Launch initial conditions: gamma0 = 89.5/180*pi; % Initial flight path angle v0 = 0; % Velocity (m/s) % Earth's Rotation considered in eq of motion. x0 = 0; % Downrange distance [km] h0 = Alt; % Launch site altitude [km] vD0 = 0; % Loss due to drag (Velocity)[m/s] vG0 = 0; % Loss due to gravity (Velocity)[m/s] state0 = [v0, gamma0, x0, h0, vD0, vG0]; % Solve initial value problem for ordinary differential equations [t,state] = ode45(@RocketDynEq,t_range,state0) ; v = state(:,1)/1000; % Velocity [km/s] gamma = state(:,2)*180/pi; % Flight path angle [deg] x = state(:,3)/1000; % Downrange distance [km] h = state(:,4)/1000; % Altitude[km] vD = -state(:,5)/1000; % Loss due to drag (Velocity)[m/s] vG = -state(:,6)/1000; % Loss due to gravity (Velocity)[m/s] plot(t,h,'b'); hold on; grid on; plot(t,h,'.b'); title('Rocket Dynamics,VEGA Rocket sub-orbital ascent profile(1st stage P80)'); xlabel('time[s]'); ylabel('Altitude[km]'); text(80,5,'smallsats.org','Color',[0 0 1], 'VerticalAlignment','middle',... 'HorizontalAlignment','left','FontSize',14 ); % VEGA Rocket: First Stage P80 fprintf('\n VEGA Rocket: First Stage P80\n') fprintf('\n Propellant mass = %4.2f [kg]',m_prop) fprintf('\n Gross mass = %4.2f [kg]',m_stage_gross(1)) fprintf('\n Isp = %4.2f [s]',Isp) fprintf('\n Thrust(mean) = %4.2f [kN]',T/1000) fprintf('\n Initial flight path angle = %4.2f [deg]',gamma0*180/pi) fprintf('\n Final speed = %4.2f [km/s]',v(end)) fprintf('\n Final flight path angle = %4.2f [deg]',gamma(end)) fprintf('\n Altitude = %4.2f [km]',h(end)) fprintf('\n Downrange distance = %4.2f [km]',x(end)) fprintf('\n Drag loss = %4.2f [km/s]',vD(end)) fprintf('\n Gravity loss = %4.2f [km/s]',vG(end)) fprintf('\n');

VEGA Rocket: First Stage P80 Propellant mass = 88365.00 [kg] Gross mass = 95796.00 [kg] Isp = 280.00 [s] Thrust(mean) = 2272.67 [kN] Initial flight path angle = 89.50 [deg] Final speed = 1.76 [km/s] Final flight path angle = 80.11 [deg] Altitude = 71.51 [km] Downrange distance = 54.55 [km] Drag loss = 0.05 [km/s] Gravity loss = 1.04 [km/s]

RocketDynEq.m

function dfdt = RocketDynEq(t,y) global m0 g0 T A Cd rh0 H0 Re hgr_turn md v = y(1); % Velocity gm = y(2); % Flight path angle x = y(3); % Downrange distance h = y(4); % Altitude vD = y(5); % Velocity loss due to drag vG = y(6); % Velocity loss due to gravity % Equations of motion of a gravity turn trajectory m = m0 - md*t; % Vehicle mass % else % m = mf; % Burnout mass % T = 0; % No more thrust is generated % end g = g0/(1 + h/Re)^2; % Gravitational variation with altitude rh = rh0*exp(-h/H0); % Atmospheric density exponential model D = 1/2 * rh*v^2 * A * Cd; % Drag force % Rocket starts the gravity turn when h = hgr_turn if h <= hgr_turn % Vertical flight dv_dt = T/m - D/m - g; dgm_dt = 0; dx_dt = 0; dh_dt = v; dvG_dt = -g; else % Gravity turn dv_dt = T/m - D/m - g*sin(gm); dgm_dt = -1/v*(g - v^2/(Re + h))*cos(gm); dx_dt = Re/(Re + h)*v*cos(gm) + 463*sin(gm); dh_dt = v*sin(gm) + 463*cos(gm); % Adding earth's rotation speed dvG_dt = -g*sin(gm); % Gravity loss rate [m/s^2] end dvD_dt = -D/m; % Drag loss rate [m/s^2] dfdt = [ dv_dt,dgm_dt, dx_dt,dh_dt, dvD_dt, dvG_dt]'; return

When do you show “The full sub-orbital ascent profile simulation coming soon !!” ?

Thank you

Good question…don’t know yet.. it took a bit longer back then so I did not finish…

I run your code, and i plot(x,h) and appear that you did something wrong, because it’s doing a gravity turn upside down., i.e. the x and y axis are changed!

You right. I will double check and modify the post.Thanks for pointing this out !!

You welcome. Say me something when you modify the code. Thank you

you can’t add earth rotation in gravity turn equations. if you remove them you will have correct results

I’m not sure about that. I think it should be considered somehow. If you launch from equator the speed relative to inertial frame 463 m/s should be there. Thats why most of launch sites are close to earth’s equator. I also checked the code again. The axis seems correct. Also refer to the http://www.arianespace.com/launch-services-vega/VEGAUsersManual.pdf may be it will helpful. See the picture i uploaded above.

The rocket data after the burn of first stage.

Calculations:

Altitude = 71.51 [km]

Downrange distance = 54.55 [km]

Final speed = 1.76 [km/s]

Their value

VEGA manual

Altitude = 44 [km]

Downrange distance = 63 [km]

Final speed = 1.87 [km/s]

I assume the difference could be the drag coefficient. I just assumed Cd = 0.5 a constant value.

What you think..?

Hi,Thank you for your program.

during the run of program it occure a error at line 74,75,79 and 85.

” ??? Error: File: RocketDynEq.m Line: 74 Column: 39

Unbalanced or unexpected parenthesis or bracket.

??? Error: File: RocketDynEq.m Line: 74 Column: 39

Unbalanced or unexpected parenthesis or bracket.”

please help me

it is strange Sahep. Several people run it without an issue including me :). Did you created a RocketDynEq.m function in a same folder ?

hi . thanks for your web.

i need some help to simulating reentry or reusable vehicles. for example Apollo and …

can you help me?

To give you some boost try this books, Atmospheric and Space Flight Dynamics,Modeling and Simulation with MATLAB and Simulink and Claus Weiland, Computational Space Flight Mechanics…

They have some examples and codes to look at …I’m quite busy lately but let me know how it goes .. i can try to be helpful !