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Rocket Dynamics, VEGA Rocket sub-orbital ascent profile(1st stage P80)
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