4. Propulsion
4.1 Launch Vehicle
The Daedalus, in both the MSP and OSP configurations, will use the Delta IV-Heavy as its launch vehicle. NASA’s OSP request for proposal demanded the use of either the Delta-IV or Atlas-V Evolved Expendable Launch Vehicle. Both vehicles have been designed in more than one configuration based on customers’ lifting performance requirements. Along with the Lockheed’s Atlas V-H configuration, Rocketdyne’s Delta IV-H possesses the largest payload lifting capability of any currently manufactured launch vehicle. It was quickly agreed upon that in order to maximize the capabilities of the spaceplane a maximum weight limit would be needed. Table 4.1 compares the two vehicles. The slightly better lifting performance of the Delta IV-H coupled with its higher technological standard made it the best candidate for launching Daedalus.
Atlas V Heavy / Delta IV HeavyPayload to 90 / 19,000 kg / 20,800 kg
Cost / $140-170 million / $140-170 million
Launch Sites / Vandenberg/
Cape Canaveral / Vandenberg/
Cape Canaveral
Booster Stage / RD-180 (2) / RS-68 (2)
First Stage / RD-180 / RS-68
Second Stage / RL10A-4-2 / RL10B-2
Table 4.1: Comparison of Atlas V and Delta IV EELV
Thestringent reaction and turn-around times demanded by the Department of Defense for the MSP design would automatically disqualify both the Atlas and Delta rockets, which rely on cryogenic propellants in all their stages. The only storable heavy launch vehicle with the requisite capabilities is Lockheed’s Titan IV-B. The vehicle is currently out of production and only a handful remains to be launched. Zero boil-off technologies in cryogenics are in a state of advancement. However, most research has been done on scales much smaller than that of an all-cryogenic vehicle. Due to this dilemma it was decided that, for the sake of trajectory analyses, the Delta IV-H would also be used to boost the MSP mission.
4.2 Boost Trajectory
The trajectory analysis was done for two different target orbits: for the OSP, the orbit of the International Space Station and for the MSP, a 185km polar orbit. The computer code that was used to perform the analysis was written in FORTRAN. Although codes from previous semesters were available for modification, the Daedalus code was written from scratch in order to gain a better understanding of the launch trajectory analysis.
4.2.1 Vehicle Equation’s of Motion
The boost trajectory for the Delta IV-H is based on the equations for flight derived in Chapter 2 of Vihn’s Hypersonics (Appendix D1: Eqs 1-6). Vihn defines the vehicle’s trajectory with respect to a rotating Earth, the axes through the center of rotation of which form the inertial frame of reference. The method for determining the boost trajectory consists of plotting an uncontrolled trajectory or ‘ideal trajectory’ from the launch point to the target orbit with the assumption of no cross-winds. The plotting of this trajectory requires a systematic variation of variables until the required solution to Vihn’s equations is reached. The cross-winds are then factored into these equations and the thrust of the vehicle is vectored to negate any deviation from the ideal trajectory. Vihn divides his equations into two groups of three, namely the force equations and the kinematic equations. The force equations (Appendix D1: Eqs 1-3 ) define the rate of change of velocity, flight path angle and heading angle which are dependent on the dominant forces acting on the vehicle specifically the vehicle’s thrust, aerodynamic forces of lift and drag, and the force of gravity. For the case of a rocket, the coefficient of lift is negligible and thus also the force of lift. Also, thrust vectoring is assumed only to be a means for counteracting trajectory disturbances due to winds during the low altitude portion of flight. Hence they are neglected until the factoring in of cross winds. The terms containing the angular velocity of the Earth are the Coriolis terms and, although negligible for most cases, they should be retained for an accurate analysis.The kinematic equations (Appendix D1: Eqs 4-6) relate the motion of the vehicle to spherical coordinates. They allow the position of the vehicle to be tracked as a function of its altitude, latitude, and longitude. To account for mass variation as propellant leaves the vehicle, the mass flow rate equation is added to the system (Appendix D1: Eq 7)
The simultaneous solutions of these ordinary differential equations with respect to time provide the values necessary to define the trajectory of the vehicle. These equations are solved in FORTRAN with the help of a subroutine entitled ‘RK4.F’, which is based on the Runge-Kutta numerical method of integration. Before an accurate solution can be obtained from these equations, the proper initial conditions of the trajectory must be defined.
4.2.2 Initial Conditions
The simultaneous solution of the differential equations of motion requires the use of proper initial conditions. As the ideal trajectory must have no control interference, the path of the vehicle is dependent solely on the user’s choice of initial conditions. While initial conditions of latitude and longitude are preset by the location of the launch site, the vehicle’s initial flight path angle and heading angle must be varied in order to achieve the required final solutions. Also, a proper variation in the vehicle’s thrust profile is integral to attaining the required orbit.
4.2.3 Final Solutions
The target orbit is assumed to be circular at the required altitude. Hence the velocity needed to remain in the orbit can be calculated simply from the altitude and Appendix D: Eq 8. The inclination of the orbit is given by Appendix D1: Eq 9, which is dependent on the vehicle’s heading angle and latitude. Both these angles are output from the code. Hence, the goal of the trajectory code is to end up with the proper velocity, altitude, and inclination angle.
4.2.4 Constant Flight Path Trajectory
As general equations of atmospheric reentry flight, Vihn’s equations were derived for the purpose of defining the trajectory of a body moving with a ‘high’ velocity for its entire trajectory. A ‘high’ velocity refers to one which prevents the negative gravity term in Appendix D1: Eq 2 from dominating the equation for the rate of change of flight path angle. Neglecting aerodynamic forces and the Coriolis terms, Appendix D1: Eq 2 reduces to Eq 10,
Eq 10
For the case of a zero initial velocity, analytically we see that the gravity term in Eq 10 tends toward negative infinity resulting in an instantaneous decrease of the flight path angle immediately after launch. By the time the velocity is high enough to counteract the gravity term, the direction of the velocity vector is already reversed and pointing toward the Earth. To avoid this unintentional low-altitude gravity turn it is necessary to define the trajectory of the vehicle through alternative means up to a velocity that Vihn’s equations can handle.
Vihn derives a linear equation to determine the velocity of a vehicle at the end of a specified burn time. This equation is based on the familiar ‘rocket equation’ (Appendix D1: Eq 11) and assumes a constant flight path angle while neglecting Coriolis effects. Although Eq 11 is a simple approximation, it is used for a very small portion of the trajectory and thus will not affect the overall trajectory by much. Various burn times were used to obtain a velocity that would serve as a feasible initial condition to Vihn’s equations. The altitude at the end of the burn time is given byAppendix D1: Eq 12.
These equations are used in the start of the main program of the Daedalus boost codes.The minimum time of vertical flight was determined as 30 sec, which is comparable to the 15 sec of vertical flight used in previous boost codes. For a realistic simulation, the constant flight path angle was set to 90°. The velocity and altitude thus obtained are initial conditions for Vihn’s equations.
4.2.5 Variable Flight Path Trajectory
After the completion of the linear trajectory, Vihn’s equations are implemented up to the orbital insertion point. The orbital insertion point, which forms the final conditions for the solution to Vihn’s equations, is defined by a velocity, an altitude, and an inclination angle. These were defined earlier. Vihn’s equations are represented in state variable form in a subroutine entitled ‘boostraj’. This subroutine calculates thederivatives of the velocity, flight path and heading angles, altitude, latitude, longitude, vehicle mass, and vehicle stagnation temperature with respect to time. These derivatives are then called by the subroutine ‘RK4’ to be numerically integrated. However, as Appendix D1: Eqs 1-7 indicate, there are many other terms in Vihn’s equations that must be defined before the equations can be solved.
Environmental Data
The boost codes take into account the variation of acceleration due to gravity with altitude using Appendix D: Eq 13. The density and temperature of the atmosphere is defined in the file ‘atmo76.f’, which was provided by Prof. Gustafson, formerly of PurdueUniversity, in the fall of 1997. It contains numerical values for Earth’s atmospheric density and temperature from sea-level up to an altitude of 121.4km and is based on the 1976 standard atmosphere. When called, the program outputs ambient temperature and density with altitude input. It also outputs speed of sound values from ambient temperature using Appendix D1: Eq 14. By assuming an ideal atmosphere, ambient pressure is determined from the perfect gas law(Appendix D1: Eq 15). With the vehicle’s environment defined, the boost code then defines the aerodynamics of the launch vehicle.
Vehicle Aerodynamics
As described earlier, the coefficient of lift term in Vihn’s equations is neglected. However, the vehicle’s drag is too large to be neglected in the ideal trajectory, and is even more necessary when computing the forces due to winds. The subroutine ‘codrag’ calculates the drag coefficient of the launch vehicle with inputs of current velocity, altitude, and atmospheric speed of sound. Appendix D1: Eqs 16-18 determine the drag coefficient from the vehicle Mach number, which is computed from the velocity and speed of sound. They are equations for the drag on a cone and were obtained by Prof. Gustafson from Shapiro’s ‘The Dynamics and Thermodynamics of Fluid Flow’.
Vehicle Surface Areas
The base area of the vehicle is required for calculating its drag. It is computed simply as the largest circular area of the launch vehicle stack. This value is obtained from Isakowitz’s ‘International Space Launch Systems’. For the Delta IV-H this number would be multiplied by three since the two booster stages are identical to the vehicle first stage. The side area is the longitudinal planform area of the rocket components. To analyze the largest wind forces, the largest planform area of the vehicle is used.This is simply three times the planform area of the vehicle components. The tumbling drag of the discarded stages, which will be discussed later, is also dependent on the side areas of each stage.
Thrust Profile
The throttling capabilities of the Delta IV’s first stage motor give us a parameter that can be used to passively control the vehicle. Varying the burn times of each stage and their throttle has a large effect on all the parameters, specifically the rates of change of velocity and flight path angle. The variation of thrust with atmospheric pressure is governed by Eq 19 from G. P.Sutton’s ‘Rocket Propulsion Elements’,
Eq-19
Assuming negligible exit pressure, the first term represents the vacuum thrust of the engine, which is a constant parameter. Hence, the formula used to vary thrust becomes Eq-20,
Eq-20
Total Acceleration
The vehicle’s total acceleration throughout its trajectory can be resolved into a tangential and a normal component. The tangential component is the rate of change of tangential velocity, or in other words, the value of the velocity derivative from Vihn’s equations. The normal component of acceleration is given by Appendix D1: Eq-21, which was obtained from the Fall 2000 code.
Wind Forces
The winds and gimbal angle are set to zero for the ideal trajectory. However, in reality, the forces due to winds can have a large effect on trajectory as they occur at low altitudes. The vehicle is moving relatively slow at these altitudes and is thus more susceptible to deviation. The subroutine ‘windy’ is used to calculate the tangential and normal forces due to winds at the launch site. These forces are then included in the force equations of Vihn’s equations. The subroutine is a modified version of a subroutine of the same name taken from Prof. Gustafson’s ‘boost.for’ (Summer 1998) and last modified by Shannon Fitzpatrick (Fall 2000). It requires the definition of the variation of wind velocity with altitude over the launch site. Wind velocity at various altitudes over Cape Canaveral and Vandenberg is contained in a report provided by Prof. Gustafson, although the source of the report is currently unknown.This data is curve fit into a quadratic equation using the Matlab file ‘winds.m’ (Appendix D2). For higher order polynomials the equation is much less accurate. At each increment of the non-ideal trajectory ‘windy’ is given the altitude of the vehicle along with its flight path and heading angles. Using this data, along with the drag coefficient and side area of the vehicle, the subroutine determines the normal and tangential wind forces on the craft. Along with the ideal solutions, the wind-affected solutions with zero gimbal angle are determined.
Gimbal Control
The thrust vectoring of the first and booster engines of the Delta IV-H must be implemented to counteract the effects of the wind forces. The ideal trajectory is maintained by stabilizing the flight path angle of the vehicle. This is achieved by using a proportional-derivative feedback controller as in Fig 4.1.
Fig 4.2: Proportional-Derivative gimbal control
In this case, the control force is the gimbal angle and the error is the difference between the ideal and non-ideal flight path angles. The proportional and derivative gains must be varied until a minimum error is attained throughout the wind-affected portion of flight. Information regarding the maximum gimballing angle for the Delta IV’s first stage could not be found. However, the angles must be kept to a minimum in order to ensure that the tangential thrust does not get depleted and the vehicle does not lose velocity. A range of 3-4° will typically prevent too much thrust from being lost.
Discarded Stage Trajectory
The trajectories of the discarded stages are determined much in the same way as the launch vehicle. The boost code keeps track of the solutions to Vihn’s equations for the times at which the stages separate from the main vehicle. These solutions form the initial conditions for the discard trajectories. The thrust for the discard stages is assumed to be zero while the mass is assumed to be constant, i.e. the stages remain intact. Also, the trajectories of the two boost engines are assumed to be identical. The drag on the inert stage component is a combination of the regular base drag as well as the side drag of the stage. The drag is analyzed using a crude tumbling model given by Appendix D: Eq 22. The code then solves Vihn’s equations for each discarded stage.
4.2.6 Boost Trajectory Analysis
Boost trajectories were determined for the two different OSP configurations and the MSP configuration using the defined methods. The mission target parameters for the vehicles are outlined in Table-4.2.
Daedalus 1 / Daedalus 2 / Daedalus 1MVehicle Mass (kg) / 22,214 / 23,203 / 20,787
Inclination (°) / 51.6 (ISS) / 51.6 (ISS) / 90.0
Velocity (m/s) / 7,670 / 7,670 / 7,790
Altitude (m) / 400,000 / 400,000 / 185,000
Table 4.2: Boost trajectory missions
Daedalus 1 Results
Daedalus 2 Results
Daedalus 1M Results
4.3Reaction Control System
4.3.1 RCS Thrusters
According to G. P. Sutton’s ‘Rocket Propulsion Elements’, a total of 12 thrusters are required to ensure complete maneuverability using the application of pure torques about three axes. The RFP requires double redundancy in the RCS, and as a result, the OSP is required to be fitted with a total of 24 thrusters. The selection and position of the reaction control thrusters is ultimately dependent on their ability to turn the spacecraft 180° along its yaw axis within the time period between the end of the de-orbit burn and the time at which the craft reaches the Earth’s atmosphere. This time period is given by Appendix D1: Eq 23. The force term in this equation is the thrust of a pair of rockets involved in the turning maneuver separated by the length term.The moment of inertia term is along the axis of rotation and is an output of the CAD program used to model the spacecraft and its components.By varying the separation distance, the time taken for a pair of thrusters to turn the vehicle about an axis can be determined. The Matlab script ‘RCS.m’ was used to plot the variation of rotational time periods with thruster separation distance for Rocketdyne’s RS-25 thruster. This thruster has a comparatively high thrust output (111 N) for a small weight (0.96 kg). The shuttle RCS thrusters were found to be heavier with a high thrust output beyond the needs of the Daedalus.