Abstract
The two dimensional gravity turn problem is addressed allowing for the effects of variable rocket mass due to propellant consumption, thrust and thrust vector angle, lift and drag forces at an angle-of-attack and atmospheric mass density varying with altitude; Coriolis and centrifugal forces are neglected. Three distinct analytical solutions are obtained for constant: propellant flow rate, thrust, thrust vector angle, angle-of-attack and acceleration of gravity; the lift and drag are assumed to be proportional to the square of velocity, and the mass density is assumed to decrease exponentially with altitude. The method III uses power series of time for the horizontal (downrange) and vertical (altitude) coordinates; the method II replaces the altitude as variable by the atmospheric mass density and method I by its inverse. Thus the three solutions have distinct properties, e.g., I and III converge best close to lift-off and II close to burn-out. The three solutions: I, II, III, can be applied in isolation (or matched in combination) to the single-point boundary-value problem (SPBVP) of finding the trajectory with given initial conditions at launch. They can also be used as pairs in six distinct ways (I + II, I + III, II + III or reverse orders) to solve the two-point boundary-value problem (TPBVP), viz.: from given conditions at launch achieve one (not more) specified condition at burn-out, e.g., ã desired horizontal velocity for payload release. Each of the six distinct combinations of methods of addressing the TPBVP shares three features: (i) it can determine if there is a solution, viz. if the rocket has enough performance to reach the desired burn-out condition; (ii) if the desired burn-out condition is achievable it can calculate the complete trajectory from launch to burn-out; (iii) it can determine the range of achievable burn-out conditions, e.g., the minimum and maximum possible horizontal velocity at burn-out for given initial conditions at launch.
Highlights
In an earlier paper [21] the calculation of rocket trajectories in the atmosphere was considered taking into account the following effects: (i) the variable mass associated with propellant consumption; (ii) the effect of thrust at an angle to the rocket axis; (iii) the lift and drag for flight at an angle-of-attack; (iv) the proportionality of the aerodynamic forces on the square of the velocity and on the atmospheric
Aerospace 2020, 7, 131 mass density; (v) the dependence of the atmospheric mass density on the altitude. The latter effect has not been considered in most of the literature on analytic calculation of rocket trajectories [22,23,24,25,26,27,28,29]: in particular, if time is eliminated to express the trajectory in terms of velocity and flight path angle, the inclusion of the dependence of aerodynamic forces on altitude through the atmospheric mass density, introduces altitude explicitly as a third variable, related to the velocity and flight path angle by an integral relation
In the earlier paper [21] the equations of motion were written in an earth fixed reference frame, which has two advantages: (i) there is no singularity even for zero initial velocity, which is simpler than starting with initial motion, that case is obviously included; (ii) the dependence of the aerodynamic forces on altitude through the atmospheric mass density does not add a new variable to the problem
Summary
The optimization [1] and reconstruction [2] of rocket trajectories is a research area with renewed interest for small satellite launchers [3,4], including dynamics [5] and control [6] aspects. In the earlier paper [21] the equations of motion were written in an earth fixed reference frame, which has two advantages: (i) there is no singularity even for zero initial velocity, which is simpler than starting with initial motion, that case is obviously included; (ii) the dependence of the aerodynamic forces on altitude through the atmospheric mass density does not add a new variable to the problem. In a preceding paper [21] four methods of trajectory calculation were discussed in detail, with a minimum of algebraic clutter, by considering the simplest case of a vertical climb, without lift and with thrust aligned to the rocket axis; since account was taken of mass dependence on time, and drag dependence on velocity and altitude, similar methods apply in the more general case considered in the present paper: a gravity turn, with lift and drag allowing for an angle-of-attack and thrust vector at an angle to the rocket axis. By choosing the matching time it is possible to specify one (not, more) burn-out condition (Section 7), e.g., the desired horizontal velocity for payload launch
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