Mixed-mode propulsion: Optimum burn profile accounting for gravity and drag

Abstract

Combining different propulsion modes in an advanced stage promises improved vehicle performance, cost and operating characteristics, and may make a reusable one-stage-to-orbit shuttle feasible. It has been shown theoretically that in a two-mode system, maximization of ideal velocity gain requires purely sequential operation of the two modes, with the higher density mode operating first. The question addressed in this analysis, is whether some form of overlapping burn profile may be required to maximize real velocity gain in cases such as ascent from earth-to-orbit where losses to gravity and drag are important factors. The analysis takes as a point of departure a generalized form of the ideal rocket equation where propellant density and specific impulse are no longer constant as in the more familiar classical single mode case, but are allowed to vary arbitrarily. An expression is derived which relates the trust levels for the two modes in terms of the density, the specific impulses and the throttle settings for the two modes. The optimum burn profile is determined by applying the calculus of variations to maximize velocity gain in the generalized ideal rocket equation. Solution of the Euler—LaGrange equations for this case produces an identity. An extremal solution is obtained and velocity gain is maximized, if, and only if, this identity is satisfied at every point in the burn history. Results are shown of computer runs which satisfy this identity at each point in an earth-to-orbit ascent trajectory as computed by a standard trajectory optimization program. Acceleration constraints are included, as are propellant volume constraints associated with the particular vehicle design assumed. The results are compared to those obtained for the same vehicle with pure sequential burn, and it is shown that overlapping sequential burn does, in fact, lead to sizable performance gains. A sample computer trajectory run is shown for an optimum burn profile.