Simple numerical model for calculation of entry vehicle trim response

Abstract

T HE influence of mass and configurational asymmetries on the dynamics of slender, spinning entry vehicles is an important aspect of vehicle design and performance assessment. High-altitude roll resonance and lower-altitude roll/trim effects influence vehicle loading and attendant impact miss. Descriptions of aerodynamic trim effects on reentry vehicle (RV) angle of attack for vehicles with linear aerodynamics are well known. 1 ' 4 Influence of nonlinear aerodynamics on resonant and nonresonant response have been considered by Murphy 5 and by Nayfeh and Saric. 6 ' 7 Although these solutions are useful in characterizing angle-ofattack response, it is still generally necessary to numerically integrate the six-degree-of-freedom equations of motion in order to quantify the resulting trim-induced dispersion for a given re-entry system. This is often done using a Monte Carlo approach, necessitating significant amounts of computer time.Presented in this Note is a simple trim response model which is numerically efficient and which can be incorporated into standard point-mass trajectory simulators for calculation of trim-induced dispersion and angle-of-attack/load behavior. This model is based on the fact that the coupling of trim and body fixed low-frequency (in the sense of Nelson 8 ) oscillatory motion component dominates the trim-induced response. The high-frequency component/trim coupling is not significant, except in cases where sudden trim changes are encountered, e.g., due to rapid configurational change. By using standard asymptotic solutions to linear differential equations with time varying coefficients, 9 the two secondorder equations in angle-of-attack a and sideslip 0 can be reduced to two first-order equations in which the highfrequency contribution is eliminated. This allows for a much coarser integration increment in calculating the trim-induced response.In the present model, time variations in roll rate, asymmetries, and aerodynamic coefficients are taken into account. Longitudinal principal axis misalignments in pitch and yaw are also included, which was not done in the previously cited work. l ' 7 In addition to the reduced equations of motion, results are obtained for the effect of roll acceleration and socalled density damping on the steady-state trim angle relations. The simplified model is shown to provide adequate accuracy in calculating the combined low-frequency/trim contribution to angle-of-attack.The angle-of-attack behavior is formulated in body fixed coordinates as used by Nelson. 8 For small angles of attack, linear aerodynamics, and with inclusion of products of inertia and aerodynamic asymmetries for an otherwise symmetric vehicle, the equations of rotational motion may be written as follows:. j3= -r+pa-(<7oo5C La /mK)]3(1)Here, a and 0 are angles of attack and sideslip, respectively; /?, q, and r are roll, pitch, and yaw rates, respectively; A and B are roll and pitch moments of inertia (pitch and yaw moments of inertia are assumed equal); vehicle mass and aerodynamic reference area and ...