Effect of lift with roll rate variation on re-entry vehicle impact

The effect of lift with roll rate variation on the impact of re-entry vehicles is treated analytically. The equations of motion of a vehicle experiencing lunar motion at small angle of attack are integrated assuming the roll rate history can be approximated with a series of linear segments. Each segment occurs over a sufficiently short interval of time that the trajectory properties can be assumed constant. The result is a closed-form solution expressing the resulting impact deviation from the zero-lift case as a function of the zero-lift trajectory characteristics and the roll rate history. The effect of a single linear roll rate variation between arbitrary initial and final roll rates on impact is derived from this general result. Roll through zero between roll rates of large magnitude but opposite sign, and roll acceleration of infinite magnitude between arbitrary roll rates are presented as special cases. An example problem involving a single linear variation is treated and compared with results obtained using a sixdegree-of-freedom trajectory simulation program. These results compare to within 5% typically, with none of the differences exceeding 13%.