for a suitably large sample so the distribution of resulting impact points will, for all
practical purposes, represent all possible impact points, irrespective of the actual nature
of the failure.
Depending on vehicle breakup characteristics and failure time, a vehicle that
experiences a random-attitude failure may break up at the instant of failure, or after a
few seconds into the tum, or not at all. In making the calculations, three separate
breakup thresholds and a no-breakup case were investigated. With respect to vehicle
breakup, the assumption was made that the vehicle would break up if qa. exceeded a
specified constant limit, where q is the dynamic pressure and a. is the total angle of
attack. Although the breakup qa may well be a complicated function of Mach number
and other parameters, this simplistic approach was taken.

Random-attitude-failure calculations were made individually for Atlas, Delta, Titan,
and LLVl starting shortly after pitchover and continuing to some convenient time such
as a stage burnout when the vehicle could no longer endanger the launch area.
Theoretically, the Mode-5 impact density function extends downrange until the
instantaneous impact point vanishes. Since this study is concerned with evaluation of ·
density-function parameters for launch-area risk analysis, the random-attitude
calculations were _stopped at a staging event when the vehicle no· longer had sufficient
energy to return the impact point to the launch area. Using trajectory data for each
vehicle, program RAFIP was run to generate 10,000 impact-point samples at each
starting time. Calculations were made at ten-second intervals.
6.1.2 Slow-Turn Failures
Certain types of guidance and control failures can cause the thrusting engine to gimbal
to null or a near-null position: Such failures can produce what is herein called a slow
tum. For various reasons, after an engine is commanded to null it may not thrust
precisely through the center of gravity, e.g., structural misalignments, shifting center of
gravity, canted nozzles. Since, like random-attitude failures, slow ·turns constitute a
subset of Mode-5 failure responses, they have been investigated using RTI program
RAFIP. The following assumptions have been made in making the calculations:
 (1) The effective thrust offset of a "nulled" engine is normally distributed with a zero
     mean and a standard deviation of 0.1 °.
 (2) A fixed thrust offset results in a constant angular acceleration of the airframe, and
     thus a constant angular acceleration of the thrust vector.
 (3) For small thrust misalignments, the angular acceleration of the airframe is
     proportional to the angular thrust misalignment.
At each time point, the angular acceleration produced by small thrust offsets was
estimated from the malfunction turn data provided to the safety office by the range
user. Malfunction turns for the Atlas IIAS were provided for three gimbal angles, the
smallest being one degree. For each gimbal angle, the results were plotted as


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