These equations can be used to determine the horizontal components of V, Vxi, and Vx 2 in terms of accelerometer outputs and the desired platform torque rates. These torque rates, which must also be supplied to the gyro torquers, can be computed as follows.Define the east, north, and vertical axes, yi, y^ and yz, respectively, as in Fig. 2. The axes of y\ and y% are principal directions of curvature on the ellipsoid (east and north). The components of UEP rates in principal axes arewhere Pm = principal radius of curvature of the reference ellipsoid in the meridian plane, sometimes written as p, i.e, Pm = a(l -6 2 )/(l -e 2 sin 2 0) 3/2 p p = principal radius of curvature in the vertical eastwest plane, called the prime radius of curvature and sometimes written as 77, i.e., p p = a/(le^in 2^)1 / 2 By resolving these equations into platform axes and performing some algebraic manipulation,(3)
Pp -Pm(pp + )(p where Pa = radius of curvature at an azimuth a(l/p a ) = (sinWpp) + (cos 2 a/Pm) p a + 90 = radius of curvature at an azimuth a + 90 °P P -Pm = (ae 2 cos 2 <£)/(l -e 2 sin 2 <£) 3/2 Equations (3) are the exact expressions for the torque rates of the platform relative to earth. The inertial torque rate is UEP + 12 = W/P. These are not power series approximations.Note that the torque rate about the Xi axis is not merely the x 2 component of velocity divided by the radius of curvature in the #2 -#3 plane except at cardinal headings. At all other headings, a small correction, proportional to V x \, must be added. This correction is of the order (e 2 /2)(7/a), or about 0.1 deg/hr at 1800 knots. In geometric terms, the platform rotates around the velocity vector as well as around the binormal.Equations (3) can be expanded hi a power series in (h/a) and e, which is accurate to 10 ~~4 12(0.0015 deg/hr) for speeds up to 1800 knots and at altitudes below 25 naut miles :
JThe calculation of position from the three components of velocity, as found in Eq. (1), depends on the coordinate frame chosen. For example, if a = 0 so that the platform is north-seeking, then the latitude and longitude A can be found from -/ Jo dt Nomenclature freestream Mach number body radius (= nose radius of curvature for a sphere) Reynolds number based on body diameter and conditions immediately downstream of a normal shock reservoir temperature of gas body wall temperature shock detachment distance measured from the body to the shock leading edge along the axis of symmetryT HE shock detachment distance in front of blunt bodies at high Reynolds numbers has been studied both theoretically and experimentally. Van Dyke and Gordon 1 have presented a theoretical analysis for a series of perfect gases having specific heat ratios of 1, 3-, and f and a Mach number range from 1.2 to infinity. For a sphere, their analysis indicates that the shock detachment distance is primarily a function of the density ratio across the normal part of the shock. In such studies, it is implied that the shock and boundary-layer thicknesses are small compared with the shoc...
