Experimental identification of shock tunnel flow regimes.

Abstract

TECHNICAL NOTES 365 7 = -Now let F8 = F3 4-and define P(0 as sin0 + a 6 (h/p)(e + cos0) (25) F Q = 3£ F 7 = 3/Z0-3 (26)where A is a column vector of the six constants of integration.For a given set of initial conditions for R, V, \, and \, the integration constants can be found from Eq. (29) : Equations (7) and (8) now can be written asOnce A has been determined, X and X can be evaluated fromEqs. (7) and (8). State transition matrixA perturbation in the state at some reference time t 0 is propagated through the state transition matrix $(t, fo) as (3o) The solution to Eq. (6) can be written as Substitution of Eq. (29) into Eq. (28) gives |M = P0)p-i0 0 )f^° 1 (32) L^J L^oJ Comparison of Eqs. (31) and (32) reveals that the state transition matrix can be expressed asAn analytic inversion of the matrix P(t) was determined by calculating the determinant and the co-factors. The details are omitted here for brevity but the result is given by Eqs. (34) and (35) It is interesting to note that the determinant of P(t) was found to be h 6 . SummaryA simple analytic solution for the Lagrange multipliers has been derived in terms of flight time, Cartesian position, velocity and angular momentum vectors. The solution is valid for all types of conic arcs, except rectilinear, with a special form of the expression for one function for parabolic trajectories. The solution for the Lagrange multipliers was used to find the state transition matrix.