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Unformatted text preview: ASE 167M: Programming Assignment # 2 Numerical Integration of the Equations of Motion for Gliding Flight The equations of motion for nonsteady gliding flight in the vertical plane can be written as x = V cos h = V sin V = g W ( D + W sin ) = g WV ( L W cos ) Using the following data for the Lear 23 W = 11 , 000[ lbs ] S W = 232 ft 2 / C D = 0 . 02 K = 0 . 07 t = 0[ s ] x = 0[ ft ] h = 10 , 000[ ft ] g = 32 . 2 ft/s 2 / the assignment is 1. Assuming a flat glide angle (   1) and quasi steadystate glide ( V = 0, = 0), derive analytical expressions for h ( x ), V ( x ), and ( x ) along the maximum range path. lt is not necessary to explicitly state the functional relationship between altitude and air density (i.e. ( h ( x )) is good enough). 2. Write a computer program to integrate the equations of motion given above along the max imum range path from h o to sea level ( h = 0) using a RungeKutta integration subroutine, and the standard atmosphere subroutine to compute air density. For the initial velocity and flight path angle ( V o and o ) use the analytical results from Step 1. Use a stepsize of t = 2 seconds, and find t f and x f by iteration (tolerance  h f  < 10 4 ). Plot x ( t ), h ( t ), V ( t ), and ( t ). 3. Repeat the above procedure using a canned integration algorithm such as MATLABs ode45 . Compare the results with your RungeKutta solution. 4. Plot h ( t ), V ( t ), and ( t ) vs. x ( t ) for both the analytical solution from Step 1 and the numerical solution from Step 2 on the same graph for each variable. Comment on the differ ences/similarities of the transient and steadystate behavior between the two cases. RungeKutta Integration Scheme This section details some of the specifics regarding integration with the RungeKutta equations. These equations were fully developed in lecture, so some parts may be omitted for brevity. Recall that the RungeKutta equations, as given, are z n +1 = z n + 1 6 ( k 1 + 2 k 2 + 2 k 3 + k 4 ) + O ( t 6 ) For our purposes, we truncate this expression and neglect the 5 th and higher order terms and use the RungeKutta update as z n +1 = z n + 1 6 ( k 1 + 2 k 2 + 2 k 3 + k 4 ) where k 1 = f ( t n ,z n ) t k 2 = f t n + 1 2 t,z n + 1 2 k 1 t k 3 = f t n + 1 2 t,z n + 1 2 k 2 t k 4 = f ( t n + t,z n + k 3 ) t Under this formulation, we use the notation that f ( t,z ) represents the equations of motion for the...
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This note was uploaded on 09/18/2011 for the course ASE 167M taught by Professor Staff during the Spring '10 term at University of Texas at Austin.
 Spring '10
 Staff

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