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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 steady-state 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 Runge-Kutta 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 Runge-Kutta 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 steady-state behavior between the two cases. Runge-Kutta Integration Scheme This section details some of the specifics regarding integration with the Runge-Kutta equations. These equations were fully developed in lecture, so some parts may be omitted for brevity. Recall that the Runge-Kutta 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 Runge-Kutta 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