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Unformatted text preview: AERSP 304 R.G. Melton Numerical Integration Suppose we have a set of ordinary diff. eqs. (linear or nonlinear). And we need a numerical solution. If linear eqs. then could use Laplace xform method, and get exact solution analytically. But if nonlinear, or if linear but forcing fcn is complicated, then a numerical solution may be easier (and adequate for engineering purposes) WARNING: numerical integration is only an approximation . Never confuse this with the true solution (although it may be a pretty good approx!) NEVER unquestioningly trust the result of numerical integration (or any computer output for that matter). Look for some means to test the result for accuracy: Are there physical constants such as energy or momentum that can be checked? Is there a known analytical solution for a special case of your problem? (e.g., when the forcing is zero or when the damping is very small). Converting Diff. Eqs. into Proper Form for Numerical Integration With few exceptions * , diff. eqs. must be converted into 1 st-order form before they can be integrated numerically. The standard form, shown below, is called state-variable or state-space form, and each equation is called a state equation ) , , , , ( ) , , , , ( ) , , , , ( 2 1 2 1 2 2 2 1 1 1 t x x x g x t x x x g x t x x x g x n n n n n " # " " = = = In this form, the left-hand sides contain only first derivatives of the state variables x 1 , , x n and the right-hand sides contain only state variables and time (plus some constants), but no derivatives of the state variables. Example 1 ) ( t f kx x c x m = + + where m, c, k are constants, and f ( t ) is the forcing function. x x x x x x x Let = = = = 2 1 2 1 * The most famous of these is Gauss-Jackson, which was developed to integrate the 2 nd-order equations of orbital motion. It directly integrates acceleration to give position, but to get velocity, the result must be numerically differentiated....
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