numerical_integration_notes

numerical_integration_notes - AERSP 304 R.G. Melton...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

Page1 / 5

numerical_integration_notes - AERSP 304 R.G. Melton...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online