Sigrid_LectWk9

Sigrid_LectWk9 - Sigrid Leyendecker Computational Dynamics...

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Unformatted text preview: Sigrid Leyendecker Computational Dynamics for Mechanical Systems 1 Computational Dynamics for Mechanical Systems two lectures of course CDS 140b: Introduction to Dynamics Tuesday March 4 and Thursday March 6 2008, 10:30am-11:55am, Steele 214 Sigrid Leyendecker sleye@caltech.edu 1 Numerics of ODEs 1.1 Introduction P. Deuflhard and F. Bornemann: Scientific computing with ordinary differential equations . Springer, 2002. Consider the planar pendulum as a motivating example. One is interested in the points in space, where the point mass m is located at a certain time, if the pendulum is released at an initial configuration q ( t ) with a certain initial velocity q ( t ) . Let q denote the angle measured against the vertical as depicted in Figure 1. Then the trajectory q ( t ) R n (in this case n = 1) yields the evolution of this angle. With the fixed length l , the positions of the point mass in the plane can then be computed. Figure 1: Planar pendulum. The evolution of the angle q ( t ) in the time interval [ a, b ] consists of infinitely many values. Since computers can handle only finite sets of data, the solution is approximated on a time- grid = { t , . . ., t N | a = t < t 1 < . . . < t N = b } Let k = t k +1 t k denote the time-step and = max k<N k Sigrid Leyendecker Computational Dynamics for Mechanical Systems 2 the maximal step-size of the mesh . A time-stepping method yields a sequence of discrete configurations { q k } N k =0 that approximate the real trajectory q k q ( t k ) . Intuitively speaking: if the approximate solution get closer and closer to the real motion for decreasing time-steps as in Figure 2, then the method is converging. 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8-6-5-4-3-2-1 1 2 3 t q reference t = 0.05 t = 0.1 t = 0.2 Figure 2: Convergence of approximations to a reference solution for decreasing time-steps. Two important questions on a numerical scheme: Does it converge? Does it yield realistic solutions? Non-converging methods do not make sense. For a converging method, unrealistic behaviour (like artificial energy gain or dissipation) improves for decreasing time-steps. However there are methods, that yield realistic behaviour even for relatively large time-steps, e.g. the me- chanical integrators in Section 2 and 3. 1.2 Example of an explicit one-step scheme: forward Euler Approximate the solution x C 1 ([ a, b ] , R n ) of the initial value problem x = f ( t, x ) x ( t ) = x (1) by the recursive iteration x k +1 = x k + k f ( t k , x k ) (2) in this case, the derivative x has been replaced by a forward difference quotient x k +1 x k k x ( t k ) = f ( t k , x k ) It is called an explicit scheme, since knowing x k , one can directly compute x k +1 . The geometric interpretation of (2) is that during one time-interval, the curve is approximated by the tangent at the beginning of the time-interval, see Figure 3. Sigrid Leyendecker Computational Dynamics for Mechanical Systems...
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Sigrid_LectWk9 - Sigrid Leyendecker Computational Dynamics...

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