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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, 10:30am11:55am, 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 timestep and = max k<N k Sigrid Leyendecker Computational Dynamics for Mechanical Systems 2 the maximal stepsize of the mesh . A timestepping 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 timesteps as in Figure 2, then the method is converging. 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8654321 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 timesteps. Two important questions on a numerical scheme: Does it converge? Does it yield realistic solutions? Nonconverging methods do not make sense. For a converging method, unrealistic behaviour (like artificial energy gain or dissipation) improves for decreasing timesteps. However there are methods, that yield realistic behaviour even for relatively large timesteps, e.g. the me chanical integrators in Section 2 and 3. 1.2 Example of an explicit onestep 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 timeinterval, the curve is approximated by the tangent at the beginning of the timeinterval, see Figure 3. Sigrid Leyendecker Computational Dynamics for Mechanical Systems...
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