By Biringen S., Chow C.-Y.

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To approximate the second-order derivative of f , we show only a centraldifference formula . 9) h having an error O(h 2 fii v ). 5) to eliminate fi . 2). 1). 9).

12) the solution can be obtained using the fourth-order Runge-Kutta method. 13) x = x + 16 ( 1x +2 2x +2 3x + 4x ) y = y + 16 ( 1y +2 2y +2 3y + 4 y) This numerical integration procedure is programmed in a subprogram named in which the names XNEXT and YNEXT are used, respectively, for x and y . As an alternative, we also implement ODE45 MATLAB initial value solver for this problem. Plots of numerical results are obtained by using MATLAB plotting routines. 11) are respectively called U(XI, YI) and V(XI, YI) for obvious reasons, and are defined in two separate function subprograms.

Velocity increments, DIZ, D2Z, etc. Displacement increments, G Gravitational acceleration, g, m/s H Time increment, h, s NU Kinematic viscosity of fluid, ν, m2 /s PI π RE Reynolds number vd ν RHO Density of body, ρ, kg/m3 RHOBAR Density ratio, ρ, or ρf ρ 1 vi , 2 vi , 1 zi , etc. 2 zi , 2 etc. 2 Motion of a Simple Pendulum (continued) Program Symbol Definition Y, Y0 Circumferential displacement, y or l θ, and its initial value, y0 , respectively, m YV Circumferential displacement in vacuum, yv , m (Subprogram RUNGE) D1P, D2P, etc.

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