Usingdsolvefornumericalintegrationofdifferentialequations.docVIP

Solving ordinary differential equations: In general, we would like to have the capability to solve differential equations. This is your main job in mechanics--write down Newtons laws, get the acceleration, and then solve for the position as function of time. Nonlinear Pendulum Lets solve for the motion of a pendulum, but not necessarily with small angles. This consists of a small ball (mass M) at the end of a light rod (massless, length L). Newtons laws yield the following equation where ? is the angle measured from the vertical with ????? straight down. To solve this numerically we need to turn our second-order equation into two first-order equations. They look like this , and . ************ Be sure you understand this process of turning a single second-order equation into two first-order equations! ************ Along with the differential equations we must provide the initial conditions. Lets put the pendulum at an initial angle of ???????? = ?/2, this is horizontal, and let it go from rest, so ?0=0. Lets also set a few parameters. Lets take g = 9.8 m/s2 and L =1.0 m. Matlab’s tour de force is its ODE solver. But, to use it you have to understand the interface that Matlab expects. (You can try to bypass this by using Matlab’s Simulink GUI to solve problems, but this isn’t real programming and it will prove at least as frustrating as the programming solution.) So here goes. The following can be put in a reusable function called pendulumODE.m: function derivs = pendulumODE(t, currentValues, L); % Nonlinear Pendulum: (odefun) % % dTheta/dt = omega % dOmega/dt = -gsin(theta)/L % % Our job is to set: dervis = [dTheta/dt; dOmega/dt] % using the currentValues = [theta; omega]. g = 9.8; % m/s^2 theta = currentValues(1); omega = currentValues(2); dTheta = omega; dOmega = -g*sin(theta)/L; derivs = [dTheta; dOmega]; We have a 2nd order ODE so the currentValues are an array of [????]. Based on the current value of ??and ?, we return the time derivat