Question

Consider the pendulum, y " + sin(y) = 0. Using at least a 41th order Runge-Kutta method: Compute the motion for a variety of amplitudes. Keep the amplitudes to 3 or less. For each amplitude, determine the corresponding period of motion. Plot the period as a function of amplitude.

Answer 1

a)The Runge-Kutta method is afourth-order method-it can be proved that the cumulative error on a bounded interval

[a, b] with a = Xo is of order h4.

(Thus, It is also called the fourth-order Runge-Kutta method because it is possible to develop Runge-Kutta methods of other orders. That is,

½y(Xn) - yn½≤ Ch^4

y(t+dt) = y(t) + 1/6 * (k1 + 2k2 + 2k3 + k4)
k1 = dt * f(t,y)
k2 = dt * f(t+dt/2, y(t)+k1/2)
k3 = dt * f(t+dt/2, y(t) + k2/2)
k4 = dt * f(t+dt, y(t)+k3)

b)

Plot = 3 Sin (2x)

x = 0 to 4π

Arc length of curve:

∫o ^4π √ 19 + 18 Cos (4x) dx ≈ 50.4465

∫o ^4π √ 19 + 18 Cos (4x) dx = 8√37 E(36/37) ≈ 50.4465

1. plot period as function of amplitude y”+sin(y) = 0

Amplitude = y = - sin(y) = -0.712097697

Z = -0.653423149

Period = 2π/3