.GNU_ScientificLibrary.Examples.specfunc.LargeAmplitudePendulum

Pendulum with pi/2 Amplitude and Period from Elliptic Integral

Information

This example involves a pendulum with pi/2 (=theta0) amplitude. A mass (m,Icm) is suspended a distance L from a pivot. From the conservation of total (kinetic + potential) energy, one finds (parallel-axis theorem: I=Icm+mL2):

dtheta/dt = sqrt[(2mgL/I)(cos(theta) - cos(theta0))] 

After differentiating and doing some algebra, one arrives at the following DE:

d2theta/dt2 + (mgL/I) sin(theta) = 0 

The period can be found by integrating the dt/dtheta expression above, performing a change of integration variable, and using a trig half-angle formula. The result is a complete elliptic integral, K(k):

T = 4 sqrt(I/mgL) K(sin(theta0/2)) 

Plotting the period, T, as a function of starting angle (assuming dtheta/dt=0 at theta0):

Near theta0=0, one finds the solution (T0~2s) for the small-angle approximation: sin(theta) ~ theta.

The period of the swinging-body simulation is found to match the calculation of the period from the max angle:
Keep in mind that this acts as a check for the former, not the latter.

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