Pendulum with Finite Amplitude of Swing

In equation (1) if the angle of swing is finite then () and equation (1) becomes,

To simplify the procedure and deal exclusively with swing amplitude, the viscous term [] will be neglected reducing the equation to,

a solution to this equation gives the following expression for period of oscillation,

where (K()) is a function of () known as the complete elliptic integral of the first kind. It can be expanded into a series of sines of  giving,

When () is fairly small, [] can be replaced by [] in the second term and subsequent terms can be neglected; consequently,