...............................................................................(2.1.9)

Or, ..................................................................(2.1.10)

It may be noted that for higher power of θ, the coefficient become very small and hence the higher order terms can be neglected.

Keeping up to 5^th order, the equation can be written as

........................................................................................... (2.1.11)

which is a form of Duffing equation with cubic and quintic nonlinearities.

One may derive the same equation using the fact that the moment of a force about a fixed point is equal to the time rate of change of the angular momentum about poin . In mathematical form it can be written as . Refereeing to Figure 2.1.2(b)

..................................................................................................................(2.1.12)

Or, .............................................................(2.1.13)

Now, ..................................................................................(2.1.14)

Or, ...........................................................................................(2.1.15)

Or , ............................................................................................ (2.1.16)

Or, ...................................................................................................... (2.1.17)

Keeping up to cubic nonlinearity Eq. (2.1.17) can be written as

..................................................................................................... (2.1.18)

Taking the length of the pendulum 1 m and acceleration due to gravity as 10 m/s² the equation of motion can be written as

..............................................................................................(2.1.19)

It may be noted that the coefficient for the cubic order term is very less than that of the linear term. A MatLab code is given below to obtain the variation of restoring force with θ which will give an idea regarding the approximation one has to take while writing the equation of motion.

The equation is similar to a Duffing equations with soft type cubic nonlinearity