where ΔL is the change in length of the spring with stiffness K₁ due to the motion X of the mass. The coefficient of the cubic nonlinear term is assumed to be . The negative sign is due to the soft spring assumption. For , from Fig. 2.5.3 one may write

........................................................................(2.5.44)

......... (2.5.45)

Hence the expression for potential energy is

Figure 2.5.3

.............(2.5.46)

Taking the generalized coordinate , the Lagrangian of the system can be written as

(2.5.47)

As no external force is acting on the system, the Lagrange Equation can be given by .......................................................................................................(2.5.48)

Neglecting the two higher order terms marked in blue in Eq. (2.5.47) and applying (2.5.48) one can get the following equation.

............ (2.5.49)

Or, ................................................. (2.5.50)

Or, .........................................................(2.5.51)

Or, ........................................................(2.5.52)

Or, ..............................................................(2.5.53)