Figure 2.1.3 shows the restoring force for actual, linear, cubic, 5th order and 7th order approximation. It may clearly be noted that depending on the range of θ one may take the approximation accordingly.
Example 2.1.2: Derive equation of motion for a nonlinear spring-mass-damper system as shown below. Consider the spring force in the form of 
and damping force equal to 
Figure 2.1.4 (a) Nonlinear spring-mass-damper system (b) free body diagram
Solution: Taking unit vector along positive X direction as 
, if a small displacement 
is given to the mass m , as shown in Fig. 2.1.4(b), spring force, damping force and inertia force will act in a direction opposite to that of the external force 
. Now applying d' Alembert's principle one can write the following equation.
.......................................................................(2.1.20)
Or, 
......................................................................(2.1.21)
Or, 
...........................................................................(2.1.22)