Where
...(2.5.37)

For k =1

............................ (2.5.38)

For k =2

...............(2.5.39)

..................(2.5.40)

Example 2.5.2: Using Lagrange Principle find the equation of motion of the system shown in Figure 2.5.2 . Spring K₁ is under pretension T₀ for small amplitude of vertical Oscillation i.e., . Spring K₂ is a soft spring with cubic nonlinearity.

Figure 2.5.2: Vibration of a spring mass system with additional pre-tensioned horizontal spring.

Solution

As spring is under pretension which is produced by an initial extension of the spring by an amount , one may write

......................................................................................................................(2.5.41)

The kinetic energy of the system is .............................................................(2.5.42)

The potential energy of the system is due to the potential energy of the nonlinear spring and due to the linear spring . Considering oscillations about the static equilibrium position, the potential energy can be obtained as follows.

.............................................................. (2.5.43)