Undamped Free Vibration:

There are many methods to analyse an undapmed system.

Methodology

Method depend on Newton's II Law

According to this, the rate of change of linear momentum is proportional to the force impressed on it

                     (d /dt) (mv) α Net force in direction of the velocity

By using   v = dx /dt = x

∴ dx /dt= (m x ) = k ∑ F

or         m x  = k ∑ F

For proper units in a system k = 1

               m x  = ∑ F

The direction of forces m ? and ∑ F are similar. A model that represents undamped single degree of freedom system will have two elements, that mean helical spring and mass. The mass is only constrained to move in one direction as illustrated in Figure 2. The mass is in static condition in Figure 2(i). The free body diagram of the mass is illustrated in Figure 2(ii). The body is in equilibrium under the action of the two forces. Here 'Δ' refers to the extension of the spring after suspension of the mass on the spring.

Hence,                  k Δ= mg                         . . . (1)

(i)            (ii) Static Condition     (iii) Dynamic Condition

        Figure 2

Figure 2(iii) shows the dynamic condition of the body. In this, the body is moving down having acceleration ' x ' also in downward direction, thus,

            mx  = ∑ F in direction of   x

or         mx = mg - k ( x + Δ)                         . . . (2)

 Incorporating Equation (1) in Equation (2)

mx= - k x

or         m x + k x = 0                                     . . . (3)