Fig.1 gives the different stages of a freely vibrating system without damping. In Fig.1(c) the displacement ‘z’ from the position of static equilibrium at a certain time ‘t’ is shown. The static deflection is given by,

———-(1)

Fig.1 (f) shows the free body diagram of the system. Applying Newton’s second law we get,

———- (2)

Or, ———- (3)

Or, ———- (4)

Or, ———- (5)

Put z = e^{Î»t} ———- (6)

Then e^{Î»t} ———- (7)

By substation the value from equation (6) and (7) in equation (5),

Î»^{2} Ã— e^{Î»t} + e^{Î»t} = 0 ———- (8)

Or, = 0 ———- (9)

Or, ———- (10)

Or, ———- (11)

By substitution Î» from equation (11) into equation (6) we have,

z = e^{} ———- (12)

z = A Ã— e^{ }+ B Ã— e^{–}^{} ———- (13)

Substituting the value of e in equation (13) we get,

———- (14)

Where, ———- (15)

and is called the undamped natural angular velocity of the vibrating system. f_{n} is the undamped natural frequency. Hence, we can write the equation for natural frequency as,

Cycles/sec ———- (16)

And the time period is given by

———- (17)

By substituting, and ---------- (18)

And ---------- (19)

The above equations clearly shows that the un-damped single degree of freedom vibrating system is harmonic at a natural frequency . The amplitude of vibration can be obtained by putting boundary conditions. At t = 0, z = z_{0} and , hence by substituting this condition in equation (14) we get,

and

Therefore equation (14) may be expressed as,

---------- (20)