It is observed experimentally that the amplitude of free vibration of a certain structure modelled as a single degree of freedom system, decreases from 1.0 to 0.4 in 10 cycles. What is the percentage of critical damping?

(Take ln 2 = 0.693 and ln 10 = 2.303)

TB Solution

Given Data:

The number of cycles (k = 10) in which the amplitude changes is as follows:

Initial stage amplitude of free vibration of a certain structure(u_i) = 1.0

Final stage amplitude of free vibration of a certain structure(u_k) = 0.4

Explanation:

Now use the relationship between logarithmic decrement of displacement(\(δ\)), number of cycles(k), and changes in amplitudes

k \(\times\)δ  = ln(ui/uk)  ⇒ 10 \(\times\)δ = ln(1/0.4)

δ = 0.09162

So the damping ratio

\(\xi\) = (δ / 2π ) 

\(\xi\) = 0.01458

Now the % Ciritical Daming = \(\xi\) \(\times\) 100%

 % Ciritical Daming = 1.46%

Additional InformationCritical Damping Constant(c_c):

c_c = 2\( √(\)km)

Damping Ratio(\(\xi\)):

\(\xi\) = c / c_c 

where k= spring constant, m= total mass, c= damping constant

Case-I: Underdamped system

\(\xi\) < 1 or c<c_c or [(c/2m) < \(\sqrt (\)k /m)]

Case-II: Critically damped system

\(\xi\) = 1 or c = cc or [(c/2m) = \(\sqrt(\)k /m)]

Case-III: Overdamped system

\(\xi\) > 1 or c > cc or [(c/2m) > \(\sqrt (\)k /m)]