A mass-spring-damper system has a mass of 100 kg. Its free response amplitude decays such that the amplitude of the 30th cycle is 20% of the amplitude of the 1st cycle. It takes 60 s to complete 30 cycles. Estimate the damping constant c and the spring constant k.

the value for the damping constant is c = 5.3408Ns/m and the spring constant is k is 987N/m

Explanation:

Consider the formula for the logarithmic decrement (δ)

δ =

Consider the formula for damping ratio  (C)

C =δ / √4π² + ² ----------(2)

The given amplitude of the 30th cycle is 20% amplitude of the 1st cycle

The ratio of last peak to first peak is

%--------------(3)

where

n is the number of cycle

Re arrange eqn (3)

The given value for n is 30

substitute 30 for n and eqn (4) in eqn(1)

δ =

substitute  0.0536 for δ in eqn (2)

Consider the formula for damping ratio  (C)

C =

where

c is the damping constant

m is the mass

substitute 100 for m  in eqn(6)

compute eqn(5) and eqn(7)

The given time to complete 30 cycles in 60s is

period = 60 / 30

= 2s

Formula for spring constant

where

is the undamped natural frequency

consider the formula for damped natural frequency

rearrange

substitute eqn (10) to eqn (9)

consider the formula for the damped natural frequency

consider eqn (12) in eqn (11)

substitute 100 for m, 2 for P, 0.0085 for C in the eqn above

substitute 987 for k in eqn (8)

Thus, the value for the damping constant is c = 5.3408Ns/m and the spring constant is k is 987N/m