Question

A certain vibrating system satisfies the equation u'' + γu' + u = 0. Find the value of the damping coefficient γ for which the quasi period of the damped motion is 40% greater than the period of the corresponding undamped motion.

Answer 1

Explanation:

Using the formula for the ratio of period of damped motion to the period of undamped motion

`(Td)/(T) = (1 - (γ^(2) )/(4km) )^(-1/2)`

`(140)/(100) = (1 - (γ^(2) )/(4) ) ^(-1/2)`

`(7)/(5) = (1 - (γ^(2) )/(4) ) ^(-1/2)`

Solving for damping coefficient γ,

`((5)/(7))^(2) = 1 - (γ^(2) )/(4)`

`0.5102 = 1 - (γ^(2) )/(4)`

`γ^(2) = (1 - 0.5102) *4 \\`

γ =
`√(1.9592)` = 1.4