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

Answer:

Step-by-step explanation:

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

$\frac{Td}{T} = (1 - \frac{γ^{2} }{4km} )^{-1/2}$

$\frac{140}{100} = (1 - \frac{γ^{2} }{4} ) ^{-1/2}$

$\frac{7}{5} = (1 - \frac{γ^{2} }{4} ) ^{-1/2}$

Solving for damping coefficient γ,

$(\frac{5}{7})^{2} = 1 - \frac{γ^{2} }{4}$

$0.5102 = 1 - \frac{γ^{2} }{4}$

$γ^{2} = (1 - 0.5102) *4$

γ = $\sqrt{1.9592}$ = 1.4