Question

A certain vibrating system satisfies the equation . Find the value of the damping coefficient for which the quasi period of the damped motion is greater than the period of the corresponding undamped motion.

Answer 1

Answer:

Answer is 1.49071

Step-by-step explanation:

See the picture for the complete details

Answer 2

Question:

A certain vibrating system satisfies the equation u''+γu'+u=0. Find the value of the damping coefficientγfor which the quasi period of the damped motion is 50% greater than the period of the corresponding undamped motion.

Answer: y = √(20/9) = √20/3 = 1.49071

Step-by-step explanation:

u''+γu'+u=0

m =1, k =1, w• = √ (k/m) = 1

The period of undamped motion T, is given by T = 2π/w•, T = 2π/1 = 2π

The quasi period Tq = 2π/quasi frequency

Quasi frequency = ((4km - y^2)^1/2)/2m

Therefore the quasi period Tq = 4πm/((4km - y^2)^1/2)

From the question the quasi period is 50% greater than the period of undamped motion

Therefore Tq = T + (1/2)T = (3/2)T

Thus,

4πm/((4km - y^2)^1/2) = (3/2)(2π)

Where, k =1, m=1,

4π/((4 - y^2)^1/2) = 3π,

(4 - y^2)^1/2 = 4π/3π,

(4 - y^2) = (4/3)^2,

4 - y^2 = 16/9,

y^2 =4 - 16/9,

y^2 = 20/9,

y = √(20/9)