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)
