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 50% greater than the period of the corresponding undamped motion.

Answer 1

The characteristic equation for this is r^2 + yr + 1 = 0. Its roots are r = y/2 +/- isqrt(1-y^2/4). Finding the general solution to this equation gives us y = sqrt(20/9), which is the value of the damping coefficient y for which the quasi period of the damped motion is 50% greater.