The following 4 differential equations describe mass spring

The following 4 differential equations describe mass spring systems. Identify each system as either undamped, overdamped, underdamped, or critically damped (Justify your answer). Determine the angular frequency and period for the undamped and the underdamped cases. a) 25y” + y = 0 b) 2y” + 2y’ + 5y = 0 c) 2y” + 16y’ + 32y = 0 d) 2y” + 7y’ + 3y = 0

Solution

a) 25y” + y = 0

Undamped

angular frequency = (1/25)^0.5 = 1/5 = 0.2

b) 2y” + 2y’ + 5y = 0

2*2 - 4*5*2 = -36 => underdamped

angular frequency = (36)^0.5 / (2*2) = 1.5

c) 2y” + 16y’ + 32y = 0

16*16 - 4*2*32 = 0 => critically damped

d) 2y” + 7y’ + 3y = 0

7*7 - 4*3*2 = 25 => overdamped