Question

Consider a mass-spring-dashpot system in which the mass is m = 4 lb-sec^2/ft, the damping constant is c =24 lb-sec/ft, and the spring constant is k=52lb/ft. The motion is free damped motion and the mass is set in motion with initial position x0=5ft and the initial velocity v0= -7ft/sec. Find the position function x(t) and determine whether the motion is overdamped, critically damped, or underdamped.

Answer 1

Consider a mass-spring-dashpot system in which the mass is m = 4 lb-sec^2/ft, the damping constant is c =24 lb-sec/ft, and the spring constant is k=52lb/ft.

Given that m= 416 - see?lft C: 24 lb-scelft k: 365 blft x - 5ft . Vo = =+ft/sec X(t) will be ce- pt cos (wt) + Ge-pt Esin (cut) C:00) p 늙 24 2(4) P:3. w.fw ²-p² -1824)2-03) 169-1 1160 w: 12.64 (critically dimped)

-30 a(t): ce-3t+sexo 31-0 dta + 3. x(t) = cest + ca tehzt x (0) = 5 5:6, x'(t) = 3c, est + s (+(-3) e 3t to - 7 = 30, + Ca : - 7 + 3x5 : 15-7 ₂:8 so x (4) = 58 e 3t+ 8 te-3t