m = 4 Kg
k = 10,000 N/m
c = 8 Ns/m
The damping ratio is zeta = c/(2*sqrt(m*k)) = 8/(2*sqrt(40,000)) = 8/(2(200) = 1/50
Since zeta < 1 then the system is underdamped
Differential equation of motion
The DE is:
mx'' + cx' + kx = 0
4x'' + 8x' + 10000x = 0
x'' + 2x' + 2500 = 0
auxiliary equation
r^2 + 2r + 3500 = 0
roots are: r1, r2 = -1 +/- 7sqrt(51)*i
(general) eqn of motion is: x(t) = e^(-t){A*cos(7*sqrt(51)*t) + B*sin(7*sqrt(51)*t)}
initial conditions
x'(t) = -e^(-t){ [A-7Bsqrt(51)]*cos(7*sqrt(51)*t) + [B+7Asqrt(51)]*sin(7*sqrt(51)*t) }
x(0) = 0.1, x'(0) = 0 (mass is at rest)
At t= 0: x(0) = 0.1 = 1{A*1 + 0} -> A = 0.1
At t= 0, x'(0) = 0 = -1{[0.1-7Bsqrt(51)]*1 +{B+7*0.1*sqrt(51)*0} = -{0.1 - 7B*sqrt(51) -> B = 1/(70*sqrt(51))
eqn of motion is: x(t) = e^(-t){0.1*cos(7*sqrt(51)*t) + 1/(70*sqrt(51))*sin(7*sqrt(51)*t)}