Question

A mass M=4kg is connected to a spring-dashpot system with spring constant k=10000N/m and a
damper with coefficient c=8Ns/m. The acceleration of gravity is g=9.8m/s. Determine if the system is
un-damped, under-damped, critically damped, or over-damped and find the equation of motion y(t) if
the mass is initially held at rest a distance of 0.1m from the equilibrium position.

Answer 1

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)}