Question

6. A spring-mass system is oscillating under no external force, with a damping constant equal to 2 and spring constant equal to 1. If at t = 0 we release (x'(0) = 0) the mass from a stretched position at a displacement of x(0) = 3. Write down the modelling ODE and solve it to find the solution. What will happen eventually to the system? Is it overdamped or undedamped? Will it oscillate?

Answer 1

let mass is 1 kg

then differential equation is

x''+2x'+x=0

taking laplce transform

s²X(s)-Sx(0)-x'(0)+2(X(s)-x(0))+X(s)=0

(s²+2s+1)X(s) = 3s+6

X(s)= (3s+6)/(s+1)^2

X(s)= 3*( s+1+1)/(s+1)^2

X(s)=3/(s+1)+3/(s+1)^2

taking laplace inverse

x(t)= 3e^-t+3te^-t=3(1+t)e^-t

since exp(-t) term is present hence eventually output oscillation reduced to zero.

Above system is critically damped . It will not oscillate