A weight of 10 ib is suspended from a spring having a stiffn

A weight of 10 ib is suspended from a spring having a stiffness of 30 lb/in, and the system is critically damped. The weight is given an initial displacement of 2 in from its equilibrium position, held there momentarily, and suddenly released with zero initial velocity. Determine the damping coefficient c of the dashpot and the displacement of the mass x(t) from its equilibrium position after 0.01, 0.1 and 1.0 sec. Plot the response.

Solution

Solution:-

Given

M= 10 lb

K =30 lb/inch

Angular frequency() = (K/M)

                                       = (30/10)

                                       = 1.73 radian/second

Damping coefficient at critical condition:-

C =C_cr = 2(KM)

           = 2(30*10)

    C =34.64 lb second/inch     Answer

We know that

x = x₀cos(t) +x^®₀/ *sin(t)

x₀ =Initial displacement = 2 inch

x^®₀ = initial velocity =0

Then

x = 2cos(1.73t)

Dispalcement at t=0.01 second

x₁ =2cos(1.73*0.01)

    = 1.99 inch Answer

Displacement at t =0.1 second

x₂ = 2cos(1.72*0.1)

= 1.99 inch      Answer

Displacement at t=1 second

x₃ =2cos(1.73*1)

     = 1.999 inch   Answer