Question

An engineer designing a vibration isolation system for an instrument console models the consoleand isolation system as a 1 degree-of-freedom spring-mass-damper system. The modeled mass is 2kg, the spring constant is 8 N/m, and the damping constant is 1 Ns/m. The system is exposed to atime-varying force acting on the mass. This force measured is approximated by:F(t) = 20 sin(4t) [N]. Determine the steady-state response of the system to this input.

Answer 1

Given :

m = 2 kg

k = 8 N/m

C= 1 N-s/m

F(t) = 20 sin (4t)

`$F_0= 20 \ N$`

ω = 4 rad/s

`$\omega_n = \sqrt{(k)/(m)}$`

`$\omega_n = \sqrt{(8)/(2)}$`

= 2 rad/s

Therefore,

`$\epsilon = (C)/(2 m \omega_n)$`

`$\epsilon = (1)/(2 * 2 * 2)$`

=
`$(1)/(8)$`

= 0.125

So, r =
`$(\omega)/(\omega_n)$`

=
`$(4)/(2)$`

= 2

Steady state response is given by

`$(A)=(F_0 / k)/(√((1-r^r)^2+(2 \epsilon r)^r))$`

`$(A)=(20 / 8)/(√((1-2^r)^2+(2 * 0.125 * 2)^r))$`

A = 0.82 m