A single stage vibration isolation system consists of a heavy granite slab (100 kg) sitting on legs which act as a vertical damped spring. The "Q" (Quality Factor) of the system is 8.0, and the velocity damping coefficient is 80 kg/s. What is the natural frequency of oscillation of the slab? Write down the equation of motion for the slab assuming that the ground is moving up and down like like A*cos(wt). I used w, but it's really an omega. At what frequency is this ground motion attenuated by a factor of 1000? (In other words at what frequency the slab moves with an amplitude of A/1000?)
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mass, m = 100 kg
Q = 8.0
velocity, damping coefficient, c = 80 kg/s
Natural frequency, fo = ?
Because, Q = 1/2r
r = daming ratio = c/c_cr
c_cr = critical damping coefficient = 2*m*wo
wo = fo*2*pi
=>c_cr = 2*m*wo = 2*m*fo*2*pi = 4*pi*m*fo
=> r = c/c_cr = c/(4*pi*m*fo)
=> Q = 1/(2*r) = 4*pi*m*fo/(2*c) = 2*pi*m*fo/c
=> fo = Q*c/(2*pi*m) = ...
A slab is oscillating and transferring the oscillations (or vibrations to the ground). These are forced as well as damped oscillations. Here we estimate corresponding Q value and a frequency where amplitude is attenuated.