# damped forced oscillations

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?)

I've never used this before, but please understand that I will not be upset if a solution is not 100% correct. I will be happy with your best answer based on the information provided, as I am sure that it will be significantly better than a solution I come up with. I've done well in this class (Classical Mechanics) up to this point, but every question is 10x harder than anything we have encountered before. If they were like previous problems we encountered, I would be more than happy and excited to solve them myself, but my confidence is taking a major beating.

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#### Solution Preview

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) = ...

#### Solution Summary

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.

Oscillator Force

2) A force Fext(t) = F0[ 1?exp(?((alpha)(t)) ] acts, for time t > 0, on an oscillator which is at rest at x=0 at time 0. The mass is m; the spring constant is k; and the damping force is ?b[(x)dot]. The parameters satisfy these relations:

b = m*q , k = 4*m*q^2 (where q is a constant with units of inverse time)

Find the motion. Determine x(t); and hand in a qualitatively correct graph of x(t).

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