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    A mass m is supported by a spring with spring constant σ and a natural length xo. The mass is immersed in a viscous liquid such that there is a resistive force F = -nv acting on it where v is the velocity of m and n is a constant. Gravity acts downward. The block to which the spring is attached oscillates up and down with amplitude A and frequency f.
    (a) Using Newton's laws, show that the differential equation describing the motion of m can be written
    mz'' + nz' +σz = σAcos(ft)
    where z is the displacement of m downwards from its equilibrium position, being the position of the mass when the support is not vibrating.
    I said that according to Newton's 2nd law F = ma = mz''. The gravitational force is simply F = mg (downwards) and Hooke's spring force is F = -σz. We are told the resistive force is F = -nv = -nz'. The force from the oscillating platform is represented by F = σAcos(ft)
    Putting this all together you get the equation of motion:
    mz'' + nz' +σz = σAcos(ft)

    (b) At what frequency would the system vibrate if the support were motionless? State an explicit criterion for the existence of oscillatory behaviour.
    If the support were motionless then A = 0, so σAcos(ft) = 0, hence mz'' + nz' +σz = 0. I don't know how to get the frequency from here? The "explicit criterion" of oscillatory behaviour has to mean at what point will it definately oscillate. ie: The system is not over damped. The system is overdamped if σ/2m >> f so I guess the criterion must be for σ/2m < f
    (c) How does the magnitude of the motion of m depend on the magnitude of A?
    Directly proportional?
    (d) Explain in words what happens to the motion of m in the limits:
    (i) f -> 0
    (ii) f -> ∞
    (iii) f -> resonant frequency

    It's been a while since I did this stuff, so I'm not totally confident about my comments. Anyway, hope it helps.
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  1. File Type: pdf Forced damped vibration.pdf (43.4 KB, 299 views)
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Updated: August 31, 2005

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