Automobile shock absorbers can be modelled as a mass–spring–damper system.
(i) Distinguish between under-damped, critically damped, and over-damped motions in terms of the relative magnitudes of β and the natural angular frequency ω0 of the system.
(ii) For the under-damped case, obtain an expression for the time te required for the amplitude to fall to eA0 of its initial value A0.
(iii) A particular shock absorber has ω0=4πrads−1 and β=0.30s−1. State whether the motion is oscillatory. If it is, calculate the number of complete oscillations that occur before the amplitude becomes half of its initial value.