Mathematics

Problem

Automobile shock absorbers can be modelled as a mass–spring–damper system. a. (i) Distinguish between under-damped, critically damped, and over-damped motions in terms of the relative magnitudes of

\beta

and the natural angular frequency

\omega_0

of the system. b. (ii) For the under-damped case, obtain an expression for the time

t_{e}

required for the amplitude to fall to

\dfrac{A_0}{e}

of its initial value

A_0

. c. (iii) A particular shock absorber has

\omega_0 = 4\pi\,\text{rad\,s}^{-1}

and

\beta = 0.30\,\text{s}^{-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.

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