This chapter considers the simplest system capable of vibratory motion: the single degree of freedom (SDOF) system, which, in its discrete parameter form, is often called the harmonic oscillator. Despite its apparent simplicity, this system contains and exhibits many of the essential features of vibrating systems and its analysis is a necessary prerequisite for any further investigation in both theory and practice. Moreover, its importance is also due to the fact that, in certain (special, but not rare) circumstances, many vibrating systems do behave and can be modelled as SDOF systems – at least in a rst approximation. When this is the case, clearly, both procedures of measurement and analysis can be considerably simplied.