The object of this thesis is to investigate the validity of a popular approximate procedure which is used to select the optimum deeming for a Given mechanical system. Since this cannot be done in general, a particular system is chosen. Me exact determination of the optimum damping for a very simple mechanical system is performed. This value of optimum damping is then compared with the value obtained by the popular, so-called "fixed point" method. Since the fixed point solution is obtained independently of the initial dumping of the systems, the exact solution is worked out for sets of values of initial damping, which are typical of practical mechanical devices, to determined the influence of the initial damping on the optimum damping. It is found that for this particular system the fixed point method end the exact method give the same value for optima damping then there is no initial clamping in the system. As more and more initial clamping is added to the system the value for optimum clomping increases significantly, but it becomes a much less serious to use a value different from the optimum value.