The Equivalence of Continuous and Discrete Mass Distributions in Certain Vibration Problems

Several numerical methods for determining the natural frequencies of non-uniform shafts in torsional and transverse vibration require the replacement of any continuous distribution of matter by an ‘equivalent system’, in which the mass or inertia is concentrated at particular points. This paper discusses the error introduced by this process by considering a few simple cases, in which both the continuous and the discrete systems can be treated exactly. Following a brief statement of the results obtained by previous authors, an expression is derived for the frequency error in the case of a uniform simply supported beam in transverse vibration. It is shown that if the continuous mass distribution of the beam is replaced by a set of equally spaced concentrated masses, the errors in the natural frequencies are proportional to the inverse fourth power of the number of segments into which the beam is divided.