The dynamic analysis of the running hardware of reciprocating machines is complex and is usually dealt with by using a number of simplifying assumptions. Usually an ''equivalent dynamic system'' is built for performing the torsional analysis: such equivalent system has inertial properties which are assumed to be constant and the variation of the actual configuration is taken into account only by adding suitable ''inertia torques'' to the driving torques. The aim of the present paper is that of studying the torsional vibration of crankshafts with account taken of the variation of the geometry of the system with the crank angle; both the free behaviour and the response to external excitation are dealt with. The analysis is linearized and a mathematical model having the form of a set of linear differential equations with periodic coefficients is obtained. The solution of the free behaviour is obtained through a formulation similar to Hill's infinite determinant, whose truncated forms yield an approximated solution of accuracy increasing with the number of harmonics which are retained. A similar method allows the forced response to be computed. Two examples, one related to a simplified single-cylinder machine and one to an actual aircraft engine, conclude the work.