Equations of Finite Differences Applied to Torsional Oscillations of Crankshafts

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Abstract
From the viewpoint of torsional oscillations an internal combustion engine with a long crankshaft is generally considered to be equivalent to a uniform shaft carrying equidistant identical disks. It is here shown that advantage can be taken of the regularity of such a system to simplify the calculation of torsional oscillations. This is done by applying a mathematical method known as the calculus of finite differences. The procedure leads to a frequency equation (2.7) of remarkable symmetry in which appear as parameter the number n of cylinders in line and two simple functions K1 and K2 of the frequency which characterize the dynamical properties of the machines coupled at both ends of the crankshaft. These characteristic functions are of the nature of mechanical impedances, but due to their physical interpretation as a spring modulus (or spring constant) generalized to dynamic phenomena, the appellation dynamic modulus is being preferably used in the present paper. The concept of dynamic modulus is briefly introduced in the first section, while the second deals with the establishment of the frequency equation and an artifice for its rapid graphical solution avoiding the necessity of plotting an oscillatory function. Numerical applications to Diesel engines are treated in the last section. An example is also given of an extreme case where the fundamental frequency has a very low value and a special method is used for the calculation of this frequency.

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