A new closed-form method for inertia force and moment calculation in reciprocating piston engine design

Abstract

The piston crank mechanism is an important component of a reciprocating piston engine. It is an inherent vibration system, and as such, the calculation of unbalance quantity is a critical procedure in balancing mechanism design, which is adopted to balance inertia loading. The traditional method usually applies a Taylor series expansion with the crank-conrod ratio, then a Fourier transform with the crank angle. The Taylor expansion generally ignores the influence on calculations resulting from the high order terms. However, the high order terms of the Taylor expansion will also contribute to the low order terms in the Fourier series. This will induce poor precision in the inertia loading calculation, especially in a high crank-conrod ratio engine. Thus, this paper proposes a new closed-form method, which only adopts a Fourier transformation for the calculation. The coefficients of the Fourier transformation terms contain the contributions of all order terms of the crank-conrod ratio. Therefore, we named it as a closed-form method. Compared with the traditional method, the closed-form method improves the numerical accuracy of the secondary reciprocating inertia force by 1.5%–4%, when the crank-conrod ratio varies from 0.25 to 0.4. Using this new closed-form method to design a balancing mechanism, the primary and secondary reciprocating inertia forces can be completely balanced. For an engine, where the primary and secondary inertia forces are balanced, the ratio of the residual inertia force to the total inertia force using the traditional method is 1.5%, while the ratio decreases to 0.5% using the closed-form method. The closed-form method is independent of engine configurations, including centric and eccentric engines, and single and multi-cylinder engines. Examples of applications using the proposed method are provided.