Analysis of Two-Dimensional Bending Vibration of an Internal Combustion Engine’s Shafting System Based on Wave Equation

Abstract

Reciprocating machines are widely used in many mechanical industries. Bending vibration of spinning components, such as the shaft system of a high-speed power engine (vehicles, ships, helicopters, etc.), turbine engines and generators usually requires the integration of both design and analysis. The general method to analyze vibration is the discretization method. The bending/torsion/axial vibration of these systems is often solved using the lumped parameter method, in which small diameter long shafts are neglected by concentrated masses. Well known since the Holzer’s tabulating methods, many researchers proposed both transfer matrix and extended transfer matrix methods to solve free and/or forced vibration under dampened and/or undampened vibration based on discrete methods. In many situations, however, the results from a discrete method couldn’t exactly agree with actual results. In particular, in bending vibration analysis there is no correction and precision method to calculate bending vibration of multi-stepped spin shafts because the shafts are often treated as one dimensional vibration. We should therefore develop a new precision method to analyze two-dimensional bending vibration of shaft systems. In this paper, a new precise approach is proposed to study and calculate the bending vibration of multi-step systems with N-stepped changes of their property under rotary conditions, supported by N+1 springs and dampers governed by a two-dimensional wave equation. At first, the author analyzed the source of bending vibration. When a crankshaft is excited by external forces which can be divided into vertical and horizontal forces, the shaft will deform in two planes. Then strain and stress equations of the spin shaft are established, applied to the Hamilton Principle, and a two-dimensional wave equation is established. After that, to simplify, the author deduced the calculation formula of one dimensional bending vibration — a sine exciting force acted on arbitrary concentrated masses between point No.1 and No.N+1. By using this formula, the total dynamic response of additional force can be obtained through a general procedure of superposition. And then, the influence of the coupled product of inertia is discussed and the two-dimensional dynamic response is obtained. To demonstrate the theoretic analysis, we checked the prediction using the bending vibration of a highspeed diesel engine’s crankshaft. By using a newly-designed axial/ torsion/ bending vibration testing device, we measured the bending vibration at the front free-end of the crankshaft. When the calculated results were compared with measured results, they agreed with each other. This verified that the author’s proposed new method to study bending vibration is warranted and correct.