A 3D Free Vibration Analysis of the Horn-Gear System through Chebyshev–Ritz Method in Ultrasonic Gear Honing

Abstract

Applying the ultrasonic machining in gear honing can improve honing speed, reduce cutting force, and avoid blocking. There are two problems leading to the decrease of calculation accuracy in the traditional nonresonant theory of the ultrasonic gear honing. One is that one-dimensional longitudinal vibration theory and two-dimensional theory cannot reflect the vibration characteristics of ultrasonic horn and gear comprehensively. And, the other one is that the difference of the analysis dimension between the two theories leads to mismatch of the coupling condition dimension between ultrasonic horn and gear. A free vibration analysis through Chebyshev–Ritz method based on three-dimensional elasticity theory was presented to analyze the eigenfrequencies of the horn-gear system in ultrasonic gear honing. In the method, the model of the horn-gear system was divided into four parts: a solid circular plate, an annular plate, a solid cylinder, and a cone with hole. The eigenvalue equations were derived by using displacement coupling condition between each part under completely free boundary condition. It was found that the eigenfrequencies were highly convergent through convergence study. The hammering method for a modal experiment was used to test the horn-gear systems’ eigenfrequencies. And, the finite element method was also applied to solve the eigenfrequencies. Through a comparative analysis of the frequencies obtained by these three methods, it showed that the results achieved by the Chebyshev–Ritz method were close to those obtained from the experiment and finite element method. Thus, it was feasible to use the Chebyshev–Ritz method to solve the eigenfrequencies of the horn-gear system in ultrasonic gear honing.