Abstract
The main object of this paper is the mathematical study of the vibration behavior in ultrasonic machining (USM) described by non-linear differential equations. The ultrasonic machining (USM) consists of the tool holder and the absor-bers representing the tools. This leads to four-degree-of-freedom system subject to multi-external excitation forces. The aim of this project is the reduction of the vibrations in the tool holder and have reasonable amplitudes for the tools represented by the multi-absorbers. Multiple scale perturbation method is applied to obtain the solution up to the second order approximation and to study the stability of the steady state solution near different simultaneous resonance cases. The resulting different resonance cases are reported and studied numerically. The stability of the steady state solution near the selected resonance cases is studied applying both frequency response equations and phase-plane technique. The effects of the different parameters of the system and the absorbers on the system behavior are studied numerically. Optimum working conditions for the tools were obtained. Comparison with the available published work is reported.
Highlights
Ultrasonic machining (USM) is of particular interest for the machining of non-conductive, brittle materials such as engineering ceramics
The main object of this paper is the mathematical study of the vibration behavior in ultrasonic machining (USM) described by non-linear differential equations
Rupinder and Aspinwall [1,2] introduced a review for the fundamental principles of stationary ultrasonic machining, the material removal mechanisms involved and the effect of operating parameters on material removal rate, tool wear rate, and work piece surface finish of titanium and its alloys for application in manufacturing industry
Summary
Ultrasonic machining (USM) is of particular interest for the machining of non-conductive, brittle materials such as engineering ceramics. Amer [4] investigated the coupling of two non-linear oscillators of the main system and absorber representing ultrasonic cutting process subjected to parametric excitation forces. A threshold value of main system linear damping has been obtained, where vibration can be reduced dramatically This threshold value can be used effectively for passive vibration control, if it is economical. El-Bassiouny and Jaensch [17-23] showed how effective is the active control in vibration reduction at resonance at different modes of vibration. They demonstrated the advantages of active control over the passive one. The system represents the vibration of a single-degree-of-freedom cantilever or the wing of an aircraft They reported the occurrence of saturation phenomena at different parameters values. El Ganaini et al [27-29] studied USM model subject to multi-external or AM
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