Abstract
Vibrations are usually undesired phenomena as they may cause damage or destruction of the system. However, sometimes they are desirable, as in ultrasonic machining (USM). In such case, the problem is a complicated one, as it is required to reduce the vibration of the machine head and have reasonable amplitude for the tool. In the present work, the coupling of two non-linear oscillators of the tool holder and tool representing ultrasonic cutting process is investigated. This leads to a two-degree-of-freedom system subjected to multi-external excitation force. The aim of this work is to control the tool holder behavior at simultaneous primary and internal resonance condition and have high amplitude for the tool. Multiple scale perturbation method is applied to obtain a solution up to the second order approximations. Other different resonance cases are reported and studied numerically. The stability of the system is investigated applying both phase-plane and frequency response techniques. The effects of the different parameters of the tool on the system behavior are studied numerically. Comparison with the available published work is reported.
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This paper was recommended for publication in revised form by Associate Editor Eung-Soo Shin
M. M. Kamel received his B.S. degree in Mathematics from Ain Shams University, EGYPT, in 1979. He then received his M.S.c degrees from Ain Shams University, in 1986 and Ph.D. degrees from Menofia University, in 1994. Dr. M. M. Kamel is currently an Associate Professor of Mathematics at the Department of Engineering Mathematics, Faculty of Electronic Engineering Menofia University, Egypt. Dr. M. M. Kamel research interests include Differential equations, Numerical Analysis, and Vibration control.
W. A. A. El-Ganini received her B.S. degree in Mathematics from Ain Shams University, EGYPT, in 1980. She then received her M.S.c and Ph.D. degrees from Suez Canal University, in 1984 and 1989, respectively. Dr. W. A. A. El-Ganini is currently an Assistant Professor of Mathematics at the Department of Engineering Mathematics, Faculty of Electronic Engineering Menofia University, Egypt. Dr. W. A. A. El-Ganaini research interests include Differential equations, Numerical Analysis, and Vibration control.
Y. S. Hamed received his B.S. degree in Mathematics from Menofia University, EGYPT, in 1998. He then received his M.S.c and Ph.D. degrees from Menofia University, in 2005 and 2009, respectively. Dr. Y. S. Hamed is currently an Assistant Professor of Pure Mathematics at the Department of Engineering Mathematics, Faculty of Electronic Engineering Menofia University, Egypt. Dr. Y. S. Hamed research interests include Differential equations, Numerical Analysis, and Vibration control.
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Kamel, M.M., El-Ganaini, W.A.A. & Hamed, Y.S. Vibration suppression in ultrasonic machining described by non-linear differential equations. J Mech Sci Technol 23, 2038–2050 (2009). https://doi.org/10.1007/s12206-009-1208-9
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DOI: https://doi.org/10.1007/s12206-009-1208-9