Study of Stability and Vibration Reduction in Multi-Tool Ultrasonic Machining under Simultaneous Primary and Internal Resonance

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.

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.

Cite this paper

Y. Hamed, M. Elkader and H. Genedi, "Study of Stability and Vibration Reduction in Multi-Tool Ultrasonic Machining under Simultaneous Primary and Internal Resonance,"*Applied Mathematics*, Vol. 3 No. 1, 2012, pp. 1-11. doi: 10.4236/am.2012.31001.

Y. Hamed, M. Elkader and H. Genedi, "Study of Stability and Vibration Reduction in Multi-Tool Ultrasonic Machining under Simultaneous Primary and Internal Resonance,"

References

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[2] T. B. Thoe, D. K. Aspinwall and M. L. H Wise, “Review on Ultrasonic Machining,” Internationat Journal of Machine Tools and Manufactory, Vol. 38, No. 4, 1998, pp. 239-255.

[3] F. C. Lim, M. P. Cartmell, A. Cardoni and M. Lucas, “A Preliminary Investigation into Optimizing the Response of Vibrating Systems Used for Ultrasonic Cutting,” Journal of Sound Vibration, Vol. 272, No. 3-5, 2004, pp. 1047-1069. doi:10.1016/j.jsv.2003.03.011

[4] Y. A. Amer, “Vibration Control of Ultrasonic Cutting via Dynamic Absorber,” Chaos, Solitons & Fractals, Vol. 34, No. 2, 2007, pp. 1328-1345. doi:10.1016/j.chaos.2006.04.040

[5] K. R. Asfar, “Effect of Non-Linearities in Elastomeric Material Dampers on Torsional Vibration Control,” International Journal of Non-linear Mechanics, Vol. 27, No. 6, 1992, pp. 947-954. doi:10.1016/0020-7462(92)90047-B

[6] M. Eissa and H. M. Abdelhafez, “Stability and Control of Non-Linear Torsional Vibrating Systems,” Faculty of Engineering Alexandria University, Egypt, Vol. 41, No. 2, 2002, pp. 343-253.

[7] A. F. El-Bassiouny, “Effect of Non-Linearities in Elastomeric Material Dampers on Torsional Oscillation Control,” Journal of Applied Mathematics and Computation, Vol. 162, No. 2, 2005, pp. 835-854. doi:10.1016/j.amc.2003.12.142

[8] M. Eissa and W. El-Ganaini, “Multi-Absorbers for Vibration Control of Non-Linear Structures to Harmonic Excitations, Part I,” Proceedings of ISMV Conference, Is- lamabad, 2000.

[9] M. Eissa and W. El-Ganaini, “Multi-Absorbers for Vibration Control of Non-linear Structures to Harmonic Excitations, Part II,” Proceedings of ISMV Conference, Is- lamabad, 2000.

[10] M. Eissa and M. Sayed, “A Comparison between Active and passive Vibration Control of Non-Linear Simple Pendulum, Part I: Transversally Tuned Absorber and Negative Feedback,” Mathematical and Computational Applications, Vol. 11, No. 2, 2006, pp. 137-149.

[11] M. Eissa and M. Sayed, “A Comparison between Active and Passive Vibration Control of Non-Linear Simple Pendulum, Part II: Longitudinal Tuned Absorber and Negative Feedback,” Mathematical and Computa- tional Applications, Vol. 11, No. 2, 2006, pp. 151-162.

[12] M. Eissa and M. Sayed, “Vibration Reduction of a Three DOF Non-Linear Spring Pendulum,” Communications in Nonlinear Science and Numerical Simulation, Vol. 13, No. 2, 2008, pp. 465-488. doi:10.1016/j.cnsns.2006.04.001

[13] A. F. El-Bassiouny, “Three-to-One Internal Resonance in the Non linear Oscillation of Shallow Arch,” Physica Scripta, Vol. 72, No. 6, 2005, pp. 439-450. doi:10.1088/0031-8949/72/6/003

[14] M. Eissa and A. F. El-Bassiouny, “Response of Three-Degree-of-Freedom System with Cubic Non-Linearities to Harmonic Excitations,” Physica Scripta, Vol. 59, No. 3, 1999, pp. 183-194. doi:10.1238/Physica.Regular.059a00183

[15] M. V. Shitikova and Y. U. Rossikhin, “Analysis of Free Non-Linear Vibrations of a Viscoelastic Plate under the Conditions of Different Internal Resonances,” International Journal of Non-Linear Mechanics, Vol. 41, No. 2, 2006, pp. 313-325. doi:10.1016/j.ijnonlinmec.2005.08.002

[16] Y. A. Amer and A. T. El-Sayed, “Vibration Suppression of Non-Linear System via Non-Linear Absorber,” Communications in Nonlinear Science and Numerical Simulation, Vol. 13, No. 9, 2008, pp. 1948-1963. doi:10.1016/j.cnsns.2007.04.018

[17] A. F. El-Bassiouny, “Internal Resonance of a Nonlinear Vibration Absorber,” Physica Scripta, Vol. 72, No. 2-3, 2005, pp. 203-211. doi:10.1238/Physica.Regular.072a00203

[18] M. Eissa, S. EL-Serafi, H. El-Sherbiny and T. H. El-Ghareeb, “Comparison between Passive and Active Control of Non-Linear Dynamical System,” Japan Journal of Industrial and Applied Mathematics, Vol. 23, No. 2, 2006, pp. 139-161. doi:10.1007/BF03167548

[19] M. Eissa, S. El-Serafi, H. El-Sherbiny and T. H. El-Ghareeb, “On Passive and Active Control of Vibrating System,” International Journal of Applied Mathemetics, Vol. 18, No. 4, 2005, pp. 515-537.

[20] M. Eissa, S. EL-Serafi, H. El-Sherbiny and T. H. El-Ghareeb, “1:4 Internal Resonance Active Absorber for Non- Linear Vibrating System” International Journal of Pure and Applied Mathematics, Vol. 28, No. 1, 2006, pp. 515-537.

[21] M. Jaensch and M. U. Lampérth, “Development of a Multi-Degree-of-Freedom Micropositioning, Vibration Isolation and Vibration Suppression System,” Smart Material and Structure, Vol. 16, No. 2, 2007, pp. 409-417. doi:10.1088/0964-1726/16/2/020

[22] M. Eissa, Y. A. Amer and H. S. Bauomey, “Active Control of an Aircraft Tail Subject to Harmonic Excitation,” Acta Mechanica Sinica, Vol. 23, No. 4, 2007, pp. 451-462. doi:10.1007/s10409-007-0077-2

[23] Y. A. Amer and H. S. Bauomey, “Vibration Reduction in a 2DOF Twin-Tail System to Parametric Excitations,” Communications in Nonlinear Science and Numerical Si- mulation, Vol. 14, No. 2, 2009, pp. 560-573. doi:10.1016/j.cnsns.2007.10.005

[24] M. Eissa, W. El-Ganaini and Y. S. Hamed, “Saturation, Stability and Resonance of Non-Linear Systems” Physica A, Vol. 356, No. 2-4, 2005, pp. 341-358. doi:10.1016/j.physa.2005.01.058

[25] M. Eissa, W. El-Ganaini and Y. S. Hamed, “Optimum Working Conditions of a Non-Linear SDOF System to Harmonic and Multi-Parametric Excitations,” Scientific Bulletin, Part III: Mechanical Engineering and Physics & Mathematics, Vol. 40, No. 1, 2005, pp. 1113-1127.

[26] M. Eissa, W. El-Ganaini and Y. S. Hamed, “On the Saturation Phenomena and Resonance of Non-Linear Differential Equations,” Minufiya Journal of Electronic Engineering Research, Vol. 15, No. 1, 2005, pp. 73-84.

[27] M. M. Kamel W. El-Ganaini and Y. S. Hamed, “Vibration Suppression in Ultrasonic Machining Described by Non-Linear Differential Equations,” Journal of Mechanical Science and Technology, Vol. 23, No. 8, 2009, pp. 2038-2050.

[28] M. M. Kamel W. El-Ganaini and Y. S. Hamed, “Vibration Suppression in Multi-Tool Ultrasonic Machining to Multi-External and Parametric Excitations,” Acta Mechanica Sinica, Vol. 25, No. 3, 2009, pp. 403-415. doi:10.1007/s10409-009-0229-7

[29] M. M. Kamel W. El-Ganaini and Y. S. Hamed, “Vibration Reduction in Ultrasonic Machine to External and Tuned Excitation Forces,” Applied Mathematical Modeling, Vol. 33, No. 6, 2009, pp. 2853-2863. doi:10.1016/j.apm.2008.08.020

[1] R. Singh and J. S. Khambab, “Ultrasonic Machining of Titanium and Its Alloys: A Review,” Journal of Materials Processing Technology, Vol. 173, No. 2, 2006, pp. 125-135.

[2] T. B. Thoe, D. K. Aspinwall and M. L. H Wise, “Review on Ultrasonic Machining,” Internationat Journal of Machine Tools and Manufactory, Vol. 38, No. 4, 1998, pp. 239-255.

[3] F. C. Lim, M. P. Cartmell, A. Cardoni and M. Lucas, “A Preliminary Investigation into Optimizing the Response of Vibrating Systems Used for Ultrasonic Cutting,” Journal of Sound Vibration, Vol. 272, No. 3-5, 2004, pp. 1047-1069. doi:10.1016/j.jsv.2003.03.011

[4] Y. A. Amer, “Vibration Control of Ultrasonic Cutting via Dynamic Absorber,” Chaos, Solitons & Fractals, Vol. 34, No. 2, 2007, pp. 1328-1345. doi:10.1016/j.chaos.2006.04.040

[5] K. R. Asfar, “Effect of Non-Linearities in Elastomeric Material Dampers on Torsional Vibration Control,” International Journal of Non-linear Mechanics, Vol. 27, No. 6, 1992, pp. 947-954. doi:10.1016/0020-7462(92)90047-B

[6] M. Eissa and H. M. Abdelhafez, “Stability and Control of Non-Linear Torsional Vibrating Systems,” Faculty of Engineering Alexandria University, Egypt, Vol. 41, No. 2, 2002, pp. 343-253.

[7] A. F. El-Bassiouny, “Effect of Non-Linearities in Elastomeric Material Dampers on Torsional Oscillation Control,” Journal of Applied Mathematics and Computation, Vol. 162, No. 2, 2005, pp. 835-854. doi:10.1016/j.amc.2003.12.142

[8] M. Eissa and W. El-Ganaini, “Multi-Absorbers for Vibration Control of Non-Linear Structures to Harmonic Excitations, Part I,” Proceedings of ISMV Conference, Is- lamabad, 2000.

[9] M. Eissa and W. El-Ganaini, “Multi-Absorbers for Vibration Control of Non-linear Structures to Harmonic Excitations, Part II,” Proceedings of ISMV Conference, Is- lamabad, 2000.

[10] M. Eissa and M. Sayed, “A Comparison between Active and passive Vibration Control of Non-Linear Simple Pendulum, Part I: Transversally Tuned Absorber and Negative Feedback,” Mathematical and Computational Applications, Vol. 11, No. 2, 2006, pp. 137-149.

[11] M. Eissa and M. Sayed, “A Comparison between Active and Passive Vibration Control of Non-Linear Simple Pendulum, Part II: Longitudinal Tuned Absorber and Negative Feedback,” Mathematical and Computa- tional Applications, Vol. 11, No. 2, 2006, pp. 151-162.

[12] M. Eissa and M. Sayed, “Vibration Reduction of a Three DOF Non-Linear Spring Pendulum,” Communications in Nonlinear Science and Numerical Simulation, Vol. 13, No. 2, 2008, pp. 465-488. doi:10.1016/j.cnsns.2006.04.001

[13] A. F. El-Bassiouny, “Three-to-One Internal Resonance in the Non linear Oscillation of Shallow Arch,” Physica Scripta, Vol. 72, No. 6, 2005, pp. 439-450. doi:10.1088/0031-8949/72/6/003

[14] M. Eissa and A. F. El-Bassiouny, “Response of Three-Degree-of-Freedom System with Cubic Non-Linearities to Harmonic Excitations,” Physica Scripta, Vol. 59, No. 3, 1999, pp. 183-194. doi:10.1238/Physica.Regular.059a00183

[15] M. V. Shitikova and Y. U. Rossikhin, “Analysis of Free Non-Linear Vibrations of a Viscoelastic Plate under the Conditions of Different Internal Resonances,” International Journal of Non-Linear Mechanics, Vol. 41, No. 2, 2006, pp. 313-325. doi:10.1016/j.ijnonlinmec.2005.08.002

[16] Y. A. Amer and A. T. El-Sayed, “Vibration Suppression of Non-Linear System via Non-Linear Absorber,” Communications in Nonlinear Science and Numerical Simulation, Vol. 13, No. 9, 2008, pp. 1948-1963. doi:10.1016/j.cnsns.2007.04.018

[17] A. F. El-Bassiouny, “Internal Resonance of a Nonlinear Vibration Absorber,” Physica Scripta, Vol. 72, No. 2-3, 2005, pp. 203-211. doi:10.1238/Physica.Regular.072a00203

[18] M. Eissa, S. EL-Serafi, H. El-Sherbiny and T. H. El-Ghareeb, “Comparison between Passive and Active Control of Non-Linear Dynamical System,” Japan Journal of Industrial and Applied Mathematics, Vol. 23, No. 2, 2006, pp. 139-161. doi:10.1007/BF03167548

[19] M. Eissa, S. El-Serafi, H. El-Sherbiny and T. H. El-Ghareeb, “On Passive and Active Control of Vibrating System,” International Journal of Applied Mathemetics, Vol. 18, No. 4, 2005, pp. 515-537.

[20] M. Eissa, S. EL-Serafi, H. El-Sherbiny and T. H. El-Ghareeb, “1:4 Internal Resonance Active Absorber for Non- Linear Vibrating System” International Journal of Pure and Applied Mathematics, Vol. 28, No. 1, 2006, pp. 515-537.

[21] M. Jaensch and M. U. Lampérth, “Development of a Multi-Degree-of-Freedom Micropositioning, Vibration Isolation and Vibration Suppression System,” Smart Material and Structure, Vol. 16, No. 2, 2007, pp. 409-417. doi:10.1088/0964-1726/16/2/020

[22] M. Eissa, Y. A. Amer and H. S. Bauomey, “Active Control of an Aircraft Tail Subject to Harmonic Excitation,” Acta Mechanica Sinica, Vol. 23, No. 4, 2007, pp. 451-462. doi:10.1007/s10409-007-0077-2

[23] Y. A. Amer and H. S. Bauomey, “Vibration Reduction in a 2DOF Twin-Tail System to Parametric Excitations,” Communications in Nonlinear Science and Numerical Si- mulation, Vol. 14, No. 2, 2009, pp. 560-573. doi:10.1016/j.cnsns.2007.10.005

[24] M. Eissa, W. El-Ganaini and Y. S. Hamed, “Saturation, Stability and Resonance of Non-Linear Systems” Physica A, Vol. 356, No. 2-4, 2005, pp. 341-358. doi:10.1016/j.physa.2005.01.058

[25] M. Eissa, W. El-Ganaini and Y. S. Hamed, “Optimum Working Conditions of a Non-Linear SDOF System to Harmonic and Multi-Parametric Excitations,” Scientific Bulletin, Part III: Mechanical Engineering and Physics & Mathematics, Vol. 40, No. 1, 2005, pp. 1113-1127.

[26] M. Eissa, W. El-Ganaini and Y. S. Hamed, “On the Saturation Phenomena and Resonance of Non-Linear Differential Equations,” Minufiya Journal of Electronic Engineering Research, Vol. 15, No. 1, 2005, pp. 73-84.

[27] M. M. Kamel W. El-Ganaini and Y. S. Hamed, “Vibration Suppression in Ultrasonic Machining Described by Non-Linear Differential Equations,” Journal of Mechanical Science and Technology, Vol. 23, No. 8, 2009, pp. 2038-2050.

[28] M. M. Kamel W. El-Ganaini and Y. S. Hamed, “Vibration Suppression in Multi-Tool Ultrasonic Machining to Multi-External and Parametric Excitations,” Acta Mechanica Sinica, Vol. 25, No. 3, 2009, pp. 403-415. doi:10.1007/s10409-009-0229-7

[29] M. M. Kamel W. El-Ganaini and Y. S. Hamed, “Vibration Reduction in Ultrasonic Machine to External and Tuned Excitation Forces,” Applied Mathematical Modeling, Vol. 33, No. 6, 2009, pp. 2853-2863. doi:10.1016/j.apm.2008.08.020