Non-Linear Analysis of Vibrations of Non-Linear System Subjected to Multi-Excitation Forces via a Non-Linear Absorber

ABSTRACT

The dynamic response of mechanical and civil structures subjected to high-amplitude vibration is often dangerous and undesirable. Sometimes controlled vibration is desirable as in the machinery used in the formation of rigid hard material as Ceramics and diamond. This process is done via the passive control methods. The main purpose of this paper is to how reduction of vibration of nonlinear system subjected to multi-excitation forces via a nonlinear absorber. The nonlinear differential equations describing the model, which describe ultrasonic cutting machine are solved by using perturbation method. The effects of different parameter on the response of the system are studied. The stability of the numerical solution is investigated by using frequency response equations and phase-plane method. The simulation results are achieved using Matlab and Maple programs. A comparison is made with the available published work.

The dynamic response of mechanical and civil structures subjected to high-amplitude vibration is often dangerous and undesirable. Sometimes controlled vibration is desirable as in the machinery used in the formation of rigid hard material as Ceramics and diamond. This process is done via the passive control methods. The main purpose of this paper is to how reduction of vibration of nonlinear system subjected to multi-excitation forces via a nonlinear absorber. The nonlinear differential equations describing the model, which describe ultrasonic cutting machine are solved by using perturbation method. The effects of different parameter on the response of the system are studied. The stability of the numerical solution is investigated by using frequency response equations and phase-plane method. The simulation results are achieved using Matlab and Maple programs. A comparison is made with the available published work.

Cite this paper

T. El-Ghareeb, Y. Hamed and M. Abd Elkader, "Non-Linear Analysis of Vibrations of Non-Linear System Subjected to Multi-Excitation Forces via a Non-Linear Absorber,"*Applied Mathematics*, Vol. 3 No. 1, 2012, pp. 64-72. doi: 10.4236/am.2012.31011.

T. El-Ghareeb, Y. Hamed and M. Abd Elkader, "Non-Linear Analysis of Vibrations of Non-Linear System Subjected to Multi-Excitation Forces via a Non-Linear Absorber,"

References

[1] S. S. Oueini and A. H. Nayfeh, “Single-Mode Control of a Cantilever Beam under Principal Parametric Excitation,” Journal of Sound and Vibration, Vol. 224, No. 1, 1999, pp. 33-47. doi:10.1006/jsvi.1998.2028

[2] 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

[3] M. Eissa, “Vibration Control of Non-Linear Mechanical Systems via Neutralizers,” Electronic Engineering Bulletin, No. 18, 1999.

[4] M. Eissa, “Vibration and Chaos Control in I.C Engines Subject to Harmonic Torque via Non-Linear Absorbers,” Proceedings of ISMV Conference, Islamabad, 2000.

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

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

[7] M. M. Kamel and Y. A. Amer, “Response of Parametrically Excited One-Degree-of-Freedom System with Non-Linear Damping and Stiffness,” Physica Scripta, Vol. 66, No. 6, 2002, pp. 410-416. doi:10.1238/Physica.Regular.066a00410

[8] Y. Song, H. Sato, Y. Iwata and T. Komatsuzaki, “The Response of a Dynamic Vibration Absorber System with a Parametrically Excited Pendulum,” Journal of Sound and Vibration, Vol. 259, No. 4, 2003, pp. 747-759. doi:10.1006/jsvi.2002.5112

[9] A. Soom and M. Lee, “Optimal Design of Linear and Non-Linear Vibration Absorbers for Damped System,” Journal of Vibration, Acoustic Stress, and Reliability in Design, Vol. 105, No. 4, 1983, pp. 112-119. doi:10.1115/1.3269054

[10] I. N. Jordanov and B. I. Cheshankov, “Optimal Design of Linear and Non-Linear Dynamic Vibration Absorbers,” Journal of Sound and Vibration, Vol. 123, No. 1, 1988, pp. 157-170. doi:10.1016/S0022-460X(88)80085-3

[11] H. J. Rice, “Combinational Instability of the Non-Linear Vibration Absorber,” Journal of Sound and Vibration, Vol. 108, No. 4, 1986, pp. 526-532. doi:10.1016/S0022-460X(86)80046-3

[12] J. Shaw, S. W. Shaw and A. G. Haddow, “On the Response of the Non-Linear Vibration Absorber,” International Journal of Non-Linear Mechanics, Vol. 24, No. 4, 1989, pp. 281-293. doi:10.1016/0020-7462(89)90046-2

[13] S. Natsiavas, “Steady State Oscillations and Stability of Non-Linear Dynamic Vibration Absorbers,” Journal of Sound and Vibration, Vol. 156, No. 2, 1992, pp. 227-245. doi:10.1016/0022-460X(92)90695-T

[14] S. J. Zhu, Y. F. Zheng and Y. M. Fu, “Analysis of Non-Linear Dynamics of a Two-Degree-of-Freedom Vibration System with Non-Linear Damping and Non-Linear Spring,” Journal of Sound and Vibration, Vol. 271, No. 1-2, 2004, pp. 15-24. doi:10.1016/S0022-460X(03)00249-9

[15] A. H. Nayfeh, “Resolving Controversies in the Application of the Method of Multiple Scales and the Generalized Method of Averaging,” Nonlinear Dynamics, Vol. 40, No. 1, 2005, pp. 61-102. doi:10.1007/s11071-005-3937-y

[16] Y. A. Amer, “Vibration Control of Ultrasonic Cutting via Dynamic Absorber,” Chaos, Solutions and Fractals, Vol. 33, No. 5, 2007, pp. 1703-1710. doi:10.1016/j.chaos.2006.03.038

[17] M. Eissa and M. Sayed, “A Comparison between Passive and Active Control of Non-Linear Simple Pendulum Part-I,” Mathematical and Computational Applications, Vol. 11, No. 2, 2006, pp. 137-149.

[18] M. Eissa and M. Sayed, “A Comparison between Passive and Active control of Non-Linear Simple Pendulum Part-II,” Mathematical and Computational Applications Vol. 11, No. 2, 2006, pp. 151-162.

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

[20] M. Sayed, “Improving the Mathematical Solutions of Nonlinear Differential Equations Using Different Control Methods,” Ph.D. Thesis, Menofia University, Shebin El-Koom, 2006.

[21] Y. S. Hamed, W. El-Ganaini and M. M. Kamel, “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. doi:10.1007/s12206-009-1208-9

[22] Y. S. Hamed, W. El-Ganaini and M. M. Kamel, “Vibration Suppression in Multitool 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

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

[24] M. Sayed and Y. S. Hamed, “Stability and Response of a Nonlinear Coupled Pitch-Roll Ship Model under Parametric and Harmonic Excitations,” Nonlinear Dynamics, Vol. 64, No. 3, 2011, pp. 207-220. doi:10.1007/s11071-010-9841-0

[25] M. Sayed and M. Kamel, “Stability Study and Control of Helicopter Blade Flapping Vibrations,” Applied Mathematical Modelling, Vol. 35, No. 6, 2011, pp. 2820-2837. doi:10.1016/j.apm.2010.12.002

[26] M. Sayed and M. Kamel, “1:2 and 1:3 Internal Resonance Active Absorber for Non-Linear Vibrating System,” Applied Mathematical Modelling, Vol. 36, No. 1, 2012, pp. 310-332. doi:10.1016/j.apm.2011.05.057

[27] Y. A. Amer and M. Sayed, “Stability at Principal Resonance of Multi-Parametrically and Externally Excited Mechanical System,” Advances in Theoretical and Applied Mechanics, Vol. 4, No. 1, 2011, pp. 1-14.

[28] A. H. Nayfeh, “Introduction to Perturbation Techniques,” John Wiley & Sons, New York, 1981.

[1] S. S. Oueini and A. H. Nayfeh, “Single-Mode Control of a Cantilever Beam under Principal Parametric Excitation,” Journal of Sound and Vibration, Vol. 224, No. 1, 1999, pp. 33-47. doi:10.1006/jsvi.1998.2028

[2] 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

[3] M. Eissa, “Vibration Control of Non-Linear Mechanical Systems via Neutralizers,” Electronic Engineering Bulletin, No. 18, 1999.

[4] M. Eissa, “Vibration and Chaos Control in I.C Engines Subject to Harmonic Torque via Non-Linear Absorbers,” Proceedings of ISMV Conference, Islamabad, 2000.

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

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

[7] M. M. Kamel and Y. A. Amer, “Response of Parametrically Excited One-Degree-of-Freedom System with Non-Linear Damping and Stiffness,” Physica Scripta, Vol. 66, No. 6, 2002, pp. 410-416. doi:10.1238/Physica.Regular.066a00410

[8] Y. Song, H. Sato, Y. Iwata and T. Komatsuzaki, “The Response of a Dynamic Vibration Absorber System with a Parametrically Excited Pendulum,” Journal of Sound and Vibration, Vol. 259, No. 4, 2003, pp. 747-759. doi:10.1006/jsvi.2002.5112

[9] A. Soom and M. Lee, “Optimal Design of Linear and Non-Linear Vibration Absorbers for Damped System,” Journal of Vibration, Acoustic Stress, and Reliability in Design, Vol. 105, No. 4, 1983, pp. 112-119. doi:10.1115/1.3269054

[10] I. N. Jordanov and B. I. Cheshankov, “Optimal Design of Linear and Non-Linear Dynamic Vibration Absorbers,” Journal of Sound and Vibration, Vol. 123, No. 1, 1988, pp. 157-170. doi:10.1016/S0022-460X(88)80085-3

[11] H. J. Rice, “Combinational Instability of the Non-Linear Vibration Absorber,” Journal of Sound and Vibration, Vol. 108, No. 4, 1986, pp. 526-532. doi:10.1016/S0022-460X(86)80046-3

[12] J. Shaw, S. W. Shaw and A. G. Haddow, “On the Response of the Non-Linear Vibration Absorber,” International Journal of Non-Linear Mechanics, Vol. 24, No. 4, 1989, pp. 281-293. doi:10.1016/0020-7462(89)90046-2

[13] S. Natsiavas, “Steady State Oscillations and Stability of Non-Linear Dynamic Vibration Absorbers,” Journal of Sound and Vibration, Vol. 156, No. 2, 1992, pp. 227-245. doi:10.1016/0022-460X(92)90695-T

[14] S. J. Zhu, Y. F. Zheng and Y. M. Fu, “Analysis of Non-Linear Dynamics of a Two-Degree-of-Freedom Vibration System with Non-Linear Damping and Non-Linear Spring,” Journal of Sound and Vibration, Vol. 271, No. 1-2, 2004, pp. 15-24. doi:10.1016/S0022-460X(03)00249-9

[15] A. H. Nayfeh, “Resolving Controversies in the Application of the Method of Multiple Scales and the Generalized Method of Averaging,” Nonlinear Dynamics, Vol. 40, No. 1, 2005, pp. 61-102. doi:10.1007/s11071-005-3937-y

[16] Y. A. Amer, “Vibration Control of Ultrasonic Cutting via Dynamic Absorber,” Chaos, Solutions and Fractals, Vol. 33, No. 5, 2007, pp. 1703-1710. doi:10.1016/j.chaos.2006.03.038

[17] M. Eissa and M. Sayed, “A Comparison between Passive and Active Control of Non-Linear Simple Pendulum Part-I,” Mathematical and Computational Applications, Vol. 11, No. 2, 2006, pp. 137-149.

[18] M. Eissa and M. Sayed, “A Comparison between Passive and Active control of Non-Linear Simple Pendulum Part-II,” Mathematical and Computational Applications Vol. 11, No. 2, 2006, pp. 151-162.

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

[20] M. Sayed, “Improving the Mathematical Solutions of Nonlinear Differential Equations Using Different Control Methods,” Ph.D. Thesis, Menofia University, Shebin El-Koom, 2006.

[21] Y. S. Hamed, W. El-Ganaini and M. M. Kamel, “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. doi:10.1007/s12206-009-1208-9

[22] Y. S. Hamed, W. El-Ganaini and M. M. Kamel, “Vibration Suppression in Multitool 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

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

[24] M. Sayed and Y. S. Hamed, “Stability and Response of a Nonlinear Coupled Pitch-Roll Ship Model under Parametric and Harmonic Excitations,” Nonlinear Dynamics, Vol. 64, No. 3, 2011, pp. 207-220. doi:10.1007/s11071-010-9841-0

[25] M. Sayed and M. Kamel, “Stability Study and Control of Helicopter Blade Flapping Vibrations,” Applied Mathematical Modelling, Vol. 35, No. 6, 2011, pp. 2820-2837. doi:10.1016/j.apm.2010.12.002

[26] M. Sayed and M. Kamel, “1:2 and 1:3 Internal Resonance Active Absorber for Non-Linear Vibrating System,” Applied Mathematical Modelling, Vol. 36, No. 1, 2012, pp. 310-332. doi:10.1016/j.apm.2011.05.057

[27] Y. A. Amer and M. Sayed, “Stability at Principal Resonance of Multi-Parametrically and Externally Excited Mechanical System,” Advances in Theoretical and Applied Mechanics, Vol. 4, No. 1, 2011, pp. 1-14.

[28] A. H. Nayfeh, “Introduction to Perturbation Techniques,” John Wiley & Sons, New York, 1981.