The Analytical and Numerical Solutions of Differential Equations Describing of an Inclined Cable Subjected to External and Parametric Excitation Forces

Author(s)
Mohamed S. Abd Elkader

ABSTRACT

The analytical and numerical solutions of the response of an inclined cable subjected to external and parametric excitation forces is studied. The method of perturbation technique are applied to obtained the periodic response equation near the simultaneous principal parametric resonance in the presence of 2:1 internal resonance of the system. All different resonance cases are extracted. The effects of different parameters and worst resonance case on the vibrating system are investigated. The stability of the system are studied by using frequency response equations and phase-plane method. Variation of the parameters α2, α3, β2, γ2, η2, γ3, η3, f2 leads to multi-valued amplitudes and hence to jump phenomena. The simulation results are achieved using MATLAB 7.6 programs.

The analytical and numerical solutions of the response of an inclined cable subjected to external and parametric excitation forces is studied. The method of perturbation technique are applied to obtained the periodic response equation near the simultaneous principal parametric resonance in the presence of 2:1 internal resonance of the system. All different resonance cases are extracted. The effects of different parameters and worst resonance case on the vibrating system are investigated. The stability of the system are studied by using frequency response equations and phase-plane method. Variation of the parameters α2, α3, β2, γ2, η2, γ3, η3, f2 leads to multi-valued amplitudes and hence to jump phenomena. The simulation results are achieved using MATLAB 7.6 programs.

Cite this paper

nullM. Elkader, "The Analytical and Numerical Solutions of Differential Equations Describing of an Inclined Cable Subjected to External and Parametric Excitation Forces,"*Applied Mathematics*, Vol. 2 No. 12, 2011, pp. 1469-1478. doi: 10.4236/am.2011.212209.

nullM. Elkader, "The Analytical and Numerical Solutions of Differential Equations Describing of an Inclined Cable Subjected to External and Parametric Excitation Forces,"

References

[1] H. N. Arafat and A. H. Nayfeh, “Non-Linear Responses of Suspended Cables to Primary Resonance Excitations,” Journal of Sound and Vibration, Vol. 266, No. 2, 2003, pp. 325-354. doi:10.1016/S0022-460X(02)01393-7

[2] G. Rega, “Non-Linear Vibrations of Suspended Cables; Part I: Modeling and Analysis,” Journal of Applied Mechanics Review, Vol. 57, No. 6, 2004, pp. 443-478. doi:10.1115/1.1777224

[3] G. Rega, “Non-Linear Vibrations of Suspended Cables; Part II: Deterministic Phenomena,” Journal of Applied Mechanics Review, Vol. 57, No. 6, 2004, pp. 479-514. doi:10.1115/1.1777225

[4] S. R. Nielsen and P. H. Kirkegaard, “Super and Combinatorial Harmonic Response of Flexible Inclined Cables with Small Sag,” Journal of Sound and Vibration, Vol. 251, No. 1, 2002, pp. 79-102. doi:10.1006/jsvi.2001.3979

[5] G. Zheng, J. M. Ko and Y. O. Ni, “Super-Harmonic and Internal Resonances of a Suspended Cable with Nearly Commensurable Natural Frequencies,” Nonlinear Dynamics, Vol. 30, No. 1, 2002, pp. 55-70. doi:10.1023/A:1020395922392

[6] W. Zhang and Y. Tang, “Global Dynamics of the Cable under Combined Parametrical and External Excitations,” International Journal of Non-Linear Mechanics, Vol. 37, No. 3, 2002, pp. 505-526. doi:10.1016/S0020-7462(01)00026-9

[7] A. H. Nayfeh, H. Arafat, C. M. Chin and W. Lacarbonara, “Multimode Interactions in Suspended Cables,” Journal of Vibration and Control, Vol. 8, No. 3, 2002, pp. 337-387. doi:10.1177/107754602023687

[8] H. Chen and Q. Xu, “Bifurcation and Chaos of an Inclined Cable,” Nonlinear Dynamics, Vol. 57, No. 2-3, 2009, pp. 37-55. doi:10.1007/s11071-008-9418-3

[9] M. M. Kamel and Y. S. Hamed, “Non-Linear Analysis of an Inclined Cable under Harmonic Excitation,” Acta Mechanica, Vol. 214, No. 3-4, 2010, pp. 315-325. doi:10.1007/s00707-010-0293-x

[10] A. Abe, “Validity and Accuracy of Solutions for Nonlinear Vibration Analyses of Suspended Cables with One-To-One Internal Resonance,” Nonlinear Analysis: Real World Applications, Vol. 11, No. 4, 2010, pp. 2594-2602. doi:10.1016/j.nonrwa.2009.09.006

[11] N. Srinil, G. Rega and S. Chucheepsakul, “Two-To-One Resonant Multi-Modal Dynamics of Horizontal/Inclined Cables. Part I: Theoretical Formulation and Model Validation,” Nonlinear Dynamics, Vol. 48, No. 3, 2007, pp. 231-252. doi:10.1007/s11071-006-9086-0

[12] N. Srinil and G. Rega, “Two-To-One Resonant Multi-Modal Dynamics of Horizontal/Inclined Cables. Part II: Internal Resonance Activation Reduced-Order Models and Nonlinear Normal Modes,” Nonlinear Dynamics, Vol. 48, No. 3, 2007, pp. 253-274. doi:10.1007/s11071-006-9087-z

[13] R. Alaggio and G. Rega, “Characterizing Bifurcations and Classes of Motion in the Transition to Chaos through 3D-Tori of a Continuous Experimental System in Solid Mechanics,” Physica D, Vol. 137, No. 1, 2000, pp. 70-93. doi:10.1016/S0167-2789(99)00169-4

[14] G. Rega and R. Alaggio, “Spatio-Temporal Dimensionality in the Overall Complex Dynamics of an Experimental Cable/Mass System,” International Journal of Solids and Structures, Vol. 38, No. 10-13, 2001, pp. 2049-2068. doi:10.1016/S0020-7683(00)00152-9

[15] A. Gonzalez-Buelga, S. A. Neild, D. J. Wagg and J. H. G. Macdonald, “Modal Stability of Inclined Cables Subjected to Vertical Support Excitation,” Journal of Sound and Vibration, Vol. 318, No. 3, 2008, pp. 565-579. doi:10.1016/j.jsv.2008.04.031

[16] N. C. Perkins, “Modal Interactions in the Non-Linear Response of Inclined Cables under Parametric/External Excitation,” International Journal of Non-linear Mechanics, Vol. 27, No. 2, 1992, pp. 233-250. doi:10.1016/0020-7462(92)90083-J

[17] C. L. Lee and N. C. Perkins, “Nonlinear Oscillations of Suspended Cables Containing a Two-To-One Internal Resonance,” Nonlinear Dynamics, Vol. 3, 1992, pp. 465-490.

[18] C. L. Lee and N. C. Perkins, “Three-Dimensional Oscillations of Suspended Cables Involving Simultaneous Internal Resonance,” Proceedings of ASME Winter Annual Meeting AMD-14, 1992, pp. 59-67.

[19] 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.

[20] 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.

[21] 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

[22] M. Sayed, “Improving the Mathematical Solutions of Nonlinear Differential Equations Using Different Control Methods,” Ph.D. Thesis, Menofia University, Egypt, November 2006.

[23] 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

[24] 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

[25] 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

[26] 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.

[27] M. Sayed, Y. S. Hamed and Y. A. Amer, “Vibration Reduction and Stability of Non-Linear System Subjected to External and Parametric Excitation Forces under a Non- linear Absorber,” International Journal of Contemporary Mathematical Sciences, Vol. 6, No. 22, 2011, pp. 1051-1070.

[28] A. H. Nayfeh, “Non-linear Interactions”, Wiley/Inter-Science, New York, 2000.

[1] H. N. Arafat and A. H. Nayfeh, “Non-Linear Responses of Suspended Cables to Primary Resonance Excitations,” Journal of Sound and Vibration, Vol. 266, No. 2, 2003, pp. 325-354. doi:10.1016/S0022-460X(02)01393-7

[2] G. Rega, “Non-Linear Vibrations of Suspended Cables; Part I: Modeling and Analysis,” Journal of Applied Mechanics Review, Vol. 57, No. 6, 2004, pp. 443-478. doi:10.1115/1.1777224

[3] G. Rega, “Non-Linear Vibrations of Suspended Cables; Part II: Deterministic Phenomena,” Journal of Applied Mechanics Review, Vol. 57, No. 6, 2004, pp. 479-514. doi:10.1115/1.1777225

[4] S. R. Nielsen and P. H. Kirkegaard, “Super and Combinatorial Harmonic Response of Flexible Inclined Cables with Small Sag,” Journal of Sound and Vibration, Vol. 251, No. 1, 2002, pp. 79-102. doi:10.1006/jsvi.2001.3979

[5] G. Zheng, J. M. Ko and Y. O. Ni, “Super-Harmonic and Internal Resonances of a Suspended Cable with Nearly Commensurable Natural Frequencies,” Nonlinear Dynamics, Vol. 30, No. 1, 2002, pp. 55-70. doi:10.1023/A:1020395922392

[6] W. Zhang and Y. Tang, “Global Dynamics of the Cable under Combined Parametrical and External Excitations,” International Journal of Non-Linear Mechanics, Vol. 37, No. 3, 2002, pp. 505-526. doi:10.1016/S0020-7462(01)00026-9

[7] A. H. Nayfeh, H. Arafat, C. M. Chin and W. Lacarbonara, “Multimode Interactions in Suspended Cables,” Journal of Vibration and Control, Vol. 8, No. 3, 2002, pp. 337-387. doi:10.1177/107754602023687

[8] H. Chen and Q. Xu, “Bifurcation and Chaos of an Inclined Cable,” Nonlinear Dynamics, Vol. 57, No. 2-3, 2009, pp. 37-55. doi:10.1007/s11071-008-9418-3

[9] M. M. Kamel and Y. S. Hamed, “Non-Linear Analysis of an Inclined Cable under Harmonic Excitation,” Acta Mechanica, Vol. 214, No. 3-4, 2010, pp. 315-325. doi:10.1007/s00707-010-0293-x

[10] A. Abe, “Validity and Accuracy of Solutions for Nonlinear Vibration Analyses of Suspended Cables with One-To-One Internal Resonance,” Nonlinear Analysis: Real World Applications, Vol. 11, No. 4, 2010, pp. 2594-2602. doi:10.1016/j.nonrwa.2009.09.006

[11] N. Srinil, G. Rega and S. Chucheepsakul, “Two-To-One Resonant Multi-Modal Dynamics of Horizontal/Inclined Cables. Part I: Theoretical Formulation and Model Validation,” Nonlinear Dynamics, Vol. 48, No. 3, 2007, pp. 231-252. doi:10.1007/s11071-006-9086-0

[12] N. Srinil and G. Rega, “Two-To-One Resonant Multi-Modal Dynamics of Horizontal/Inclined Cables. Part II: Internal Resonance Activation Reduced-Order Models and Nonlinear Normal Modes,” Nonlinear Dynamics, Vol. 48, No. 3, 2007, pp. 253-274. doi:10.1007/s11071-006-9087-z

[13] R. Alaggio and G. Rega, “Characterizing Bifurcations and Classes of Motion in the Transition to Chaos through 3D-Tori of a Continuous Experimental System in Solid Mechanics,” Physica D, Vol. 137, No. 1, 2000, pp. 70-93. doi:10.1016/S0167-2789(99)00169-4

[14] G. Rega and R. Alaggio, “Spatio-Temporal Dimensionality in the Overall Complex Dynamics of an Experimental Cable/Mass System,” International Journal of Solids and Structures, Vol. 38, No. 10-13, 2001, pp. 2049-2068. doi:10.1016/S0020-7683(00)00152-9

[15] A. Gonzalez-Buelga, S. A. Neild, D. J. Wagg and J. H. G. Macdonald, “Modal Stability of Inclined Cables Subjected to Vertical Support Excitation,” Journal of Sound and Vibration, Vol. 318, No. 3, 2008, pp. 565-579. doi:10.1016/j.jsv.2008.04.031

[16] N. C. Perkins, “Modal Interactions in the Non-Linear Response of Inclined Cables under Parametric/External Excitation,” International Journal of Non-linear Mechanics, Vol. 27, No. 2, 1992, pp. 233-250. doi:10.1016/0020-7462(92)90083-J

[17] C. L. Lee and N. C. Perkins, “Nonlinear Oscillations of Suspended Cables Containing a Two-To-One Internal Resonance,” Nonlinear Dynamics, Vol. 3, 1992, pp. 465-490.

[18] C. L. Lee and N. C. Perkins, “Three-Dimensional Oscillations of Suspended Cables Involving Simultaneous Internal Resonance,” Proceedings of ASME Winter Annual Meeting AMD-14, 1992, pp. 59-67.

[19] 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.

[20] 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.

[21] 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

[22] M. Sayed, “Improving the Mathematical Solutions of Nonlinear Differential Equations Using Different Control Methods,” Ph.D. Thesis, Menofia University, Egypt, November 2006.

[23] 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

[24] 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

[25] 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

[26] 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.

[27] M. Sayed, Y. S. Hamed and Y. A. Amer, “Vibration Reduction and Stability of Non-Linear System Subjected to External and Parametric Excitation Forces under a Non- linear Absorber,” International Journal of Contemporary Mathematical Sciences, Vol. 6, No. 22, 2011, pp. 1051-1070.

[28] A. H. Nayfeh, “Non-linear Interactions”, Wiley/Inter-Science, New York, 2000.