Identification of Nonlinear Modal Interactions in a Beam-Mass-Spring-Damper System based on Mono-Frequency Vibration Response
Authors
Abstract
In this paper, nonlinear modal interactions caused by one-to-three internal resonance in a beam-mass-spring-damper system are investigated based on nonlinear system identification. For this purpose, the equations governing the transverse vibrations of the beam and mass are analyzed via the multiple scale method and the vibration response of the system under primary resonance is extracted. Then, the frequency behavior of the vibration response is studied by Fourier and Morlet wavelet transforms. In order to perform the nonparametric identification of the time response, mono-frequency intrinsic mode functions are derived by the advanced empirical mode decomposition. In this approach, masking signals are utilized in order to avoid mode mixing caused by modal interaction. After analyzing the frequency behavior of each mode function, slow flow dynamics of the system is established and intrinsic modal oscillators for reconstructing the corresponding intrinsic mode are extracted. Finally, by analyzing the beating phenomenon in a simple one-degree-of-freedom system, it is shown that the internal resonance causes beating only under the circumstance that the slope of the logarithmic amplitude of oscillator force is nonzero. The results, therefore, show that depending on the periodic, pseudo-periodic, and chaotic behavior of the response, modal interactions might be stationary or non-stationary. Moreover, the chaotic behavior occurs mostly in the vibration mode excited by the internal resonance mechanism
Keywords
1. Laura, P., Susemihl, E., Pombo, J., Luisoni, L., and Gelos, R., “On the Dynamic Behaviour of Structural Elements Carrying Elastically Mounted, Concentrated Masses”, Applied Acoustics, Vol. 10, No. 2, pp. 121-145, 1977.
2. Hijmissen, J., Van den Heuvel, N., and Van Horssen, W., “On the Effect of the Bending Stiffness on the Damping Properties of a Tensioned Cable with an Attached Tuned-Mass-Damper”, Engineering Structures, Vol. 31, No. 5, pp. 1276-1285, 2009.
3. Sadeghi, M. H., and Lotfan, S., “Nonparametric System Identification of a Cantilever Beam Model with Local Nonlinearity in the Presence of Artificial Noise”, Modares Mechanical Engineering, Vol. 16, No. 11, pp. 177-186, 2017.
4. Wu, J.-S., and Chen, D.-W., “Dynamic Analysis of a Uniform Cantilever Beam Carrying a Number of Elastically Mounted Point Masses with Dampers”, Journal of Sound and Vibration, Vol. 229, No. 3, pp. 549-578, 2000.
5. Hamdan, M., and Jubran, B., “Free and Forced Vibrations of a Restrained Uniform Beam Carrying an Intermediate Lumped Mass and a Rotary Inertia”, Journal of Sound and Vibration, Vol. 150, No. 2, pp. 203-216, 1991.
6. Bambill, D., and Rossit, C., “Forced Vibrations of a Beam Elastically Restrained Against Rotation and Carrying a Spring-Mass System”, Ocean Engineering, Vol. 29, No. 6, pp. 605-626, 2002.
7. Kukla, S., and Posiadala, B., “Free Vibrations of Beams with Elastically Mounted Masses”, Journal of Sound and Vibration, Vol. 175, No. 4, pp. 557-564, 1994.
8. Wu, J.-S., and Chen, D.-W. “Free Vibration Analysis of a Timoshenko Beam Carrying Multiple Spring-Mass Systems by using the Numerical Assembly Technique”, International Journal for Numerical Methods in Engineering, Vol. 50, No. 5, pp. 1039-1058, 2001.
9. Yesilce, Y., and Demirdag, O., “Effect of Axial Force on Free Vibration of Timoshenko Multi-Span Beam Carrying Multiple Spring-Mass Systems”, International Journal of Mechanical Sciences, Vol. 50, No. 6, pp. 995-1003, 2008.
10. Yesilce, Y., “Effect of Axial Force on the Free Vibration of Reddy-Bickford Multi-Span Beam Carrying Multiple Spring-Mass Systems”, Journal of Vibration and Control, 2009.
11. Zhang, Z., Chen, F., Zhang, Z., and Hua, H., “Vibration Analysis of Non-uniform Timoshenko Beams Coupled with Flexible Attachments and Multiple Discontinuities”, International Journal of Mechanical Sciences, Vol. 80, pp. 131-143, 2014.
12. Yesilce, Y., Demirdag, O., and Catal, S., “Free Vibrations of a Multi-Span Timoshenko Beam Carrying Multiple Spring-Mass Systems”, Sadhana, Vol. 33, No. 4, pp. 385-401, 2008.
13. Lin, H. -Y., and Tsai, Y.-C., “Free Vibration Analysis of a Uniform Multi-Span Beam Carrying Multiple Spring-Mass Systems”, Journal of Sound and Vibration, Vol. 302, No. 3, pp. 442-456, 2007.
14. Nicholson, J. W., and Bergman, L. A., “Free Vibration of Combined Dynamical Systems”, Journal of Engineering Mechanics, Vol. 112, No. 1, pp. 1-13, 1986.
15. Pakdemirli, M., and Nayfeh, A., “Nonlinear Vibrations of a Beam-Spring-Mass System”, Journal of Vibration and Acoustics, Vol. 116, No. 4, pp. 433-439, 1994.
16. Ghayesh, M. H., Kazemirad, S., and Darabi, M. A., “A General Solution Procedure for Vibrations of Systems with Cubic Nonlinearities and Nonlinear/Time-Dependent Internal Boundary Conditions, Journal of Sound and Vibration, Vol. 330, No. 22, pp. 5382-5400, 2011.
17. Eftekhari, M., Ziaei-Rad, S., and Mahzoon, M., “Vibration Suppression of a Symmetrically Cantilever Composite Beam using Internal Resonance under Chordwise Base Excitation”, International Journal of Non-Linear Mechanics, Vol. 48, pp. 86-100, 2013.
18. Barry, O., Oguamanam, D., and Zu, J., “Nonlinear Vibration of an Axially Loaded Beam Carrying Multiple Mass-Spring-Damper Systems”, Nonlinear Dynamics, Vol. 77, No. 4, pp. 1597-1608, 2014.
19. Wang, Y. -R., and Liang, T. -W., Application of Lumped-Mass Vibration Absorber on the Vibration Reduction of a Nonlinear Beam-Spring-Mass System with Internal Resonances”, Journal of Sound and Vibration, Vol. 350, pp. 140-170, 2015.
20. Ebrahimi Mamaghani, A., and Esameilzadeh Khadem, S., “Vibration Analysis of a Beam under External Periodic Excitation using a Nonlinear Energy Sink”, Modares Mechanical Engineering, Vol. 16, No. 9, pp.186-194, 2016
21. Sadeghi M. H., and Lotfan S., “Stability and Bifurcation Analysis of a Beam-Mass-Spring-Damper System under Primary and One-to-Three Internal Resonances”, Modares Mechanical Engineering, Vol. 17, No. 2, pp.166-176, 2017.
22. Lotfan, S., and Sadeghi, M. H., “Large Amplitude Free Vibration of a Viscoelastic Beam Carrying a Lumped Mass-Spring-Damper”, Nonlinear Dynamics, Vol. 90, No. 2, pp. 1053-1075, 2017.
23. Huang, N. E., Shen, Z., Long, S. R., Wu, M. C., Shih, H. H., Zheng, Q., Yen, N.-C., Tung, C. C., and Liu, H. H., “The Empirical Mode Decomposition and the Hilbert Spectrum for Nonlinear and Non-stationary Time Series Analysis”, Proceeding of The Royal Society, pp. 903-995, 1998.
24. Yan, J., and Deller, J., “NARMAX Model Identification using a Set-Theoretic Evolutionary Approach, Signal Processing, Vol. 123, pp. 30-41, 2016.
25. De Filippis, G., Noël, J.-P., Kerschen, G., Soria, L. and Stephan, C., Experimental Nonlinear Identification of an Aircraft with Bolted Connections, in: Nonlinear Dynamics, Volume 1, Eds., Springer, pp. 263-278, 2016.
26. Staszewski, W., “Analysis of Non-linear Systems using Wavelets”, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, Vol. 214, No. 11, pp. 1339-1353, 2000.
27. Billings, S., Jamaluddin, H., and Chen, S., “Properties of Neural Networks with Applications to Modelling Non-Linear Dynamical Systems”, International Journal of Control, Vol. 55, No. 1, pp. 193-224, 1992.
28. Kerschen, G., Worden, K., Vakakis, A. F., and Golinval, J. -C., “Past, Present and Future of Nonlinear System Identification in Structural Dynamics”, Mechanical Systems and Signal Processing, Vol. 20, No. 3, pp. 505-592, 2006.
29. Noël, J. -P., and Kerschen, G., “Nonlinear System Identification in Structural Dynamics: 10 More Years of Progress”, Mechanical Systems and Signal Processing, Vol. 83, pp. 2-35, 2017.
30. Lee, Y. S., Tsakirtzis, S., Vakakis, A. F., Bergman, L. A., and McFarland, D. M., “Physics-Based Foundation for Empirical Mode Decomposition”, AIAA Journal, Vol. 47, No. 12, pp. 2938-2963, 2009.
31. Lee, Y. S., Tsakirtzis, S., Vakakis, A. F., Bergman, L. A., and McFarland, D. M., “A Time-Domain Nonlinear System Identification Method Based on Multiscale Dynamic Partitions”, Meccanica, Vol. 46, No. 4, pp. 625-649, 2011.
32. Tsakirtzis, S., Lee, Y., Vakakis, A., Bergman, L., and McFarland, D., “Modelling of Nonlinear Modal Interactions in the Transient Dynamics of an Elastic Rod with an Essentially Nonlinear Attachment, Communications in Nonlinear Science and Numerical Simulation, Vol. 15, No. 9, pp. 2617-2633, 2010.
33. Kurt, M., Chen, H., Lee, Y. S., McFarland, D. M., Bergman, L. A., and Vakakis, A. F., “Nonlinear System Identification of the Dynamics of a Vibro-Impact Beam: Numerical Results, Archive of Applied Mechanics, Vol. 82, No. 10-11, pp. 1461-1479, 2012.
34. Eriten, M., Kurt, M., Luo, G., McFarland, D. M., Bergman, L. A., and Vakakis, A. F., “Nonlinear System Identification of Frictional Effects in a Beam with a Bolted Joint Connection”, Mechanical Systems and Signal Processing, Vol. 39, No. 1, pp. 245-264, 2013.
35. Chen, H., Kurt, M., Lee, Y. S., McFarland, D. M., Bergman, L. A., and Vakakis, A. F., “Experimental System Identification of the Dynamics of a Vibro-Impact Beam with a View Towards Structural Health Monitoring and Damage Detection”, Mechanical Systems and Signal Processing, Vol. 46, No. 1, pp. 91-113, 2014.
36. Kurt, M., Eriten, M., McFarland, D. M., Bergman, L. A., and Vakakis, A. F., “Strongly Nonlinear Beats in the Dynamics of an Elastic System with a Strong Local Stiffness Nonlinearity: Analysis and Identification”, Journal of Sound and Vibration, Vol. 333, No. 7, pp. 2054-2072, 2014.
37. Ghayesh, M. H., Kazemirad, S., Darabi, M. A., and Woo, P., “Thermo-Mechanical Nonlinear Vibration Analysis of a Spring-Mass-Beam System”, Archive of Applied Mechanics, Vol. 82, No. 3, pp. 317-331, 2012.
38. Rezaee, M., and Lotfan, S., “Non-Linear Nonlocal Vibration and Stability Analysis of Axially Moving Nanoscale Beams with Time-Dependent Velocity”, International Journal of Mechanical Sciences, Vol. 96, pp. 36-46, 2015.
39. Huang, J., Su, R., Li, W., and Chen, S., “Stability and Bifurcation of an Axially Moving Beam Tuned to Three-to-One Internal Resonances”, Journal of Sound and Vibration, Vol. 330, No. 3, pp. 471-485, 2011.
40. Chen, L. -Q., Zhang, G. -C., and Ding, H., “Internal Resonance in Forced Vibration of Coupled Cantilevers Subjected to Magnetic Interaction”, Journal of Sound and Vibration, Vol. 354, pp. 196-218, 2015.
41. Hoover, W. G., Sprott, J. C., and Hoover, C. G., “Adaptive Runge-Kutta Integration for Stiff Systems: Comparing Nosé and Nosé–Hoover Dynamics for the Harmonic Oscillator”, American Journal of Physics, Vol. 84, No. 10, pp. 786-794, 2016.
42. Manevitch, L. I., Complex Representation of Dynamics of Coupled Nonlinear Oscillators, in: Mathematical Models of Non-linear Excitations, Ttransfer, Dynamics, and Control in Condensed Systems and other Media, Eds., Springer, pp. 269-300, 1999.
43. Manevitch, L., “The Description of Localized Normal Modes in a Chain of Nonlinear Coupled Oscillators using Complex Variables”, Nonlinear Dynamics, Vol. 25, No. 1-3, pp. 95-109, 2001.
44. Chen, Y., and Feng, M. Q., “A Technique to Improve the Empirical Mode Decomposition in the Hilbert-Huang Transform”, Earthquake Engineering and Engineering Vibration, Vol. 2, No. 1, pp. 75-85, 2003.
45. Rato, R., Ortigueira, M., and Batista, A., “On the HHT, Its Problems, and Some Solutions”, Mechanical Systems and Signal Processing, Vol. 22, No. 6, pp. 1374-1394, 2008.
46. Vakakis, A., Bergman, L., McFarland, D., Lee, Y., and Kurt, M., “Current Efforts Towards a Non-linear System Identification Methodology of Broad Applicability”, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, p. 0954406211417217, 2011.
47. Sadeghi, M. H., and Lotfan, S., “Identification of Non-Linear Parameter of a Cantilever Beam Model with Boundary Condition Non-linearity in the Presence of Noise: an NSI-and ANN-Based Approach”, Acta Mechanica, Vol. 228, No. 12, pp. 4451-4469, 2017.
48. Eberhart, R. C., and Shi, Y., “Particle Swarm Optimization: Developments, Applications and Resources, Proceeding of, IEEE, pp. 81-86, 2001
2. Hijmissen, J., Van den Heuvel, N., and Van Horssen, W., “On the Effect of the Bending Stiffness on the Damping Properties of a Tensioned Cable with an Attached Tuned-Mass-Damper”, Engineering Structures, Vol. 31, No. 5, pp. 1276-1285, 2009.
3. Sadeghi, M. H., and Lotfan, S., “Nonparametric System Identification of a Cantilever Beam Model with Local Nonlinearity in the Presence of Artificial Noise”, Modares Mechanical Engineering, Vol. 16, No. 11, pp. 177-186, 2017.
4. Wu, J.-S., and Chen, D.-W., “Dynamic Analysis of a Uniform Cantilever Beam Carrying a Number of Elastically Mounted Point Masses with Dampers”, Journal of Sound and Vibration, Vol. 229, No. 3, pp. 549-578, 2000.
5. Hamdan, M., and Jubran, B., “Free and Forced Vibrations of a Restrained Uniform Beam Carrying an Intermediate Lumped Mass and a Rotary Inertia”, Journal of Sound and Vibration, Vol. 150, No. 2, pp. 203-216, 1991.
6. Bambill, D., and Rossit, C., “Forced Vibrations of a Beam Elastically Restrained Against Rotation and Carrying a Spring-Mass System”, Ocean Engineering, Vol. 29, No. 6, pp. 605-626, 2002.
7. Kukla, S., and Posiadala, B., “Free Vibrations of Beams with Elastically Mounted Masses”, Journal of Sound and Vibration, Vol. 175, No. 4, pp. 557-564, 1994.
8. Wu, J.-S., and Chen, D.-W. “Free Vibration Analysis of a Timoshenko Beam Carrying Multiple Spring-Mass Systems by using the Numerical Assembly Technique”, International Journal for Numerical Methods in Engineering, Vol. 50, No. 5, pp. 1039-1058, 2001.
9. Yesilce, Y., and Demirdag, O., “Effect of Axial Force on Free Vibration of Timoshenko Multi-Span Beam Carrying Multiple Spring-Mass Systems”, International Journal of Mechanical Sciences, Vol. 50, No. 6, pp. 995-1003, 2008.
10. Yesilce, Y., “Effect of Axial Force on the Free Vibration of Reddy-Bickford Multi-Span Beam Carrying Multiple Spring-Mass Systems”, Journal of Vibration and Control, 2009.
11. Zhang, Z., Chen, F., Zhang, Z., and Hua, H., “Vibration Analysis of Non-uniform Timoshenko Beams Coupled with Flexible Attachments and Multiple Discontinuities”, International Journal of Mechanical Sciences, Vol. 80, pp. 131-143, 2014.
12. Yesilce, Y., Demirdag, O., and Catal, S., “Free Vibrations of a Multi-Span Timoshenko Beam Carrying Multiple Spring-Mass Systems”, Sadhana, Vol. 33, No. 4, pp. 385-401, 2008.
13. Lin, H. -Y., and Tsai, Y.-C., “Free Vibration Analysis of a Uniform Multi-Span Beam Carrying Multiple Spring-Mass Systems”, Journal of Sound and Vibration, Vol. 302, No. 3, pp. 442-456, 2007.
14. Nicholson, J. W., and Bergman, L. A., “Free Vibration of Combined Dynamical Systems”, Journal of Engineering Mechanics, Vol. 112, No. 1, pp. 1-13, 1986.
15. Pakdemirli, M., and Nayfeh, A., “Nonlinear Vibrations of a Beam-Spring-Mass System”, Journal of Vibration and Acoustics, Vol. 116, No. 4, pp. 433-439, 1994.
16. Ghayesh, M. H., Kazemirad, S., and Darabi, M. A., “A General Solution Procedure for Vibrations of Systems with Cubic Nonlinearities and Nonlinear/Time-Dependent Internal Boundary Conditions, Journal of Sound and Vibration, Vol. 330, No. 22, pp. 5382-5400, 2011.
17. Eftekhari, M., Ziaei-Rad, S., and Mahzoon, M., “Vibration Suppression of a Symmetrically Cantilever Composite Beam using Internal Resonance under Chordwise Base Excitation”, International Journal of Non-Linear Mechanics, Vol. 48, pp. 86-100, 2013.
18. Barry, O., Oguamanam, D., and Zu, J., “Nonlinear Vibration of an Axially Loaded Beam Carrying Multiple Mass-Spring-Damper Systems”, Nonlinear Dynamics, Vol. 77, No. 4, pp. 1597-1608, 2014.
19. Wang, Y. -R., and Liang, T. -W., Application of Lumped-Mass Vibration Absorber on the Vibration Reduction of a Nonlinear Beam-Spring-Mass System with Internal Resonances”, Journal of Sound and Vibration, Vol. 350, pp. 140-170, 2015.
20. Ebrahimi Mamaghani, A., and Esameilzadeh Khadem, S., “Vibration Analysis of a Beam under External Periodic Excitation using a Nonlinear Energy Sink”, Modares Mechanical Engineering, Vol. 16, No. 9, pp.186-194, 2016
21. Sadeghi M. H., and Lotfan S., “Stability and Bifurcation Analysis of a Beam-Mass-Spring-Damper System under Primary and One-to-Three Internal Resonances”, Modares Mechanical Engineering, Vol. 17, No. 2, pp.166-176, 2017.
22. Lotfan, S., and Sadeghi, M. H., “Large Amplitude Free Vibration of a Viscoelastic Beam Carrying a Lumped Mass-Spring-Damper”, Nonlinear Dynamics, Vol. 90, No. 2, pp. 1053-1075, 2017.
23. Huang, N. E., Shen, Z., Long, S. R., Wu, M. C., Shih, H. H., Zheng, Q., Yen, N.-C., Tung, C. C., and Liu, H. H., “The Empirical Mode Decomposition and the Hilbert Spectrum for Nonlinear and Non-stationary Time Series Analysis”, Proceeding of The Royal Society, pp. 903-995, 1998.
24. Yan, J., and Deller, J., “NARMAX Model Identification using a Set-Theoretic Evolutionary Approach, Signal Processing, Vol. 123, pp. 30-41, 2016.
25. De Filippis, G., Noël, J.-P., Kerschen, G., Soria, L. and Stephan, C., Experimental Nonlinear Identification of an Aircraft with Bolted Connections, in: Nonlinear Dynamics, Volume 1, Eds., Springer, pp. 263-278, 2016.
26. Staszewski, W., “Analysis of Non-linear Systems using Wavelets”, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, Vol. 214, No. 11, pp. 1339-1353, 2000.
27. Billings, S., Jamaluddin, H., and Chen, S., “Properties of Neural Networks with Applications to Modelling Non-Linear Dynamical Systems”, International Journal of Control, Vol. 55, No. 1, pp. 193-224, 1992.
28. Kerschen, G., Worden, K., Vakakis, A. F., and Golinval, J. -C., “Past, Present and Future of Nonlinear System Identification in Structural Dynamics”, Mechanical Systems and Signal Processing, Vol. 20, No. 3, pp. 505-592, 2006.
29. Noël, J. -P., and Kerschen, G., “Nonlinear System Identification in Structural Dynamics: 10 More Years of Progress”, Mechanical Systems and Signal Processing, Vol. 83, pp. 2-35, 2017.
30. Lee, Y. S., Tsakirtzis, S., Vakakis, A. F., Bergman, L. A., and McFarland, D. M., “Physics-Based Foundation for Empirical Mode Decomposition”, AIAA Journal, Vol. 47, No. 12, pp. 2938-2963, 2009.
31. Lee, Y. S., Tsakirtzis, S., Vakakis, A. F., Bergman, L. A., and McFarland, D. M., “A Time-Domain Nonlinear System Identification Method Based on Multiscale Dynamic Partitions”, Meccanica, Vol. 46, No. 4, pp. 625-649, 2011.
32. Tsakirtzis, S., Lee, Y., Vakakis, A., Bergman, L., and McFarland, D., “Modelling of Nonlinear Modal Interactions in the Transient Dynamics of an Elastic Rod with an Essentially Nonlinear Attachment, Communications in Nonlinear Science and Numerical Simulation, Vol. 15, No. 9, pp. 2617-2633, 2010.
33. Kurt, M., Chen, H., Lee, Y. S., McFarland, D. M., Bergman, L. A., and Vakakis, A. F., “Nonlinear System Identification of the Dynamics of a Vibro-Impact Beam: Numerical Results, Archive of Applied Mechanics, Vol. 82, No. 10-11, pp. 1461-1479, 2012.
34. Eriten, M., Kurt, M., Luo, G., McFarland, D. M., Bergman, L. A., and Vakakis, A. F., “Nonlinear System Identification of Frictional Effects in a Beam with a Bolted Joint Connection”, Mechanical Systems and Signal Processing, Vol. 39, No. 1, pp. 245-264, 2013.
35. Chen, H., Kurt, M., Lee, Y. S., McFarland, D. M., Bergman, L. A., and Vakakis, A. F., “Experimental System Identification of the Dynamics of a Vibro-Impact Beam with a View Towards Structural Health Monitoring and Damage Detection”, Mechanical Systems and Signal Processing, Vol. 46, No. 1, pp. 91-113, 2014.
36. Kurt, M., Eriten, M., McFarland, D. M., Bergman, L. A., and Vakakis, A. F., “Strongly Nonlinear Beats in the Dynamics of an Elastic System with a Strong Local Stiffness Nonlinearity: Analysis and Identification”, Journal of Sound and Vibration, Vol. 333, No. 7, pp. 2054-2072, 2014.
37. Ghayesh, M. H., Kazemirad, S., Darabi, M. A., and Woo, P., “Thermo-Mechanical Nonlinear Vibration Analysis of a Spring-Mass-Beam System”, Archive of Applied Mechanics, Vol. 82, No. 3, pp. 317-331, 2012.
38. Rezaee, M., and Lotfan, S., “Non-Linear Nonlocal Vibration and Stability Analysis of Axially Moving Nanoscale Beams with Time-Dependent Velocity”, International Journal of Mechanical Sciences, Vol. 96, pp. 36-46, 2015.
39. Huang, J., Su, R., Li, W., and Chen, S., “Stability and Bifurcation of an Axially Moving Beam Tuned to Three-to-One Internal Resonances”, Journal of Sound and Vibration, Vol. 330, No. 3, pp. 471-485, 2011.
40. Chen, L. -Q., Zhang, G. -C., and Ding, H., “Internal Resonance in Forced Vibration of Coupled Cantilevers Subjected to Magnetic Interaction”, Journal of Sound and Vibration, Vol. 354, pp. 196-218, 2015.
41. Hoover, W. G., Sprott, J. C., and Hoover, C. G., “Adaptive Runge-Kutta Integration for Stiff Systems: Comparing Nosé and Nosé–Hoover Dynamics for the Harmonic Oscillator”, American Journal of Physics, Vol. 84, No. 10, pp. 786-794, 2016.
42. Manevitch, L. I., Complex Representation of Dynamics of Coupled Nonlinear Oscillators, in: Mathematical Models of Non-linear Excitations, Ttransfer, Dynamics, and Control in Condensed Systems and other Media, Eds., Springer, pp. 269-300, 1999.
43. Manevitch, L., “The Description of Localized Normal Modes in a Chain of Nonlinear Coupled Oscillators using Complex Variables”, Nonlinear Dynamics, Vol. 25, No. 1-3, pp. 95-109, 2001.
44. Chen, Y., and Feng, M. Q., “A Technique to Improve the Empirical Mode Decomposition in the Hilbert-Huang Transform”, Earthquake Engineering and Engineering Vibration, Vol. 2, No. 1, pp. 75-85, 2003.
45. Rato, R., Ortigueira, M., and Batista, A., “On the HHT, Its Problems, and Some Solutions”, Mechanical Systems and Signal Processing, Vol. 22, No. 6, pp. 1374-1394, 2008.
46. Vakakis, A., Bergman, L., McFarland, D., Lee, Y., and Kurt, M., “Current Efforts Towards a Non-linear System Identification Methodology of Broad Applicability”, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, p. 0954406211417217, 2011.
47. Sadeghi, M. H., and Lotfan, S., “Identification of Non-Linear Parameter of a Cantilever Beam Model with Boundary Condition Non-linearity in the Presence of Noise: an NSI-and ANN-Based Approach”, Acta Mechanica, Vol. 228, No. 12, pp. 4451-4469, 2017.
48. Eberhart, R. C., and Shi, Y., “Particle Swarm Optimization: Developments, Applications and Resources, Proceeding of, IEEE, pp. 81-86, 2001
)