Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-27T11:24:24.541Z Has data issue: false hasContentIssue false

Targeted energy transfer in stochastically excited system with nonlinear energy sink

Published online by Cambridge University Press:  18 September 2018

P. KUMAR
Affiliation:
Dynamic Analysis Group, Bharat Heavy Electrical Limited, Nagpur 440001, India email: [email protected]
S. NARAYANAN
Affiliation:
Department of Mechanical Engineering, Indian Institute of Information Technology (Design and Manufacturing), Kancheepuram, India email: [email protected]
S. GUPTA
Affiliation:
Department of Applied Mechanics, Indian Institute of Technology Madras, Chennai 600036, India email: [email protected]

Abstract

This study investigates the phenomenon of targeted energy transfer (TET) from a linear oscillator to a nonlinear attachment behaving as a nonlinear energy sink for both transient and stochastic excitations. First, the dynamics of the underlying Hamiltonian system under deterministic transient loading is studied. Assuming that the transient dynamics can be partitioned into slow and fast components, the governing equations of motion corresponding to the slow flow dynamics are derived and the behaviour of the system is analysed. Subsequently, the effect of noise on the slow flow dynamics of the system is investigated. The Itô stochastic differential equations for the noisy system are derived and the corresponding Fokker–Planck equations are numerically solved to gain insights into the behaviour of the system on TET. The effects of the system parameters as well as noise intensity on the optimal regime of TET are studied. The analysis reveals that the interaction of nonlinearities and noise enhances the optimal TET regime as predicted in deterministic analysis.

Type
Papers
Copyright
© Cambridge University Press 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alevras, P. & Yurchenko, D. (2015) GPU computing for accelerating the numerical path integration approach. Comput. Struct. 171, 4653.CrossRefGoogle Scholar
Arnold, L. (1988) Dynamical Systems III. Encyclopedia of Mathematical Sciences, Springer-Verlag, Berlin Heidelberg.CrossRefGoogle Scholar
Atkinson, J. D. (1973) Eigenfunction expansions for randomly excited non-linear systems. J. Sound Vib. 30, 153172.CrossRefGoogle Scholar
Blum, T. & McKane, A. J. (1996) Variational schemes in the Fokker-Planck equation. J. Phys. A: Math. Gen. 29, 18591872.CrossRefGoogle Scholar
Devathi, H. & Sarkar, S. (2016) Study of a stall induced dynamical system under gust using the probability density evolution technique. Comput. Struct. 162, 3846.CrossRefGoogle Scholar
Gendelman, O. V. (2001) Transition of energy to a nonlinear localized mode in a highly asymmetric system of two oscillators. Nonlinear Dyn. 25, 237253.CrossRefGoogle Scholar
Gendelman, O. V., Starosvetsky, Y. & Feldman, M. (2008) Attractors of harmonically forced linear oscillator with attached nonlinear energy sink I: description of response regimes. Nonlinear Dyn. 51(12), 3146.CrossRefGoogle Scholar
Gourdon, E. & Lamarque, C. H. (2005) Energy pumping with various nonlinear structures: numerical evidences. Nonlinear Dyn. 40, 281307.CrossRefGoogle Scholar
Gourdon, E., Lamarque, C. H. & Pernot, S. (2007) Contribution to efficiency of irreversible passive energy pumping with a strong nonlinear attachment. Nonlinear Dyn. 50, 793808.CrossRefGoogle Scholar
Huai, X., Xianren, K., Zhenguo, Y. & Yuan, L. (2015) Response regimes of narrow-band stochastic excited linear oscillator coupled to nonlinear energy sink. Chin. J. Aeronaut. 28(2), 457468.Google Scholar
Jenkins, R., Daniels, G. & Andrews, D. (2004) Quantum pathways for resonance energy transfer. J. Chem. Phys. 120, 1144211448.CrossRefGoogle ScholarPubMed
Kloeden, P. E. & Platen, E. (1992) Numerical Solution of Stochastic Differential Equation, Springer-Verlag, Berlin Heidelberg.CrossRefGoogle Scholar
Kougioumtzoglou, I. A., Di Matteo, A., Spanos, P. D., Pirrotta, A. & Di Paola, M. (2015) An efficient Wiener path integral technique formulation for stochastic response determination of nonlinear MDOF systems. ASME J. Appl. Mech. 82, 101005:1101005:7.CrossRefGoogle Scholar
Kumar, P. & Narayanan, S. (2006) Solution of Fokker-Planck equation by finite element and finite difference methods for nonlinear system. Sadhana 31(4), 455473.CrossRefGoogle Scholar
Kumar, P. & Narayanan, S. (2012) Numerical solutions of Fokker-Planck equation of nonlinear systems subjected to random and harmonic excitations. Probab. Eng. Mech. 27, 3546.Google Scholar
Kumar, P., Narayanan, S. & Gupta, S. (2014) Finite element solution of Fokker-Planck equation of nonlinear oscillators subjected to colored non-Gaussian noise. Probab. Eng. Mech. 38, 143155.CrossRefGoogle Scholar
Lee, Y. S., Vakakis, A. F., Bergman, L. A., McFarland, D. M. & Kerschen, G. (2007) Suppression of aeroelastic instability using broadband passive targeted energy transfer, part 1: theory. AIAA J. 45(3), 693711.CrossRefGoogle Scholar
Li, J. & Chen, J. (2006) The probability density evolution method for dynamic response analysis of nonlinear stochastic structures. Int. J. Numer. Methods Eng. 65, 882903.CrossRefGoogle Scholar
Li, J. & Chen, J. (2009) Stochastic Dynamics of Structures, John Wiley and Sons, Singapore.CrossRefGoogle Scholar
Manevitch, L. I., Gendelman, O. V., Moussienko, A. I., Vakakis, A. F. & Bergman, L. A. (2003) Dynamic interaction of a semi-infinite linear chain of coupled oscillators with a strongly nonlinear end attachment. Physica D 178, 118.CrossRefGoogle Scholar
Masud, A. & Bergman, L. A. (2005) Solution of the four dimensional Fokker-Planck equation: Still a challenge. In: G, Augusti, G. I, Schuller and M, Ciampoli (editors), Proceedings of the ICOSSAR 2005 Conference, Rome, Italy, June 22–26, 2005. Millpress, Rotterdam, pp. 19111916.Google Scholar
Muscolino, G., Ricciardi, G. & Vasta, M. (1997) Stationary and non-stationary probability density function for non-linear oscillators. Int. J. Non-Linear Mech. 32, 10511064.CrossRefGoogle Scholar
Naess, A. & Moe, V. (2000) Efficient path integration method for nonlinear dynamics system. Probab. Eng. Mech. 15, 221231.CrossRefGoogle Scholar
Quinn, D., Gendelman, O. V., Kerschen, G., Sapsis, T. P., Bergman, L. A. & Vakakis, A. F. (2008) Efficiency of targeted energy transfer in coupled oscillators associated with 1:1 resonance captures: part I. J. Sound Vib. 311, 12281248.CrossRefGoogle Scholar
Risken, H. (1989) The Fokker-Planck Equation: Methods of Solution and Applications, Springer-Verlag, New York.CrossRefGoogle Scholar
Sapsis, T. P., Vakakis, A. F. & Bergman, L. A. (2011) Effect of stochasticity on targeted energy transfer from a linear medium to a strongly nonlinear attachment. Probab. Eng. Mech. 26, 119133.CrossRefGoogle Scholar
Sapsis, T. P., Vakakis, A. F., Gendelman, O. V., Bergman, L. A., Kerschen, G. & Quinn, D. (2009) Efficiency of targeted energy transfer in coupled oscillators associated with 1:1 resonance captures: part II, analytical study. J. Sound Vib. 325, 297320.CrossRefGoogle Scholar
Sobczyk, K. & Trȩbicki, J. (1993) Maximum entropy principle and nonlinear stochastic oscillators. Phys. A: Stat. Mech. Appl. 193, 448468.CrossRefGoogle Scholar
Stephen, N. (2006) On energy harvesting from ambient vibration. J. Sound Vib. 293, 409425.CrossRefGoogle Scholar
Vakakis, A. (2001) Inducing passive nonlinear energy sinks in vibrating systems. J. Vib. Acoust. 123, 324332.CrossRefGoogle Scholar
Vakakis, A. F., Gendelman, O. V., Bergman, L. A., McFarland, D. M., Kerschen, G. & Lee, Y. S. (2009) Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems, Springer Science & Business Media, The Netherlands.Google Scholar
Wedig, W. V. (1997) Iterative schemes for stability problems with non-singular Fokker-Planck equations. Int. J. Non-Linear Mech. 31, 707715.CrossRefGoogle Scholar
Yu, J. S., Cai, G. Q. & Lin, Y. K. (1997) A new path integration procedure based on Gauss-Legendre scheme. Int. J. Non-Linear Mech. 32, 759768.CrossRefGoogle Scholar