Abstract
In this chapter, the performance of a novel method to determine the stability of periodic solutions is analyzed. Using the Koopman framework, the linear time-periodic nonautonomous perturbed dynamics around a periodic solution can be approximated by a linear autonomous system of higher order, whose system matrix is the well-known Hill matrix. The evaluation of the closed-form solution reveals that the monodromy matrix can be approximated by the Hill matrix, using only a matrix exponential and a projection instead of solving a large eigenvalue problem or integrating numerically over one period. There does not exist a unique choice for the projection of the Hill matrix to the monodromy matrix. In this chapter, various ways to obtain a suitable projection are discussed. The computational efficiency of the novel method is analyzed for the vertically excited multiple pendulum, comparing the effect of the considered projection options.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
The number n in the name 2n-pass refers to the number of DOFs of the mechanical system, called \({n_{\mathrm {p}}}\) below, and should not be confused with the number of states n.
References
Krack, M., Gross, J.: Harmonic Balance for Nonlinear Vibration Problems. Springer, Cham (2019)
Detroux, T., Renson, L., Masset, L., Kerschen, G.: The harmonic balance method for bifurcation analysis of large-scale nonlinear mechanical systems. Comput. Methods Appl. Mech. Eng. 296, 18–38 (2015)
Lazarus, A., Thomas, O.: A harmonic-based method for computing the stability of periodic solutions of dynamical systems. C. R. Mec. 338(9), 510–517 (2010)
Peletan, L., Baguet, S., Torkhani, M., Jacquet-Richardet, G.: A comparison of stability computational methods for periodic solution of nonlinear problems with application to rotordynamics. Nonlinear Dyn. 72(3), 671–682 (2013)
Zhou, J., Hagiwara, T., Araki, M.: Spectral characteristics and eigenvalues computation of the harmonic state operators in continuous-time periodic systems. Syst. Control Lett. 53(2), 141–155 (2004)
Wu, J., Hong, L., Jiang, J.: A robust and efficient stability analysis of periodic solutions based on harmonic balance method and Floquet-Hill formulation. Mech. Syst. Signal Proces. 173, 109057 (2022)
Bayer, F., Leine, R.I.: Sorting-free Hill-based stability analysis of periodic solutions through Koopman analysis. Nonlinear Dyn. 111, 8439–8466 (2023) https://doi.org/10.1007/s11071-023-08247-7
Teschl, G.: Ordinary Differential Equations and Dynamical Systems. No. 140 in Graduate Studies in Mathematics, American Mathematical Society, Providence (2012)
Müller, P.C.: Calculation of Lyapunov exponents for dynamic systems with discontinuities. Chaos, Solitons Fractals 5(9), 1671–1681 (1995)
Mélot, A., Benaïcha, Y., Rigaud, E., Perret-Liaudet, J., Thouverez, F.: Effect of gear topology discontinuities on the nonlinear dynamic response of a multi-degree-of-freedom gear train. J. Sound Vib. 516, 116495 (2022)
Mauroy, A., Mezić, I., Susuki, Y. (eds.): The Koopman Operator in Systems and Control. Concepts, Methodologies and Applications. No. 484 in Lecture Notes in Control and Information Sciences. Springer, Cham (2020)
Al-Mohy, A.H., Higham, N.J.: Computing the action of the matrix exponential, with an application to exponential integrators. SIAM J. Sci. Comput. 33(2), 488–511 (2011)
Guillot, L., Lazarus, A., Thomas, O., Vergez, C., Cochelin, B.: A purely frequency based Floquet-Hill formulation for the efficient stability computation of periodic solutions of ordinary differential systems. J. Comput. Phys. 416, 109477 (2020)
Debeurre, M., Grolet, A., Mattei, P.O., Cochelin, B., Thomas, O.: Nonlinear modes of cantilever beams at extreme amplitudes: numerical computation and experiments. In: Brake, M.R.W., Renson, L., Kuether, R.J., Tiso, P. (eds.) Nonlinear Structures & Systems, vol. 1. pp. 245–248. Springer International Publishing, Cham (2023)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2024 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
Bayer, F., Leine, R.I. (2024). Optimal Projection in a Koopman-Based Sorting-Free Hill Method. In: Lacarbonara, W. (eds) Advances in Nonlinear Dynamics, Volume I. ICNDA 2023. NODYCON Conference Proceedings Series. Springer, Cham. https://doi.org/10.1007/978-3-031-50631-4_35
Download citation
DOI: https://doi.org/10.1007/978-3-031-50631-4_35
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-50630-7
Online ISBN: 978-3-031-50631-4
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)