NLvib

Capabilities

• Harmonic Balance (HB): Alternating Frequency–Time HB.
• numerical integration: unconditionally stable Newmark integrator, shooting method.
• numerical solution and path continuation: predictor-corrector method with Newton-type solver (MATLAB’s fsolve) and analytical gradients.
• types of nonlinearities: local generic nonlinear elements; distributed polynomial stiffness nonlinearity.
• analysis types: frequency response analysis; nonlinear modal analysis

Basic examples included in the package

• Duffing oscillator
• frequency response of a two-degree-of-freedom system with cubic spring showing a double peak with both turning points and Neimark-Sacker bifurcations
• frequency response of a two-degree-of-freedom system with unilateral spring showing interesting types of stability loss
• nonlinear modes of a two-degree-of-freedom system with cubic spring (see e.g. [2])
• nonlinear modes of a two-degree-of-freedom system with regularized Coulomb dry friction law [3]
• mass attached to two springs undergoing geometric nonlinearity [4]
• 1D finite element beam model with different nonlinear elements (cubic spring [5], dry friction)

Book on Harmonic Balance and NLvib

A Springer book on Harmonic Balance was published, covering the theoretical basis, its application to mechanical systems, and its computational implementation. It also includes several solved exercises and homework problems relying on NLvib, and a guide to get started with NLvib in an appendix.

Downloadable files

• The code is available via https://github.com/maltekrack/NLvib/.
• Presentation on Harmonic Balance, with detailed appendix on NLvib HB_NLvib_presentation.pdf

NLvib was developed by Malte Krack and Johann Gross. If you use NLvib, please refer to our book [1].

NLvib comes with ABSOLUTELY NO WARRANTY. NLvib is free software, you can redistribute and/or modify it under the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. For details on license and warranty, see http://www.gnu.org/licenses.

References

[1] M. Krack, J. Gross: Harmonic Balance for Nonlinear Vibration Problems. Springer (2019) DOI 10.1007/978-3-030-14023-6

[2] Kerschen, G., Peeters, M., Golinval, J.C., Vakakis, A.F.: Nonlinear normal modes, part i: A useful framework for the structural dynamicist:
Special issue: Non-linear structural dynamics. Mechanical Systems and Signal Processing 23(1), 170-194 (2009)

[3] Laxalde, D., Thouverez, F.: Complex non-linear modal analysis for mechanical systems: application to turbomachinery bladings with friction interfaces. Journal of Sound and Vibration 322(4-5), 1009-1025 (2009)

[4] Touzé, C., Amabili, M.: Nonlinear normal modes for damped geometrically nonlinear systems: Application to reduced-order modelling of harmonically forced structures. Journal of Sound and Vibration 298(4-5), 958-981 (2006)

[5] Peeters, M., Viguié, R., Sérandour, G., Kerschen, G., Golinval,
J.C.: Nonlinear normal modes, part ii: Toward a practical computation using numerical continuation techniques: Special issue: Non-linear structural dynamics. Mechanical Systems and Signal Processing 23(1),195-216 (2009)