A comparative study of the harmonic balance method and the orthogonal collocation method on stiff nonlinear systems

Sami Karkar (1) , Bruno Cochelin (2) , Christophe Vergez (3)

1 LMA - Laboratoire de Mécanique et d'Acoustique [Marseille]
2 M&S - Matériaux et Structures
3 Sons

The high-order purely frequency-based harmonic balance method (HBM) presented by Cochelin and Vergez (2009) [1] and extended by Karkar et al. (2013) [2] now allows to follow the periodic solutions of regularized non-smooth systems (stiff systems). This paper compares its convergence property to a reference method in applied mathematics: orthogonal collocation with piecewise polynomials. A first test is conducted on a nonlinear smooth 2 degree-of-freedom spring mass system, showing better convergence of the HBM. The second test is conducted on a one degree-of-freedom vibro-impact system with a very stiff regularization of the impact law. The HBM continuation of the nonlinear mode was found to be very robust, even with a very large number of harmonics. Surprisingly, the HBM was found to have a better convergence than the collocation method for this vibro-impact system.

Mots clés: Numerical continuation; periodic solutions; nonlinear dynamics; periocdic solutions; harmonic balance method; orthogonal colocation; impact

Domaines: Acoustique [physics.class-ph]; Acoustique [physics.class-ph]; Vibrations [physics.class-ph]; Vibrations [physics.class-ph]; Systèmes dynamiques [math.DS]

Sami Karkar, Bruno Cochelin, Christophe Vergez. A comparative study of the harmonic balance method and the orthogonal collocation method on stiff nonlinear systems. Journal of Sound and Vibration, 2014, 333 (12), pp.2554-2567. ⟨10.1016/j.jsv.2014.01.019⟩. ⟨hal-01065672⟩