142 T. Breunung and G. Haller the validity of such a reduction. We then continue with a given autonomous parameterization of the SSM and the reduced dynamics under the addition of small forcing terms. For near-resonant forcing, we eliminate the small denominators arising in the parameterization of the SSM. We give an explicit parameterization of the non-autonomous SSM and the reduced dynamics. Through a judicious choice of the parameterization, we simplify the reduced dynamics significantly, such that we can solve for forced-periodic responses analytically. Having derived condensed formulas for the forced response, we solve for the forcing frequency at which the response amplitude is maximal. Such amplitude-frequency pairs form a one dimensional curve (i.e. the backbone curve) as the forcing amplitude is varied as a parameter. Furthermore, we can analytically compute stability regions of the forced response. We show the application of our SSM-based analytic results on forced responses and backbone curves in numerical examples. On these examples, we compare our results with an approach from Neild and Wagg [5], with a normal-form type method from Touzé and Amabili [11] and with direct numerical continuation. References 1. Kerschen, G., Peeters, M., Golinval, J.C., Vakakis, A.: Nonlinear normal modes, Part I: a useful framework for the structural dynamicest. Mech. Syst. Signal Process. 23, 170–194 (2009) 2. Jiang, D., Pierre, C., Shaw, S.: Nonlinear normal modes for vibratory systems under harmonic excitation. J. Sound Vib. 288, 791–812 (2005) 3. Gabale, A., Sinha, S.: Model reduction of nonlinear systems with external periodic excitations via construction of invariant manifolds. J. Sound Vib. 330, 2596–2607 (2011) 4. Jezequel, L., Lamarque, C.H.: Analysis of non-linear dynamical systems by the normal form theory. J. Sound Vib. 149(3), 429–459 (1991) 5. Neild, S., Wagg, D.: Applying the method of normal forms to second-order nonlinear vibration problems. Philos. Trans. R. Soc. A Math. Phys. Eng. Sci. 467, 1141–1163 (2015) 6. Peeters, M., Kerschen, G., Golinval, J.-C.: Dynamic testing of nonlinear vibrating structures using nonlinear normal modes. J. Sound Vib. 330(3), 486–509 (2011) 7. Haller, G., Ponsioen, S.: Nonlinear normal modes and spectral submanifolds: existence, uniqueness and use in model reduction. Nonlinear Dyn. 86(3), 1493–1534 (2016) 8. Cabré, X., Fontich, E., de la Llave, R.: The parameterization method for invariant manifolds i: manifolds associated to non-resonant subspaces. Univ. Indiana Math. J. 52, 283–328 (2003) 9. Haro, A., de la Llave, R.: A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: rigorous results. J. Differ. Equ. 228(2), 530–579 (2006) 10. Szalai, R., Ehrhardt, D., Haller, G.: Nonlinear model identification and spectral submanifolds for multi-degree-of-freedom mechanical vibrations. Proc. R. Soc. A. 473, (2017) 11. Touzé, C., Amabili, M.: Nonlinear normal modes for damped geometrically nonlinear systems: application to reduced-order modelling of harmonically forced structures. J. Sound Vib. 298(4), 958–981 (2006)
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