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Dynamic analysis of externally excited NES-controlled systems via a mixed Multiple Scale/Harmonic Balance algorithm

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Abstract

A general, nonlinear, multi-d.o.f. structure, excited by harmonic external force in 1:1 resonance with one of the modes of the system, is considered. The structure is attached to an essentially nonlinear oscillator, with small mass and damping (Nonlinear Energy Sink, NES). The scope of the NES is to passively control the amplitude of vibrations of the main structure. A mixed Multiple Scale/Harmonic Balance Method (MSHBM) is proposed to get the differential equations describing the slow- and fast-flow dynamics of the whole structure. The main advantage of the procedure is that no complexification-averaging is required, so that the analysis is reconducted in the framework of the classical perturbation techniques.

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Authors and Affiliations

  1. DISAT—University of L’Aquila, Via Giovanni Gronchi, 18, 67100, L’Aquila, AQ, Italy

    Angelo Luongo & Daniele Zulli

Authors
  1. Angelo Luongo
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  2. Daniele Zulli
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Correspondence to Angelo Luongo.

Appendix: Coefficients of the equations

Appendix: Coefficients of the equations

The mode u is assumed normalized to get unitary modal mass (u T Mu=1). The expression of the coefficients of Eq. (19) are:

$$ \everymath{\displaystyle }\begin{array}{@{}l} c_1\!= -\frac{1}{2}\mathbf {u}^T\hspace{-0.6pt}\mathbf {C}\mathbf {u},\qquad c_2=-\frac{1}{\!2\omega\!}\mathbf {u}^T\hspace{-0.6pt}\mathbf {K}_1\mathbf {u},\qquad c_3=-\frac{\xi}{2}r \\ \noalign {\vspace {6pt}} c_4= \frac{3\kappa r}{2\omega},\qquad c_5=\frac{1}{2\omega} \mathbf {u}^T\hspace{-0.6pt}\mathbf {n}(\mathbf {u},\mathbf {u},\bar{\mathbf {u}}),\qquad c_6=\frac{1}{4\omega} \end{array} $$
(47)

In Eq. (20), the column matrices w j (j=1,…,6) are the solutions of the following singular algebraic problems in which, however, compatibility is satisfied:

(48)
(49)
(50)
(51)
(52)
(53)

The solution is made unique by the normalization condition \(\mathbf {w}_{j}^{T}\mathbf {u}=0\).

Moreover, w j (j=7,8) are the solutions of the following non-singular algebraic:

(54)
(55)

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Luongo, A., Zulli, D. Dynamic analysis of externally excited NES-controlled systems via a mixed Multiple Scale/Harmonic Balance algorithm. Nonlinear Dyn 70, 2049–2061 (2012). https://doi.org/10.1007/s11071-012-0597-6

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  • DOI: https://doi.org/10.1007/s11071-012-0597-6

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