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Nonlinear energy sink to control elastic strings: the internal resonance case

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Abstract

The use of nonlinear energy sink as a passive control device is extended here to a nonlinear elastic string, in internal resonance conditions, excited by an external harmonic force. The Multiple Scale/Harmonic Balance Method is directly applied to the partial differential equations ruling the dynamics of the system. The internal resonance condition of the string involves a rich response containing essentially both the resonant, directly excited, mode and a superharmonic one. Numerical results on a case study are presented.

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Authors and Affiliations

  1. M&MoCS - 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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Corresponding author

Correspondence to Angelo Luongo.

Additional information

This work has been supported by the Italian Ministry of University (MIUR) through a PRIN 2010–2012 program (2010MBJK5B).

Appendix: Coefficients of the equations

Appendix: Coefficients of the equations

The expression of the right-hand sides of Eq. (31) is:

$$\begin{aligned} \mathcal {F}_1= & {} -\,\frac{\zeta }{2}a_r-\xi b\cos (\gamma -\vartheta )\sin (\omega _r x_C)+\frac{p_r}{\omega _r}\sin (\gamma ) \\&-\,\frac{3}{4\omega }\kappa b^3\sin (\gamma -\vartheta )\sin (\omega _r x_C) \end{aligned}$$
$$\begin{aligned} \mathcal {F}_2= & {} -\frac{\zeta }{2}a_s-\frac{1}{4\omega }\kappa b^3\sin (\omega _r x_C)\sin (3\gamma -3\vartheta -\chi ) \\ \mathcal {F}_3= & {} -\frac{1}{2}\xi (\omega _r\sigma ) b-\frac{1}{2}m x_C\xi \omega _r^2b\sin (\omega _r x_C)\cos (\omega _r x_C) \\&+\,\frac{1}{2}m\xi \omega _r b\sin (\omega _r x_C)^2 \\&+\,m\omega _r(\omega _r+\sigma )a\sin (\omega _r x_C)\sin (\gamma -\vartheta )\\&-\,\frac{1}{2}mp_r\left[ 1 +(1-x_C+x_C\cos (\omega _r))\cos (\omega _r x_C)\right. \\&\left. +\,\frac{1}{2\omega _r}(1-\cos (\omega _r))\sin (\omega _r x_C)\right] \sin (\vartheta ) \\&+\,\frac{m\omega _r^4\omega _s^2}{32\omega _r^3-32\omega _r\omega _s^2}(\omega _s-\omega _r)a_r^2a_s\\&\times \,\sin (\omega _r x_C)\sin (\gamma -\vartheta -\chi ) \end{aligned}$$
$$\begin{aligned} \mathcal {F}_4= & {} a_r\sigma -\frac{3}{32}\eta \omega _r^3a_r^3-\frac{\eta }{16}\omega _r\omega _s^2a_ra_s^2+\frac{ p_r}{\omega _r}\cos (\gamma )\nonumber \\&-\,\frac{3}{4\omega _r}\kappa b^3\sin (\omega _r x_C)\cos (\gamma -\vartheta )\nonumber \\&+\,\xi b\sin \omega _r x_C)\sin (\gamma -\vartheta )\nonumber \\ \mathcal {F}_5= & {} -3\sigma a_s+\frac{\eta }{16}\omega _r\omega _s^2\_r^2 a_s+\frac{3}{32\omega _r}\eta \omega _s^4a_s^3\nonumber \\&+\frac{\kappa }{4\omega _r}b^3\sin (\omega _r x_C)\cos (3\gamma -3\vartheta -\chi )\nonumber \\ \mathcal {F}_6= & {} -m\omega _r\left( \frac{\omega _r}{2}+\sigma \right) b+\frac{3}{8}\kappa b^3\nonumber \\&-\frac{1}{2}m p_r x_C\left[ 1-(1-x_C+x_C\cos (\omega _r))\right. \nonumber \\&\left. \times \cos (\omega _r x_C)-(1-\cos (\omega _r)\frac{\sin (\omega _r x_C)}{2\omega _r}\right] \cos (\vartheta )\nonumber \\&+\,\frac{3}{8}m\kappa \omega _r x_C b^3\cos (\omega _r x_C)\sin (\omega _r x_C)\nonumber \\&+\,m\left( \sigma +\frac{\omega _r}{2}\right) \omega _ra_r\sin (\omega _r x_C)\cos (\gamma -\vartheta )\nonumber \\&+\frac{m\omega _r^4\omega _s^2}{32\omega _r^3-32\omega _r\omega _s^2}(\omega _s-\omega _r)a_r^2a_s\sin (\omega _r x_C)\nonumber \\&\times \cos (\gamma -\vartheta -\chi )-\frac{3}{8}\kappa m b^3\sin (\omega _r x_C)^2 \end{aligned}$$
(34)

The mass matrix of Eq. (31) is \(\mathcal {M}(\mathbf {u})=\)

$$\begin{aligned} \begin{bmatrix} 1&\quad 0&\quad 0&\quad 0&\quad 0&\quad 0\\ 0&\quad 1&\quad 0&\quad 0&\quad 0&\quad 0\\ -m\omega _r\sin (\omega _rx_C)\cos (\gamma -\vartheta )&\quad 0&\quad m\omega _r&\quad m\omega _r a_r\sin (\omega _r x_C)\sin (\gamma -\vartheta )&\quad 0&\quad -\frac{1}{2}\xi b\\ 0&\quad 0&\quad 0&\quad a_r&\quad 0&\quad 0\\ 0&\quad 0&\quad 0&\quad -3a_s&\quad a_s&\quad 0\\ -m\omega _r\sin (\omega _rx_C)\sin (\gamma -\vartheta )&\quad 0&\quad \frac{1}{2}\xi&\quad -m\omega _r a_r\sin (\omega _r x_C)\cos (\gamma -\vartheta )&\quad 0&\quad m\omega _rb \end{bmatrix} \end{aligned}$$
(35)

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Luongo, A., Zulli, D. Nonlinear energy sink to control elastic strings: the internal resonance case. Nonlinear Dyn 81, 425–435 (2015). https://doi.org/10.1007/s11071-015-2002-8

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