Vibration control of a cantilever beam coupled with magnetic tri-stable nonlinear energy sink

Abstract

In response to limitations in vibration suppression performance of traditional linear tuned mass damper due to energy threshold constraints and narrow vibration bands, this study proposes a magnetic tri-stable NES (MTNES) formed by combining a linear spring and magnets. Compared to the conventional nonlinear energy sink (NES), the magnetic tri-stable NES (MTNES) incorporates magnetism to enhance the nonlinear stiffness. Firstly, the mechanism of the MTNES is introduced in this study, which reveals the existence of the three stable points in the system. Subsequently, the equations of motion of the coupled system with MTNES attached to the cantilever beam are derived, and the optimal parameter combination for MTNES is determined using a global optimization method. Furthermore, the influence of MTNES parameter variations on vibration suppression efficiency is studied through parameter analysis. Then, the restoring force of the MTNES is simplified into polynomial form, and the system response is analyzed using the harmonic balance method and Runge–Kutta method. Finally, experimental studies on the coupled system are conducted. The results indicate that MTNES can effectively suppress the resonance of the host structure within a wide frequency band, with the highest vibration suppression rate of up to 66% under strong modulated response. Additionally, the results of numerical calculations and theoretical analysis are in good agreement with that of the experiment.

Appendices

Appendix A. Stability criterion of SIM

Linearizing the second equation of Eq. (22) at the fixed point $B_{eq} = B - \Delta_{B}$ obtained from the solutions of Eq. (23), the stability of the solutions on the SIM is described as,

$$ \left[ {\begin{array}{*{20}c} {\dot{\Delta }_{B} } \\ {\dot{\Delta }_{B}^{ * } } \\ \end{array} } \right] = \underbrace {{\left[ {\begin{array}{*{20}c} {a_{11} } & {a_{12} } \\ {a_{21} } & {a_{22} } \\ \end{array} } \right]}}_{\Theta }\left[ {\begin{array}{*{20}c} {\Delta_{B} } \\ {\Delta_{B}^{ * } } \\ \end{array} } \right]. $$

(A-1)

where $\Delta_{B} = B - B_{eq}$ and

$$ \begin{aligned} & a_{11} = a^{ * }_{22} = - iX - \xi_{n} \sqrt X + \left. {\sum\limits_{\gamma = 1}^{9} {\frac{\gamma + 1}{2}k_{\gamma } } C_{\gamma }^{{\frac{\gamma + 1}{2}}} B^{{\frac{\gamma - 1}{2}}} B^{{ * \frac{\gamma - 1}{2}}} } \right|_{{B = B_{eq} }} \quad (\gamma = 1,3,5,7,9), \hfill \\ & a_{12} = a^{ * }_{21} = \left. {i\sum\limits_{\gamma = 1}^{9} {\frac{\gamma - 1}{2}k_{\gamma } } C_{\gamma }^{{\frac{\gamma - 1}{2}}} B^{{\frac{\gamma + 1}{2}}} B^{{ * \frac{\gamma - 3}{2}}} } \right|_{{B = B_{eq} }} \quad (\gamma = 1,3,5,7,9). \hfill \\ \end{aligned} $$

(A-2)

The eigenvalues of the Jacobian matrix $\Theta$ dictate the stability. Positive real eigenvalues indicate instability of the fixed point.

Appendix B. Stability criterion under harmonic load

The calculation of the stability for the harmonic balanced Eq. (21) involves assessing the linear stability around the equilibrium points A and B, where $\Delta_{A} = A - A_{eq}$, $\Delta_{B} = B - B_{eq}$ and

$$ i2\sqrt X \left[ {\begin{array}{*{20}c} {\dot{\Delta }_{A} } \\ {\dot{\Delta }_{A}^{ * } } \\ {\dot{\Delta }_{B} } \\ {\dot{\Delta }_{B}^{ * } } \\ \end{array} } \right] = \underbrace {{\left[ {\begin{array}{*{20}c} {a_{11} } & {a_{12} } & {a_{13} } & {a_{14} } \\ {a_{21} } & {a_{22} } & {a_{23} } & {a_{24} } \\ {a_{31} } & {a_{32} } & {a_{33} } & {a_{34} } \\ {a_{41} } & {a_{42} } & {a_{43} } & {a_{44} } \\ \end{array} } \right]}}_{\Psi }\left[ {\begin{array}{*{20}c} {\Delta_{A} } \\ {\Delta_{A}^{ * } } \\ {\Delta_{B} } \\ {\Delta_{B}^{ * } } \\ \end{array} } \right], $$

(B-1)

$$ a_{12} = a_{21} = a_{32} = a_{41} = 0, $$

(B-2)

$$ a_{11} = - a^{ * }_{22} = - \varepsilon \sigma - i\varepsilon \xi \sqrt X , $$

(B-3)

$$ a_{13} = - a_{24}^{ * } = i\varepsilon \xi_{n} \sqrt X + \left. {\varepsilon \sum\limits_{\gamma = 1}^{9} {\overline{{k_{\gamma } }} } \frac{\gamma + 1}{2}C_{\gamma }^{{\frac{\gamma + 1}{2}}} B^{{\frac{\gamma - 1}{2}}} B^{{ * \frac{\gamma - 1}{2}}} } \right|_{{B = B_{eq} }} (\gamma = 1,3,5,7,9), $$

(B-4)

$$ a_{14} = - a_{23}^{ * } = \varepsilon \left. {\sum\limits_{\gamma = 1}^{9} {\overline{{k_{\gamma } }} } \frac{\gamma - 1}{2}C_{\gamma }^{{\frac{\gamma + 1}{2}}} B^{{\frac{\gamma + 1}{2}}} B^{{ * \frac{\gamma - 3}{2}}} } \right|_{{B = B_{eq} }} \quad (\gamma = 1,3,5,7,9), $$

(B-5)

$$ a_{31} = - a_{42}^{ * } = 1 + i\varepsilon \xi \sqrt X , $$

(B-6)

$$ a_{33} = - a_{44}^{ * } = X - \frac{1 + \varepsilon }{\varepsilon }a_{13} , $$

(B-7)

$$ a_{34} = - a_{43}^{ * } = - \frac{1 + \varepsilon }{\varepsilon }a_{14} . $$

(B-8)

If any eigenvalue of matrix Ψ possesses a real part greater than zero, the point is deemed unstable.