Low-frequency multi-mode vibration suppression of a metastructure beam with two-stage high-static-low-dynamic stiffness oscillators

Abstract

Metastructures with periodic local resonators are effective in attenuating waves in a special frequency range due to their band gap properties. A low-frequency multi-mode resonator based on a two-stage high-static-low-dynamic stiffness oscillator is proposed in this paper to create multiple low-frequency band gaps of flexural waves in a metastructure beam. The theoretical models of infinite and finite metastructure beams are established, respectively. The band structures obtained by the plane wave expansion method are thereafter verified by the mode superstition method. This demonstrates such metastructure features with multiple low-frequency band gaps. Multiple band gaps property is discussed, and a dynamic analysis of a simplified cell shows that locations of band gaps are determined by the natural frequencies of the resonator, which can be adjusted by configuring the physical parameters of the oscillators. The influence of mass ratio and damping on band gaps is numerically analysed, and the results show that mass ratio has an optimal value and damping can suppress the resonance effectively. Finally, a case study is performed, and the results show that the proposed metastructure has good performance in vibration suppression both around concerned multiple low-frequency modes and high-frequency range. The general design procedure of a metastructure beam is also summarized.

Appendix

In Eq. (10), the parameters in detail are

$$\begin{aligned} \mathbf{K}= & {} \left[ {{\begin{array}{ccccc} {EIa\left[ {\mathbf{q}_\mathbf{m} } \right] +k_1 \left[ \mathbf{U} \right] }&{}\quad {-k_1 \left[ \mathbf{P} \right] }&{}\quad &{}\quad &{}\quad \\ {-k_1 \left[ \mathbf{P} \right] ^{\mathrm{T}}}&{}\quad {k_1 +k_2 +\cdots +k_g }&{}\quad {-k_2 }&{}\quad \cdots &{}\quad {-k_g } \\ &{}\quad {-k_2 }&{}\quad {k_2 }&{}\quad &{}\quad \\ &{}\quad \vdots &{}\quad &{}\quad \ddots &{}\quad \\ &{}\quad {-k_g }&{}\quad &{}\quad &{}\quad {k_g } \\ \end{array} }} \right] ,\nonumber \\ \mathbf{M}= & {} \left[ {{\begin{array}{ccccc} {\rho Aa\mathbf{I}_{\bar{{M}}\times \bar{{M}}} }&{}\quad &{}\quad &{}\quad &{}\quad \\ &{}\quad {m_1 }&{}\quad &{}\quad &{}\quad \\ &{}\quad &{}\quad {m_2 }&{}\quad &{}\quad \\ &{}\quad &{}\quad &{}\quad \ddots &{}\quad \\ &{}\quad &{}\quad &{}\quad &{}\quad {m_g } \\ \end{array} }} \right] , \\ \mathbf{q}_{\mathbf{m}}= & {} \left[ {{\begin{array}{cccc} {\left[ {q-{2M\pi }/a} \right] ^{4}}&{}\quad &{}\quad &{}\quad \\ &{}\quad {\left[ {{\left( {q-2\left( {M-1} \right) } \right) }/a} \right] ^{4}}&{}\quad &{}\quad \\ &{}\quad &{}\quad \ddots &{}\quad \\ &{}\quad &{}\quad &{}\quad {\left[ {q+{2M\pi }/a} \right] ^{4}} \\ \end{array} }} \right] , \mathbf{P}=\left( {{\begin{array}{l} 1 \\ 1 \\ \vdots \\ 1 \\ \end{array} }} \right) _{\bar{{M}}\times \bar{{M}}} , \mathbf{U}=\mathbf{PP}^{\mathrm{T}}. \end{aligned}$$

In Eq. (15), the parameters in detail are

$$\begin{aligned} {\tilde{\mathbf{K}}}= & {} \left[ {{\begin{array}{ccccc} {EI\left[ {\varvec{{\upbeta }}} \right] +k_1 \left[ \mathbf{V} \right] }&{}\quad {-k_1 \left[ \mathbf{Q} \right] }&{}\quad &{}\quad &{}\quad \\ {-k_1 \left[ \mathbf{Q} \right] ^{\mathrm{T}}}&{}\quad {\left( {k_1 +k_2 +\cdots +k_g } \right) \mathbf{I}_{N\times N} }&{}\quad {-k_2 \mathbf{I}_{N\times N} }&{}\quad \cdots &{}\quad {-k_g \mathbf{I}_{N\times N} } \\ &{}\quad {-k_2 \mathbf{I}_{N\times N} }&{}\quad {k_2 \mathbf{I}_{N\times N} }&{}\quad &{}\quad \\ &{}\quad \vdots &{}\quad &{}\quad \ddots &{}\quad \\ &{}\quad {-k_g \mathbf{I}_{N\times N} }&{}\quad &{}\quad &{}\quad {k_g \mathbf{I}_{N\times N} } \\ \end{array} }} \right] ,\\ {\tilde{\mathbf{M}}}= & {} \left[ {{\begin{array}{ccccc} {\rho A\mathbf{I}_{S\times S} }&{}\quad &{}\quad &{}\quad &{}\quad \\ &{}\quad {m_1 \mathbf{I}_{N\times N} }&{}\quad &{}\quad &{}\quad \\ &{}\quad &{}\quad {m_2 \mathbf{I}_{N\times N} }&{}\quad &{}\quad \\ &{}\quad &{}\quad &{}\quad \ddots &{}\quad \\ &{}\quad &{}\quad &{}\quad &{}\quad {m_g \mathbf{I}_{N\times N} } \\ \end{array} }} \right] ,\\ {\varvec{\upbeta }}= & {} \left[ {{\begin{array}{llll} {\beta _1 }&{}\quad &{}\quad &{}\quad \\ &{}\quad {\beta _2 }&{}\quad &{}\quad \\ &{}\quad &{}\quad \ddots &{}\quad \\ &{}\quad &{}\quad &{}\quad {\beta _S } \\ \end{array} }} \right] ,\quad \mathbf{Q}=\left[ {{\begin{array}{llll} {W_{01} \left( {x_1 } \right) }&{}\quad {W_{01} \left( {x_2 } \right) }&{}\quad \cdots &{}\quad {W_{01} \left( {x_N } \right) } \\ {W_{02} \left( {x_1 } \right) }&{}\quad {W_{02} \left( {x_2 } \right) }&{}\quad \cdots &{}\quad {W_{02} \left( {x_N } \right) } \\ \vdots &{}\quad \vdots &{}\quad &{}\quad \vdots \\ {W_{0S} \left( {x_1 } \right) }&{}\quad {W_{0S} \left( {x_2 } \right) }&{}\quad \cdots &{}\quad {W_{0S} \left( {x_N } \right) } \\ \end{array} }} \right] , \mathbf{V}=\mathbf{QQ}^{\mathrm{T}},\\ {\tilde{\mathbf{C}}}= & {} \left[ {C_{0,1} \,\cdots \,C_{0,S} \quad \,C_{1,1} \quad \cdots \quad C_{1,N} \quad C_{2,1} \cdots \quad C_{g,N} } \right] ^{\mathrm{T}}, \quad {\tilde{\mathbf{F}}}=\left[ {F_1 \quad F_2 \quad \cdots \quad F_S \quad \mathbf{0}_{1\times Ng} } \right] ^{\mathrm{T}}, \quad F_s =W_{0s} \left( {x_0 } \right) . \end{aligned}$$