The Low-Frequency Broadband Mechanism of Nonlinear Elastic Metamaterials With Gaps
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摘要:
揭示了基于非线性混沌理论含间隙的非线性局域共振结构的低频宽带形成机理，提出了一类含间隙非线性局域共振结构设计的新理念。在该间隙非线性局域共振系统中，产生了非线性混沌现象，且这种非线性运动可以成功地改变振动噪声中的频谱结构，当系统运动进入混沌状态时，线性谱能量大大削弱，变成了一个连续的宽频谱，进而有效隔离低频线谱。有限元计算结果表明，正是这个间隙引起的非线性混沌现象导致了低频宽带的产生，且理论分析和有限元分析结果高度一致。因此，这类含间隙非线性局域共振弹性超材料结构的设计新思想为局域共振弹性超材料的发展开辟了新天地，且基于非线性混沌理论的低频带隙的形成机理为减振降噪应用研究奠定了非常重要的理论基础。
Abstract:A new formation mechanism of the low-frequency broadband within gapped nonlinear local resonance structures was revealed based on the nonlinear chaos theory, and a novel concept for designing nonlinear local resonant structures with small gaps was further proposed. Due to the small gaps, the nonlinear chaos phenomenon occurs in the local resonance system, which can change the spectrum structure in vibration noise successfully, and the linear spectral energy greatly weakens and a continuous broad spectrum forms after chaotic motion, to effectively isolate the low-frequency spectrum. Most importantly, the finite element results show that, the nonlinearity of the small gap indeed leads to the low-frequency band-gap within the nonlinear local resonance. Therefore, the new idea for designing the nonlinear local resonance structure makes a new way to the development of local resonant elastic metamaterials, and the formation mechanism of low-frequency band-gap based on the nonlinear chaos theory lays a very important theoretical basis for vibration and noise reduction.
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图 2 不同间隙下m2的相图及基体m1的位移响应频谱图：(a) 间隙为0.065 mm，m2相图；(b) 间隙为0.065 mm，m1频谱图；(c) 间隙为0.062 mm，m2相图；(d) 间隙为0.062 mm，m1频谱图；(e) 间隙为0.05 mm，m2相图；(f) 间隙为0.05 mm，m1频谱图；(g) 间隙为0.013 5 mm，m2相图；(h) 间隙为0.013 5 mm，m1频谱图
Figure 2. Phase diagrams of m2 and displacement response spectra of matrix m1 under different gaps: (a) gap is 0.065 mm, the m2 phase diagram; (b) gap is 0.065 mm, the m1 displacement response spectrum; (c) gap is 0.062 mm, the m2 phase diagram; (d) gap is 0.062 mm, the m1 displacement response spectrum; (e) gap is 0.05 mm, the m2 phase diagram; (f) gap is 0.05 mm, the m1 displacement response spectrum; (g) gap is 0.013 5 mm, the m2 phase diagram; (h) gap is 0.013 5 mm, the m1 displacement response spectrum
图 4 基体m1的位移响应频谱图：(a) 线性系统基体m1的位移响应频谱图；(b) 未混沌状态下非线性系统m1的位移响应频谱图；(c) 混沌状态下非线性系统m1的位移响应频谱图
Figure 4. Displacement response spectrograms of matrix m1: (a) the linear system m1 displacement response spectrum; (b) the displacement response spectrogram of nonlinear system m1 in the unchaotic state; (c) the displacement response spectrogram of nonlinear system m1 in the chaotic state
图 5 局域共振弹性超材料结构：(a) 无间隙局域共振结构单元；(b) 含间隙局域共振结构单元
Figure 5. The local resonance elastic metamaterial structure: (a) the gapless LR structural unit; (b) the gapped LR structural unit
图 7 局域共振结构带隙: (a) 无间隙局域共振结构的带隙；(b) 含间隙非线性局域共振结构的带隙；(c) 图(b)中带隙下界对应的放大图
Figure 7. Local resonance structure band gaps: (a) band gaps of gapless local resonance structures; (b) band gaps of gapped local resonance structures; (c) the enlarged view corresponding to the lower bound of the band gaps in fig. (b)
图 9 非线性局域共振结构的带隙及传输率: (a) 弹性波的透射率；(b) 带隙
Figure 9. Band gaps and transmission rates of nonlinear local resonance structures: (a) the transmissivity of elastic waves; (b) band gaps
图 10 非线性局域共振结构各个点的振型: (a) A点的模态；(b) B点的模态；(c) C点的模态；(d) D点的模态；(e) E点的模态；(f) F点的模态
注 为了解释图中的颜色，读者可以参考本文的电子网页版本。
Figure 10. Mode shapes at various points of nonlinear LR structures: (a) point A modal; (b) point B modal; (c) point C modal; (d) point D modal; (e) point E modal; (f) point F modal
图 11 间隙对非线性局域共振结构带隙的影响
Figure 11. Influences of gaps on the band gaps of nonlinear LR structures
表 1 材料参数
Table 1. Material parameter
material Young’s modulus E/GPa Poisson’s ratio ν density ρ/(kg/m3) silicone rubber 1.2 0.47 1300 plumbum 4.35 0.368 11600 perspex 0.2 0.389 1142 aluminum 7.2 0.35 2730 
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