## 留言板 X构型张力非线性系统共振激励下的拍振现象

 引用本文: 齐子涵，吴志强，焦云雷，贾文文. X构型张力非线性系统共振激励下的拍振现象 [J]. 应用数学和力学，2022，43（6）：597-607 QI Zihan, WU Zhiqiang, JIAO Yunlei, JIA Wenwen. Beat Vibration of X-Configuration Tension Nonlinear Systems Under Resonance Excitation[J]. Applied Mathematics and Mechanics, 2022, 43(6): 597-607. doi: 10.21656/1000-0887.420326
 Citation: QI Zihan, WU Zhiqiang, JIAO Yunlei, JIA Wenwen. Beat Vibration of X-Configuration Tension Nonlinear Systems Under Resonance Excitation[J]. Applied Mathematics and Mechanics, 2022, 43(6): 597-607. • 中图分类号: O322

## Beat Vibration of X-Configuration Tension Nonlinear Systems Under Resonance Excitation

• 摘要:

圆形太阳翼因收纳比高、供电能力强等特点受到人们的广泛重视。作为大尺寸薄膜结构，为了调节薄膜张力，通常会引入由绳和弹簧组成的张力调节装置，其力学特性具有强非线性特征，但目前还未有研究讨论其影响。该文提出了一种研究张力影响的机理模型，并利用Lagrange能量法建立了系统二自由度非线性动力学方程，以某工程样机为例，研究了张力机构出现肋板不对称时系统在共振激励下的响应。研究表明，激励幅值变化对系统拍振响应特点有重要影响，使其出现了混沌、概周期以及多倍周期等现象。这些结果对张力机构参数设计有重要参考作用。

• 图  1  张力非线性结构模型示意图

Figure  1.  Schematic diagram of the tension nonlinear symmetric model

图  2  激励幅值为0.3线性系统的响应：(a) 肋板1时间历程图；(b) 肋板2时间历程图；(c) 肋板1频谱图；(d) 肋板2频谱图

Figure  2.  With an excitation amplitude of 0.3, the linear system responses：(a) the time history diagram of rib 1; (b) the time history diagram of rib 2; (c) the spectrum diagram of rib 1; (d) the spectrum diagram of rib 2

图  3  激励幅值为0.3非线性系统的响应：(a) 肋板1时间历程图；(b) 肋板2时间历程图；(c) 肋板1 Poincaré截面图；(d) 肋板2 Poincaré截面图；(e) 肋板1频谱图；(f) 肋板2频谱图；(g) 肋板1 Hilbert包络谱；(h) 肋板2 Hilbert包络谱

Figure  3.  With an excitation amplitude of 0.3, the nonlinear system responses: (a) the time history diagram of rib 1; (b) the time history diagram of rib 2; (c) the Poincaré section diagram of rib 1; (d) the Poincaré section diagram of rib 2; (e) the spectrum diagram of rib 1; (f) the spectrum diagram of rib 2; (g) the Hilbert envelope spectrum diagram of rib 1; (h) the Hilbert envelope spectrum diagram of rib 2

图  4  激励幅值改变下肋板分岔图：(a) 肋板1分岔图；(b) 肋板2分岔图

Figure  4.  With a changing excitation amplitude, the bifurcation diagrams of the lower ribs: (a) the bifurcation diagram of rib 1; (b) the bifurcation diagram of rib 2

图  5  激励幅值为0.67结构非对称系统的响应：(a) 肋板1时间历程图；(b) 肋板2时间历程图；(c) 肋板1 Poincaré截面图；(d) 肋板2 Poincaré截面图；(e) 肋板1频谱图；(f) 肋板2频谱图；(g) 肋板1 Hilbert包络谱；(h) 肋板2 Hilbert包络谱

Figure  5.  With an excitation amplitude of 0.67, the asymmetric-structure system responses: (a) the time history diagram of rib 1; (b) the time history diagram of rib 2; (c) the Poincaré section diagram of rib 1; (d) the Poincaré section diagram of rib 2; (e) the spectrum diagram of rib 1; (f) the spectrum diagram of rib 2; (g) the Hilbert envelope spectrum diagram of rib 1; (h) the Hilbert envelope spectrum diagram of rib 2

图  6  激励幅值为0.72结构非对称系统的响应：(a) 肋板1时间历程图；(b) 肋板2时间历程图；(c) 肋板1 Poincaré截面图；(d) 肋板2 Poincaré截面图；(e) 肋板1频谱图；(f) 肋板2频谱图；(g) 肋板1 Hilbert包络谱；(h) 肋板2 Hilbert包络谱

Figure  6.  With an excitation amplitude of 0.72, the asymmetric-structure system responses: (a) the time history diagram of rib 1; (b) the time history diagram of rib 2; (c) the Poincaré section diagram of rib 1; (d) the Poincaré section diagram of rib 2; (e) the spectrum diagram of rib 1; (f) the spectrum diagram of rib 2; (g) the Hilbert envelope spectrum diagram of rib 1; (h) the Hilbert envelope spectrum diagram of rib 2

图  7  激励幅值为2.66结构非对称系统的响应：(a) 肋板1时间历程图；(b) 肋板2时间历程图；(c) 肋板1 Poincaré截面图；(d) 肋板2 Poincaré截面图；(e) 肋板1频谱图；(f) 肋板2频谱图；(g) 肋板1 Hilbert包络谱；(h) 肋板2 Hilbert包络谱

Figure  7.  With an excitation amplitude of 2.66, the asymmetric-structure system responses: (a) the time history diagram of rib 1; (b) the time history diagram of rib 2; (c) the Poincaré section diagram of rib 1; (d) the Poincaré section diagram of rib 2; (e) the spectrum diagram of rib 1; (f) the spectrum diagram of rib 2; (g) the Hilbert envelope spectrum diagram of rib 1; (h) the Hilbert envelope spectrum diagram of rib 2

表  1  模型参数取值

Table  1.   Values of model parameters

 parameter value rib plate length l /m 2.9 elastic modulus E /Pa 7.2 × 1010 rib plate width e /m 0.005 rib plate 1 height h1 /m 0.04 rib plate 2 height h2 /m 0.05 material density ρ/(kg/m3) 435 tension spring stiffness k0/(N/m) 5.59 grounding spring stiffness k1/(N/m) 2900 structural damping C /(N·s·m-1) 0.366
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##### 出版历程
• 收稿日期:  2021-10-28
• 修回日期:  2021-11-16
• 网络出版日期:  2022-05-23
• 刊出日期:  2022-06-30

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