Beat Vibration of X-Configuration Tension Nonlinear Systems Under Resonance Excitation
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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。
Abstract:The circular membrane solar array has attracted extensive attention due to its high storage ratio and strong power supply capability. In order to adjust the tension of large film structures, a tension adjusting device composed of ropes and springs is usually introduced, and its mechanical characteristics are highly nonlinear, with the effects rarely studied yet. Aimed at the tension adjustment, a mechanism model was proposed. The nonlinear dynamics equation for the 2DOF system was established with the Lagrangian energy method. With an engineering prototype as the example, the responses of the tension mechanism with unsymmetrical ribs under resonance excitation were studied. The results show that, the change of the excitation amplitude has an important influence on the characteristics of the beat response of the system. Consequently, the responses of the system exhibit chaotic, almost periodic and multifold periodic phenomena. The research makes an important reference to the parameter design of tension mechanisms.
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Key words:
- tension adjusting unit /
- nonlinearity /
- resonance /
- beat vibration
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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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