INTERVAL UNCERTAINTY ANALYSIS METHODS FOR MULTIBODY SYSTEMS BASED ON SIGNAL DECOMPOSITION AND CHEBYSHEV POLYNOMIALS
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摘要: 参数的不确定性会对多体系统动力学响应产生显著影响, 区间分析方法只需根据不确定性参数的边界信息, 便可实现在多体系统动力学分析中考虑参数的不确定性. 考虑区间参数不确定性, 采用切比雪夫区间方法(Chebyshev interval method, CIM)在分析多体系统动力学响应时, 随着时间的增大, 响应边界精度会越来越低. 为解决CIM的这一问题, 本文将信号分解技术与切比雪夫多项式结合, 采用切比雪夫多项式分别对HHT变换(Hilbert-Huang transform, HHT)和局域均值分解(local mean decomposition, LMD)得到的瞬时幅值和瞬时相位近似, 提出CIM-HHT方法和CIM-LMD方法, 以获得含区间参数的长周期动力学响应边界. HHT和LMD分解能够将多体系统的多分量响应分解为多个单分量和一个趋势分量(残余分量)之和, CIM-HHT和CIM-LMD对每个分量的瞬时幅值和瞬时相位、和趋势分量采用切比雪夫多项式近似, 进而建立系统响应的耦合模型, 可以得到系统的动力学响应边界. 最后, 考虑单摆和曲柄滑块机构中的参数不确定性, 验证了CIM-HHT和CIM-LMD方法的有效性. 结果表明, 相比CIM, 在长周期区间动力学响应分析中CIM-HHT和CIM-LMD能够获得较准确的结果. 此外, 相比CIM-HHT, CIM-LMD具有更弱的末端效应, 计算精度更高.Abstract: Uncertainty inherited in the parameters of multibody systems will induce significant deviation on the dynamic responses. The interval analysis method, which only need the information of lower and upper bounds of the interval uncertain parameters, can efficiently consider uncertainties in the dynamics analysis of multibody systems. The bounds of responses obtained by the CIM (Chebyshev interval method) for multibody systems in the presence of interval uncertainty would deteriorate with the increase of time history. To circumvent this problem, two novel methods CIM-HHT (Hilbert-Huang transform) and CIM-LMD (local mean decomposition), which combine signal decomposition technique and Chebyshev polynomials, are developed in this paper to accurately envelope the long period interval responses of system under interval uncertainty. The HHT and LMD are combined, respectively, with the Chebyshev polynomials to approximate the instantaneous amplitude and phase obtained by signal decomposition. HHT and LMD can decompose the multicomponent responses of multibody system into the sum of several monocomponent and a trend component. Then, the instantaneous amplitude and instantaneous phase of the monocomponent, and the trend component can be employed to construct corresponding surrogate model by the Chebyshev polynomials, respectively. Based on the surrogate models for the instantaneous amplitude, instantaneous phase and trend component, the coupling entire surrogate model for the system can be established and the upper bound and lower bound of the system responses can be calculated subsequently. To verify the accuracy and effectiveness of the proposed methods, a simple pendulum and a crank slider under interval uncertainty are presented. Numerical results demonstrated that the CIM-HHT and CIM-LMD present desirable computational accuracy in the procedure of long period interval dynamic analysis of multibody systems. Furthermore, compared with CIM-HHT, the CIM-LMD is characterized with weaker end effect and high computational accuracy in the long period interval dynamic analysis of multibody systems.
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Key words:
- interval uncertainty /
- multibody system /
- dynamics /
- signal decomposition /
- Chebyshev polynomial
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图 3 末端位置和速度的HHT和LMD分解
Figure 3. HHT and LMD decomposition of position and velocity response of end-tip
图 4 不同样本的末端位置响应分解
Figure 4. Decomposition of position response of end tip under different samples
图 7 不同方法得到的位置响应边界
Figure 7. Bounds for position responses of end-tip obtained by different methods
图 9 滑块位移和速度响应的HHT和LMD分解
Figure 9. HHT and LMD decomposition for position and velocity response of slider
图 14 不同方法得到的滑块速度响应边界
Figure 14. Bounds for response of slider velocity using different methods
图 15 不确定度为5%时CIM-LMD和RS-LMD得到的响应边界
Figure 15. Bounds of responses using CIM-LMD and RS-LMD under 5% uncertainty level
表 1 曲柄机构参数
Table 1. Parameters for crank slider
Parameter Value length of crank/m 0.15 mass of crank/kg 0.37 length of connector/m 0.56 mass of connector/kg 0.77 mass of slider/kg 0.45 τ/(N·m) −0.5 c/(N·m·s−1) 1 k/(N·m−1) 5 
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