正交多项式在不确定转子动态响应计算中的应用及对比分析

为研究航空发动机转子系统中的区间不确定性对系统动态响应的影响，提出利用正交多项式求解不确定转子响应的非嵌入式区间分析法，克服了传统概率方法需要参数先验概率分布的苛刻要求。用有限元法建立了悬臂转子的确定性运动方程，阐述了Chebyshev和Legendre两种正交多项式建立响应代理模型的原理和计算步骤。通过与Monte Carlo抽样对比，验证方法的可行性和精度。对照Monte Carlo方法500样本的计算结果，两种多项式区间法计算结果都具有较高精度，误差均小于1%，而计算时间则分别为Monte Carlo法的2.5%和5.4%，Chebyshev多项式方法具有更高的计算效率。分析了不同不确定参数在不同不确定水平下，系统的响应范围。研究表明，正交多项式区间分析法可高精度高效率地计算转子系统区间响应范围，不确定性对该转子系统动力特性影响很大，多源不确定性传播可引起转子系统大幅振动。 

Application and comparative analysis of orthogonal polynomials inuncertain rotor dynamic response calculation

In order to quantify the influence of interval uncertainties on the dynamic response of a rotor system, a non-intrusive interval analysis procedure was proposed using orthogonal polynomials, so the strict requirements of probabilistic distribution in stochastic methods were met. The deterministic model and equations of motion were established via the finite element method. The surrogate models of the uncertain response were derived by using Chebyshev polynomials and Legendre polynomials and the general solution procedure was illustrated. The classic Monte Carlo simulation was used to demonstrate the accuracy and efficiency of the two series based interval methods. Compared with the results of Monte Carlo simulation with 500 samples, the errors in the interval methods were less than 1% and the computing time were 2.5% and 5.4%, respectively. In the problem under study, the Chebyshev polynomials based method was more efficient. Then the dynamic response curves of the rotor under different uncertain parameters with different uncertain degrees were given. The results show that the orthogonal polynomials can be successfully applied to uncertain rotordynamics with high efficiency and accuracy. Uncertainty has significant effects on the dynamics of the rotor system and multiple uncertainties propagation may lead to heavy vibration.