改进的Neumann-LU混合法在失谐整体叶盘随机分析中的应用
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摘要
为了降低失谐整体叶盘的随机分析矩阵求逆的计算成本,提出一种基于Neumann级数和LU分解的改进Neumann-LU混合法。一方面,在非共振频率附近的大区域采用Neumann级数频域展开式以提高计算效率;另一方面,在共振频率附近小区域采用LU分解法以提高计算计算精度;根据Neumann级数展开项数来选定自适应阀值,实现两种矩阵求逆方法的自动转换而不用消耗额外的计算资源。选取具有失谐特征的整体叶盘为研究对象,分别运用改进的Neumann-LU混合法和单一LU分解法求解叶片在阶次激励下的频域响应,并从计算效率和计算精度方面进行比较。研究结果表明,在保证计算精度误差不大于0.1%的前提下,改进的Neumann-LU混合法可以提高失谐整体叶盘随机分析大约55%的计算效率。
For decreasing computational cost of matrix inversion in the stochastic analysis of mistuned blisk,an improved hybrid Neumann-LU approach was proposed based on Neumann expansion coupled with LU factorization. On the one hand,the Neumann expansion in the frequency domain was used to enhance computational efficiency outside the region of resonant frequencies; on the other hand,the LU factorization in the neighborhood of the resonant frequencies was utilized for improving computational accuracy; in which,the number of the expansion terms was regarded as an adaptive indicator to select the matrix inversion technique to be used without introducing any additional computational cost. Finally,according to different mistuning patterns via stiffness or mass perturbations under the excitation provided by the engine orders,a mistuned blisk was chosen as study case to evaluate the frequency responses of the blades,and compared with the single LU factorization method by both accuracy of calculation and computational effort. Results reveal that the improved hybrid Neumann-LU approach can improve the 55% computational efficiency approximately for the stochastic analysis of mistuned blisk,in the premise of ensuring that errors are not greater than 0.1%.
For decreasing computational cost of matrix inversion in the stochastic analysis of mistuned blisk,an improved hybrid Neumann-LU approach was proposed based on Neumann expansion coupled with LU factorization. On the one hand,the Neumann expansion in the frequency domain was used to enhance computational efficiency outside the region of resonant frequencies; on the other hand,the LU factorization in the neighborhood of the resonant frequencies was utilized for improving computational accuracy; in which,the number of the expansion terms was regarded as an adaptive indicator to select the matrix inversion technique to be used without introducing any additional computational cost. Finally,according to different mistuning patterns via stiffness or mass perturbations under the excitation provided by the engine orders,a mistuned blisk was chosen as study case to evaluate the frequency responses of the blades,and compared with the single LU factorization method by both accuracy of calculation and computational effort. Results reveal that the improved hybrid Neumann-LU approach can improve the 55% computational efficiency approximately for the stochastic analysis of mistuned blisk,in the premise of ensuring that errors are not greater than 0.1%.
引文
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[18]Petrov E P,Sanliturk K Y,Ewins D J.A New Method for Dynamic Analysis of Mistuned Bladed Disks Based on the Exact Relationship between Tuned and Mistuned Systems[J].Journal of Engineering for Gas Turbines&Power,2002,124(3):586-597.
[19]Malkus D S,Plesha M E.Zero and Negative Masses in Finite Element Vibration and Transient Analysis[J].Computer Methods in Applied Mechanics and Engineering,1986,59(3):281-306.
[20]Castanier M P,Pierre C.Modeling and Analysis of Mistuned Bladed Disk Vibration:Status and Emerging Directions[J].Journal of Propulsion and Power,2006,22(2):384-396.
[2]余学冉,陆山.整体叶盘结构强度减振一体化设计方法[J].航空动力学报,2013,28(10):2235-2239.
[3]白斌,白广忱,童晓晨,等.整体叶盘结构失谐振动的国内外研究状况[J].航空动力学报,2014,29(1):91-103.
[4]Lee S Y,Castanier M,Pierre C.Assessment of Probabilistic Methods for Mistuned Bladed Disk Vibration[R].AIAA 2005-1990.
[5]Jacquelin E,Adhikari S,Sinou J J,et al.Polynomial Chaos Expansion and Steady-State Response of a Class of Random Dynamical Systems[J].Journal of Engineering Mechanics,2014,141(4):04014145.
[6]Kundu A,Diazdelao F A,Adhikari S,et al.A hybrid Spectral and Metamodeling Approach for the Stochastic Finite Element Analysis of Structural Dynamic Systems[J].Computer Methods in Applied Mechanics and Engineering,2014,270(3):201-219.
[7]Castanier M P,Pierre C.Modeling and Analysis of Mistuned Bladed Disk Vibration:Current Status and Emerging Directions[J].Journal of Propulsion and Power,2006,22(2):384-396.
[8]Yuan J,Allegri G,Scarpa F,et al.Probabilistic Dynamics of Mistuned Bladed Disc Systems Using Subset Simulation[J].Journal of Sound and Vibration,2015,350:185-198.
[9]Aquino R.J.Stochastic Finite Element Modeling and Its Applications in Heat Conduction and Elastostatics Problems[D].UK:Swansea University,2009.
[10]Xiangyu Wang,Song Cen,Chenfeng Li.Generalized Neumann Expansion and Its Application in Stochastic Finite Element Methods[J].Mathematical Problems in Engineering,2013,(1):1-13.
[11]Sinou J J,Jacquelin E.Influence of Polynomial Chaos Expansion Order on an Uncertain Asymmetric Rotor System response[J].Mechanical Systems and Signal Processing,2015,50-51:718-731.
[12]Yamazaki F,Member A,Shinozuka M,et al.Neumann Expansion for Stochastic Finite Element Analysis[J].Journal of Engineering Mechanics,1988,114(8):1335-1354.
[13]Yuan J,Allegri G,Scarpa F,et al.Numerical Assessment of Using Sherman-Morrison,Neumann Expansion Techniques for Stochastic Analysis of Mistuned Bladed Disc Systems[C].Montreal:Proceedings of ASME Turbo Expo,2015.
[14]Yuan J,Allegri G,Scarpa F,et al.Novel Parametric Reduced Order Model for Aeroengine Blade Dynamics[J].Mechanical Systems and Signal Processing,2015,62-63:235-253.
[15]Choi J,Dongarra J J,Ostrouchov L S,et al.Design and Implementation of the Sca LAPACK LU,QR,and Cholesky Factorization Routines[J].Scientific Programming,1996,5(3):173-184.
[16]姚建尧,王建军,李其汉.基于振型节径谱的失谐叶盘结构动态特性评价[J].推进技术,2011,32(5):645-653.(YAO Jian-yao,WANG Jian-jun,LI Qihan.Dynamic Characteristics Assessment of Mistuned Bladed Disk Using Nodal Diameter Spectrum of Mode Shapes[J].Journal of Propulsion Technology,2011,32(5):645-653.)
[17]王建军,于长波,姚建尧,等.失谐叶盘振动模态局部化定量描述方法[J].推进技术,2009,30(4):457-461.(WANG Jian-jun,YU Chang-bo,YAO Jian-yao et al.Vibratory Mode Localization Factors of Mistuned Bladed Disk Assemblies[J].Journal of Propulsion Technology,2009,30(4):457-461.)
[18]Petrov E P,Sanliturk K Y,Ewins D J.A New Method for Dynamic Analysis of Mistuned Bladed Disks Based on the Exact Relationship between Tuned and Mistuned Systems[J].Journal of Engineering for Gas Turbines&Power,2002,124(3):586-597.
[19]Malkus D S,Plesha M E.Zero and Negative Masses in Finite Element Vibration and Transient Analysis[J].Computer Methods in Applied Mechanics and Engineering,1986,59(3):281-306.
[20]Castanier M P,Pierre C.Modeling and Analysis of Mistuned Bladed Disk Vibration:Status and Emerging Directions[J].Journal of Propulsion and Power,2006,22(2):384-396.