基于本征正交分解的网格加筋筒壳模型降阶方法
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摘要
针对网格加筋筒壳结构动力响应分析效率低的问题,论文提出了一种基于本征正交分解技术的模型降阶方法.基本思路是通过静力分析获得原模型的节点位移场并组装成快照矩阵,利用本征正交分解技术提取快照矩阵的主成分作为转换矩阵,实现模型降阶.通过算例对比验证了论文提出的降阶模型具有较高的计算精度及效率,降阶模型的低阶频率计算结果与全阶模型十分吻合,高阶频率误差仅为1.01%,而计算时间为全阶模型的0.03%.最后以自由-固支的网格加筋筒为例,采用降阶模型计算其在不同激励下的振动响应,降阶模型的计算结果与全阶模型非常吻合,计算效率有明显提升.
Stiffened shells have been widely utilized in fuel tanks and launch vehicles because of their high specific stiffness and strength.Since these structures are subjected to complex dynamic loads,reliable prediction of natural vibration characteristics is essential in preventing excessive vibration levels,which may result in failure or very high noise levels.The natural frequency of a system is calculated via high fidelity models such as the finite element(FE)method.However,the detailed FE model has a very high computational cost.In order to obtain the frequencies of these complex structures accurately and efficiently,reduced-order models(ROMs)are used instead to lower down the computational expense.This paper presents a model-reduction method based on the proper orthogonal decomposition(POD)technique.The basic idea is to extract the principal component as the transformation matrix from the correlation matrix assembled by the nodal displacement field of full-order models(FOMs)subjected to different cross-sectional loads.The relationship between the capability of ROMs in predicting the mode shapes of interest and the cross-sectional loads is investigated by a clamped-clamped stiffened shell,and the accuracy and efficiency of the proposed model-reduction method are also validated by this example.Numerical results show that the maximum frequency error is about 1.01%,and the computational time is only 0.03% of the associated FOM.Finally,a clamp-free stiffened shell subjected to harmonic external force is studied,and the frequency response function is calculated through FOM and the associated ROM.The number of degrees of freedom reduces from 48960 to 480,and the average calculation time for each frequency point is only 0.04 s for ROM and 4.65 sfor the associated FOM.The displacement response in frequency range shows good agreement between the FOM and ROM.
Stiffened shells have been widely utilized in fuel tanks and launch vehicles because of their high specific stiffness and strength.Since these structures are subjected to complex dynamic loads,reliable prediction of natural vibration characteristics is essential in preventing excessive vibration levels,which may result in failure or very high noise levels.The natural frequency of a system is calculated via high fidelity models such as the finite element(FE)method.However,the detailed FE model has a very high computational cost.In order to obtain the frequencies of these complex structures accurately and efficiently,reduced-order models(ROMs)are used instead to lower down the computational expense.This paper presents a model-reduction method based on the proper orthogonal decomposition(POD)technique.The basic idea is to extract the principal component as the transformation matrix from the correlation matrix assembled by the nodal displacement field of full-order models(FOMs)subjected to different cross-sectional loads.The relationship between the capability of ROMs in predicting the mode shapes of interest and the cross-sectional loads is investigated by a clamped-clamped stiffened shell,and the accuracy and efficiency of the proposed model-reduction method are also validated by this example.Numerical results show that the maximum frequency error is about 1.01%,and the computational time is only 0.03% of the associated FOM.Finally,a clamp-free stiffened shell subjected to harmonic external force is studied,and the frequency response function is calculated through FOM and the associated ROM.The number of degrees of freedom reduces from 48960 to 480,and the average calculation time for each frequency point is only 0.04 s for ROM and 4.65 sfor the associated FOM.The displacement response in frequency range shows good agreement between the FOM and ROM.
引文
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[3] 潘忠文,邢誉峰,杨阳.蒙皮加筋圆柱壳扭转频率的三种计算模型[J].北京航空航天大学学报,2011,37(9):1156-1159.(Pan Z W,Xing Y F,Yang Y.Three analysis models of torsional frequency of stiffening cylindrical shell[J].Journal of Beijing University of Aeronautics and Astronautics,2011,37(9):1156-1159.(in Chinese))
[4] 王博,田阔,郑岩冰,等.超大直径网格加筋筒壳快速屈曲分析方法[J].航空学报,2017,38(2):220379-09.(Wang B,Tian K,Zheng Y B,et al.A rapid buckling analysis method for large-scale grid-stiffened cylinderical shells[J].Acta Aeronautica et Astronautica Sinica,2017,38(2):220379-09.(in Chinese))
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[7] 王文胜,程耿东,郝鹏.基于超梁降阶模型的蒙皮加筋圆柱壳频率分析[J].固体火箭技术,2015,38(3):401-406.(Wang W S,Cheng G D,Hao P.Frequency analysis of stiffened cylindrical shell based on reduced super beam model[J].Journal of Solid Rocket Technology,2015,38(3):401-406.(in Chinise))
[8] Wang W S,Cheng G D,Li Q H.Fast dynamic performance optimization of complicated beam-type structures based on two new reduced physical models[J]. Engineering Optimization,2013,45(7):835-850.
[9] Cheng G D,Wang W S.Fast dynamic analysis of complicated beam-type structure based on reduced super beam model[J].AIAA Journal,2014,52(5):952-963.
[10] Carrera E,Pagani A.Analysis of reinforced and thinwalled structures by multi-line refined 1D/beam models[J].International Journal of Mechanical Sciences,2013,75:278-287.
[11] Carrera E,Zappino E,Filippi M.Free vibration analysis of thin-walled cylinders reinforced with longitudinal and transversal stiffeners[J].Journal of Vibration and Acoustics,2013,135:011019-11.
[12] Pagani A,Boscolo M,Banerjee J R,et al.Exact dynamic stiffness elements based on one-dimensional higher-order theories for free vibration analysis of solid and thin-walled structures[J].Journal of Sound and Vibration,2013,332:6104-6127.
[13] Wang B,Li Y W,Hao P,et al.Free vibration analysis of beam-type structures based on novel reducedorder model[J].AIAA Journal,2017,55(9):3143-3152.
[14] Li Y W,Wang B,Hao P,et al.Fast dynamic analysis of beam-type structures based on reduced-order model[C].Proceedings of the 12th World Congress of Structural and Multidisciplinary Optimisation,Springer International Publishing,2018:590-596.
[15] 李玉韦,王博,程耿东.基于新型降阶模型的网格加筋圆柱壳频率分析[J].宇航总体技术,2017,1(1):41-48.(Li Y W,Wang B,Cheng G D.Frequency analysis of stiffened cylinder based on novel reduced order model[J].Astronautical Systems Engineering Technology,2017,1(1):41-48.(in Chinese))
[16] Feeny B F,Kappagantu R.On the physical interpretation of proper orthogonal modes in vibrations[J].Journal of Sound and Vibration,1998,211(4):607-616.
[17] Liang Y C,Lee H P,Lim S P,et al.Proper orthogonal decomposition and its applications-Part I:Theory[J].Journal of Sound and Vibration,2002,252(3):527-544.
[18] Feeny B F,Liang Y.Interpreting proper orthogonal modes of randomly excited vibration systems[J].Journal of Sound and Vibration,2003,265:953-966.
[19] Qu Z Q.Model order reduction techniques:with application in finite element analysis[C].London,Springer-Verlar London Berlin,2004:96-106.
[20] Guyan R J.Reduction of stiffness and mass matrices[J].AIAA Journal,1965,3(2):380
[21] O’Callahan J,Avitabile P,Riemer R.System equivalent reduction expansion process(SEREP)[C].Proc,7th Int Modal Analysis Conf,Union College,Schenectady,New York,1989,29-37.
[22] Zhang D W,Wei F S.Complete mode-type reduction(CMR)for structural dynamic system[C].Proceedings of the 13th International Modal Analysis Conference(Nashville,Tennessee),Society for Experimental Mechanics,Inc,Bethel,CT,1995.
[2] 邢誉峰,潘忠文,杨阳.蒙皮加筋圆柱壳弯曲频率的三种计算模型[J].北京航空航天大学学报,2012,38(4):438-443.(Xing Y F,Pan Z W,Yang Y.Three solution models of lateral frequencies of stiffening cylindrical shell[J].Journal of Beijing University of Aeronautics and Astronautics,2012,38(4):438-443.(in Chinese))
[3] 潘忠文,邢誉峰,杨阳.蒙皮加筋圆柱壳扭转频率的三种计算模型[J].北京航空航天大学学报,2011,37(9):1156-1159.(Pan Z W,Xing Y F,Yang Y.Three analysis models of torsional frequency of stiffening cylindrical shell[J].Journal of Beijing University of Aeronautics and Astronautics,2011,37(9):1156-1159.(in Chinese))
[4] 王博,田阔,郑岩冰,等.超大直径网格加筋筒壳快速屈曲分析方法[J].航空学报,2017,38(2):220379-09.(Wang B,Tian K,Zheng Y B,et al.A rapid buckling analysis method for large-scale grid-stiffened cylinderical shells[J].Acta Aeronautica et Astronautica Sinica,2017,38(2):220379-09.(in Chinese))
[5] Hao P,Wang B,Tian K,et al.Fast procedure for non-uniform optimum design of stiffened shells under buckling constraint[J].Structural and Multidisciplinary Optimization,2017,55(4):1503-1516.
[6] Wang B,Tian K,Hao P,et al.Numerical-based smeared stiffener method for global buckling analysis of grid-stiffened composite cylindrical shells[J].Composite Structures,2016,152:807-815.
[7] 王文胜,程耿东,郝鹏.基于超梁降阶模型的蒙皮加筋圆柱壳频率分析[J].固体火箭技术,2015,38(3):401-406.(Wang W S,Cheng G D,Hao P.Frequency analysis of stiffened cylindrical shell based on reduced super beam model[J].Journal of Solid Rocket Technology,2015,38(3):401-406.(in Chinise))
[8] Wang W S,Cheng G D,Li Q H.Fast dynamic performance optimization of complicated beam-type structures based on two new reduced physical models[J]. Engineering Optimization,2013,45(7):835-850.
[9] Cheng G D,Wang W S.Fast dynamic analysis of complicated beam-type structure based on reduced super beam model[J].AIAA Journal,2014,52(5):952-963.
[10] Carrera E,Pagani A.Analysis of reinforced and thinwalled structures by multi-line refined 1D/beam models[J].International Journal of Mechanical Sciences,2013,75:278-287.
[11] Carrera E,Zappino E,Filippi M.Free vibration analysis of thin-walled cylinders reinforced with longitudinal and transversal stiffeners[J].Journal of Vibration and Acoustics,2013,135:011019-11.
[12] Pagani A,Boscolo M,Banerjee J R,et al.Exact dynamic stiffness elements based on one-dimensional higher-order theories for free vibration analysis of solid and thin-walled structures[J].Journal of Sound and Vibration,2013,332:6104-6127.
[13] Wang B,Li Y W,Hao P,et al.Free vibration analysis of beam-type structures based on novel reducedorder model[J].AIAA Journal,2017,55(9):3143-3152.
[14] Li Y W,Wang B,Hao P,et al.Fast dynamic analysis of beam-type structures based on reduced-order model[C].Proceedings of the 12th World Congress of Structural and Multidisciplinary Optimisation,Springer International Publishing,2018:590-596.
[15] 李玉韦,王博,程耿东.基于新型降阶模型的网格加筋圆柱壳频率分析[J].宇航总体技术,2017,1(1):41-48.(Li Y W,Wang B,Cheng G D.Frequency analysis of stiffened cylinder based on novel reduced order model[J].Astronautical Systems Engineering Technology,2017,1(1):41-48.(in Chinese))
[16] Feeny B F,Kappagantu R.On the physical interpretation of proper orthogonal modes in vibrations[J].Journal of Sound and Vibration,1998,211(4):607-616.
[17] Liang Y C,Lee H P,Lim S P,et al.Proper orthogonal decomposition and its applications-Part I:Theory[J].Journal of Sound and Vibration,2002,252(3):527-544.
[18] Feeny B F,Liang Y.Interpreting proper orthogonal modes of randomly excited vibration systems[J].Journal of Sound and Vibration,2003,265:953-966.
[19] Qu Z Q.Model order reduction techniques:with application in finite element analysis[C].London,Springer-Verlar London Berlin,2004:96-106.
[20] Guyan R J.Reduction of stiffness and mass matrices[J].AIAA Journal,1965,3(2):380
[21] O’Callahan J,Avitabile P,Riemer R.System equivalent reduction expansion process(SEREP)[C].Proc,7th Int Modal Analysis Conf,Union College,Schenectady,New York,1989,29-37.
[22] Zhang D W,Wei F S.Complete mode-type reduction(CMR)for structural dynamic system[C].Proceedings of the 13th International Modal Analysis Conference(Nashville,Tennessee),Society for Experimental Mechanics,Inc,Bethel,CT,1995.
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