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Article(id=1148011786416157467, tenantId=1146029695717560320, journalId=1146119989267898375, issueId=1149298830442578865, articleNumber=null, orderNo=null, doi=10.7654/j.issn.2097-1974.20240304, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=research-article, receivedDate=1651680000000, receivedDateStr=2022-05-05, revisedDate=1701273600000, revisedDateStr=2023-11-30, acceptedDate=null, acceptedDateStr=null, onlineDate=1751636938909, onlineDateStr=2025-07-04, pubDate=1719244800000, pubDateStr=2024-06-25, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1751636938909, onlineIssueDateStr=2025-07-04, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1751636938909, creator=13701087609, updateTime=1751636938909, updator=13701087609, issue=Issue{id=1149298830442578865, tenantId=1146029695717560320, journalId=1146119989267898375, year='2024', volume='47', issue='3', pageStart='1', pageEnd='106', issueExtLink='null', onlineDate='null', pubDate='null', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=null, createTime=1751943794116, creator=13701087609, updateTime=1754895897628, updator=13701087609, preIssue=null, nextIssue=null, ext={EN=IssueExt(id=1161680850048344508, tenantId=1146029695717560320, journalId=1146119989267898375, issueId=1149298830442578865, language=EN, specialIssueTitle=, coverIllustrator=, specialIssueEditor=, specialIssueAbout=), CN=IssueExt(id=1161680850048344509, tenantId=1146029695717560320, journalId=1146119989267898375, issueId=1149298830442578865, language=CN, specialIssueTitle=, coverIllustrator=, specialIssueEditor=, specialIssueAbout=)}, issueFiles=null}, startPage=21, endPage=28, ext={EN=ArticleExt(id=1148011786625872693, articleId=1148011786416157467, tenantId=1146029695717560320, journalId=1146119989267898375, language=EN, title=Multi-state Model Updating based on Simplified Rocket Model, columnId=1154057568293999177, journalTitle=Missiles and Space Vehicles, columnName=Launch Vehicle and Missile, runingTitle=null, highlight=null, articleAbstract=

The continuous reduction of fuel during the rocket launch causes the rocket structure to change continuously, and the natural frequency and modal shape of the structure also change accordingly. Aiming at the inaccuracy of the finite element model, the moment of inertia of the section is used as the correction parameter, and the minimum error between the numerical simulation results and the experimental test results under different states is the objective function, and an optimization model for multi-state model correction is established. The objective function is explicitly processed by sensitivity analysis and first-order Taylor expansion, and the optimal solution of design variables is obtained by sequential linear programming. At the same time, for the multi-state structure, taking the multi-section simply supported beam model as an example, the simply supported beam is tested by the hammering method, and the magnetic attraction weight scheme is designed to simulate the multi-state model. An optimization model is established for the objective function, and the multi-section simply supported beam model is revised to a single-section simply supported beam model after modification, and the ratio of the moment of inertia of the section to the area is met, which verifies the accuracy of the multi-state model modification method.

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火箭发射过程中燃料的不断减少导致火箭结构不断变化,结构的固有频率和模态振型也会随之变化。针对有限元模型的不准确性,以截面惯性矩为修正参数,不同状态下数值仿真结果与试验测试结果的误差最小为目标函数,建立多状态模型修正的优化模型。通过灵敏度分析和一阶泰勒展开对目标函数进行显式化处理,并利用序列线性规划求解得到设计变量的最优解。同时对于多状态结构,以多截面简支梁模型为例,利用锤击法对简支梁进行试验,设计磁吸重物方案模拟多状态模型,以单截面简支梁试验的前4阶频率为目标函数建立优化模型,经过修正将多截面简支梁模型修正为单截面简支梁模型,并且符合截面惯性矩与面积的比例,验证了多状态模型修正方法的准确性。

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杜家政(1975—),男,博士,副教授,主要研究方向为损伤拓扑优化、模型修正和在役检测。

王哲(1997—),男,硕士研究生,主要研究方向为模型修正和在役检测。

李星辰(2000—),男,硕士研究生,主要研究方向为模型修正和在役检测。

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杜家政(1975—),男,博士,副教授,主要研究方向为损伤拓扑优化、模型修正和在役检测。

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王哲(1997—),男,硕士研究生,主要研究方向为模型修正和在役检测。

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Mechanical Systems and Signal Processing, 2019(1): 193-210., articleTitle=Dynamic tests and model updating of nonlinear beam structures with bolted joints, refAbstract=null)], funds=[Fund(id=1197273053181821335, tenantId=1146029695717560320, journalId=1146119989267898375, articleId=1148011786416157467, awardId=11872080, language=CN, fundingSource=国家自然科学基金资助项目(11872080), fundOrder=null, country=null)], companyList=[AuthorCompany(id=1197273037050528081, tenantId=1146029695717560320, journalId=1146119989267898375, articleId=1148011786416157467, xref=1, ext=[AuthorCompanyExt(id=1197273037058916690, tenantId=1146029695717560320, journalId=1146119989267898375, articleId=1148011786416157467, companyId=1197273037050528081, language=EN, country=null, province=null, city=null, postcode=null, companyName=null, departmentName=null, remark=1Beijing University of Technology,Beijing,100124), AuthorCompanyExt(id=1197273037071499603, tenantId=1146029695717560320, journalId=1146119989267898375, 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articleId=1148011786416157467, language=CN, label=图7, caption=多状态试验装置, figureFileSmall=/zwTmWv7UQw1VEc4fU5ztg==, figureFileBig=HVH6e9KOQuFY9TT2xC2cZw==, tableContent=null), ArticleFig(id=1197273048345788807, tenantId=1146029695717560320, journalId=1146119989267898375, articleId=1148011786416157467, language=EN, label=Fig. 8, caption=Simplified modeling of simply supported beam structures, figureFileSmall=gobRzY3vv4vbTNokm4+O5Q==, figureFileBig=nXt2J4B7OhGga/tavUKdkg==, tableContent=null), ArticleFig(id=1197273048517755272, tenantId=1146029695717560320, journalId=1146119989267898375, articleId=1148011786416157467, language=CN, label=图8, caption=简支梁结构简化建模, figureFileSmall=gobRzY3vv4vbTNokm4+O5Q==, figureFileBig=nXt2J4B7OhGga/tavUKdkg==, tableContent=null), ArticleFig(id=1197273048798773641, tenantId=1146029695717560320, journalId=1146119989267898375, articleId=1148011786416157467, language=EN, label=Tab. 1, caption=Sectional dimension, figureFileSmall=null, figureFileBig=null, tableContent=
部件外径/m内径/m${\mathbf{I}}_{1}/{\mathrm{m}}^{4}$${\mathbf{I}}_{2}/{\mathrm{m}}^{4}$
一级a3.3525.3974.318
一级$\mathrm{b}$3.351.85.6674.534
二级3.352.25.0324.026
三级32.42.3471.878
整流罩42.610.3238.259
), ArticleFig(id=1197273049029460362, tenantId=1146029695717560320, journalId=1146119989267898375, articleId=1148011786416157467, language=CN, label=表1, caption=截面尺寸, figureFileSmall=null, figureFileBig=null, tableContent=
部件外径/m内径/m${\mathbf{I}}_{1}/{\mathrm{m}}^{4}$${\mathbf{I}}_{2}/{\mathrm{m}}^{4}$
一级a3.3525.3974.318
一级$\mathrm{b}$3.351.85.6674.534
二级3.352.25.0324.026
三级32.42.3471.878
整流罩42.610.3238.259
), ArticleFig(id=1197273049247564171, tenantId=1146029695717560320, journalId=1146119989267898375, articleId=1148011786416157467, language=EN, label=Tab. 2, caption=The mass of mass points in each state, figureFileSmall=null, figureFileBig=null, tableContent=
节点状态1质量/t状态2质量/t状态3质量/t
1300200100
10300300300
19200200200
28200200200
37100100100
46100100100
), ArticleFig(id=1197273049524388236, tenantId=1146029695717560320, journalId=1146119989267898375, articleId=1148011786416157467, language=CN, label=表2, caption=各个状态集中质量点的质量, figureFileSmall=null, figureFileBig=null, tableContent=
节点状态1质量/t状态2质量/t状态3质量/t
1300200100
10300300300
19200200200
28200200200
37100100100
46100100100
), ArticleFig(id=1197273049633440141, tenantId=1146029695717560320, journalId=1146119989267898375, articleId=1148011786416157467, language=EN, label=Tab. 3, caption=The initial and iterative values of cross-sectional moment of inertia, figureFileSmall=null, figureFileBig=null, tableContent=
梁段号截面惯性矩修正前/${\mathrm{m}}^{4}$试验值/${\mathrm{m}}^{4}$修正后/${\mathrm{m}}^{4}$相对误差/%
1${\mathbf{I}}_{1}$55.39695.39830.0259
${\mathbf{I}}_{2}$44.31754.31830.0185
2${\mathbf{I}}_{1}$55.66705.66840.0247
${\mathbf{I}}_{2}$44.53364.53540.0397
3${\mathbf{I}}_{1}$55.03245.02800.0874
${\mathbf{I}}_{2}$44.02594.02230.0894
4${\mathbf{I}}_{1}$22.34752.34800.0213
${\mathbf{I}}_{2}$21.87801.87900.0532
5${\mathbf{I}}_{1}$1010.323210.34210.1831
${\mathbf{I}}_{2}$88.25868.25120.0896
), ArticleFig(id=1197273049906069902, tenantId=1146029695717560320, journalId=1146119989267898375, articleId=1148011786416157467, language=CN, label=表3, caption=截面惯性矩初始值及其迭代后的值对比, figureFileSmall=null, figureFileBig=null, tableContent=
梁段号截面惯性矩修正前/${\mathrm{m}}^{4}$试验值/${\mathrm{m}}^{4}$修正后/${\mathrm{m}}^{4}$相对误差/%
1${\mathbf{I}}_{1}$55.39695.39830.0259
${\mathbf{I}}_{2}$44.31754.31830.0185
2${\mathbf{I}}_{1}$55.66705.66840.0247
${\mathbf{I}}_{2}$44.53364.53540.0397
3${\mathbf{I}}_{1}$55.03245.02800.0874
${\mathbf{I}}_{2}$44.02594.02230.0894
4${\mathbf{I}}_{1}$22.34752.34800.0213
${\mathbf{I}}_{2}$21.87801.87900.0532
5${\mathbf{I}}_{1}$1010.323210.34210.1831
${\mathbf{I}}_{2}$88.25868.25120.0896
), ArticleFig(id=1197273050178699663, tenantId=1146029695717560320, journalId=1146119989267898375, articleId=1148011786416157467, language=EN, label=Tab. 4, caption=Frequency comparison before and after optimization, figureFileSmall=null, figureFileBig=null, tableContent=
状态模态阶数修正前/Hz试验值/Hz修正后/Hz相对误差$/\%$
状态11阶2.2208892.2308672.2308370.0013
2阶2.3767042.4941872.4940090.0071
3阶6.1966566.3221866.3231220.0148
4阶6.6584227.0684347.0692130.0110
状态21阶2.3212262.3271832.3271540.0013
2阶2.4783782.6018732.6016840.0073
3阶6.4198216.5564596.5573970.0143
4阶6.9080267.3303607.3311450.0107
状态31阶2.4771362.4758192.4757910.0011
2阶2.6348782.7680532.7678500.0073
3阶6.8622057.0155067.0164250.0131
4阶7.3980987.8435947.8443650.0098
), ArticleFig(id=1197273050413580688, tenantId=1146029695717560320, journalId=1146119989267898375, articleId=1148011786416157467, language=CN, label=表4, caption=修正前后频率对比表, figureFileSmall=null, figureFileBig=null, tableContent=
状态模态阶数修正前/Hz试验值/Hz修正后/Hz相对误差$/\%$
状态11阶2.2208892.2308672.2308370.0013
2阶2.3767042.4941872.4940090.0071
3阶6.1966566.3221866.3231220.0148
4阶6.6584227.0684347.0692130.0110
状态21阶2.3212262.3271832.3271540.0013
2阶2.4783782.6018732.6016840.0073
3阶6.4198216.5564596.5573970.0143
4阶6.9080267.3303607.3311450.0107
状态31阶2.4771362.4758192.4757910.0011
2阶2.6348782.7680532.7678500.0073
3阶6.8622057.0155067.0164250.0131
4阶7.3980987.8435947.8443650.0098
), ArticleFig(id=1197273050627490193, tenantId=1146029695717560320, journalId=1146119989267898375, articleId=1148011786416157467, language=EN, label=Tab. 5, caption=Experimental frequency results, figureFileSmall=null, figureFileBig=null, tableContent=
状态1阶/Hz2阶/Hz3阶/Hz4阶/Hz
状态139.631143.824329.906527.673
状态241.247153.220338.987548.822
状态341.910161.092369.687584.607
), ArticleFig(id=1197273050824622482, tenantId=1146029695717560320, journalId=1146119989267898375, articleId=1148011786416157467, language=CN, label=表5, caption=试验频率结果, figureFileSmall=null, figureFileBig=null, tableContent=
状态1阶/Hz2阶/Hz3阶/Hz4阶/Hz
状态139.631143.824329.906527.673
状态241.247153.220338.987548.822
状态341.910161.092369.687584.607
), ArticleFig(id=1197273050950451603, tenantId=1146029695717560320, journalId=1146119989267898375, articleId=1148011786416157467, language=EN, label=Tab. 6, caption=Comparison of cross-sectional moment of inertia before and after updating, figureFileSmall=null, figureFileBig=null, tableContent=
截面惯性矩理论值$/{\mathrm{m}}^{4}$修正前$/{\mathrm{m}}^{4}$修正后$/{\mathrm{m}}^{4}$相对误差$/\%$
${\mathbf{I}}_{1}$2133.35000.02195.82.9273
${\mathbf{I}}_{2}$83333.350000.080625.23.2497
), ArticleFig(id=1197273051214692756, tenantId=1146029695717560320, journalId=1146119989267898375, articleId=1148011786416157467, language=CN, label=表6, caption=截面惯性矩设定初始值及其迭代后的值对比, figureFileSmall=null, figureFileBig=null, tableContent=
截面惯性矩理论值$/{\mathrm{m}}^{4}$修正前$/{\mathrm{m}}^{4}$修正后$/{\mathrm{m}}^{4}$相对误差$/\%$
${\mathbf{I}}_{1}$2133.35000.02195.82.9273
${\mathbf{I}}_{2}$83333.350000.080625.23.2497
), ArticleFig(id=1197273051369882005, tenantId=1146029695717560320, journalId=1146119989267898375, articleId=1148011786416157467, language=EN, label=Tab. 7, caption=Frequency comparison before and after updating, figureFileSmall=null, figureFileBig=null, tableContent=
状态模态阶数试验数据/$\mathrm{{Hz}}$修正前/$\mathrm{{Hz}}$修正后/$\mathrm{{Hz}}$相对误差/%
状态11阶39.63144.765737.51605.3368
2阶143.824178.2374142.41580.9791
3阶329.906408.2067348.14165.5275
4阶527.673459.9814531.94460.8095
状态21阶41.24746.248739.10245.1995
2阶153.220189.0948151.07311.4012
3阶338.987418.9007355.30064.8125
4阶548.822479.5579550.62260.3281
状态31阶41.91047.857340.86822.4858
2阶161.092203.7950163.48571.4859
3阶369.687437.4271367.81850.5054
4阶584.607500.5157570.13802.4750
), ArticleFig(id=1197273051617345942, tenantId=1146029695717560320, journalId=1146119989267898375, articleId=1148011786416157467, language=CN, label=表7, caption=修正前后频率对比, figureFileSmall=null, figureFileBig=null, tableContent=
状态模态阶数试验数据/$\mathrm{{Hz}}$修正前/$\mathrm{{Hz}}$修正后/$\mathrm{{Hz}}$相对误差/%
状态11阶39.63144.765737.51605.3368
2阶143.824178.2374142.41580.9791
3阶329.906408.2067348.14165.5275
4阶527.673459.9814531.94460.8095
状态21阶41.24746.248739.10245.1995
2阶153.220189.0948151.07311.4012
3阶338.987418.9007355.30064.8125
4阶548.822479.5579550.62260.3281
状态31阶41.91047.857340.86822.4858
2阶161.092203.7950163.48571.4859
3阶369.687437.4271367.81850.5054
4阶584.607500.5157570.13802.4750
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基于火箭简化模型的多状态模型修正研究
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杜家政 1 , 王哲 1, 2 , 李星辰 1
导弹与航天运载技术 | 运载器及导弹总体技术 2024,47(3): 21-28
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导弹与航天运载技术 | 运载器及导弹总体技术 2024, 47(3): 21-28
基于火箭简化模型的多状态模型修正研究
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杜家政1, 王哲1, 2, 李星辰1
作者信息
  • 1北京工业大学,北京,100124
  • 2中国电子科技集团公司第四十五研究所,北京,100176

    • 杜家政(1975—),男,博士,副教授,主要研究方向为损伤拓扑优化、模型修正和在役检测。

    王哲(1997—),男,硕士研究生,主要研究方向为模型修正和在役检测。

    李星辰(2000—),男,硕士研究生,主要研究方向为模型修正和在役检测。

Multi-state Model Updating based on Simplified Rocket Model
Jiazheng DU1, Zhe WANG1, 2, Xingchen LI1
Affiliations
  • 1Beijing University of Technology,Beijing,100124
  • 2The 45th Research Institute of China Electronics Technology Group Corporation,Beijing,100176
出版时间: 2024-06-25 doi: 10.7654/j.issn.2097-1974.20240304
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摘要
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火箭发射过程中燃料的不断减少导致火箭结构不断变化,结构的固有频率和模态振型也会随之变化。针对有限元模型的不准确性,以截面惯性矩为修正参数,不同状态下数值仿真结果与试验测试结果的误差最小为目标函数,建立多状态模型修正的优化模型。通过灵敏度分析和一阶泰勒展开对目标函数进行显式化处理,并利用序列线性规划求解得到设计变量的最优解。同时对于多状态结构,以多截面简支梁模型为例,利用锤击法对简支梁进行试验,设计磁吸重物方案模拟多状态模型,以单截面简支梁试验的前4阶频率为目标函数建立优化模型,经过修正将多截面简支梁模型修正为单截面简支梁模型,并且符合截面惯性矩与面积的比例,验证了多状态模型修正方法的准确性。

多状态模型修正  /  截面惯性矩  /  优化设计  /  火箭简化模型  /  有限元分析

The continuous reduction of fuel during the rocket launch causes the rocket structure to change continuously, and the natural frequency and modal shape of the structure also change accordingly. Aiming at the inaccuracy of the finite element model, the moment of inertia of the section is used as the correction parameter, and the minimum error between the numerical simulation results and the experimental test results under different states is the objective function, and an optimization model for multi-state model correction is established. The objective function is explicitly processed by sensitivity analysis and first-order Taylor expansion, and the optimal solution of design variables is obtained by sequential linear programming. At the same time, for the multi-state structure, taking the multi-section simply supported beam model as an example, the simply supported beam is tested by the hammering method, and the magnetic attraction weight scheme is designed to simulate the multi-state model. An optimization model is established for the objective function, and the multi-section simply supported beam model is revised to a single-section simply supported beam model after modification, and the ratio of the moment of inertia of the section to the area is met, which verifies the accuracy of the multi-state model modification method.

multi-state model updating  /  section moment of inertia  /  optimal design  /  simplified rocket model  /  finite element analysis
杜家政, 王哲, 李星辰. 基于火箭简化模型的多状态模型修正研究. 导弹与航天运载技术, 2024 , 47 (3) : 21 -28 . DOI: 10.7654/j.issn.2097-1974.20240304
Jiazheng DU, Zhe WANG, Xingchen LI. Multi-state Model Updating based on Simplified Rocket Model[J]. Missiles and Space Vehicles, 2024 , 47 (3) : 21 -28 . DOI: 10.7654/j.issn.2097-1974.20240304
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0 引言
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在工程实际中, 飞行器、火箭、船舶等, 各种机械装备、建筑物等都处在振动环境下并以不同的方式进行振动。如果可以了解、掌控和应用这些振动,对国防建设、工业发展等研究将有重大意义。随着计算机技术突飞猛进地发展,有限元建模、仿真分析在结构动力学分析过程中的应用日益广泛, 要了解、掌控和应用系统的振动, 就要建立有效的有限元模型, 这样才能更好地表现出物体的动态特性[1]。
有限元分析过程中, 要先将结构连续模型进行离散化处理, 对结构的形状、边界条件作力学上的简化, 利用有限元分析概念里的梁、杆、板单元取代符合这些单元特征的实际结构, 利用理想的边界固定条件取代实际的连接方式, 这种模型简化带来的误差是不可避免的[2]。有限元分析的不确定因素主要来源于有限元模型简化带来的误差、模型数据的不准确带来的误差和工况简化带来的误差[3],很难一次性准确地确定模型参数。在工程分析中建立仿真与试验相匹配的有限元模型是亟待解决的问题。为使有限元模型更加精确, 需要对有限元模型参数进行修正, 如果只依靠工程技术人员试验测试再修正是很难完成的, 需要通过有限元模型修正方法来分析过程中结果误差较大的问题。
有限元模型修正技术已经发展了很多年, 追溯到20世纪60年代中期, Rodden[4]提出了一种从有效自由地面振动试验找出任意约束系统结构影响系数矩阵的方法。许多学者一直致力于利用试验结果数据来修正结构的有限元模型参数, Berman[5]、Baruch等[6]提出了利用试验振型和固有频率来提高结构分析质量和修正刚度矩阵模型的方法, 奠定了这一领域的基础。Mottershead和Friswell[7]对有限元模型修正方法进行了系统的总结与介绍。Lauwagie等[8]采用多模型修正方法, 对层合板试样各层的平面弹性特性进行了评估, 得出的结论是, 为了保证得到性质的唯一性,需要采用多模型方法。Link等[9]采用多模型修正方法, 利用时域数据对损伤参数进行定位, 同时对损伤模型和未损伤模型进行更新, 在不考虑建模误差的情况下识别损伤参数。Li等[10]对钢导管架模型利用模态试验识别的模态参数进行了有限元模型修正, 通过求解模态能量进行灵敏度分析, 利用不同的参数进行两步更新, 每一步采用交叉模型更新方法, 将有限元模型修正为更具有物理意义的模型。张忠等[11]通过多状态模型修正方法对火箭卫星耦合系统模型的误差诊断和定位、修正参数选取方法等进行讨论。李元生等[12]利用两层修正系统对有限元模型进行同步修正, 使得设计变量最终收敛于最优解。魏锦辉等[13]构造出结合结构的动静力有限元模型修正方法的目标函数, 利用响应面方法构建桥梁静动力响应的代理模型进行有限元模型修正。方圣恩等[14]微量表述不确定性参数与响应间复杂的隐式函数关系, 建立了基于概率配点法和回归分析的随机响应面模型, 对参数的均值和标准差进行依次修正, 简化了随机修正过程, 提高了修正效率。王亮等[15]提出采用灵敏度分析的多目标多状态模态修正方法。Yuan等[16]针对高温环境下结构有限元模型修正问题, 研究了一种多状态模型修正方法, 设计了特定的约束矩阵, 采用改进的序列二次规划法对不同温度边界条件下板的有限元模型进行了修正。Batou[17]提出了一个替代算法,通过求解分离单变量的逆问题去顺序识别每个模型参数, 输出通过识别的每一个参数灵敏度并用Sobol指数进行量化得到最优解。Yuan等[18]采用线性梁单元和Iwan梁单元模拟螺栓悬臂梁结构, 利用正演增量位移中心差分法计算相应的结构动力响应, 结合基于构件主动力响应瞬时特性的非线性有限元模型修正方法, 对螺栓悬臂梁结构进行了修正。
为保证寻优过程中不超出近似模型的有效范围, 在近似模型中对设计变量施加适当的限制, 称之为运动极限。当前的一些研究中, 虽然有部分方法能做到多状态模型修正, 但是对于目标函数与设计变量之间的关系表示并不明确, 设计变量的运动极限设置可能并不合理, 为了弥补这些不足, 本文提出一种基于灵敏度分析调节运动极限的多状态模型修正方法, 通过根据设计变量变化的情况不断调整运动极限, 再通过灵敏度分析将目标函数显式化后计算出所有状态的最优解, 并进行验证。
1 简化火箭模型
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针对火箭的简化模型, 主要考虑火箭的整体性能, 采用集中质量模拟燃料的质量, 通过集中质量的变化模拟液体的减少, 采用梁单元模拟火箭壳体, 通过调整梁单元的刚度修正火箭壳体和级间段的刚度, 确保试验数据和仿真数据的匹配, 提高简化模型的仿真精度。
在火箭发射过程中需建立与火箭相符程度较高的有限元模型来模拟发射过程火箭的姿态。火箭可分为助推器、一级火箭、二级火箭、三级火箭以及整流罩, 将整个结构划分为5个梁段, 每个梁段分别代表各级火箭, 因一级火箭实际情况下长度较长, 将助推器与一级火箭分为一级$\mathrm{a}$火箭、一级$\mathrm{b}$火箭,各级火箭的抗弯刚度均不相同,各级火箭均含有燃料。随着火箭的升空, 燃料的不断减少, 质量会不断减少。将火箭模型简化为一根不同截面的圆形梁, 如图1所示。
2 优化模型的分析
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2.1 优化模型的建立
为使火箭简化模型与试验结构的固有频率和结构振型一致, 需进行结构优化。建立关于有限元分析结果与相应试验结果之间的残差函数, 其中固有频率数值相对较大采用相对误差, 振型数值相对较小采用绝对误差。以火箭简化模型的截面惯性矩为设计变量。
$\left\{\begin{array}{l}\text{ find }\mathbf{x}\\\text{ make }\\ F\left(\mathbf{x}\right)= \mathop{\sum }\limits_{{i = 1}}^{a}\left\lbrack {{w}_{i1}{\left(\frac{{f}_{i}\left(\mathbf{x}\right)- {f}_{i}}{{f}_{i}}\right)}^{2}+ }\right.\\\left.{{w}_{i2}\mathop{\sum }\limits_{{j = 1}}^{J}{\left({u}_{j}\left(\mathbf{x}\right)- {u}_{j}\right)}^{2}}\right\rbrack \rightarrow \min \end{array}\right.$
式中$x$为火箭简化模型的截面惯性矩,是设计变量;$a$表示选取$a$阶模态；$J$为试验点的个数；${w}_{i1}$为第$i$阶频率的权重系数;${w}_{i2}$为第$i$阶振型的权重系数;${f}_{i}$为第$i$阶频率的试验值;${f}_{i}\left(\mathbf{x}\right)$为第$i$阶频率仿真计算值;${u}_{j}$为第$j$个节点振型的试验测量值;${u}_{j}\left(\mathbf{x}\right)$为第$j$个节点有限元模型的振型计算值。
随着火箭燃料的不断消耗, 火箭质量也会不断减小, 导致火箭模型不断变化, 需要通过多状态模型修正方法来修正火箭模型, 为了将非线性问题转化为线性问题, 并且使结果更加精确, 本文将目标函数进行一阶泰勒展开,通过式(2)求出灵敏度${S}_{i}$。
${S}_{i}= \mathop{\sum }\limits_{{m = 1}}^{M}\frac{\partial {F}_{\mathrm{m}}\left({x}_{0}\right)}{\partial {x}_{i}}\;\left({i = 1,\cdots , n}\right)$
式中$M$为选取的状态数;${x}_{0}$为设计变量的初始值。
将隐式的非线性问题近似地转化为显式的线性规划, 通过线性规划求出运动极限范围内最优解, 建立数学模型如式(3)所示。
$\left\{\begin{array}{ll}\text{ find }& \mathbf{x}= {\left\lbrack {x}_{1},{x}_{2},\cdots ,{x}_{n}\right\rbrack }^{\mathrm{T}}\\\text{ make }& \mathop{\sum }\limits_{{i = 1}}^{n}{S}_{i}{x}_{i}\rightarrow \min \\\text{ s.t.} &{x}_{i0}\cdot \left({1 - d}\right)\leq {x}_{i}\leq {x}_{i0}\cdot \left({1 + d}\right)\\& \left({i = 1,\cdots , n}\right)\end{array}\right.$
式中$d$为初始运动极限因子;${x}_{i0}$为第$i$个设计变量的初始值。
由此通过线性规划求出的最优解${x}_{1},{x}_{2},\cdots ,{x}_{n}$作为新的设计变量初始值, 为使近似的目标函数在一定范围内, 收敛精度可以被接受, 在所有设计变量都发生振荡时, 把寻优的空间缩小至原来的0.1倍, 再次建立新的运动极限进行寻优, 经过不断迭代后, 运动极限达到某一值时迭代收敛, 最终求出理想结果。
2.2 多状态模型修正的方法流程
根据上述公式通过MSC.Nastran对输入文件进行灵敏度分析, 提取出灵敏度后, 将灵敏度、设计变量、运动极限利用Python中的scipy模块进行线性规划, 求出新的设计变量, 再通过迭代法求出最优解, 多状态模型修正方法程序流程如图2所示。
3 算例1
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利用多截面梁对多状态模型修正方法进行验证, 将一根泊松比$v ={0.3}$、弹性模量$E ={70}\mathrm{{GPa}}$、密度$\rho =$${2700}\mathrm{\;{kg}}/{\mathrm{m}}^{3}$的${60}\mathrm{\;m}$梁分成5段进行分析,为了避免在模型修正中发生模态交换, 将梁圆形截面的截面惯性矩${\mathbf{I}}_{1}\text{、}{\mathbf{I}}_{2}$采取不同值,取${\mathbf{I}}_{2}= {0.8}{\mathbf{I}}_{1}$进行分析,截面尺寸如表1所示。
本文利用有限元分析软件MSC.Patran&Nastran对建立的火箭有限元模型进行求解, 使用一维梁单元对该结构进行网格划分。将网格沿$y$方向划分为45个梁单元, 节点编号如图3所示, 每个梁段由9个相邻梁单元组成, 对各个梁单元赋予不同的截面属性, 有限元模型如图4所示。
在各级梁段边缘节点分别是节点1、10、19、28、 37、46处添加集中质量点代表火箭燃料质量。通过改变各个状态下质量集中点的质量代替燃料的消耗, 各个状态集中质量点的质量如表2所示。
对建立的3个火箭发射状态进行模态分析, 选取其中前4阶模态分析结果作为试验数据进行多状态模型修正。权重系数在满足式(4)的情况下可以根据需求任意设定, 本文对前4阶模态参数的权重系数依次取0.4,0.3,0.2,0.1。选取合适的设计变量初始值, 选取好收敛条件, 开始迭代。
$\mathop{\sum }\limits_{{m = 1}}^{M}{w}_{\mathrm{m}}= 1 $
经过724次迭代后得到设计变量的收敛结果如表3所示, 由修正前设计变量与试验值的最大相对误差17.4% 缩小至修正后最大相对误差0.18%,修正效果明显。试验值与修正后模态分析的频率对比误差如表4所示, 由修正前设计变量与试验值的最大相对误差5.8% 缩小至修正后最大相对误差0.0148%,修正效果明显, 整体迭代效果较为理想。在对比振型时选择了对比较为明显的15到30节点的阵型图进行对比, 可以看出试验值与修正前模态分析的振型对比有一定差距, 试验值与经过修正后模态分析的振型对比基本保持一致, 迭代效果较为理想。状态1、状态2、状态3前4阶振型相似, 采用状态1前4阶振型为代表, 如图5所示,其中$\mathrm{a}$、$\mathrm{b}$、$\mathrm{c}$、$\mathrm{d}$分别为各状态的1阶振型、2阶振型、3阶振型、4阶振型,图中黑色线代表修正前振型曲线, 红色线代表修正后振型曲线, 蓝色线代表试验振型曲线。
图5振型图对比结果说明: 对基于本文理论和程序得出修正后的截面惯性矩进行模态分析, 分析得到的模态数据与试验值都在允许的误差范围内, 符合工程要求, 能够为实际工程设计人员提供一定的参考依据。目标函数迭代曲线如图6所示, 经过不断迭代目标函数趋于0 , 迭代效果达到理想状态。
由图6可知, 3个状态目标函数迭代情况良好, 表明所有状态均能收敛, 该种多状态模型修正方法可以通过其他状态较好的收敛将该状态的收敛误差减小。最终计算出所有状态修正后的修正参数与试验值误差较小, 各个状态的频率误差接近相同, 证明了这种多状态模型修正方法的可行性。
4 算例2
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利用 “力锤法” 试验测量得到简支梁的固有频率, 通过多状态模型修正方法修正简支梁的截面尺寸以验证该方法的合理性, 利用带有磁铁的重物来模拟集中质量点, 建立多截面简支梁有限元模型, 利用单截面简支梁试验所得数据修正截面尺寸, 若将多截面尺寸修正为单截面尺寸, 则修正方法有效, 3个状态试验装置如图7所示。
4.1 优化模型的建立
被测工件材料采用合金钢,弹性模量$E =$${200}\mathrm{{GPa}}$,泊松比$v ={0.33}$,密度$\rho ={7850}\mathrm{\;{kg}}/{\mathrm{m}}^{3}$。多截面简支梁分为两段,长为${680}\mathrm{\;{mm}}$,第1段宽${50}\mathrm{\;{mm}}$,高$8\mathrm{\;{mm}}$,则${I}_{1}= {2133.3}{\mathrm{\;m}}^{4}$,${I}_{2}= {83333.3}{\mathrm{\;m}}^{4}$,设第2段${I}_{1}=$${5000.0}{\mathrm{\;m}}^{4},{I}_{2}= {50000.0}{\mathrm{\;m}}^{4}$,则根据截面惯性矩公式得到截面的宽为${37.11}\mathrm{\;{mm}}$,高为${11.74}\mathrm{\;{mm}}$,利用Beam单元对梁划分节点, 将简支梁划分为68个梁单元, 每个梁段由34个相邻梁单元组成。在节点52处添加集中质量点, 根据多次测量得到带有磁性的重物的质量为${196.5}\mathrm{\;g}$,则3个状态下集中质量点的质量分别为${393}\mathrm{\;g}\text{、}{196.5}\mathrm{\;g}\text{、}0\mathrm{\;g}$。在简支梁两端点处赋予约束, 只保留$y$方向的转动自由度,简支梁结构的简化建模如图8所示。
设第2段梁的截面惯性矩为设计变量, 根据试验结果修正截面尺寸, 以前4阶固有频率的相对误差最小为目标函数, 建立优化模型, 如式(5)所示。
$\left\{\begin{array}{ll}\text{ find }& {\mathbf{I}}_{1},{\mathbf{I}}_{2}\\\text{ make }& F\left({{\mathbf{I}}_{1},{\mathbf{I}}_{2}}\right)= \mathop{\sum }\limits_{{i = 1}}^{a}\left\lbrack {{w}_{i1}{\left(\frac{{f}_{i}\left({{\mathbf{I}}_{1},{\mathbf{I}}_{2}}\right)- {f}_{i}^{t}}{{f}_{i}^{t}}\right)}^{2}\rightarrow \min }\right.\end{array}\right.$
式中${\mathbf{I}}_{1}$为对于$y$轴的面积惯性矩;${\mathbf{I}}_{2}$为对于$z$轴的面积惯性矩;$a$表示选取$a$阶模态;${f}_{i}^{t}$为第$i$阶固有频率的试验测量值;${f}_{i}\left({{\mathbf{I}}_{1},{\mathbf{I}}_{2}}\right)$为第$i$阶固有频率有限元分析结果。
因选取的是4阶固有频率为目标函数,即$a = 4$。 4阶固有频率权重系数取相同,即${w}_{i1}= {w}_{i2}= {w}_{i3}= {w}_{i4}= {0.25}$, 经过试验, 测得被测工件的前4阶频率如表5所示, 建立目标函数, 进行迭代求解。
4.2 试验结果分析
经过172次迭代后设计变量收敛结果如表6所示, 误差在5% 以内, 试验数据与修正前后基频对比如表7所示。
修正后的截面惯性矩${\mathbf{I}}_{1}= 2\;{195.78}{\mathrm{\;m}}^{4},{\mathbf{I}}_{2}=$${80625.20}{\mathrm{\;m}}^{4}$,带入式(6)和式(7)中联立求得截面的尺寸长$b ={49.21}\mathrm{\;{mm}}$,高$h ={8.12}\mathrm{\;{mm}}$,面积$A =$${399.58}{\mathrm{\;{mm}}}^{2}$,经过修正将不成比例的初始值优化为成比例的迭代值, 完成修正目标。
${\mathbf{I}}_{1}= \frac{b{h}^{3}}{12}$
${\mathbf{I}}_{2}= \frac{{b}^{3}h}{12}$
5 结论
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根据火箭不同时段动力学特性的不同, 提出多状态有限元模型修正, 将多个状态下的优化模型组合到同一优化模型下进行求解, 因缺少火箭试验数据, 故采用有限元分析结果作为试验数据进行模型修正, 又建立多截面梁有限元模型通过测得试验数据对其进行模型修正, 验证方法的可行性, 得出如下结论:
a)以截面惯性矩为设计变量,固有频率和模态振型的误差为目标函数, 设置运动极限为约束条件, 建立多状态模型修正方法的优化模型。对各个状态的模态参数给与相应的权重系数, 对目标函数进行灵敏度分析, 将每个状态灵敏度加权求和组成新的灵敏度矩阵, 通过一阶泰勒展开完成对目标函数的显式化处理, 利用线性规划求解得到迭代后设计变量的最优解, 达到将多状态数学模型放到一个数学模型中求解的效果, 将得到的最优解作为下一次迭代的初始值。 当所有设计变量处于振荡状态时, 调整运动极限, 使设计变量的变化范围更小, 进行下一次迭代, 完成多状态模型修正, 以有限元分析数据作为试验数据, 最终求解出的最优解与试验值的误差控制在0.2% 以下, 验证了该方法的有效性。
b)利用锤击法对简支梁进行试验, 设计磁吸重物方案模拟多状态模型, 利用加速度传感器测得被测工件前4阶频率,经过修正将多截面简支梁修正为单截面简支梁, 并且符合截面惯性矩与面积比例, 验证了多状态模型修正方法的准确性。
研究结果对于多状态有限元模型修正误差的进一步减小提供了可借鉴的思路, 能够为结构设计优化工程提供一定的参考价值。
基金
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  • 国家自然科学基金资助项目(11872080)
文献
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参考文献 引证文献
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2024年第47卷第3期
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doi: 10.7654/j.issn.2097-1974.20240304
  • 接收时间：2022-05-05
  • 首发时间：2025-07-04
  • 出版时间：2024-06-25
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  • 收稿日期：2022-05-05
  • 修回日期：2023-11-30
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国家自然科学基金资助项目(11872080)
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    1北京工业大学,北京,100124
    2中国电子科技集团公司第四十五研究所,北京,100176
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2种不同金属材料的力学参数
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Number of
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Number of
species		 占总种数比例
Percentage of
total species (%)		 属
Genus		 种数
Number of
species		 占总种数比例
Percentage of total
species (%)
鹅膏菌科Amanitaceae 2 11 5.26 鹅膏菌属 Amanita 10 4.78
小菇科 Mycenaceae 2 12 5.74 丝盖伞属 Inocybe 5 2.39
多孔菌科 Polyporaceae 8 14 6.70 蜡蘑属 Laccaria 5 2.39
红菇科 Russulaceae 3 23 11.00 小皮伞属 Marasmius 6 2.87
小菇属 Mycena 11 5.26
光柄菇属 Pluteus 5 2.39
红菇属 Russula 17 8.13
栓菌属 Trametes 5 2.39
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