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Article(id=1228295811621122453, tenantId=1146029695717560320, journalId=1225147924628267009, issueId=1228295801890336965, articleNumber=null, orderNo=null, doi=10.16385/j.cnki.issn.1004-4523.2025.02.002, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=null, receivedDate=1679932800000, receivedDateStr=2023-03-28, revisedDate=1683648000000, revisedDateStr=2023-05-10, acceptedDate=null, acceptedDateStr=null, onlineDate=1770778142122, onlineDateStr=2026-02-11, pubDate=1739116800000, pubDateStr=2025-02-10, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1770778142122, onlineIssueDateStr=2026-02-11, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1770778142122, creator=13701087609, updateTime=1770778142122, updator=13701087609, issue=Issue{id=1228295801890336965, tenantId=1146029695717560320, journalId=1225147924628267009, year='2025', volume='38', issue='2', pageStart='223', pageEnd='448', issueExtLink='null', onlineDate='null', pubDate='null', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=null, createTime=1770778139803, creator=13701087609, updateTime=1770949100774, updator=13701087609, preIssue=null, nextIssue=null, ext={EN=IssueExt(id=1229012864237760763, tenantId=1146029695717560320, journalId=1225147924628267009, issueId=1228295801890336965, language=EN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=), CN=IssueExt(id=1229012864237760764, tenantId=1146029695717560320, journalId=1225147924628267009, issueId=1228295801890336965, language=CN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=)}, issueFiles=null}, startPage=232, endPage=241, ext={EN=ArticleExt(id=1228295811885363618, articleId=1228295811621122453, tenantId=1146029695717560320, journalId=1225147924628267009, language=EN, title=Theoretical and experimental study on the vibration characteristics of rotating pre-twisted beams, columnId=null, journalTitle=Journal of Vibration Engineering, columnName=null, runingTitle=null, highlight=null, articleAbstract=

Aiming at the vibration problem of rotating pre-twisted beams in engineering, an effective modeling method is proposed to study their vibration characteristics. Rotating experiments are designed to verify the accuracy of theoretical research. By adjusting the boundary spring stiffness, different boundary conditions are simulated. The displacement field is expanded using the modified Fourier series method, and the motion equation of the rotating beam is derived using the Rayleigh-Litz method. Based on the theoretical research, vibration tests of rotating straight beams and pre-twisted beams with different sizes are designed. The accuracy of this method is verified by comparing the theoretical calculations with finite element simulations and experimental results. The possibility of elastic boundaries is also verified through error analysis. The results show that the natural frequency of the beam increases with the increase in rotating speed and thickness. The increase in pre-twist angle has a minimal effect on the first-order natural frequency but significantly reduces the second-order natural frequency.

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针对工程中的旋转预扭梁振动问题，提出了一种有效的建模方法研究其振动特性，并设计了旋转实验验证理论研究的准确性。通过调节边界弹簧刚度模拟不同边界约束，采用修正傅里叶级数法对位移场进行展开，利用瑞利-里兹法推导旋转梁的运动方程。基于理论研究设计不同尺寸的旋转直梁和预扭梁的振动实验，将理论计算与有限元仿真、实验结果对比验证了本文方法的准确性，通过误差分析验证了存在弹性边界的可能。结果表明：转速和厚度增大使梁的固有频率增大；预扭角增大对1阶固有频率影响较小，可显著减小2阶固有频率。

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宿柱（1988—），男，博士，副研究员。E-mail：
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熊星星（1998—），男，硕士研究生。E-mail：

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熊星星（1998—），男，硕士研究生。E-mail：

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熊星星（1998—），男，硕士研究生。E-mail：

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Journal of Vibration Engineering，2022,35(3):577-584., articleTitle=Vibration properties of rectangular plates with elastic boundary constraints：theory，finite element and experiments, refAbstract=null)], funds=[Fund(id=1228313053960273966, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295811621122453, awardId=12272165, language=CN, fundingSource=国家自然科学基金资助项目(12272165), fundOrder=null, country=null), Fund(id=1228313054069325875, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295811621122453, awardId=51805250, language=CN, fundingSource=国家自然科学基金资助项目(51805250), fundOrder=null, country=null)], companyList=[AuthorCompany(id=1228313042908283540, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295811621122453, xref=null, ext=[AuthorCompanyExt(id=1228313042912477847, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295811621122453, companyId=1228313042908283540, language=EN, 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journalId=1225147924628267009, articleId=1228295811621122453, language=EN, label=Fig. 5, caption=Acceleration signal of 1 mm thick straight beam at different rotational speeds, figureFileSmall=jHkd5a7vsilxvc8KxIfETw==, figureFileBig=CEXs0ejhyMDYNg0tnPPCzQ==, tableContent=null), ArticleFig(id=1228313047261971271, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295811621122453, language=CN, label=图5, caption=不同转速下1 mm厚直梁加速度信号, figureFileSmall=jHkd5a7vsilxvc8KxIfETw==, figureFileBig=CEXs0ejhyMDYNg0tnPPCzQ==, tableContent=null), ArticleFig(id=1228313047379411793, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295811621122453, language=EN, label=Fig. 6, caption=Frequency response curve of 1 mm thick straight beam at different rotational speeds, figureFileSmall=MTm3vcV8PDTZ2IhB9MYWOA==, figureFileBig=DEJmGfIZwXx4QSCL6f4Y+A==, tableContent=null), ArticleFig(id=1228313047492658011, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295811621122453, language=CN, label=图6, caption=不同转速下1 mm厚直梁频响曲线, figureFileSmall=MTm3vcV8PDTZ2IhB9MYWOA==, figureFileBig=DEJmGfIZwXx4QSCL6f4Y+A==, tableContent=null), ArticleFig(id=1228313047614292834, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295811621122453, language=EN, label=Fig. 7, caption=Acceleration signal of 1.5 mm thick straight beam at different rotational speeds, figureFileSmall=vwhSgUYGmgSuhdaoBYXLvw==, figureFileBig=GXUyZuyhndQLETvj30V4fg==, tableContent=null), ArticleFig(id=1228313047714956139, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295811621122453, language=CN, label=图7, caption=不同转速下1.5 mm厚直梁加速度信号, figureFileSmall=vwhSgUYGmgSuhdaoBYXLvw==, figureFileBig=GXUyZuyhndQLETvj30V4fg==, tableContent=null), ArticleFig(id=1228313047836590962, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295811621122453, language=EN, label=Fig. 8, caption=Frequency response curve of 1.5 mm thick straight beam at different rotational speeds, figureFileSmall=CICyw15PFeSZqO0bfD07WA==, figureFileBig=AxJrOO7XS7hxvNTVuX67tg==, tableContent=null), ArticleFig(id=1228313047966614393, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295811621122453, language=CN, label=图8, caption=不同转速下1.5 mm厚直梁频响曲线, figureFileSmall=CICyw15PFeSZqO0bfD07WA==, figureFileBig=AxJrOO7XS7hxvNTVuX67tg==, tableContent=null), ArticleFig(id=1228313049371706242, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295811621122453, language=EN, label=Fig. 9, caption=Acceleration signal of 1.7 mm thick，30° pre-twisted angle beam at different rotational speeds, figureFileSmall=O7Si4kLJ7KpOcIErfXrF3Q==, figureFileBig=qeKhk14iRMWhnWpciZjcLQ==, tableContent=null), ArticleFig(id=1228313049493341069, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295811621122453, language=CN, label=图9, 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figureFileSmall=hFTxq1npYnZD+sUO4fcmew==, figureFileBig=02JKr9VCuZ5cUDkWWuyt4w==, tableContent=null), ArticleFig(id=1228313050218955692, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295811621122453, language=EN, label=Fig. 13, caption=Acceleration signal of 1.5 mm thick，15° pre-twisted angle beam at different rotational speeds, figureFileSmall=vUZn7b6adLpXep3iktUgTw==, figureFileBig=ezKScBJm5Zo7QdyBu+5TFQ==, tableContent=null), ArticleFig(id=1228313050353173425, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295811621122453, language=CN, label=图13, caption=不同转速下厚1.5 mm、预扭角15°梁的加速度信号, figureFileSmall=vUZn7b6adLpXep3iktUgTw==, figureFileBig=ezKScBJm5Zo7QdyBu+5TFQ==, tableContent=null), ArticleFig(id=1228313050449642426, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295811621122453, language=EN, label=Fig. 14, caption=Frequency response curve of 1.5 mm thick，15° pretwisted angle beam at different rotational speeds, figureFileSmall=8H9pp/7QOHYan4cZAU5pSQ==, figureFileBig=0sutNfNyCIzgNnNpNPsrbQ==, tableContent=null), ArticleFig(id=1228313050579665859, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295811621122453, language=CN, label=图14, caption=不同转速下厚1.5 mm、预扭角15°梁的频响曲线, figureFileSmall=8H9pp/7QOHYan4cZAU5pSQ==, figureFileBig=0sutNfNyCIzgNnNpNPsrbQ==, tableContent=null), ArticleFig(id=1228313050688717771, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295811621122453, language=EN, label=Fig. 15, caption=Comparison of theoretical and experimental errors under different boundary conditions （kv=kw=Kv=1010）, figureFileSmall=Poa03hKhH2+z/pmPXoK05g==, figureFileBig=B7lAJ9AZX56CKabUgK0Z0Q==, tableContent=null), ArticleFig(id=1228313050768409554, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295811621122453, language=CN, label=图15, caption=不同边界条件下理论与实验误差对比（kv=kw=Kv=1010）, figureFileSmall=Poa03hKhH2+z/pmPXoK05g==, figureFileBig=B7lAJ9AZX56CKabUgK0Z0Q==, tableContent=null), ArticleFig(id=1228313050873267159, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295811621122453, language=EN, label=Tab. 1, caption=

The first five order natural frequencies under different truncation numbers （α=45°，κ=0.25，δ=1，γ=5）

, figureFileSmall=null, figureFileBig=null, tableContent=
M固有频率
1阶2阶3阶4阶5阶
97.3998.32122.33728.36945.757
117.3968.31622.33428.36545.733
137.3958.31422.33328.36345.722
157.3948.31022.33228.36245.712
177.3948.31322.33228.36245.711
), ArticleFig(id=1228313050982319072, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295811621122453, language=CN, label=表1, caption=

不同截断数下前5阶固有频率（α=45°、κ=0.25、δ=1、γ=5）

, figureFileSmall=null, figureFileBig=null, tableContent=
M固有频率
1阶2阶3阶4阶5阶
97.3998.32122.33728.36945.757
117.3968.31622.33428.36545.733
137.3958.31422.33328.36345.722
157.3948.31022.33228.36245.712
177.3948.31322.33228.36245.711
), ArticleFig(id=1228313051108148196, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295811621122453, language=EN, label=Tab. 2, caption=

Material parameters of two kinds of beam specimens

, figureFileSmall=null, figureFileBig=null, tableContent=
梁试件材料名称密度/(g·cm-3)杨氏模量/GPa泊松比
直梁65Mn弹簧钢7.81198.60.26
预扭梁MS1模具钢8.00170.00.26
), ArticleFig(id=1228313051200422888, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295811621122453, language=CN, label=表2, caption=

两种梁试件的材料参数

, figureFileSmall=null, figureFileBig=null, tableContent=
梁试件材料名称密度/(g·cm-3)杨氏模量/GPa泊松比
直梁65Mn弹簧钢7.81198.60.26
预扭梁MS1模具钢8.00170.00.26
), ArticleFig(id=1228313051313669102, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295811621122453, language=EN, label=Tab. 3, caption=

Experimental and theoretical results comparison of the first two order natural frequencies of a rotating straight beam at different rotational speeds （Thickness is 1 mm）

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不同转速下旋转直梁的前两阶固有频率实验和理论结果对比（厚度为1 mm）

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Experimental and theoretical results comparison of the first two order natural frequencies of a rotating straight beam at different rotational speeds （Thickness is 1.5 mm）

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不同转速下旋转直梁的前两阶固有频率实验和理论结果对比（厚度为1.5 mm）

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Experimental and theoretical results comparison of the first two order natural frequencies of the pre-twisted beam at different rotational speeds （Thickness is 1.7 mm，pre-twisted angle is 30°）

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不同转速下预扭梁的前两阶固有频率实验和理论结果对比（厚度为1.7 mm、预扭角为30°）

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20029.84432.005632.2848.24177.500173.9092175.482.02
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Experimental and theoretical results comparison of the first two order natural frequencies of the pre-twisted beam at different rotational speeds （Thickness is 1.7 mm，pre-twisted angle is 15°）

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不同转速下预扭梁的前两阶固有频率实验和理论结果对比（厚度为1.7 mm、预扭角为15°）

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Experimental and theoretical results comparison of the first two order natural frequencies of the pre-twisted beam at different rotational speeds （Thickness is 1.5mm，pre-twisted angle is 15°）

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不同转速下预扭梁的前两阶固有频率实验和理论结果对比（厚度为1.5 mm、预扭角为15°）

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旋转预扭梁振动特性理论与实验研究
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熊星星 , 宿柱 , 马小虎
振动工程学报 | 2025,38(2): 232-241
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振动工程学报 | 2025, 38(2): 232-241
旋转预扭梁振动特性理论与实验研究
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熊星星 , 宿柱 , 马小虎
作者信息
  • 南京航空航天大学航空航天结构力学及控制全国重点实验室，江苏 南京 210016

    • 熊星星（1998—），男，硕士研究生。E-mail：

通讯作者:

宿柱（1988—），男，博士，副研究员。E-mail：
Theoretical and experimental study on the vibration characteristics of rotating pre-twisted beams
Xingxing XIONG , Zhu SU , Xiaohu MA
Affiliations
  • State Key Laboratory of Mechanics and Control for Aerospace Structures, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China
出版时间: 2025-02-10 doi: 10.16385/j.cnki.issn.1004-4523.2025.02.002
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摘要
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针对工程中的旋转预扭梁振动问题，提出了一种有效的建模方法研究其振动特性，并设计了旋转实验验证理论研究的准确性。通过调节边界弹簧刚度模拟不同边界约束，采用修正傅里叶级数法对位移场进行展开，利用瑞利-里兹法推导旋转梁的运动方程。基于理论研究设计不同尺寸的旋转直梁和预扭梁的振动实验，将理论计算与有限元仿真、实验结果对比验证了本文方法的准确性，通过误差分析验证了存在弹性边界的可能。结果表明：转速和厚度增大使梁的固有频率增大；预扭角增大对1阶固有频率影响较小，可显著减小2阶固有频率。

旋转梁  /  弹性约束  /  修正傅里叶级数  /  旋转实验

Aiming at the vibration problem of rotating pre-twisted beams in engineering, an effective modeling method is proposed to study their vibration characteristics. Rotating experiments are designed to verify the accuracy of theoretical research. By adjusting the boundary spring stiffness, different boundary conditions are simulated. The displacement field is expanded using the modified Fourier series method, and the motion equation of the rotating beam is derived using the Rayleigh-Litz method. Based on the theoretical research, vibration tests of rotating straight beams and pre-twisted beams with different sizes are designed. The accuracy of this method is verified by comparing the theoretical calculations with finite element simulations and experimental results. The possibility of elastic boundaries is also verified through error analysis. The results show that the natural frequency of the beam increases with the increase in rotating speed and thickness. The increase in pre-twist angle has a minimal effect on the first-order natural frequency but significantly reduces the second-order natural frequency.

rotating beam  /  elastic constraint  /  modified Fourier series  /  rotation experiment
熊星星, 宿柱, 马小虎. 旋转预扭梁振动特性理论与实验研究. 振动工程学报, 2025 , 38 (2) : 232 -241 . DOI: 10.16385/j.cnki.issn.1004-4523.2025.02.002
Xingxing XIONG, Zhu SU, Xiaohu MA. Theoretical and experimental study on the vibration characteristics of rotating pre-twisted beams[J]. Journal of Vibration Engineering, 2025 , 38 (2) : 232 -241 . DOI: 10.16385/j.cnki.issn.1004-4523.2025.02.002
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旋翼和叶片作为动力系统的组成部分在整个机械系统中起着至关重要的作用，若其损坏将导致事故的发生。因此，研究这类结构的振动特性具有重要工程意义。
很多旋转梁的研究都建立在经典边界条件下。YOO等[1]通过一组混合变形变量考虑旋转产生的离心力影响，推导了旋转梁轴向、弦向和垂向的运动方程。基于von Karman非线性应变和相应的线应力，KIM等[2]利用Hamilton原理推导了旋转梁的运动方程，通过将固有频率与其他文献进行比较证明了这种方法的可靠性。RAMESH等[3]研究了旋转预扭曲功能梯度悬臂梁的振动固有频率，系统地分析了系统参数对旋转悬臂梁固有频率的影响。但由于制造工艺和装配技术的限制，且随着长期服役，导致边界约束弱化，工程上很难实现经典边界。这类边界条件若等效为弹性边界显然更加合适。ADAIR等[4]引入阿德米安修正分解法（Adomian modified decomposition method）对在弹性边界下水平旋转欧拉梁的振动问题进行研究。SU等[5]基于Timoshenko梁理论，采用修正傅里叶级数法对功能梯度压电曲梁进行了自由振动与瞬态振动分析，通过与现有文献结果及有限元结果进行比较验证了该方法的准确性。LI等[6]采用正交多项式作为容许函数，建立了一种具有弹性约束的变厚度预扭叶片的动力学模型，得到了旋转预扭叶片的运动控制方程和模态。LI等[7]利用切比雪夫多项式理论推导了具有干摩擦支承边界条件的旋转梁的动力学微分方程。XU等[8]通过将容许位移函数统一扩展为雅可比多项式的形式，解决了旋转梁在不同边界条件下的自由振动问题。
在现有研究中涉及旋转梁振动模态的实验较少，旋转实验的一个难点在于对旋转件的激励方式，常见的激励方式有力锤敲击和激振器激励。但一方面，敲击高速旋转的梁结构难度大、危险系数高。另一方面，由于旋转产生的离心力，若直接与激振器连接极容易对梁结构造成损坏。因此，必须寻找一个更加安全可靠的激励方式。祝发荣等[9]建立了一个简单的旋转悬臂梁实验模型，从实验结果可以看出随着转速的增大，耦合梁的振动频率有增大的趋势，但由于旋转圆盘不固定和旋转机械造成的非线性影响等因素，实验结果并不明显。MAHMOODI等[10]使用电磁惯性致动器对梁进行扫频分析，使用激光位移传感器获得了梁的非线性振动频率，实验结果与多尺度法求得的理论结果吻合较好。QIAN等[11]利用基于相机的三维运动分析系统对旋转轴进行了振动实验，提出了一种共轭分解法对动态响应进行时频分析。张春芳[12]基于双目立体视觉原理和三维重构原理，用相机对悬臂梁的振动过程进行记录并对图片进行后处理，得到了梁上多个测点的位移曲线，相机测量结果与激光测振仪测量结果基本一致。景帅帅[13]用拉拽绳子的方式给旋翼实验台一个未知激励，以测量铝合金旋翼的固有模态。LEI等[14]通过扬声器产生的正弦声信号作用于微梁以获取梁的共振响应，结果表明：随着微梁厚度的减小，梁的无量纲固有频率和无量纲弯曲刚度随之增大。黄志诚等[15]设计了悬臂黏弹夹芯梁的振动实验，从响应图谱中直接读出了梁的固有频率，并利用半功率法获得相应的损耗因子。柴玉阳等[16]设计了不同边界组合下矩形板的振动实验，通过调节理论模型中边界约束弹簧的刚度模拟不同边界，验证其理论模型的正确性。
基于上述研究，本文建立一个弹性约束下的旋转预扭梁模型，采用修正傅里叶级数法对梁的位移场进行展开，利用瑞利-里兹法推导旋转梁的运动方程，求解运动方程可获得不同边界约束下旋转梁的固有频率及振型。通过调整虚拟弹簧刚度分析其对旋转梁振动特性的影响。设计旋转直梁和预扭梁的振动实验，并将理论结果与有限元结果、实验结果对比以验证理论分析的准确性，通过误差分析验证了存在弹性边界的可能。
1　理论模型
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1.1　弹性边界下旋转预扭梁
图1（a）为一预扭梁与旋转轮毂弹性连接的示意图，轮毂旋转带动梁旋转，转速为Ω，轮毂半径为r。梁的原长为l，密度为ρ，材料杨氏模量为E，梁截面的弦向和垂向惯性矩分别为Iy和Iz，截面积为A，梁的预扭角随着轴线方向均匀增大，最大预扭角为α。为便于表达在系统中建立了三个坐标系：轮毂坐标系XYZ、梁根部的坐标系xyz和端部的坐标系abc。轴X、x和a共线，Z轴是圆柱轮毂的中心轴，z轴与Z轴平行，c'轴和b'轴分别为c轴和b轴在yNz平面的投影。对于细长梁而言，弯曲与拉伸之间的耦合对系统的振动结果影响较小，所以在本文的研究中忽略与轴向运动相关的项。v和w分别表示梁中轴线上任一点P变形后在轴y、z上的投影。梁边界的连接方式如图1（b）所示，由两个线弹簧kv和kw和两个扭簧Kv和Kw表示弹性约束。
对于Euler-Bernoulli梁，旋转预扭梁的应变能可表示为：
式中，I2、I3为梁任意界面的惯性矩；I23为梁任意界面的惯性积。与扭转角的关系可表示为：
其中，θ为任意截面与根部截面的夹角，θ=αx/l。
将边界等效为由弹簧支撑的弹性边界，则产生的弹性势能为：
由离心力引起的轴向收缩势能可表示为：
其中，Fs为离心力，表达式为：
则系统的总势能为：
梁内任一点P的绝对速度为：
式中，i、j、k分别为x、y、z向的单位向量。则梁的动能可表示为：
1.2　修正傅里叶级数法
为解决弹性边界问题，采用修正傅里叶级数法将位移函数展开为下列形式：
式中，M为截断数；λm=mπ/l；Bm和Cm为傅里叶级数展开式的系数；P1、P2、P3和P4为辅助函数，所选辅助函数需在区间[0，l]内封闭，在梁内任意一点处三阶可导且连续，以消除边界上的不连续项；bn和cn（n=1、2、3、4）表示辅助函数的系数。本文可选择以下形式的辅助函数：
对于自由振动分析，梁的拉格朗日函数可用系统的势能和动能表示为：
将由修正傅里叶级数展开式表示的动能和势能代入到式（11）中，对位移展开式中的未知系数求导并设为0：
可整理出共2（M+5）个方程，将上式整理为下列矩阵形式：
式中，ω表示梁的固有频率；K为刚度矩阵；M为质量矩阵；G为所有傅里叶级数未知系数组成的向量：
通过求解式（13）的特征方程可以获得弹性约束下旋转预扭梁固有频率以及对应的模态振型。
2　数值分析
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2.1　弹性边界的影响
为了便于比较，令所有等效边界弹簧刚度相等，即kv =Kv=kw=Kw=K，数值结果采用无量纲参数的形式表示，无量纲参数如下：
通过调节边界弹簧的刚度，可以实现任意边界条件的约束。图2给出了不同条件下旋转预扭梁前4阶固有频率随弹簧刚度的变化趋势，从图中频率的变化趋势可以看出，随着弹簧刚度的增大，前4阶固有频率的变化可分为3个区间：第一区间各阶固有频率较小甚至为0，这时边界约束较小，这一区间可以看作是自由边界；第二区间各阶固有频率随着弹簧刚度的增大而增大，这一区间可以看作是弹性边界；第三区间即K≥108，这一区间随着弹簧刚度的持续增大，固有频率基本不变，说明这一阶段弹簧刚度的变化对固有频率影响不大，此时约束可近似看作固定边界。
2.2　收敛性分析
为了验证本文所建模型的稳定性与准确性，进行收敛性分析。表1给出了当K=1010时，取不同截断数M时旋转预扭梁的前5阶固有频率，由表可知，随着所取截断数的不断增大，固有频率逐渐收敛，表现出良好的收敛性。考虑计算效率，截断数M取为15。
3　实验研究
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3.1　实验装置
旋转梁的振动实验中由于梁在旋转，其激励方式一直是困扰学者的一个问题。传统的激励方式有力锤敲击和激振器激励，敲击能够高效地激发出梁的多阶共振频率，激振器激励则能精准地确定梁的共振频率。但这两种方式在旋转梁中运用都比较困难。
本文采用一种利用电磁铁激励的激振方式，将电磁铁置于梁下方合适位置，当梁旋转至电磁铁上方时按下开关使磁铁产生瞬时的吸力，梁受到吸力产生振动。多次实验后证明这种方法能够有效激起旋转梁的各阶共振频率。图3所示为旋转梁实验平台，为了模拟边界约束，设计了铝合金夹具，并在梁上方固定另一根等质量的梁防止旋转偏心以引起剧烈振动，在等质量梁与夹具间夹一层塑料以免夹具与试件发生耦合。
为了能取得较好的振动效果，所选取的梁试件必须有较大的刚度和相对小的阻尼，使梁在受激励后振动的时间尽可能长。图4分别为利用机加工和3D打印技术制作的直梁和预扭梁试件，直梁采用65Mn弹簧钢材料，预扭梁采用MS1模具钢材料，具体材料参数如表2所示。
3.2　旋转直梁实验结果
在搭好实验平台和完成准备工作后，分别测量旋转直梁在转速为0、200、400和600 r/min时的固有频率。为确保实验的普适性和准确性，使实验结果更加明显，在理论分析的基础上，直梁的旋转实验中选取梁长200 mm、宽20 mm，厚度分别为1和1.5 mm的梁作为对照实验。
厚度为1 mm的直梁试件在不同转速下的加速度信号和对应的频响曲线如图5和6所示。
厚度为1.5 mm的直梁试件在不同转速下的加速度信号和对应的频响曲线如图7和8所示。
图5和7分别为不同转速下厚度为1和1.5 mm的旋转直梁的时域信号，在开始记录时有一个明显的激励信号，说明电磁铁激励有明显且稳定的效果，当转速较高时可以发现加速度信号并没有消失，由于实验平台的旋转机械并不是完全同心的，旋转引起的偏心问题带动实验台振动，但这个振动频率并不与旋转梁发生共振且相对于共振响应较小，所以可以忽略它的影响。图6和8分别为不同转速下厚度为1和1.5 mm的旋转直梁的前两阶固有频率的频响曲线，可以发现随着转速的加快会出现一些杂波，这是由旋转偏心引起的，相较于共振波峰这些杂波较小且不稳定。在实验过程中进行多次测量以保证获得准确的不同转速下的旋转梁固有频率。结果表明：随着转速的增大，各阶固有频率增大；随着厚度的增大，各阶固有频率增大。
表3和4分别为不同转速下厚度为1和1.5 mm旋转直梁的前两阶固有频率实验结果与本文基于固支边界下旋转梁的理论结果和COMSOL仿真结果的对比，理论结果是将边界刚度调整为等效固支边界条件（取K=1010）下获得的，COMSOL仿真采用自由四面体网格，单元个数为42684个。理论结果（ft）与实验结果（fe）之间的误差计算公式如下式所示：
从表3和4中可以看出，理论结果与COMSOL仿真结果基本一致，最大误差不超过0.55%，证明了理论结果的正确性；同时，理论和仿真结果都与实验结果有一定误差，最大可达4.35%，说明实验中夹具并不能使边界达到理想的固支条件，边界约束弱化令旋转梁固有频率偏小，进一步证明了分析弹性边界问题的必要性和本文理论方法的准确性。
3.3　旋转预扭梁实验结果
分别测量旋转预扭梁在转速为0、200、400和600 r/min时的固有频率。为确保实验的普适性和准确性，使实验结果更加明显，在理论分析的基础上，预扭梁的旋转实验中选取梁长200 mm、宽20 mm，厚1.5 mm、预扭角15°，厚1.7 mm、预扭角15°和厚1.7 mm、预扭角30°的三根梁作为对照试验。
图9~14分别为厚1.7 mm、预扭角30°，厚1.7 mm、预扭角15°和厚1.5 mm、预扭角15°的旋转预扭梁在不同转速下的时域信号和频响曲线，通过多次测量以确保得到准确的不同转速下的前两阶固有频率。从结果分析中可以发现：随着旋转速度加快，各阶固有频率升高；随着厚度的增大，各阶固有频率升高；预扭角增大对1阶固有频率的影响较小，使2阶固有频率明显减小。
表5~7所示分别为不同转速下厚1.7 mm、预扭角30°，厚1.7 mm、预扭角15°和厚1.5 mm、预扭角15°三根梁的前两阶固有频率实验结果与本文基于固支边界旋转梁理论结果和COMSOL仿真结果的对比，理论结果是将边界刚度调整为等效固支边界条件（取K=1010）下获得的，COMSOL仿真单元设置和误差定义与直梁一致。从表5~7中可以看出，理论结果与COMSOL仿真结果吻合良好，误差不超过1.85%；理论结果与实验结果最大误差为8.69%。同时，实验结果与理论结果相比普遍偏小，这是由于夹具并不能令旋转梁系统的边界条件达到固支条件。相对于直梁，预扭梁实验与理论结果之间的误差更大，说明对于预扭梁模型，弹性边界对旋转梁振动特性的影响更大。
根据前文分析可知，理论上在边界虚拟弹簧刚度取无穷大时边界可等效为固支，随着刚度减小边界变为弹性边界约束，各阶频率降低。图15为预扭梁边界约束分别为弹性边界和固支边界时1阶频率的理论结果与实验结果的误差对比图，圆点表示弹性边界，方点表示等效固支边界，通过将边界弹簧刚度调至弹性边界区间，1阶频率的绝对误差可显著减小，说明实验中可能存在由于约束不完全，边界处于弹性边界区域的情况。
4　结论
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本文采用修正傅里叶级数法对位移函数进行展开，建立了弹性边界下旋转梁的动力学模型，利用瑞利-里兹法推导了旋转梁的运动方程。通过调节边界约束弹簧刚度模拟不同的边界条件，根据边界约束条件将边界分为自由、弹性、固支三个区间。数值计算结果表明该方法具有良好的收敛性。
在理论分析的基础上设计了直梁和预扭梁的旋转实验，利用电磁铁对梁试件进行激励，能充分激起梁的前两阶固有频率。将本文理论部分的边界条件等效为固支边界，直梁的理论结果与COMSOL仿真结果误差不超过0.55%，与实验结果的误差最大为4.35%。预扭梁理论结果与COMSOL仿真结果误差不超过1.85%，验证了本文理论模型的准确性，但与实验结果误差达8.69%。可发现实验结果与理论结果相比普遍偏小，且通过减小理论中边界弹簧刚度可使1阶频率误差明显减小，说明实验中可能存在由于约束不完全，边界处于弹性边界区域的情况，进一步证明了研究弹性边界问题的必要性和本文理论方法的准确性。实验结果分析表明：随着转速、厚度的增大，各阶固有频率增大；预扭角增大对1阶固有频率的影响较小，但能显著降低2阶固有频率。
基金
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  • 国家自然科学基金资助项目(12272165)
  • 国家自然科学基金资助项目(51805250)
文献
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参考文献 引证文献
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2025年第38卷第2期
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doi: 10.16385/j.cnki.issn.1004-4523.2025.02.002
  • 接收时间：2023-03-28
  • 首发时间：2026-02-11
  • 出版时间：2025-02-10
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  • 收稿日期：2023-03-28
  • 修回日期：2023-05-10
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国家自然科学基金资助项目(12272165)
国家自然科学基金资助项目(51805250)
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    南京航空航天大学航空航天结构力学及控制全国重点实验室，江苏 南京 210016

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宿柱（1988—），男，博士，副研究员。E-mail：
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2种不同金属材料的力学参数
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Family		 属数
Number of
genus		 种数
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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