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Article(id=1228653712009261061, tenantId=1146029695717560320, journalId=1225147924628267009, issueId=1228653708687377017, articleNumber=null, orderNo=null, doi=10.16385/j.cnki.issn.1004-4523.2024.11.001, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=null, receivedDate=1667577600000, receivedDateStr=2022-11-05, revisedDate=1678723200000, revisedDateStr=2023-03-14, acceptedDate=null, acceptedDateStr=null, onlineDate=1770863472225, onlineDateStr=2026-02-12, pubDate=1732723200000, pubDateStr=2024-11-28, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1770863472225, onlineIssueDateStr=2026-02-12, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1770863472225, creator=13701087609, updateTime=1770863472225, updator=13701087609, issue=Issue{id=1228653708687377017, tenantId=1146029695717560320, journalId=1225147924628267009, year='2024', volume='37', issue='11', pageStart='1803', pageEnd='1992', issueExtLink='null', onlineDate='null', pubDate='null', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=null, createTime=1770863471433, creator=13701087609, updateTime=1770863902026, updator=13701087609, preIssue=null, nextIssue=null, ext={EN=IssueExt(id=1228655514792427773, tenantId=1146029695717560320, journalId=1225147924628267009, issueId=1228653708687377017, language=EN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=), CN=IssueExt(id=1228655514792427774, tenantId=1146029695717560320, journalId=1225147924628267009, issueId=1228653708687377017, language=CN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=)}, issueFiles=null}, startPage=1803, endPage=1810, ext={EN=ArticleExt(id=1228653714311933975, articleId=1228653712009261061, tenantId=1146029695717560320, journalId=1225147924628267009, language=EN, title=Active control of forced vibration in spindle milling with integrated electromagnetic actuator, columnId=null, journalTitle=Journal of Vibration Engineering, columnName=null, runingTitle=null, highlight=null, articleAbstract=

Milling vibration problems can increase tool wear and reduce part machining accuracy，the suppression of which is a hot research topic in the field of milling. In order to suppress forced vibration in milling and enhance the stability of the machining system，a spindle-tool system is proposed based on electromagnetic actuators technology. The electromagnetic actuators are mounted in the spindle-tool system and acts on the toolholder position，using its controlled electromagnetic force for vibration suppression. Firstly，the structure and working principle of the system is described and the milling dynamics of the system is modelled. Secondly，based on the working principle of electromagnetic actuators，a PD controller is designed and the dynamic model and controller are simulated and verified. Finally，machine vibration signals are collected and external excitation experiments are carried out on a rotor test bench simulating tool process. Simulation and experimental results show that the spindle-tool system based on electromagnetic actuators under PD control law can effectively suppress the forced vibrations during the milling process.

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铣削加工振动会加剧刀具磨损，降低零件加工精度，振动抑制是铣削加工领域的研究热点。为抑制铣削强迫振动，提高加工系统的稳定性，提出一种集成电磁作动器的主轴-刀具系统，将电磁作动器装夹于主轴-刀具系统中，并作用于刀柄位置，利用电磁作动器的可控电磁力进行振动抑制。介绍了系统的结构和工作原理，并建立了系统的铣削动力学模型。基于电磁作动器的工作原理，设计了PD控制器，并对所搭建的动力学模型和控制器进行仿真验证。采集机床振动信号，在模拟刀具工作的转子实验台上进行外部激振实验。仿真和实验结果表明，PD控制下，基于电磁作动器的主轴-刀具系统可有效抑制铣削加工过程中的强迫振动。

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周瑾（1972—），女，博士，教授。E-mail：。
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崔刚（1995—），男，博士研究生。E-mail：。

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崔刚（1995—），男，博士研究生。E-mail：。

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崔刚（1995—），男，博士研究生。E-mail：。

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CIRP Journal of Manufacturing Science and Technology， 2008， 1（1）： 35-44., articleTitle=Chatter stability of milling in frequency and discrete time domain, refAbstract=null)], funds=[Fund(id=1228653732968198732, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228653712009261061, awardId=J2019-Ⅶ-0001-0141, language=CN, fundingSource=国家重大科技专项资助项目(J2019-Ⅶ-0001-0141), fundOrder=null, country=null)], companyList=[AuthorCompany(id=1228653725313593647, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228653712009261061, xref=null, ext=[AuthorCompanyExt(id=1228653725321982256, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228653712009261061, companyId=1228653725313593647, language=EN, country=null, province=null, city=null, postcode=null, companyName=null, departmentName=null, remark=College of Mechanical and Electrical Engineering，Nanjing University of Aeronautics and Astronautics，Nanjing 210016，China), 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tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228653712009261061, language=EN, label=Fig.2, caption=2-DOF milling vibration model, figureFileSmall=5abKralwa4Jy+FpkH4IwKA==, figureFileBig=zlSBRt/r2nS8YIO2Tz36+A==, tableContent=null), ArticleFig(id=1228653729289794005, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228653712009261061, language=CN, label=图2, caption=两自由度铣削加工振动模型, figureFileSmall=5abKralwa4Jy+FpkH4IwKA==, figureFileBig=zlSBRt/r2nS8YIO2Tz36+A==, tableContent=null), ArticleFig(id=1228653729390457304, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228653712009261061, language=EN, label=Fig.3, caption=Schematic diagram of the 1-DOF magnetic bearing, figureFileSmall=tkdHs3puvxUGKuRP2Z84Dw==, figureFileBig=SFVbQSr6k4HUX1rfKGjsEQ==, tableContent=null), ArticleFig(id=1228653729461760476, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228653712009261061, language=CN, label=图3, caption=单自由度磁悬浮轴承结构示意图, figureFileSmall=tkdHs3puvxUGKuRP2Z84Dw==, figureFileBig=SFVbQSr6k4HUX1rfKGjsEQ==, tableContent=null), ArticleFig(id=1228653729579200994, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228653712009261061, language=EN, label=Fig.4, caption=The control block diagram of milling process, figureFileSmall=L5fVKJRus4TXWY40c5UAAw==, figureFileBig=2nuiznZXjGshs9qRI0grPA==, tableContent=null), ArticleFig(id=1228653729667281382, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228653712009261061, language=CN, label=图4, caption=铣削加工控制框图, figureFileSmall=L5fVKJRus4TXWY40c5UAAw==, figureFileBig=2nuiznZXjGshs9qRI0grPA==, tableContent=null), ArticleFig(id=1228653729742778859, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228653712009261061, language=EN, label=Fig.5, caption=The milling dynamics and control block diagram, figureFileSmall=1uZoQ2Wist4vvueUHEN29Q==, figureFileBig=g/IbYOsz+7jjGFm+vy/yhA==, tableContent=null), ArticleFig(id=1228653731093344754, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228653712009261061, language=CN, label=图5, caption=铣削动力学和控制框图, figureFileSmall=1uZoQ2Wist4vvueUHEN29Q==, figureFileBig=g/IbYOsz+7jjGFm+vy/yhA==, tableContent=null), ArticleFig(id=1228653731168842231, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228653712009261061, language=EN, label=Fig.6, caption=The stability lobe diagram, figureFileSmall=vgU5aESX2JjQjRHNAvwDoQ==, figureFileBig=SRUQHrH1N3uUTgbHnOXyfg==, tableContent=null), ArticleFig(id=1228653731240145405, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228653712009261061, language=CN, label=图6, caption=稳定性叶瓣图, figureFileSmall=vgU5aESX2JjQjRHNAvwDoQ==, figureFileBig=SRUQHrH1N3uUTgbHnOXyfg==, tableContent=null), ArticleFig(id=1228653731328225792, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228653712009261061, language=EN, label=Fig.7, caption=The results of milling time domain simulation, figureFileSmall=TXQUF8W0YXjAeL/GDymHwg==, figureFileBig=lF376vXT/GFAnPP09avO2A==, tableContent=null), ArticleFig(id=1228653731399528964, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228653712009261061, language=CN, label=图7, caption=铣削时域仿真结果, figureFileSmall=TXQUF8W0YXjAeL/GDymHwg==, figureFileBig=lF376vXT/GFAnPP09avO2A==, tableContent=null), ArticleFig(id=1228653731487609353, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228653712009261061, language=EN, label=Fig.8, caption=The milling time domain control simulation, figureFileSmall=MQvgs206GVYs43NJ9hkchQ==, figureFileBig=JiKUb2HvPwtZWviKFIsbag==, tableContent=null), ArticleFig(id=1228653731567301134, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228653712009261061, language=CN, label=图8, caption=铣削时域控制仿真, figureFileSmall=MQvgs206GVYs43NJ9hkchQ==, figureFileBig=JiKUb2HvPwtZWviKFIsbag==, 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Spindle-tool system dynamics parameters

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参数数值
主轴转速n/（r⋅min-1）1800
铣刀进给量fd/（mm⋅min-1）400
刀具齿数N*3
切向切削力系数Kt/（109 N⋅m-2）1.8698
径向切削力系数Kr/（109 N⋅m-2）0.9154
x/y方向刀具固有频率wn/（rad⋅s-1）2193
x/y方向刀具阻尼比ζ0.03
刀具质量m/kg1.24
刀具直径D/mm10
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主轴-刀具系统动力学参数

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参数数值
主轴转速n/（r⋅min-1）1800
铣刀进给量fd/（mm⋅min-1）400
刀具齿数N*3
切向切削力系数Kt/（109 N⋅m-2）1.8698
径向切削力系数Kr/（109 N⋅m-2）0.9154
x/y方向刀具固有频率wn/（rad⋅s-1）2193
x/y方向刀具阻尼比ζ0.03
刀具质量m/kg1.24
刀具直径D/mm10
), ArticleFig(id=1228653732703957566, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228653712009261061, language=EN, label=Tab.2, caption=

The parameters of electromagnetic actuators

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参数数值
偏置电流i0/A1.2
线圈匝数N150
磁极面积A/mm22×10-4
单边气隙x0/mm0.25
磁极夹角α/（°）22.5
电流刚度系数/（N⋅A-1）39.18
位移刚度系数/（N⋅m-1）1.18×105
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电磁作动器主要参数

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参数数值
偏置电流i0/A1.2
线圈匝数N150
磁极面积A/mm22×10-4
单边气隙x0/mm0.25
磁极夹角α/（°）22.5
电流刚度系数/（N⋅A-1）39.18
位移刚度系数/（N⋅m-1）1.18×105
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集成电磁作动器的主轴铣削强迫振动主动控制
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崔刚 , 周瑾 , 徐九华 , 徐园平 , 曾磊
振动工程学报 | 2024,37(11): 1803-1810
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振动工程学报 | 2024, 37(11): 1803-1810
集成电磁作动器的主轴铣削强迫振动主动控制
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崔刚 , 周瑾 , 徐九华, 徐园平, 曾磊
作者信息
  • 南京航空航天大学机电学院，江苏　南京　210016

    • 崔刚（1995—），男，博士研究生。E-mail：。

通讯作者:

周瑾（1972—），女，博士，教授。E-mail：。
Active control of forced vibration in spindle milling with integrated electromagnetic actuator
Gang CUI , Jin ZHOU , Jiu-hua XU, Yuan-ping XU, Lei ZENG
Affiliations
  • College of Mechanical and Electrical Engineering，Nanjing University of Aeronautics and Astronautics，Nanjing 210016，China
出版时间: 2024-11-28 doi: 10.16385/j.cnki.issn.1004-4523.2024.11.001
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摘要
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铣削加工振动会加剧刀具磨损，降低零件加工精度，振动抑制是铣削加工领域的研究热点。为抑制铣削强迫振动，提高加工系统的稳定性，提出一种集成电磁作动器的主轴-刀具系统，将电磁作动器装夹于主轴-刀具系统中，并作用于刀柄位置，利用电磁作动器的可控电磁力进行振动抑制。介绍了系统的结构和工作原理，并建立了系统的铣削动力学模型。基于电磁作动器的工作原理，设计了PD控制器，并对所搭建的动力学模型和控制器进行仿真验证。采集机床振动信号，在模拟刀具工作的转子实验台上进行外部激振实验。仿真和实验结果表明，PD控制下，基于电磁作动器的主轴-刀具系统可有效抑制铣削加工过程中的强迫振动。

铣削加工  /  强迫振动  /  振动抑制  /  电磁作动器

Milling vibration problems can increase tool wear and reduce part machining accuracy，the suppression of which is a hot research topic in the field of milling. In order to suppress forced vibration in milling and enhance the stability of the machining system，a spindle-tool system is proposed based on electromagnetic actuators technology. The electromagnetic actuators are mounted in the spindle-tool system and acts on the toolholder position，using its controlled electromagnetic force for vibration suppression. Firstly，the structure and working principle of the system is described and the milling dynamics of the system is modelled. Secondly，based on the working principle of electromagnetic actuators，a PD controller is designed and the dynamic model and controller are simulated and verified. Finally，machine vibration signals are collected and external excitation experiments are carried out on a rotor test bench simulating tool process. Simulation and experimental results show that the spindle-tool system based on electromagnetic actuators under PD control law can effectively suppress the forced vibrations during the milling process.

milling process  /  forced vibration  /  vibration suppression  /  electromagnetic actuators
崔刚, 周瑾, 徐九华, 徐园平, 曾磊. 集成电磁作动器的主轴铣削强迫振动主动控制. 振动工程学报, 2024 , 37 (11) : 1803 -1810 . DOI: 10.16385/j.cnki.issn.1004-4523.2024.11.001
Gang CUI, Jin ZHOU, Jiu-hua XU, Yuan-ping XU, Lei ZENG. Active control of forced vibration in spindle milling with integrated electromagnetic actuator[J]. Journal of Vibration Engineering, 2024 , 37 (11) : 1803 -1810 . DOI: 10.16385/j.cnki.issn.1004-4523.2024.11.001
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引言
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铣削加工振动会加剧刀具磨损、制约零件加工精度，降低加工效率［1］，铣削加工中90%以上的振动由颤振和强迫振动引起［2］。颤振是由于系统本身加工特性所引起的一种自激振动，可分为再生型颤振、振型耦合型颤振、摩擦型颤振等［3］；其中，再生型颤振由刀齿通过周期的时变性产生，是颤振产生的主要原因。而铣削过程中的强迫振动主要由刀齿与工件表面切削的断续性或切屑形成的不连续性引起［4］。为提高加工系统的效率和稳定性，铣削振动抑制技术得到了国内外学者的广泛研究。
铣削加工过程中，经常通过绘制稳定性叶瓣图（Stability Lobe Diagram，SLD）来选择加工参数以避免再生型颤振问题［5］。为提高SLD的计算精度，国内外学者考虑了主轴转速［6］、工件材料［7］和过程阻尼［8］等因素，对SLD进行了进一步完善，但保守的工艺参数降低了机床的加工效率。
为进一步提升材料的去除率，提高铣削稳定域，国内外学者提出了不同的控制方法抑制铣削过程中的振动问题，主要分为被动抑制和主动抑制两种方式。
被动抑制主要是通过添加阻尼装置或改变铣床主轴的切削参数提高加工系统的稳定性，典型的抑制方法主要有调谐质量阻尼器［9-10］、变螺旋铣刀［11］和变主轴转速［12］。该类振动抑制方法操作简单，成本较低，但需根据刀具加工过程中的动力学变化实时做出调整，效率较低，未从根本解决问题。主动抑制方法通过实时检测刀具振动位移，改变主轴-刀具的动力学参数对刀具或工件进行振动抑制，可从根本上解决刀具加工过程中的振动问题。目前，铣床主轴-刀具振动的主动控制大多数采用压电作动器作为执行机构［13-15］，通过对主轴轴承外圈施加控制力间接进行刀具的振动抑制。该作用装置体积小，易于集成，但无法直接作用于旋转工作的刀具，此外，压电作动器的非线性以及时滞问题也使控制器的设计变得较为复杂。
为解决上述问题，国内外学者基于磁悬浮轴承阻尼、刚度可控和主动振动抑制的优点，设计了利用磁悬浮轴承支承的电主轴［16-17］，该装置可直接监测主轴-刀具振动信号，实时反馈给控制系统，对主轴进行振动抑制，可有效解决加工过程中的振动问题。但磁悬浮轴承支承刚度较低，不能承受大的切削载荷，现阶段只适合应用于高速小径向铣削当中。为结合机械轴承高支承刚度和磁悬浮轴承主动振动抑制的优点，在传统机械主轴的基础上，将磁悬浮轴承作为电磁作动器，进行主轴-刀具的振动抑制。文献［18-19］提出将电磁作动器用于镗削加工，作动器作用于镗刀，实时监测镗刀振动，并利用电磁力进行振动抑制，但该方法局限于镗削加工。为解决上述问题，WAN等［20］提出一种将电磁作动器用于铣削加工的方案，将电磁作动器集成于主轴内部，检测主轴端部振动，并进行铣削颤振抑制。该方案为定制主轴，主轴刀柄采用一体式结构，无法满足不同的铣削环境要求。
综上所述，国内外研究人员对铣削颤振抑制做了大量研究，而在铣削过程中即使是在不发生颤振的条件下，其强迫振动的存在，仍然会影响加工零件的表面精度。为抑制铣削强迫振动，本文提出一种集成电磁作动器的主轴-刀具系统，将作动器装夹于主轴-刀具系统前端，并作用于刀柄位置，利用其可控电磁力对刀柄进行振动抑制。首先，介绍了主轴-刀具系统结构并建立了动力学模型。然后，基于电磁作动器工作原理设计了控制器并进行仿真计算。最后，采集机床振动信号，并在转子实验台上进行外部激振实验，验证所提出的结构和控制方法能有效抑制铣削过程中的强迫振动。
1　电磁作动器主轴-刀具系统建模
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1.1　主轴-刀具系统结构
集成电磁作动器的主轴-刀具系统结构如图1所示。主要由主轴系统和作动系统组成。主轴系统主要包括：主轴轴心、电机、机械轴承、主轴外壳、刀柄和刀具；作动系统主要包括：电磁作动器、位移传感器和夹持装置。其中，主轴轴心由机械轴承支承，以保证其刚度和回转精度，电机驱动机床主轴高速旋转并提供加工扭矩。作动系统通过夹持装置固定于机床主轴系统，电磁作动器作用于刀柄位置，利用可控电磁力改变刀柄的阻尼和刚度从而对铣削加工进行振动抑制。
作动系统的控制机理为：铣削加工过程中，刀具产生振动时，均布于刀柄周向的四个位移传感器精确检测刀柄振动位移，并将位移传感器检测到的振动位移传递给电磁作动器控制器，控制器输出控制电压传输至功率放大器，功率放大器将控制电压转换为控制电流，电磁作动器利用控制电流改变电磁力大小对刀柄进行实时振动控制。
1.2　动力学建模
主轴和刀具在加工过程中可视为相互正交的二自由度弹簧-阻尼振动系统［21］，其两自由度振动模型如图2所示。图2中，c（x）和k（x）分别为刀具x方向的阻尼和刚度，c（y）和k（y）分别为刀具y方向的阻尼和刚度。铣削加工过程中第j个刀齿的瞬时切削厚度可表示为：
式中　 fz为每齿进给量；为沿y轴顺时针测量的刀齿瞬时切入角；qj（t）和qj（t-τ）分别为当前刀齿和前一刀齿的动态位移，且qj=-xsin-ycos，τ=60/（N*·Ω）为铣削时滞量（N*为刀具齿数，Ω为主轴转速）；g（）为窗函数，用于判断刀齿是否参与切削，其表达式为：
式中　 ϕst为刀具切入角；ϕex为刀具切出角。
作用在第j个刀齿上的瞬时切向切削力Ftj和瞬时径向切削力Frj可表示为：
式中　 a为轴向切深；Kt为切向切削力系数；Kr为径向切削力系数。将切削力在x和y方向分解，并进行相加，可得到铣削总切削力为：
联立式（1）~（4）整理得到：
式中　 Fs（t）为静态切削力；Fd（t）为动态切削力；H（t）是随刀具切入和切出角时变的系数矩阵，为方便后续控制器设计，可通过傅里叶变换，近似为时不变系数矩阵［22］，可表示为：
本文采用8极C型主动磁悬浮轴承作为作动器施加在刀柄位置，并在刀柄处的x和y方向分别布置一对磁极，相同方向上的磁极采用偏置电流加控制电流的方式进行差动控制。以x方向一对磁极为例，建立电磁作动器电磁力模型，如图3所示。
图3中，o为作动器定子几何中心，o'为刀柄几何中心。刀柄因铣削振动发生偏移时，根据麦克斯韦电磁吸力方程，在x方向所受的电磁力可表示为：
式中　 fx1和fx2分别为x+和x-方向的电磁力；μ0为真空磁导率；A为单个磁极截面积；N为线圈匝数；i0为偏置电流；ix为控制电流；x0为转子几何中心与定子几何中心重合时磁极之间的单边气隙；xcosα为作动器定子和刀柄之间气隙变化量。
由式（7）可知，磁悬浮轴承的电磁力是位移和电流的非线性函数，为便于控制器的设计和分析计算，需对其进行线性化。将式（7）在x=0处进行泰勒展开，并忽略高阶小量，可得到磁悬浮轴承电磁力线性化计算公式：
式中　 kx和ki分别为电磁作动器的位移刚度系数和电流刚度系数。同理，由电磁作动器的对称性，可得到y方向的电磁力fy。
综合上述分析，可得到基于磁悬浮轴承技术的主轴-刀具系统动力学方程：
式中　 q=［x（t） y（t）］T为位移向量；M=diag（m（x），m（y）），C=diag（c（x），c（y））和K=diag（k（x），k（y））分别为主轴-刀具系统的质量矩阵，阻尼矩阵和刚度矩阵；Fa=［fx fy］T为电磁作动器的电磁力。
2　控制器设计
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2.1　PD控制器
由上述动力学分析进行基于电磁作动器的主轴-刀具控制器设计。PID控制原理简单，适应性强，广泛应用于工业控制系统。由于本文采用的电磁作动器仅需对刀柄进行振动抑制，仅需控制刚度和阻尼，不存在稳态误差问题，因此，采用PD控制器对切削过程中的强迫振动进行振动抑制，控制系统框图如图4所示。
电磁作动器的支承特性可由等效刚度ks和等效阻尼ds来表示，将其视为线性的质量弹簧阻尼系统，其动力学方程可表示为：
式中　 z为位移向量；P为外部干扰力。对其进行Laplace变换，令s=jω可得到系统的频率特性方程：
为方便控制器的设计，对电磁作动器控制系统采用电流放大器，其传递函数可表示为：
式中　 C（s）为PD控制器的传递函数；为输入信号的Laplace变换。
对式（8）进行Laplace变换，并代入式（12），可得到电磁作动器系统的频率特性方程：
式中　 =Re［］+Im［］= Kp+Kds，令式（11）等于式（13）可得到：
式中　 Kp为比例反馈系数；Kd为微分反馈系数。
由式（14）可知，可分别调节Kp和Kd改变电磁作动器的刚度和阻尼，进而改变电磁力的大小，经控制回路对铣削过程中产生的强迫振动进行振动抑制。
2.2　鲁棒性分析
系统鲁棒性是指系统抵抗不确定参数摄动和外界干扰的能力，闭环系统的鲁棒性可由灵敏度函数表示。由于电磁作动器的作动对象是主轴刀柄，为方便分析，将主轴-刀具产生的振动视为干扰力P，并将该干扰力以扰动的形式添加到作动器的控制系统中，由此进行控制器的鲁棒性分析。定义灵敏度函数S为扰动P到刀柄位移x的传递函数：
灵敏度函数的幅值代表了系统抵抗外界干扰的能力，其幅值越小，系统抗干扰能力越强。对应于电磁作动器，其幅值越小抑制振动效果越明显。式（15）的特征方程为：
根据劳斯判据，系统稳定的充要条件为：
由于电磁作动器不起支承作用，因此无需要求高刚度，刚度选取为“自然”刚度即可，即广义刚度与电磁作动器的负刚度处于同一数量级。同理，广义阻尼的选取也应与处于相同数量级，选取阻尼比ξ=0.1~1。由此设计控制器的参数Kp和Kd以达到所需控制效果。
3　仿真分析
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为验证动力学模型和控制器效果，基于上述分析，结合MATLAB/Simulink对铣削加工进行时域仿真计算。通过模态实验和铣削加工工艺，得到主轴-刀具系统的动力学参数如表1所示。
结合表1的动力学参数，利用MATLAB/ Simulink搭建铣削加工动力学和控制系统仿真框图如图5所示。
图5的基本计算逻辑为：将参数变量作为输入端添加到铣削系统中，经公式（5）计算后，分别输出x和y方向的铣削力F（x）和F（y），并添加到主轴-刀具系统中；主轴-刀具系统模块经计算后，输出刀柄的振动位移x和y，以及由刀齿通过频率引起的Δx和Δy（Δx和Δy反馈给铣削系统用于计算动态铣削力Fd）进入电磁作动器控制系统；控制系统经控制器、功率放大器等模块输出电磁力Fax和Fay，并反馈给主轴-刀具系统从而抑制刀柄振动。
由表1动力学参数进行铣削加工稳定性分析，绘制SLD，结果如图6所示。
图6中曲线波谷的下方代表绝对稳定切削区域，在该区域内，铣削绝对稳定，且主轴转速不对其产生影响。该曲线的上方代表不稳定区域，两区域之间为稳定区，该区域内需调节主轴转速以达到稳定切削目的。在主轴转速为2000 r/min时，选取图6中三个点，进行铣削时域仿真计算，在不同的轴向切深下，未施加控制时的铣削时域仿真结果如图7所示。
图7（a）为轴向切深为0.2 mm时的时域仿真结果。图中曲线在开始的波动后位移趋于0附近，即刀具加工过程中无颤振和强迫振动，在实际加工过程中，该转速下，刀具轴向切深为0.2 mm时可实现稳定铣削加工；图7（b）为临界稳定铣削，图示曲线趋于稳态波动，对应图6曲线上的B点。图7（c）为轴向切深0.6 mm时的时域仿真结果，图中曲线逐渐发散，并趋于无穷大（实际加工过程中，由于主轴-刀具结构之间的阻尼作用，振动不会趋于无穷大）。该切深状态下，刀具发生强迫振动和颤振，严重时会产生断刀现象。
为验证PD控制器的控制效果，在模型中添加控制后选取0.6 mm轴向切深，对其进行时域仿真。其仿真结果如图8所示。
由图8仿真结果可知，施加控制后，0.6 mm轴向切深下的铣削时域仿真结果由发散变为收敛，铣削由不稳定转换为稳定。为进一步验证控制器效果，进行稳定性分析，并绘制施加控制后的SLD，其计算结果如图9所示。
由图9可知，对系统施加控制后，由于改变了主轴-刀具系统的刚度、阻尼和固有频率，SLD整体向上并向右移动，稳定铣削区域明显增加。
4　实验论证
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为验证理论和仿真分析结果，在所搭建的转子实验台上进行激振实验。首先采集铣削加工振动信号，图10为在主轴转速1800 r/min，刀具进给量400 mm/min，轴向切深0.8 mm时的刀具振动加速度时域信号，以及在10 s时对应的频域信号。由于工件在不同切削位置处的刚度有所变化，导致其时域加速度信号在不同时间段略有不同。频域图中，348 Hz为颤振频率，f0（90 Hz）为刀齿通过频率的主频，为强迫振动频率。
转子激振实验如图11所示，转子由机械轴承支承，其悬臂为模拟刀具端，实验过程中，只有左径向磁悬浮轴承处于工作状态并充当电磁作动器，其主要参数如表2所示。
将图10所示的加速度信号经PC1输入至动态信号分析仪（OROS），OROS将振动信号输送给功率放大器，HEV-50激振器接收功率放大器发出的信号，并由连杆激振转子悬臂端。实验过程中，磁悬浮轴承控制系统实时监测转子悬臂端振动，并进行振动抑制，实验过程中的振动信号由PC2实时记录。
在图10所示的加速度信号激振下，控制前/后的转子振动时域响应信号如图12所示。图12（a）中，转子在仅由机械轴承的支承状态下，其振动位移响应最大幅值约为50 μm，施加电磁作动器控制后，转子位移响应最大幅值约为10 μm。振动幅值降低80%。由图12（b）可知，施加控制后，颤振频率和强迫振动频率处的位移都得到了一定程度的衰减，且对刀齿通过频率的3倍频和5倍频抑制更为显著。从外部激振实验结果可知，PD控制下，电磁作动器可降低转子悬臂端振动，减振效果良好。
综上所述，在转子实验平台上进行的转子外部激振振动抑制实验表明，电磁作动器可明显降低转子悬臂端的振动。由此推断，集成电磁作动器的主轴-刀具系统可降低铣削加工过程中的强迫振动，提高加工系统的稳定性。
5　结论
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为抑制铣削强迫振动，提高加工系统的稳定性，本文提出了一种集成电磁作动器的主轴-刀具系统，利用电磁作动器的可控电磁力对刀柄进行振动抑制。首先，介绍了其主要结构和工作原理，建立了基于电磁作动器的主轴-刀具系统动力学模型，并设计了PD控制器；其次，利用MATLAB/Simulink对所搭建的动力学模型进行了铣削时域仿真分析，验证了控制器的抑制效果；最后，在实验室所搭建的转子实验台上进行了铣削振动信号的激振抑制实验，并对振动信号进行了有效抑制。
所提出的集成电磁作动器的主轴-刀具系统在PD控制下能有效抑制铣削强迫振动。后续将考虑主轴-刀具系统的动力学特性，并针对模型设计控制器以抑制铣削加工过程中的颤振。
基金
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  • 国家重大科技专项资助项目(J2019-Ⅶ-0001-0141)
文献
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2024年第37卷第11期
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doi: 10.16385/j.cnki.issn.1004-4523.2024.11.001
  • 接收时间：2022-11-05
  • 首发时间：2026-02-12
  • 出版时间：2024-11-28
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  • 收稿日期：2022-11-05
  • 修回日期：2023-03-14
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国家重大科技专项资助项目(J2019-Ⅶ-0001-0141)
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    南京航空航天大学机电学院，江苏　南京　210016

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周瑾（1972—），女，博士，教授。E-mail：。
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2种不同金属材料的力学参数
科
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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