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Article(id=1228048675486761839, tenantId=1146029695717560320, journalId=1225147924628267009, issueId=1228048667874095618, articleNumber=null, orderNo=null, doi=10.16385/j.cnki.issn.1004-4523.2024.03.009, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=null, receivedDate=1655740800000, receivedDateStr=2022-06-21, revisedDate=1666713600000, revisedDateStr=2022-10-26, acceptedDate=null, acceptedDateStr=null, onlineDate=1770719220276, onlineDateStr=2026-02-10, pubDate=1711555200000, pubDateStr=2024-03-28, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1770719220276, onlineIssueDateStr=2026-02-10, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1770719220276, creator=13701087609, updateTime=1770719220276, updator=13701087609, issue=Issue{id=1228048667874095618, tenantId=1146029695717560320, journalId=1225147924628267009, year='2024', volume='37', issue='3', pageStart='365', pageEnd='538', issueExtLink='null', onlineDate='null', pubDate='null', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=null, createTime=1770719218462, creator=13701087609, updateTime=1770795476854, updator=13701087609, preIssue=null, nextIssue=null, ext={EN=IssueExt(id=1228368518803030940, tenantId=1146029695717560320, journalId=1225147924628267009, issueId=1228048667874095618, language=EN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=), CN=IssueExt(id=1228368518803030941, tenantId=1146029695717560320, journalId=1225147924628267009, issueId=1228048667874095618, language=CN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=)}, issueFiles=null}, startPage=448, endPage=456, ext={EN=ArticleExt(id=1228048675788751738, articleId=1228048675486761839, tenantId=1146029695717560320, journalId=1225147924628267009, language=EN, title=Vibration characteristics analysis of multi-dimensional vibration isolation system with joint clearance, columnId=null, journalTitle=Journal of Vibration Engineering, columnName=null, runingTitle=null, highlight=null, articleAbstract=

Aiming at reducing multi-dimensional vibration experienced by vehicle-mounted precise instrument，a multi-dimensional passive vibration isolator is built based on parallel mechanism with joint clearance. The 4-PUU parallel mechanism with axes 45° offset value，which exhibits three translations and one rotation characteristics is synthesized by GF set type synthesis theory. The springs and viscous dampers are added on the active joints，meanwhile the kinematics and dynamics of the multi-dimensional vibration isolator are established. The vibration isolation capability with different values of joint clearance under harmonic and road random excitations is addressed. The results demonstrate that the proposed multi-dimensional isolator with joint clearance can inhibit multi-dimensional vibration in time and frequency domain significantly. As the value of joint clearance increasing，the vibration isolation capability degenerates，especially in x direction. Meanwhile，the first order resonance peak is sensitive to joint clearance，which shifts to low frequency range.

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为隔离车载精密仪器承受的多维振动，引入基于并联机构且考虑机构关节间隙的多维被动隔振系统。通过集 （Generalized Function Set） 型综合理论，得到一种具有三平移一转动运动特性的4-PUU （移动副-虎克副-虎克副） 并联机构，在主动关节处添加弹簧和黏滞阻尼器，建立考虑关节间隙的多维隔振系统运动学和动力学方程，研究存在关节间隙的多维隔振系统在谐波、随机路面激励下的时域、频域隔振能力。研究结果表明：存在关节间隙的多维隔振系统能够实现多维振动在时域和频域中的有效衰减；随着关节间隙数值逐渐增大，隔振能力沿x轴方向变化最敏感，沿y轴方向变化最不敏感；系统第一阶共振峰对关节间隙变化敏感，向低频区域移动，隔振能力恶化。

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高翔（1989—），男，博士，讲师。E-mail： 。

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高翔（1989—），男，博士，讲师。E-mail： 。

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高翔（1989—），男，博士，讲师。E-mail： 。

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3T1R mechanism configuration with symmetrical structure

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机构构型
4-PPPR4-PR1PR1R2
4-PPRR4-R1R1R1R2R2
4-PRRR4-R1PR1R2R2
4-PR1R2PP4-PUR1R2
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具有对称结构的三平移一转动机构构型

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机构构型
4-PPPR4-PR1PR1R2
4-PPRR4-R1R1R1R2R2
4-PRRR4-R1PR1R2R2
4-PR1R2PP4-PUR1R2
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Parameters and values of the multi-dimentional isolation system［28］

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参数取值
动平台质量/kg15.8
绕轴转动惯量/()
定平台边长/m0.4
定平台边长/m0.4
动平台边长/m0.3
动平台边长/m0.3
无关节间隙连杆长/m0.1
弹簧刚度/()
黏滞阻尼系数/()
轴的杨氏模量/GPa210
轴套的杨氏模量/GPa200
轴的泊松比0.3
轴套的泊松比0.1
下截止频率/
参考空间频率/
路面不平度系数/
车速/()
), ArticleFig(id=1228048726506275379, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228048675486761839, language=CN, label=表2, caption=

多维隔振系统的主要参数和数值［28］

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参数取值
动平台质量/kg15.8
绕轴转动惯量/()
定平台边长/m0.4
定平台边长/m0.4
动平台边长/m0.3
动平台边长/m0.3
无关节间隙连杆长/m0.1
弹簧刚度/()
黏滞阻尼系数/()
轴的杨氏模量/GPa210
轴套的杨氏模量/GPa200
轴的泊松比0.3
轴套的泊松比0.1
下截止频率/
参考空间频率/
路面不平度系数/
车速/()
), ArticleFig(id=1228048726581772853, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228048675486761839, language=EN, label=Tab.3, caption=

RMS values of output velocity in all directions with joint clearance

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关节间隙/mm沿轴/(m·s-1)沿轴/(m·s-1)沿轴/(m·s-1)绕轴/(rad·s-1)
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存在关节间隙时各方向的输出速度均方根值

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关节间隙/mm沿轴/(m·s-1)沿轴/(m·s-1)沿轴/(m·s-1)绕轴/(rad·s-1)
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考虑关节间隙的多维隔振系统振动特性分析
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高翔 1 , 石峰 2 , 牛军川 3 , 董瑞春 1
振动工程学报 | 2024,37(3): 448-456
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振动工程学报 | 2024, 37(3): 448-456
考虑关节间隙的多维隔振系统振动特性分析
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高翔1 , 石峰2, 牛军川3, 董瑞春1
作者信息
  • 1山东理工大学机械工程学院, 山东　淄博　255000
  • 2中铝万成山东建设有限公司, 山东　淄博　255000
  • 3山东大学机械工程学院, 山东　济南　250061

    • 高翔（1989—），男，博士，讲师。E-mail： 。

Vibration characteristics analysis of multi-dimensional vibration isolation system with joint clearance
Xiang GAO1 , Feng SHI2, Jun-chuan NIU3, Rui-chun DONG1
Affiliations
  • 1School of Mechanical Engineering, Shandong University of Technology, Zibo 255000, China
  • 2CHINALCO Wancheng Shandong Construction Co., Ltd., Zibo 255000, China
  • 3School of Mechanical Engineering, Shandong University, Jinan 250061, China
出版时间: 2024-03-28 doi: 10.16385/j.cnki.issn.1004-4523.2024.03.009
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摘要
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为隔离车载精密仪器承受的多维振动，引入基于并联机构且考虑机构关节间隙的多维被动隔振系统。通过集 （Generalized Function Set） 型综合理论，得到一种具有三平移一转动运动特性的4-PUU （移动副-虎克副-虎克副） 并联机构，在主动关节处添加弹簧和黏滞阻尼器，建立考虑关节间隙的多维隔振系统运动学和动力学方程，研究存在关节间隙的多维隔振系统在谐波、随机路面激励下的时域、频域隔振能力。研究结果表明：存在关节间隙的多维隔振系统能够实现多维振动在时域和频域中的有效衰减；随着关节间隙数值逐渐增大，隔振能力沿x轴方向变化最敏感，沿y轴方向变化最不敏感；系统第一阶共振峰对关节间隙变化敏感，向低频区域移动，隔振能力恶化。

多维隔振系统  /  并联机构  /  关节间隙  /  隔振性能

Aiming at reducing multi-dimensional vibration experienced by vehicle-mounted precise instrument，a multi-dimensional passive vibration isolator is built based on parallel mechanism with joint clearance. The 4-PUU parallel mechanism with axes 45° offset value，which exhibits three translations and one rotation characteristics is synthesized by GF set type synthesis theory. The springs and viscous dampers are added on the active joints，meanwhile the kinematics and dynamics of the multi-dimensional vibration isolator are established. The vibration isolation capability with different values of joint clearance under harmonic and road random excitations is addressed. The results demonstrate that the proposed multi-dimensional isolator with joint clearance can inhibit multi-dimensional vibration in time and frequency domain significantly. As the value of joint clearance increasing，the vibration isolation capability degenerates，especially in x direction. Meanwhile，the first order resonance peak is sensitive to joint clearance，which shifts to low frequency range.

multi-dimensional vibration isolation system  /  parallel mechanism  /  joint clearance  /  vibration isolation performance
高翔, 石峰, 牛军川, 董瑞春. 考虑关节间隙的多维隔振系统振动特性分析. 振动工程学报, 2024 , 37 (3) : 448 -456 . DOI: 10.16385/j.cnki.issn.1004-4523.2024.03.009
Xiang GAO, Feng SHI, Jun-chuan NIU, Rui-chun DONG. Vibration characteristics analysis of multi-dimensional vibration isolation system with joint clearance[J]. Journal of Vibration Engineering, 2024 , 37 (3) : 448 -456 . DOI: 10.16385/j.cnki.issn.1004-4523.2024.03.009
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引言
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多维振动同时发生于多个方向，兼有移动和转动两种形式，在工程中普遍存在。以车载精密仪器为例，由于路面不平度和驾驶员的操作，车辆可能会同时产生横向、纵向、垂向的平移振动，以及侧倾、俯仰和横摆的旋转振动［1］。多维振动严重影响车载精密仪器的精度和可靠性，需构建具有适当自由度的多维隔振系统对其进行有效隔离。机械组合分层隔振［2］、浮筏［3］和磁浮隔振［4］等现有的车载设备多维隔振技术能够实现多维振动的衰减，同时也存在动力学特性不随外界激励的变化而改变、隔振元件变形小、一般仅隔离沿隔振器轴线方向的振动、结构复杂等不足。
并联机构因其具有刚度高、承载能力强、动力学特性与位姿密切相关、采用最少的主动关节实现多维运动输出等特点，成为构建多维隔振系统的优势对象［5-8］。YANG等［9］建立考虑派生刚度的六自由度多维隔振系统动力学方程，控制力由支链反力和位姿反馈综合产生，设计新型PI子控制器，从而实现隔振带宽和主动阻尼系数的独立调节。TANG等［10］采用Newton-Euler法建立Stewart并联隔振机构的关节动力学方程，基于分散控制理论，设计准线性控制器，实现了六轴多维振动的有效衰减。上述多维振动主动控制策略可实现多维振动的有效衰减，但同时也增加了设计和维护成本，如果失电，系统稳定性难以保证。多维振动被动控制方法因其具有低能耗、结构简单和良好的稳定性等优点得到了众多学者的广泛关注。WU等［11］基于Hamilton原理，建立“X”型多维隔振系统运动学模型，研究系统的等效刚度和位移传递率。牛军川等［12］设计了一种基于变胞并联机构的多维被动隔振装置，通过改变铰链轴线的配置方式调整机构的自由度，从而实现三自由度或四自由度多维减振。刘乃军等［13］通过闭环矢量法建立了基于3-PRC 机构的多维被动减振平台模型，采用自适应遗传算法对弹簧刚度进行优化，从而实现减振平台的调频目标。文献［13］中采用的3-PRC机构共有3条对称支链，自定平台至动平台，运动副构成形式依次为移动副 （Prismatic Pair，P副），转动副 （Revolute Joint，R副） 和圆柱副 （Cylindrical Pair，C副）。文中其他有关并联机构的简写具有类似的表达方式，为表达简洁，后续仅给出并联机构运动副的中文解释。
上述研究表明基于并联机构的多维隔振系统的理论可行性。然而，在并联机构制造和装配等实际环节中，引入的关节间隙不能忽略。关节间隙对多维隔振系统振动特性的影响规律尚不明确。针对存在机构关节间隙的并联机构运动学和动力学模型，已有学者展开研究，可为本文含间隙隔振系统的建模和振动特性分析提供参考。
王见等［14］建立了同时考虑关节间隙和连杆柔性的3-CPaRR （圆柱副-复合移动副-转动副-转动副）并联机构动力学模型，轴和轴套间的法向和切向碰撞力分别由Lankarani-Nikravesh模型和Coulomb摩擦模型计算得到。研究表明，关节间隙和连杆柔性对并联机构运动学特性影响显著，导致动平台输出速度存在波动现象。ZHANG等［15］提出考虑关节间隙润滑的3-PRR （移动副-转动副-转动副） 并联机构动力学响应评估方法，轴和轴套间的碰撞力由Pinkus-Sternlicht模型计算得到。因存在关节间隙，轴的中心运动轨迹为不规则的圆形，且轴和轴套之间的碰撞力实时可变。CHEN等［16］建立了存在关节间隙的冗余驱动并联机构动力学模型，研究球铰关节间隙对并联机构动力学特性的影响规律。研究表明，随着球铰关节间隙的增大，动平台输出速度的波动现象愈发明显，输出运动精度不断降低。
本文考虑基于并联机构的多维隔振系统制造和装配环节中的关节间隙问题，结合间隙关节的运动学和动力学模型以及并联机构的输入、输出特性，建立考虑关节间隙的多维被动隔振系统运动学和动力学方程，分别在时域和频域研究关节间隙对隔振性能的影响规律，研究结果更加符合实际情况，可为基于并联机构的多维隔振系统设计提供参考。
1　含关节间隙的多维隔振系统建模
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1.1　多维隔振系统模型
实际车辆振动难以同时兼具六个自由度，通常为车辆全部振动状态中一项或者几项的叠加。在此选择具有代表性的横向、纵向、垂向和俯仰方向的多维振动作为外界激励，基于集型综合理论得到具有三平移一转动 （Three Translations and One Rotation，3T1R） 运动特性的少自由度并联机构作为多维隔振系统的主体。并联机构的末端运动特征可由各支链的运动特征求交计算得到［17］，即：
式中　 为并联机构的末端运动特征，为每条支链的运动特征，。
由于并联机构的主动副可安装在不同支链上，因此，需对并联机构进行数综合。数综合方程［18-19］可表示为：
式中　 为集的维数；和分别为支链数和主动支链数；为被动支链数；为第条主动支链上的主动副个数。
由于所选机构的运动特性为3T1R，且无被动支链和冗余驱动，其自由度为4，式 （2） 中的具体参数可表示为：
考虑到机构承载能力稳定和便于制造安装等因素，在此选择具有对称支链结构的并联机构作为多维隔振系统主体。那么根据集求交规则，可得到具有四条对称支链的3T1R并联机构末端特征为：
式中　 表示平动；下标 分别为任意两坐标轴不平行且不共面的沿三条坐标轴的平动。表示转动；下标分别为相交于同一点的绕三条坐标轴的转动。
根据式 （4），部分具有四条对称支链的机构构型如表1所示。
表1中，P表示移动副，R1和R2表示回转轴线不同的转动副，U表示虎克副。为说明表1中各机构构型的含义，以4-PPPR机构为例，4表示机构共有四条对称支链，自定平台至动平台，支链中的运动副依次为移动副-移动副-移动副-转动副。根据空间轴迁移定律，支链中的转动副可任意布置，其派生机构未列出。对于多维隔振系统，需在移动副上安装弹簧和黏滞阻尼器，在此选择机构构型为4-PUR1R2。虎克副由两个不同轴线的转动副组合而成，为叙述方便，定义所选机构为4-PUU （移动副-虎克副-虎克副）机构。
多维隔振系统的主体分别由动平台、定平台和四条对称支链构成。定平台和动平台均为矩形，其边长分别为和，为动平台绕轴的转角。从定平台到动平台，每条支链依次由移动副-虎克副-虎克副构成，每个虎克副中的两条转动轴线呈45°偏置。在移动副上安装弹簧和黏滞阻尼器，分别在动平台和定平台建立动坐标系和定坐标系，构造的多维隔振系统如图1（a）所示。图1（a）中，分别为与阻尼器相连接的虎克副，分别为与动平台相连接的虎克副。为更清晰地展示各支链关节的位置关系，将图1（a）中的Ⅰ，Ⅱ两处细节做局部放大，得到图1（b）。如前所述，虎克副可视为两个转动副的合成形式：图1（b）中Ⅰ处，转动副轴线和呈45°夹角，转动副轴线和呈45°夹角，转动副轴线和沿竖直方向；图1（b）中Ⅱ处，转动副轴线和呈45°夹角，转动副轴线和呈45°夹角。上述转动副两两之间均存在关节间隙。由于机构的对称特性，另外两条支链的布置形式与Ⅰ，Ⅱ两处相同。阻尼器为普通黏滞阻尼器，弹簧与阻尼器呈串联关系。基于4-PUU并联机构的多维隔振系统自由度为4，可通过修正的G-K公式［20］进行验算。由于虎克副为轴线偏置的两转动副组合而成，在下文中重点考虑具有转动副间隙的多维隔振系统运动学和动力学模型。
1.2　存在关节间隙的多维隔振系统运动学
为简化计算，通常认为轴和轴套在初始位置同心，关节间隙限定了轴的运动轨迹位于轴套的边界之内［21］。那么在初始位置，转动副关节间隙为：
式中　 和分别表示轴和轴套的半径向量。
在工作状态，轴和轴套产生相对随机运动，直到两者发生碰撞。由于轴和轴套的轴向长度接近，在此忽略轴线方向的倾斜碰撞。定义轴和轴套径向局部坐标系和径向总体坐标系分别为和，和分别为轴和轴套的中心，那么轴和轴套的相对运动示意图如图2所示。
在总体坐标系中，轴和轴套中心的偏心量可表示为：
式中　 和分别为轴和轴套在总体坐标系中的位姿向量。
那么，轴和轴套之间的相对运动关系可由撞入深度表示：当时，表示自由运行模式；当时，表示碰撞或者分离的临界状态；当时，表示碰撞模式。
当多维隔振系统动平台处于某一特定位姿时，分别表示点在动坐标系和定坐标系中的坐标，二者具有如下关系：
式中　 为绕轴的变换矩阵，具体形式为：
那么不考虑关节间隙，点和点之间的距离可表示为：
于是可得到该并联机构的运动学逆解：
式中　 。
对于上述多维隔振系统，点通过转动副 （虎克副） 与动平台连接，则根据图2，在发生碰撞时，点的位姿向量可表示为：
式中　 表示偏心量的单位向量，为碰撞的法向。
式（11）两端对时间求导，得到碰撞时的速度向量：
那么法向和切向的速度向量可分别表示为：
式中　 ， 为碰撞的切向，是单位向量。
对于另一连接点，位姿向量（），速度向量（）与式（11）和（12）具有类似的表达形式。那么考虑关节间隙时，点和点之间的实际距离为：
根据并联机构的输入、输出关系，计算得到存在关节间隙的机构Jacobian逆矩阵：
式中　 ，。
1.3　存在关节间隙的多维隔振系统动力学
当轴和轴套在无润滑的情况下发生碰撞时，产生的碰撞力将传递到机械系统［22］。碰撞力可表示为撞入深度的分段函数［23］：
式中　 和分别表示法向力向量和切向力向量。
法向力的幅值可由Lankarani-Nikravesh（L-N）模型表述［24-25］：
式中　 为相对撞入速度；为初始碰撞速度；为恢复系数；对于大多数金属材料碰撞，指数取1.5；为碰撞刚度，依赖于材料和碰撞体的几何参数，具体可表示为：
其中，系数的具体形式分别为：
式中　 和分别表示轴和轴套的泊松比和杨氏模量。
切向力的幅值可由修正的Coulomb摩擦力模型表示［26-27］：
式中　 为摩擦系数；为修正系数，可表示为：
式中　 和为切向速度的边界值。
由于并联机构的耦合特性，在建立并求解隔振系统动力学方程时应作如下简化：（1）动平台在平衡位置附近做微幅振动；（2）忽略各连杆的质量，仅考虑动平台的质量。那么存在关节间隙的多维隔振系统动力学模型可由Lagrange方程得到：
式中　 为系统动能，为系统弹性势能，为系统阻尼耗散能；为广义坐标；为含有碰撞力的广义力向量。为系统的惯量矩阵，其中和分别为动平台的质量和绕轴的转动惯量；和分别为隔振系统的刚度矩阵和阻尼矩阵，其中和分别为弹簧刚度和黏滞阻尼系数；（Jacobian逆矩阵的转置） 为Jacobian力矩阵。
考虑关节间隙的多维隔振系统动力学方程可由式（22）整理得到：
2　含关节间隙的多维隔振系统仿真
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存在关节间隙的多维隔振系统仿真分析中所用到的主要参数和数值如表2所示。
2.1　存在关节间隙的多维隔振系统时域响应
在此定义为多维隔振系统的状态变量，式 （23） 可改写为状态空间形式：
式中　 A，Ba和B分别为系统矩阵、碰撞力传递矩阵和输入矩阵，分别表示为：
为研究关节间隙变化对时域隔振性能的影响规律，避免未知频率成分对时域隔振性能的影响，选取具有确定频率成分的谐波激励作为复合外界振动激励，加速度方程为，作用方向沿轴和绕轴。式（24）可由Runge-Kuta方法求解，存在关节间隙的多维隔振系统在复合谐波激励下的时间历程如图3所示。
图3为存在关节间隙时动平台在复合谐波激励下各方向的时域速度输出。为清晰地表示速度输出特性，谐波激励图像未添加到图3中。由图3可得，存在关节间隙时的输出速度变化趋势与无关节间隙时的输出速度变化趋势基本一致。随着关节间隙逐渐增大，动平台的速度输出幅值同步增大。该现象表明，多维隔振系统的隔振能力随着关节间隙的增大而逐渐恶化。由碰撞产生的法向力和切向力传递到多维隔振系统，此时，附加的碰撞力会抵消一部分阻尼力，这是导致隔振性能退化的重要原因。在各速度峰值处，此时轴和轴套的动能也处于极大值，将产生较大的法向和切向碰撞力，导致在速度峰值处产生明显的波动现象。由于轴和轴套之间的随机运动，碰撞位置和碰撞力具有显著的不确定性。
为量化不同关节间隙大小对隔振性能的影响程度，选取动平台输出速度的均方根（RMS）值作为评价指标。
表3为存在关节间隙时动平台在不同方向的输出速度均方根值。由于并联机构的耦合特性，在某一方向激励，在所有方向均会产生输出，但其余方向的输出幅值均显著小于激励方向的输出幅值。因此，在评价多维振动响应时，可忽略由于耦合效应引起的其余方向的输出值。对振动加速度激励进行积分计算，得到速度均方根值为或0.543 。与振动激励速度均方根值相比，考虑和未考虑关节间隙的多维隔振系统在各方向上均能够实现多维振动的有效减弱。在不同关节间隙数值影响下，经过计算，沿轴线方向和绕轴方向的速度均方根值最大变化率分别为80%，31.4%，48.2%和53.3%，该变化率可表示隔振能力随关节间隙变化的灵敏度。上述结果表明，隔振性能随关节间隙的变化沿轴方向最敏感，沿轴方向最稳定。
2.2　存在关节间隙的多维隔振系统频域响应
假设多维隔振系统固定于车辆中，定平台承受由路面不平度产生的随机激励［29］：
式中　 为路面随机激励幅值；为路面不平度系数；为车辆行驶速度；和分别为下截止频率和参考空间频率；为均值为零的高斯白噪声。参数的具体取值如表2所示。
定义矩阵，对式（24）取Fourier变换，那么存在关节间隙的多维隔振系统动平台频域位移输出可表示为：
如果不考虑关节碰撞力，即忽略关节间隙的影响，那么式 （27） 将变为理想的被动隔振系统动平台频域位移输出形式：
式中　 为随机路面激励的频域形式。
图4为存在不同关节间隙时动平台的频域位移输出特性。为研究关节间隙对隔振能力的频域影响规律，同时给出无关节间隙的理想被动隔振系统频域输出曲线。结合表2中的数据，计算得到多维隔振系统的四阶固有频率分别为f1=3.82 Hz，f2=9.53 Hz，f3=16.98 Hz，f4=62.12 Hz。图4中，当黏滞阻尼系数为零时，多维隔振系统的四阶共振峰被完全激发，其横坐标值恰好与四阶固有频率对应，表明了理论计算的正确性。在理想的被动隔振模式下，系统的四阶共振峰均被显著抑制。随着关节间隙增大，对第一阶共振峰的抑制能力逐渐变弱，同时，所有方向的第一阶共振峰 （除沿向） 均有向低频区域移动的现象，其余三阶共振峰变化不明显。由于机构关节间隙的存在，Jacobian矩阵和关节碰撞力实时改变，致使第一阶共振峰对关节间隙变化较为敏感。关节间隙逐步增大对多维隔振系统隔振能力的影响规律，与增加隔振系统等效惯量或降低隔振系统等效刚度的规律大体一致。
3　结论
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本文建立了存在关节间隙的多维隔振系统运动学和动力学模型，由关节间隙引起的法向和切向碰撞力通过Jacobian力矩阵作用于多维隔振系统，分别在时域和频域内研究存在关节间隙的多维隔振系统隔振能力的变化规律，可得到如下结论：
（1） 基于4-PUU并联机构的多维隔振系统，在考虑和未考虑关节间隙影响时，均能够实现对多维振动的有效衰减。当存在关节间隙且关节间隙逐步增大时：隔振能力沿轴变化最为敏感，沿轴最稳定；第一阶共振峰对关节间隙变化最敏感，逐步向低频区域移动。
（2） 在速度峰值处，由于系统动能为极大值，导致关节碰撞力也为极大值，在速度峰值处具有明显的波动现象。由关节间隙导致的关节碰撞力通过时变Jacobian力矩阵添加到多维隔振系统中，抵消了部分阻尼力，这是导致存在关节间隙的多维隔振系统隔振能力退化的重要因素。
（3） 本文提出的考虑关节间隙的多维隔振系统建模和分析方法更加贴合实际情况，不仅适用于车载精密仪器减振，也可以推广到舰船设备、空天设备减振等其他多维振动抑制领域，为多维隔振系统的研究和应用提供了参考。
基金
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  • 山东省自然科学基金资助项目(ZR2021QE203)
  • 国家自然科学基金资助项目(52305119)
  • 国家自然科学基金资助项目(52075294)
  • 国家自然科学基金资助项目(52005305)
  • 国家自然科学基金资助项目(U22A20204)
文献
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2024年第37卷第3期
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doi: 10.16385/j.cnki.issn.1004-4523.2024.03.009
  • 接收时间：2022-06-21
  • 首发时间：2026-02-10
  • 出版时间：2024-03-28
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  • 收稿日期：2022-06-21
  • 修回日期：2022-10-26
基金
山东省自然科学基金资助项目(ZR2021QE203)
国家自然科学基金资助项目(52305119)
国家自然科学基金资助项目(52075294)
国家自然科学基金资助项目(52005305)
国家自然科学基金资助项目(U22A20204)
作者信息
    1山东理工大学机械工程学院, 山东　淄博　255000
    2中铝万成山东建设有限公司, 山东　淄博　255000
    3山东大学机械工程学院, 山东　济南　250061
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