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单、双径向轴承安装时三叉式万向联轴器的附加弯矩分析
李松梅,李帅帅,常德功     
青岛科技大学 机电工程学院, 山东 青岛 266061
摘要: 为找出三叉式万向联轴器在安装和使用过程中出现振动、噪声的原因,分析采用单、双径向轴承安装三叉式万向联轴器输出轴时联轴器附加弯矩随输入轴转角及偏转角的分布规律。首先建立2种安装方式下联轴器的运动简图,应用空间机构坐标变换技术建立输入轴与输出轴的关系方程,推导出系统存在偏转角时输出轴的运动方程,表明输出轴采用单径向轴承安装时三叉式万向联轴器为准等角速传动,采用双径向轴承安装时其具有等角速传动特性。其次分析这2种安装方式下作用在输入轴和输出轴上的附加弯矩分量,根据虚位移原理确定三叉式万向联轴器输出轴在2种安装方式下系统均出现附加弯矩。通过分析发现:采用单径向轴承安装时,存在偏转弯矩,且一个运动周期内附加弯矩的波动频率是输入轴的3倍,呈正弦曲线变化,且附加弯矩的变化趋势为随偏转角的增大呈直线上升;双径向轴承安装时,偏转弯矩为零,附加弯矩随偏转角的增大而逐渐增大,未出现波动。结果表明:2种安装方式下三叉式万向联轴器上均产生附加弯矩,从而使系统产生振动效应。该结果对研究系统产生振动的原因、确定系统非线性动力学行为具有重要的意义。
关键词: 三叉式万向联轴器     单、双径向轴承     偏转弯矩     附加弯矩    

基金项目: 国家自然科学基金资助项目(50975147);山东省科技发展计划项目(2013GGX10305);山东省自然科学基金资助项目(ZR2015EM037,ZR2016EEP12)
Additional bending moment analysis of the trigeminal universal joint installed by single or twin radial bearing
LI Song-mei, LI Shuai-shuai, CHANG De-gong     
College of Electromechanical Engineering, Qingdao University of Science and Technology, Qingdao 266061, China
Abstract: To find out the causes of vibration and noise of the trigeminal universal joint which appeared in the installation and use process, the additional moment distribution with the varies of the input shaft angular and the deflection angle was analyzed. Kinematics diagrams of the trigeminal universal joint when the output shaft installed by single radial bearing or twin radial bearing were established. The system coordinates were established under the two installation ways, and the motion equations were established by the direction cosine matrix tools. It indicated that the trigeminal universal joint installed by the single radial bearing was an approximately isometric speed transmission, and was a constant velocity transmission with twin radial bearing installed. Furthermore, the additional moment component on the input shaft and the output shaft was analyzed under the two installation ways. According to the virtual displacement principle, the additional moment on the tripod universal joint when the output shaft installed by twin radial bearing and single radial bearing were determined. When installed by single radial bearing, the deflection moment exists and the vibration frequency of additional moment was three times of the input shaft as a sine curve. The variation trend of the additional moment was straight up with the increase of deflection angle. When installed by twin radial bearing, the deflection moment was zero and the additional moment increased gradually with the increase of deflection angle but did not fluctuate. The analysis revealed that additional moment existed in the trigeminal universal joint system under both installation ways would produce bending vibration. This study is of great significance to study the causes of vibration and determine the nonlinear dynamics of the system.
Key words: trigeminal universal joint     single & twin radial bearing     deflection moment     additional bending moment    

三叉式万向联轴器的结构组成简单、紧凑,安装维修便利,同步性高,传输能力大,易于热处理,加工制造成本低廉。其不仅能起到连接各个转子,传递扭矩和补偿轴向、角向、径向位移的作用,且在输入轴和输出轴存在偏转角时传输能力随偏转角增大而降低,实现平稳传动。

三叉式万向联轴器是一种新生结构,国内外学者虽只进行了初步的理论分析,但已看到了其广阔的应用前景,因此为推广应用需要对其进行深入的理论研究,尤其是在不同安装方式下的运动学、动力学分析仍缺乏系统的理论指导。国外学者Mariot和Serveto等[1-2]分析了三柱销式万向节的运动学特性与动力学特性,得出当输出轴轴线方向不变时三柱销式万向节属于等角速万向联轴器。Desmidt[3]对非等角速万向联轴器传动系统的稳定性进行了理论分析。Cai等[4]建立了三柱销式万向联轴器的简化动力学模型,对产生的轴向力的变化进行了分析。Sai等[5]将三球销式万向联轴器作为空间机构对组件间相对运动特性进行了理论分析,同时开发出一种数值分析程序模拟了套筒的法向力及轴向力,并进行了实验验证。朱拥勇等[6]采用传递矩阵法对万向联轴器传动系统进行理论分析和数值求解,得到了存在偏转角时系统的振动特性。王学锋等[7-9]利用数值仿真分析方法对滑移型万向联轴器在调心球轴承安装时进行了部分理论分析和数值求解。

三叉式万向联轴器的输出轴采用双径向轴承安装是解决三叉式万向联轴器准等角速问题的一个切实可行的方案[10]。本文研究了三叉式万向联轴器存在偏转角时,采用单径向轴承安装和双径向轴承安装时产生的附加弯矩,找出三叉式万向联轴器在安装过程中出现振动、噪声的原因,确定系统的动力学行为。

1 三叉式万向联轴器运动分析

三叉式万向联轴器包括输入轴、三叉杆、输出轴、轴套、滑块组件等[10],轴套上均布有3个平行的导向槽,轴套的一端连接输入轴,另一端通过3组滑块组件连接三叉杆,三叉杆中心与输出轴固定连接[11-12],其三维结构如图 1所示。

1—输入轴;2—轴套;3—滑块组件;4—三叉杆;5—输出轴 图 1 三叉式万向联轴器的三维结构图 Fig.1 The three-dimensional structure diagram of the trigeminal universal joint

建立采用单径向轴承和双径向轴承安装三叉式万向联轴器输出轴时系统的运动简图, 如图 2图 3所示。

图 2 采用单径向轴承安装时三叉式万向联轴器的运动简图 Fig.2 The kinematic diagram of the trigeminal universal joint installed by the single radial bearing
图 3 采用双径向轴承安装时三叉式万向联轴器的运动简图 Fig.3 The kinematic diagram of the trigeminal universal joint installed by twin radial bearing

采用空间机构坐标变换技术[13-16]建立三叉式万向联轴器输入轴、输出轴的运动关系方程,推导出系统存在偏转角时输出轴的运动方程[17-18]φi为输入轴转角,φo为输出轴转角,输入轴和输出轴之间的偏转角为γ,得到单径向轴承安装输出轴时系统的转角差方程为:

$ {\varphi _{\rm{o}}}-{\varphi _{\rm{i}}} \approx \frac{\lambda }{{2L}}{\rm{ta}}{{\rm{n}}^2}\frac{\gamma }{2}{\rm{tan}}\gamma {\rm{cos}}3{\varphi _{\rm{i}}} $ (1)

双径向轴承安装输出轴时系统的转角差为:

$ {\varphi _{\rm{o}}}-{\varphi _{\rm{i}}} = 0 $ (2)

当采用单径向轴承安装三叉式万向联轴器的输出轴时,系统出现了转角差,可见该安装方式下联轴器传动不属于等角速传动,但因为转角差值很小,特别是当存在微小的偏转角时,转角差几乎为零。所以,输出轴采用单径向轴承安装时三叉式万向联轴器输出轴绕中心轴做微小的圆锥运动,这时的传动可以称为准等角速传动。当采用双径向轴承安装三叉式万向联轴器的输出轴时,未出现转角差,输出轴绕中心轴做微小的圆柱运动,表明采用这种安装方式的三叉式万向联轴器具有等角速传动特性。

2 三叉式万向联轴器附加弯矩

三叉式万向联轴器的输入轴和输出轴之间的偏转角为γ,假设在输入轴Wi上作用的扭矩为Ti,作用在输出轴Wo上的扭矩为To,当γ≠0时,其扭矩的合矢量一般不等于零,因而在输入和输出轴上出现附加弯矩。当给定输入转矩Ti后,便可求出三叉式万向联轴器输入轴及输出轴某横截面S1S2上作用的附加弯矩分量U1V1U2V2,如图 4所示。

图 4 三叉式万向联轴器的附加弯矩示意图 Fig.4 Additional bending moment diagram of the trigeminal universal joint

由于外部只作用纯扭矩,所以作用在横截面S1S2上的各力矩由三叉式万向联轴器的平衡条件可知:

$ \left\{ \begin{array}{l} {X_{\rm{o}}}:{\mathit{\boldsymbol{U}}_1} + {\mathit{\boldsymbol{U}}_2} = 0\\ {Y_{\rm{o}}}:-{\mathit{\boldsymbol{T}}_{\rm{i}}}{\rm{sin}}\gamma + {\mathit{\boldsymbol{V}}_1}{\rm{cos}}\gamma + {\mathit{\boldsymbol{V}}_2} = 0\\ {Z_{\rm{o}}}:{\mathit{\boldsymbol{T}}_{\rm{i}}}cos\gamma + {\mathit{\boldsymbol{V}}_1}sin\gamma + {\mathit{\boldsymbol{T}}_{\rm{o}}} = 0 \end{array} \right. $ (3)

从而得到:

$ \left\{ \begin{array}{l} {\mathit{\boldsymbol{V}}_1} = \frac{{-{\mathit{\boldsymbol{T}}_{\rm{o}}}-{\mathit{\boldsymbol{T}}_{\rm{i}}}{\rm{cos}}\gamma }}{{{\rm{sin}}\gamma }}\\ {\mathit{\boldsymbol{V}}_2} = {\mathit{\boldsymbol{T}}_{\rm{i}}}{\rm{sin}}\gamma + \frac{{{\mathit{\boldsymbol{T}}_{\rm{o}}} + {\mathit{\boldsymbol{T}}_{\rm{i}}}{\rm{cos}}\gamma }}{{{\rm{sin}}\gamma }}{\rm{cos}}\gamma \end{array} \right. $ (4)

根据虚位移原理确定U1U2,假想将三叉式万向联轴器输出轴Wo固定,而将输入轴WiOoZo轴转过dγ角,并绕自身转过dβ角,此时由于矢量V1Ti与dγ互相垂直,故力矩TiV1对dγ角所作的功为零。因为只有U1作功,其大小为Ti·dβ。在不考虑机械损失的情况下,总的虚功和为0,即-U1·dγ+Ti·dβ=0,所以:

$ \left\{ \begin{array}{l} {\mathit{\boldsymbol{U}}_1} = {\mathit{\boldsymbol{T}}_{\rm{i}}}\cdot\frac{{{\rm{d}}\beta }}{{{\rm{d}}\gamma }}\\ {\mathit{\boldsymbol{U}}_2} =-{\mathit{\boldsymbol{T}}_{\rm{i}}}\cdot\frac{{{\rm{d}}\beta }}{{{\rm{d}}\gamma }} \end{array} \right. $ (5)
2.1 单径向轴承安装时的附加弯矩

输出轴采用单径向轴承安装时,有:

$ \frac{{{\rm{d}}{\varphi _{\rm{o}}}}}{{{\rm{d}}{\varphi _{\rm{i}}}}} = \frac{{{\omega _{\rm{o}}}}}{{{\omega _{\rm{i}}}}} = 1-\frac{{3\lambda }}{{2L}}{\rm{ta}}{{\rm{n}}^2}\frac{\gamma }{2}{\rm{tan}}\gamma {\rm{sin}}3{\varphi _{\rm{i}}} $ (6)
$ {\varphi _{\rm{o}}} = {\varphi _{\rm{i}}} + \frac{\lambda }{{2L}}{\rm{ta}}{{\rm{n}}^2}\frac{\gamma }{2}{\rm{tan}}\gamma {\rm{cos}}3{\varphi _{\rm{i}}} $ (7)

在不计摩擦损失的情况下,三叉式万向联轴器输入扭矩Ti和输出扭矩To在两轴转过dφ时所作的功相等,在任一时刻均有:

$ {\boldsymbol{T}_{\rm{o}}}\cdot{\rm{d}}{\varphi _{\rm{o}}} =-{\boldsymbol{T}_{\rm{i}}}\cdot{\rm{d}}{\varphi _{\rm{i}}} $ (8)
$ \frac{{{\boldsymbol{T}_{\rm{o}}}}}{{{\boldsymbol{T}_{\rm{i}}}}} = \frac{{-2L}}{{2L-3\lambda {\rm{ta}}{{\rm{n}}^2}\gamma 2{\rm{tan}}\gamma {\rm{cos}}3{\varphi _{\rm{i}}}}} $ (9)

代入式 (4),得:

$ {\boldsymbol{V}_1} = \frac{{2L({\rm{tan}}-{\rm{sin}}\gamma ) + 3\lambda {\rm{ta}}{{\rm{n}}^2}\gamma 2{\rm{tan}}\gamma {\rm{sin}}\gamma {\rm{sin}}3{\varphi _{\rm{i}}}}}{{2L{\rm{sin}}\gamma {\rm{tan}}\gamma + 3\lambda {\rm{ta}}{{\rm{n}}^2}\gamma 2{\rm{ta}}{{\rm{n}}^2}\gamma {\rm{sin}}\gamma {\rm{sin}}3{\varphi _{\rm{i}}}}}\cdot{\boldsymbol{T}_{\rm{i}}} $ (10)

根据输出轴采用单径向轴承安装时的转角误差公式可以得到:

$ \begin{array}{l} \frac{{{\rm{d}}\beta }}{{{\rm{d}}\gamma }} = \frac{{{\rm{d}}\left( {\frac{\lambda }{{2L}}{\rm{ta}}{{\rm{n}}^2}\frac{\gamma }{2}{\rm{tan}}\gamma {\rm{cos}}3{\varphi _{\rm{i}}}} \right)}}{{{\rm{d}}\gamma }} = \\ \frac{{\lambda {\rm{cos}}3{\varphi _{\rm{i}}}{\rm{tan}}\gamma {\rm{sin}}\gamma (2 + {\rm{sec}}\gamma )}}{{2L{{(1 + {\rm{cos}}\gamma )}^2}}} \end{array} $ (11)

将式 (11) 代入式 (5),得:

$ {\boldsymbol{U}_1} = \frac{{\lambda {\rm{cos}}3{\varphi _{\rm{i}}}{\rm{tan}}\gamma {\rm{sin}}\gamma (2 + {\rm{sec}}\gamma )}}{{2L{{(1 + {\rm{cos}}\gamma )}^2}}}\cdot{\boldsymbol{T}_{\rm{i}}} $ (12)

V1U1合成后得到采用单径向轴承安装时输入轴的附加弯矩为:

$ \begin{array}{l} {\boldsymbol{M}_1} = \sqrt {\boldsymbol{V}_1^2 + \boldsymbol{U}_1^2} = {\boldsymbol{T}_{\rm{i}}}\cdot\\ \sqrt {{{\left[ {\frac{{2L({\rm{tan}}\gamma - {\rm{sin}}\gamma ) + 3\lambda {\rm{ta}}{{\rm{n}}^2}\gamma 2{\rm{tan}}\gamma {\rm{sin}}\gamma {\rm{sin}}3{\varphi _{\rm{i}}}}}{{2L{\rm{sin}}\gamma {\rm{tan}}\gamma + 3\lambda {\rm{ta}}{{\rm{n}}^2}\gamma 2{\rm{ta}}{{\rm{n}}^2}\gamma {\rm{sin}}\gamma {\rm{sin}}3{\varphi _{\rm{i}}}}}} \right]}^2} + {{\left[ {\frac{{\lambda {\rm{cos}}3{\varphi _{\rm{i}}}{\rm{tan}}\gamma {\rm{sin}}\gamma (2 + {\rm{sec}}\gamma )}}{{2L{{(1 + {\rm{cos}}\gamma )}^2}}}} \right]}^2}} \end{array} $ (13)

假设输入扭矩为200 N·m,λ=30 mm,L=600 mm,γ=0°~40°,在输入轴旋转1个周期内,通过数值分析得到采用单径向轴承安装三叉式万向联轴器输出轴时输入轴上的附加弯矩如图 5所示。

图 5 输出轴采用单径向轴承安装时输入轴上的附加弯矩 Fig.5 Additional bending moment of the input shaft in the situation of output shaft installed by single radial bearing

图 5可以看出,偏转弯矩不为零,附加弯矩作用在矢量V1U1合成矢量与联轴器轴线构成的平面内,且1个运动周期内附加弯矩的波动频率是输入轴的3倍,呈正弦曲线变化,附加弯矩的变化趋势随偏转角的增大呈直线上升。

2.2 双径向轴承安装时的附加弯矩

系统输出轴采用双径向轴承安装,在不计摩擦损失的条件下,联轴器输入扭矩Ti和输出扭矩To在两轴转过dφ时所作的功相等,即Ti·dφi=-To·dφo,由于dφi=dφo所以Ti=-To。因此采用双径向轴承安装三叉式万向联轴器输出轴时,得到:

$ \left\{ \begin{array}{l} {\mathit{\boldsymbol{V}}_1} = \frac{{-{\mathit{\boldsymbol{T}}_i}-{\mathit{\boldsymbol{T}}_i}{\rm{cos}}\gamma }}{{{\rm{sin}}\gamma }} = {\mathit{\boldsymbol{T}}_i}{\rm{tan}}\frac{\gamma }{2}\\ {\mathit{\boldsymbol{V}}_2} = {\mathit{\boldsymbol{T}}_i}{\rm{sin}}\gamma + \frac{{{\mathit{\boldsymbol{T}}_i} + {\mathit{\boldsymbol{T}}_i}{\rm{cos}}\gamma }}{{{\rm{sin}}\gamma }}{\rm{cos}}\gamma = {\mathit{\boldsymbol{T}}_i}{\rm{tan}}\frac{\gamma }{2} \end{array} \right. $ (14)

根据三叉式万向联轴器输出轴采用双径向轴承安装时的等角速传动特性,当偏转角γ发生变化时,转角差恒为零,即dβ/dγ=0,因此:

$ \left\{ \begin{array}{l} {\boldsymbol{U}_1} = 0\\ {\boldsymbol{U}_2} = 0 \end{array} \right. $ (15)

所以当采用双径向轴承安装三叉式万向联轴器输出轴时,三叉式万向联轴器的附加弯矩位于输入轴和输出轴所在的平面内,弯矩矢量V1V2分别垂直于输入、输出轴,偏转弯矩U1U2等于零。

V1U1合成后得到采用双径向轴承安装时输入轴的附加弯矩:

$ {\boldsymbol{M}_1} = \sqrt {\boldsymbol{V}_1^2 + \boldsymbol{U}_1^2} = {\boldsymbol{T}_{\rm{i}}}{\rm{tan}}\frac{\gamma }{2} $ (16)

假设输入扭矩为200 N·m,λ=30 mm,L=600 mm,γ=0°~40°,在输入轴旋转1个周期内,通过数值分析得到采用双径向轴承安装时输入轴的附加弯矩如图 6所示。

图 6 输出轴采用双径向轴承安装时输入轴上的附加弯矩 Fig.6 Additional bending moment of the input shaft in the situation of output shaft installed by twin radial bearing

图 6可以看出,采用双径向轴承安装三叉式万向联轴器的输出轴时,联轴器的附加弯矩随偏转角的增大呈线性升高,在输入轴旋转的1个周期内未出现波动。

比较输出轴采用单径向轴承和双径向轴承两种安装方式时三叉式万向联轴器的附加弯矩,可以发现:采用双径向轴承安装时,联轴器的附加弯矩矢量作用在联轴器两轴线构成的平面内,偏转弯矩为零,附加弯矩随偏转角的增大而逐渐升高,未出现波动;而在采用单径向轴承安装时,联轴器的附加弯矩作用在矢量V1U1的合矢量与联轴器轴线构成的平面内,存在偏转弯矩,且1个运动周期内附加弯矩的波动频率是输入轴的3倍,呈正弦曲线变化,且附加弯矩的变化趋势随偏转角的增大呈直线上升。

3 结论

1) 研究发现三叉式万向联轴器输出轴在采用单径向轴承安装和双径向轴承安装时系统均产生附加弯矩。采用单径向轴承安装时,存在偏转弯矩,且1个运动周期内附加弯矩的波动频率是输入轴的3倍,呈正弦曲线变化,且附加弯矩的变化趋势随偏转角的增大呈直线上升;采用双径向轴承安装时,偏转弯矩为零,附加弯矩随偏转角的增大而逐渐升高,未出现波动。

2) 采用单径向轴承安装和双径向轴承安装时,三叉式万向联轴器上均产生附加弯矩,从而使传动系统产生振动效应。本文对在不同安装方式下联轴器的附加弯矩进行分析,为将来设计和使用三叉式万向联轴器提供重要的理论指导和参考价值,有望加速我国在汽车工业、橡胶工业、纺织工业、化工机械工业等相关领域内联轴器的更新换代,从而取得重大的经济效益和社会效益。

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http://dx.doi.org/10.3785/j.issn.1006-754X.2017.02.009
教育部主管,浙江大学和中国机械工程学会主办
0

文章信息

李松梅, 李帅帅, 常德功
LI Song-mei, LI Shuai-shuai, CHANG De-gong
单、双径向轴承安装时三叉式万向联轴器的附加弯矩分析
Additional bending moment analysis of the trigeminal universal joint installed by single or twin radial bearing
工程设计学报, 2017, 24(2): 182-186.
Chinese Journal of Engineering Design, 2017, 24(2): 182-186.
http://dx.doi.org/10.3785/j.issn.1006-754X.2017.02.009

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收稿日期: 2016-05-16

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