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行星减速器 由于 因为 对称性 独立 摆动

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行星减速器由于其对称性而独立摆动,行星减速器,由于,因为,对称性,独立,摆动

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【中文 2865 字】行星减速器由于其对称性而独立摆动俄罗斯莫斯科机械工程研究所俄罗斯科学院摘要：行星减速器的行星齿轮是对称的机械系统。适用于对称群的代表理论的振荡分析是属于广义的机械系统。结果发现,由于减速器的对称性,衍生出了振荡分解。减速器有独立的振荡类别，例如，太阳轮和本轮卫星轮角振荡阶段；太阳轮和本轮卫星横向振荡反阶段。太阳轮和本轮振荡中的一个阶段不依赖与角行星轮振荡。关键词：行星减速器，对称性，组代表性的理论，独立振荡一、导言众所周知，运作的行星减速器振荡的因素有太阳轮，本轮，和行星轮，这些因素基本上不利于减速器的运行，在某些情况下可能会导致其曲率的破坏。有大量的专门研究减速器齿轮动力学分析的文件。基本上都是理论研究，在已有的文件中介绍了研究减速器动力学的分析方法。行星减速器具有高度的对称性。因此，这个结论被广泛引用，组代表性的理论也适用。这一理论的应用允许用对称性对其展开深入的动态分析。为此，必须有一个能考虑到减速器各要素之间刚度联系的动力学模型。对称组代表性的理轮的数学仪器被广泛应用在量子力学，晶体，光谱【2，3，4】 。这种方法的优点是很难估算的。有他的帮助能够确定详尽完备的动力特性，使用结构对称的系统仅仅不能确定运动方程。然而，这一经典力学方法也不能被广泛使用。这是因为一些特殊的机械系统所具有的特点。首先，目前需要一个有 6 个自由度的刚体。不清楚的是要怎样放置这个刚体才能使系统的对称性稳定。 。第二，真正可能的是技术设计的错误和安装上的错误。所以即使一个小小的不对称也可能导致系统成为准对称系统。有各种对称组的多个子系统组成了机械系统。在这方面必须有方法，来分析有各种分系统和固体组成的对称机械系统和准对称机械系统。在取得了一些初步的进展后，包括数学仪器的机械系统可以使用。为此，我们提出申请广义操作。这些操作有适当的命令的矩阵，而不是表在物理。利用广义操作可以考虑到所有上述功能的机械系统。对初步的刚度矩阵实施这些操作，导致其分解为独立的模块，每个模块独立对应与自己的振荡级。考虑到固体对称相当与点输入：这些点选择在固体上，因此，它们彼此独立又互相关联，并且组成所有系统的对称组。这些做法也可以用于有限元模型。二、动态模型的行星减速器，刚度矩阵。该模型的行星减速器第一步如图 1 所示。该步骤包括由太阳轮，其质量和半径等于 M1，r1，它的周围有 3 个行星轮（它们的质量和半径均相等，都等于 m2,r2） 。行星轮和它们连接，并由它带动。齿轮传动装置的太阳轮，行星轮的刚度等于 h1 后，r 是角传动装置。图 1行星减速器的第一步。s-太阳轮，-本轮 1，2，3行星轮。我们再次考虑所有行星减速器振荡的步骤：横向（x,y）,角（j）振荡（不包括套管） 。一个刚度矩阵可以代表一个块这里主要有角刚度（3x3）采取适当的内容，和外面的主对角线的刚度，这些要素之间有联系。有 15 个广义坐标具体根据这些区块提附录。因此矩阵 k 是（15x15） ，一个惯性矩阵 m 是对角线矩阵。三、介绍相当于点动态模型。对称性操作凭借对称性行星轮紧固本轮系统已对称，如 3 架c（三角形） 。揭示对称 3 架 c 移动太阳齿轮 s 和本轮 Ep 我们将进入坐标L1，L2，L3 ,行星轮固定在太阳轮上的 s 点如图 2.将行星轮 1，2，3 分别固定在图上太阳轮所示位置。它们是等分点，它们的坐标分别是：或以矩阵形式写类似于本轮上的等分点，但是在图 2 中 r1 必须等于 r3。它们将协调太阳轮和本轮的 x,y,j。之后整个坐标系应对称与 c3。因此有可能适用于所有有 s,Ep,和三颗行星轮（i=1，2，3） 。 （如图 3）该邻正常投影算子克对称性点组 c3 被称为【2】它是对于整个系统的操作必须射影派块对角矩阵这里每个分矩阵对应与 s,Ep,和三个行星轮（i=1,2,3） 。因为我们有三个相同的行星轮并且它们有三个自由度，因此有必要深入每一步操作【3，4】 ，把 gst 当作每个元素都是对角矩阵（3x3）的快矩阵，它可以代表每一个元素。因此太阳轮和本轮最初的坐标（x,y,）E,Ep 可以互换 A 和 G.。由此产生的变化是最初的矩阵 K 等于新产生的矩阵 GA,它们看起来很像。通过应用这一转变从矩阵 K1 中我们可以得到因此，调节力和力相应的转变为：最初的矩阵 K(15x15)分为 3 个独立块（5x5） ，它们看上去很像惯性矩阵 M 仍然为对角线矩阵，因为 GA 是正交的，因此独立的振振荡类型只定义为矩阵 K*。四、揭示自然振荡和强迫振荡的独立运动类型A、自然振荡从矩阵（6）中可以看到，由于系统的对称性，对原始矩阵 K 进行分解，因此振荡类型和空间参数都各不相同。具体的关系在矩阵（6）中表明，有以下的振荡类型：第一，太阳轮和本轮角振荡+行星轮振荡阶段，其确定参数是：第二，太阳轮和本轮横向振荡+行星轮振荡反阶段。产生了俩个相同的矩阵 （5x5）这意味着在系统中有 5 个平等频率。其确定的参数是：因此考虑到只有性能的对称性有可能获得足够深的动态分析性能的系统的行星减速器，他除了能简化也能优化系统的过程。B、强迫振荡强迫振荡中独立振荡的使用仅适用于两种情况;a,如果点的适用外部有相同类型的对称性；例如作为设计的。b,如果它们要根据独立振荡的类型处理，真的，然后变换（5）把力 F *换成含零元素或已次子或二次子。减速器实际装载力的分析表明，它是有效的，如果系统是不平横的，同样的:a,相同的行星轮不平衡+本轮不平衡；b,相同的行星轮不平衡+太阳轮不平衡。五、进一步的动作分解进一步分解（6）中子一和二只要有附加条件，使这类系统成为对称系统是可能的。一些特殊的条件可以使太阳轮和本轮有对称性，例如：1、S 和 EP 有相同的传动刚度，即2、S 和 EP 有相同的部分频率和角速度，即因此，若满足条件 1，2 则会出现反对称，这一对称足组的操作 是符合实施这些操作后矩阵 K*会出现相应的变化，真的它们可能有太阳轮和本轮的振荡对称和反振荡对称。因此，坐标变换是：和这个坐标变换出以下独立运动的类型具体的关系表明这些矩阵有以下独立的振荡类型：I 子（矩阵 ）太阳轮 S 的角振荡和本轮 EP 的相+行星轮的振荡轴线 X*在第一阶段II 子（矩阵 ）太阳轮 S 的角振荡和本轮 EP 的反相+行星轮的振荡的轴线 Y*的阶段。同样发生分解子二和矩阵 而是 S 和 EP 的横向振荡沿轴 X*,(Y*)和振荡中的行星轮有个反阶段。作为显示分析矩阵 的振荡 S 和 EP 中的一个阶段并不取决于角振荡行星轮。从分析 和 我们注意到，h7=h8=h6=0 可能出现。这种振荡类型是指太阳轮和本轮，行星轮的自由振荡。A、强迫振荡。根据这些振荡类型不诱导其它振荡类型去掉外力，因为它们是相互正交的。冲击力提供了一个子二独立的对称与反对称性振荡 S 和 EP 如果它们同事适用 S 和 EP 有平等的价值。然后转化为外部力量 表格 加载的外部力量这些加载的外力不诱导反对称振荡类型。六、结论有规定，行星齿轮减速器由于其对称性其振荡分解增加。有独立的振荡，如太阳轮和本轮的角振荡+行星轮振荡阶段；太阳轮和本轮的横向振荡+行星轮振荡反阶段。平等的部分频率的太阳轮和本轮的振荡阶段并不取决与行星轮的角振荡。自由振荡的行星轮的一个特定的参数的选择是独立于太阳轮和本轮的角振荡的。根据这些诱导振荡的振荡类型去掉外力，因为振荡类型正交对方。这些结果是正确的行星减速器齿轮在参数改变时给出对称性。12th IFToMM World Congress, Besanon (France), June18-21, 2007 Revealing of Independent Oscillations in Planetary Reducer Gear owing to its symmetry L.Banakh* Yu. Fedoseev+ Mechanical Engineering Research Institute of Russian Academy of Sciences Moscow, Russia Abstract - The planetary reducer11 gear is a symmetric system. For its oscillation analysis there is applied the symmetry group representation theory, which was generalized for mechanical systems. It was found that due to reducer symmetry the oscillations decomposition has arisen. There are independent oscillations classes, such as: angular oscillations of solar gear and epicycle - satellites oscillations in phase; transversal oscillations of solar gear and epicycle - satellites oscillations in anti phase. Solar gear and epicycle oscillations in a phase do not depend on angular satellites oscillations. Keywords: planetary reducer, symmetry, group representation theory, independent oscillations I. Introduction It is well known that at the operation of planetary reducer there are oscillations of its elements, such as solar gear, epicycle and satellites. This factor essentially worsens a quality of reducer operation, and in some cases can result in their curvature and breakage. A plenty of papers are devoted to the dynamic analysis of gear reducers 1. Basically there are computational researches. In the given paper the analytical approaches for investigation of reducer dynamics is presented. The planetary reducer has a high degree of symmetry. So this property was used and the group representation theory was applied. Application of this theory allows carrying out deep enough dynamic analysis, using symmetry properties only. For this purpose it is necessary to have the dynamical model which is taking into account stiffness characteristics in linkages between reducer elements. The mathematical apparatus of the symmetry groups representation theory is widely used in the quantum mechanics, crystallographic, spectroscopy 2, 3, 4. The advantages of this approach are difficult for overestimating. With its help it is possible to define with exhaustive completeness the dynamic properties, using structure symmetry of system only without solving of motion equations. However in the classical mechanics this approach is not widely used. It is result from some particular features of mechanical systems. First, there is an * E-mail: banlinbox.ru availability of solids with 6-th degrees of freedom. It is unclear to what symmetry group to relate a solid in order that system symmetry may be retained. Second at real designs may be technological errors and mistakes at assembly, so there is a small asymmetry and the system becomes quasi symmetric Further the mechanical systems consist from various subsystems with various symmetry groups. In this connection it is necessary to have methods for the analysis as symmetric and quasi symmetric mechanical systems consisting of various subsystems and solids. Having made some generalizations, this mathematical apparatus for mechanical systems may be used. For this purpose we propose to apply the generalized projective operators 5. These operators are matrixes of the appropriate order instead of scalar as in physics. The use of generalized projective operators allows taking into account all above mentioned features of mechanical systems. The application of these operators to initial stiffness matrix leads to its decomposition on independent blocks each of them corresponds to own oscillation class in independent subspaces. To account for the solids symmetry the equivalent points were entered: these points are chosen on solid so that their displacements were compatible to connections and corresponded to group of symmetry of all system. These operators enable also may be applied with the finite elements models (FEM). II. Dynamic model of planetary reducer. Stiffness matrix. The model of a planetary reducer step is submitted on fig. 1 6. The step consists from solar gear S, its mass and radius are equal to 1 1,m r . It engages into mesh with three satellites Sti (i=1,2,3) (its masses and radius are identical and equal to 2 2,m r ). Satellites in turn are engaged into mesh with epicycle Ep ( 3 3,m r ) and they are fasten on carrier by elastic support with rigidity h6. The rigidity of gearing solar gear-satellites is equal to 1h , the gearing epicycle-satellites is 3h , is angle of gearing. 12th IFToMM World Congress, Besanon (France), June18-21, 2007 Fig.1 Planetary reducer step. S- solar gear, - epicycle, 1,2,3 satellites (St). Lets consider all over again the plane oscillations of planetary reducer step: transversal (x, y) and angular ( ) oscillations (without the casing). A stiffness matrix may be represented in a block view K =1 2 31 2 3123S SSt SSt SSt St St StStStSt K K K KK K K KKKK(1) Here on the main diagonal there are the stiffness submatrixes (3x3) for appropriate elements, and outside of the main diagonal there are stiffness submatrixes of connection between these elements. There are 15 generalized coordinates: X=( * * * * * *, , ; , , ,S S S Ep Ep Epx y x y ; 1 1 1 3, , .St St St Stx y ) The concrete view of these blocks is submitted in Appendix. Thus, the total order of matrix K is (15x15). An inertia matrix M is diagonal. III. Introduction of equivalent points in dynamic model. Operators of symmetry. By virtue of symmetry of satellites fastening this subsystem has symmetry such as 3C (as triangle). To reveal symmetry 3C at moving of solar gear S and epicycle Ep we shall enter the coordinates 1 2 3, ,l l l on solar gear S in points of satellites fastening (fig. 2.). Fig.2 Equivalent points on solar gear S. 1,2,3,- satellites They are “equivalent points”. Their coordinates are: 1 1 1 12 2 2 13 3 3 12 ( 1)cos3;2 ( 1)sin31, 2,3.S S iS SiS Sir Xr iripi pi = = = =(2) or in matrix form L=AX And analogues relations for“ equivalent points” on epicycle, but instead 1r in (2) must be written 3r . And later on these coordinate of solar gear and epicycle will be used instead (x, y) and ( ). After that it is possible to count, that all coordinates of system should vary according to symmetry group 3C and, hence, it is possible to apply the projective operator of symmetry to all system elements: S, Ep, and also to three satellites St i (i =1, 2, 3).(fig.3) The ortho-normal projective operator g of symmetry for point group 3C is known as 2 . It is g =1 1 13 3 31 2 16 6 61 102 2 (3) For the whole system the projective operator must be represented as block-diagonal matrix 12th IFToMM World Congress, Besanon (France), June18-21, 2007 G=St ggg(4) Here each sub matrix corresponds to S, Ep, and also to three satellites St i (i=1, 2, 3). So Because we have three identical satellites and each of them has 3 degrees of freedom ( ,i iSt Stx y and angular .i iSt St ), therefore it is necessary to enter generalize operator (3) 3,4 and to consider Stg as block matrix where the each element is diagonal matrix (33), that is it is possible to present each element as Stg =1, 11 = Eg E EEThus to initial coordinates ,( , , )S Epx y of solar gear and epicycle consistently two transformations are applied: A and G . And resulting transformation of an initial matrix K equals to product of operators G A. This orthogonal transformation and it looks like G=St gAgAg, where gA= 2 23 30 0 100 + By applying of this transformation to matrix (1), we shall receive *= (G)()(G)tr So the corresponding transformations of coordinates and forces are X*=(G), F*=(G)trF (5) As a result the initial matrix K (1515) is divided on 3 independent blocks (5x5) and, looking like, * *(1)*(2) = IIIIIKK KK(6) The inertia matrix M remains diagonal because matrix G A is orthogonal; therefore the independence of oscillation classes defines matrix * only. IV. Revealing of independent motions classes at for natural and forced oscillations A. Natural oscillations From the view of matrix (6) it is seen, that owing to system symmetry there is a decomposition of initial matrix K, and, hence, division of oscillation classes and as well as space of parameters. The concrete relations for submatrixes in (6) show that there are following independent oscillations classes: I-st class (subspace I- submatrix *IK ): angular oscillation of solar gear and epicycle + oscillations of satellites in a phase. Dimension of this subspace is equal to 5. Its determining parameters are: 1 2 3 1 3 6 12 13 9, , , , , , , , .r r r h h h h r h 2-nd class (subspace II- submatrixes *(1)IIK (2)*IIK ): transversal oscillations of solar gear and epicycle + oscillations of satellites in an anti phase. Subspace II breaks up to two identical submatrixes *(1)IIK and (2)*IIK (55). It means that in system there are 5 equal frequencies. Its determining parameters are: 2 1 3 6 7 9, , , , , , .r h h h h h Thus, taking into account only properties of symmetry it is possible to receive deep enough analysis of dynamic properties of system of a planetary reducer. Besides it is possible to simplify also process of system optimization. B. Forced oscillations At the forced oscillations the use of the independent oscillation classes is suitable only in two cases: a) if the points of application of the external forces have the same type of symmetry, as a design has, or b) if they are disposed according to the independent classes of oscillations. Really, then transformation (5) bring a forces vector F* into a form containing zero elements or in 1-st, or 2-th subspaces. The analysis of the real loadings forces on a reducer, shows, that it is valid if elements disbalances are the same: ) identical satellites disbalances + disbalance of epicycle; ) identical satellites disbalances + disbalance of solar gear. V. The further motions decomposition. The further decomposition of subspaces I and II in (6) is possible only if there are additional conditions raising a type of system symmetry. 12th IFToMM World Congress, Besanon (France), June18-21, 2007 These conditions, in particular, can be received from similarity symmetry of solar gear and epicycle. They look like: 1. Equality of gearing stiffness with S and Ep, i.e. 1 2h h= , 2. Equality of partial frequencies for angular motions S and Ep ( ) ( )S Ep = , whence: 7 8h h= , or 3. Equality of partial frequencies at tranversal motions of S and Ep ( , ) ( , )S Epx y x y = , whence: h7=2h9. So by fulfillment of conditions 1,2 (or 1,3) the additional symmetry type 2C is appeared (reflection symmetry). To this symmetry group the operator 2G (or 2G ) is corresponded 2G =1 13 11 1 13 ;13 1 1r hr h 2G =1111 13213 1 12hh The application of these operators to matrix K* permit to achieve the further decomposition of corresponding matrixes and appropriate motions. Really they may have symmetric and anti symmetric oscillation classes for solar gear and epicycle. Thus the coordinate transformation is: 1 1 1 * *3 1 311 1 1 * *3 1 31S Epr rhS Epr rh = + = X X XX X XAnd 1 * *3121 * *312S EphS Eph = + = X X XX X XBy this coordinate transformation the following independent motion types are arisen ( )*( ) IIIKKK The concrete relations for these submatrixes show that there are following independent oscillations classes: I subspace (matrix *IK ): -angular oscillations of solar gear S and epicycle Ep in phase + satellites Sti oscillation along axis *x in phase), II subspace(matrix *IK ): -angular oscillations of solar gear S and epicycle Ep in anti phase + satellites St i oscillation along axis *y and around axis in phase. Similarly occurs decomposition of subspaces II and matrixes *(1)IIK , (2)*IIK but instead of angular oscillations S and Ep there are their transversal oscillation along an axis x * (or y *), and oscillation of satellites in an anti phase. As shows the analysis of matrix *IK the oscillations of S and Ep in a phase do not depend on angular oscillation of satellites ( St ). From analysis of *IK and *IIK we notice, that at h7=h8=h6 =0 a zero root may be arisen. This oscillation type means the free oscillations of satellites (navigation) at angular (or translation) oscillations solar gear and epicycle. A. Forced oscillations. The choice of exited forces according one of these oscillation types do not induce the other oscillation types because they are orthogonal each other. The exciting forces acting in subspace II provide an independence of symmetric and anti symmetric oscillations S and Ep if they are applied simultaneously to S and Ep and are equal on value. Then transformation of external forces F * looks like, submitted in table. 12th IFToMM World Congress, Besanon (France), June18-21, 2007 Table. The distribution of external forces 2 F = g F = subspace IIK IIK Sx 2 1 1f h 0 Sy 3 1 1f h 0 xSt 1 12 cosf h 0 And this distribution of external forces do not induce the anti symmetric oscillations classes. VI. Conclusions There was established that in planetary reducer gear due to its symmetry the decomposition of oscillations arised. There are independent oscillations, such as: -angular oscillations of solar gear and epicycle + satellites oscillations in phase; -transversal oscillations of solar gear and epicycle + satellites oscillation in anti phase. At equality of partial frequencies of solar