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12th IFToMM World Congress, Besançon (France), June18-21, 2007 The Displacement Analysis of a Concentric Double Universal Joint C.-C. Lin* W.-K. Liao Department of Mechanical and Mechatronic Engineering National Taiwan Ocean University Keelung, Taiwan Abstract— The displacement analysis of a concentric A comprehensive list of references about constant- double universal joint is studied. Based on the geometric velocity joint can be found in the book by Duditza [2]. characteristic of the angle bisector mechanism, the proposed The constant velocity ratio behaviour of a coupling method divides the main loop into two artificial loops, and then between two shafts was first clarified using screw theory the loop equations are formulated and solved. All the joint displacements can be expressed in the closed-form functions. by Hunt [3] in 1973. In that paper he proposed the concept From the displacement analysis result, it is concluded that the of the homokinetic plane or bisecting plane, and proved Thompson CV coupling is a true constant velocity shaft coupling the validity of the constant-velocity effect when the with all revolute joint in it. intersection (or the midpoint of the common normal) of two shaft axes always maintains on the bisecting plane. Keywords: double universal joint, constant velocity, shaft The CV couplings such as 7R (double universal joint), coupling, homokinetic plane, shaft misalignment Tracta, Clemens, Myard, and Altmann couplings are spatial linkages with revolute joints only or combinations of revolute, prismatic, cylindrical, planar, and spherical I. Introduction joints. Some researchers have investigated the kinematics of the above CV couplings in the past four decades. It is well understood that a single universal joint can not Wallace, et al [4] studied the displacement analysis and maintain a constant velocity ratio between two shafts if parameter sensitivity analysis of Tracta and Clemens they are misaligned. A common method to solve the couplings. Fischer [5] introduced dual number method to above problem is to combine two universal joints back-to- derive the displacement equation for each joint of the back at two ends of an intermediate link, forming a so- Tracta coupling. Jun Nango, et al [6] considered the called double universal joint, and then the variation of the displacement and static problems of a 7R CV coupling velocity ratio of each universal joint can be cancelled out. based on dual number method, and the effects of velocity Double universal joints have been widely used in the ratio variation due to manufacturing and assembly errors. automotive industry for transmitting power between two Chen [7] studied the kinematics and statics problems of intersecting or parallel shafts which require a constant 7R, Tracta, and Altmann couplings, and derived the velocity ratio between them. The main drawback of a closed-form formulation of the joint displacement and double universal joint is that the mechanism is relatively joint torque. Unlike the above mechanisms with one loop bulky compared to other CV (constant-velocity) couplings only, the Thompson CV coupling considered here is a such as Rzeppa, Bendix-Weiss, tripod, etc. The spatial spherical mechanism with up to twelve links and five constraint is in general a major concern for a designer in loops. Thus the displacement analysis problem is much considering which type of mechanisms should be adopted. more complicated than that of a single-loop mechanism. For this reason, the Rzeppa or tripod couplings are most Wampler [8] presented a modification of Sylvester’s widely adopted between the shafts in an automotive front- dialytic elimination process to solve the displacement wheel drive system. However, Rzeppa or tripod couplings problem of spherical mechanisms with up to three loops, also suffer the disadvantage in that considerable amount and such formulations are without any extraneous roots. of friction loss or heat dissipation will be generated from In this paper we shall consider the Thompson CV the roll-slide motion between the balls and the groove. To coupling (a five-loop spherical mechanism) and perform overcome the disadvantages, a new type of CV coupling the displacement analysis of the mechanism. called Thompson CV coupling was invented recently [1]. The Thompson CV coupling is essentially a concentric II. Formulization of joint displacements double universal joint which is a spherical linkage A. Modelling of the concentric double universal joint comprises all revolute joints in it. The mechanism is relatively compact in size and generates less friction loss Figure 1(a) shows the computer solid model of a and wear during operation. Thompson CV coupling. The spherical mechanism has _________ twelve links interconnected by sixteen revolute joints, * E-mail: cclin@euler.me.ntou.edu.tw resulting in five independent loops and a mobility of one. 12th IFToMM World Congress, Besançon (France), June18-21, 2007 We designate link 2 as the input and link 12 as the output. (intersection of all joint axes) of the mechanism for each Both input and output shafts pivot about their joint axes joint. We first attach a global coordinate system x1y1z1 to (a) (a) (b) Fig. 1. (a)The solid model of a Thompson CV coupling (b)The angle bisector mechanism (double spherical parallelogram) fixed on the ground link (link 1). The angular misalignment between the two joint axes is (π − α1). In Fig. 1(b), we have intentionally hidden the links 10, 3, and (b) 11 to show the angle bisector mechanism. The angle bisector mechanism comprises six links (links 6, 4, 9, 7, 5, Fig. 2. (a)Graph representation (b)Structural representation of the mechanism (in planar form) 8), forming a double spherical parallelogram. Links 6, 7, 8, and 9 have equal twist angle, which is also one-half of the the ground link, where z1-axis is aligned with the joint twist angle of link 4 or 5. The angle bisector mechanism is axis of link 2 about link 1, and x1-axis is chosen arbitrarily pin-supported by links 2, 12, and 3 at points A, B, C, as long as it is perpendicular to z1-axis. All local respectively. All joint axes intersect at the stationary coordinates are then defined subsequently as we traverse sphere centre O. It can be observed from Fig. 1(b) that the around each loop. For example, as shown in Fig 2(b), angle between OA and OC always equal to the angle starting from link 2, a clockwise path is traversed along R2-R10-R4-R6-R7-R2. The z2-axis is defined along the R2 between OB and OC at any configuration. Therefore joint axis, and x2-axis is obtained by the cross product OC will bisect the angle between z1 (input axis) and z12 z1 × z2 . Then z10-axis and x10-axis are defined in the same (output axis). By Hunt’s definition in ref. [3], OC is the manner. The subsequent local coordinate systems in the loop, x4y4z4, x6y6z6, and x7y7z7 can also be defined normal of homokinetic plane (bisecting plane). Figures accordingly. Figure 3 shows only some local coordinate 2(a) and 2(b) show the graph representation and structural systems due to limited space. Figure 4 shows the layout of representation of the mechanism, respectively. Links 1, 2, the input (z1), output (z12), bisecting vector (z4), and the 10, 3, 11, and 12 form a closed-loop kinematic chain homokinetic plane for the configuration shown in Fig. (main loop); in which there are two combined universal 1(a). The DH parameters of each link can be defined joints. To model the multiloop spherical mechanism, a set of local coordinate system are defined at the sphere centre according to the convention. For examples, θ2 is the angle required to rotate the x2-axis into alignment with the x10- 12th IFToMM World Congress, Besançon (France), June18-21, 2007 axis about the positive z2-axis; α10 (or α10,2) is the angle make the problem solvable, a geometric characteristic of required to rotate the z2-axis into alignment with the z10- the double parallelogram is utilized. That is, joint axis z4 axis about the positive x10-axis. Since there are some can be considered fixed when the angular misalignment multiple-jointed links in the CV coupling, more than one (π-α1) is kept fixed. As depicted in the previous section twist angles (αia, αib, ...) can be defined for those links. about the angle bisector mechanism, the joint axis z4 (along OC ) bisects the angle between z1 and z12 axes. As shown in Fig. 5, the homokinetic plane bisects the main loop into two artificial loops L1 and L2. Since z4-axis can be regarded as being attached to the ground when α1 is fixed, the twist angle between z1 and z4 axes is also a constant (α1/2). Hence we can solve the loop equation of L1 easily. The solution process begins by giving the input angular displacement θ1. Then from the loop equations of the four-link chain L1, the rest three angular displacements (θ2, θ10, θ4g) can be obtained in closed form; where θ2 and θ10 represent the angular displacements at joints R2 and R10, respectively; and θ4g denotes the angular displacement of link 3 with respect to the ground link about joint R4. Fig. 3. The local coordinate system for some joints Fig. 4 The bisecting axis of the angle between input and output axes Fig. 5. The loops of the mechanism. The main (bottom) loop is divided into two virtual loops L1 and L2 by the homokinetic plane. B. Solution scheme of displacement analysis Next from loop equations of L3, θ4 can be derived first, Figure 5 shows that there are five loops inside the and then θ6 and θ7 can be obtained by back substituting θ4 Thompson CV coupling. The displacement analysis into the loop equations. Next from loop equations of the problem will become very complicated and unfeasible to be formulated if we try to establish the nonlinear loop four-link chain L5, θ5 can be derived first, and then θ13 and equations from the five loops. The most important reason θ15 can be obtained by back substituting θ5 into the loop is that there are too many variables in the main (bottom) equations. The above process complete the solution of all loop comprised by six links (1, 2, 10, 3, 11, 12), and the the angular displacements of the left-hand side loops (L1, variable elimination process fails in reducing the number L3, L5). The rest angular displacements of the right-hand of variables. To simplify the formulation process and side loops can be derived in the same manner as the above process. The solution process starts from the loop L6, and 12th IFToMM World Congress, Besançon (France), June18-21, 2007 then L4, and finally L2. Note that the above solution L3 : Ζ7 X 2b Ζ 2 X 10 Ζ10 X 3a Ζ 4 X 4a Ζ6 X 6a = Ι (7) scheme is based on the premise of the function of the angle bisector mechanism. ′ L4 Ζ 11 X 12b Ζ 8 X 9a Ζ 9 X 4b Ζ 4 X 3b Ζ 3 X 11 = Ι (8) C. Loop equation formulation ′ ′ L5 Ζ 13 X 6b Ζ 6 X 4c Ζ 5 X 5a Ζ 15 X 7 = Ι (9) In [8] Wampler described a procedure to derive the displacement equation from each loop equation. Two joint ′ L6 Ζ16 X 5b Ζ 5 X 4d Ζ 9 X 9b Ζ 14 X 8 = Ι (10) variables are eliminated during the process. As shown in Fig. 5, the input joint axis is z1, and the loop L1 is a four where Z i' denotes the inverse of the matrix Zi. The link chain. The loop equation of L1 can be expressed as occurrence of the inverse matrix is because the paths of traverse result in opposite sign of a joint angle common to Ζ 4g X 1a Ζ 1 X 2a Ζ 2 X 10 Ζ 10 X 3a = Ι (1) two neighboring loops. In the same manner, the rest angular displacements can be where Xn is the transformation matrix representing the solved one by one for each loop by back substituting the side rotation about the xn -axis, which is defined by previously solved variables to the loop equation (starting 1 0 0 from L3, and then L5, L6, L4, L2), finally the output X n = 0 cα n − sα n , (2) displacement can be obtained. To verify the result, a set of DH parameters shown in 0 sα n cα n Table I was assigned to each link of the Thompson CV coupling. The resulting angular displacements of all joints and Zn is the transformation matrix representing the joint in the main loop are shown in Fig. 6 to Fig. 10, where the angle rotation about the zn -axis, which is defined by abscissa is the input joint angle, θ1, which ranges from 0° cθ n − sθ n 0 to 360°, and the shaft misalignment (π – α1) ranges from Z n = sθ n cθ n 0 (3a) 0° to 30°. 0 0 1 Link i twist angle αi Joint angle θi 1a α1/2 θ4 g and 1 1 − t n − 2tn 2 0 1b α1/2 θ12 (output) ˆ = θ1 (0°~360°) Zn 2tn 1 − tn 2 0 , (3b) 2 2a 90˚ 0 0 1 + tn 2 2b -45˚ θ7 let sin θ n and cos θ n in Eq.(3a) be replaced by the tangent 3a 90˚ θ10 3 θ n ). 3b 90˚ - θ4 half-angle function tn (= tan 2 4a -30˚ θ4 Let z = [0 0 1]T, the relations z T Z n = z T and Z n z = z 4b -30˚ θ9 4 can be obtained. By applying the above relations and Eq. 4c -30˚ - θ6 (3b) to Eq. (1), a scalar equation containing only one variable t2 can be derived as follows 4d -30˚ θ5 - θ5 [ z T X 1a Ζ 1 X 2a Ζ 2 X 10 − X 3a (1 + t12 )(1 + t 2 ) z = 0 (4) ˆ ˆ T 2 ] 5 5a 5b 30˚ 30˚ θ16 After solving t2 from Eq. (4), the joint angle θ2 can be 6a -30˚ θ6 obtained by the equation θ 2 = 2 tan −1 (t2 ) . Pre-multiply Eq. 6 6b -30˚ θ13 (1) by zT, again we obtain a vector equation containing 7 7 30˚ θ15 only one variable θ10 as follows (since θ2 has been solved). 8 8 30˚ θ14 z T Ζ 4g X 1a Ζ 1 X 2a Ζ 2 X 10 Ζ 10 X 3a = z T I (5) 9a -30˚ θ8 9 Finally, by substituting θ2 and θ10 back into Eq. (1), the 9b -30˚ - θ9 last unsolved variable θ4g in loop L1 can be solved. 10 10 90˚ θ2 According to the path of traverse, the loop equations of L2 11 11 90˚ θ3 through L6 can be established below. 12a 90˚ θ11 12 ′ L2 Ζ12 X 1b Ζ 4g X 3b Ζ 3 X 11 Ζ 11 X 12c = Ι (6) 12b -45˚ θ11 TABLE I. The DH parameters of the Thompson CV coupling 12th IFToMM World Congress, Besançon (France), June18-21, 2007 Fig. 6. The output joint angle θ12 vs input angle as the misalignment Fig. 10.The joint angle θ3 vs. input angle as the misalignment changes changes (four overlapping loci) III. Result Figure 6 indicates the angular displacement of the output shaft is identical to the corresponding input angular displacement except for a different sign due to the definition of the local coordinate systems. The four loci corresponding to four different values of misalignment (π - α1 = 0°, 10°, 20°, 30°) completely overlap, indicating the output angular velocity is not affected by the shaft misalignment. Hence the Thompson CV coupling is verified theoretically as a true constant velocity shaft coupling. Some observations from the resulting figures (Figs. 6~10) are stated below. Fig. 7. The joint angle θ2 vs. input angle as the misalignment changes 1. The angular displacements of the joints which are symmetric about the homokinetic plane are identical. The symmetric angular displacements include: θ1 and θ12, θ2 and θ11, θ10 and θ3. 2. When the shaft misalignment is zero, all joint angles are fixed constants except the output joint angleθ12. In other words, the whole mechanism rotates as a single rigid body, i.e., no relative motion between inner links. 3. As the shaft misalignment increases, the peak-to-peak values of the angular displacements also increase. 4. The function of the six-link double parallelogram mechanisms (angle bisector mechanism) is to provide the plane of symmetry for the two universal joints. This is achieved by guaranteeing the equality of the angle Fig. 8. The joint angle θ11 vs. input angle as the misalignment changes between OA and OC and the angle between OB and OC at any configuration (Fig. 1(b)). 5. The maximum misalignment in this report is 30°, provided that no interference occurs. However, the amount of the misalignment can further increase with a proper design of the link lengths. Fig. 9. The joint angle θ10 vs. input angle as the misalignment changes 12th IFToMM World Congress, Besançon (France), June18-21, 2007 IV. Conclusion The displacement analysis problem of a concentric double universal joint is investigated. Due to the complex structure of the CV coupling, the solution by using the formulation of the total five loops is not feasible. Hence the geometric characteristic of the angle bisector mechanism is utilized. That is, when the angular misalignment (π-α1) is kept fixed, the z4-axis of the angle bisector mechanism can be regarded as a fixed joint axis. Using the proposed method, the main loop of the coupling is divided into two artificial loops, and then the loop equations are formulated and solved. All the joint displacements can be expressed in the closed-form functions. From the analysis result, it is concluded that the Thompson CV coupling is a true constant velocity shaft coupling with all revolute joint in it. Further study has been done to optimize the link geometry [9] in order to allow a larger range of shaft misalignment, and smaller peak-to-peak values of the joint displacements (under the same angular misalignment). References [1] Thompson CV Couplings website, http://cvcoupling.com [2] Duditza F. L. Cuplaje mobile homocinetice. Editura Teknica, Bucharest, 1974. [3] Hunt K. H. Constant-velocity shaft couplings: a general theory. Trans. ASME J. Eng. Ind., 95:455-464, 1973. [4] Wallace D. M., and Freudenstein F. The displacement analysis of the generalized Tracta coupling. Trans. ASME J. Appl. Mechanics, 37(3):713-719, 1970. [5] Fischer I. S. Numerical analysis of displacements in a Tracta coupling. Engineering with Computers, 15:334-344, 1999. [6] Nango J., and Watanabe K. Static analysis of spatial 7R link constant-velocity joints extended shaft angle. In Proc. ASME, Des. Eng. Tech. Conf, Montreal, Canada, 2002. [7] Chen C. K. The kinematic and static analysis of constant velocity coupling. Master thesis, Department of mechanical and mechatronic engineering, National Taiwan Ocean University, 2004. (in Chinese) [8] Wampler C. W. Displacement analysis of spherical mechanisms having three or fewer loops. Trans. ASME J. Mech.. Des., 126:93-100, 2004 [9] Liao W. K. The displacement analysis of multi-loop constant velocity shaft couplings. Master thesis, Department of mechanical and mechatronic engineering, National Taiwan Ocean University, 2006. (in Chinese)

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