Mechanism and Machine Theory 44 (2009) 1807–1825 Contents lists available at ScienceDirect Mechanism and Machine Theory journal homepage: www.elsevier.com/locate/mechmt On grinding manufacture technique and tooth contact and stress analysis of ring-involute spherical gears Li Ting *, Pan Cunyun School of Mechatronics and Automation, National University of Defense Technology, Changsha 410073, China a r t i c l e i n f o Article history: Received 29 April 2008 Received in revised form March 2009 Accepted 17 March 2009 Available online 21 May 2009 Keywords: Spherical gear Generating method Plate grinding wheel Contact analysis Contact point path a b s t r a c t The spherical gear is a new gear-driven mechanism with two degrees of freedom (DOF), which can transfer spatial motion and power A grinding machine is designed for manufacturing spherical gear with a plate grinding wheel by using generating method Based on the mathematical models of spherical gears, the tooth contact analysis (TCA) of spherical gear pairs is performed, and the contact point paths on tooth surfaces of spherical gears are investigated Finally, the tooth contact and bending stress analysis are simulated using Finite element method (FEM) Ó 2009 Elsevier Ltd All rights reserved Introductions The spherical gears with continuous ring-involute teeth in Fig were invented by Pan [1] in 1990 As a new DOF geardriven mechanism, it overcomes the two defects of Trallfa spherical gear [2] with discrete cone tooth: error of transmission principle and difficulty in machining The spherical gear can be used for robot arms and wrists, the seeker mechanism of missile, the flexible wrist mechanism of spatial robot in Fig 2, the control system of antenna’s attitude on satellite in Fig 3, the lunar rover with complex function of wheel and leg, the attitude control system of solar array on spacecraft, and so on Although the manufacturing method of spherical gears has already been developed, yet few investigations are on spherical gear, until Yang et al introduced a spherical gear with discrete arc teeth [3] and a spherical gear with discrete ring-involute teeth [4] Tsai et al discussed the application of the rapid prototyping and manufacturing technology to form a spherical gear with skew axes [5] This new type of spherical gears is different from spur involute gears and spherical involute conical gears, so its manufacture methods are also different from that of traditional gears In Fig 1, the spherical gears are machined by milling cutter and can satisfy the low-precision requirements However, in some fields, the spherical gears cannot be applied because of its low-precision Therefore, in this paper, a grinding manufacture method is put forward to improve the machining precision of spherical gears, and the machining theory comes from the mesh of a spherical gear and a teeth turner [6,7] The generation of the teeth turner is similar with that of a rack formed by a gear: when the teeth number of the spherical gear is infinite, the radius of its reference sphere will also be infinite, and the spherical gear becomes a teeth turner Fig 4a shows the meshing of a spherical gear and a teeth turner, and Fig 4b is its section drawing Since the spherical gear can move in three-dimensional * Corresponding author Tel.: +86 0731 4574932 E-mail address: bee.lt@163.com (L Ting) 0094-114X/$ - see front matter Ó 2009 Elsevier Ltd All rights reserved doi:10.1016/j.mechmachtheory.2009.03.005 1808 L Ting, P Cunyun / Mechanism and Machine Theory 44 (2009) 1807–1825 Fig Spherical gears Fig Flexible transmission shaft with spherical gear pairs space and the teeth turner allows plane movement, the spherical gear and teeth turner mechanism can achieve a shift in movement between spatial and plane movement The teeth turner can be machined plate grinding wheel, and then the plate grinding wheel can grind the spherical gear according to the meshing movement relationship between a spherical gear and a teeth turner The tooth contact analysis (TCA) is an important factor which can evaluate gear transmission characteristics In Refs [8–10], the TCA of spur gears, bevel gears and worm gears are investigated, but the meshing theory and tooth surface characteristics of the spherical gear are different from those of any other gears Up to now, few or no investigation about the TCA analysis and stress analysis of spherical gears can be found So, the efforts of this paper are to build the mathematical models of spherical gears with continuous convex ring-involute tooth and concave ring-involute tooth With these mathematical models, the meshing model and tooth contact analysis are performed, and the instantaneous contact points can be obtained In addition, the contact and bending stress of spherical gear drive is performed These play the important L Ting, P Cunyun / Mechanism and Machine Theory 44 (2009) 1807–1825 1809 Fig Installed spherical gear mechanism Fig Spherical gear and teeth turner mechanism roles in studying the transmission characteristic of spherical gear mechanism, and also are useful to the practical applications of spherical gear pairs Design of grinding machine A grinding machine for the generation of spherical gears must have DOFs Four of these DOFs are necessary for the control of the related motions of the plate grinding wheel and a spherical gear The fifth DOF is required to provide the desired velocity of grinding and is not related to the process of surface generation 2.1 Coordinate systems applied for grinding machine According to meshing theory [6] of spherical gear pairs and the freedom requirement, the grinding machine (Fig 5) was designed for generating spherical gears The machine is provided with five DOF for three rotational motions, and two translational motions The translational motions are performed in two perpendicular directions Two rotational motions are provided as rotation of the spherical gear and the rotation that enables the machine to change the angle between the axes of spherical gear and the plate grinding wheel The third rotational motion is provided as rotation of plate grinding wheel 1810 L Ting, P Cunyun / Mechanism and Machine Theory 44 (2009) 1807–1825 Fig Schematic of grinding machine x1 yd ym xn xm yf ye φ β xd Og zn zg Om On ze z yn z1 d y1 xe xf ( Od ) ze θ O Od yg xg zm ( Om ) xf zf Of O e z (fOm ) Fig Coordinate systems applied for grinding machine and generally is not related to the process of generation The motions of the other DOF are provided as related motions in the process of surface generation As shown in Fig 6, the coordinate systems S1 ðx1 ; y1 ; z1 Þ and Sg ðxg ; yg ; zg Þ are rigidly connected to the spherical gear and the plate grinding wheel, respectively For further discussion, we distinguish three reference frames designated in Fig as À, a and b The reference frame / is the fixed one that is the housing of the machine, which is rigidly connected to coordinate system Sf ðxf ; yf ; zf Þ Reference frames // and /// perform translations in two perpendicular directions Coordinate system Se ðxe ; ye ; ze Þ performs rotational motion with respect to Sf about the yf axis with an angle h Coordinate system ðO Þ Sd ðxd ; yd ; zd Þ and Se are parallel to each other and the location of Sd with respect to Se is represented by ð0; 0; ze d Þ And then coordinate system S1 performs rotational motion with respect to Sd about the zd axis with an angle b Coordinate system ðO Þ ðO Þ Sm ðxm ; ym ; zm Þ and Sf are parallel to each other and the location of Sm with respect to Sf is represented by ðxf m ; 0; zf m Þ Coordinate system Sn ðxn ; yn ; zn Þ performs rotational motion with respect to Sm about the xm axis with 180° And then, coordinate system Sg performs rotational motion with respect to Sn about the zn axis with an angle / According to the relationship of coordinate transformation in Fig 6, the transformation matrix from coordinate system Sg to S1 is as follows: 1811 L Ting, P Cunyun / Mechanism and Machine Theory 44 (2009) 1807–1825 x1 xf Convex tooth in position x1 Plate grinding wheel in position Reference line z1 xg D zg O1 O O f O1 θ Og Plate grinding wheel in position z1 zf rb Convex tooth in position r α Reference line Fig Kinematic meshing relationship between spherical gear and plate grinding wheel M1g ¼ M1d Mde Mef Mfm Mmn Mng ðO Þ ðO Þ xf m cos b cos h À zf m cos b sin h cos b cos h cos u À sin b sin u À cos u sin b À cos b cos h sin u cos b sin h 7 ðO Þ mÞ À cos h cos u sin b À cos b sin u À cos b cos u þ cos h sin b sin u À sin b sin h ÀxðO cos h sin b þ zf m sin b sin h f ¼6 ðOd Þ ðOm Þ ðOm Þ u sin h À sin h sin u À cos h x sin h þ z cos h À z cos e f f 0 ð1Þ Here, M1d ; Mde ; Mef ; Mfm ; Mmn and Mng are the transformation matrixes from coordinate system Sd to S1 , from coordinate system Se to Sd , from coordinate system Sf to Se , from coordinate system Sm to Sf , from coordinate system Sn to Sm , and from coordinate system Sg to Sn , respectively 2.2 Kinematic meshing relationship between spherical gear and plate grinding wheel According to the motion relationship of the machine (in Fig 5), we can simplify the coordinate systems in the xf zf plane as shown in Fig 7, because the motion for generation of ring-direction tooth shape of spherical gear only requires the rotational motion of spherical gear with respect to its polar axis In the xf zf plane, if the spherical gear rotates about its spherical center O1 , the meshing motion between the spherical gear and the plate grinding wheel is the same as that between a spur involute gear and a rack However, in Fig 5, the spherical gear is installed on the rotating floor When the rotating floor rotates about ðO Þ the yf axis (in Fig 6), the spherical gear is driven and sways There is a distance D (equal to ze d in Fig 6) between the spherical center O1 and the swaying center Of The advantage of the machine design is that it is easy to install spherical gear since its spherical center O1 need not be fixed with rotational center, and the DOF of the machine does not increase Consider now that the initial position is the position in which the polar axis z1 of spherical gear coincides with the zg axis of plate grinding wheel r (in Fig 7) is the radius of reference sphere of the spherical gear When the rotating floor rotates about the yf axis with an angle h, spherical center O1 also rotates about the origin Of with the equal angle h Accordingly, the plate grinding wheel must move not only about the xf axis, but also about the zf axis Based on the meshing movement rule of a gear and a rack, the following equations can be obtained: & Dxf ¼ D sin h þ rh Dzf ¼ ÀDð1 À cos hÞ ð2Þ Mathematical models of plate grinding wheels The spherical gear pair for the meshing simulation comprises a convex-tooth spherical gear and a concave-tooth spherical gear Correspondingly, the grinding tools are a concave-tooth plate grinding wheel and a convex-tooth plate grinding wheel P P Assume that the concave-tooth plate grinding wheel surfaces v and convex-tooth plate grinding wheel surfaces c genP P erate the convex-tooth spherical gear surfaces and concave-tooth spherical gear surfaces , respectively The rotation 1812 L Ting, P Cunyun / Mechanism and Machine Theory 44 (2009) 1807–1825 xg yg α zg Og A A0 m Fig Section of plate grinding wheel of the plate grinding wheel is used to provide the desired velocity of grinding and is not related to the process of surface generation, so its surface equation can be represented by its generatrix equation Assume that the pressure angle and the module of rack section of plate grinding wheel are a and m, respectively According to Fig 8, the section of plate grinding wheel consists of two straight edges Therefore, the mathematical models of two straight edges of the concave-tooth plate grinding wheel can be represented in coordinate system Sf by 6 Rgðv Þ ¼ kv pm Æ ð0:25pm þ m tan a À lv sin aÞ m À lv cos a 7 ð3Þ     where the design parameter lv ¼ A0 A represents the distance between the initial point A0 and the moving point A The upper and lower signs of Eq (3) represent the cutter surface equation of the inner and outer side on the concave-tooth plate grinding wheel kv ðkv ¼ 0; 1; and 3Þ represents the tooth number of the concave-tooth plate grinding wheel The unit normal to the concave-tooth plate grinding wheel surface can be gotten by ngðv Þ ¼ ½Ç sin a0 À cos aŠT ð4Þ Similarly, the mathematical models of these two straight edges of the convex-tooth plate grinding wheel can be represented in coordinate system Sf by kc pm Æ ð0:25pm À m tan a þ lc sin aÞ 6 RðcÞ g ¼ m À lc cos a 7 ð5Þ where kc ðkc ¼ 0; and 2Þ represents the tooth number of convex-tooth plate grinding wheel The unit normal to the convex-tooth plate grinding wheel surface can be gotten by T nðcÞ g ¼ ½Æ sin a0 À cos aŠ ð6Þ Mathematical models of spherical gears As shown in Fig 5, when spherical gear is grinded, its tooth shape along the involute-direction is generated by its rotating motion about the yf axis and the translational motions of the plate grinding wheel according to Eq (2) Except that they are meshing in linear contact in the initial position, in any other position they are meshing in point contact In the generation process, the plate grinding wheel performs translational motion with a velocity V, while the spherical gear rotates with an angular velocity x According to the theory of gearing [8], the plane axode of the plate grinding wheel and the pitch sphere of spherical gear roll over each other without sliding on the pitch plate Therefore, with Eqs (3) and (4) and concave-tooth plate P grinding wheel surfaces v , the mathematical model of the generated convex-tooth spherical gear surface can be represented in coordinate system Sf as follows: R1 ¼ M1g Rgðv Þ ð7Þ 1813 L Ting, P Cunyun / Mechanism and Machine Theory 44 (2009) 1807–1825 The unit normal vector of the generated convex-tooth spherical gear can be gained by n1 ¼ L1g ngðv Þ ð8Þ Similarly, with Eqs (5) and (6) and the same generation mechanism, the mathematical model of concave-tooth spherical P gear surface generated by convex-tooth plate grinding wheel surfaces c can be expressed in coordinate system Sf as follows: R2 ¼ M1g RðcÞ g ð9Þ The unit normal vector of the generated concave-tooth spherical gear also can be obtained by n2 ¼ L1g ngðcÞ ð10Þ Here, L1g is the rotating transformation matrix from coordinate system Sg to S1 Meshing model of spherical gear pairs Fig shows the simplified model of the spherical gear mechanism The origin O0 of the fixed coordinate system S0 ðx0 ; y0 ; z0 Þ coincides with the origin O1 of coordinate system S1 (fixed with convex-tooth spherical gear) The coordinate systems S1 ðx1 ; y1 ; z1 Þ and S2 ðx2 ; y2 ; z2 Þ are fixed with the convex-tooth spherical gear and the concave-tooth spherical gear, respectively Since the tooth contact form of meshing spherical gears is point contact between a convexity and a saddle surface, according to the tooth contact analysis method and the tooth surface characteristic of spherical gear, the position vectors and unit normal vectors of both convex-tooth spherical gear and concave-tooth spherical gear should be represented in the same coordinate system S0 The instantaneous common contact point of convex-tooth spherical gear and concave-tooth spherical gear is the same point in coordinate system S0 Furthermore, the unit normal vectors of convex-tooth spherical gear and concave-tooth spherical gear should be collinear to each other Therefore, the following equations should hold at the point of tangency of the meshing spherical gear pair: R0 À R0 ¼ ð1Þ ð2Þ ð11Þ ð1Þ n0 ð2Þ n0 ð12Þ ð1Þ Â ¼0 ð2Þ ð1Þ Here, R0 and R0 are the position vectors of the convex-tooth spherical gear and concave-tooth spherical gear, while n0 and ð2Þ ð1Þ ð2Þ n0 represent the unit normal vectors, respectively, in coordinate system S0 Since jn0 j ¼ jn0 j ¼ 1, Eqs (11) and (12) yield five independent nonlinear equations with six independent parameters b1 ; lv ; h01 ; b2 ; lc and h02 The vector n, the swaying axis, is perpendicular to the swaying plane in which two spherical gears mesh Any meshing position of the spherical gear pair can be gotten by making the convex-tooth spherical gear rotate about the vector n with an angle h01 from the initial position in which two spherical gears’ polar axes z1 and z2 coincide with to each other, and the concave-tooth spherical gear rotate about the vector p going through the center O2 with an angle h02 If the input rotation angle h01 of the convex-tooth spherical gear is known, other five independent parameters can be solved by nonlinear solver Therefore, a common contact point of two tooth surfaces can be obtained z0 O20 x20 x2 O2 θ2 p y20 y2 y0 z1 z20 z O0 O1 θ1 y1 n x1 x0 Fig Coordinate systems of spherical gear mechanism 1814 L Ting, P Cunyun / Mechanism and Machine Theory 44 (2009) 1807–1825 Transmission error model of spherical gear mechanism According to the movement characteristic of spherical gear mechanism [7], its transmission error can be described by error of the swaying angle and the azimuth angle of the mechanism’s output axis (polar axis z2 of the concave-tooth spherical gear in Fig 9) In Fig 10, because of the assembly errors, the z2 axis does not coincide with the ideal z20 axis (no assembly and hzz20 represent the swaying angles of the z20 axis and z2 axis, respectively error) of the concave-tooth spherical gear hzz20 z20 z2 nz0 and nz0 represent the unit normal vectors of the plane / and plane //, respectively The azimuth angle az20 of the z20 axis is and the y0 axis, and the azimuth angle az2 of the z2 axis is equal to the equal to the angle between the normal vector nzz20 angle between the normal vector nzz20 and the y0 axis Therefore, the error of swaying angle and azimuth angle can be obtained as follows: Dh ¼ hzz20 À hzz20 ð13Þ Da ¼ az2 À az20 ð14Þ Contact point path of spherical gear pairs The contact path of spherical gears is complex There are two reasons for it First, since every tooth of traditional gear (spur gear, bevel gear, spiral bevel gear, and so on) is the same, the contact path on every tooth is uniform whenever the teeth are in point contact or linear contact However, the teeth of spherical gear pairs are diverse Second, the spherical gear mechanism has two DOF, and the spherical gear can rotate about any alterable axis through its spherical center (for example, the vector n in Fig 9) Therefore, the transmission characteristic of spherical gear pair is more complex and flexible than that of the traditional gear pairs, and the tooth contact paths of spherical gears are also alterable and complex because of the above two reasons The following research will be focused on the path of tooth contact point of spherical gear pair and will be divided into three cases to be investigated 7.1 Contact points path of tooth meshing in a fixed swaying plane When convex-tooth spherical gear rotates about a fixed swaying axis vector n with an angle h01 , the meshing movement of two spherical gears in the swaying plane (passes through the origin O1 and O2 and is perpendicular to the vector n) is the same as that of the a spur involute gear pair z0 p2 θ zz θ zz 20 x2 O2 y2 z20 y0 α2 z2 x0 O0 α 20 n zz020 n zz02 Fig 10 Transmission error model of spherical gear mechanism Table Design parameters of spherical gear Type of gears Convex-tooth spherical gear Convex-tooth spherical gear Module (mm/teeth) Number of teeth Pressure angle (°) 17 20 17 20 1815 L Ting, P Cunyun / Mechanism and Machine Theory 44 (2009) 1807–1825 Fig 11 Contact point paths of spherical gear pair with convex tooth and concave tooth Example The major spherical gear parameters are given in Table Based on the above developed mathematical model of spherical gears, the three-dimensional tooth profiles of convex-tooth and concave-tooth spherical gears can be plotted Fig 11 shows the convex-tooth and concave-tooth spherical gears Assume that the swaying axis n is the y0 axis, and h01 ¼ À60 to 60°, the contact paths on the tooth surfaces of two spherical gears are shown in Fig 11 7.2 Contact points path of tooth meshing in alterable swing planes In the initial position, the polar axes of two spherical gears are not collinear to each other Assume that the convex-tooth spherical gear has rotated about a fixed axis vector p10 with an angle hp10 From this position, the convex-tooth spherical gear rotates about another fixed axis vector p1 with an angle hp1 Along with the increase of hp1 , the contact paths on the tooth surfaces of two spherical gears are the special curves In Fig 9, to get the transformation matrixes from coordinate system S1 and S2 to coordinate system S0 , it can be simulated by using the following procedures: Step 1: In initial position, the angles between the axis vector p10 and three axes (x0 ; y0 and z0 ) of the coordinate system S0 are a; b and c, respectively To describe the coordinate transformation from S1 to S0 in the initial position, we use coordinate system S10 ðx10 ; y10 ; z10 Þ coinciding with the convex-tooth spherical gear in this position Then the transformation matrix from the coordinate system S10 to S0 can be obtained by: M010 ¼ M010 ða; b; c; hp10 Þ cos2 að1 À cos hp10 Þ þ cos hp10 6 cos a cos bð1 À cos hp10 Þ þ cos c sin hp10 ¼6 cos a cos cð1 À cos h Þ À cos b sin h p10 p10 cos a cos bð1 À cos hp10 Þ À cos c sin hp10 cos a cos cð1 À cos hp10 Þ þ cos b sin hp10 cos2 bð1 À cos hp10 Þ þ cos hp10 cos b cos cð1 À cos hp10 Þ À cos a sin hp10 cos b cos cð1 À cos hp10 Þ þ cos a sin hp10 cos2 cð1 À cos hp10 Þ þ cos hp10 0 0 07 7 07 ð15Þ Step 2: If the angles between the axis vector p1 and three axes (x0 ; y0 and z0 ) of the coordinate system S0 are a0 ; b0 and c0 , the angles between p1 and the three axes (x10 ; y10 and z10 Þ of the coordinate system S10 are a1 ; b1 and c1 , respectively a1 ; b1 and c1 can be calculated by: a1 ¼ ð10Þ Áp C Bx cosÀ1 @  A; ð10Þ x  jp1 j ð10Þ ð10Þ b1 ¼ ð10Þ Áp C By cosÀ1 @  A; ð10Þ y  jp1 j c1 ¼ ð10Þ Áp C Bz cosÀ1 @  A ð10Þ z  jp1 j ð16Þ ð10Þ Here, the axes x0 ; y0 and z0 are the representation of the three coordinate axes of S10 in S0 And then, when the convextooth spherical gear rotates about p1 with hp1 , the transformation matrix from the coordinate system S1 to S10 can be obtained by: M101 ¼ M010 ða1 ; b1 ; c1 ; hp1 Þ ð17Þ Step 3: The transformation matrix from the coordinate system S1 to S0 is that: M01 ¼ M010 M101 ð18Þ Step 4: According to the transmission characteristic of spherical gear pairs, when the convex-tooth spherical gear rotates about p1 with an angle hp1 , to simulate the meshing movement of the spherical gear pair, it can be regarded as the convex-tooth spherical gear rotating a fixed axis n with an angle h01 and the concave-tooth spherical gear rotating a fixed axis p(parallel to the vector unit n in ideal condition [7]) with an angle h02 The unit vector n and angle h01 can be calculated by: 1816 L Ting, P Cunyun / Mechanism and Machine Theory 44 (2009) 1807–1825 nx ð1Þ n ¼ ny ¼ z0  z0 nz h01 ¼ ð19Þ ð1Þ B z0 Á z C cosÀ1 @  A ð1Þ jz0 jz0  ð20Þ Assume that an ; bn and cn represent the angles between the unit vector n and three coordinate axes x0 ; y0 and z0 of S0 , we can obtain the following calculations: n n n x y z C B C B C an ¼ cosÀ1 B @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA; bn ¼ cosÀ1 @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA; cn ¼ cosÀ1 @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA n2x þ n2y þ n2z n2x þ n2y þ n2z n2x þ n2y þ n2z ð21Þ In Fig 9, the origin O20 of the fixed coordinate system S20 ðx20 ; y20 ; z20 Þ coincides with the origin O2 of the coordinate system S2 , and the coordinate axes of S20 and S0 are parallel to each other From the above discussion, we can get that the angles between p and three coordinate axes x20 ; y20 and z20 of S20 is equal to 180 —an ; bn and cn , respectively Therefore, the transformation matrix from the coordinate system S2 to S20 is that: À Á M202 ¼ M010 180 À an ; bn ; cn ; h02 ð22Þ In Fig 9, assume that C represents the center distance between two spherical gears We obtain the transformation matrix from the coordinate system S20 to S0 0 À1 0 7 M020 ¼ 0 À1 C 0 ð23Þ Step 5: the transformation matrix from the coordinate system S2 to S0 is that: M02 ¼ M020 M202 ð24Þ Lastly, according to step step 5, we can obtain the mathematical model and the unit normal vector of the convex-tooth spherical gear in S0 ð1Þ R0 ¼ M01 R1 ð25Þ ð1Þ n0 ð26Þ ¼ L01 n1 Similarly, the mathematical model and the unit normal vector of the concave-tooth spherical gear in S0 are as follows: ð2Þ R0 ¼ M02 R2 ð27Þ ð2Þ n0 ð28Þ ¼ L02 n2 where, L01 and L02 are the rotational transformation matrixes from coordinate systems S1 and S2 to S0 , respectively With the Eqs (25)–(28), Eqs (11) and (12), the contact point position of two spherical gear tooth surfaces can be obtained by using nonlinear solver Example Assume that the p10 is the x0 axis, p1 is the y0 axis and hp10 ¼ 0; 0:2 ; 2 and 20 Along with the change of hp1 , the contact paths on the tooth surfaces of two spherical gears are shown in Fig 12 The curves with the same color represent Fig 12 Contact point paths of spherical gear pair with convex tooth and concave tooth L Ting, P Cunyun / Mechanism and Machine Theory 44 (2009) 1807–1825 1817 z1 The convex tooth surface The ring direction The involute direction O1 y1 x1 Fig 13 Convex tooth surface of spherical gear the same meshing contact path from the same initial position (the same hp10 ) We can find that, when hp10 is equal to 0.2° and 2°, the contact paths on convex tooth surface and concave tooth surface are discontinuous on middle parts of the two teeth The reason is that two spherical gears on the middle parts of the convex tooth and concave tooth on which two spherical gears’ polar axes are collinear to each other are in line contact When meshing is near to this position, two spherical gears are separated from one side of tooth surface along with ring-direction 90° (in Fig 13) and begin to contact on anther side of tooth surface along with the ring-direction 270° 7.3 Contact points path of tooth meshing under the condition of assembly errors Fig 14 shows the three-dimensional model of the spherical gear mechanism It comprises the convex-tooth spherical gear, the concave-tooth spherical gear, the cross 1, the cross 2, and the tie-bar frame The cross and cross are installed on the tie-bar frame which is fixed, and two spherical gears are jointed with cross and cross 2, respectively Consider now that the mechanism has assembly errors which affect the transmission performance of the spherical gear set The assembly errors include the errors of center distance and axial misalignment According to Fig 14, we set up the coordinate systems as shown in Fig 15 The coordinate system S0 ðx0 ; y0 ; z0 Þ is fixed with the tie-bar frame (the same as S0 in Fig 9), and the coordinate system Sb ðxb ; yb ; zb Þ is fixed with the cross The coordinate systems S1 and S2 are fixed with the convex-tooth spherical gear and the concave-tooth spherical gear, respectively To describe the initial position of the spherical gear mechanism with assembly errors, the coordinate system S10 ðx10 ; y10 ; z10 Þ can be set up in Fig 15, which coincides with the initial position of convex-tooth spherical gear Because of the error of center distance, the spherical center O10 does not coincided with the symmetry center Ob (that is O0 ) of Fig 14 Three-dimensional model of spherical gear mechanism 1818 L Ting, P Cunyun / Mechanism and Machine Theory 44 (2009) 1807–1825 p2 O2 x2 y2 zb yb θ2 θ yb z0 z2 C y0 θ x0 y10 Δβ1b z1 θ x0 O0 x0 xb Ob z10 y1 θ1 x1 r10 r1 O1 O10 x10 Fig 15 Coordinate system set of spherical gear mechanism with assembly error the cross In initial condition, the spherical center of the convex-tooth spherical gear can be represented by the vector r10 ¼ ða0 ; b0 ; c0 Þ To simulate the error of axial misalignment, it can be performed by rotating the coordinate system S10 about axis xb with a misaligned angle Db1b According to the above kinematic transformation relationship of the spherical gear mechanism, the movement locus of the convex-tooth spherical center lies on the spherical surface, and the spherical center and radius are O0 and jr10 j, respectively In any instantaneous meshing, the meshing movement rule between two spherical gears in the meshing swaying plane (the shadow field in Fig 15) is the same as that of traditional spur involute gear pair Since the spherical gear mechanism has DOF, two spherical gears can rotate about two vertical axes to each other, respectively When the convex-tooth spherical gear rotates about the x0 axis with an angle hx0 and the yb axis with an angle hyb , respectively, the coordinate systems S1 of convex-tooth spherical gear and coordinate system S2 of concave-tooth spherical gear are shown in Fig 15 To get the transformation matrixes from coordinate system S1 and S2 to fixed coordinate system S0 , the following procedures can be used: Step 1: the position vector r1 of the spherical O1 in S0 can be calculated by 3 a0 cos hyb þ c0 sin hyb xr1 7 r1 ¼ b0 cos hx0 À c0 cos hyb sin hx0 þ a0 sin hx0 sin hyb ¼ yr1 c0 cos hx0 cos hyb þ b0 sin hx0 À a0 cos hx0 sin hyb zr1 ð29Þ Therefore, we obtain the translational transformation matrix from S1 to S0 0 xr 60 y r1 S01 ¼ 0 zr1 0 ð30Þ Step 2: When the convex-tooth spherical gear rotates about the x0 axis with hx0 , the transformation matrix from Sb to S0 is as follows: 0 cos h x0 M0b ¼ sin hx0 À sin hx0 0 cos hx0 07 7 05 ð31Þ In initial position, the transformation matrix from S10 to Sb is that cos Db 1b Mb10 ¼ sin Db1b 0 À sin Db1b cos Db1b 0 07 7 05 ð32Þ L Ting, P Cunyun / Mechanism and Machine Theory 44 (2009) 1807–1825 1819 The angles between the yb axis and coordinate axes of S10 are 90°, Db1b and 90°-Db1b When the convex-tooth spherical gear rotates about the yb axis with hyb , the transformation matrix from S1 to S10 is as follows: M101 ¼ M010 ð90 ; Db1b ; 90 À Db1b ; hyb Þ ð33Þ Therefore, we can obtain the transformation matrix from S1 to S0 M01 ¼ S01 M0b Mb10 M101 Step 3: Assume that ! h01 ð34Þ ! is the angle between the z1 axis and O1 O2 , the following equations are obvious: ! O1 O2 ¼ O0 O2 Àr1    !  O1 O2       !  O1 C  ¼  cos h0    !  !  ð1Þ  O1 C ¼ O1 C z0 ð35Þ ð36Þ ð37Þ According to the meshing theory of spherical gear pairs, the swaying plane is the shadow field surrounded by the z1 axis, the ! z2 axis and O1 O2 , as shown in Fig 15 The following equation can be gotten: ! ! ! xO2 C O C ¼ O C À O O ¼ y O2 C zO C ð2Þ z ð1Þ ! 0ð2Þ ð2Þ z0 ¼ O2 C ¼ z ð2Þ jO2 Cj ð2Þ z0 ð3Þ ð38Þ ð39Þ In Fig 15, since the y2 axis is always in the y0 z0 plane and is also perpendicular to the z2 axis, it is the intersecting line between the y0 z0 plane and the plane that passes through the origin O2 and is perpendicular to the z2 axis The unit vectors of y2 axis and x2 axis vector in S0 are calculated by the following formulas: ð2Þ y0 ð1Þ C ð2Þ ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi À2r yO2 C ¼ y0 ð2Þ 2 zO C ð2Þ 2r 2r y22 þ y0 ð3Þ ð2Þ y0 zO O2 C ð2Þ x0 ð1Þ ð40Þ ð2Þ ð2Þ ð2Þ ð2Þ x0 ¼ y0  z0 ¼ x0 ð2Þ ð2Þ x0 ð3Þ ð41Þ Finally, the transformation matrix from S2 to S0 is as follows: ð2Þ ð2Þ ð2Þ x0 ð1Þ y0 ð1Þ z0 ð1Þ ð2Þ ð2Þ x0 ð2Þ yð2Þ z0 ð2Þ ð2Þ M02 ¼ ð2Þ ð2Þ x0 ð3Þ yð2Þ ð3Þ z ð3Þ C 0 0 ð42Þ Import the Eqs (34) and (42) into Eq (25)–(28), the mathematical models and two meshing spherical gears in coordinate system S0 can be obtained And then, according to Eqs (11) and (12), the contact point position of two spherical gear tooth surfaces can be obtained by using nonlinear solver Example The assembly conditions are that r10 ¼ ða0 ; b0 ; c0 Þ ¼ ð1 mm;1 mm; À mmÞ; Db1b ¼ 0; hx0 ¼ 0, and hyb ¼ À45 to 60 By using the nonlinear solver, we can obtain the parameters: b1 ; lv ; b2 ; lc and h02 Fig 16 shows the contact point paths on tooth surfaces when two spherical gears mesh The blue curves on every tooth surface are the contact paths with no assembly errors (the same as in Fig 11) Fig 17 shows the contact point paths of the convex tooth surface and the concave tooth surface, respectively The discontinuous contact paths on the middle parts of the convex and concave tooth surfaces are caused by the assembly errors More detailed explanation is that: On the middle parts of the convex tooth and the concave tooth on which two spherical gears’ polar axes are collinear to each other, two spherical gears are in line contact; When spherical gear rotates near to this position, the convex tooth and concave tooth are separated from one side along with ring-direction 90° (in Fig 13) and begin to contact on anther side along with ring-direction 270° 1820 L Ting, P Cunyun / Mechanism and Machine Theory 44 (2009) 1807–1825 Fig 16 Contact point paths of spherical gear pair with convex tooth and concave tooth Fig 17 Contact point paths of convex tooth and concave tooth Fig 18 shows how with the change of hyb ðÀ45 to 60 Þ, error of center distance of the spherical gear set varies Fig 19 indicates that swaying angle error Dh of the polar axis z2 (output axis of the spherical gear pair) of concave-tooth spherical gear is smaller near to the position of hyb ¼ than on any other positions In Fig 20, the blue curve shows the azimuth angle of the z2 axis in the assembly error condition, and the red curve shows the azimuth angle of the z2 axis in the ideal assembly condition Unit: mm 1.5 0.5 -40 -20 20 40 60 θ yb(deg.) Fig 18 Error of center distance Unit: deg -2 -4 -6 -40 -20 20 θ yb(deg.) 40 60 Fig 19 Swaying angle error of output axis z2 L Ting, P Cunyun / Mechanism and Machine Theory 44 (2009) 1807–1825 ‘ Unit: deg 150 ’ Case ‘ _’ Case 100 50 -40 -20 20 θ yb(deg.) 40 60 Fig 20 Azimuth angle of output axis z2 1.5 Unit: mm ‘ _’ Example ‘ -’ Example 0.5 -0.5 -40 -20 20 40 60 θ yb(deg.) Fig 21 Error of center distance ‘ _’ Example ‘ -’ Example Unit: deg -2 -4 -6 -40 -20 20 40 60 θ yb(deg.) Fig 22 Swaying angle error of output axis z2 200 ‘ _’ Example ‘ -’ Example Unit: deg 150 ‘ _’ Example 100 50 -40 -20 20 θ yb(deg.) 40 Fig 23 Azimuth angle of output axis z2 60 1821 1822 L Ting, P Cunyun / Mechanism and Machine Theory 44 (2009) 1807–1825 Example the assembly conditions is that r10 ¼ ða0 ; b0 ; c0 Þ ¼ ð0:1 mm;0:1 mm; À 0:1 mmÞ, and other parameters of initial assembly condition are invariable Similarly, we can get the contact point paths on tooth surfaces when two spherical gears mesh, and the contact paths are very near to the ideal contact paths because of the little assembly errors Fig 21 shows the errors of center distance in example and example Fig 22 describes the swaying angle errors Dh of the polar axis z2 in example and example Fig 23 draws the azimuth angles of the polar axis z2 in example 1, example and example 4, respectively Stress analysis of spherical gear set In this study, the goal of stress analysis is to determine the contact and bending stresses of spherical gear pair in meshing drive with the finite element (FE) method 8.1 Finite element model of spherical gears The FE model of gear mechanism is the key to the FE calculation, and different model selections and different boundary conditions have different influences on the results The contact intensity analysis of meshing gear pair with FE method has been investigated in references [11–14] However, the spherical gear is different, and means of restrictions and loads are more complex So we divide the complex solid models into simple sub-models and control the discretization of these sub-models into finite elements Fig 24 shows the meshing model of a spherical gear pair Fig 24 FE model of meshing spherical gears Fig 25 FE model with restriction and concentrated load L Ting, P Cunyun / Mechanism and Machine Theory 44 (2009) 1807–1825 1823 8.2 Stress analysis According to the characteristics of spherical gear teeth, bending resistance of convex tooth is weakest in the process of spherical gears drive, so this section focuses on the contact and bending stress analysis in the meshing contact of convex tooth and concave tooth To improve the accuracy of stress analysis, and reduce the number of elements and nodes, and save the computing time, the solid models of convex tooth and concave tooth are meshed more densely than other part of solid model of spherical gear set Fig 25 shows the FE model of whole meshing spherical gear pair with restriction and concentrated load The tooth parameters of spherical gears are given in Table 1; and the total number of elements is 84,370 with 112,554 nodes The material is alloy steel with properties of Young’s Modules E ¼  105 MPa and Poisson’s ratio of 0.3 A concentrated force of 100 N has been applied to the axis of convex spherical gear Fig 26 Stress distribution of convex tooth and concave tooth with mid-axis superposition Fig 27 Stress distribution of convex tooth and concave tooth about their spherical centers with a rotating angle À8° Fig 28 Stress distribution of convex tooth and concave tooth about their spherical centers with a rotating angle 15° 1824 L Ting, P Cunyun / Mechanism and Machine Theory 44 (2009) 1807–1825 Contact stress (MPa) 800 600 400 200 -15 -10 -5 Rotating angle(deg.) Fig 29 Contact stresses along the contact path Bending stress (MPa) 50 40 30 20 10 -15 -10 -5 Rotating angle(deg.) Fig 30 Bending stresses along the contact path Fig 26a and Fig 26b show the distribution of contact and bending stresses on the convex tooth surface and concave tooth surface, respectively, when the mid-axis of convex gear coincides with the mid-axis of concave gear Fig 27a and Fig 27b show the distribution of contact and bending stresses on the convex tooth surface and concave tooth surface, respectively, when two spherical gears rotate about their spherical centers with an angle À8° Fig 28a and Fig 28b show the distribution of contact and bending stresses on the convex tooth surface and concave tooth surface, respectively, when two spherical gears rotate about their spherical centers with an angle 15° Figs 29 and 30 show the variation of contact and bending stresses along the path of contact in a chosen swaying plane From Figs 26–30, we can find that the contact area of spherical gear tooth surface at a chosen point presents an ellipse, and the contact ellipse in the mid-position of tooth is larger than that in other position and contact and bending stresses is less than that in other position Conclusions In this paper, a grinding manufacture method of spherical gear pairs is put forward, and a grinding machine is designed Based on the mathematical models of grinding tools, the mathematical models of convex-tooth and concave-tooth spherical gear are obtained, respectively Furthermore, TCA of the spherical gear pair is performed Finally, the contact and bending stress are computed by FEM Computer simulations indicate that: Since the spherical gear pair is a gear-driven mechanism with DOFs and the teeth are different, the contact point paths on tooth surfaces of two spherical gears are complex and various When two spherical gears mesh from the position in which polar axes of two spherical gears are collinear to each other, their meshing principle is the same as that of the traditional spur involute gears Furthermore, the contact paths on tooth surfaces of spherical gears and spur involute gears are similar When two spherical gears mesh from the position where there is an angle between polar axes of two spherical gears, the contact paths belong to spatial curves, and distribution of contact point paths on tooth surfaces are complex; The assembly errors not only affect the contact point path of spherical gear set, but also have an effect to the transmission performance, and different assembly errors have different effects The variation of convex tooth contact and bending stresses is researched: contact stress on tooth mid-position is less than that on tooth base and tooth tip; bending stress on tooth mid-position is more than that on tooth base and tooth tip L Ting, P Cunyun / Mechanism and Machine Theory 44 (2009) 1807–1825 1825 Now, the design of grinding machine for the generation of spherical gears has been finished, and the main components, such as principal axis of the plate grinding wheel, the rotating floor and the guideway and so on, have been purchased The installation and test of the grinding machine will be finished in the end of 2009 And then, the grinding experiment for the generation of spherical gears will be performed Acknowledgement This research has been supported by the national 863 Project under Grant No 2006AA09Z235 References [1] Pan Cunyun, Shang Jianzhong, Transmission theory and kinematic analysis of spherical gear, Mach Design Res (in Chinese) (1996) 14–16 [2] Z.Q Li, G.X Liu, H.M Li, Research on cone tooth spherical gear transmission of robot flexible joint, ASME 26 (5) (1990) 56–60 [3] Shyue-Cheng Yang, Chao-Kuang Chen, Ke-Yang Li, A geometric model of a spherical gear with a double degree of freedom, J Mater Process Technol 123 (2002) 219–224 [4] Shyue-Cheng Yang, A rack-cutter surface used to generate a spherical gear with discrete ring-involute teeth, Int J Adv Manuf Technol 27 (2005) 14– 20 [5] Y.C Tsai, W.K Jehng, Rapid prototyping and manufacturing technology and applied to forming of spherical gear sets with skew axes, J Mater Process Technol 95 (1999) 169–179 [6] Pan Cunyun, Research on transmission theory and manufacture method of spherical gear, Natl Univ Defense Technol (in Chinese) (2001) [7] Pan Cun-yun, Wen Xi-sen, Research on transmission principle and kinematics analysis for involute spherical gear, Frontiers Mech Eng China (2006) 183–193 [8] Faydor L Litvin, Alfonso Fuentes, Gear Geometry and Applied Theory, Cambridge University Press, 2004 [9] Vilmos Simon, Computer simulation of tooth contact analysis of mismatched spiral bevel gears, Mech Mach Theory 42 (2007) 365–381 [10] Chia-Chang Liu, Jia-Hong Chen, Chung-Biau Tsay, et al, Meshing simulations of the worm gear cut by a straight-edged flyblade and the ZK-type worm with a non-90° crossing angle, Mech Mach Theory 41 (2006) 987–1002 [11] Faydor L Litvin, Alfonso Fuentes, Gear Geometry and Applied Theory, Cambridge University Press, 2004 [12] F.L Litvin, Alfonso Fuentes, Ignacio Gonzalez-Perez, et al, Modified involute helical gears: computerized design, simulation of meshing and stress analysis, Comput Methods Appl Mech Eng 192 (2003) 3619–3655 [13] Shuting Li, Finite element analyses for contact strength and bending strength of a pair of spur gears with machining errors assembly error and tooth modifications, Mech Mach Theory 42 (2007) 88–114 [14] Tengjiao Lin, H Ou, Runfang Li, A finite element method for 3D static and dynamic contact/impact analysis of gear drives, Comput Methods Appl Mech Eng 196 (2007) 1716–1728