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Gedong Jiang* and Xuesong Mei ... drive system often works as a reducer, it consists of three ... conditions as a certain engagement range, but he did not.
Proceedings of 2013 IEEE International Conference on Mechatronics and Automation August 4 - 7, Takamatsu, Japan

Deformation and Stress Analysis of Short Flexspline in the Harmonic Drive System with Load Chuang Zou and Tao Tao

Gedong Jiang* and Xuesong Mei

Xi’an Jiaotong University Xi'an, Shaanxi, 710049, PR China

School of Mechanical Engineering State Key Laboratory for Manufacturing Systems Engineering Xi’an Jiaotong University Xi'an, Shaanxi, 710049, PR China

[email protected], [email protected]

[email protected], [email protected]

School of Mechanical Engineering

Abstract - A detailed surface to surface contact finite element model of harmonic drive system is established in this paper to reveal the stress and deformation states of short flexspline. The boundary condition of the gear teeth meshing is evaluated by experimental formula. The stress and deformation of flexspline are solved and their relationships with varying loads are analyzed. It is found that the deformation and stress at the flexspline gear cross section change geometrically with heightened load, but the distributions of the deformation and stress increments remain unchanged. The solution results are compatible with the cases of flexspline destruction and axial stress distribution under load. This approach can help to optimize the structure and manufacturing process of harmonic reducer and increase the reliability of related automation equipment.

transmission ratio, high precision, small backlash, stable transmission and small volume and weight [1]. Harmonic drive system often works as a reducer, it consists of three components which are an elliptical wave generator(WG), a non-rigid flexspline(FS) and a rigid circular spline(CS), as shown in Fig. 1 [2]. Circular spline has an inner teeth and flexspline outer teeth. Periodic elastic deformation of the flexspline by the wave generator causes a relative rotation in the opposite rotation direction of the wave generator in the circular spline. The harmonic reducer’s main failure forms are fatigue fracture of roots and wear of tooth surface. However, the complicated load conditions of robots may have a tremendous impact on the harmonic reducer and even increase the possibility of its failures. So the deformation and stress calculation and analysis of short flexspline with different loads are of great significance. The finite element method is an effective method to study harmonic drive. KIKUCHI et al calculated the stress on flexible bearing and flexspline, but did not consider the impact of the loads on the calculation results [3]. Zhang Shimin let different sizes of force couples act on two symmetric teeth of flexspline to simulate the different meshing forces, but this method cannot embody the multi-tooth meshing process of harmonic drive [4]. Bo Shuxin chose the better boundary conditions as a certain engagement range, but he did not analyse the stress state of flexspline with different loads [5]. Wu Weiguo et al established meshing teeth model between flexspline and circular spline, but the contact was defined as a flexible body to flexible body contact [6]. OSTAPSKI et al carried out finite element analysis and comparison on different styles of flexsplines, but the flexspline was equivalent to a simple ring by him [7-8]. Gao Haibo et al analysed the sensitivity of structure parameters and temperature on the stress state of flexspline, but the influence of loads on stress was not taken into account [9].

Index Terms - Harmonic drive system; short flexspline; finite element model

NOMENCLATURE daR Diameter of addendum circle. dWR Diameter of cylinder’s outer circle. dRB Diameter of cylinder’s inner circle. Width of section in front of gear. bl Width of gear. bR Radius of fillet between gear front-end and cylinder. R1 R2 Radius of fillet between cylinder and flange. R3 Radius of fillet between gear back-end and cylinder. Length of flexspline. L Thickness of flange. H Diameter of flange’s outer circle. dT Thickness of cylinder. δ Angle between the symmetry axis CC' of the meshing ϕ1 range and the major axis AA' of wave generator. Angles standing for the left meshing range. ϕ2

ϕ3 qt qr q dg

Angles standing for the right meshing range. Tangential component of meshing force. Radial component of meshing force. Meshing force. Diameter of flexspline’s reference circle. I. INTRODUCTION

Fig. 1 Component of harmonic reducer

Harmonic drive system is widely used in many automation equipment such as radar, moment controlled gyroscope, industrial robot joint because of its large

978-1-4673-5558-2/13/$31.00 ©2013 IEEE

In summary, few studies have been done on the analysis of the short harmonic reducer’s deformation and stress states

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δ dWR H bl dT R2 R1 R3 daR

with different loads, and the impact of actual robot conditions on the performance of harmonic reducer has not been reflected. So, firstly, we established the contact finite element model between the short flexspline and wave generator, evaluated the boundary conditions of meshing force distribution between circular spline and flexspline with different loads, and solved the stress and deformation of flexspline. Then, the relationship and the variation between states of flexspline and different loads were revealed and analysed.

0.48 60.96 4 2.4 36 3 3 6 62.622

C. Boundary Condition Definition We established a surface-surface contact element model to analyse the contact between flexspline and wave generator. We defined the surface of wave generator’s outer ring as the contact surface with target170 element, and defined the surface of flexspline’s inner ring as the contact surface with target174 element, and the friction coefficient was set to 0.01 [10]. At the back-end of flexspline, the flange’s degrees of freedom in three directions were set to zero. The meshing force between flexspline and circular spline is difficult to evaluate. It’s not only relative to the load acting on the out shaft of the reducer, but also relative to the range of meshing. Shown in Fig. 3, the meshing force distribution could be obtained from the experimental results [11-12].

II. FINITE ELEMENT MODELING AND SOLUTION A. Geometric modeling of Harmonic Reducer The structure of short flexspline is shown in Fig. 2, and the geometrical parameters are shown in Table I. The length to diameter ratio of flexspline is 0.5, the speed ratio of reducer is 100, the module of gear is 0.3mm, material of flexspline is 30CrMnSiA, modulus of elasticity is 206GPa, Poisson’s ratio is 0.3. We chose an elliptical cam as a wave generator. Its width is 10mm, radius of its major axis is 30.2mm, radius of its minor axis is 29.8mm. Firstly, we established the geometric models of flexspline and wave generator in PRO/E5.0. Then we put the geometric models into ANSYS for the calculation of flexspline’s deformation and stress and further analysis of their relationship with loads. B. Mesh Generation The amount of flexspline’s deformation caused by wave generator and the width of cylinder are in the same order of magnitude, so we chose the elements which can simulate large deformation in ANSYS. To reduce the number of meshes, we divided flexspline into two parts: cylinder part and gear part. The cylinder part was meshed in the style of sweep because of its regularity, and the element was chosen as 8 nodes solid185 hexahedral element; the gear part was meshed in the style of freedom, and the element was chosen as 10 nodes solid187 tetrahedral elements. The whole flexspline consists of 106907 elements.

Fig. 3 Meshing force distribution

According to experimental results, if we consider ϕ2 is approximately equal to ϕ3 , the meshing force in the range of

ϕ2 could be evaluated in the form ⎧⎪qt = qt max cos ⎡⎣π (ϕ − ϕ1 ) / 2ϕ2 ⎤⎦ ⎨ qr = qt tan α ⎪⎩

(1)

In the same way, the meshing force in the range of could be evaluated in the form:

⎧⎪qt = qt max cos ⎡⎣π (ϕ − ϕ1 ) / 2ϕ3 ⎤⎦ ⎨ qr = qt tan α ⎪⎩

3

(2)

If the moment which stands for the load acting on the output of reducer is T, the relationship between T and qt max could be represented as

Fig. 2 Structure of Short Flexspline Table I Geometrical parameters of flexspline Parameter Value(mm) dRB 60 bR 12 L 30

T = 4∫

ϕ2

ϕ1

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2

⎛ dg ⎞ bR ⎜ ⎟ qt max cos ⎡⎣π (ϕ − ϕ1 ) / ( 2ϕ2 ) ⎤⎦ dϕ ⎝ 2 ⎠

(3)

(

)

qt max = πT / 2ϕ 2 d g2 bR (4) Actually, the engaging force only acts on one side of gear, so the continuous distribution of the engaging force shown in equation (1) to equation (4) was changed into discrete distribution in the style of piecewise integral and the integration step is the teeth space. We considered that the two sides of one gear respectively corresponded to angel ϕi and ϕi +1 , so the tangential force and radius force acting on this gear tooth was as follows ϕi +1 π (ϕ − ϕ1 ) ⎛ dg ⎞ (5) f t = ∫ bR ⎜ ⎟ qt max cos dϕ ϕi 2ϕ 2 ⎝ 2 ⎠ (6) f r = f t tan α We got an approximation of the meshing range size according to the wave generator outer circle figure and the flexspline inner circle figure, so the ϕ2 was set to 40°. We made the forces shown in equation (5) and (6) to act on every node of flexspline, and then the whole model of flexspline in ANSYS is shown as Fig. 4.

15mm away from the rim, and the three cross sections are perpendicular to the axial lead of cylinder, as shown in Fig. 2. Without load, the axial distribution of flexspline cylinder deformation is shown in Fig. 5. The maximum deformation point is at the rim, and the maximum deflection is 0.2852mm. It could be found that there is an approximate linear relationship between the deflection Ck of the flexspline cylinder and the distance s to the rim of flexspline, and this result matches up to the experimental result [12]. And the deflection of flexspline cylinder at the major axis of wave generator is larger than at the minor axis and at the middle axis between the major axis and the minor axis of wave generator. With load, the axial distribution change of cylinder deflection at the major axis of wave generator is shown in Fig. 6. On the whole, the deflection ΔCf of cylinder decreases with the increase of load T. The average increment is less than 1% of the deflection without load. The order of magnitude of the increment is so little that it could be ignored.

D. Deformation and Stress Solution In order to decrease the scale of solution and improve the computational speed and precision, we selected the precondition conjugate gradient method in ANSYS, and set the size of solution step enough large. In harmonic reducer, flexspline is the weakest link, so we mainly analyse the deformation and stress of flexspline with different loads in this article. Because the deformation and stress of flexspline were caused by both the stretching effect of wave generator and the meshing force between flexspline and circular spline, we firstly solved the deformation and stress of flexspline only with the action of wave generator (without load), then we made the mesh force which could simulate the load on the output of the reducer to act on the teeth of flexspline for further solution and analysis (with load).

Fig. 5 Axial distribution of cylinder deflection without load

Fig. 6 Axial distribution of cylinder deflection increment at the major axis of WG with load

And then we analysed the deflection of three cross sections to study the radial distribution of flexspline cylinder deflection. As Fig. 7, we could find that the three figures of flexspline cylinder deflection without load are all like cruciate flowers, the maximum deflection point is at the major axis of wave generator and the minimum deflection point is at the point 45° away from minor axis of wave generator. The front cross section deflection is larger than other two sections, and the maximum is 0.2773mm. The rear cross section deflection is less, and the minimum is 0.1607mm.

Fig. 4 Finite element model of flexspline

III. RESULTS AND ANALYSIS A. Deformation Analysis Firstly, we analysed the deformation of flexspline both without load and with load. According to the observation of all the solution results, we defined the front cross section that is 1mm away from the rim of flexspline, the gear cross section that is 11mm away from the rim, the rear cross section that is

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Fig. 7 Radial distribution of flexspline cylinder deflection without load

With Load, the deflection distribution of flexspline at gear cross section rotates clockwise about 45°, in the same direction of tangential component of meshing force, as shown in Fig. 8. It is found that the deflection increment increases geometrically with heightened load, but the distribution of the increment remains unchanged. We divided the increment figure into four parts by boundary of four zero points in this figure. In part 1 and part 3, the increment increases with the increase of load, however, in part 2 and part 4, the increment decreases with the increase of load. B. Stress Analysis Next, we analysed the stress of flexspline both without load and with load. The axial distribution of cylinder deflection without load was shown in Fig. 9. The stress of cylinder Pk at the major axis of wave generator is obviously larger than at other two sections. The maximum stress of cylinder is 220.8MPa at the rear section of gear circle where the failure of flexspline appears. This result is consistent with the case of destruction of flexspline in reference [14]. And stress fluctuation appears at the rear section of gear circle because of the concentration effort from the edge of wave generator and the boundary effect of filet. As Fig. 10, with load, the stress increment ΔPf of flexspline at the major axis of wave generator appears fluctuation. The change trend of flexspline cylinder stress is consistent with the experimental result in reference [15]. It could be found that the fluctuation increases with the heighted load, so the impact of load on the stress fluctuation cannot be ignored. But the average stress increment is less than 5% of the stress without load, so the effect of wave generator has a major contribution to the stress amplitude. And then we analysed the stress of three cross sections to study the radial distribution of flexspline cylinder stress. As Fig. 11, the stress of flexspline cylinder shows dramatic fluctuation. The amplitude is largest and most volatile at the gear circle of cylinder, and the maximum stress is 256.5MPa. The reason was that the flexspline cylinder was deformed as many prisms by wave generator which has a less curvature radius than the cylinder at the major axis, and the concentration effect appeared at the junction of those prisms [12]. So it’s further translated that it’s most possible that destruction of flexspline happened at gear cross section, and the impact of load on the stress of flexspline could not be ignored.

2

1

3

4

Fig. 8 Radial distribution of flexspline cylinder deflection increment at gear cross section with load

Fig. 9 Axial distribution of cylinder deflection without load

Fig. 10 Axial distribution of cylinder deflection increment at the major axis of WG with load

As Fig. 12, in the meshing range, serrated stress increment distribution appears at the gear cross section of flexspline. The stress with load increases in 40°and decreases in 40°to 90° clockwise from the major axis of wave generator, but the stress decreases in 40°and increases in 40°to 90° anticlockwise from the major axis of wave generator. The stress distribution of flexspline at gear cross section rotates clockwise about 20 °, in the same direction of tangential component of meshing force. But it could be found that the increment increases geometrically with heightened load, but the distribution of the increment remains unchanged. The impact of the change in load on the stress of the flexspline should be taken into account.

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ACKNOWLEDGEMENTS This study is supported by the National Natural Science Foundation of China program, under number: 51175400 and the Program for Changjiang Scholars and Innovative Research Team in University, under number: IRT1172. REFERENCES [1] R. Slatter and G. Mackrell, “Harmonic Drives in Tune with Robots,” Industrial Robot, vol. 21, no. 3, pp. 24-25, 1994. [2] Harmonic Drive LLC, Comprehensive Dictionary of Harmonic Drive Combination Production, Harmonic Drive LLC, pp. 01-03, 2011. [3] M. Kikuchi, R. Nitta and Y. Kiyosawa, “Stress Analysis of Cup Type Strain Wave Gearing,” Key Engineering Materials, pp. 129-134, 2003. [4] S. Zhang and C. Li, “ANSYS Based Analysis of Flexspline in Harmonic Drive,” Drive System Technique, vol. 23, no. 1, pp. 13-17, 2009. [5] S. Bo, “Analysis of Flexspline Stress and Tooth Wear in Harmonic Gear Drive,” Thesis of Harbin Institute of Technology, 2008. [6] W. Wu, Y. Zhang and F. Liang, “Stablishment and Analysis of FEM Model for Harmonic Drive with Contact Pairs between Meshing Tooth Surface,” Journal of Mechanical Transmission, vol. 35, no. 12, pp. 3941, 2011. [7] W. OSTAPSKI and I. MUKHA, “Stress State Analysis of Harmonic Drive Elements by FEM,” Technical Sciences, vol. 55, no. 1, pp. 115123, 2007. [8] W. OSTAPSKI, “Analysis of the Stress State in the Harmonic Drive Generator-flexspline System in Relation to Selected Structural Parameters and Manufacturing Deviations,” Technical Sciences, vol. 58, no. 4, pp. 683-698, 2010. [9] H. Gao, Z. Li and Z. Deng. “Sensitivity Analysis of Cup-Shaped Flexible Gear Parameters to Its Stress Based on ANSYS,” Journal of Mechanical Engineering, vol. 46, no. 5, pp. 2-6, 2010. [10] G. Xiang, “Analysis and Study on the Flexible Wheel of Harmonic Gear Drive by Finite Element Method,” Thesis of Sichuan University, 2005. [11] O. KAYABASI and F. ERZINCANLI. “Shape Optimization of Tooth Profile of a Flexspline for a Harmonic Drive by Finite Element Modeling,” Materials and Design, pp. 441-447, 2007. [12] M. Ivanov, Harmonic gear drives, Moscow, Visajas Kola Press, 1981. [13] H. Dong. “Study of Kinematics and Meshing Characteristic of Harmonic Gear Drives Based on the Deformation Function of the Flexspline,” Thesis of Dalian Unversity of Technology, 2008. [14] C. Guan and S. Yuan. “A Stress Calculation on Flexspline of the Harmonic Drive,” Journal of Mechanical Strength, vol. 16, no. 1, pp. 29-30, 1994. [15] Y. Shen and Q. Ye, The Theory and Design of Harmonic Gear Drive, Beijing, China Machine Press, 1985.

Fig. 11 Radial distribution of flexspline cylinder stress without load

Fig. 12 Radial distribution of flexspline cylinder stress increment at gear cross section with load

IV. CONCLUSIONS In this paper, the stress and deformation of flexspline in harmonic drive system under different loads were solved and their relationship with varying load was analysed, the main conclusions were as follows 1) There was large stress amplitude and stress fluctuation at the rear section of gear circle, so the gear cross section was a weak point of the flexspline in harmonic drive system. 2) The effect of wave generator has a major contribution to the deformation and stress amplitude of flexspline cylinder, but the impact of load and the change of load on the failure of the harmonic drive system could not be ignored. 3) With load, the deformation and stress increments increased geometrically with heightened load, but the distribution of the increments remain unchanged. The approach proposed in this paper can be used to solve the stress and deformation of flexspline. The designer could change their designs according to the calculation results by the method proposed in this paper to ensure that the stress or deformation of the harmonic drive components are properly low and change stably. It is hoped that this current study would generate more interests in the new structure and manufacturing process of the harmonic drive system which will be helpful to increase the reliability or other performances of related automation equipment.

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