Mech. Sci., 12, 185–192, 2021https://doi.org/10.5194/ms-12-185-2021© Author(s) 2021. This work is distributed underthe Creative Commons Attribution 4.0 License.

Contact dynamics analysis of nutation drive withdouble circular-arc spiral bevel gear based on

mathematical modeling and numerical simulation

Yujing Su, Ligang Yao, and Jun ZhangSchool of Mechanical Engineering and Automation, Fuzhou University, Fuzhou 350116, China

Correspondence: Ligang Yao (ylgyao@fzu.edu.cn)

Received: 1 November 2020 – Revised: 11 January 2021 – Accepted: 16 January 2021 – Published: 18 February 2021

Abstract. This paper describes a dynamic mathematical model of a new type two-stage nutation drive systemwith double circular-arc bevel gears. The dynamic displacement–vibration coupling model takes into account thegyro torque and side clearance of the nutating gear. A numerical analysis geometric model of the nutation drivesystem is developed. The geometric model considers the time-varying and contact deformation of nutation gearmeshing. Subsequently, the model is analyzed using the ANSYS/LS-DYNA software, and the dynamic contactcharacteristics of the nutation gear are calculated. The simulation results revealed that the nutation transmissionof the double circular-arc spiral bevel gear is reliable. Moreover, the mathematical verification showed that thesimulation is feasible and accurate. The research has important significance for improving the tooth load capacityand transmission stability of nutation drives.

1 Introduction

Nutation drive is a new type of motion that has been widelyinvestigated in recent years. Compared to traditional geardrive, nutation gear transmission has characteristics such ashigh transmission efficiency, large transmission ratio, andstable transmission. Nutation drive has been widely used inmechanical transmission. Kemper (1981) proposed a nuta-tion traction device. Green (2008) developed the kinemat-ics model of a nutation mechanism and performed dynamicanalysis. Uzuka et al. (2009) presented and optimized a newnutation pneumatic motor with small size and large torquebased on the nutation principle (Oda et al., 2010). In peri-cyclic mechanical transmission, the tooth load capacity andnutation gear meshing efficiency are investigated (Saribayand Bill, 2013). The kinematics, mechanical efficiency, anddynamic balance of nutation drive have been analyzed basedon a multi-body model (Fanghella et al., 2015, 2016). Yaoet al. (2010) designed a double circular-arc spiral bevelgear nutation transmission based on the spiral theory, andHong et al. (2015) developed a kinematics model of nu-tation drive. To dynamically analyze nutation drive, Su etal. (2019) conducted nonlinear finite-element analysis on the

dynamic meshing of a double circular-arc spiral bevel gearnutation drive. Lin et al. (2020) developed a dynamic modelof double-sided meshing nutation drive with double circular-arc spiral bevel gears and analyzed its corresponding vi-bration modes. Concerning magnetic nutation transmission,Huang et al. (2017) built a magnetic nutation gear reducermodel and analyzed its torque, while Lou and Yao (2020) es-tablished a mathematical model of magnetic nutation trans-mission and performed numerical simulations.

With respect to gear meshing contact, the development ofintegrated computer finite-element analysis (FEA) programspromotes the contact analysis of spiral bevel gear transmis-sion (Simon, 2007). Lin et al. (2007) demonstrated a finite-element method for 3D static and dynamic contact/impactanalysis of gear drives, where the dynamic response of thegear can be obtained by performing FEA. Under nutationgear mesh contact, the nutation gear is subjected to largeloads and impacts, and the contact dynamic characteristicsaffect its transmission performance. However, the main bodyof research focuses on the structural design, vibration analy-sis, and static analysis of nutation gears, with the research ondynamic contact analysis being relatively limited.

Published by Copernicus Publications.

186 Y. Su et al.: Contact dynamics analysis of nutation drive

In this paper, a two-stage double circular spiral bevel gearnutation reducer (Fig. 1) was investigated, and a multi-DOFcoupling vibration dynamic model of double circular spiralbevel gear nutation transmission was established using theconcentrated parameter method. The dynamic model takesinto account the nutation gyro moment, meshing transmis-sion error, and gear tooth side clearance. In this paper, thenonlinear FEA method is used for the numerical simulations,the contact deformation and time-varying meshing are takeninto consideration, and the dynamics and contact characteris-tics during meshing are solved simultaneously. The dynamiccontact meshing stress and strain, vibration displacement, vi-bration speed, and dynamic meshing force of the nutationdrive are determined, which are important for the reductionof nutation drive vibration and shock as well as for improvingthe transmission performance.

2 Dynamic modeling of the nutation drive

2.1 Mathematical model

A cross section of the two-stage nutation drive reducer struc-ture is shown in Fig. 2. Gear 1 and gear 3 are externalbevel gears, which are fastened by screws to perform nuta-tion movement. Gear 2 and gear 4 are internal bevel gears.The rotation speed is transmitted to gear 1 and gear 3 by theinput shaft. Gear 1 meshes with gear 4 and gear 3 mesheswith gear 2. Gear 2 is fixed to the frame, and torque is out-put from gear 4. In this paper, gear 1 and gear 4 are callednutation external bevel gear and output internal bevel gear,respectively.

Figure 3 illustrates the dynamic model of the two-stagenutation gear system, whereO–xyz represents the global co-ordinate system, O–x0y0z0 is a nutation coordinate system,and origin O is on the conical point of the nutation externalbevel gear. The gears are simulated by the lumped parametermethod, the axle is simulated by mass-free rigid elements,and the bearing is elastically supported by a spring and adamper. Subscripts i = 1, 3, 2 and 4 represent the externalbevel gears 1 and 3 and the internal bevel gears 2 and 4.

2.2 Dynamic differential equations

In Fig. 4, the force on the nutation external bevel gear is an-alyzed. The meshing forces on the nutation bevel gear alongeach coordinate direction are expressed as Eq. (1):Fnj = kmj (t)f (λnj )+Cmj λnj ,Fix =−Fnj (cosδi sinαn+ sinδi sinβ cosαn),Fiy = Fnj (cosβ cosαn),Fiz =−Fnj (cosδi sinβ cosαn− sinδi sinαn),

(1)

where j = 1, 2 represents the gear pairs 1 and 2, kmj (t) rep-resents the time-varying meshing stiffness of the gear pairs,and f (λnj ) is the clearance function.

The backlash can be expressed as

fm(λni)=

λni − b λni > bi/2,0 |λni | ≤ bi/2,λni + b λni <−bi/2,

(2)

where bi (i = 1, 2) is the tooth backlash of each gear pair, andλni is the dynamic relative displacement between the mesh-ing gear teeth due to vibration and error during the bevel gearmeshing process. The backlash function is shown in Fig. 5.

According to the Hamilton principle, the dynamic differ-ential equations of the two-stage nutation drive system canbe obtained as follows.

m2x2+C2x x2+ k2xx2+F2x = 0m2y2+C2y y2+ k2yy2+F2y = 0m2z2+C2zz2+ k2zz2+F2z = 0J2θ2 = T1−Fn2 · rm2m13x13+ 2C13x x13+ 2k13xx13−F2x +F4x = 0m13y13+ 2C13y y13+ 2k13yy13−F2y +F4y = 0m13z13+ 2C13zz13+ 2k13zz13−F2z+F4z

=L01rm1+L03rm3

J13θ13 = T13−Fn2+Fn4m4x4+C4x x4+ k4xx4−F4x = 0m4y4+C4y y4+ k4yy4−F4y = 0m4z4+C4zz4+ k4zz4−F4z = 0J4θ4 =−T4+Fn4rm4

(3)

Subsequently, the dynamic differential equation of the nuta-tion system can be written as

Mu+Cu+Ku= Rt , (4)

u= [x2,y2,z2,θ2,x13,y13,z13,θ13,x4,y4,z4,θ4]T. (5)

3 Computer simulation

The dynamic differential equation of the nutation system(Eq. 4) can be solved numerically. The geometric model wasdeveloped in the HYPERMESH software (Fig. 6) and nu-merically solved using the ANSYS/LS-DYNA software. Thegear parameters are shown in Table 1. In the two-stage nu-tation drive system, the nutation external and output inter-nal bevel gears are the main transmission components, whichwere set as elastic bodies, the material of the double circular-arc spiral bevel gear was defined as structural steel with anelastic modulus E of 206 GPa, a Poisson ratio of 0.29, anda mass density of 7850 kg m−3, while the remaining compo-nents were set as rigid bodies. To obtain a higher accuracy,eight-node hexahedral elements were used to mesh the ex-ternal nutation and output internal bevel gears. The meshingresult, consisting of a total of 116 704 elements and 140 126nodes, is shown in Fig. 7.

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Figure 1. 3D model of the two-stage nutation reducer.

Figure 2. Cross-sectional diagram of the nutation reducer structure.

Table 1. Main gear parameters.

Parameter Nutation Outputexternal internal

bevel gear bevel gear

Spiral angle β 25◦ 25◦

Normal pressure angle 24◦ 24◦

Nutation angle ϕ 5◦ 5◦

Number of teeth 26 28Pitch cone angle δ 47.19◦ 127.8◦

Module 2 2Cone distance R 50 mm 50 mm

4 Simulation analysis of the nutation drive system

4.1 Stress distribution

In order to investigate the tooth surface meshing condition,the nutation drive system was simulated under an input speedof 1050 r min−1, and the stress distributions on the nuta-tion external bevel (gear 1) and output internal bevel gears(gear 4) are shown in Fig. 8. In the figure, it can be seen

that the stress was distributed along the tooth line, and dur-ing transmission, there were three teeth under stress, whichimproved the stability of the transmission.

4.2 Equivalent stress

The plots of the equivalent stress over time of the nutationgear are shown in Figs. 9 and 10. The input speed was set as1050 r min−1 and the simulation time was 0.2 s. Accordingto the results, the equivalent stress on the gear tooth root wasabout 180 MPa and that on the tooth top was about 70 MPa.When the nutation gears meshed, the equivalent stress at thetooth top was smaller than that at the tooth root, exhibiting aperiodic change.

4.3 Contact stress

Due to the existence of the nutation angle, the nutation gearis periodically impacted during transmission. The dynamicmeshing contact stress (impact stress) of the nutation drive isplotted in Fig. 11: it can be seen that the impact on the gearis periodic.

4.4 Vibration displacement

Considering that the nutation transmission gear produces vi-bration and shock during meshing, the vibration displace-ment of the nutation external bevel was investigated. The in-put speed was set to 1000, 1200, 1500, and 2000 r min−1.A point at the center of the tooth with equal distance fromthe tooth root and the tooth top was selected as the samplingpoint. Based on the different input speeds, the displacementof the nutation gear along the z axis during transmission wasanalyzed. It is shown as Fig. 12. According to Fig. 12, thez-axis direction vibration displacement amplitude at differ-ent speeds was stable, while the vibration displacement fre-quency increased with increasing input speed.

4.5 Vibration velocity

Figure 13 shows the vibration velocity of the nutation exter-nal bevel gear over time under different input speeds. Again,

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188 Y. Su et al.: Contact dynamics analysis of nutation drive

Figure 3. Dynamic analysis model of the two-stage nutation drive system.

Figure 4. Tooth-meshing force of the double circular spiral bevelgears.

Figure 5. Backlash function.

Figure 6. Geometric dynamic analysis model.

the sampling point was at the center of the tooth with equaldistance from the tooth root and the tooth top. According tothe results, the dynamic response of the vibration velocityincreased with increasing input speed; consequently, the vi-bration amplitude in the nutation system was increased.

4.6 Dynamic meshing force

The dynamic meshing force generated between the input ex-ternal bevel gear and the output internal gear under differ-ent input speeds is presented in Fig. 14. It can be seen thatwhen the load of the reducer was empty, the dynamic mesh-

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Figure 7. Finite-element dynamic analysis model.

Figure 8. Stress distributions on the double circular-arc spiral bevelgears.

ing force changed due to the different input speeds and thecorresponding contact meshing force decreased significantlywith increasing input speed.

5 Mathematical verification

The gear dynamic meshing force in the x, y, and z directionsunder an input speed of 1050 r min−1 is shown in Fig. 15. Themeshing force in the x and y directions is shown in Fig. 15a–b, respectively. The dynamic meshing force along the z axisis shown in Fig. 15c. The meshing force between the inputexternal bevel gear and the output internal gear is presentedin Fig. 15d, according to which the average meshing forcewas 6523.12◦ N.

When the input speed is 1050 r min−1, the input torque Tis 1.7× 105 N, and according to the load, the mechanical ef-ficiency η is set as 0.85; then, the torque of the output bevelgear T4 is

T4 =T

η=

1.7× 105

0.85N mm= 2× 105 N mm. (6)

Figure 9. Equivalent stress of dynamic meshing on the nutationexternal bevel gear.

Figure 10. Equivalent stress of dynamic meshing on the output in-ternal bevel gear.

Figure 11. Dynamic contact stress on the double circular spiralbevel gears.

Then, the gear circumferential force can be expressed as

Ft =T4

R sinδ=

2× 105

50sin(47.19◦)N mm= 5924.02N. (7)

Subsequently, the meshing force can be represented as

Fr =Ft

cosβ cosαn=

5942.02(cos25◦)(cos24◦)

= 6896.40N. (8)

Based on the calculations, the theoretical value of the con-tact force of the nutation gear was 6896.40 N and the simu-lated one was 6523.12 N. According to Table 2, the error was5.41 %, which is within a reasonable range.

6 Conclusions

A dynamic model of the nutation transmission system hasbeen developed using the lumped parameter method. A FEA

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190 Y. Su et al.: Contact dynamics analysis of nutation drive

Figure 12. Vibration displacement under different input speeds.

Figure 13. Vibration velocity under different input speeds.

model of the double circular-arc spiral bevel gear nuta-tion drive was built, which considered the nutation geartime-varying meshing and contact deformation. The geomet-ric model of the nutation drive system was simulated us-ing ANSYS/LS-DYNA, and the mathematical verificationproved the accuracy of the model. Basic dynamic contactcharacteristics of double circular-arc spiral bevel gears, in-cluding dynamic contact stress and vibration displacementmeshing force, have been obtained for the first time. Theresults revealed that the overall transmission of the nuta-tion reducer is stable, and the meshing force of the dou-ble circular-arc spiral bevel nutation gear is between 4 and9 kN. Through the comparison under different input speeds,it was found that, at high speeds, the dynamic response vi-bration frequency increases, while the contact meshing forcedecreases. These features are conducive to future structuraloptimization and performance improvement.

Figure 14. Dynamic meshing force under different input speeds.

Figure 15. Dynamic contact force in 1050 r min−1 on differentaxes.

Table 2. Comparison between simulated and theoretical values.

Item Theoretical Simulation Errorvalue (N) value (N) (%)

Meshing force 6896.40 6523.12 5.41

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Appendix A: Nomenclature

mi Equivalent concentrated mass of each gearkix,kiy,kiz Bearing stiffnessCix,Ciy,Ciz Bearing dampingrmi Radius of the meshing point of the gear pairJi The moment of inertia of each componentL01,L03 Gyro torque generated by the nutation of two external bevel gearsu System displacementM Mass matrixC The damping matrixK Stiffness matrixRt Gear-meshing external loadDijk Strain conditionLei Element-equivalent lengthCr Velocity of elastic material wave

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Data availability. All data included in this study are availableupon request by contact with the corresponding author.

Author contributions. LY proposed the idea and methodology;YS and JZ applied the methodology; YS conducted the simulations,surveyed the literature, and wrote the paper.

Competing interests. The authors declare that they have no con-flict of interest.

Acknowledgements. This research has been supported by theNational Science Foundation of China (grant nos. 51775114 and51875105) and the Fujian Provincial Industrial Robot Basic Com-ponents Technology Research and Development Center, Fujian, PRChina (grant no. 2014H21010011).

Financial support. This research has been supported by the Na-tional Natural Science Foundation of China (grant nos. 51775114and 51875105) and the Fujian Provincial Industrial Robot BasicComponents Technology Research and Development Center (grantno. 2014H21010011).

Review statement. This paper was edited by Dario Richiedei andreviewed by Luca Bruzzone and one anonymous referee.

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