Investigation of the load transmission in the toothing of a flat wheel harmonic gear drive

The function principle of the flat wheel harmonic gear drive is similar to the basic principle of the classical harmonic drives. The flexible and the rigid gear of the drive are coaxial flat wheels. The rotating wave generator deforms periodically and elastically different portions of an annular face gear on the flexible member in axial direction into engagement with teeth on an annular face gear on the rigid member. The numbers of teeth of the face gears are different. An analytical method is introduced, that investigates backlash conditions of flat-wheel harmonic drives. The tooth flanks were approximated by planes. The numbers of the connected teeth taking part in the loadtransmission and the tangential component of the acting force on them in the range of tooth engagement are dependent on backlash. Finite element investigations were launched at different levels of torques even in case of preloaded flexible gear.


Introduction
For the dimensioning and endurance determination of a flat wheel harmonic drive it is necessary to know the amplitude of force acting on teeth and the number of tooth-pairs taking part in the load transmission of the drive in case of a prescribed loading torque.One possible way to determine this force is the finite element analysis (FEA), the other one is an analytical approximation.This paper reports a method for backlash calculation and investigates load transmission in the toothing in case of preloaded flexible gear, which is used for reducing clearance of the gear.

Description and operation of a flat wheel harmonic gear drive
As for the function principle of the flat wheel harmonic drive (Fig. 1), it is similar to the basic principle of the classical harmonic drives; it could be regarded as a special variety of them.The flexible and the rigid gear of the drive are coaxial flat wheels.The cam-type wave generator (G) consisting of a flexible axial bearing and a disc with cams, deforms the flexible gear (1) in axial direction periodically and elastically, the toothing of which comes into mesh with the toothing of the rigid gear (2).Fig. 2 shows the schematic representation of the engagement, where φ is the polar angle from the top of the deformation wave made of the wave generator, φ = 0˚means the symmetry plane of the deformation wave.Since the flexible and rigid gears have a different number of teeth, there will be a relative rotational motion between the flexible and the solid gear.
The kinematic ratio of the investigated harmonic drive, in case of rigid gear is fixed, flexible wheel connects to the output shaft, wave generator connects to the input shaft, can be calculated as the following: where z 1 describes the number of teeth of the flexible wheel, z 2 describes the number of teeth on the rigid wheel.3 Applied coordinate systems for the backlash calculation Fig. 3 shows the meshing of teething in radial and tangential direction; dashed line indicates the deformed shape.Coordinate system [x 2, y 2, z 2 ] is fixed to the rigid wheel; its centre point (O 2 ) is on the outer diameter (d k ) of the pitch cone in the symmetry plane of the investigated tooth groove; x 2 is the axis of tangential; y 2 is the axis of radial and z 2 is the axis of axial direction.
Coordinate system [x 1, y 1, z 1 ] is fixed to the flexible wheel; its centre point (O 1 ) is on the outer diameter (d k ) of the neutral plane of the wheel in the symmetry plane of the investigated tooth; x 1 is the axis of tangential; y 1 is the axis of radial and z 1 is the axis of axial direction (hereafter index '1' means the flexible, '2' the rigid wheel).

Coordinates of the investigated points of the toothing
Backlashes were determined by calculation of the distances between the addendum points of one gear and the tooth flank of the other gear.Coordinates of the addendum and dedendum points of the investigated tooth of the rigid gear in coordinate system [x 2 ,y 2 ,z 2 ] on the outer and inner diameter of the toothing (Figure 3/b and 3/c) (hereafter superscript "2" means the rigid gear superscript "1" the flexible gear coordinate system): Coordinates of the points in the inner diameter (superscript "b" means the inner diameter): Coordinates of the points of the investigated tooth of the flexible gear in coordinate system [x 1 ,y 1 ,z 1 ] on the outer and inner diameter of the toothing in case of the flexible gear is not deformed by the wave generator (Fig. 3/b): On the inner diameter: Coordinates of the investigated points of the flexible gear tooth in deformed state of the flexible gear against φ in coordinate In the relation:  Matrix of transformation between the coordinate systems: Coordinates of the investigated points of the flexible gear tooth in deformed state of the flexible gear against φ in coordinate system [x 2 ,y 2 ,z 2 ]: On the inner diameter: where: Distance between outer addendum edge endpoint of flexible gear (P a1 ) and tooth flank plane of the solid wheel: Per. Pol.Mech.Eng.

Load transmission in the toothing
Fig. 5 shows the distribution of backlash between the teeth in an examined drive against φg in no-load-state of the drive.
Main sizes and toothing parameters of the examined drive were determined according to non-fictional recommendations Results of backlash examinations showed, that the minimumbacklash appears in areas -25˚< φg < -15˚and 15˚< φg < 25˚.
Test bench measurements of an experimental drive [3] indicated, that similarly to classical harmonic gear boxes [4,5,6] increasing loading torque causes increasing number of teeth taking part in load transmission.It was indicated moreover, that teeth taking part in load transmission are in the -φ area.6 shows the load-transmission process in the gear.Continuous thick lines show backlash distribution against φ gin no-load state of the drive, where minimum of backlash appears at -gφ n in the n-th pair of teeth from the top of the deformation wave.Backlash and its location can be influenced by changing tooth parameters [1].In case of a given backlash-curve backlashes of pairs of teeth in the area of φ n can be ranged into increasing order, for example: In case of loading torque acting the output shaft of the drive, backlash will decrease in the -gφ garea and increase in +φ area.In Fig. 6 thin lines show different load cases.In that range, where backlash curves are below the horizontal axis, so backlash would be negative, teeth in engagement deform each other in tangential direction.Dashed lines show the common deformation (f) of the tooth pairs.
In load case 1 backlash of the minimum-backlash tooth pair n (j n ) becomes zero and then teeth of this tooth pair deform each other.Increasing the loading torque, if the common deformation of the teeth of the tooth-pair n comes at j 1 = j n−1 -j n , the next where T is the loading torque acting on the output shaft of the drive, d is the outer or inner diameter of the toothing, depending on the place F i is acting.

The preloaded state of the gear
For reducing clearance of the gear, the flexible gear can be preloaded by the wave generator in axial direction.In this case backlash will be zero and the teeth will deform each other and initial load will appear on both sides of the deformation wave.Fig. 7 shows the case of axial preload without any loading torque acting on the output shaft.
In case of loading torque acts on the output shaft of the drive it is supposed that rotation of the flexible gear will decrease backlash in the -φ area, that means tooth load will increase here.On the other side of the deformation wave (+φ area) backlash will increase and tooth load will decrease (Fig. 8).

Analyzing of the distribution of tangential force on tooth-pairs with FEA
Amplitude of tangential forces on the teeth of the gear and force distribution in the load-transmission area was analyzed with FEA, Fig. 9 shows the finite element model of the basic parts of the drive.
The assembly contains a whole model of the flexible wheel, because in case of loading there are no symmetry conditions.On the inner border of the annular-shaped plate fix constraint and prescribed rotation was applied.The flexible axial bearing of the wave generator was eliminated, the working surfaces of the cam of the wave generator was modelized in the range of the teeth are in mesh.Prescribed displacement constraint in axial direction was located on these surfaces of wave generator that deflect the flexible wheel in meshing area.Toothing of the rigid wheel was shaped only in the load transmission area; teeth were fixed below their root separated from each other.Between touching faces gap elements were placed and element size was refined.Loading torque was generated by prescribed rotation of the inner border of the flexible wheel, results were the reaction force amplitudes on teeth taking part in load transmission.Analyzes were launched with large displacement calculation method.

FEA results
Fig. 10 shows the distribution of the tangential force of the teeth against φ calculated with FEA and analytical method, in case of two different preloads and loading torque for the toothing and geometric parameters described above.

Concluding remarks
An approximating analytical method was introduced for calculating backlash in the toothing of a flat wheel harmonic gear drive.Tooth load was also analyzed with FEA in case of preloaded flexible gear.

Fig. 1 .
Fig. 1.Basic parts of flat-wheel harmonic gear drive and their photos

Fig. 2 .
Fig. 2. Schematic representation of the engagement in flat-wheel harmonic gear

Fig. 3 .
Fig. 3. Applied coordinate systems and the investigated points tgβ − deformation of the flexible gear against φgn the investigated diameter -d means d k or d b -of the toothing in case of the working surfaces of the cam of the wave generator are planes that shows Fig. 4.

Fig. 4 .Fig. 5 .
Fig. 4. Working surface of the cam type wave generator tgβ g − deformation of the flexible gear against φgnd on the inner diameter of the toothing.Normal vector of the solid wheel tooth flank:

Fig. 6 .
Fig. 6.Backlash and tooth deformation in different load cases Fig.5shows the distribution of backlash between the teeth in an examined drive against φg in no-load-state of the drive.Main sizes and toothing parameters of the examined drive were determined according to non-fictional recommendations [2]: d k = 190 mm -outer diameter of toothing, d b = 160 mminner diameter of toothing, v = 1.5 mm -thickness of flexible wheel, w 0 = 3 mm -deformation.Number of teeth of the flexible wheel: z 1 = 190, that of the rigid wheel z 2 = 188, tooth profile parameters at the outer diameter: addendum heights: h a1 = h a2 = 0.75 mm; dedendum heights: h f 1 = h f 2 = 0.75 mm; clearance: c 1 =c 2 = 0.25 mm, profile angles: α x g,gα y = 30˚, thickness at pitch line: s 1 , s 2 are halves of pitch.Cam angle of the wave generator: βg = g3˚, angle of toothing of solid wheel: β y g = g3˚,g ax = 0.25 mm-axial clearance between the toothings at φ = 0˚.Results of backlash examinations showed, that the minimumbacklash appears in areas -25˚< φg < -15˚and 15˚< φg < 25˚.Test bench measurements of an experimental drive[3] indicated, that similarly to classical harmonic gear boxes[4,5,6] increasing loading torque causes increasing number of teeth taking part in load transmission.It was indicated moreover, that teeth taking part in load transmission are in the -φ area.

Fig. 10 .
Fig. 10.Distribution of the tangential force on teeth against φ calculated with FEA