Meshing stiffness property and meshing status simulation of harmonic drive under transmission loading

Abstract

The multitooth meshing state of harmonic drive (HD) is an important basic characteristic of its high transformation precision and high bearing capacity. Meshing force distribution affects the load sharing of the tooth during meshing, and theoretical research remains insufficient at present. To calculate the spatial distributed meshing forces and loading backlashes along the axial direction, an iterative algorithm and finite element model (FEM) is proposed to investigate the meshing state under varied transmission loading. The displacement formulae of meshing point under tangential force are derived according to the torsion of the flexspline cylinder and the bending of the tooth. Based on the relationship of meshing forces and circumferential displacements, meshing forces and loading backlashes in three cross-sections are calculated with the algorithm under gradually increased rotation angles of circular spline, and the results are compared with FEM. Owing to the taper deformation of the cup-shaped flexspline, the smallest initial backlash and the earliest meshing point appear in the front cross-section far from the cup bottom, and then the teeth in the middle cross-section of the tooth rim enter the meshing and carry most of the loading. Theoretical and numerical research show that the flexibility is quite different for varied meshing points and tangential force amplitude because of the change of contact status between the flexspline and the wave generator. The meshing forces and torsional stiffness of the HD are nonlinear with the torsional angle.

Graphical abstract

Keywords

harmonic drive / meshing flexibility matrix / meshing force / loading backlash / flexspline / contact analysis

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Acknowledgements

The authors were grateful for support from the National Natural Science Foundation of China (Grant No. 51575390), and the Natural Science Key Foundation of Tianjin, China (Grant Nos. 19JCZDJC38700 and 18JCZDJC39000).

Nomenclature

b	Width of tooth rim
ci	Minimum initial backlash
dij (j = 1, 2, …, n)	Circumferential displacement on tooth j produced by the tangential force on tooth i
drim	Hole diameter of the disk model
D	Meshing flexibility matrix
{d}	Circumferential displacement array
e	Eccentric distance of the disk
E	Young’s modulus
F	Tangential force applied on meshing point
{F}	Meshing force array
G	Shear modulus of elasticity
GAP	Loading backlash in finite element model
haf	Tooth height from tooth tip to neutral curve
i	Number of a tooth
Ip	Inertia moment of the cross-section
Jt	Backlash between FS and CS
k	Meshing tooth number
K1, K2	Meshing point on FS tooth and CS tooth, respectively
K	Meshing stiffness matrix
l	Cylinder length from the diaphragm to the open edge of cup-shaped FS
lc	Length of the cylinder
m	Gear modulus
n	Number of the meshing points
{N}	Array of tooth number in the meshing region
PRES	Normal contact pressure in finite element model
r	Any radius on the disk model
r1	Radius of FS pitch circle
raf	Radius of the FS addendum circle
ri	Inner radius of the cylinder
rm	Radius of the tooth rim neutral circle
R	Radius of the disk in double-disk WG
Rd	Out radius of disk model
sh	Tooth thickness at x = haf/2
sp	Tooth thickness on pitch circle
Sc, Sf, Sw	Coordinate system connected with CS, FS, and WG, respectively
stp	Rotate step
tw	Width of the disk
T	Torsional moment applied on the disk model
u	Radial displacement of the neutral line
u0	Maximum radial displacement of the neutral line
u1, u2	Radial displacements of the neutral line within wrap angle and out of wrap angle, respectively
v	Circumferential displacement of the neutral line
v1, v2	Circumferential displacements of the neutral line within wrap angle and out of wrap angle, respectively
vaf	Circumferential displacement of meshing point due to the rotation of the tooth root
vb	Circumferential displacement of the cup bottom at the outside
vc	Circumferential displacement corresponding to the twist of cylinder
vr	Circumferential displacement at any radius r
vR	Circumferential displacement of meshing point due to torsional deformation of tooth rim
vT	Circumferential displacement of meshing point under bending moment of tangential force
vθ	Circumferential displacement of the disk with torsional moment
x1, x2	Tooth modification coefficients of FS and CS, respectively
z0	Tooth number of the inserted blade
z1, z2	Tooth numbers of the FS and the CS, respectively
Tk	Total torsional torque
α0	Tooth profile angle
αk	Pressure angle at any point of involute
αG2	Cutting angle of the slotting cutter when machining the CSTP
β	Wrap angle of the tooth rim to the disk
γ	Shear strain at any radius r of the disk model
λ	Involute parameter
δ1	Thickness of the tooth rim
δ2	Thickness of the cylinder
δr	Thickness of disk model
ε	Iteration accuracy
θ	Deflection angle of the neutral line normal
θ1, θ2	Deflection angles of the neutral line normal within wrap angle and out of wrap angle
θk	Circumferential displacement on the meshing point of CS
θr	Rotation angle at the tooth root
ρ(φ, z)	Polar radius of neutral layer
τ	Shear stress at any radius r of the disk model
φ	Polar angle between the major axis of WG and the point on undeformed neutral circle
φ1	Polar angle between the major axis of WG and the point on deformed neutral line
φc	Twist angle of the cylinder
φf	Rotation angle of OwOf in Fig. 5
φF	Rotation angle of FS
φi	Index angle of the CS tooth symmetry line
φw	Rotation angle of WG
ψ	Half of the corresponding center angle of a pitch
Δ	Coefficient of tooth thickness on pitch circle
Φ	Angle between yf and yc in Fig. 5
μ	Angle between OwOf (polar radius ρ) and axis yc

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