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

doi:10.1007/s11465-022-0674-6

Front. Mech. Eng.

2022, Vol. 17

Issue (2) : 18 https://doi.org/10.1007/s11465-022-0674-6

RESEARCH ARTICLE

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

Xiaoxia CHEN^1,²(

), Yunpeng YAO¹, Jingzhong XING^1,²

¹. School of Mechanical Engineering, Tiangong University, Tianjin 300387, China
². Tianjin Key Laboratory of Modern Mechatronics Equipment Technology, Tiangong University, Tianjin 300387, China

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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.

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

Corresponding Author(s): Xiaoxia CHEN

About author:

Tongcan Cui and Yizhe Hou contributed equally to this work.

Just Accepted Date: 08 April 2022 Issue Date: 10 June 2022

Cite this article:

Xiaoxia CHEN,Yunpeng YAO,Jingzhong XING. Meshing stiffness property and meshing status simulation of harmonic drive under transmission loading[J]. Front. Mech. Eng., 2022, 17(2): 18.

URL:

https://academic.hep.com.cn/fme/EN/10.1007/s11465-022-0674-6
https://academic.hep.com.cn/fme/EN/Y2022/V17/I2/18

Fig.1 Physical model of cup-shaped FS under a couple of tangential forces: (a) FS under a couple of tangential forces, (b) four components of FS.

Fig.2 Calculation model of deformed components under tangential force: (a) shear deformation of cup bottom, (b) torsion of cylinder and tooth rim, (c) displacement due to rotation of tooth, and (d) bending deformation of tooth.

Fig.3 Tapered deformation in the cup-shaped FS.

Fig.4 Neutral curve under double-disk WG [36].

Fig.5 Geometric relation of engaged TPs on FS and CS.

Fig.6 Meshing backlash of involute HD.

Fig.7 Flow chart of iterative algorithm of meshing force.

Tab.1 Basic geometric parameters of FEM of HD

Fig.8 Radial displacement in middle cross-section of FS tooth rim in assembly state.

Tab.2 Comparison of theoretical circumferential displacement components and FEM results

Fig.9 Flexibilities of meshing points on FS in three cross-sections under a couple of unit tangential forces: (a) front cross-section, (b) middle cross-section, and (c) back cross-section.

Fig.10 Variation of flexibility coefficients of tangential forces with different magnitudes.

Fig.11 Flexibility of meshing points on FS in three cross-sections under 130 N meshing force: (a) front cross-section, (b) middle cross-section, and (c) back cross-section.

Fig.12 Initial backlashes in three cross-sections.

Fig.13 Loading backlashes of algorithm in three cross-sections under different transmission loads: (a) front cross-section, (b) middle cross-section, and (c) back cross-section.

Fig.14 Loading backlashes of FEM in three cross-sections under different transmission loads: (a) front cross-section, (b) middle cross-section, and (c) back cross-section.

Fig.15 Meshing forces of iterative algorithm in three cross-sections under different transmission loads: (a) front cross-section, (b) middle cross-section, and (c) back cross-section.

Fig.16 Meshing forces on FEM in three cross-sections under different transmission loads: (a) front cross-section, (b) middle cross-section, and (c) back cross-section.

Fig.17 Comparison of loading backlashes in three cross-sections under maximum load.

Fig.18 Comparison of meshing forces in three cross-sections under maximum torque.

Fig.19 Comparison of meshing forces on two teeth.

Fig.20 Torsional stiffness of HD represented by torque and rotation angle.

b	Width of tooth rim
c_i	Minimum initial backlash
d_ij (j = 1, 2, …, n)	Circumferential displacement on tooth j produced by the tangential force on tooth i
d_rim	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
h_af	Tooth height from tooth tip to neutral curve
i	Number of a tooth
I_p	Inertia moment of the cross-section
J_t	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
l_c	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
r_af	Radius of the FS addendum circle
r_i	Inner radius of the cylinder
r_m	Radius of the tooth rim neutral circle
R	Radius of the disk in double-disk WG
R_d	Out radius of disk model
s_h	Tooth thickness at x = h_af/2
s_p	Tooth thickness on pitch circle
S_c, S_f, S_w	Coordinate system connected with CS, FS, and WG, respectively
stp	Rotate step
t_w	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
v_af	Circumferential displacement of meshing point due to the rotation of the tooth root
v_b	Circumferential displacement of the cup bottom at the outside
v_c	Circumferential displacement corresponding to the twist of cylinder
v_r	Circumferential displacement at any radius r
v_R	Circumferential displacement of meshing point due to torsional deformation of tooth rim
v_T	Circumferential displacement of meshing point under bending moment of tangential force
v_θ	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
T_k	Total torsional torque
α₀	Tooth profile angle
α_k	Pressure angle at any point of involute
α_G2	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
δ_r	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
θ_k	Circumferential displacement on the meshing point of CS
θ_r	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
φ_c	Twist angle of the cylinder
φ_f	Rotation angle of O_wO_f in Fig. 5
φ_F	Rotation angle of FS
φ_i	Index angle of the CS tooth symmetry line
φ_w	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 y_f and y_c in Fig. 5
μ	Angle between O_wO_f (polar radius ρ) and axis y_c

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