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Dynamic characteristic analysis of spur gear system considering tooth contact state caused by shaft misalignment

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Abstract

The effect of gear contact state change due to shaft misalignment on meshing stiffness is usually neglected in the traditional stiffness calculation model with misalignment error, the further influence mechanism of shaft misalignment on gear dynamic characteristics is also unclear. To address these shortcomings, a new mesh stiffness calculation model with misaligned gear considering the effects of tooth contact state is proposed by combining the improved loaded tooth contact analysis (LTCA) model. Then the effects of tooth contact state changes aroused by shaft misalignment on the meshing stiffness excitation are investigated. Moreover, a dynamic model of the misaligned gear system with 8 degrees of freedom (DOF) is established, and based on which the dynamic characteristics of the gear system are investigated and verified by experiment. The study results indicate that the proposed model can be used to evaluate the stiffness excitation and dynamic characteristics of the misaligned gear system with the tooth contact state taken into consideration. This study can provide a theoretical method for evaluating and identifying shaft misalignment errors.

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Data availability

The datasets generated and analyzed during the current study are available from the corresponding author on reasonable request.

Abbreviations

α :

Gear central angle corresponding to the micro-section at x from the dedendum circle

α 1 :

Gear central angle corresponding to the force point

α r/α q :

Gear central angle corresponding to the force point on the tooth #r/#q (r = 1, 2; q = 1, 2, r ≠ q)

α 3 :

Half of the tooth angle on base circle

A x/A x ’ :

Cross-sectional area of the aligned/misaligned gear

A :

Deformation influence coefficient matrix

A γ :

Deformation influence coefficient matrix of tooth pair #1 (γ = 1) or #2 (γ = 2)

c λx/c λy/c λz :

Bearing damping of driving gear (λ = 1) or driven gear (λ = 2) in x/y/z direction

d :

Distance between tooth force point and dedendum circle

E :

Elasticity modulus of gear

f H , kj /f B , kj /f S , kj :

Influence coefficient of the contact/bending/shear deformation of the grid unit k generated by the applied force on the grid unit j

F :

Normal force

F m :

Dynamic mesh force

F γ :

Mesh force of tooth pair #1 (γ = 1) or #2 (γ = 2)

F a/F b :

Horizontal/vertical component of mesh force of aligned gear

F a ’/F b ’/F t :

Horizontal/vertical/axial component of mesh force of misaligned gear

F mx/F my/F mz :

Horizontal/vertical/axial component of dynamic mesh force of misaligned gear

G :

Shear modulus of gear

h x/h :

Half tooth height at the mesh position/the position of the micro-section at x from dedendum circle

H :

Clearance matrix corresponding to unit grid of common tangent plane of tooth pair

H γ :

Clearance matrix corresponding to unit grid of common tangent plane of tooth pair #1 (γ = 1) or #2 (γ = 2)

H f r :

Clearance matrix of tooth #r (r = 1, 2) caused by the fillet foundation deformation under the meshing force on the tooth #r (r = 1, 2)

H f rq :

Clearance matrix of tooth #r caused by fillet foundation deformation under the meshing force on the adjacent tooth #q (r = 1, 2; q = 1, 2, r ≠ q)

I :

Unit vector matrix

I x/I x ’ :

Section moment of inertia of the aligned/misaligned gear tooth

I p x :

Polar moment of inertia of the micro-section

J λ :

Rotational inertia of driving gear (λ = 1) or driven gear (λ = 2)

k m :

Mesh stiffness

k h/k b/k s/k a/k f/k t :

Stiffness of contact/bending/shear/axial compression/fillet foundation/torsion

k λx/k λy/k λz :

Bearing stiffness of driving gear (λ = 1) or driven gear (λ = 2) in x/y/z direction

L 0 :

Effective contact line length

L eq :

Distance between applied point of equivalent force and gear tooth center

m λ :

Mass of driving gear (λ = 1) or driven gear (λ = 2)

n :

Number of unit grid in tooth common tangent plane

p j :

Applied force at unit grid j

P :

Pressure matrix of common tangent plane of contact tooth pair

P γ :

Pressure matrix of common tangent plane of tooth pair #1 (γ = 1) or #2 (γ = 2)

r b :

Base circle radius of gear

r bλ :

Base circle radius of driving gear (λ = 1) or driven gear (λ = 2)

s :

Area of unit grid of tooth common tangent plane

T :

Torsional torque of misaligned gear tooth

T λ :

Torque of driving gear (λ = 1) or driven gear (λ = 2)

U b/U s/U a/U t :

Deformation energy of bending/shear/axial compression/torsion

w k :

Deformation of unit grid node k

W :

Tooth width

\(x_{\lambda } /\dot{x}_{\lambda } /\ddot{x}_{\lambda } {\kern 1pt}\) :

Lateral displacement/velocity/acceleration of driving gear (λ = 1) or driven gear (λ = 2) along x direction

\(y_{\lambda } /\dot{y}_{\lambda } /\ddot{y}_{\lambda } {\kern 1pt}\) :

Lateral displacement/velocity/acceleration of driving gear (λ = 1) or driven gear (λ = 2) along y direction

\(z_{\lambda } /\dot{z}_{\lambda } /\ddot{z}_{\lambda }\) :

Lateral displacement/velocity/acceleration of driving gear (λ = 1) or driven gear (λ = 2) along z direction

\(\theta_{\lambda } /\dot{\theta }_{\lambda } /\ddot{\theta }_{\lambda }\) :

Angular displacement/velocity/acceleration of driving gear (λ = 1) or driven gear (λ = 2)

δ mis :

Comprehensive deformation of gear

Δθ x/Δθ y :

Misalignment error in x/y direction

υ :

Poisson's ratio

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Acknowledgments

The authors are grateful for the National Natural Science Foundation of China (Grant Nos. 51905053, 52035002), the Graduate Research and Innovation Foundation of Chongqing, China (Grant No. CYB21014) and the Chongqing Science and Technology Innovation and Application Development Special Project, China (cstc2020jscx-msxmX0194).

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Authors and Affiliations

  1. State Key Laboratory of Mechanical Transmission, Chongqing University, Chongqing, 400044, China

    Lantao Yang, Qiang Zeng, Liming Wang, Guorong Long, Xiaoxi Ding & Yimin Shao

  2. College of Mechanical and Vehicle Engineering, Chongqing University, Chongqing, 400044, China

    Lantao Yang, Qiang Zeng, Liming Wang & Xiaoxi Ding

  3. Chongqing Gear Box Co. LTD, Chongqing, 400044, China

    Haishi Yang

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  1. Lantao Yang
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Correspondence to Liming Wang.

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Yang, L., Zeng, Q., Yang, H. et al. Dynamic characteristic analysis of spur gear system considering tooth contact state caused by shaft misalignment. Nonlinear Dyn 109, 1591–1615 (2022). https://doi.org/10.1007/s11071-022-07519-y

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