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Study on multi-clearance nonlinear dynamic characteristics of herringbone gear transmission system under optimal 3d modification

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Abstract

The tooth surface 3d modification technology is the most effective method to realize the reduction of vibration and noise of gear system (GS) by comprehensively considering tooth profile modification and axial modification. Moreover, the system nonlinear factors such as backlash and bearing clearance will make the motion of GS have complex variability. Therefore, we will combine 3d modification technology with contact performance and dynamic characteristics to optimize the meshing characteristics of herringbone gear pair through tooth contact analysis technology, loaded tooth contact analysis technology and antlion optimizer with error minimum amplitude as optimization objective, determine three nonlinear dynamic model of herringbone gear for numerical solution. Finally, this paper studies the influence of different system parameters on the system nonlinear dynamic characteristics under 3d modification and non-modification from both local vibration characteristics on the diagrams of time history, phase, spectrum and Poincare map and global vibration characteristics on the diagrams of system bifurcation and maximum Lyapunov exponent. The results show: After optimizing modification, the system vibration is obviously reduced. With the increase of input power, input speed, backlash and static transmission error (STE), the system has jump, while there is no jump with the change of bearing clearance and damping ratio. Compared with other parameters, the changes of input speed and STE make system have complex bifurcation characteristics. The 3d modification can eliminate the jump and make system motion more regular.

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Data availability

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

Abbreviations

GS:

Gear system

TCA:

Tooth contact analysis

LTCA:

Loaded tooth contact analysis

GTE:

Geometric transmission error

LTE:

Loaded transmission error

ALO:

Antlion optimizer

STE:

Static transmission error

MLE:

Maximum Lyapunov exponent

RMS:

Root mean square

∑1 :

3D modified tooth surface

∑2 :

The theoretical involute tooth surface

u o, u p, u q and u a :

The involute expansion angles at tooth profile modification boundary line.

θ o, θ n, θ m and θ max :

The gear rotation angle for axial modification boundary line

y 1 and y 3 :

Maximum modification amount at root and top of tooth

y 2 and y 4 :

Modification length at root and top of tooth.

y 5 and y 6 :

Maximum modification amount of the lower and upper of tooth width.

z hf and z ha :

Modification lengths of the lower and upper of tooth width

y 7 :

The length without modification in axial.

δ n, δ m, δ p and δ q :

Modification amounts of the top, root, lower and upper of tooth

a ms :

Modification coefficient of tooth profile

u 1 :

Involute expansion angle of whole tooth profile

u 0, θ 0 :

Starting point of different modified areas

λ 0 :

The half angle of base circular groove

E 0 :

The center distance between grinding wheel and gear center

\(E^{\prime}_{0}\) :

Standard center distance

a mk :

Axial modification coefficient

θ 1 :

Gear rotation angle

S m :

The coordinate system fixed on gear mounting table

S 1 :

The coordinate system fixed on gear blank

S t :

The coordinate system fixed on grinding wheel

“ ± ” and “ ∓ ”:

The upper symbol indicates right tooth surface of the groove, and the lower symbol indicates the left tooth surface of the groove.

δ :

The axis non-parallelism error

ΔE :

Center distance error of driven gears

M :

Coordinate system transformation matrix

L :

The submatrix removing the last row and last column of M.

w ik :

Initial clearance of ellipse center point i before contact

w jk :

Tooth surface clearance at any point j before contact

p ik and p jk :

Unit normal load of contact point of the long axis of ellipse

d jk :

Tooth surface clearance after deformation at point j under load

[w]k :

Initial clearance of tooth surface before being loaded

[d]k :

Tooth surface clearance after being loaded.

[Z]:

Tooth rigid body displacement

[L]k [P]k :

Elastic deformation obtained by multiplying normal flexibility matrix and normal load

[p]:

The discrete point load

M :

Input torque

X j :

An artificial variable.

T :

Output torque

r b 2 :

Base radius of passive gear

α n :

Normal pressure angle

β :

The helix angle

ΔT e :

The normal loaded comprehensive deformation obtained by LTCA

Δδ :

The normal GTE obtained by TCA.

v s :

Speed difference of mesh-in point

δ s :

Maximum impact deformation

k s :

Meshing stiffness of mesh-in point

J 1 and J 2 :

Moment of inertia

e m :

The average value of errors of gear pair

e r :

Error amplitude of gear pair

f s 1(t) and f s 2(t):

The meshing impact excitation of left and right helical gear pairs of herringbone gear.

k 12 L(t) and k 12 R(t):

The normal time-varying meshing stiffness of left and right helical gear pairs of herringbone gear

c 12 L and c 12 R :

Normal meshing damping of left and right helical gear pairs of herringbone gear.

e 12 L(t) and e 12 R(t):

The STEs of left and right helical gear pairs of herringbone gear

2b 12 L and 2b 12 R :

The backlash between left and right helical gear pairs

2γ 1 L, 2γ 1 R,2γ 2 L and 2γ 2 R :

Each radial bearing clearance.

ζ :

The damping ratio

k m :

The average meshing stiffness

m 1 and m 2 :

The equivalent meshing mass of left and right helical gear pairs

M D, C D, K D, q and F D :

Mass matrix, damping matrix, stiffness matrix, displacement vector and force vector.

\({\overline{\mathbf{C}}}_{{\varvec{D}}}\) :

The dimensionless damping matrix

\({\overline{\mathbf{K}}}_{{\varvec{D}}}\) :

The dimensionless stiffness matrix

\(\overline{\user2{F}}_{{\varvec{D}}}\) :

The dimensionless load array.

I :

A unit matrix

P :

The input power

N :

The input speed

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Acknowledgements

First and foremost, I would like to show my deepest gratitude to my supervisor, who has walked me through all the stages of the writing of this article. My sincere appreciation also goes to all my friends, especially my three lovely roommates, for their encouragement and support.

Funding

This study was supported by the Special Fund for Civil Machinery of China (Grant No. MJ-2016-D-28).

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Authors and Affiliations

  1. School of Mechanical Engineering, Northwestern Polytechnical University, Xi’an, 710072, China

    Zhibin Li, Sanmin Wang, Linlin Li, Fei Li, Linlin Liu & Haoran Zou

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  1. Zhibin Li
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Contributions

All authors contributed to the study conception and design. The overall idea and structure of the paper were determined by Professor [Sanmin Wang]. Material preparation, data collection Programming, and analysis were performed by [Zhibin Li], [Linlin Li], [Fei Li], [Linlin Liu] and [Haoran Zou]. The first draft of the manuscript was written by [Zhibin Li] and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

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Correspondence to Zhibin Li.

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Li, Z., Wang, S., Li, L. et al. Study on multi-clearance nonlinear dynamic characteristics of herringbone gear transmission system under optimal 3d modification. Nonlinear Dyn 111, 4237–4266 (2023). https://doi.org/10.1007/s11071-022-08083-1

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