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Calculation method of mesh stiffness for helical gear pair with manufacturing errors, assembly errors and tooth modifications

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Abstract

For the accurate calculation of the time-varying mesh stiffness (TVMS) of helical gear pairs, a novel method is proposed in this paper. This proposed method can predict the TVMS based on the gear accuracy grade or the measurement coordinates of the tooth surface. The abnormal meshing phenomena caused by manufacturing errors (MEs), assembly errors (AEs), and tooth modifications (TMs), such as the loss of contact of tooth pairs, out-of-line meshing, and eccentric loads, are considered in the calculation process. The proposed method was verified to be effective for both spur and helical gear pairs. The effects of MEs, AEs, and TMs on the TVMS of helical gear pairs were also investigated. The results showed that the pitch, helix, and misalignment deviations were the main influencing factors of the TVMS in MEs and AEs. Both profile modification and lead crowning reduced the mean of the TVMS. The proposed method is expected to provide accurate TVMS excitation data of gear transmission systems for dynamic analysis.

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Abbreviations

a :

Standard center distance between two gears

A x :

Area of section

B, B i :

Width of tooth and slice

C a, C b :

Maximum modification amounts of profile modification and lead crowning

C F :

Correction factor considering coupling effect between adjacent slices

dx :

Coefficient defined in Fig. 7

\(D_{{\text{g}}}^{\Delta Z}\) :

Minimum distance from arc \(\Gamma_{{\text{g}}}^{\Delta Z}\) center to arc \(\Gamma_{{\text{p}}}^{\Delta Z}\)

\(D_{{\text{p}}}^{\Delta Z}\) :

Minimum distance from arc \(\Gamma_{{\text{p}}}^{\Delta Z}\) center to arc \(\Gamma_{{\text{g}}}^{\Delta Z}\)

\(D_{{{\text{gp}}}}^{\Delta Z}\) :

Minimum distance between arc \(\Gamma_{{\text{g}}}^{\Delta Z}\) and \(\Gamma_{{\text{p}}}^{\Delta Z}\)

e :

Calculation accuracy

e c :

Coefficient of modification curve

\(e_{{\text{g}}}^{i}\), \(e_{{\text{p}}}^{i}\) :

Initial gaps between axial slices

E :

Young’s modulus

E sni, E sns :

Lower and upper deviations of tooth thickness

f pt :

Single pitch deviation

F p :

Total cumulative pitch deviation

F α :

Total profile deviation

F β :

Helix deviation

G :

Shear modulus

h, h x :

Coefficients defined in Fig. 7

i :

Index of discrete elements along tooth width

i c :

Index of contacted tooth pairs

i g, i p :

Index of slices where Hertzian contact center is located on gear and pinion

I x :

Area moment of inertia

j :

Index of discrete elements along tooth profile

\(k_{{\text{a}}}^{ij}\) :

Axial compressive stiffness

\(k_{{\text{b}}}^{ij}\) :

Bending stiffness

\(k_{{\text{f}}}^{ij}\) :

Stiffness considering gear fillet-foundation deflection

\(k_{{\text{f}}}^{ * }\), \(k_{{\text{f}}}^{i * }\) :

Correction foundation stiffnesses of gear and slice

\(k_{{{\text{h\_}}i{\text{c}}}}^{t}\) :

Hertzian contact stiffness

k mesh :

Time-varying mesh stiffness

\(k_{{\text{s}}}^{ij}\) :

Shear stiffness

\(k_{{{\text{tooth\_p,g\_}}i{\text{c}}}}^{t}\) :

Tooth stiffness of nominal slice \(S_{{{\text{N\_}}ic}}^{t}\) on tooth pair ic

L :

Coefficient defined in Fig. 7

L 1, L 2 :

Profile modification length and half of lead crowning length

L *, M *, P *, Q * :

Coefficients in Eq. (27)

n :

Index of teeth

n 1, n 2 :

Number of discrete elements along tooth width and profile

n 3 :

Number of contact tooth pairs

n 4 :

Number of nominal slices on contact tooth pair ic

n lg, n lp :

Half of number of uncontacted slices between current and last contact slices on gear and pinion

n ng, n np :

Half of number of uncontacted slices between current and next contact slices on gear and pinion

R :

Reference radius

r b :

Radius of base circle

\(r_{{\text{g}}}^{\Delta Z}\), \(r_{{\text{p}}}^{\Delta Z}\) :

Radii of arcs \(\Gamma_{{\text{g}}}^{\Delta Z}\) and \(\Gamma_{{\text{p}}}^{\Delta Z}\)

\(r_{{\text{L}}}^{nij}\), \(r_{{\text{R}}}^{nij}\) :

Radii of circles where control points of elements \(\Sigma_{{\text{L}}}^{nij}\) and \(\Sigma_{{\text{R}}}^{nij}\) are located

R pt(1), R Es(1), R α(1), R β(1):

Variables with a value range of [0, 1] and obey different probability distributions

S 1, S 2 :

Modification distances of profile modification and lead crowning

S f :

Coefficient defined in Fig. 7

S i, \(S_{{\text{g}}}^{i}\), \(S_{{\text{p}}}^{i}\) :

Axial slices

\(S_{{\text{N}}}^{t}\) :

Nominal slice

t :

Index of contacted nominal slices

T g :

Output torque of gear

u f :

Coefficient defined in Fig. 7

\(W_{{\text{F}}}^{i}\) :

Weighting factor of the slice Si

W k :

Base tangent length

x :

Coefficient defined in Fig. 7

z :

Number of teeth

\(X_{{\text{c}}}^{ij}\), \(Y_{{\text{c}}}^{ij}\), \(Z_{{\text{c}}}^{ij}\) :

Coordinates of center of curvature corresponding to control point of discrete element

\(X_{L}^{{{1}ij}}\), \(Y_{{\text{L}}}^{1ij}\), \(Z_{{\text{L}}}^{1ij}\), \(X_{{\text{R}}}^{{{1}ij}}\), \(Y_{{\text{R}}}^{{{1}ij}}\), \(Z_{{\text{R}}}^{{{1}ij}}\) :

Theoretical coordinates of element control point on surfaces of first tooth

\(X_{{{\text{MEL}}}}^{nij}\), \(Y_{{{\text{MEL}}}}^{nij}\), \(Z_{{{\text{MEL}}}}^{nij}\), \(X_{{{\text{MER}}}}^{nij}\), \(Y_{{{\text{MER}}}}^{nij}\), \(Z_{{{\text{MER}}}}^{nij}\) :

Coordinates of element control point considering manufacturing errors (MEs)

\(X_{{{\text{TML}}}}^{nij}\), \(Y_{{{\text{TML}}}}^{nij}\), \(Z_{{{\text{TML}}}}^{nij}\), \(X_{{{\text{TMR}}}}^{nij}\), \(Y_{{{\text{TMR}}}}^{nij}\), \(Z_{{{\text{TMR}}}}^{nij}\) :

Coordinates of element control point considering MEs and tooth modifications (TMs)

\(X_{{{\text{AEL}}}}^{nij}\), \(Y_{{{\text{AEL}}}}^{nij}\), \(Z_{{{\text{AEL}}}}^{nij}\), \(X_{{{\text{AER}}}}^{nij}\), \(Y_{{{\text{AER}}}}^{nij}\), \(Z_{{{\text{AER}}}}^{nij}\) :

Coordinates of element control point considering MEs, TMs, and assembly errors (AEs)

β :

Helix angle of reference circle

β j :

Coefficient defined in Fig. 7

δ f, δ g :

Foundation deflections

\(\delta_{{{\text{N\_}}i_{{\text{c}}} }}^{t}\) :

Comprehensive deflection of nominal slice

\(\delta_{{i_{{\text{c}}} }}^{t}\) :

Total deflection of contact center of nominal slice

Δ1, Δ2 :

Modification amounts of profile modification and lead crowning

Δsum :

Total amount of TMs

Δx, Δy :

Center distance deviations along X-axis and Y-axis

ΔZ :

Distance between centers of arc \(\Gamma^{\Delta Z}\) and arc corresponding to element control point

Δθ 1, Δθ 2 :

Rotation steps of gear and pinion

Δθ c :

Angle parameter used for calculating center coordinates of arc \(\Gamma^{\Delta Z}\)

\(\Delta \theta_{{{\text{MEL}}}}^{nij}\), \(\Delta \theta_{{{\text{MER}}}}^{nij}\) :

Angle parameters used for generating tooth surfaces with MEs in Eqs. (5) and (6)

\(\Delta \theta_{{{\text{TML}}}}^{nij}\), \(\Delta \theta_{{{\text{TMR}}}}^{nij}\) :

Angle parameters used to generate tooth surfaces with TMs in Eqs. (12) and (13)

\(\Sigma_{{\text{L}}}^{nij}\), \(\Sigma_{{\text{R}}}^{nij}\) :

Discrete elements on tooth surfaces

\(\Sigma^{\Delta Z}\) :

Plane perpendicular to Z-axis

\(\Gamma^{\Delta Z}\) :

Arc on discrete element

υ :

Poisson’s ratio

θ 1, θ 2 :

Rotation angles of gear and pinion during iterative process for calculating time-varying mesh stiffness (TVMS)

θ f :

Coefficient defined in Fig. 7

φ, γ :

Misalignment deviations on XBpZ-plane and YBpZ-plane

η, λ :

Correction coefficients of foundation stiffness

AEs:

Assembly errors

FEM:

Finite element method

MEs:

Manufacturing errors

TCA:

Tooth contact analysis

TMs:

Tooth modifications

TVMS:

Time-varying mesh stiffness

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Acknowledgements

The authors are grateful for the financial support from the NSFC. This research was funded by the National Natural Science Foundation of China (No. 51675061).

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  1. State Key Laboratory of Mechanical Transmission, Chongqing University, Chongqing, 400044, People’s Republic of China

    Chang Liu, Wankai Shi & Kun Liu

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  1. Chang Liu
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Correspondence to Wankai Shi.

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Liu, C., Shi, W. & Liu, K. Calculation method of mesh stiffness for helical gear pair with manufacturing errors, assembly errors and tooth modifications. Meccanica 57, 541–565 (2022). https://doi.org/10.1007/s11012-022-01479-8

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