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Effect of installation errors on beveloid gears’ precision ground by cone-shape worm wheel

Auswirkung von Montagefehlern auf den Präzisionsschliff der Kegelräder durch ein kegelförmiges Schneckenrad

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Abstract

Both the mathematical model of a cone-shape worm wheel and the mathematical model of cone-shape worm wheel grinding beveloid gears with installation errors of a gear workpiece and a cone-shape worm wheel are developed. The effect of the installation errors including gears’ incline error, eccentric error, mounting angle error of a cone-shape worm wheel and coupling error of installation errors mentioned on gears’ surface deviation is investigated. The sensitivity of tooth surface deviation to installation errors mentioned is analyzed. The results show that the helix slope deviation, helix total deviation and tooth cumulative pitch deviation are more sensitive to gears’ incline error. The tooth cumulative pitch deviation, profile total deviation, profile slope deviation are more sensitive to gears’ eccentric error. The effect of gears’ incline error and eccentric error on cumulative pitch deviation and individual cumulative pitch deviation is inverse which lead that coupling error including gears’ incline error and eccentric error causes smaller cumulative pitch deviation and individual cumulative pitch deviation compared to gears’ incline error or eccentric error. Moreover Effect of the mounting angle error of cone-shape worm wheel on gears’ surface deviation is very limited.

Zusammenfassung

Mathematisches Modell der konischen rad gebaut wurde. Das mathematische Modell des Kegelradschleifens von Kegelrädern mit Einbaufehlern eines Zahnradwerkstücks und eines Kegelradschneckenrads wird entwickelt. Fehler bei der Installation auf variable Dicke klamotten Genauigkeit untersucht werden. Gangsensibilität für die Fehler analysiert werden. Die Ergebnisse zeigen: Helix kippen der abweichung, der Helixabweichung und kumulative abweichungempfindlicher für die Abweichung der Schrägstellung des Zahnrads. Kumulative Abweichung-, Profil-Hang abweichung und Profil der abweichung empfindlicher auf den gang der exzentrische abweichung. Die Auswirkung des Neigungsfehlers und des exzentrischen Fehlers der Zahnräder auf die kumulative Teilungsabweichung und die individuelle kumulative Teilungsabweichung ist umgekehrt. Darüber hinaus ist die Auswirkung des Montagewinkelfehlers des kegelförmigen Schneckenrads auf die Oberflächenabweichung der Zahnräder sehr begrenzt.

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Abbreviations

b :

Beveloid gear width

d mi, d ma :

Small end and big end diameter of reference cone of worm wheel

L :

Cone-shape worm wheel length

M 4T, M W5, M 56, M 64, M 21, M T2 :

Coordinate transformation matrix

n 4 :

Cone-shape worm wheel surface’s normal vector in S4

p :

Cone-shape worm wheel lead

R :

Beveloid gear’s pitch circle radius

R T, R 4 :

Cone-shape worm wheel surface’ position vectors in ST and S4

R W :

Beveloid gear surface’ position vectors in SW

S T, S W :

Worm wheel and beveloid gears modeling coordinate system

S 1, S 2, S 4 :

Moveable coordinate systems

S 5, S 6 :

Reference coordinate systems

S 0, S j :

Gear theoretical installation posture and posture with installation error coordinate system

\(\overset{\rightarrow }{\nu _{4}^{TW}}\) :

Relative velocity between worm wheel and beveloid gear in S4

x 4, y 4, z 4 :

Coordinate of any point in cone-shape worm wheel surfaces in the S4

x o5, y o5, z o5 :

S6 origin’s coordinate in S5

z :

Tooth number of beveloid gear

α l, α r :

Pressure angle of worm wheel axial section left and right profile

β :

Installation angle of cone-shape worm wheel

β 1 :

Beveloid gear’s helix angle

Δd :

Offset deviation

ΔX, ΔY :

Coordinate of Sj’s origin in S0

Δα, Δβ, Δγ :

Gear incline error along X, Y, Z axis

Δθ :

Mounting angle error of cone-shape worm wheel

Δz :

S4 origin’s Z axis coordinate in S6

γ :

Cone-shape worm wheel cone angle

λ :

Cone-shape worm wheel lead angle

u :

Length parameter

ϕ, ϕ s :

Rotation angle

ϕ t, ϕ W :

Rotation angle of cone-shape worm wheel and beveloid along self axis in processing

δ :

Cone angle of a beveloid gear

ω T, ν T :

Worm wheel rotation velocity along self axis and moving speed

ω w :

Beveloid gear total rotation velocity along self axis

ω w1, ω w2 :

Beveloid gear rotation velocity components along self axis generated by ωT and νT

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Acknowledgements

The authors are grateful for the support from the National Natural Science Foundation, China (grant no. 51875066) and Chongqing Research Program of Major Research (No.cstc2017zdcy-zdzxX0005).

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Authors and Affiliations

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

    Guolong Li

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  1. Bing Cao
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Correspondence to Guolong Li.

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Cao, B., Li, G. Effect of installation errors on beveloid gears’ precision ground by cone-shape worm wheel. Forsch Ingenieurwes 83, 727–739 (2019). https://doi.org/10.1007/s10010-019-00330-z

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