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Load sharing by tooth pairs in involute toothed harmonic drive with conventional wave generator cam

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Abstract

A harmonic drive is a two gear epicyclic drive with a gear set of circular ring gear (RG), a flex rimmed external toothed gear (FG) and an oval cam. FG, with oval cam inside, takes non-circular gear shape encounters improper teeth mating with RG, having only two teeth difference. Consequently, interferences occur at several tooth pairs even at no load. These are inherent and obvious. Overcoming such interferences and further with applied load estimation of load sharing by tooth pairs poses a complex problem. In solving it, first, tooth stiffness of internal gear and external gear are derived in the present investigation. A method of estimating the load shared by the multiple tooth pairs in contact is proposed. The load distribution pattern in proportion to the tooth deformation is considered. Load shared by contacting tooth pairs is estimated and stresses in FG cup are found out using FEM. Finally, such results are compared with experimental results, which have good agreement.

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Abbreviations

B:

Width of gear tooth (mm)

E:

Young’s modulus (N/mm2)

F:

Force acting on a tooth (N)

\({\text{F}}_{{{\text{Inter}},{\text{RG}}}} ,{\text{F}}_{{{\text{Inter}},{\text{FG}}}}\) :

Load shared by RG and FG tooth in order to avoid interference before application of external load (N)

\({\text{F}}_{{{\text{RG}},{\text{net}},{\text{i}}}} ,{\text{F}}_{{{\text{FG}},{\text{net}},{\text{i}}}}\) :

Load shared by RG and FG tooth while externally force F is applied to the assembled FG–RG teeth (N)

Ki :

Stiffness of a tooth (N/mm)

\({\text{K}}_{{{\text{FG}},{\text{i}}}} \,{\text{and}}\,{\text{K}}_{{{\text{RG}},{\text{i}}}}\) :

Stiffness of flex-gear and ring gear tooth correspond to \(\updelta_{{{\text{FG}},{\text{i}}}}\) and \(\updelta_{{{\text{RG}},{\text{i}}}}\) (N/mm)

\({\text{K}}_{{{\text{FG}} - {\text{RG}},{\text{i}}}}\) :

Combined stiffness of a meshing tooth pair correspond to \({\text{K}}_{{{\text{FG}},{\text{i}}}}\) and \({\text{K}}_{{{\text{RG}},{\text{i}}}}\) (N/mm)

Y,Yi,Y1,y, Xa, Xb, ψ:

Tooth related parameter of external gear tooth related to Fig. 5

Z:

Number of teeth in general

ZFG/Zp :

Teeth number in flexspline/flex-gear

ZRG/Zg :

Teeth number in circular spline/ring gear

a:

Semi major axis (mm)

af :

Addendum factor

b:

Semi minor axis (mm)

df :

Dedendum factor

hm, tF :

Tooth related parameter of internal gear tooth related to Fig. 6

m:

Module of the gear (mm)

ri :

Inner cup radius (mm)

rb :

Radius of base circle of undeformed FG cup tooth (mm)

rd :

Radius of dedendum circle of undeformed FG cup tooth (mm)

\({\text{t}}_{\text{p}}\) :

Pitch circle thickness of FG cup tooth corresponds to correction factor (mm)

x:

Correction/profile modification factor for FG cup teeth

α:

Pressure angle (deg)

ν:

Poisson’s ratio

\(\uprho_{1} ,\,\uprho_{2}\) :

Radius of curvature at the point of contact of external (FG) and internal (RG) tooth respectively (mm)

\(\updelta_{{{\text{T}}_{\text{E}} }} =\updelta_{{{\text{FG}},{\text{i}}}} ,\updelta_{{{\text{T}}_{\text{i}} }} =\updelta_{{{\text{RG}},{\text{i}}}}\) :

Total deflection of external and internal tooth respectively (mm)

\(\updelta_{{{\text{B}}_{\text{E}} }} ,\updelta_{{{\text{C}}_{\text{E}} }} ,\updelta_{{{\text{S}}_{\text{E}} }} ,\updelta_{{{\text{G}}_{\text{E}} }}\) :

Bending, contact, shearing and foundation deformation of external gear tooth (mm)

\(\updelta_{{{\text{t}}_{\text{i}} }}\) :

Combined deformation due to bending, shearing and foundation of internal tooth (mm)

\(\updelta_{{{\text{C}}_{\text{i}} }}\) :

Contact deformation of internal tooth (mm)

\(\updelta_{{{\text{Inter}},{\text{RG}}}} ,\updelta_{{{\text{Inter}},{\text{FG}}}}\) :

Deflection shared by RG and FG tooth in order to avoid interference (mm)

\(\updelta \uptheta _{{{\text{Inter}},{\text{RG}}}} ,\updelta \uptheta _{{{\text{Inter}},{\text{FG}}}}\) :

Angular deflection at the centre of flex-gear correspond to \(\updelta_{{{\text{Inter}},{\text{RG}}}}\), \(\updelta_{{{\text{Inter}},{\text{FG}}}}\)(deg)

\(\updelta_{{{\text{RG}},{\text{F}},{\text{i}}}} ,\updelta_{{{\text{FG}},{\text{F}},{\text{i}}}}\) :

Deflection of RG and FG tooth due to externally applied force F only (mm)

\(\updelta \uptheta _{{{\text{RG}},{\text{F}},{\text{i}}}} ,\updelta \uptheta _{{{\text{FG}},{\text{F}},{\text{i}}}}\) :

Angular deflection at the centre of flex-gear correspond to \(\updelta_{{{\text{RG}},{\text{F}},{\text{i}}}}\), \(\updelta_{{{\text{FG}},{\text{F}},{\text{i}}}}\)(deg)

\(\updelta_{{{\text{RG}},{\text{net}},{\text{i}}}} ,\updelta_{{{\text{FG}},{\text{net}},{\text{i}}}}\) :

Deflection of RG and FG tooth while externally force F is applied to the assembled FG–RG teeth (mm)

\(\updelta \uptheta _{{{\text{RG}},{\text{net}},{\text{i}}}} ,\updelta \uptheta _{{{\text{FG}},{\text{net}},{\text{i}}}}\) :

Angular deflection at the centre of flex-gear correspond to \(\updelta_{{{\text{RG}},{\text{net}},{\text{i}}}}\),\(\updelta_{{{\text{FG}},{\text{net}},{\text{i}}}}\)(deg)

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Acknowledgements

This research work is an outcome of the general PhD programme in the authors’ Institute, IIT Kharagpur, India. There is no specific financial grant for this investigation.

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  1. Mechanical Engineering Department, Indian Institute of Technology, Kharagpur, 721302, India

    Vineet Sahoo & Rathindranath Maiti

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  1. Vineet Sahoo
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Correspondence to Rathindranath Maiti.

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Sahoo, V., Maiti, R. Load sharing by tooth pairs in involute toothed harmonic drive with conventional wave generator cam. Meccanica 53, 373–394 (2018). https://doi.org/10.1007/s11012-017-0698-x

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