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Research Papers: Design of Direct Contact Systems

A Stiffness Formulation for Spline Joints

[+] Author and Article Information
J. Hong, A. Kahraman

Department of Mechanical and
Aerospace Engineering,
The Ohio State University,
201 West 19th Avenue,
Columbus, OH 43210

D. Talbot

Department of Mechanical and
Aerospace Engineering,
The Ohio State University,
201 West 19th Avenue,
Columbus, OH 43210
e-mail: talbot.11@osu.edu

1Corresponding author.

Contributed by the Power Transmission and Gearing Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received August 24, 2015; final manuscript received December 11, 2015; published online February 19, 2016. Assoc. Editor: Qi Fan.

J. Mech. Des 138(4), 043301 (Feb 19, 2016) (8 pages) Paper No: MD-15-1592; doi: 10.1115/1.4032631 History: Received August 24, 2015; Revised December 11, 2015

Due to the lack of knowledge in terms of their flexibility and deformation, spline joints are typically assumed to be rigid in dynamic models of gearboxes, transmissions, and drivetrains. As various dynamic phenomena are associated with the stiffness of a spline joint, any high-fidelity dynamic model of drivetrains must properly capture the stiffness of spline joints. In this study, a general analytical stiffness formulation for spline joints is proposed based on a semi-analytical spline load distribution model. This formulation defines a fully populated stiffness matrix of a spline joint including radial, tilting, and torsional stiffness values as well as off-diagonal coupling terms. A blockwise inversion method is proposed and implemented with this analytical formulation to reduce computational time required. At the end, a detailed parametric study is presented to demonstrate the sensitivity of the spline stiffness matrix to torque level, tooth modifications, misalignments, and tooth indexing errors.

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References

Figures

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Fig. 2

Flowchart of the procedure for stiffness calculation using the analytical stiffness formula

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Fig. 4

Effect of lead crown modification on (a) torsional stiffness, (b) radial stiffness, and (c) tilting stiffness of the example spline under pure torsion at different torque levels

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Fig. 5

Effect of misalignment about the x axis, θx, on (a) torsional stiffness, (b) radial stiffness along the x axis, (c) radial stiffness along the y axis, (d) tilting stiffness about the x axis, and (e) tilting stiffness about the y axis, for the example spline at different torque levels

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Fig. 1

Schematic representation of (a) reaction load components of a spline joint and (b) relative rigid body displacements between the external and internal spline

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Fig. 3

Effect of profile crown modification on (a) torsional stiffness, (b) radial stiffness, and (c) tilting stiffness of the example spline under pure torsion at different torque levels

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Fig. 6

Load distribution of the example spline having random indexing errors at different rotational positions for the torque Mz=4520 N⋅m, misalignment θx = 0.04 deg, profile crown 5 μm, and lead crown 10 μm

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Fig. 7

Effects of random tooth indexing error sequence on (a) torsional stiffness, (b) radial stiffness along the x axis, (c) radial stiffness along the y axis, (d) tilting stiffness about the x axis, and (e) tilting stiffness about the y axis of the example spline having 0.04 deg misalignment at Mz=4520 N⋅m at different rotational positions (profile crown 5 μm and lead crown 10 μm)

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