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Research Papers

Analysis of Elliptical Rolling Contact Joints in Compression

[+] Author and Article Information
Jacob R. Montierth

Product Planning, Ford Motor Company, Dearborn, MI 48124jmontier@ford.com

Robert H. Todd

Department of Mechanical Engineering, Brigham Young University, Provo, UT 84602todd@byu.edu

Larry L. Howell

Department of Mechanical Engineering, Brigham Young University, Provo, UT 84602lhowell@byu.edu

J. Mech. Des 133(3), 031001 (Feb 22, 2011) (10 pages) doi:10.1115/1.4003499 History: Received December 30, 2009; Revised January 01, 2011; Published February 22, 2011; Online February 22, 2011

This paper presents elliptical rolling contact joints in compression as an alternative to circular rolling contact and conventional revolute joints where high quality force transmission—low friction and backlash—with variable output are desired. Parameters specific to the joint and its position are defined in terms of relative link angles and elliptical surface geometry. These relationships allow elliptical rolling contact joints to be incorporated in vector loop summations used in kinematic analysis. Notably, elliptical rolling contact is developed as the more general case of which circular rolling contact is a subset. Elliptical rolling contact joints are shown to offer several benefits over circular rolling contact, including reduced Hertz contact stresses, variable output velocity, maximum use of contact interface by distributing small rotations across surfaces of small curvature, reduced forces on constraining members, and no-slip pure rolling provided by either connecting links or flexures, without the need for gear teeth or friction.

Copyright © 2011 by American Society of Mechanical Engineers
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References

Figures

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Figure 1

Undesirable characteristics of circular rolling contact joints in compression: (a) large Hertz contact stresses due to small radii, (b) large bending stresses when flexures are used, and (c) gear teeth or other no-slip provision required for tractive rolling

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Figure 2

Although the links in both examples experience the same relative rotation, (a) will have much larger contact angles (ψc) because the normal line through the contact rotates away from the force rather than with the force

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Figure 3

(a) Basic elements and dimensions of an ellipse. (b) The motion of the elliptical gear is derived from the antiparallelogram, where d(F1A,F2B)=d(F1B,F2A)=2a and d(F1A,F2A)=d(F1B,F2B)=2c.

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Figure 4

Vector loop path through elliptical rolling contact joint (εj=0.762)

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Figure 5

Geometric parameters defining a point P on adjacent elliptical rolling links

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Figure 6

Motion path of ellipse center Cj and point P(u=0.5a, v=0.7a) with εj=0.762

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Figure 7

Dimensions used to derive the polar coordinates from the ellipse center

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Figure 8

Curvature for the elliptical contact surfaces shown in Fig. 4(εj=0.762, aj=27 mm) as a function of the polar angle at the ellipse center

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Figure 9

Dimensions and forces required to calculate the component loads in the stabilizing flexures or connecting links (shown here) of an elliptical rolling contact joint

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Figure 10

The tangential force is divided by cos γ to obtain the component in the direction of the connecting link. Notably, cos γ=εj (in this case εj=0.762) when θj=0

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Figure 11

Tensile stress in connecting links increases with increasing relative link angle θj and decreasing eccentricity ε and contact friction (in this case, μ=0.1). Note that tensile stress in flexures is not a function of eccentricity.

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Figure 12

Mechanical brake concept with elliptical rolling contact joints and connecting links (εj=0.333, aj=4.76 mm)

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Figure 13

The kinematics of elliptical rolling link mechanisms with toggle linkages in series (here εj=0.333, aj=4.76 mm, and Lj=31.75 mm) results in significant mechanical advantage near the toggle point

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Figure 14

Relationship between total output stroke and average mechanical advantage for toggle linkages as a function of eccentricity (where aj=4.76 mm and Lj=31.75 mm)

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Figure 15

Hertz contact deflections Δy can be calculated using the half-width of the contact patch x=aHz and the equation of an ellipse

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Figure 16

Joint deflection is insensitive to changes in eccentricity (and radius of contact surface) while Hertz stress decreases with increased eccentricity. Note in this case the selected link size (aj=4.76 mm, LHz=9.5 mm) is insufficient to support the applied loads.

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