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RESEARCH PAPERS

# On Spatial Euler–Savary Equations for Envelopes

[+] Author and Article Information
David B. Dooner

Department of Mechanical Engineering, University of Puerto Rico-Mayagüez, Mayagüez, PR 00681-9045ḏdooner@me.uprm.edu

Michael W. Griffis

The Eigenpoint Company, PO Box 1708, High Springs, FL 32655

The above sign convention is based on Ref. 1

The relative angular speed $ω$ between the fixed and moving centrode is $ω=ωf(1+g)=ωf+ωm$ and the speed $vp$ of the pitch point is $vP=ωfRf=ωmRm$.

There is no agreement on the spatial analog to the osculation circle. It is recognized that these hyperboloids are not tangent to the ruled surfaces and do not osculate in a fashion analogous to the planar scenario. However, the term “hyperboloid of osculation” is used here to identify these hyperboloids as there is no known established terminology.

The formulation of the third law of gearing presented in Ref. 22 improperly accounts for the angle $ϕ$ between the curvature $K$ and the tooth normal $Ŝl$.

The distance $r$ and pitch $h$ of the Disteli axis as defined in terms of curvature $κ$ and torsion $τ$ of geodesic $tθ$ are $u=κ∕κ2+τ2$ and $h=τ∕κ2+τ2$.

The relative angular speed $ω$ is $ω=ωf1+g2−2gcosΣ$ and the speed of any point $p$ along the common generator is $vp=ωfuf2cos2αf+wf2sin2αf=ωmum2cos2αm+wm2sin2αm$.

J. Mech. Des 129(8), 865-875 (Jul 13, 2006) (11 pages) doi:10.1115/1.2735339 History: Received July 20, 2005; Revised July 13, 2006

## Abstract

Presented are three equations that are believed to be original and new to the kinematics community. These three equations are extensions of the planar Euler–Savary relations (for envelopes) to spatial relations. All three spatial forms parallel the existing well established planar Euler–Savary equations. The genesis of this work is rooted in a system of cylindroidal coordinates specifically developed to parameterize the kinematic geometry of generalized spatial gearing and consequently a brief discussion of such coordinates is provided. Hyperboloids of osculation are introduced by considering an instantaneously equivalent gear pair. These analog equations establish a relation between the kinematic geometry of hyperboloids of osculation in mesh (viz., second-order approximation to the axode motion) to the relative curvature of conjugate surfaces in direct contact (gear teeth). Planar Euler–Savary equations are presented first along with a discussion on the terms in each equation. This presentation provides the basis for the proposed spatial Euler–Savary analog equations. A lot of effort has been directed to establishing generalized spatial Euler–Savary equations resulting in many different expressions depending on the interpretation of the planar Euler–Savary equation. This work deals with the interpretation where contacting surfaces are taken as the spatial analog to the contacting planar curves.

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## Figures

Figure 1

Centrodes, inflection circle, and pitch circles for planar motion

Figure 2

Displacement of moving osculation circle

Figure 3

Anatomy of skew axis gear pair

Figure 4

Enlarged view of hyperboloidal pitch surface

Figure 5

System of cylindroidal coordinates

Figure 6

Pitch surfaces are determined by any generator of the cylindroid

Figure 7

Axode with hyperboloid of osculation

Figure 8

Cylindroid of torsure defined by the radius of torsure for the geodesic tθ

Figure 9

Relative curvature between conjugate surfaces as parameterized using cylindroidal coordinates

## Errata

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