1 Hertzian Contact Stresses December 2011 Nicholas LeCain OPTI 521 Optomechanical Engineering.

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Hertzian Contact Stresses

December 2011Nicholas LeCain

OPTI 521 Optomechanical Engineering

• Overview• Hertzian Contact Stresses• Non-Hertzian Contact Stresses• Failure modes• Implications in Opto-Mechanics• Summary

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OPTI 521 Optomechanical Engineering

• Contact stresses– Stress developed from

two radii in contact– Stress

• σ=F/A• Force is constant• Area is infinitely small• Stress approaches infinity

– Deformation occurs until area is large enough to reduce stress to below elastic limit of parts.

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Overview

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OPTI 521 Optomechanical Engineering

Ball with no contact force Deformation Caused by Hertzian

Contact Stresses

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Hertzian Contact StressesOPTI 521

Optomechanical Engineering

• Hendrick Hertz first published his work on contact stresses in 1881.

• Work was based on a few assumptions. – Frictionless – Elastic bodies– Isotropic materials– Homogeneous materials– No external shear stress

• Without these assumptions the equations get out of hand pretty quickly and an FEA approach to analysis is required.

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Hertzian Contact Stresses

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OPTI 521 Optomechanical Engineering

• Spherical Equations

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Hertzian Contact StressesSpherical Bodies

Radius of deformed contact area

Maximum pressure from force applied Note: For a flat surface R would equal

infinity and for a concave surface like a spherical hole R would be negative

OPTI 521 Optomechanical Engineering

• Principle and Shear Stresses

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Hertzian Contact Stresses Spherical Bodies cont.

OPTI 521 Optomechanical Engineering

• For Cylindrical contacts instead of a circular contact area an elliptical contact area is produced. The equations below cover this change.

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Hertzian Contact StressesCylindrical Bodies

OPTI 521 Optomechanical Engineering

• Note: In the cylindrical case the principle stresses are not constant. For more detailed information on this see Mechanical Engineering Design, Shigley 2004

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Hertzian Contact StressesCylindrical Bodies cont.

OPTI 521 Optomechanical Engineering

• Applications where the assumptions listed in the previous slide do not apply fall under Non-Hertzian contact stresses. – These applications must be handled with finite

element analysis or with the Smith-Liu equations

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Non-Hertzian Contact StressesOPTI 521 Optomechanical Engineering

• Permanent Plastic Deformation of parts• Fatigue damage

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Failure Modes

Fatigue damage on bearing.

http://www.vibanalysis.co.uk/vibcases/vibch13/vibch13p1.jpg

Plastic Deformation of Aluminum

OPTI 521 Optomechanical Engineering

• Weight limits for kinematic mounts

• Design limits of Sharp Edge lens seats

• Point Contacts for mirror supports

• Point Contacts of micrometers

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Implications in Opto-Mechanics

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http://www.optics.arizona.edu/optomech/Fall11/Notes/27%20Mounting%20of%20lenses.pdf

OPTI 521 Optomechanical Engineering

• Hertzian equations apply to contact stresses created by the contact of radii.

• Hertzian Equations work well if the stated above assumptions apply

• If there are exceptions to the assumptions FEA or more complicated equations must be used.

• In the field of opto-mechanics Hertzian equations work well for the analysis of Kinematic mounts and lens seats.

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SummaryOPTI 521 Optomechanical Engineering