OPTI 521 Optomechanical Engineering Hertzian Contact Stresses December

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

OPTI 521 Optomechanical Engineering • • • Overview Hertzian Contact Stresses Non-Hertzian Contact Stresses Failure modes Implications in Opto-Mechanics Summary 2

OPTI 521 Optomechanical Engineering Overview • 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. L-3 Insight Technology http: //t 0. gstatic. com/images? q=tbn: ANd 9 Gc. T 30 f 7 Op. Emou Xx. Tq. MSO 4 X 0 Mx 0 HIx 3 t. WEQq. XLnnap 3

OPTI 521 Optomechanical Engineering Hertzian Contact Stresses Ball with no contact force Deformation Caused by Hertzian Contact Stresses L-3 Insight Technology 4

OPTI 521 Optomechanical Engineering Hertzian Contact Stresses • 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. L-3 Insight Technology http: //t 3. gstatic. com/images? q=tbn: ANd 9 Gc. QFr. TP 2 mp T 7 mzxryu 0 OSUB 0 if. FE 5 vh 8 P 2 ILc. Htfo 9 dx 6 Cjcf. YB 8 CQ 5

OPTI 521 Optomechanical Engineering Hertzian Contact Stresses Spherical Bodies • Spherical Equations 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 L-3 Insight Technology 6

OPTI 521 Optomechanical Engineering Hertzian Contact Stresses Spherical Bodies cont. • Principle and Shear Stresses L-3 Insight Technology 7

OPTI 521 Optomechanical Engineering • Hertzian Contact Stresses Cylindrical Bodies For Cylindrical contacts instead of a circular contact area an elliptical contact area is produced. The equations below cover this change. L-3 Insight Technology 8

OPTI 521 Optomechanical Engineering • Hertzian Contact Stresses Cylindrical Bodies cont. Note: In the cylindrical case the principle stresses are not constant. For more detailed information on this see Mechanical Engineering Design, Shigley 2004 L-3 Insight Technology 9

OPTI 521 Optomechanical Engineering Non-Hertzian Contact Stresses • 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 L-3 Insight Technology 10

OPTI 521 Optomechanical Engineering Failure Modes • Permanent Plastic Deformation of parts • Fatigue damage http: //www. vibanalysis. co. uk/vibcases/vibch 13 p 1. jpg Fatigue damage on bearing. L-3 Insight Technology Plastic Deformation of Aluminum 11

OPTI 521 Optomechanical Engineering Implications in Opto-Mechanics • Weight limits for kinematic mounts • Design limits of Sharp Edge lens seats • Point Contacts for mirror supports • Point Contacts of micrometers L-3 Insight Technology http: //assets. newport. com/web 900 w-EN/images/8566. jpg http: //www. optics. arizona. edu/optomech/Fall 11/Notes/27%20 Mounting%20 of%20 lenses. pdf 12

OPTI 521 Optomechanical Engineering Summary • 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. L-3 Insight Technology 13