Hertzian Contact Stresses December 2011 Nicholas LeCain

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

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

3. 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. http://t0.gstatic.com/images?q=tbn:ANd9GcT30f7OpEmouXxTqMSO4X0Mx0HIx3tWEQqXLnnap L-3 Insight Technology

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

5. 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. http://t3.gstatic.com/images?q=tbn:ANd9GcQFrTP2mpT7mzxryu0OSUB0ifFE5vh8P2ILcHtfo9dx6CjcfYB8CQ L-3 Insight Technology

6. 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

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

8. 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

9. 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

10. 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

11. OPTI 521 Optomechanical Engineering Failure Modes • Permanent Plastic Deformation of parts • Fatigue damage http://www.vibanalysis.co.uk/vibcases/vibch13/vibch13p1.jpg Fatigue damage on bearing. Plastic Deformation of Aluminum L-3 Insight Technology

12. 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 http://assets.newport.com/web900w-EN/images/8566.jpg http://www.optics.arizona.edu/optomech/Fall11/Notes/27%20Mounting%20of%20lenses.pdf L-3 Insight Technology

13. 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