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

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

3. 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 Technology3 Overview http://t0.gstatic.com/images?q=tbn:ANd9GcT30f7OpEmou XxTqMSO4X0Mx0HIx3tWEQqXLnnap OPTI 521 Optomechanical Engineering

4. Ball with no contact force Deformation Caused by Hertzian Contact Stresses L-3 Insight Technology4 Hertzian Contact Stresses OPTI 521 Optomechanical Engineering

5. 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 Technology5 Hertzian Contact Stresses http://t3.gstatic.com/images?q=tbn:ANd9GcQFrTP2mp T7mzxryu0OSUB0ifFE5vh8P2ILcHtfo9dx6CjcfYB8CQ OPTI 521 Optomechanical Engineering

6. Spherical Equations L-3 Insight Technology6 Hertzian Contact Stresses Spherical 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

7. Principle and Shear Stresses L-3 Insight Technology7 Hertzian Contact Stresses Spherical Bodies cont. OPTI 521 Optomechanical Engineering

8. For Cylindrical contacts instead of a circular contact area an elliptical contact area is produced. The equations below cover this change. L-3 Insight Technology8 Hertzian Contact Stresses Cylindrical Bodies OPTI 521 Optomechanical Engineering

9. 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 Technology9 Hertzian Contact Stresses Cylindrical Bodies cont. OPTI 521 Optomechanical Engineering

10. 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 Technology10 Non-Hertzian Contact Stresses OPTI 521 Optomechanical Engineering

11. Permanent Plastic Deformation of parts Fatigue damage L-3 Insight Technology11 Failure Modes Fatigue damage on bearing. http://www.vibanalysis.co.uk/vibcases/vibch13/vibch13p1.jpg Plastic Deformation of Aluminum OPTI 521 Optomechanical Engineering

12. Weight limits for kinematic mounts Design limits of Sharp Edge lens seats Point Contacts for mirror supports Point Contacts of micrometers L-3 Insight Technology12 Implications in Opto-Mechanics http://assets.newport.com/web900w-EN/images/8566.jpg http://www.optics.arizona.edu/optomech/Fall11/Notes/27%20 Mounting%20of%20lenses.pdf OPTI 521 Optomechanical Engineering

13. 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 Technology13 Summary OPTI 521 Optomechanical Engineering