Simple Is Beautiful – Refreshing thinking in engineering modeling and beyond

1. Simple Is Beautiful –Refreshing thinking in engineering modelingand beyond Liming Chang Professor Penn State University Guest Professor National Chung Cheng University

2. Implications of Simplicity • Deep understanding leads to simple approaches to problem solving • Simple solutions often generate time-lasting significance • Ability to solve a complex problem simply is the highest level of competency Three examples…….

3. I. An Analytical Model for the Basic Design Calculations of Journal Bearings R. K. Naffin and L. Chang http://www.mne.psu.edu/chang/me462/finite-journal.pdf

4. A basic journal bearing

5. Long-bearing model (L/D > 3)

6. Short-bearing model (L/D < 1/4)

7. A finite-bearing model Define a dimensionless load: Then for short bearings for long bearings

8. Take log: Or, short bearings long bearings

9. Approximate finite bearings by:

10. II. A Theory for the Design ofCentrally-Pivoted Thrust Bearings L. Chang http://www.mne.psu.edu/chang/me462/JOT_slider.pdf

11. Centrally-pivoted plane-pad thrust bearing

12. Classical lubrication theory fails to predict

13. Potential mechanisms of lubrication • Viscosity-temperature thermal effect

14. Load capacity by thermal effect

15. A simple thermal-lubrication model: assumptions • Infinitely wide pad • Conduction heat transfer negligible • Convection heat transfer at cross-film average velocity • Uniform shear-strain rate

16. A simple thermal-lubrication model: equations Reynolds equation: Pad equilibrium: Temperature equation: Oil h ~ T relation:

17. Temperature distribution Temperature rise Dimensionless variables:

18. Pressure distribution Pressure Pad equilibrium Givensolve for  and

19. Bearing dimensionless load parameter, Wth Load and dimensionless load Bearing load parameter b = viscosity-temperature coefficient ~ 0.04 oC-1 r = lubricant density ~ 900 kg/m3 c = lubricant specific heat ~ 2000 J/kg-oC w/B = bearing working pressure ~ 5.0 MPa 

20. One-to-one relation between Cth and Wth

21. Bearing film thickness, ho hmax = outlet film thickness under isothermal maximum-load-capacity condition (X = .58 )

22. Verification with numerical results for square pad

23. Further development of the theory for finite padsY. Yan and L. Chang – Tribology Transactions, in press Infinitely-wide padFinite-width pad

24. ho/hmax results

25. III. Research on gear meshing efficiency L. Chang and Y. R. Jeng Manuscript in review

26. Meshing of a spur gear pair Meshing loss can be less than 0.5% of input power

27. Meshing of a spur gear pair

28. Governing equations Reynolds equation Load equation Film-thickness equation Temperature equation Friction calculated by

29. Experimental repeatability scatter Repeatability amounts to 0.04% of input power

30. Well, simple is beautiful! • Hertz pressure distribution • Parallel film gap • Numerical solution of temperature equation

31. Upper surface No localization w With localization Lower surface Cross-film velocity Thermal shear localization

32. Effects of shear localization on oil shear stress

33. Effect of load on gear meshing loss

34. Effect of speed on gear meshing loss

35. Effect of gear geometry – module

36. Theory vs. experiment Experiment Theory

37. Effect of gear geometry – pressure angle

38. Effect of gear geometry – addendum length

39. Oil property – viscosity-pressure sensitivity

40. Oil property – viscosity-temperature sensitivity

41. Effect of gear thermal conductivity

42. w Shear stress reduction with one surface insulated

43. Summary • Clever simple approaches to problem solving can help reveal fundamental insights and/or produce key order-of-magnitude results/trends. • It is no small feat to develop a mathematic model that is simple and generally applicable. • The significance of a simple model of general validity can be tremendous and long lasting.