1. National Chung Cheng University
Simple Is Beautiful – Refreshing thinking in engineering modeling and 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

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 of Centrally-Pivoted Thrust Bearings
L. Chang

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
Pad equilibrium
Given solve 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. Infinitely-wide pad Finite-width pad
Further development of the theory for finite pads
Infinitely-wide pad Finite-width pad

24. ho/hmax results

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

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
Test number
Pinion speed (rpm)
Pinion toque (N-m) 1
6000
413 2
546 3
684 4
8000 5 6 7
10000 8 9
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. Thermal shear localization
Cross-film velocity
No localization
With
localization
Upper surface
Lower surface w

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 Test number Pinion speed (rpm)
Pinion speed (rpm)
Pinion toque (N-m) 1
6000
413 2
546 3
684 4
8000 5 6 7
10000 8 9
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. 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.