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

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…….

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

A basic journal bearing

Long-bearing model (L/D > 3)

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

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

Take log: Or, short bearings long bearings

Approximate finite bearings by:

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

Centrally-pivoted plane-pad thrust bearing

Classical lubrication theory fails to predict

Potential mechanisms of lubrication Viscosity-temperature thermal effect

Load capacity by thermal effect

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

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

Temperature distribution Temperature rise Dimensionless variables:

Pressure distribution Pad equilibrium Given solve for  and

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 

One-to-one relation between Cth and Wth

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

Verification with numerical results for square pad

Infinitely-wide pad Finite-width pad Further development of the theory for finite pads http://www.tandfonline.com/doi/abs/10.1080/10402004.2012.700765 Infinitely-wide pad Finite-width pad

ho/hmax results

III. Research on gear meshing efficiency L. Chang and Y. R. Jeng http://tribology.asmedigitalcollection.asme.org/article.aspx?articleid=1656917

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

Meshing of a spur gear pair

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

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

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

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

Effects of shear localization on oil shear stress

Effect of load on gear meshing loss

Effect of speed on gear meshing loss

Effect of gear geometry – module

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

Effect of gear geometry – pressure angle

Effect of gear geometry – addendum length

Oil property – viscosity-pressure sensitivity

Oil property – viscosity-temperature sensitivity

Effect of gear thermal conductivity

Shear stress reduction with one surface insulated

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.