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Simple Is Beautiful – Refreshing thinking in engineering modeling and beyond Liming Chang Professor Penn State University Guest Professor National Chung Cheng University

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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…….

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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

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A basic journal bearing

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Long-bearing model (L/D > 3)

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

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

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Take log: Or, short bearings long bearings

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Approximate finite bearings by:

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

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Centrally-pivoted plane-pad thrust bearing

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Classical lubrication theory fails to predict

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Potential mechanisms of lubrication Viscosity-temperature thermal effect

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Load capacity by thermal effect

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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

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

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Temperature distribution Temperature rise Dimensionless variables:

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Pressure distribution Pressure Pad equilibrium Given solve for and

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Bearing dimensionless load parameter, W th Load and dimensionless load Bearing load parameter = viscosity-temperature coefficient ~ 0.04 o C -1 = lubricant density ~ 900 kg/m 3 c = lubricant specific heat ~ 2000 J/kg- o C w/B = bearing working pressure ~ 5.0 MPa

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One-to-one relation between C th and W th

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Bearing film thickness, h o h max = outlet film thickness under isothermal maximum-load-capacity condition (X =.58 )

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Verification with numerical results for square pad

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

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h o /h max results

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

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

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Meshing of a spur gear pair

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

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Experimental repeatability scatter Test number Pinion speed (rpm) Pinion toque (N- m) 16000413 26000546 36000684 48000413 58000546 68000684 710000413 810000546 910000684 Repeatability amounts to 0.04% of input power

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

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

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Effects of shear localization on oil shear stress

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Effect of load on gear meshing loss

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Effect of speed on gear meshing loss

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Effect of gear geometry – module

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Theory vs. experiment Theory Experiment Test number Pinion speed (rpm) Pinion toque (N-m) 16000413 26000546 36000684 48000413 58000546 68000684 710000413 810000546 910000684

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Effect of gear geometry – pressure angle

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Effect of gear geometry – addendum length

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Oil property – viscosity-pressure sensitivity

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Oil property – viscosity-temperature sensitivity

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Effect of gear thermal conductivity

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w Shear stress reduction with one surface insulated

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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.

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DESIGN OF MACHINE ELEMENTS

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