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

dx

dhU

z

ph

zx

ph

x633

Long-bearing model (L/D > 3)

)1)(2(

)4(

4

322

2/12222

2

3

c

LDW

dx

dhU

z

ph

zx

ph

x633

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

dx

dhU

z

ph

zx

ph

x633

22

2/12

2

3

)1(

)162.0(

8

c

DLW

A finite-bearing model

Define a dimensionless load:

Then

WD

cW

4

2

3

22

2/12

)1(8

)162.0(

D

LW

D

LW

)1)(2(4

)4(322

2/12222

for short bearings

for long bearings

Take log:

Or,

short bearings

long bearings

D

LW log3

)1(8

)162.0(loglog

22

2/12

D

LW log

)1)(2(4

)4(3loglog

22

2/12222

XfY S 3)(

XfY L )(

Approximate finite bearings by:

ocXcXcXcXfY 12

23

3),(

XfY S 3)(

XfY L )(

II. A Theory for the Design ofCentrally-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

dxpxpxdxBB

c 00dx

dhU

y

ph

yx

ph

x633

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 ~ T relation:

dx

dhU

dx

dph

dx

d6

3

02/)(2

2

oi hh

U

dx

dTUc

)( oTToe

dxppxdxBB

00

5.0

Temperature distribution

Temperature rise

Dimensionless variables:

XH

CT th

2)1(

81ln

ch

UBC

o

oth

2

oi hhH /

BxX /

0.10 X

TT

Pressure distribution

Pressure

Pad equilibrium

Given solve for and

21 )()()( cXBcXAXp

2

2)1(

)1(

81

6)(

XHHXH

C

dXXA

th

3

2)1(

)1(

81

)(

XHHXH

C

dXXB

th

dXpXdXp 0.1

0

0.1

05.0

pBU

hp

o

o

2

ch

UBC

o

oth

2

)(Xp oi hhH /

0.10 X

Bearing dimensionless load parameter, Wth

Load and dimensionless load

Bearing load parameter

= viscosity-temperature coefficient ~ 0.04 oC-1

= lubricant density ~ 900 kg/m3 c = lubricant specific heat ~ 2000 J/kg-oCw/B = bearing working pressure ~ 5.0 MPa

wBU

hBxdp

BU

hdXpw

o

oB

o

o2

2

0

20.1

0)/(

tho

o

o

oth W

B

w

cw

BU

h

ch

UBwC

2

2

2

1.0~thW

One-to-one relation between Cth and Wth

Bearing film thickness, ho

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

max65.0 hho

Verification with numerical results for square pad

max65.0 h

05.0thW

17.0thW

max6.0 h

Further development of the theory for finite padshttp://www.tandfonline.com/doi/abs/10.1080/10402004.2012.700765

Infinitely-wide pad Finite-width pad

dx

dhU

dx

dph

dx

d6

3

dx

dhU

z

ph

zx

ph

x633

ho/hmax results

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.160

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Bearing load parameter, Wth

Re

lativ

e fi

lm th

ickn

ess

, ho/h

iso

N=2.0

N=1.0N=0.5

N=0

III. Research on gear meshing efficiency

L. Chang and Y. R. Jenghttp://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

t

h

x

huu

x

ph

x

21221

3

dss

xstsp

Etxrtxgthtxh

o

i

x

xo

2

ln),('

2),(),()(),(

dstsptwo

i

x

x ),()(

02

2

x

Tuc

z

Tk ffff

dxtzxtfo

i

x

x z 0|),,()(

Experimental repeatability scatter

Test number

 Pinion speed

(rpm)Pinion toque (N-

m)

1 6000 413

2 6000 546

3 6000 684

4 8000 413

5 8000 546

6 8000 684

7 10000 413

8 10000 546

9 10000 684

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

0.0

0.2

0.4

0.6

0.8

1.0

1.90 1.95 2.00 2.05 2.10

Velocity, m/s

Z

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

Theory

ExperimentTest

number 

Pinion speed (rpm)

Pinion toque (N-m)

1 6000 413

2 6000 546

3 6000 684

4 8000 413

5 8000 546

6 8000 6847 10000 413

8 10000 546

9 10000 684

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

w

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.