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.
