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
  3 p    3 p 
dh
h
  h
  6U
x  x  z  z 
dx
Long-bearing model (L/D > 3)
  3 p    3 p 
dh
h
  h
  6U
x  x  z  z 
dx
3D 3 L  (4 2   2   2 2 )1 / 2
W
4c 2
(2   2 )(1   2 )
Short-bearing model (L/D < 1/4)
  3 p    3 p 
dh
h
  h
  6U
x  x  z  z 
dx
DL  (0.62  1)
W
8c 2
(1   2 ) 2
3
2
1/ 2
A finite-bearing model
Define a dimensionless load:
c2
W
W
4
D
Then
 (0.62 2  1)1 / 2
W 
8(1   2 ) 2
L
 
D
3
3 (4 2   2   2 2 )1 / 2  L 
W
 
2
2
4(2   )(1   )
 D
for short bearings
for long bearings
Take log:
 
  (0.62 2  1)1 / 2 
L

log W  log

3
log
 
2 2

8(1   )
D


 
 3 (4 2   2   2 2 )1 / 2 
L

log W  log

log
 
2
2

4(2   )(1   )
D


Or,
Y  f S ( )  3 X
short bearings
Y  f L ( )  X
long bearings
Approximate finite bearings by:
Y  f ( , X )  c3 X 3  c2 X 2  c1 X  co
Y  f L ( )  X
Y  f S ( )  3 X
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
  3 p    3 p 
dh


h

h

6

U


x  x  y  y 
dx

B
0
B
pxdx  xc  pdx
0
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:
d  h 3 dp 
dh

  6U
dx   dx 
dx
B
B
Pad equilibrium:

Temperature equation:


U dT
U
  0
c
  
2 dx
 (hi  ho ) / 2 
Oil  ~ T relation:
  oe  (T T )
0
pxdx  0.5 pdx
0
2
o
Temperature distribution
Temperature rise


8Cth
T  ln1 
X 
2
 (1  H )

Dimensionless variables:
T  T
X  x/B
H  hi / ho
UBo 
Cth  2
ho c
0  X  1.0
Pressure distribution
Pressure p( X )  A( X )  c1 B( X )  c2
p
ho2
Uo B
A( X )  6
p
B( X )  
0  X  1.0
dX


8Cth
2
1

X
 (1  H ) 2 H  ( H  1) X 


dX


8Cth
3
1

X
 (1  H ) 2 H  ( H  1) X 


1.0
Pad equilibrium
UBo 
Given Cth  2
ho c

0
1.0
pXdX  0.5 pdX
0
solve for  p(X ) and H  hi / ho
Bearing dimensionless load parameter, Wth
Load and dimensionless load
1.0
B
0
0
w   pdX  
 ho2

 U o B

ho2
p d ( x / B) 
w
2
U o B

Bearing load parameter
 UBo   ho2
  w

Cth w   2
w  
 Wth
2
 ho c  Uo B  c B
 = viscosity-temperature coefficient ~ 0.04 oC-1
 = lubricant density ~ 900 kg/m3
c = lubricant specific heat ~ 2000 J/kg-oC
w/B = bearing working pressure ~ 5.0 MPa 
Wth ~ 0.1
One-to-one relation between Cth and Wth
Bearing film thickness, ho
ho  0.65hmax
hmax = outlet film thickness under isothermal
maximum-load-capacity condition (X = .58 )
Verification with numerical results for square pad
Wth  0.05
0.65hmax
0.6hmax
Wth  0.17
Further development of the theory for finite pads
Y. Yan and L. Chang – Tribology Transactions, in press
Infinitely-wide pad
d  h 3 dp 
dh

  6U
dx   dx 
dx
Finite-width pad
  3 p    3 p 
dh
h
  h
  6U
x  x  z  z 
dx
ho/hmax results
0.7
N=0
o
Relative film thickness, h /h
iso
0.6
N=0.5
N=1.0
0.5
0.4
0.3
N=2.0
0.2
0.1
0
0
0.02
0.04
0.06
0.08
0.1
0.12
Bearing load parameter, W
th
0.14
0.16
III. Research on gear meshing efficiency
L. Chang and Y. R. Jeng
Manuscript in review
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
  h3 p  u1  u2 h h

 

x  12 x 
2 x t
Load equation
xo
w(t )   p(s, t )ds
xi
Film-thickness equation
2 xo
sx
h( x, t )  ho (t )  g ( x, t )  r ( x, t ) 
p
(
s
,
t
)
ln

 ds

x
E ' i
 s 
2
Temperature equation
 2T
T
kf


c
u
   0
f f f
2
z
x
xo
Friction calculated by f (t )  x  ( x, z, t ) |z 0dx
i
Experimental repeatability scatter
Test
number
Pinion speed
(rpm)
Pinion toque (Nm)
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
Upper surface
w
1.0
0.8
No localization
Z
0.6
0.4
0.2
With
localization
0.0
1.90
1.95
2.00
2.05
2.10
Velocity, m/s
Lower surface
Cross-film velocity
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
Test
number
1
2
3
4
5
6
7
8
9
Pinion
speed (rpm)
6000
Pinion toque
(N-m)
413
6000
6000
8000
8000
546
684
413
546
8000
10000
684
413
10000
10000
546
684
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
w
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.