Simple Is Beautiful – Refreshing thinking in engineering modeling and beyond Liming Chang...
-
Upload
cali-tillison -
Category
Documents
-
view
220 -
download
0
Transcript of Simple Is Beautiful – Refreshing thinking in engineering modeling and beyond Liming Chang...
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 padsY. Yan and L. Chang – Tribology Transactions, in press
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. 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
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.