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ILLATIONS
cantilever
δ
In the absence of any externally applied forces [e.g. far away
from the cantilever surface], a high resolution force tranducer
will oscillate at its natural resonant frequency (maximum
displacement of the amplitude of the oscillations) due to a
non-zero thermal energy, kBT (room temperature)... | https://ocw.mit.edu/courses/3-052-nanomechanics-of-materials-and-biomaterials-spring-2007/13bf43445b0222923489ee0ef37f7aaf_lec2.pdf |
interparticle forces
10-4 10-3
mN
protein unfolding
actin filament
extension
antibody-
antigen
covalent
bond
protein-
protein
molecular
motors
DNA conformational
transition
cell contraction
9
3.052 Nanomechanics of Materials and Biomaterials Thursday 02/08/06
Prof. C. Ortiz, MIT-DMSE
BIOSEN... | https://ocw.mit.edu/courses/3-052-nanomechanics-of-materials-and-biomaterials-spring-2007/13bf43445b0222923489ee0ef37f7aaf_lec2.pdf |
Lecture 5
Amortization
6.046J
Spring 2015
Lecture 5: Amortization
Amortized analysis is a powerful technique for data structure analysis, involving
the total runtime of a sequence of operations, which is often what we really care
about. This lecture covers:
• Different techniques of amortized analysis
– aggre... | https://ocw.mit.edu/courses/6-046j-design-and-analysis-of-algorithms-spring-2015/13e2c7d165259712327af0af312a068e_MIT6_046JS15_lec05.pdf |
assign any
amortized cost to each operation, as long as they “preserve the total cost”, i.e., for
any sequence of operations,
�
�
amortized cost ≥
actual cost
where the sum is taken over all operations.
For example, we can say a 2-3 tree achieves O(1) amortized cost per create, O(lg n ∗)
amortized cost per ins... | https://ocw.mit.edu/courses/6-046j-design-and-analysis-of-algorithms-spring-2015/13e2c7d165259712327af0af312a068e_MIT6_046JS15_lec05.pdf |
below.
– amortized cost for table doubling: O(n) − c · n/2 = 0 for large enough c.
– amortized cost per insertion: 1 + c = O(1).
2
Lecture 5
Amortization
6.046J
Spring 2015
an element
a unused coin
table doubling due to the next insert
2-3 trees
Now let’s try the accounting method on 2-3 trees. Our goal... | https://ocw.mit.edu/courses/6-046j-design-and-analysis-of-algorithms-spring-2015/13e2c7d165259712327af0af312a068e_MIT6_046JS15_lec05.pdf |
cost. This way, the table is half full again after
any resize (doubling or shrinking). Now each table doubling still has ≥ m/2 insert
operations to charge to, and each table halving has ≥ m/4 delete operations to charge
to. So the amortized cost per insert or delete is still Θ(1).
3
Lecture 5
Amortization
6.04... | https://ocw.mit.edu/courses/6-046j-design-and-analysis-of-algorithms-spring-2015/13e2c7d165259712327af0af312a068e_MIT6_046JS15_lec05.pdf |
0, which is usually the case, then Φ should never go
negative (intuitively, we cannot ”owe the bank”).
Relation to accounting method
In accounting method, we specify ΔΦ, while in potential method, we specify Φ. One
determines the other, so the two methods are equivalent. But sometimes one is more
intuitive than th... | https://ocw.mit.edu/courses/6-046j-design-and-analysis-of-algorithms-spring-2015/13e2c7d165259712327af0af312a068e_MIT6_046JS15_lec05.pdf |
.
The above analysis holds for any (a, b)-tree, if we define Φ to be the number of
b-nodes.
If we consider both insertion and deletion in 2-3 trees, can we claim both O(1) splits
for insert, and O(1) merges for delete? The answer is no, because a split creates two
2-nodes, which are bad for merge. In the worse case... | https://ocw.mit.edu/courses/6-046j-design-and-analysis-of-algorithms-spring-2015/13e2c7d165259712327af0af312a068e_MIT6_046JS15_lec05.pdf |
b-nodes plus the
number of a nodes.
Note: The potential examples could also be done with the accounting method by
placing coins on 1s (binary counter) or 2/5-nodes ((2, 5)-trees).
6
MIT OpenCourseWare
http://ocw.mit.edu
6.046J / 18.410J Design and Analysis of Algorithms
Spring 2015
For information about citing t... | https://ocw.mit.edu/courses/6-046j-design-and-analysis-of-algorithms-spring-2015/13e2c7d165259712327af0af312a068e_MIT6_046JS15_lec05.pdf |
Lecture # 16
Thermomechanical Conversion II
Two-Phase Cycles and Combined Cycles
Ahmed Ghoniem
April 1, 2020
Rankine Cycle: two phase region
Superheat and Ultra-superheat Cycles. Reheating. Recuperation.
Supercritical Cycles. Hypercritical Cycles (CO2 as working fluid)
Water requirements.
© Ahmed F. Ghoniem
1
... | https://ocw.mit.edu/courses/2-60j-fundamentals-of-advanced-energy-conversion-spring-2020/1437d5925093b270c871160ec861fd12_MIT2_60s20_lec16.pdf |
= h3 − h2
Simple saturated cycle efficiency,
Pressure Ratio = 8,
Pump = 65%, turbine 90%.
Conventional
Tmin=20
Closed
cycle
1.23
1.12
Pmin=1atm
Open cycle
736
735
316
315
27.4 %
13.4%
30.4%
14.9%
33.9%
15.8%
0.794
0.8856
wpump
(kJ/kg)
wt
(kJ/kg)
wnet
(kJ/kg)
η
ηideal
ηcar
X4
© Ahmed F. Gho... | https://ocw.mit.edu/courses/2-60j-fundamentals-of-advanced-energy-conversion-spring-2020/1437d5925093b270c871160ec861fd12_MIT2_60s20_lec16.pdf |
Cold gases
Condensed
water
Economizer
Evaporator
Superheater
Hot gases
Drum
Superheated
steam out
From Smith and Cravalho,
Engineering Thermodynamics
Superheat
+100
Tmin=20
1.23
1.23
736
735
818
817
27.4 %
28.1%
30.4%
33.9%
46.0%
0.794
0.8517
wpump
(kJ/kg)
wt
(kJ/kg)
wnet
(kJ/kg)
η
ηideal
... | https://ocw.mit.edu/courses/2-60j-fundamentals-of-advanced-energy-conversion-spring-2020/1437d5925093b270c871160ec861fd12_MIT2_60s20_lec16.pdf |
20 +100
Reheat Cycle
+200
+300
wpump
(kJ/kg)
1.23
1.23
1.23
1.23
wt (kJ/kg)
736
947.2
1086
1400
wnet (kJ/kg)
735
946
1085
1398
η
ηideal
ηcar
X6
27.4 %
28.1% 30.3%
35.5%
30.4%
33.9%
0.794
46.0% 54.4%
60.6%
0.9583 Vapor Vapor
Better efficiency and steam
quality at end of expansion
© Ahmed... | https://ocw.mit.edu/courses/2-60j-fundamentals-of-advanced-energy-conversion-spring-2020/1437d5925093b270c871160ec861fd12_MIT2_60s20_lec16.pdf |
800
750
700
650
600
550
500
450
400
350
300
250
4
3
2
1
0
1
2
3
From reheat
7
Second stage
turbine
5
First stage
turbine
To Reheat
6
Open
feedwater
heater
G
8
1
8
7
8
9
10
4
2
3
Boiler feedwater
pump (BFP)
Condensate
pump (CP)
4
5
Entropy [kJ/kg.K]
6
Best feedwater heater ar... | https://ocw.mit.edu/courses/2-60j-fundamentals-of-advanced-energy-conversion-spring-2020/1437d5925093b270c871160ec861fd12_MIT2_60s20_lec16.pdf |
reheat
7
Second stage
turbine
5
First stage
turbine
To Reheat
6
Closed
feedwater
heater
3
2
G
8
1
BFP
4
9
Throttle
valve
Less efficient because of throttling and
some heat rejection in condenser,
but only one pump is required .
© Ahmed F. Ghoniem
12
3. Cascading Forward,
Closed Feedwate... | https://ocw.mit.edu/courses/2-60j-fundamentals-of-advanced-energy-conversion-spring-2020/1437d5925093b270c871160ec861fd12_MIT2_60s20_lec16.pdf |
of supercritical steam cycle with reheat
(3.Büki G.,Magyar Energiatechnika 1998;6:33-42)
Coal plans are less efficient than NG plants
because of exhaust gas clean up
© Source unknown. All rights reserved. This content is excluded from our Creative Commons license. For more information, see https://ocw.mit.edu/fairus... | https://ocw.mit.edu/courses/2-60j-fundamentals-of-advanced-energy-conversion-spring-2020/1437d5925093b270c871160ec861fd12_MIT2_60s20_lec16.pdf |
https://ocw.mit.edu/fairuse.
• Supercritical CO2 are “hypercritical” cycles.
© Ahmed F. Ghoniem
17
Rankine cycles:
1. Fuel flexible, works well with coal
and other dirty fuels (closed cycle).
2. Have high efficiency, low pumping
power.
3. Require lower flow rate (latent
enthalpy).
4. Run at lower high T ... | https://ocw.mit.edu/courses/2-60j-fundamentals-of-advanced-energy-conversion-spring-2020/1437d5925093b270c871160ec861fd12_MIT2_60s20_lec16.pdf |
,
ηGT = 0.38, and ηST = 0.40,
ηCC = 0.55
ηCC = 0.5
ηCC = 0.535
ηCC = 0.628
© Ahmed F. Ghoniem
Gas Cycle
3
4
8'
8
Steam Cycle
2
5
9
9'
1
2
1
Air
Fuel
3
Combustion
products
GT
4
ST
8
4
5
Entropy [kJ/kg.K]
6
7
8
9
10
!
5
7
9
6
19
Mass flow rates are not a... | https://ocw.mit.edu/courses/2-60j-fundamentals-of-advanced-energy-conversion-spring-2020/1437d5925093b270c871160ec861fd12_MIT2_60s20_lec16.pdf |
et tower cooling,
consumption
All of these •
also depend on local water
and weather conditions!
exit T is the dew point of water
at its partial p in the exit air.
23
Cooling system types
Image: EPRI Journal Summer 2007 (cid:1)Running Dry at the Power Plant(cid:2)
• Simple, low-cost
... | https://ocw.mit.edu/courses/2-60j-fundamentals-of-advanced-energy-conversion-spring-2020/1437d5925093b270c871160ec861fd12_MIT2_60s20_lec16.pdf |
see https://ocw.mit.edu/fairuse.
26
Working fluids requirements:
1. High Tc for efficiency but low pc for simplicity
2. Large enthalpy of evaporation
3. Non toxic, non flammable, non corrosive, cheap ..
Water: pc=22.088 MPa Tc=374 C, most common
CO2: pc=7.39 MPa, Tc=30.4C (low p... | https://ocw.mit.edu/courses/2-60j-fundamentals-of-advanced-energy-conversion-spring-2020/1437d5925093b270c871160ec861fd12_MIT2_60s20_lec16.pdf |
Fast Fourier Transform:
VLSI Architectures
Lecture 10
Vladimir Stojanović
6.973 Communication System Design – Spring 2006
Massachusetts Institute of Technology
Cite as: Vladimir Stojanovic, course materials for 6.973 Communication System Design, Spring 2006.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Inst... | https://ocw.mit.edu/courses/6-973-communication-system-design-spring-2006/1460f43d3993b7c956d4bb8ee03d1fb0_lecture_10.pdf |
256)
Figure by MIT OpenCourseWare.
Cite as: Vladimir Stojanovic, course materials for 6.973 Communication System Design, Spring 2006.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology.
Downloaded on [DD Month YYYY].
6.973 Communication System Design
2
Radix-2 Multi-path Delay Commutato... | https://ocw.mit.edu/courses/6-973-communication-system-design-spring-2006/1460f43d3993b7c956d4bb8ee03d1fb0_lecture_10.pdf |
x4
x8
x12
x15
DFT
4
DFT
4
DFT
4
DFT
4
X0
X12
X1
X13
X2
X14
X3
X15
Figure by MIT OpenCourseWare.
x(n)
x(n+ )
N
4
x(n+ )
N
2
x(n+ )
3N
4
0
WN
n
WN
2n
WN
3n
WN
-j
-1
-1
-1
j
-1
-j
y(n)
y(n+ )
N
4
y(n+ )
N
2
3N
y(n+ )
4
Radix-4 butterfly utilization only 25%
Figure by MIT OpenCourseWare.
Butterfly fairly complicated
... | https://ocw.mit.edu/courses/6-973-communication-system-design-spring-2006/1460f43d3993b7c956d4bb8ee03d1fb0_lecture_10.pdf |
1
-1
j
-1
-j
y(n)
y(n+ )
N
4
y(n+ )
N
2
3N
y(n+ )
4
Figure by MIT OpenCourseWare.
Cite as: Vladimir Stojanovic, course materials for 6.973 Communication System Design, Spring 2006.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology.
Downloaded on [DD Month YYYY].
6.973 Communication System... | https://ocw.mit.edu/courses/6-973-communication-system-design-spring-2006/1460f43d3993b7c956d4bb8ee03d1fb0_lecture_10.pdf |
Design
7
R4SDC commutator and butterfly details
input
Nt
Nt
Nt
Nt
1
0
Nt
Nt
mt c1
0
1
0
1
0
2
0
3
c2
1
1
0
0
c3
1
1
1
0
Figure by MIT OpenCourseWare.
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
0
0
0
0
0
1
2
3
0
2
4
6
0
3
6
9
0
0
0
0
1
1
1
1
2
2
2
2
3
3
3
3
0
1
2
3
0
1
2
3
0
1
2
3
0
1
2
3
0
4
8
12
1
5
9
13
2
6
10
14
3
7
1... | https://ocw.mit.edu/courses/6-973-communication-system-design-spring-2006/1460f43d3993b7c956d4bb8ee03d1fb0_lecture_10.pdf |
1 0
15 14 13 12 11 10 9 8 7
6 5
6 5 4 3 2 1 0
15 14 13 12 11 10 9
8 7
6 5
4
3 2 1
t'+28T
t'+12T
= 3
m
1
= 2
m
1
= 1
m
1
= 0
m
1
2:1 multiplexers
Outputs from
commutator at
stage 1
Figure by MIT OpenCourseWare.
re (0)
im (0)
re (1)
im (1)
re (2)
im (2)
re (3)
im (3)
add/sub
add/sub
add/sub
add/sub
add/sub
Re
D
add/sub
I... | https://ocw.mit.edu/courses/6-973-communication-system-design-spring-2006/1460f43d3993b7c956d4bb8ee03d1fb0_lecture_10.pdf |
Decomposition – a review
Twiddle factor is Nth primitive root of unity
With exponent evaluated modulo N
Most fast algorithms share same general
strategy
Map one-dimensional transform int a two or multi
dimensional representation
Exploit congruence property of coefficients to simplify
computation
Unl... | https://ocw.mit.edu/courses/6-973-communication-system-design-spring-2006/1460f43d3993b7c956d4bb8ee03d1fb0_lecture_10.pdf |
0)
N/4 DFT
(k1=1, k2=1)
W0
W2
W4
W6
W0
W1
W2
W3
W0
W3
W6
W9
X(0)
X(8)
X(4)
X(12)
X(2)
X(10)
X(6)
X(14)
X(1)
X(9)
X(5)
X(13)
X(3)
X(11)
X(7)
X(15)
x(0)
x(1)
x(2)
x(3)
x(4)
x(5)
x(6)
x(7)
x(8)
x(9)
x(10)
x(11)
x(12)
x(13)
x(14)
x(15)
W2
W4
W6
W1
W2
W3
W3
W6
W9
-j
-j
-j
-j
-j
-j
-j
-j
BF I
BF II
BF I
BF II
BF III
BF IV
X(... | https://ocw.mit.edu/courses/6-973-communication-system-design-spring-2006/1460f43d3993b7c956d4bb8ee03d1fb0_lecture_10.pdf |
4
2
1
x(n)
clk
BF2I
X
BF2II
t
X
+
BF2I
X
BF2II
t
X
X
+
BF2I
X
BF2II
t
X
+
BF2I
X
BF2II
t
X
X(k)
W1(n)
W2(n)
W3(n)
7
6
5
4
3
2
1
0
xr(n)
xi(n)
xr(n+N/2)
xi(n+N/2)
+
+
+
+
-
-
0
1
0
1
1
0
1
0
x
xr(n)
xi(n)
xr(n+N/2)
xi(n+N/2)
Figure by MIT OpenCourseWare.
+
+
+
+
+
-
-
-
+
0
1
0
1
1
0
1
0
zr(n+N/2)
zi(n+N/2)
zr(n)
zi(n)
... | https://ocw.mit.edu/courses/6-973-communication-system-design-spring-2006/1460f43d3993b7c956d4bb8ee03d1fb0_lecture_10.pdf |
cycles, muxes in BF1 switch to 1
Butterfly computes a 2pt DFT with incoming data and data stored in the shift registers
Output Z1(n) sent to twiddle multiplier
Output Z1(n+N/2) sent back to the shift register to be “multiplied” in next N/2 cycles,
when the first half of the next frame is loaded in
128
64
32
... | https://ocw.mit.edu/courses/6-973-communication-system-design-spring-2006/1460f43d3993b7c956d4bb8ee03d1fb0_lecture_10.pdf |
3N/2 - 2
N - 1
N - 1
5N/2 - 4
2N - 2
N - 1
simple
simple
medium
simple
complex
simple
Figure by MIT OpenCourseWare.
R22SDF has reached minimum requirement
for both multiplier and storage
Only R4SDC better in terms of adder usage
R22SDF well suited for VLSI implementations
of pipeline FFT processors
Cite as: ... | https://ocw.mit.edu/courses/6-973-communication-system-design-spring-2006/1460f43d3993b7c956d4bb8ee03d1fb0_lecture_10.pdf |
[3]
x[4]
x[5]
x[6]
x[7]
0
W
0
W
0
W
0
W
-1
-1
-1
-1
X[0]
X[4]
X[2]
X[6]
X[1]
X[5]
X[3]
X[7]
0
W
1
W
2
W
3
W
-1
-1
-1
-1
0
W
0
W
2
W
2
W
-1
-1
-1
-1
TFFT =
N
r
.
.
logrN Tr,PE
Where,
N/r = No. of butterfly per stage
logrN = No. of stage
Tr,PE = Time to calculate one butterfly
e
s
r
e
v
e
r
t
i
B
&
P
/
S
l
o
r
t
n
o
C
s... | https://ocw.mit.edu/courses/6-973-communication-system-design-spring-2006/1460f43d3993b7c956d4bb8ee03d1fb0_lecture_10.pdf |
The number of nontrivial complex multiplications is 49 (7x7)
Since the first twiddle is always 1
The number of nontrivial complex multiplications for radix-2
FFT is 66
Radix-4 (or 22) FFTs need only 52 multiplies
Important to note that for 8pt FFT (DIT) no need for
multiplies
Cite as: Vladimir Stojanovic, ... | https://ocw.mit.edu/courses/6-973-communication-system-design-spring-2006/1460f43d3993b7c956d4bb8ee03d1fb0_lecture_10.pdf |
not trivial
Cite as: Vladimir Stojanovic, course materials for 6.973 Communication System Design, Spring 2006.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology.
Downloaded on [DD Month YYYY].
6.973 Communication System Design
21
Input unit
Hard wired outputs and data shifting
To... | https://ocw.mit.edu/courses/6-973-communication-system-design-spring-2006/1460f43d3993b7c956d4bb8ee03d1fb0_lecture_10.pdf |
LAN Application Using OFDM." Solid-State Circuits 39 (2004): 484-493. Copyright 2004 IEEE. Used with permission.
Some of the coefficients requested concurrently by different FFT
outputs
Solve by adding temp registers in the input unit
~50% less power and area than 8 standard complex multipliers
Buffer unit ... | https://ocw.mit.edu/courses/6-973-communication-system-design-spring-2006/1460f43d3993b7c956d4bb8ee03d1fb0_lecture_10.pdf |
"Designing pipeline FFT processor for OFDM (de)modulation,"
Signals, Systems, and Electronics, 1998. ISSSE 98. 1998 URSI International Symposium on no. SN
-, pp. 257-262, 1998.
[2] E. Wold and Alvin M. Despain "Pipeline and Parallel-Pipeline FFT Processors for VLSI
Implementations," IEEE Trans. Computers vol. 33, n... | https://ocw.mit.edu/courses/6-973-communication-system-design-spring-2006/1460f43d3993b7c956d4bb8ee03d1fb0_lecture_10.pdf |
and P.G. Gulak "Empirical performance prediction for IFFT/FFT cores for OFDM
systems-on-a-chip," Circuits and Systems, 2002. MWSCAS-2002. The 2002 45th Midwest
Symposium on vol. 1, no. SN -, pp. I-583-6 vol.1, 2002.
Cite as: Vladimir Stojanovic, course materials for 6.973 Communication System Design, Spring 2006.
MI... | https://ocw.mit.edu/courses/6-973-communication-system-design-spring-2006/1460f43d3993b7c956d4bb8ee03d1fb0_lecture_10.pdf |
MIT OpenCourseWare
http://ocw.mit.edu
18.02 Multivariable Calculus
Fall 2007
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
18.02 Lecture 21.
– Tue, Oct 30, 2007
Test for gradient fields.
Observe: if F� = Mˆı + Nˆj is a gradient field then Nx = My. I... | https://ocw.mit.edu/courses/18-02-multivariable-calculus-fall-2007/14a8493b19c42b8fbbeb9d05b5c0fd52_lec_week9.pdf |
is a potential then so is f + c). Can also choose the
simplest curve C from (0, 0) to (x1, y1).
·
Simplest choice: take C = portion of x-axis from (0, 0) to (x1, 0), then vertical segment from
�
C
(x1, 0) to (x1, y1) (picture drawn).
�
�
Then
F� d�r =
·
(4x 2 + 8xy) dx + (3y 2 + 4x 2) dy:
C
C1+C2
Over C1, ... | https://ocw.mit.edu/courses/18-02-multivariable-calculus-fall-2007/14a8493b19c42b8fbbeb9d05b5c0fd52_lec_week9.pdf |
4 x3 + 4x2y+
integration constant (independent of x). The integration constant still depends on y, call it g(y).
3 x3 + 4x2y + g(y). Take partial w.r.t. y, to get fy = 4x2 + g�(y).
So f (x, y) = 4
Comparing this with (2), we get g�(y) = 3y2, so g(y) = y3 + c.
Plugging into above formula for f , we finally get f (x, ... | https://ocw.mit.edu/courses/18-02-multivariable-calculus-fall-2007/14a8493b19c42b8fbbeb9d05b5c0fd52_lec_week9.pdf |
(angular velocity).
18.02 Lecture 22.
– Thu, Nov 1, 2007
Handouts: PS8 solutions, PS9, practice exams 3A and 3B.
Green’s theorem.
If C is a positively oriented closed curve enclosing a region R, then
� �
�
C
F�
·
d�
r =
curl �
F dA
which means
�
C
M dx + N dy =
R
� �
(Nx − My
) dA.
R
Example (reduce a comp... | https://ocw.mit.edu/courses/18-02-multivariable-calculus-fall-2007/14a8493b19c42b8fbbeb9d05b5c0fd52_lec_week9.pdf |
this if the region contains the origin – for example, the line integral
along the unit circle is non-zero even though curl( F� ) is zero wherever it’s defined.
F dA = R 0 dA = 0. So F� is conservative.
F� is a gradient field).
��
·
d�
r = R
��
Proof of Green’s theorem. 2 preliminary remarks:
�
1) the theorem sp... | https://ocw.mit.edu/courses/18-02-multivariable-calculus-fall-2007/14a8493b19c42b8fbbeb9d05b5c0fd52_lec_week9.pdf |
3
a
a
� �
RHS: −
� b � f1(x)
My dA = −
� b
My dy dx = −
(M (x, f1(x)) − M (x, f0(x)) dx (= LHS).
R
a
f0(x)
a
3
Finally observe: any region R can be subdivided into vertically simple pieces (picture shown);
�
My dA, so by additivity C M dx = − R My dA.
��
�
for each piece Ci
�
��
M dx = − Ri
��
Si... | https://ocw.mit.edu/courses/18-02-multivariable-calculus-fall-2007/14a8493b19c42b8fbbeb9d05b5c0fd52_lec_week9.pdf |
ds, sums F� nˆ = component of F� perpendicular to
C, along the curve.
�
·
·
·
·
�
If we break C into small pieces of length Δs, the flux is
�
i(F� nˆ ) Δsi.
·
Physical interpretation: if F� is a velocity field (e.g. flow of a fluid), flux measures how much
matter passes through C per unit time.
Look at a small po... | https://ocw.mit.edu/courses/18-02-multivariable-calculus-fall-2007/14a8493b19c42b8fbbeb9d05b5c0fd52_lec_week9.pdf |
+ xˆj across C is zero (field tangent to C).
That was a geometric argument. What about the general situation when calculation of the line
integral is required?
Observe: d�r = Tˆ ds = �dx, dy�, and nˆ is Tˆ rotated 90◦ clockwise; so nˆ ds = �dy, −dx�.
So, if F� = P ˆı + Qˆj (using new letters to make things look diffe... | https://ocw.mit.edu/courses/18-02-multivariable-calculus-fall-2007/14a8493b19c42b8fbbeb9d05b5c0fd52_lec_week9.pdf |
� �
R
div( F� ) dA.
This proof by “renaming” the components is why we called the components P, Q instead of M, N .
�
� �
If we call F� = �M, N � the statement becomes −N dx + M dy =
(Mx + Ny) dA.
Example:
in the above example (xˆı + yˆj across circle), div F� = 2, so flux = R 2 dA =
2 area(R) = 2πa2 . If we tra... | https://ocw.mit.edu/courses/18-02-multivariable-calculus-fall-2007/14a8493b19c42b8fbbeb9d05b5c0fd52_lec_week9.pdf |
Space Systems Architecture
Lecture 3
Introduction to
Tradespace Exploration
Hugh McManus
Metis Design
Space Systems, Policy, and Architecture Research Consortium
A joint venture of MIT, Stanford, Caltech & the Naval War College
for the NRO
Space Systems, Policy, and Architecture Research Consortium
©2002 Massachusetts... | https://ocw.mit.edu/courses/16-892j-space-system-architecture-and-design-fall-2004/153e61478f3b4c67628d9c1deb35672a_3999lecture3v2.pdf |
craft Parameters
–
–
–
–
–
Antenna gain
communication architecture
propulsion type
power type
delta_v
Each point is
a specific
architecture
Total Lifecycle Cost
($M2002)
Assessment of the utility and cost of a large
space of possible system architectures
Number of Architectures Explored: 50488
Number of Architectures ... | https://ocw.mit.edu/courses/16-892j-space-system-architecture-and-design-fall-2004/153e61478f3b4c67628d9c1deb35672a_3999lecture3v2.pdf |
“Knowledgeable”
User Set
User-Specific
System Integration
Go/No-Go “green light”
“Warfighter”
User Set
Space Systems, Policy, and Architecture Research Consortium
©2002 Massachusetts Institute of Technology
9
• “what the decision makers need to
consider”
( and/or what the user truly cares about)
•
• Examples: Billa... | https://ocw.mit.edu/courses/16-892j-space-system-architecture-and-design-fall-2004/153e61478f3b4c67628d9c1deb35672a_3999lecture3v2.pdf |
km)
750
950
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
)
1
o
t
0
(
y
t
i
l
i
t
U
0
150
(
XUKk
i
(
i
)
1)
+
N
∏
i
1
=
Weight Factors of each Attribute
(k values)
(
XKU
1)
=+
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
y
t
i
l
i
t
U
Lifespan
Latitude
Latency
Equator
Time
Altitude
Total Lifecycle Cost ($M 2002)
Space Systems... | https://ocw.mit.edu/courses/16-892j-space-system-architecture-and-design-fall-2004/153e61478f3b4c67628d9c1deb35672a_3999lecture3v2.pdf |
p
o
r
P
n
o
i
t
a
n
i
l
c
n
I
m
e
t
s
y
S
m
m
o
C
m
e
t
s
y
s
r
e
w
o
P
i
n
a
G
.
t
n
A
o
i
r
a
n
e
c
S
n
o
i
s
s
i
M
t
c
a
p
m
I
l
a
t
o
T
Identify key
interactions
for modeling
Attributes
Data Lifespan
Sample Altitude
Diversity of Latitudes
Time at Equator
Latency
Total
Cost
Total w/Cost
9
9
0
0
3
9
9
0
6
3
21 27
9
6... | https://ocw.mit.edu/courses/16-892j-space-system-architecture-and-design-fall-2004/153e61478f3b4c67628d9c1deb35672a_3999lecture3v2.pdf |
c. $0.5M/Image
Each point is an
evaluated architecture
TPF System Trade Space
$2M/Image
$1M/Image
$0.5M/Image
)
s
n
2400
o
i
l
l
i
2200
m
$
2000
(
t
s
o
1800
C
e
1600
l
c
y
c
e
f
i
1400
L
1200
1000
800
0
$0.25M/Image
500 1000 1500 2000 2500 3000 3500 4000
Performance (total # of images)
1400
1350
1300
)
s
n
o
i
l
l... | https://ocw.mit.edu/courses/16-892j-space-system-architecture-and-design-fall-2004/153e61478f3b4c67628d9c1deb35672a_3999lecture3v2.pdf |
M/Image
$0.25M/Image
1000
1500
2000
Performance (total # of images)
2500
3000
Cost ($M)
3500
4000
Space Systems, Policy, and Architecture Research Consortium
©2002 Massachusetts Institute of Technology
19
Using the Trade Space to Evaluate
Point Designs
Designs from traditional process
TPF
• Terrestrial Planet... | https://ocw.mit.edu/courses/16-892j-space-system-architecture-and-design-fall-2004/153e61478f3b4c67628d9c1deb35672a_3999lecture3v2.pdf |
eric
measurements
• Several different
missions
y
t
i
l
i
t
U
e
d
u
t
i
t
a
L
h
g
i
H
Space Systems, Policy, and Architecture Research Consortium
Equatorial Utility
©2002 Massachusetts Institute of Technology
22
Changes in User Preferences Can be
Quickly Understood
Architecture
trade space
reevaluated
in less than... | https://ocw.mit.edu/courses/16-892j-space-system-architecture-and-design-fall-2004/153e61478f3b4c67628d9c1deb35672a_3999lecture3v2.pdf |
Attitude
Determination
and Control
Electronic
communication
between tools and
server
Verbal or online chat
between chairs
synchronizes actions
• Directed Design
Sessions allow very
fast production of
preliminary designs
• Traditionally, design
to requirements
Integration with
MATE allows utility
of designs to be
assess... | https://ocw.mit.edu/courses/16-892j-space-system-architecture-and-design-fall-2004/153e61478f3b4c67628d9c1deb35672a_3999lecture3v2.pdf |
Chapter 3
Scattering series
In this chapter we describe the nonlinearity of the map c (cid:55)→ u in terms of a
perturbation (Taylor) series. To first order, the linearization of this map is
called the Born approximation. Linearization and scattering series are the
basis of most inversion methods, both direct and iterat... | https://ocw.mit.edu/courses/18-325-topics-in-applied-mathematics-waves-and-imaging-fall-2015/15588369c29a866ba59b450518ba486a_MIT18_325F15_Chapter3.pdf |
.2)
with zero initial conditions and x ∈ Rn. Perturb m(x) as in (3.1). The
wavefield u correspondingly splits into
u(x) = u0(x) + usc(x),
where u0 solves the wave equation in the undisturbed medium m0,
m0(x)
∂2u0
∂t2
− ∆u0 = f (x, t).
(3.3)
We say u is the total field, u0 is the incident field1, and usc is the scattered fi... | https://ocw.mit.edu/courses/18-325-topics-in-applied-mathematics-waves-and-imaging-fall-2015/15588369c29a866ba59b450518ba486a_MIT18_325F15_Chapter3.pdf |
eld u can be formally2 ex-
pressed in terms of u0 by writing
(cid:20)
u = I + ε G m1
(cid:21)−1
∂2
∂t2
u0.
(3.5)
While this equation is equivalent to the original PDE, it shines a different
light on the underlying physics. It makes explicit the link between u0 and u,
as if u0 “generated” u via scattering through the med... | https://ocw.mit.edu/courses/18-325-topics-in-applied-mathematics-waves-and-imaging-fall-2015/15588369c29a866ba59b450518ba486a_MIT18_325F15_Chapter3.pdf |
ˆ
s2
Rn
G(y2, y1; s2 − s1)m1(y1)
(cid:21)
(y1, s1) dy1ds1 dy2ds2
∂2u0
∂t2
2For mathematicians, “formally” means that we are a step ahead of the rigorous ex-
position: we are only interested in inspecting the form of the result before we go about
proving it. That’s the intended meaning here. For non-mathematicians, “for... | https://ocw.mit.edu/courses/18-325-topics-in-applied-mathematics-waves-and-imaging-fall-2015/15588369c29a866ba59b450518ba486a_MIT18_325F15_Chapter3.pdf |
of u0 in place of u in the right-
hand side (and ε is gone, by choice of normalization of u1). Unlike (3.4),
equation (3.7) is explicit: it maps m1 to u1 in a linear way. The incident field
u0 is determined from m0 alone, hence “fixed” for the purpose of determining
the scattered fields.
∂2u0 .
∂t2
(3.7)
It is informative... | https://ocw.mit.edu/courses/18-325-topics-in-applied-mathematics-waves-and-imaging-fall-2015/15588369c29a866ba59b450518ba486a_MIT18_325F15_Chapter3.pdf |
ized scattered field coincide,
namely
u1 =
δF
δm
[m0] m1 = −Gm1
∂2u0 .
∂t2
This will justify the first term in the Taylor expansion above. For this pur-
pose, let us take the δ derivative of (3.2). As previously, write u = F(m)
and F = δF
δm [m]. We get the operator-valued equation
δm
∂2u
∂t2
I + m
∂2
∂t2
F − ∆F = 0.
Eva... | https://ocw.mit.edu/courses/18-325-topics-in-applied-mathematics-waves-and-imaging-fall-2015/15588369c29a866ba59b450518ba486a_MIT18_325F15_Chapter3.pdf |
(cid:105).
See one of the exercises at the end of chapter 1 to illustrate how the wave
equation can be put in precisely this form, with (cid:104)w, w(cid:48)(cid:105) the usual L2 inner
product and M a positive diagonal matrix.
Consider a background medium M0, so that M = M0 + εM1. Let w =
w0 + εw1 + . . . Calculations... | https://ocw.mit.edu/courses/18-325-topics-in-applied-mathematics-waves-and-imaging-fall-2015/15588369c29a866ba59b450518ba486a_MIT18_325F15_Chapter3.pdf |
107)∗ is precisely one of weak scat-
tering, i.e., that the primary reflected wave εw1 is weaker than the incident
wave w0.
While any induced norm over space and time in principle works for the
proof of convergence of the Neumann series, it is convenient to use
(cid:107)w(cid:107)∗ = max
0≤t≤T
(cid:112)
(cid:104)w, M0w(... | https://ocw.mit.edu/courses/18-325-topics-in-applied-mathematics-waves-and-imaging-fall-2015/15588369c29a866ba59b450518ba486a_MIT18_325F15_Chapter3.pdf |
legitimate oper-
ations. The left-hand-side is also d
M0w1(cid:107)2.
dt
By Cauchy-Schwarz, the right-hand-side is majorized by
(cid:104)w1, M0w1(cid:105) = 2(cid:107)
M0w1(cid:107)2
d
dt (cid:107)
√
√
(cid:112)
2(cid:107)
M0w1(cid:107)2 (cid:107)√
M1
M0
∂w0
∂t
(cid:107)2.
Hence
(cid:112)
(cid:107)
d
dt
M0w1(cid:107)2 ... | https://ocw.mit.edu/courses/18-325-topics-in-applied-mathematics-waves-and-imaging-fall-2015/15588369c29a866ba59b450518ba486a_MIT18_325F15_Chapter3.pdf |
umption needs to be used. We can invoke a classical result known as Bern-
stein’s inequality4, which says that (cid:107)f(cid:48)(cid:107)∞ ≤ Ω(cid:107)f(cid:107)∞ for all Ω-bandlimited f.
Then
∂t
(cid:107)w1(cid:107)∗ ≤ ΩT (cid:107)
(cid:107)∞(cid:107)w0(cid:107)∗.
M1
M0
In view of our request that ε(cid:107)w1(cid:1... | https://ocw.mit.edu/courses/18-325-topics-in-applied-mathematics-waves-and-imaging-fall-2015/15588369c29a866ba59b450518ba486a_MIT18_325F15_Chapter3.pdf |
3. CONVERGENCE OF THE BORN SERIES (PHYSICS)
63
• the remainder w −εw = w−w −εw is on the order of ε2
sc
1
0
1
2
(ΩT (cid:107)M1(cid:107)∞) .
Both claims are the subject of an exercise at the end of the chapter. The
second claim is the mathematical expression that the Born approximation
is accurate (small wsc − εw1 on t... | https://ocw.mit.edu/courses/18-325-topics-in-applied-mathematics-waves-and-imaging-fall-2015/15588369c29a866ba59b450518ba486a_MIT18_325F15_Chapter3.pdf |
+ ε)T ) − f (x − T ) will be out of phase and will not
give rise to values on the order of ε. The requirement is εT (cid:28) 2π/Ω,
i.e.,
εΩT (cid:28) 2π,
which is exactly what theorem 3 is requiring. We could have reached
the same conclusion by requiring either the first or the second term
of the Taylor expansion to be ... | https://ocw.mit.edu/courses/18-325-topics-in-applied-mathematics-waves-and-imaging-fall-2015/15588369c29a866ba59b450518ba486a_MIT18_325F15_Chapter3.pdf |
isolated scatterers in several dimensions, the
Born approximation is often very good. That’s when the interpretation
of the Born series in terms of multiple scattering is the most relevant.
Such is the case of small isolated objects in synthetic aperture radar:
double scattering from one object to another is often negl... | https://ocw.mit.edu/courses/18-325-topics-in-applied-mathematics-waves-and-imaging-fall-2015/15588369c29a866ba59b450518ba486a_MIT18_325F15_Chapter3.pdf |
position of receiver r,
• s indexes the source,
• and t is time.
The inverse problem of imaging is that of solving for m in the system of
nonlinear equations d = F[m]. No single method will convincingly solve
such a system of nonlinear equations efficiently and in all regimes.
The very prevalent least-squares framework f... | https://ocw.mit.edu/courses/18-325-topics-in-applied-mathematics-waves-and-imaging-fall-2015/15588369c29a866ba59b450518ba486a_MIT18_325F15_Chapter3.pdf |
• Conversely, the inverse problem may suffer from severe non-convexity
when the abundance of local minima, or local “valleys”, hinders the
search for the global minimum. This happens when the Hessian of
J alternates between having large positive and negative curvatures in
some direction m1.
Many inversion problems in hi... | https://ocw.mit.edu/courses/18-325-topics-in-applied-mathematics-waves-and-imaging-fall-2015/15588369c29a866ba59b450518ba486a_MIT18_325F15_Chapter3.pdf |
107)2).
We conclude by invoking (A.1).
With some care, calculations involving functional derivatives are more
efficiently done using the usual rules of calculus in Rn. For instance, the
result above is more concisely justified from
(cid:104)
δ
δm
(cid:18) 1
2
(cid:104)F[m] − d, F[m] − d(cid:105)
, m1(cid:105) = (cid:104)F... | https://ocw.mit.edu/courses/18-325-topics-in-applied-mathematics-waves-and-imaging-fall-2015/15588369c29a866ba59b450518ba486a_MIT18_325F15_Chapter3.pdf |
ation Hessian.
3.5 Exercises
1. Repeat the development of section (3.1) in the frequency domain (ω)
rather than in time.
2. Derive Born series with a multiscale expansion: write m = m0 + εm1,
u = u0 + εu1 + ε2u2 + . . ., substitute in the wave equation, and equate
like powers of ε. Find the first few equations for u0, u... | https://ocw.mit.edu/courses/18-325-topics-in-applied-mathematics-waves-and-imaging-fall-2015/15588369c29a866ba59b450518ba486a_MIT18_325F15_Chapter3.pdf |
expecting” a trial function m1. A second derivative
with respect to m(cid:48)
1 gives
δm
δm1
∂2
∂t2
δF(m)
δm(cid:48)
1
+
δm
δm(cid:48)
1
∂2
∂t2
δF(m)
δm1
(cid:18)
+
m
∂2
∂t2 − ∆
(cid:19) δ2F(m)
m1δm1
δ
(cid:48) = .
0
We now evaluate the result at the base point m = m0, and perform the
pairing with two trial functions m... | https://ocw.mit.edu/courses/18-325-topics-in-applied-mathematics-waves-and-imaging-fall-2015/15588369c29a866ba59b450518ba486a_MIT18_325F15_Chapter3.pdf |
(no decomposition of M into M0 +εM1).
Prove the following energy estimate for the solution of (3.8):
ˆ
(cid:18) t
(cid:19)2
E(t) ≤
(cid:107)f (cid:107)(s) ds
,
(3.13)
0
where E(t) = (cid:104)w, M w(cid:105) and (cid:107)f (cid:107)2 = (cid:104)f, f (cid:105). [Hint: repeat and adapt the
beginning of the proof of theore... | https://ocw.mit.edu/courses/18-325-topics-in-applied-mathematics-waves-and-imaging-fall-2015/15588369c29a866ba59b450518ba486a_MIT18_325F15_Chapter3.pdf |
the resulting gradient step is a con-
traction, i.e., the distance between successive iterates decreases mono-
tonically.
9. Consider J(m) any smooth, locally convex function of m.
(a) Show that the specific choice
α =
δm[m(k)], δJ
(cid:104) δJ
δm[m(k)], δJ 2
(cid:104) δJ
δm[m(k)](cid:105)
δm2 [m(k)] δJ [m(k)](cid:105)
... | https://ocw.mit.edu/courses/18-325-topics-in-applied-mathematics-waves-and-imaging-fall-2015/15588369c29a866ba59b450518ba486a_MIT18_325F15_Chapter3.pdf |
δ2F(m)
δm1δm(cid:48)
1
, F
(m) − d(cid:105).
This result is then evaluated at the base point m = m0, where δF (m0) =
F . The second term in the right-hand side already has the desired
form. The first term in the right-hand-side, when paired with m1 and
m(cid:48)
δm1
1, gives
(cid:104)F m1, F m(cid:48)
1(cid:105) = (cid:... | https://ocw.mit.edu/courses/18-325-topics-in-applied-mathematics-waves-and-imaging-fall-2015/15588369c29a866ba59b450518ba486a_MIT18_325F15_Chapter3.pdf |
MIT 2.852
Manufacturing Systems Analysis
Lecture 10–12
Transfer Lines – Long Lines
Stanley B. Gershwin
http://web.mit.edu/manuf-sys
Massachusetts Institute of Technology
Spring, 2010
2.852 Manufacturing Systems Analysis
1/91
Copyright (cid:13)2010 Stanley B. Gershwin.
c
Long Lines
M
1
B
1
M
2
B
2
M
3
B... | https://ocw.mit.edu/courses/2-852-manufacturing-systems-analysis-spring-2010/155e83dd472fcab7095db53005c093bb_MIT2_852S10_long_lines.pdf |
.
◮ Etc., etc. if there is time.
2.852 Manufacturing Systems Analysis
3/91
Copyright c(cid:13)2010 Stanley B. Gershwin.
Decomposition — Concept
◮ Conceptually: put an observer in a buffer, and tell him that he is in
the buffer of a two-machine line.
◮ Question: What would the observer see, and how can he be convi... | https://ocw.mit.edu/courses/2-852-manufacturing-systems-analysis-spring-2010/155e83dd472fcab7095db53005c093bb_MIT2_852S10_long_lines.pdf |
Decomposition — Concept
◮ Consider an observer in Buffer Bi .
◮ Imagine the material flow process that the observer sees entering and
the material flow process that the observer sees leaving the buffer.
◮ We construct a two-machine line L(i)
◮ (ie, we find machines Mu(i) and Md (i) with parameters ru (i), pu (i),
rd (... | https://ocw.mit.edu/courses/2-852-manufacturing-systems-analysis-spring-2010/155e83dd472fcab7095db53005c093bb_MIT2_852S10_long_lines.pdf |
pd (k − 1)
Therefore, we need
◮ 4(k − 1) equations, and
◮ an algorithm for solving those equations.
2.852 Manufacturing Systems Analysis
9/91
Copyright c(cid:13)2010 Stanley B. Gershwin.
Decomposition Equations
Overview
The decomposition equations relate ru (i), pu(i), rd (i), and pd (i) to behavior in
the r... | https://ocw.mit.edu/courses/2-852-manufacturing-systems-analysis-spring-2010/155e83dd472fcab7095db53005c093bb_MIT2_852S10_long_lines.pdf |
in parentheses.
Example: ru(i ).
◮ Items that pertain to the real line L will have i in the subscript.
Example: ri .
2.852 Manufacturing Systems Analysis
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Copyright c(cid:13)2010 Stanley B. Gershwin.
Decomposition Equations
Conservation of Flow
E (i ) = E (1), i = 2, . . . , k − 1.
◮ Recall that E (i ) ... | https://ocw.mit.edu/courses/2-852-manufacturing-systems-analysis-spring-2010/155e83dd472fcab7095db53005c093bb_MIT2_852S10_long_lines.pdf |
Analysis
14/91
Copyright c(cid:13)2010 Stanley B. Gershwin.
Decomposition Equations
Flow Rate-Idle Time
Observation:
Reason:
prob (ni −1 = 0 and ni = Ni ) ≈ 0.
M
i−1
B
i−1
M
i
0
M
i+1
B
i
N i
The only way to have ni −1 = 0 and ni = Ni is if
◮ Mi −1 is down or starved for a long time
◮ and Mi is up ... | https://ocw.mit.edu/courses/2-852-manufacturing-systems-analysis-spring-2010/155e83dd472fcab7095db53005c093bb_MIT2_852S10_long_lines.pdf |
ations
Flow Rate-Idle Time
Therefore
Note that
Ei ≈ ei [1 − prob (ni −1 = 0) − prob (ni = Ni )]
prob (ni −1 = 0) = ps (i − 1);
prob (ni = Ni ) = pb(i)
Two of the FRIT relationships in lines L(i − 1) and L(i) are
E (i) = eu(i) [1 − pb(i)] ; E (i − 1) = ed (i − 1) [1 − ps (i − 1)]
2.852 Manufacturing Systems Ana... | https://ocw.mit.edu/courses/2-852-manufacturing-systems-analysis-spring-2010/155e83dd472fcab7095db53005c093bb_MIT2_852S10_long_lines.pdf |
Systems Analysis
18/91
Copyright c(cid:13)2010 Stanley B. Gershwin.
Decomposition Equations
Flow Rate-Idle Time
Since
ed (i − 1) =
rd (i − 1)
pd (i − 1) + rd (i − 1)
;
eu(i) =
ru (i)
pu (i) + ru(i)
,
we can write
pd (i − 1)
rd (i − 1)
+
pu (i)
ru (i)
=
1
E (i)
+
1
ei
− 2, i = 2, . . . , k − 1
... | https://ocw.mit.edu/courses/2-852-manufacturing-systems-analysis-spring-2010/155e83dd472fcab7095db53005c093bb_MIT2_852S10_long_lines.pdf |
i+1
B
i+1
M
i+2
B
i+2
M
i+3
0
... or, Mi −1 may be down and Bi −1 may be empty, ...
M (i)
u
M (i)
d
2.852 Manufacturing Systems Analysis
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Copyright c(cid:13)2010 Stanley B. Gershwin.
Decomposition Equations
Resumption of Flow
M
i−4
B
i−4
M
i−3
B
i−3
M
i−2
B
i−2
M
i−1
B
i−1
M
i
B
i
M
... | https://ocw.mit.edu/courses/2-852-manufacturing-systems-analysis-spring-2010/155e83dd472fcab7095db53005c093bb_MIT2_852S10_long_lines.pdf |
2010 Stanley B. Gershwin.
Decomposition Equations
Resumption of Flow
M
i−4
B
i−4
M
i−3
B
i−3
M
i−2
B
i−2
M
i−1
B
i−1
M
i
B
i
M
i+1
B
i+1
M
i+2
B
i+2
M
i+3
0
0
0
0
M (i)
u
M (i)
d
... etc.
2.852 Manufacturing Systems Analysis
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Copyright c(cid:13)2010 Stanley B. Gershwin.
Decompositi... | https://ocw.mit.edu/courses/2-852-manufacturing-systems-analysis-spring-2010/155e83dd472fcab7095db53005c093bb_MIT2_852S10_long_lines.pdf |
B
i−3
M
i−2
B
i−2
M
i−1
B
i−1
M
i
B
i
M
i+1
B
i+1
M
i+2
B
i+2
M
i+3
0
Comparison
M (i)
u
M (i)
d
2.852 Manufacturing Systems Analysis
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Copyright c(cid:13)2010 Stanley B. Gershwin.
Decomposition Equations
Resumption of Flow
M
i−4
B
i−4
M
i−3
B
i−3
M
i−2
B
i−2
M
i−1
B
i−1
M
i
... | https://ocw.mit.edu/courses/2-852-manufacturing-systems-analysis-spring-2010/155e83dd472fcab7095db53005c093bb_MIT2_852S10_long_lines.pdf |
0
0
M (i−1)
u
M (i−1)
d
M
i−4
B
i−4
M
i−3
B
i−3
M
i−2
B
i−2
M
i−1
B
i−1
M
i
B
i
M
i+1
B
i+1
M
i+2
B
i+2
M
i+3
0
0
0
M (i)
u
M (i)
d
2.852 Manufacturing Systems Analysis
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Copyright c(cid:13)2010 Stanley B. Gershwin.
Decomposition Equations
Resumption of Flow
M
i−4
B
i−4
M
i−3
B... | https://ocw.mit.edu/courses/2-852-manufacturing-systems-analysis-spring-2010/155e83dd472fcab7095db53005c093bb_MIT2_852S10_long_lines.pdf |
(i)
d
◮ either real machine Mi is down,
M
M
i−4
i−4
B
B
i−4
i−4
M
M
i−3
i−3
B
B
i−3
i−3
M
M
i−2
i−2
B
B
i−2
i−2
M
M
i−1
i−1
B
B
i−1
i−1
M
M
i
i
B
B
i
i
M
M
i+1
i+1
B
B
i+1
i+1
M
M
i+2
i+2
B
B
i+2
i+2
M
M
i+3
i+3
◮ or Buffer Bi −1 is empty and the Line L(i − 1) observer sees a failure in
Mu(i − 1). ... | https://ocw.mit.edu/courses/2-852-manufacturing-systems-analysis-spring-2010/155e83dd472fcab7095db53005c093bb_MIT2_852S10_long_lines.pdf |
B. Gershwin.
Decomposition Equations
Resumption of Flow
Also, for the Line L(j) observer to see Mu(j) up, Mj must be up and Bj−1 must
be non-empty. Therefore,
{αu(j, τ ) = 1} ⇐⇒ {αj (τ ) = 1} and {nj−1(τ − 1) > 0}
{αu(j, τ ) = 0} ⇐⇒ {αj (τ ) = 0} or {nj−1(τ − 1) = 0}
2.852 Manufacturing Systems Analysis
31/91 ... | https://ocw.mit.edu/courses/2-852-manufacturing-systems-analysis-spring-2010/155e83dd472fcab7095db53005c093bb_MIT2_852S10_long_lines.pdf |
{αi (t) = 0}
W = {ni −1(t − 1) = 0}
Important: V and W are disjoint.
prob (V and W ) = 0.
2.852 Manufacturing Systems Analysis
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Copyright c(cid:13)2010 Stanley B. Gershwin.
Decomposition Equations
Resumption of Flow
prob (U|V or W ) =
prob (U and (V or W ))
prob (V or W )
=
prob ((U and V ) or (U and... | https://ocw.mit.edu/courses/2-852-manufacturing-systems-analysis-spring-2010/155e83dd472fcab7095db53005c093bb_MIT2_852S10_long_lines.pdf |
W |V or W ).
2.852 Manufacturing Systems Analysis
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Copyright c(cid:13)2010 Stanley B. Gershwin.
Decomposition Equations
Resumption of Flow
Then, if we plug U, V , and W from Slide 33 into this, we get
ru (i) = A(i − 1)X (i) + B(i)X
′
(i), i = 2, . . . , k − 1
where
A(i − 1) = prob (U|W )
= prob
ni −1... | https://ocw.mit.edu/courses/2-852-manufacturing-systems-analysis-spring-2010/155e83dd472fcab7095db53005c093bb_MIT2_852S10_long_lines.pdf |
0 | {ni −1(t − 1) = 0 or αi (t) = 0}] .
2.852 Manufacturing Systems Analysis
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Copyright c(cid:13)2010 Stanley B. Gershwin.
Decomposition Equations
Resumption of Flow
To evaluate
A(i − 1) = prob
»
ni −1(t) > 0 and αi (t + 1) = 1
˛
˛
˛
˛
ni −1(t − 1) = 0
–
:
Note that
◮ For Buffer i − 1 to be empty at t... | https://ocw.mit.edu/courses/2-852-manufacturing-systems-analysis-spring-2010/155e83dd472fcab7095db53005c093bb_MIT2_852S10_long_lines.pdf |
Manufacturing Systems Analysis
38/91
Copyright c(cid:13)2010 Stanley B. Gershwin.
Decomposition Equations
Resumption of Flow
A(i − 1) = prob
»
ni −1(t) > 0
˛
˛
˛
˛
ni −1(t − 1) = 0
–
◮ For Buffer i − 1 to be empty, Mi −1 must be down or starved. For Mi −1 to be
starved, Mi −2 must be down ... | https://ocw.mit.edu/courses/2-852-manufacturing-systems-analysis-spring-2010/155e83dd472fcab7095db53005c093bb_MIT2_852S10_long_lines.pdf |
,
B(i) = prob [ni −1(t) > 0 and αi (t + 1) = 1 | αi (t) = 0]
Note that if αi (t) = 0, we must have ni −1(t) > 0. Therefore
or,
so
B(i) = prob [αi (t + 1) = 1 | αi (t) = 0] ,
B(i) = ri
ru (i) = ru (i − 1)X (i) + ri X ′ (i),
2.852 Manufacturing Systems Analysis
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Copyright c(cid:13)2010 Stanley B. Gershwin.... | https://ocw.mit.edu/courses/2-852-manufacturing-systems-analysis-spring-2010/155e83dd472fcab7095db53005c093bb_MIT2_852S10_long_lines.pdf |
(i) = 1 − X (i)
2.852 Manufacturing Systems Analysis
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Copyright c(cid:13)2010 Stanley B. Gershwin.
Decomposition Equations
Resumption of Flow
X (i) = prob
ni −1(t − 1) = 0
(cid:20)
ni −1(t − 1) = 0 or αi (t) = 0
(cid:12)
(cid:12)
(cid:12)
(cid:12)
prob [ni −1(t − 1) = 0 and {ni −1(t − 1) = 0 or αi (... | https://ocw.mit.edu/courses/2-852-manufacturing-systems-analysis-spring-2010/155e83dd472fcab7095db53005c093bb_MIT2_852S10_long_lines.pdf |
= 0 or αi (t) = 0} and ni (t − 1) < Ni ] because
prob [ni −1(t − 1) = 0 and ni (t − 1) = Ni ] ≈ 0
so the denominator is, approximately,
Recall that this is equal to
prob [αu (i) = 0 and ni (t − 1) < Ni ]
pu (i)
ru (i)
prob [αu (i) = 1 and ni (t − 1) < Ni ] =
pu (i)
ru (i)
E (i)
2.852 Manufacturing Systems Ana... | https://ocw.mit.edu/courses/2-852-manufacturing-systems-analysis-spring-2010/155e83dd472fcab7095db53005c093bb_MIT2_852S10_long_lines.pdf |
a set of k − 2 equations.
We now have 4(k − 2) = 4k − 8 equations.
2.852 Manufacturing Systems Analysis
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Copyright c(cid:13)2010 Stanley B. Gershwin.
Decomposition Equations
Boundary Conditions
Md (1) is the same as M1 and Md (k − 1) is the same as Mk . Therefore
ru(1) = r1
pu(1) = p1
rd (k − 1) = rk
p... | https://ocw.mit.edu/courses/2-852-manufacturing-systems-analysis-spring-2010/155e83dd472fcab7095db53005c093bb_MIT2_852S10_long_lines.pdf |
(i + 1) + ri +1(1 − Y (i + 1)); Y (i + 1) =
pb(i + 1)rd (i)
pd (i)E (i)
.
pd (i) = rd (i)
1
E (i + 1)
„
+
1
ei +1
− 2 −
pu (i + 1)
ru (i + 1) «
2.852 Manufacturing Systems Analysis
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Copyright c(cid:13)2010 Stanley B. Gershwin.
Decomposition Equations
Algorithm
We use the conservation of flo... | https://ocw.mit.edu/courses/2-852-manufacturing-systems-analysis-spring-2010/155e83dd472fcab7095db53005c093bb_MIT2_852S10_long_lines.pdf |
91
Copyright c(cid:13)2010 Stanley B. Gershwin.
Decomposition Equations
Algorithm
Possible Termination Conditions:
◮ |E (i ) − E (1)| < ǫ for i = 2, ..., k − 1, or
◮ The change in each ru(i ), pu(i ), rd (i ), pd (i ) parameter,
i = 1, ..., k − 1 is less than ǫ, or
◮ etc.
2.852 Manufacturing Systems A... | https://ocw.mit.edu/courses/2-852-manufacturing-systems-analysis-spring-2010/155e83dd472fcab7095db53005c093bb_MIT2_852S10_long_lines.pdf |
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