insight through computing 29. design parameters optimizing gear ratio distribution
Post on 22-Dec-2015
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Insight Through Computing
29. Design Parameters
Optimizing Gear Ratio Distribution
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Insight Through Computing
An Optimal Design Problem
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Insight Through Computing
Gear Ratio Distribution
Assume 7 wheelsprockets
Assume 3pedal
sprockets
21 = 7 x 3 possible gear ratios
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Insight Through Computing
It’s a Matter of Teeth
E.g.,13 teeth
E.g.,48 teeth
E.g., Gear ratio = 48/13 = 3.692
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Insight Through Computing
Goal
Choose 3 pedal sprockets and 7 wheelsprockets so that the 21 gear ratios areas evenly distributed across the interval[1,4].
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Insight Through Computing
Notation
p(i) = #teeth on the i-th pedal sprocket, for i=1:3.
w(i) = #teeth on the i-th wheel sprocket,for i=1:7.
This is a 10—parameter design problem.
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Insight Through Computing
Things to Do
1. Define an Objective Function We need to measure the
quality of a particular gear ratio
distribution
2. Identify constraints. Sprockets are only available in
certain sizes etc.Typical activity in Engineering Design
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Insight Through Computing
The Quality of a Gear RatioDistribution
Ideal:
1 4
Good:
Poor:
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Insight Through Computing
Average Discrepancy
Sort the gear ratios:
g(1) < g(2) <… < g(21)
Compare g(i) with x(i) where
x = linspace(1,4,21).
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Insight Through Computing
function tau = ObjF(p,w);
g = [];
for i=1:3
for j=1:7
g = [g p(i)/w(j)];
end
end
g = sort(g);
dif = abs(g – linspace(1,4,21));
tau = sum(dif)/21;
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Insight Through Computing
There Are Other ReasonableObjective Functions
g = sort(g);
dif = abs(g –linspace(1,4,21));
tau = sum(dif)/21;
Replace “sum” with “max”
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Insight Through Computing
Goal
Choose p(1:3) and w(1:7) so thatobjF(p,w) is minimized.
This defines the “best bike.”
Our plan is to check all possible bikes.
A 10-fold nested loop problem…
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Insight Through Computing
A Simplification
We may assume that
p(3) < p(2) < p(1)
and
w(7)<w(6)<w(5)<w(4)<w(3<w(2)<w(1)
Relabeling the sprockets doesn’t change the
21 gear ratios.
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Insight Through Computing
How Constraints Arise
Purchasing says that pedal sprockets only
come in six sizes:
C1: p(i) is one of 52 48 42 39 32 28.
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Insight Through Computing
How Constraints Arise
Marketing says the best bike must havea maximum gear ratio exactly equal to4: C2: p(1)/w(7) = 4
This means that p(1) must be a multiple of
4.
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Insight Through Computing
How Constraints Arise
Marketing says the best bike must have
a minimum gear ratio exactly equal to 1:
C3: p(3)/w(1) = 1
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Insight Through Computing
How Constraints Arise
Purchasing says that wheel sprockets are available in 31 sizes…
C4: w(i) is one of 12, 13,…,42.
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Insight Through Computing
Choosing Pedal Sprockets
Possible values…
Front = [52 48 42 39 32 28];
Constraint C1 says that p(1) must bedivisible by 4.
Also: p(3) < p(2) < p(1).
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Insight Through Computing
The Possibilities..
52 48 42 52 39 32 48 39 2852 48 39 52 39 28 48 32 2852 48 32 52 32 28 42 39 3252 48 28 48 42 39 42 39 2852 42 39 48 42 32 42 32 2852 42 32 48 42 28 52 42 28 48 39 32
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Insight Through Computing
Front = [52 48 42 39 32 28];for i = 1:3 for j=i+1:6 for k=j+1:6 p(1) = Front(i); p(2) = Front(j); p(3) = Front(k);
The Loops..
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Insight Through Computing
Front = [52 48 42 39 32 28];for i = 1:3 for j=i+1:6 for k=j+1:6 p(1) = Front(i); p(2) = Front(j); p(3) = Front(k); w(1) = p(3); w(7) = p(1)/4;
w(1) and w(7) “for free”..
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Insight Through Computing
Front = [52 48 42 39 32 28];for i = 1:3 for j=i+1:6 for k=j+1:6 p(1) = Front(i); p(2) = Front(j); p(3) = Front(k); w(1) = p(3); w(7) = p(1)/4;
What About w(2:6)
Select w(2:6)
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Insight Through Computing
All Possibilities?
for a=12:w(1) for b = 12:a-1 for c = 12:b-1 for d = 12:c-1 for e = 12:d-1 w(2) = a; w(3) = b; etc
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Insight Through Computing
Reduce the Size of TheSearch Space
Build an environment that supportssomething better than brute forcesearch…
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Insight Through Computing