29 Design Parameters Optimizing Gear Ratio Distribution Insight

29. Design Parameters Optimizing Gear Ratio Distribution Insight Through Computing

An Optimal Design Problem Insight Through Computing

Gear Ratio Distribution Assume 7 wheel sprockets Assume 3 pedal sprockets 21 = 7 x 3 possible gear ratios 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 Insight Through Computing

Goal Choose 3 pedal sprockets and 7 wheel sprockets so that the 21 gear ratios are as evenly distributed across the interval [1, 4]. 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. 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 Insight Through Computing

The Quality of a Gear Ratio Distribution Ideal: 1 Good: Poor: Insight Through Computing 4

Average Discrepancy Sort the gear ratios: g(1) < g(2) <… < g(21) Compare g(i) with x(i) where x = linspace(1, 4, 21). Insight Through Computing

function tau = Obj. F(p, w); g = []; for i=1: 3 for j=1: 7 g = [g p(i)/w(j)]; end g = sort(g); dif = abs(g – linspace(1, 4, 21)); tau = sum(dif)/21; Insight Through Computing

There Are Other Reasonable Objective Functions g = sort(g); dif = abs(g –linspace(1, 4, 21)); tau = sum(dif)/21; Replace “sum” with “max” Insight Through Computing

Goal Choose p(1: 3) and w(1: 7) so that obj. F(p, w) is minimized. This defines the “best bike. ” Our plan is to check all possible bikes. A 10 -fold nested loop problem… 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 Insight Through Computing

How Constraints Arise Purchasing says that pedal sprockets only come in six sizes: C 1: p(i) is one of 52 48 42 39 32 28. Insight Through Computing

How Constraints Arise Marketing says the best bike must have a maximum gear ratio exactly equal to 4: C 2: p(1)/w(7) = 4 This means that p(1) must be a multiple of 4. Insight Through Computing

How Constraints Arise Marketing says the best bike must have a minimum gear ratio exactly equal to 1: C 3: p(3)/w(1) = 1 Insight Through Computing

How Constraints Arise Purchasing says that wheel sprockets are available in 31 sizes… C 4: w(i) is one of 12, 13, …, 42. Insight Through Computing

Choosing Pedal Sprockets Possible values… Front = [52 48 42 39 32 28]; Constraint C 1 says that p(1) must be divisible by 4. Also: p(3) < p(2) < p(1). Insight Through Computing

The Possibilities. . 52 52 48 48 42 42 39 32 28 Insight Through Computing 52 52 52 48 48 39 39 32 42 42 42 39 32 28 28 39 32 28 32 48 48 42 42 42 39 39 32 28 28

The Loops. . 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); Insight Through Computing

w(1) and w(7) “for free”. . 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; Insight Through Computing

What About w(2: 6) 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; Select w(2: 6) Insight Through Computing