Session #9 Critical Parameter Management & Error Budgeting

Transcription

1 ESD Systems Engineering Session #9 Critical Parameter Management & Error Budgeting Dan Frey

2 Plan for the Session Follow up on session #8 Critical Parameter Management Probability Preliminaries Error Budgeting Tolerance Process Capability Building and using error budgets Next steps

3 S - Curves Atish Banergee We first studied S-curves in technology strategy The question remained why the S-curve has the peculiar shape. Well I found the answer in system dynamics. It is a general phenomenon and not restricted to technology. It can be thought of as two curves:. The lower part of the curve is growth with acceleration The upper part of the s-curve is called a goal-seeking curve and can be thought of as growth with deceleration

4 Trends in Compressor Performance Wisler, D. C., 998, Axial Flow Compressor and Fan Aerodynamics, Handbook of Fluid Dynamics, CRC Press., ed. R. Johnson.

5 Evolution of Jet Engine Performance Ghost JT3C Cruise thrust specific fuel consumption ( lb fuel/hr lb thrust ( JT3D CJ85 JT8D-9 JT9D-3 JT9D-3A CF6-5E/C CF6-6D JT8D-7 JT8D-29 JT9D-7 JT8D-27A JT9D-59A JT9D-7A RB2-524C2 CFM56-2C CF6-5E2/C2 RB2-524D4 JT9D-7R4D (B) JT8D-5 JT8D-7A JT8D-29 TAY CF6-8A/A CFM56-5A RB2-535E4 237 CF6-8C2 CFM56-3 RB2-535C RB2-524D4 UP JT9D-7R4H CFM56-5B V RB2-524D4D CFM56-5C P&W dehavilland RR GE Certification date Adapted from Koff, B. L. "Spanning the World Through Jet Propulsion. AIAA Littlewood Lecture. 99.

6 Plan for the Session Follow up on session #8 Critical Parameter Management Probability Preliminaries Error Budgeting Tolerance Process Capability Building and using error budgets Next steps

7 Critical Parameter Management CPM provides discipline and structure Produce critical parameter documentation For example, a critical parameter drawing Traces critical parameters all the way through to manufacture and use Determines process capability (C p or C pk ) Therefore, requires probabilistic thinking

8 Plan for the Session Follow up on session #8 Critical Parameter Management Probability Preliminaries Error Budgeting Tolerance Process Capability Building and using error budgets Next steps

9 Probability Definitions Sample space a list of all possible outcomes of an experiment Finest grained Mutually exclusive Collectively exhaustive Event - A collection of points in the sample space

10 Concept Question You roll 2 dice Give an example of a single point in the sample space? How might you depict the full sample space? What is an example of an event?

11 Probability Measure Axioms For any event A, P(U)= P( A) If A B=φ, then P(AUB)=P(A)+P(B) For the case of rolling two dice: A = rolling a 7 and B = rolling a on at least one die Is it the case that P(A+B)=P(A)+P(B)?

12 Discrete Random Variables A random variable that can assume any of a set of discrete values Probability mass function p x (x o ) = probability that the random variable x will take the value x o Let s build a pmf for rolling two dice p x (x) random variable x is the total x x=

13 Continuous Random Variables Can take values anywhere within continuous ranges Probability density functions obey three rules P { L < x U} f ( x) dx f x U = L ( x) for all x x f x (x) P{ L < x U } f x ( x)dx = L U x

14 Measures of Central Tendency Expected value E( g( x)) b = a g( x) Mean µ = E(x) f x ( x) dx Arithmetic average Median Mode f x (x) n n i= x i x

15 Measures of Dispersion Variance VAR( x) 2 = σ = E(( x E( x)) 2 ) Standard deviation σ = E(( x E( x)) 2 ) Sample variance n th central moment Covariance S 2 = n E(( x n i ( x i x) E( x)) E(( x E( x))( y E( y))) n ) 2

16 Sums of Random Variables Average of the sum is the sum of the average (regardless of distribution and independence) E ( x + y) = E( x) + E( y) Variance also sums iff independent σ ( x + y) = σ ( x) σ ( y This is the origin of the RSS rule Beware of the independence restriction! ) 2

17 Concept Test A bracket holds a component as shown. The dimensions are independent random variables with standard deviations as noted. Approximately what is the standard deviation of the gap? A). B). C). σ =." σ =." gap

18 Uniform Distribution A reasonable (conservative) assumption when you know the limits of a variable but little else σ = ( U L) 2 3 L U

19 Basic Application I have two spinners x=result of blue spinner y=result of red spinner z=x+y What are the pdfs for variables x, y, and z? P { a < x b} f ( x) dx b = a x f x ( x)dx = f x ( x) for all x

20 Simulation Can Quickly Answer the Question trials=;nbins=trials/; x= random('uniform',,,trials,); y= random('uniform',,2,trials,); z=x+y; subplot(3,,); hist(x,nbins); xlim([ 3]); subplot(3,,2); hist(y,nbins); xlim([ 3]); subplot(3,,3); hist(z,nbins); xlim([ 3]);

21 Probability Distribution of Sums If z is the sum of two random variables x and y z = x + Then the probability density function of z can be computed by convolution y p z ( z) = z x( z ζ ) y( ζ ) dζ

22 Convolution p z ( z) = z x( z ζ ) y( ζ ) dζ

23 Convolution p z ( z) = z x( z ζ ) y( ζ ) dζ

24 Central Limit Theorem The mean of a sequence of n iid random variables with Finite µ E ( ) 2+ δ x E( ) < δ > i x i approximates a normal distribution in the limit of a large n.

25 Normal Distribution f ( x µ ) 2 2σ x( x) = e σ 2π 2-6σ -3σ -σ µ +σ +3σ +6σ 68.3% 99.7% -2ppb

26 Joint Normal Distribution T p( x) = exp ( x ) K ( x ) m µ µ K 2 ( 2π ) The lines of constant probability density are ellipsoids If the matrix K is diagonal, then the variables are uncorrelated and independent correlated uncorrelated

27 Independence Random variables x and y are said to be independent iff f ( x, y) = f ( x) f ( y) xy x y Or, knowledge of x provides no information to update the distribution of y

28 Expectation Shift S=E(y(x))- y(e(x)) S y(e(x)) E(y(x)) f y (y(x)) E(x) Under utility theory (DBD), S is a key difference between probabilistic and deterministic design y(x) x f x (x)

29 Plan for the Session Follow up on session #8 Critical Parameter Management Probability Preliminaries Error Budgeting Tolerance Process Capability Building and using error budgets Next steps

30 Error Budgets A tool for predicting and managing variability in an engineering system A model that propagates errors through a system Links aspects of the design and its environment to tolerance and capability Used for tolerance design, robust design, diagnosis

31 Engineering Tolerances Tolerance --The total amount by which a specified dimension is permitted to vary (ANSI Y4.5M) Every component within spec adds to the yield (Y) p(y) p(q) Y L U q y

32 Tolerance on Position >25%W Lead W Land

33 Tolerance of Form wide tolerance zone THIS ON A DRAWING MEANS THIS

34 GD&T Symbols Geometric Characteristic Symbols For Individual Features For Individual or Related Features For Related Features Type of Tolerance Form Profile Orientation Location Runout Characteristic Straightness Flatness Circularity (Roundness) Cylindricity Profile of a Line Profile of a Surface Angularity Perpendicularity Parallelism Position Concentricity Circular Runout Total Runout Symbol Arrowhead(s) may be filled in.

35 Multiple Tolerances Most products have many tolerances Tolerances are pass / fail All tolerances must be met (dominance) 35

36 Variation in Manufacture Many noise factors affect the system Some noise factors affect multiple dimensions (leads to correlation) 36

37 Process Capability Indices p(q) U 2 L µ Process Capability Index Bias factor Performance Index L U + 2 k L U U + L µ 2 ( U L)/ 2 q ( U L) C /2 p 3σ C C ( k) pk p 37

38 Concept Test Motorola s 6 sigma programs suggest that we should strive for a C p of 2.. If this is achieved but the mean is off target so that k=.5, estimate the process yield.

39 C p and k Determine Yield By definition p(q) Y = p( q)dq FT U L L Y U q If Gaussian YFT = erf Cp ( k) + erf Cp ( + k) This function to maps C p and k to yield 39

40 C p and k Determine Quality Loss L(q) A o Quality Loss = A o [( U L) / 2] 2 U + d 2 L 2 L U + 2 L U q 2 E(Quality Loss) = Ao k + 2 9C p Taguchi's quality loss function ANSI's implied quality loss function

41 Crankshafts What does a crankshaft do? How would you define the tolerances? How does variation affect performance?

42 Printed Wiring Boards What does the second level connection do? How would you define the tolerances? How does variation affect performance?

43 C p and k for the System C p = LL UL k = 8. Y FT = 983%. frequency 5 5 4

44 Producibility Analysis Rolled throughput yield (Y RT )-- The probability that all tolerances are met Motorola s approach Assumes probabilistic independence Y RT = m i= Y FT i Motorola s formula 368 Y RT =. 983 =. 2% Hughes data Y RT = 66. 7% 5

45 Surface Mount Data Side #3 Side # Side to side error Side to side error Lead number Lead number Side #2 2 4 Lead number Side to side error Side #4 2 4 Lead number 6 Side to side error

46 Plan for the Session Follow up on session #8 Critical Parameter Management Probability Preliminaries Error Budgeting Tolerance Process Capability Building and using error budgets Next steps

47 Error Sources Kinematic errors Straightness Squareness Bearings Drive related errors Thermal errors Static loading Dynamics

48 Errors in a Linear Drive Lead deviation (µm) revolution Once per revolution lead error (µm) Cumulative lead error (µm/mm) Nominal travel (mm)

49 Angular Errors ε y ε y X y y z x z x OK, so you put the error in the model. Now what will happen when the machine moves? ε y X y z x

50 A Model of a Robot 5 mm 4 mm Θ z Θ z2 6 mm Z 3 mm mm z Point p x y Base

51 Errors in the Robot Error Description µ σ ε z Drive error of joint # rad. rad ε z2 Drive error of joint #2 rad. rad δ z3 Drive error of joint #3 Z..mm ε x3 Pitch of joint #3 rad.5 rad ε y3 Yaw of joint #3 rad.5 rad xp 2 Parallelism of joint 2 in the x direction.2 rad. rad

52 A Model of a Robot The matrices describe the intended motions and the errors Can be applied to any point on the end effector + Θ Θ Θ + Θ = 6 5 ) cos( sin sin ) cos( mm xp mm xp mm mm mm mm mm mm mm z z z z z z ε ε T + Θ Θ Θ + Θ 4 ) cos( sin sin ) cos( z x y x y z z z z z z Z mm mm mm mm mm mm mm mm δ ε ε ε ε ε ε = 3 ' ' ' 3 T z y x p p p NOTE: These two should be swapped

53 Homework #5 Short answers on TRIZ and probability Error budgeting Two tasks are to be done with the robot Analyze the tasks Discuss changes to the system A Matlab file is available in the HW folder just so you don t have to re-type the matrices

54 Next Steps You can download HW #5 Error Budgetting Due 8:3AM Tues 3 July See you at Thursday s session On the topic Design of Experiments 8:3AM Thursday, 8 July Reading assignment for Thursday All of Thomke Skim Box Skim Frey