16.888- Multidisciplinary System Design Optimization Robust design Response Control Factor Prof. Dan frey Mechanical engineering and engineering systems
Robust Design Prof. Dan Frey Mechanical Engineering and Engineering Systems 16.888 – Multidisciplinary System Design Optimization Control Factor Response
Plan for the session Basic concepts in probability and statistics Review design of experiments Basics of robust design · Research topics Model-based assessment ofrd methods Faster computer-based robust design Robust invention
Plan for the Session • Basic concepts in probability and statistics • Review design of experiments • Basics of Robust Design • Research topics – Model-based assessment of RD methods – Faster computer-based robust design – Robust invention
Ball and Ramp Ball Response Ramp the time the Funnel ball remains in the funnel Causes of experimental error=?
Ball and Ramp Ball Ramp Funnel Response = the time the ball remains in the funnel Causes of experimental error = ?
Probability Measure · Axioms For any event A,P(A)≥0 P()=1 If the intersection of A and b=o, then P(A+B=P(A+P(B
Probability Measure • Axioms – For any event A, – P ( U)=1 – If the intersection of A and B =I, then P (A +B)=P(A)+P( B ) P ( A ) t 0
Continuous random variables Can take values anywhere within continuous ranges Probability density function P<xsb=」0 x)ax 0≤f(x) for all x fI(xh f(x)dx=1 a b x
Continuous Random Variables • Can take values anywhere within continuous ranges • Probability density function – – – f x x x 0 d ( ) for all ( ) d 1 ³ f f f x x x P^ ` a x b f x x b a x ( ) d ³ d x fx (x ) a b
Histograms graph of continuous data Approximates a pdf in the limit of large n Histogram of Crankpin Diameters Diameter. pin
Histograms • A graph of continuous data • Approximates a pdf in the limit of large n 0 5 Histogram of Crankpin Diameters Diameter, Pin #1 Frequency
Measures of Central Tendency Expected value E(g(x)=8(x)f (x)dx Mean =E(x) Arithmetic average Iv ∑
Measures of Central Tendency • Expected value • Mean P = E (x ) • Arithmetic average ³ S x E ( g ( x)) g ( x ) f ( x ) dx ¦ n i i x n 1 1
Measures of dispersion ariance VAR(x=o=e(x-e(x)) Standard deviation E(x-e(x) Sample variance ∑(x nth central moment E((x-E(x) nth moment about m E(x-m)")
Measures of Dispersion • Variance • Standard deviation • Sample variance • nth central moment • nth moment about m (( ( )) ) 2 V E x E x ( ) (( ( )) ) 2 2 VAR x V E x E x ¦ n i i x x n S 1 2 2 ( ) 1 1 (( ( )) ) n E x E x (( ) ) n E x m
Sums of random variables Average of the sum is the sum of the average(regardless of distribution and ndependence E(x+y=e(x+e( Variance also sums iff independent a(x+y)=a(x)2+o(y)2 This is the origin of the rss rule Beware of the independence restriction
Sums of Random Variables • Average of the sum is the sum of the average (regardless of distribution and independence) • Variance also sums iff independent • This is the origin of the RSS rule – Beware of the independence restriction! E ( x y ) E ( x ) E ( y ) 2 2 2 V ( x y ) V ( x ) V ( y )
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)0.011 O=0.0 B)0.01″ O=0.001 C)0.001″ gap
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) 0.011” B) 0.01” C) 0.001” V 0.001" V 0.01" gap