Uncertainty in mechanical components Why consider uncertainty Basics of uncertainty Uncertainty analysis for machine design Examples Conclusions 2
Uncertainty in Mechanical Components • A simply-supported beam has a diameter of 1.25 in. The deflection at 𝑥 = 10 in should be less than δ =0.00375 in. Can the requirement be met? • Everything is modeled perfectly. • In reality, the forces and dimensions are all random. • So is the deflection. 3
Where Does Uncertainty Come From? • Manufacturing impression – Dimensions of a component – Material properties
• Environment – Loading – Temperature – Different users
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Why Consider Uncertainty? • We know the true solution. • We know the effect of uncertainty. • We can make more reliable decisions.
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How Do We Model Uncertainty? • We use probability distributions to model parameters with uncertainty. 0.12 0.1 Probabilistic Design
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0.08 0.06 0.04 Baseline Design 0.02 0 50
55
60
Noise
65
70
75 6
Probability Distribution 60
• With more samples, we can draw a histogram.
50
40
30
20
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• If y-axis is frequency and the number of samples is infinity, we get a probability density function (PDF) 𝑓(𝑥). • The probability of 𝑎 ≤ 𝑋 ≤ 𝑏.
Pr 𝑎 ≤ 𝑋 ≤ 𝑏 =
𝑏 𝑓 𝑎
0 1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2
𝑥 𝑑𝑥
0
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
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Normal Distribution • 𝑋~𝑁(𝜇, 𝜎 2 ) • 𝐹 𝑥 = Pr 𝑋 < 𝑥 : cumulative distribution function (CDF) • Pr 𝑎 < 𝑋 < 𝑏 = 𝐹 𝑏 − 𝐹 𝑎 • Pr 𝑋 < 𝑥 =
𝑥−𝜇𝑌 𝜎𝑌
2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
• Pr 𝑋 > 𝑥 = 1 − Pr 𝑋 0} – 𝐗: random variables – 𝑔 ∙ : limit-state function – If 𝑔 𝐗
