1. 20-Jan-2010electrical, computer and energy engineering Prof. Subramaniam (“Subby”) D. Rajan, Prof. Narayanan Neithalath and Amie Baisley Graduate Students: Kirk Vance, Matt Aguayo, Tejas Ashani, Joseph Harrington and Canio Hoffarth Engineering 101 Linking Experiments to Models through the Bridge Design Exercise

2. What are Experiments? n Tests to determine the relationship between (input) variables and (output) responses n Example 1: What is the effect of dowel diameter on the weight of the bridge? –Model: The entire bridge system –Input Variable: Dowel diameter –Output Response: Weight of the bridge n Example 2: What is the effect of dowel diameter on the maximum deflection of the bridge deck? –Model: The entire bridge system –Input Variable: Dowel diameter –Output Response: Deflection of the bridge deck at various locations n 2

3. What are Models? n Relationship between (input) variables and (output) responses –Simple equation –Model described by one or more complex equation(s) – differential equation(s), integral equation(s), … n Example 1: What is the effect of dowel diameter on the weight of the bridge? n Example 2: What is the effect of dowel diameter on the maximum deflection of the bridge deck? –Needs a model whose solution can be described by several linear, algebraic equations 3

4. What is a System? n Dictionary definitions –a set of connected things or parts forming a complex whole, in particular –a set of principles or procedures according to which something is done; an organized scheme or method n Traits of a system –has structure, its parts or components are directly or indirectly interact with each other –has behavior (where input and output are linked) 4

5. Questions n Q1: Draw a diagram that shows the components of the bridge system, establishes the boundary and identifies the surroundings. n Q2: Describe the bridge system with particular attention to (a) its functionalities, (b) how the different components interact with each other and (c) how the bridge system behaves. 5

6. Engineering Process or Product Design 6

7. Verification and Validation n Models need to be validated and verified before they can be used with any confidence n Verification: Are you building it right? –Is the theory/principle embodied in the model implemented correctly? 7 g = 9.81 m/s 2

8. Verification and Validation n Validation: Are you building the right thing? –Do the results from the model correlate well with experimental results? 8 TrialM (kg) m (kg) Exp. a (m/s 2 ) Model a (m/s 2 ) % error 1 2 3

9. Questions n Q3: Describe what a bridge model could be, by identifying the input variables and output responses. n Q4: Identify the characteristics of each input variable. Describe how you would obtain the values of these variables. n Q5: Identify the characteristics of each output response. What is the purpose of each output response? n Q6: Give examples of engineering processes and products? n Q7: Describe the linkages between experiments and modeling. 9

10. Case Study 10

11. Case Study n Develop a model to predict the tip deflection (displacement) of a cantilever beam due to a tip load. Use experiments to validate the model. 11

12. Case Study: Basic Steps n Use a sound scientific or engineering principle to develop the model. What parameters will be a part of this model – input and output variables? n Design experiment(s) to verify the model. n Design experiment(s) to validate the model. 12

13. Case Study: Principle/Theory n Euler-Bernoulli Beam Theory (w/o derivation) 13 Differential Equation Boundary Conditions v(x): vertical displacement M(x): Bending moment E(x): Young’s modulus I(x): Moment of inertia L: length of the beam

14. Case Study: Cantilever Beam 14 Boundary Conditions Integrating twice and using the BCs

15. Case Study: The Model 15 Para.Remarks PThe applied load at the tip of the beam EMaterial property that needs to be found ICross-sectional property that needs to be computed LLength of the beam that needs to be measured xLocation where the displacement is computed

16. Case Study: Modulus of Elasticity n What is modulus of elasticity or Young’s modulus (E)? –In a one-dimensional state of stress it is constant of proportionality between the normal stress and the normal strain and has the units of stress. 16 Stress-strain curve (ductile material)

17. Case Study: Moment of Inertia n What is moment of inertia, I? –The second moment of area (or, moment of inertia) is a measure of a beam’s cross-sectional shape’s resistance to bending. 17

18. Experiment Measure the width, w, and thickness, t, of a steel plate 18

19. Raw Measurement Data 19 Measurements taken at 11 different locations

20. Raw Measurement Data 20 Histogram Plot

21. Statistical Analysis of Data 21 Caliper 1Caliper 2 Width (in)Thickness (in)Width (in)Thickness (in) # of readings (n) 11 Mean1.11380.03031.11430.0315 Median1.1140.03051.1150.031 Standard Deviation 0.00071980.00026110.0009050.00157

22. Questions n Q8: What is sample size? n Q9: What is mean? What is another name for mean? n Q10: What is median? n Q11: What is standard deviation? n Q12: Write a few sentences on the quality of the thickness and width data for the steel plate. 22

23. Normal Distribution 23 Probability Density Function * *Excel terminology: Probability Mass Function 68-95-99.7 rule: 1, 2, 3 standard deviations from mean Function whose graph is a continuous curve over a range of values that x can take. It has the units of probability rate (not probability). x is called random variable. Area under curve between x 1 and x 2 gives the probability that x lies in the interval x 1 and x 2. 66

24. Cumulative Distribution Function 24 What is the probability that a random width value is between 1.113 in and 1.114 in?

25. Questions n Q13: Normal distribution is often called bell curve. Are there other types of distribution? n Q14: Identify and rank the effect of the random variables in the equation for tip deflection. 25

26. Experiment 2 Measure the tip displacement of an aluminum cantilever beam 26

27. Raw Experimental Data 27

28. Case Study: Model Verification 28

29. Case Study: Model Validation 29 Published Elastic Modulus of Aluminum (6016-T6) = 1.01(10 7 ) psi

30. Forensic Engineering 30

31. One-Parameter Regression Analysis n Objective: Use the model and experimental data to determine the Young’s modulus of aluminum. n 31

32. References n Do an internet search using these keywords – system, model, experiment, verification, validation, statistical quantities. n Engineering Statistics: http://www.itl.nist.gov/div898/handbook/http://www.itl.nist.gov/div898/handbook/ n http://www.mathsisfun.com/links/curriculum-high-school-statistics.html http://www.mathsisfun.com/links/curriculum-high-school-statistics.html n http://www.stevespanglerscience.com/lab/experiments http://www.stevespanglerscience.com/lab/experiments n http://en.wikipedia.org/wiki/Verification_and_validation http://en.wikipedia.org/wiki/Verification_and_validation 32