1. EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Line fitting, Hooke’s Law, and Physics of Towers Introduction to Engineering Systems Lecture 4 (9/7/2009) Prof. Andrés Tovar

2. EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Announcements Laura Flynn, Career Center –http://careercenter.nd.eduhttp://careercenter.nd.edu Engineering Industry Day –When? Wednesday, September 16 5:00 – 9:00 p.m. –Where? Joyce Center Field House –http://careercenter.nd.edu/for-undergradshttp://careercenter.nd.edu/for-undergrads

3. EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Announcements Print out the Learning Center document from Concourse, read it, and bring it to your Learning Center class. Homework 1 and last week’s Learning Center deliverable is due at the start of your Learning Center class this week. Homework 2 is available online now. Line fitting, quantify error between data points and a line, spring stiffness.

4. EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame This Week’s Learning Center Learning Center –Use SAP2000 to simulate tower deflection for a series of applied loads –Deliverable: compare SAP2000 results (theoretical model) with experimental results (empirical model). Much like we did with the pendulums

5. EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame From last class m1 = 165; m2 = 48; Cr = 0.5; h2f = 9; hi1 = (h2f*(m1 + m2)^2)/(m1^2*(Cr + 1)^2) >> hi1 = 6.6658 Theoretical model Experimental model h1ih2f-1h2f-2h2f-3h2f-4h2f-5h2fe 0.0 5.06.07.58.07.06.57.0 10.08.014.012.013.012.411.9 15.012.018.023.0 18.518.9 20.019.025.531.027.031.426.8

6. EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Line fit of experimental data Experimental model h1ih2fe 0.0 5.07.0 10.011.9 15.018.9 20.026.8 collision1.xlsx

7. EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Line fit of experimental data Experimental model h1ih2feh2fp 0.0 -0.18 5.07.06.37 10.011.912.92 15.018.919.47 20.026.826.02 h1i = (0:5:20)'; h2fe = [0;7;11.9;18.9;26.8]; c = polyfit(h1i, h2fe, 1) h2fp = c(1)*h1i + c(2) plot(h1i,h2fe,'o',h1i,h2fp,'-') xlabel('h1i'); ylabel('h2f')

8. EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Experimental vs. theoretical collision models % input values h1i = (0:5:20)'; % experimental model h2fe = [0;7;11.9;18.9;26.8]; c = polyfit(h1i, h2fe, 1); h2fp = c(1)*h1i + c(2); plot(h1i,h2fe,'ro',h1i,h2fp,'r-') xlabel('h1i'); ylabel('h2f'); hold on % theoretical model m1 = 165; m2 = 48; Cr = 0.5; h2f = 9; h2ft = h1i*(m1^2*(Cr + 1)^2)/(m1 + m2)^2; plot(h1i,h2ft,'b.-') and the collision problem (mystery) has been solved!

9. EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame A new experiment x=(0:0.2:1)'; y_exp=[1.4;1.1;3.3;2.4;3.8;3.0]; plot(x,y_exp,'.','MarkerSize',20) xlabel('X'); ylabel('Y'); hold on xy_exp 0.01.4 0.21.1 0.43.3 0.62.4 0.83.8 1.03.0 System x y_exp

10. EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Equation of the Line (0,1) (1,4) y = mx + b y = 3x + 1 x=(0:0.2:1)'; y_exp=[1.4;1.1;3.3;2.4;3.8;3.0]; plot(x,y_exp,'b.','MarkerSize',20) xlabel('X'); ylabel('Y'); hold on y_mf=3*x+1 plot(x,y_mf,'r-') xy_expy_mf 0.01.41.0 0.21.11.6 0.43.32.2 0.62.42.8 0.83.83.4 1.03.04.0

11. EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Error Between the Line and the Data x=(0:0.2:1)'; y_exp=[1.4;1.1;3.3;2.4;3.8;3.0]; y_mf=3*x+1; mf_err=y_exp-y_mf sum(mf_err) mf_err2=mf_err.^2 sum(mf_err2) xy_expy_mfmf_errmf_err^2 0.01.41.0-0.40.16 0.21.11.60.50.25 0.43.32.2-1.11.21 0.62.42.80.40.16 0.83.83.4-0.40.16 1.03.04.01.01.00 sum0.02.94

12. EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Least square fit x=(0:0.2:1)'; y_exp=[1.4;1.1;3.3;2.4;3.8;3.0]; plot(x,y_exp,'b.','MarkerSize',20) hold on c=polyfit(x,y_exp,1); y_pf=c(1)*x+c(2) plot(x,y_pf,'g-') xy_expy_mfy_pfpf_err^2 0.01.41.01.40.00 0.21.11.61.80.55 0.43.32.22.31.04 0.62.42.82.70.10 0.83.83.43.20.41 1.03.04.03.60.36 sum2.46

13. EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Tabulating the error xy_expy_mfmf_errmf_err^2y_pfpf_errpf_err^2 0.01.41.0-0.40.161.40.00.00 0.21.11.60.50.251.80.70.55 0.43.32.2-1.11.212.31.04 0.62.42.80.40.162.70.30.10 0.83.83.4-0.40.163.2-0.60.41 1.03.04.01.01.003.60.60.36 sum0.02.940.02.46 xinput value y_exp experimental output y_mf=3.0*x+1.0 manual fit mf_err=y_exp-y_mf manual fit error y_pf=2.2*x + 1.4least-square fit pf_err=y_exp-y_pf least-square fit error

14. EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Gather Data Develop Model Verify Model MODEL DEVELOPMENT Investigate Designs using Model Optimize Design Predict Behavior DESIGN STAGE Construct Design Experimentally Verify Behavior CONSTRUCTION & VERIFICATION Stages of the Engineering Design Process Applicable to Tower Builder

15. EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Gather Data Develop Model Verify Model MODEL DEVELOPMENT COMPLETED WEEKS 2 & 3 WEEK 3 Figure courtesy of Dr. Kijewski-Correa Stages of the Engineering Design Process Applicable to Tower Builder

16. EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Verifying our Model ?? EXPERIMENTAL MODEL THEORETICAL MODEL Tower Force Displacement ForceDisp. 0 N 0 mm 0.5 N 6.17 mm 1.5 N 18.52 mm 2.0 N 24.69 mm ForceDisp. 0 N 0 mm 0.5 N 9.93 mm 1.5 N 20.00 mm 2.0 N 30.60 mm

17. EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame SAP2000: What is it? Finite element method (FEM) software used for structural analysis. Developed by Edward L. Wilson at the University of California at Berkeley in 1969. Commercialized by Computers & Structures, Inc. (CSI), founded in 1975. Intended for use on civil structures such as bridges, dams, stadiums, industrial structures and buildings.

18. EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame SAP2000: How does it work? Uses fundamental theories of mechanics Requires: –Understanding of the system to be modeled –Material it is made of –Geometry or layout –Types of connections used How can we use SAP 2000 for Tower Builder? –Find displacements for applied forces

19. EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame SAP2000: Features Various shapes Viewing 3D objects from different perspectives Animating behavior Quick modification

20. EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Hooke’s Law and Stiffness Hooke’s Law: The amount by which a material body is deformed linearly is related to the force causing the deformation. –Mathematically: F = force, ∆ = displacement k = stiffness coefficient Spring F 

21. EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Calculating Stiffness Constants k = slope of force vs. displacement plot Which design is stiffer (A or B)?

22. EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Calculating Stiffness Constant For multiple data points (experimental data), we can use the mean displacement

23. EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Hooke’s Law Example Each rubber band shown is pulled with equal forces. The green rubber band has a stiffness k1 = 1.5 N/mm. The beige rubber band has a stiffness k2 = 1.0 N/mm. Which rubber band will elongate more? What is the deflection of each if the rubber band is pulled on with a force F = 5 N?

24. EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame How does our Tower Behave Under Load? Gravity Loading Lateral Loading Where do these loads come from? Which loads are you most concerned with on your tower? Which type of load did you simulate in Learning Center 1?

25. EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Structural Components –Beams –Floors –Columns Transfer of Forces How Does Our Tower Transfer Load?

26. EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame What’s Happening Internally?

27. EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame What’s Happening Internally?

28. EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame When is Hooke’s Law (our model) valid?

29. EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame What’s Next? How can we use statistics to neatly summarize the variations in data (in general)? How will we use statistics to summarize the variations in the tower data? Physics of Towers