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Line fitting, Hooke’s Law, and Physics of Towers

Line fitting, Hooke’s Law, and Physics of Towers. Introduction to Engineering Systems Lecture 4 (9/7/2009). Prof. Andrés Tovar. Announcements. Laura Flynn, Career Center http://careercenter.nd. edu Engineering Industry Day When? Wednesday, September 16 5:00 – 9:00 p.m.

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Line fitting, Hooke’s Law, and Physics of Towers

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  1. Line fitting, Hooke’s Law, and Physics of Towers Introduction to Engineering Systems Lecture 4 (9/7/2009) Prof. Andrés Tovar

  2. Announcements • Laura Flynn, Career Center • http://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-undergrads

  3. 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. 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. From last class Theoretical model m1 = 165; m2 = 48; Cr = 0.5; h2f = 9; hi1 = (h2f*(m1 + m2)^2)/(m1^2*(Cr + 1)^2) >> hi1 = 6.6658 Experimental model

  6. Line fit of experimental data Experimental model collision1.xlsx

  7. Line fit of experimental data Experimental model 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. 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. 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 x y_exp System

  10. Equation of the Line (1,4) 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-') y = mx + b y = 3x + 1 (0,1)

  11. 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)

  12. 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-')

  13. Tabulating the error x input 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 + 1.4 least-square fit pf_err=y_exp-y_pf least-square fit error

  14. DESIGN STAGE CONSTRUCTION & VERIFICATION Investigate Designs using Model Construct Design Optimize Design Experimentally Verify Behavior Predict Behavior Stages of the Engineering Design Process Applicable to Tower Builder MODEL DEVELOPMENT Gather Data Develop Model Verify Model

  15. Stages of the Engineering Design Process Applicable to Tower Builder MODEL DEVELOPMENT COMPLETED Gather Data WEEKS 2 & 3 Develop Model WEEK 3 Verify Model Figure courtesy of Dr. Kijewski-Correa

  16. Verifying our Model Force Displacement Tower EXPERIMENTAL MODEL THEORETICAL MODEL ? 

  17. 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. 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. SAP2000: Features • Various shapes • Viewing 3D objects from different perspectives • Animating behavior • Quick modification

  20. Hooke’s Law and Stiffness F D Spring • 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

  21. Calculating Stiffness Constants Which design is stiffer (A or B)? k = slope of force vs. displacement plot

  22. Calculating Stiffness Constant • For multiple data points (experimental data), we can use the mean displacement

  23. Hooke’s Law Example Each rubber band shown is pulled with equal forces. The greenrubber band has a stiffnessk1 = 1.5 N/mm. The beige rubber band has a stiffnessk2 = 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 forceF = 5 N?

  24. How does our Tower Behave Under Load? • 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? • Gravity Loading • Lateral Loading

  25. How Does Our Tower Transfer Load? Floors → Beams → Columns • Structural Components • Beams • Floors • Columns • Transfer of Forces

  26. What’s Happening Internally?

  27. What’s Happening Internally?

  28. When is Hooke’s Law (our model) valid?

  29. 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

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