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Mathematics in Engineering

Mathematics in Engineering. Philip B. Bedient Civil & Environmental Engineering. Mathematics & Engineering. Engineering problems are often solved with mathematical models of physical situations

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Mathematics in Engineering

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  1. Mathematics in Engineering Philip B. Bedient Civil & Environmental Engineering

  2. Mathematics & Engineering • Engineering problems are often solved with mathematical models of physical situations • Purpose is to focus on important mathematical models & concepts to see why mathematics is important in engineering

  3. Greek Symbols

  4. Simple Linear Models • Simplest form of equations • Linear spring • F = kx where F = spring force k = spring constant x = deformation of spring • F is dependent variable • F depends on how much spring is stretched or compressed • x is independent variable

  5. Linear Models • Temperature distribution across wall • T(x) = (T2 – T1)x/L + T1 • T is dependent variable, x independent variable • Equation of a straight line • Y = ax + b • Relationship b/w Fahrenheit & Celsius scales • T(°F) = 9/5 T(°C) + 32

  6. Linear Models Have Constant Slopes

  7. Nonlinear Models • Used to describe relationships b/w dependent & independent variables • Predict actual relationships more accurately than linear models • Polynomial Functions • Laminar fluid velocity inside a pipe- how fluid velocity changes at given cross-section inside pipe

  8. Velocity distribution inside pipe given by U = Vc[1-(r/R)2] where U = U(r)= fluid velocity at the radial distance r Vc = center line velocity r = radial distance measured       from center of pipe R = radius of pipe where U = 0 Parabolic Function

  9. Velocity distribution inside pipe given by U = Vc[1-(r/R)2] Explore the rate of change of U with r dU/dr = 0 - 2Vc (r/R)-3 =2Vc /(r/R)3 dU/dr = k/r3 Thus, the rate of change is largest at small r And is smallest near the pipe wall Parabolic Functions

  10. Non-linear fluid velocity model • Velocity distribution where Vc = 0.5 m/s & R = 0.1 m • Nonlinear second-order polynomial • Greatest slope changes with r near zero

  11. Manning’s Equation • Calculates flows for uncovered channels that carry a steady uniform flow • V = (1.49/n)R2/3√(S) Where V = channel velocity n = Manning’s roughness coefficient R = hydraulic radius = A/P S = slope of channel bottom (ft/ft) A = cross sectional area of channel P = wetted perimeter of channel Area Wetted Perimeter

  12. Manning’s Equation • V = (1.49/n) R2/3 √(S) • A non-linear equation used in civil engineering - V fcn of n, R, and S • V is inverse with n, goes as S1/2 • Dependent variable V changes by different amounts • depends on values of independent variables R & S

  13. Manning’s Equation • V = (1.49/n)R2/3√(S) • Solve a simple problem here. • Show some pics

  14. Other Non-linear examples • Many other engineering situations w/ 2nd & higher order polynomials • Trajectory of projectile • Power consumption for resistive element • Drag force • Air resistance to motion of vehicle • Deflection of cantilevered beam

  15. Polynomial Model Characteristics • General form y = f(x) = a0 + a1x + a2x2 + … + anxn • Unlike linear models, 2nd & higher-order polynomials have variable slopes • Dependent variable y has zero value at points where intersects x axis • Some have real roots &/or imaginary roots

  16. Exponential Models • Value of dependent variable levels off as independent variable value gets larger • Simplest form given by f(x) = e-x

  17. Exponential Functions • f(x) = e-x2 • Symmetric fcn used in expressing probability distributions • Note bell shaped curve

  18. Logarithmic Functions • The symbol “log” reads logarithms to base-10 or common logarithm • If 10x = y, then define log y = x • Natural logarithm ln reads to base-e • If ex = y, then define ln y = x • Relationship b/w natural & common log • ln x = (ln 10)(log x) = 2.3 log x

  19. Matrix Algebra • Formulation of many engineering problems lead to set of linear algebraic equations solved together • Matrix algebra essential in formulation & solution of these models • A matrix is array of numbers, variables, or mathematical terms • Size defined by number of m rows & n columns

  20. Matrices • Describe situations that require many values (vector variables- posses both magnitude & direction) X = {x1 x2 x3 x4} Row Matrix [N] = 6 5 9 1 26 14 3X3 MATRIX -5 8 0

  21. Differential Calculus • Important in determining rate of change in engineering problems • Engineers calculate rate of change of variables to design systems & services • If differentiate a function describing speed, obtain acceleration • a = dv/dt

  22. Integral Calculus • Second moment of inertia- property of area (chap. 7) • Property that provides info on how hard to bend something (used in structure design) • For small area element A at distance x from axis y-y, area moment of inertia is Iy-y = x2A

  23. Integral Calculus • For more small area elements, area moment of inertia for system of discrete areas about y-y axis is Iy-y = x12A1 + x22A2 + x32A3

  24. Integral Calculus • 2nd moment of inertia for cross-sectional area: sum area moment of inertia for all little area elements • For continuous cross-sectional area, use integrals instead of summing x2A terms Iy-y= ∫ x2dA • Formula for cross- section about y-y axes Iy-y= 1/12 h w3

  25. Integral Calculus w/2 w/2 Iy-y= ∫ x2dA = ∫ x2hdx = h ∫ x2dx Integrating Iy-y= w3/24 + w3/24 = -w/2 -w/2 Iy-y= 1/12 h w3

  26. Integral Calculus • Used to determine force exerted by water stored behind dam • Pressure increases w/ depth of fluid according to P = rgy where P = fluid pressure distance       y below water surface r = density of fluid g = acceleration due to gravity y = distance of point below fluid surface

  27. Variation of Pressure with y • Must add pressure exerted on areas at various depths to obtain net force • Consider force acting at depth y over small area dA, or dF = PdA Then use P = rgy Assume const r and g Subst dA = w dy Integrate y = 0 to H Net F = 1/2 rgwH2

  28. Integral Calculus Example 1 Evaluate ∫ (3x2 – 20x)dx Use rules 2 & 6 from table ∫(3x2 – 20x)dx = ∫3x2dx + ∫-20xdx = 3∫x2dx - 20∫xdx = 3[1/(3)]x3 – 20[1/(2)]x2 + C Answer = x3 – 10x2 + C

  29. Differential Equations • Contain derivatives of functions or differential terms - MATH 211 • Represent balance of mass, force, energy, etc. • Boundary conditions tell what is happening physically at boundaries • Know initial conditions of system at t = 0 • Exact solutions give detailed behavior of system

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