1 / 32

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

qamar
Download Presentation

Mathematics in Engineering

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related