Mathematics in Engineering Philip B. Bedient Civil & Environmental Engineering

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

Greek Symbols

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

Linear Models Temperature distribution across wall Temperature distribution across wall T(x) = (T 2 – T 1 )x/L + T 1T(x) = (T 2 – T 1 )x/L + T 1 T is dependent variable, x independent variableT is dependent variable, x independent variable Equation of a straight line Equation of a straight line Y = ax + bY = ax + b Relationship b/w Fahrenheit & Celsius scales Relationship b/w Fahrenheit & Celsius scales T(°F) = 9/5 T(°C) + 32T(°F) = 9/5 T(°C) + 32

Linear Models HaveConstantSlopes

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

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

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

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

Manning’s Equation Calculates flows for uncovered channels that carry a steady uniform flow Calculates flows for uncovered channels that carry a steady uniform flow V = (1.49/n)R 2/3 √(S) V = (1.49/n)R 2/3 √(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 Wetted Perimeter Area

Manning’s Equation V = (1.49/n) R 2/3 √(S) V = (1.49/n) R 2/3 √(S) A non-linear equation used in civil engineering - V fcn of n, R, and S A non-linear equation used in civil engineering - V fcn of n, R, and S V is inverse with n, goes as S 1/2 V is inverse with n, goes as S 1/2 Dependent variable V changes by different amounts Dependent variable V changes by different amounts depends on values of independent variables R & Sdepends on values of independent variables R & S

Manning’s Equation V = (1.49/n)R 2/3 √(S) V = (1.49/n)R 2/3 √(S) Solve a simple problem here. Solve a simple problem here. Show some pics Show some pics

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

Polynomial Model Characteristics General form General form y = f(x) = a 0 + a 1 x + a 2 x 2 + … + a n x n Unlike linear models, 2 nd & higher- order polynomials have variable slopes Unlike linear models, 2 nd & higher- order polynomials have variable slopes Dependent variable y has zero value at points where intersects x axis Dependent variable y has zero value at points where intersects x axis Some have real roots &/or imaginary roots Some have real roots &/or imaginary roots

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

Exponential Functions f(x) = e -x 2 f(x) = e -x 2 Symmetric fcn used in expressing probability distributions Symmetric fcn used in expressing probability distributions Note bell shaped curve Note bell shaped curve

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

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

Matrices Describe situations that require many values (vector variables- posses both magnitude & direction) Describe situations that require many values (vector variables- posses both magnitude & direction) [N] = X3 MATRIX -580 X = {x 1 x 2 x 3 x 4 } Row Matrix

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

Integral Calculus Second moment of inertia- property of area (chap. 7) Second moment of inertia- property of area (chap. 7) Property that provides info on how hard to bend something (used in structure design) 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 For small area element A at distance x from axis y- y, area moment of inertia is I y-y = x 2 A

Integral Calculus For more small area elements, area moment of inertia for system of discrete areas about y-y axis is For more small area elements, area moment of inertia for system of discrete areas about y-y axis is I y-y = x 1 2 A 1 + x 2 2 A 2 + x 3 2 A 3

Integral Calculus 2 nd moment of inertia for cross-sectional area: sum area moment of inertia for all little area elements 2 nd 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 x 2 A terms For continuous cross-sectional area, use integrals instead of summing x 2 A terms I y-y = ∫ x 2 dA Formula for cross- Formula for cross- section about y-y axes I y-y = 1/12 h w 3

Integral Calculus I y-y = ∫ x 2 dA = ∫ x 2 hdx = h ∫ x 2 dx Integrating I y-y = w 3 /24 + w 3 /24 = -w/2 w/2 I y-y = 1/12 h w 3

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

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

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

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