Applications of Numerical Methods in Engineering CNS 3320

1 Applications of Numerical Methods in EngineeringCNS 3320 James T. AllisonUniversity of MichiganDepartment of Mechanical EngineeringJanuary 10, 2005 University of Michigan Department of Mechanical EngineeringApplications of Numerical Methods in EngineeringObjectives:BMotivate the study of Numerical Methods through discussion ofengineering the use ofMatlabusing simple Numerical of Michigan Department of Mechanical EngineeringJanuary 10, 2005 Lecture Overview Quantitative Engineering Activities: Analysis and Design Selected Categories of Numerical Methods and Applications Linearization Finding Roots of Functions Solving Systems of Equations Optimization Numerical Integration and Differentiation Selected Additional Applications MatlabExample: Fixed Point Iteration MatlabExample: Numerical IntegrationUniversity of Michigan Department of Mechanical EngineeringJanuary 10, 2005 Quantitative Engineering Activities: Analysis and DesignEngineering.

2 Solving practical technical problems using scientific andmathematical tools when available, and using experience and models provide a priori estimates of performance verydesirable when prototypes or experiments are problems frequently arise in which exact analytical solutionsare not solutions are normally sufficient for Engineering Applications ,allowing the use of approximate Numerical of Michigan Department of Mechanical EngineeringJanuary 10, 2005 Quantitative Engineering Activities: Analysis and DesignBAnalysisPredicting the response of a system given a fixed system design and operatingconditions.

3 0 60mph acceleration time of a vehicle (Mechanical Engineering ) Power output of an electric motor (Electrical/Mechanical Engineering ) Gain of an electromagnetic antenna (Electrical Engineering ) Maximum load a bridge can support (Civil Engineering ) Reaction time of a chemical process (Chemical Engineering ) Drag force of an airplane (Aerospace Engineering ) Expected return of a product portfolio (Industrial and Operations Engineering )BDesignDetermining an ideal system design such that a desired response is achieved. Maximizing a vehicle s fuel economy while maintaining adequate performance levelsby varying vehicle design parameters.

4 Minimizing the weight of a mountain bike while ensuring it will not fail structurallyby varying frame shape and of Michigan Department of Mechanical EngineeringJanuary 10, 2005 Categories of Numerical Methods and Applications Linearization Finding Roots of Functions Solving Systems of Equations Optimization Numerical Integration and DifferentiationUniversity of Michigan Department of Mechanical EngineeringJanuary 10, 2005 Linearization Nonlinear equations can be much more difficult to solve than linear equations. Taylor s series expansion provides a convenient way to approximate a nonlinear equationor function with a linear equation.

5 Accurate only near the expansion (x) =f(a) +f (a)(x a) +f (a)2!(x a)2+.. Linear approximation uses first two terms of the of Michigan Department of Mechanical EngineeringJanuary 10, 2005 Linearization Example: Swinging PendulumSum forces in tangential direction: Ft=Wt=mgsin =mat=md2 dt2L=m L gLsin = 0 Linearizesin :sin sin(0) +cos(0)( 0) = Equation of motion valid for smallangles: gL = 0mTW=mgmWtWrUniversity of Michigan Department of Mechanical EngineeringJanuary 10, 2005 Finding Roots of FunctionsFind the value ofxsuch thatf(x) = 0 Frequently cannot be solved analytically in Engineering Applications .

6 Transcendental equations Black-box functions May have multiple or infinite solutionsExample: static equilibrium problems must satisfy F= 0 M= 0 University of Michigan Department of Mechanical EngineeringJanuary 10, 2005 Root Finding Example- Statically Indeterminate Structural Analysisbeam 1beam 2beam 3beam 1rod 2rod nb-1 dnb d3 d2 d1 dr(nb-1) dr2 dr1 Llr1lr2lr(nb-1)..University of Michigan Department of Mechanical EngineeringJanuary 10, 2005 Root Finding Example- Statically Indeterminate Structural Analysis Force applied to lower beam known All other forces and displacements unknown Solution a guess for the force on the top the required applied force to generate this top beam to actual applied force, iterate until they matchSolve: F1(F3) F1= 0F33r2ad2fF2br1e1F1gc University of Michigan Department of Mechanical EngineeringJanuary 10, 2005 Solving Systems of EquationsSolve for the value of the vectorxthat satisfies the given system of Systems.

7 A11x1+a12x2=b1a21x1+a22x2=b2 [a11a12a21a22]{x1x2}={b1b2} Ax=bNon-Linear Systems:f1(x) =b1f2(x) =b2 f(x) =bUniversity of Michigan Department of Mechanical EngineeringJanuary 10, 2005 Linear Systems Example: Circuit AnalysisKirchhoff s sum of all voltage changes around any closed loop is zero:ne i=1 Vi= sum of all currents at any node is i=1 Ii= 0 Application of these two laws to an electrical circuit facilitates the formulation of a systemofnlinear equations whennunknown quantities of Michigan Department of Mechanical EngineeringJanuary 10, 2005 Linear Systems Example: Circuit AnalysisGiven thatR1= 2 ,R2= 4 ,R3= 1 ,E1= 6V,E2= 9V, and usingequations from Loop 1, Loop 2, and Node A we find:6 2I1 I3= 09 + 4I2 I3= 0 I1+I2+I3= 0 2 0 10 4 1 1 1 1 I1I2I3 = 6 90 Ax=bLoop 1 Loop 2 Node ANode BR1R3R2I1I3I212 University of Michigan Department of Mechanical EngineeringJanuary 10, 2005 Linear Systems Example: Circuit AnalysisMatlab ImplementationUniversity of Michigan Department of Mechanical EngineeringJanuary 10, 2005 Nonlinear Systems Example.

8 Turbine Blade Analysis Turbine blades are components ofgas-turbine engines (used for aircraftand electricity generation) Subject to high temperatures, highinertial forces, and high drag forces. Commonlyconstructedofmonocrystalline alloys such asInconel. Structural and thermal analysesmust be performed simultaneously(coupled non-linear equations).University of Michigan Department of Mechanical EngineeringJanuary 10, 2005 Nonlinear Systems Example: Turbine Blade AnalysisMethods apply to arbitrary non-linearequations (black-box functions)T(x) =f1(L)L=f2(T(x))wtL0xvg, TgfacThermalAnalysisStructuralAnalysisT( x) (temperatureprofile)L (dilated length)University of Michigan Department of Mechanical EngineeringJanuary 10, 2005 OptimizationFind the values of the input variables to a function such that the function is minimized(or maximized), possibly subject to Null Form:minxf(x)subject tog(x) 0h(x) =0 Applications .

9 Engineering Design Regression Equilibrium in NatureUniversity of Michigan Department of Mechanical EngineeringJanuary 10, 2005 Optimization- Engineering DesignMaximize performance criteria subject to failure constraints: Minimize bicycle frame weight subject to structural failure constraints by varying frameshape and cost subject to performance and failure constraints. Minimize vehicle cost subject to acceleration, top speed, handling, comfort, and safetyconstraints by varying vehicle design of Michigan Department of Mechanical EngineeringJanuary 10, 2005 Optimization- RegressionRegression: technique for approximating an unknown response surface (function).

10 Sample several points experimentally Fit an approximating function to the data points, minimizing the error between theapproximating function and the actual data for best fit:SSE=n i=1(fi fi(p))2 minpSSE(p)University of Michigan Department of Mechanical EngineeringJanuary 10, 2005 Optimization- Equilibrium in Nature Gravitational Potential Energy Objects seek position of minimum gravitational potential energy:V=mgh Bubbles Energy associated with surface area. Bubbles seek to minimize surface area spherical shape. Many small bubble coalesce to form fewer large bubbles. Atomic Spacing Atoms seek positions that minimize elastic potential energy.