Archives
Industrial Engineering Blank Course Plan Topics Reading Assigned Problems Assigned Problems Reviewed
Differential Equations for Engineers: the Essentials – Course Plan Template Date Class # # of slides Topics Reading Assigned Problems Assigned Problems Reviewed 1 2 13 14A/B 15 16 17 18 19 20 21 22 23 24 25 26 27 […]
Industrial Engineering Class Notes 1 Foundations 5 First Order Ordinary Differential Equations
Foundations (5) In first order ordinary differential equations, we are answering questions of the form: If is a given function of time and of the integral , what is the function ? That is, we are solving problems of the […]
Industrial Engineering Class Notes 14A Equation Equation The Solution The Laplace Transform
Laplace Transform Example #2 (3) Substituting Equations 10 through 13 into Equation 9 we have 134 154 )( 154)134( 2 2 ++ + = +=++ ss s sY sYss Equation 15 Rearranging and solving Equation 14: Equation 15 is the […]
Industrial Engineering Class Notes 19 Suspension System Model Equations 4 Before Solve
Vehicle Suspension System Model Definitions Position of vehicle centerline at right wheel relative to equilibrium position = 1 y = 2 y Position of vehicle centerline at left wheel relative to equilibrium position = cg y Position of vehicle center […]
Industrial Engineering Class Notes 19 Electrical circuit example with repeated eigenvalues
Differential Equations for Engineers: the Essentials Class 19 notes Class 19 Agenda Review of Homework Assignments 17 and 18 Lectures: Systems of first order ODEs: Eigenvalues and eigenvectors: Electrical circuit example Repeated eigenvalues Coordinate systems: Road vehicle suspension system example […]
Industrial Engineering Class Notes 23 Let the temperature at the ends of the bar
Differential Equations for Engineers: the Essentials Class 23 notes Agenda: Class 23 Review Homework Assignment 21 Lectures: Homework Assignment 23 Short Quiz #3 Partial Differential Equations: Heat Equation: Boundary Value Problem Example Partial Differential Equations Heat Equation: Boundary Value Example […]
Industrial Engineering Class Notes 24 Wave Equation Solution 5 Which Recognize Plane
Differential Equations for Engineers: the Essentials Class 24 notes Agenda: Class 24 Review Short Quiz #3 Review Homework Assignment 22 Lectures: Partial Differential Equations: Wave Equation Initial value problem example Boundary value problem example Partial Differential Equations Wave Equation Wave […]
Industrial Engineering Class Notes 26 The Solution The Beam Equation Derived The
Spherical Potential Equation Example (10) We can find coefficients satisfying Equation 10 because Legendre polynomials form yet another orthogonal set Equivalently n c )12/(2sin)(cos)(cos 0sin)(cos)(cos 1 1 0 += = −ndPP dPP mn mn nm […]
Industrial Engineering Class Notes 3 Noth rust Segment Going 4 After Some Algebra
Some Fundamental Questions About a Sounding Rocket How high does it go? What maximum speed does it achieve going up? What maximum speed does it achieve coming down? Deriving the Equations for Sounding Rocket Motion: Basic Approach Break the rocket […]
Industrial Engineering Class Notes 4 Equation Noth rust Segment Coming Down 6 Recalling
Differential Equations for Engineers: the Essentials Class 4 notes Agenda: Class 4 Review homework assignment 2 Lectures: Homework assignment 4 Short Quiz Separable first order ODEs: Example: Sounding rocket –phase 3 Possibility of Transforming Nonlinear to Linear Supplement: Differences Between […]
Industrial Engineering Class Notes 5 Proceeding This Way Find That
Preliminaries: Lipschitz Condition (3) Note that, in the previous example, the Lipschitz constant K was given by the point in R where This is true more generally: max ),( = y ytf If is continuous in R and […]
Industrial Engineering Class Notes 6A Density Rocket Exit Gases Subsonic Drag Coefficient
Example Engineering Application of ODEs A numerical integration project: Sounding Rocket Performance – Phase 1 Numerical Integration of ODEs Using the computer to calculate a solution time-step by time-step Theory will be examined later in the course. The engineering professional […]
Industrial Engineering Class Notes 6A Unstable Equilibrium Points Nearby Points Are Repelled
Differential Equations for Engineers: the Essentials Class 6 notes Agenda: Class 6 Reviewing Homework Assignment 4 Lectures: Qualitative Analysis Stability Revisited Sounding Rocket Performance Computing Project Phase 1 Homework Assignment 6 Qualitative Analysis Qualitative Analysis: Example Consider the ODE We […]
Industrial Engineering Class Notes 6B Homework Assignment 4 Print Out And Hand
Definition of Terms = = = V h altitude (ft) speed (ft/sec) flight path angle (radians) = = = = = = L D a C C S g m mass (slugs) acceleration of gravity (ft/sec^2) air density […]
Industrial Engineering Class Notes 6B Solution Ode Depends Initial Condition Solution
Differential Equations for Engineers: the Essentials Class 6 notes Agenda: Class 6 Reviewing Homework Assignment 4 Lectures: Qualitative Analysis Stability Revisited Aircraft Performance Computing Project – Phase 1 Homework Assignment 6 Qualitative Analysis Qualitative Analysis: Example Consider the ODE We […]
Industrial Engineering Class Notes 7 Equation Verify That Equation The Solution 2vc
LRC Circuit: Roots of Characteristic Equation Have finished Case 1: two real roots Now consider Case 2: repeated real roots 0 1 2 2 =− LCL R 0 1 2 2 − […]
Industrial Engineering Class Notes 7 Equation Which Called The Characteristic Equation For
Differential Equations for Engineers: the Essentials Class 7 notes Agenda: Class 7 Review Homework Assignment 5 Lecture: Second order linear time-invariant homogeneous ODEs: LRC circuit Characteristic equation In-class example problems Homework Assignment 7 Second Order Linear Time-Invariant Homogeneous ODEs Example: […]
Industrial Engineering Class Notes 18 Differential Equations For Engineers The Essentials Class
Differential Equations for Engineers: the Essentials Class 18 notes Agenda: Class 18 Review of Short Quiz #2 Review of Homework Assignments 14 and 16 Lectures: Complex Eigenvalues and Eigenvectors (1) Electrical circuit (2) Aircraft longitudinal dynamics Homework Assignment 21 Objective […]
Industrial Engineering Class Notes 14A More Equations For Forces Speed Sound Etc
Homework Assignment 14: Example Engineering Application of ODEs Sounding Rocket Performance Phase 2 Homework Assignment 14 Read: Refresh your memory by skimming (the guide to the instructor’s or student’s choice of numerical integration computer programs.) Work : We will use […]
Industrial Engineering Class Notes 14A Proof Apply Theorem Sequentially N1 More
Differential Equations for Engineers: the Essentials Class 14 notes Agenda: Class 14 Working on a qualitative aspect of ODEs: system stability Lectures: Introduction to Laplace transforms Laplace transforms for solving initial value problems Computing Project Phase 2 Questions on Determining […]
Industrial Engineering Class Notes 14B Equations Program 3 Graph Air Density Reference
Homework Assignment 14 Review: Refresh your memory by skimming (the guide to the instructor’s or student’s choice of numerical integration computer programs.) Work : We will use the numerical integration computer program to explore aircraft flight performance. We will add […]
Industrial Engineering Class Notes 15 Equation Response Periodic Input 4 Theorem The
Response to (non-repeated) square wave input (10) 020 T and )2()(2)()( 0200 TtyuTtyutyty TT −+−−= Eq’n 13 where, again, )sin)/(cos1( )( 1 )( 22 0tetety tt −− −− + = Eq’n 14 Summarizing, Equations 13 and 14 […]
Industrial Engineering Class Notes 16 Assignment 16 Prepare for short quiz
Systems of First Order ODEs Linear Systems in State Space Format Linearization better defined Let be the known solution of a nonlinear nth order system defined by Let be the unknown solution of a slightly different nth order nonlinear ODE […]
Industrial Engineering Class Notes 18 Horizontal Velocity Vector Fractional
Electrical Circuit Equations (13) Since must be real we see that the coefficients must be complex. If we let where are real-valued, then we have and all terms in Equation 17 are real. Therefore we see that the two real-valued […]
Industrial Engineering Class Notes 19 Repeated Eigenvalues 5 The Associated Fundamental Solutions
Systems of First Order ODEs Eigenvalues and eigenvectors: Repeated eigenvalues Repeated Eigenvalues In the ODE if the nby nmatrix is symmetric then it will always have nindependent eigenvectors, no matter the multiplicity of its eigenvalues. The fundamental solutions to the […]
Industrial Engineering Class Notes 21 Because Invertible Are Linearly Independent 1
Differential Equations for Engineers: the Essentials Class 21 notes Agenda: Class 21 Review Test #2 Review Homework Assignment 19 Lectures: Coordinate systems State transition matrix Nonhomogeneous equations: matrix kernel method Homework Assignment 21 Systems of First Order ODEs Coordinate Systems […]
Industrial Engineering Class Notes 21 Step Substituting These Eigenvalues Into Arrive 1
Linear Nonhomogeneous Systems of First Order ODEs (5) ),(),(),(),(),( 000 1 0 ttttttt == − Now, from the properties of the state transition matrix, Equations 7 and 8 duBtxtttx t t += )()(),(),()( 0 0 0 and so […]
Industrial Engineering Class Notes 22 Initial Value Problem Example 3 The Solution
If ttu =)( Then try )( dtctx += bA, given tbxAx += )()(0 )( cdAtbcA tbdtcAc cx −++= ++= = bcA −= Note: This method works only if is invertible A Solve for c cdA = Solve for […]
Industrial Engineering Class Notes 22 we can immediately write the RHSdown
Differential Equations for Engineers: the Essentials Class 22 notes Agenda: Class 22 Lectures: Systems of first order ODEs: nonhomogeneous equations – Homework Assignment 22 matrix Laplace transform method matrix method of undetermined coefficients Transition to partial differential equations Systems of […]
Industrial Engineering Class Notes 23 We state without proof that the steady state
Heat Equation: Boundary Value Example Consider a rod of length Lwhose ends are controlled to temperature 0 and to which is applied a point heat source at x = L/4, raising the q− 0=x Lx = 4/Lx = 4/3Lx = […]
Industrial Engineering Class Notes 24 It will turn out that this constant must
Cylindrical Vibrations (4) Substituting Equation 5 into Equation 2 we have Dividing through by leaves The right side of Equation 6 is a function of only and the left side a function of and only, hence both sides must equal […]
Industrial Engineering Class Notes 25 Modified Bessel Functions The Modified
Modified Bessel Functions The modified Bessel’s Equation of order mis Its solutions are the modified Bessel function of the first kind of order m which has finite and at the origin, and the modified Bessel function of the second kind […]
Industrial Engineering Class Notes 25 Power Series Solutions Part 7 Earlier The
Differential Equations for Engineers: the Essentials Class 25 notes Agenda: Class 25 Review Homework Assignment 23 Lectures: Partial Differential Equations: Power Series Solutions –Part 1 Bessel Functions Power Series Solutions Bessel function of first kind of order m Some general […]
Industrial Engineering Class Notes 26 Equation The Right Side Equation Function Only
Differential Equations for Engineers: the Essentials Class 26 notes Agenda: Class 26 Lectures: Partial Differential Equations: Potential Equation –Introduction Potential Equation –Boundary value problem example Beam Equation Partial Differential Equations Potential Equation: Introduction Potential Equation The potential equation (also called […]
Industrial Engineering Example Course Plan Midterm Exam Review Midterm Laplace Transforms Intro
Differential Equations for Engineers: the Essentials – Example Course Plan Date Class # # of slides Topics Reading Assigned Problems Assigned Problems Reviewed 1 36 Administrative matters, course objectives, importance of DEs to engineers, review of math foundations, classes of […]
Industrial Engineering Overview for Instructors Notes One File For Each Class The
Overview for Instructors This document presents a brief overview for instructors on the Powerpoint visuals package that accompanies the textbook Differential Equations for Engineers: the Essentials, by David V. Kalbaugh, published by CRC Press in 2017. The visuals, also called […]
Industrial Engineering Supplement to Class 14A We will use the numerical integration
Differential Equations for Engineers: the Essentials Supplement to Class 14A Homework Assignment 14: Example Engineering Application of ODEs Sounding Rocket Performance Phase 2 Homework Assignment 14 Read: Refresh your memory by skimming (the guide to the instructor’s or student’s choice […]
Industrial Engineering Supplement to Class 14B air density to the computer program we developed
Differential Equations for Engineers: the Essentials Supplement to Class 14B Homework Assignment 14: Example Engineering Application of ODEs Aircraft Flight Performance, Phase 2 Homework Assignment 14 Review: Refresh your memory by skimming (the guide to the instructor’s or student’s choice […]
Industrial Engineering Supplement to Class 14B Run Time Interval 600 Sec Homework Assignment
Equations to program (6): Thrust M )/( 00 aa TT = M M M 2.1 2.195.0 95.00 0.1 44.02.1 7.0 −M Equations to program (7): angle of attack (autopilot) −= −−−= D D dt […]
Industrial Engineering Supplement to Class 18 Assuming The Equilibrium Angle Attack Small Td
Differential Equations for Engineers: Math 241 Supplement to Class 18 Systems of First Order ODEs Definition of Terms and Derivation of Equations for Aircraft Longitudinal Dynamics Example Aircraft Dynamics: Equilibrium Straight and level flight equilibrium: Lift Drag Lift = Weight […]
Industrial Engineering Supplement to Class 18 Most of the pitch moment comes from the lift
Appendix: Deriving Linearized Equations of Motion for Perturbed Flight (6) Derived from Newton’s Second Law: the change of angular momentum equals torque in the pitch plane: Most of the pitch moment comes from the lift force as described in Equation […]
Industrial Engineering Supplement to Class 21 matrix A is diagonalized and the variables
Differential Equations for Engineers: the Essentials Supplement to Class 21 Notes Systems of First Order ODEs Coordinate Systems and Cascaded Systems Example Coupled and Uncoupled ODEs “Coupled” ODEs (especially of higher order) are relatively complicated to solve: […]
Industrial Engineering Supplement to Class 25 Substituting Equations Through Into Equation Arrive
Differential Equations for Engineers: the Essentials Supplement to Class 25 notes Partial Differential Equations Power Series Solutions: Bessel Function of Second Kind of Order Zero Bessel Functions: Second Solution The second fundamental solution to the Bessel equation of order m, […]
Industrial Engineering Supplement to Class 4 What The Resistor Slightly Nonlinear Instead Sint
Differential Equations for Engineers: the Essentials Supplement to Class 4 Notes Contents of Supplement Differences Between Linear and Nonlinear ODEs in Their Input / Output Response Example: Nonlinear Circuit Differences Between Linear and Nonlinear ODEs in Their Input / Output […]
Industrial Engineering Supplement to Class 5 Differential Equations For Engineers The Essentials Supplement
Differential Equations for Engineers: the Essentials Supplement to Class 6: Proof of Existence and Uniqueness Theorem Appendix The following pages contain a detailed proof of the existence and uniqueness theorem, following Coddington, An Introduction to Ordinary Differential Equations, Dover Publications […]
Industrial Engineering Supplement to Class 5 smaller term by term than the convergent
Appendix: Theorem Proof Part 4: the sequence converges Recall we are writing Consider each term in the series. We have already seen (Equation 3) that )(t n − =+−+= 1 0 10 )( n k kkn 001 […]