The end conditions on the beam are modelled by assuming the following boundary conditions on the function u: u(0, t) = u x (0, t) = 0;
|
|
- Theodora Snow
- 5 years ago
- Views:
Transcription
1 M344 - ADVANCED ENGINEERING MATHEMATICS Lecture 1: Motion of a Deflected Beam The deflection of a cantilever beam that is fixed at one end and free to move at the other can be shown to satisfy a fourth-order partial differential equation of the form u tt = c u xxxx, where u(x, t) is the deflection at the point x along the length of the beam, at time t. The positive constant c is EI, where A is the cross-sectional area of ρa the beam, ρ is its density, E is the modulus of elasticity, and I is the moment of inertia. u 1 x 1 The end conditions on the beam are modelled by assuming the following boundary conditions on the function u: u(, t) = u x (, t) = ; u xx (L, t) = u xxx (L, t) =, for all t >. Using the method of Separation of Variables, let u(x, t) = X(x)T (t). Then T XT c XT = X T c c XT X β4 for some constant β 4. This results in the two ordinary differential equations c T = X X β 4 X = and T + c β 4 T =. The solution of the second-order equation in T will have the form T n (t) = a n cos(cβ nt) + b n sin(cβ nt), where the constants β 4 n are the eigenvalues of the equation in X. 1
2 Solution of X β 4 X =, X() = X () = X (L) = X (L) = The characteristic polynomial for this fourth-order boundary-value problem is r 4 β 4 =, and it has the four roots r = ±β, ±βi; therefore, the general solution of the differential equation can be written as X(x) = A cos(βx) + B sin(βx) + C cosh(βx) + D sinh(βx), and differentiating with respect to x, X (x) = βa sin(βx) + βb cos(βx) + βc sinh(βx) + βd cosh(βx). Using the boundary conditions at x =, X() = A + C = C = A; and X () = βb + βd = β(b + D) = D = B. We can now write X(x) = A(cos(βx) cosh(βx)) + B(sin(βx) sinh(βx)) X (x) = βa( sin(βx) sinh(βx)) + βb(cos(βx) cosh(βx)) X (x) = β A( cos(βx) cosh(βx)) + β B( sin(βx) sinh(βx)) X (x) = β 3 A(sin(βx) sinh(βx)) + β 3 B( cos(βx) cosh(βx)) The final two boundary conditions, at x = L, give X (L) = β A( cos(βl) cosh(βl)) + β B( sin(βl) sinh(βl)) = X (L) = β 3 A(sin(βL) sinh(βl)) + β 3 B( cos(βl) cosh(βl)) = and these two equations can be written in matrix form as ( cos(βl) + cosh(βl) sin(βl) + sinh(βl) sin(βl) + sinh(βl) cos(βl) + cosh(βl) ) ( A B ) = ( ). (1) This system of linear equations in A and B will have a unique solution A = B = unless the determinant of the matrix is zero. This means that there will be non-zero solutions if, and only if,
3 (cos(βl)+cosh(βl))(cos(βl)+cosh(βl))+(sin(βl)+sinh(βl))(sin(βl) sinh(βl)) =. Multiplying out and using the two identities (sin(x)) + (cos(x)) 1 and (cosh(x)) (sinh(x)) 1, results in the equation (cos(βl)) + cosh(βl) cos(βl) + (cosh(βl)) + (sin(βl)) (sinh(βl)) = + cosh(βl) cos(βl) =. (Checkthis!) This means that the eigenvalues β n must satisfy the condition cosh(β n L) cos(β n L) = 1. There will be an infinite sequence of positive solutions Lβ 1 < Lβ <, tending to, which can be found by using the following MAPLE instructions: for n from 1 to 15 do fsolve(cosh(x)*cos(x)=-1., x=(n-1)*pi..n*pi); od; For n equal 1,, and 3 the solutions are Lβ , Lβ , Lβ ; and as n, the solution Lβ n approaches the value (n 1) π. The corresponding eigenfunctions X n (x) are X n (x) = A n (cos(β n x) cosh(β n x)) + B n (sin(β n x) sinh(β n x)), and Equation (1) implies that B n and A n are related by ( ) cos(βn L) + cosh(β n L) B n = A n. sin(β n L) + sinh(β n L) Letting A n = 1, we have a set of eigenfunctions of the form ( ) cos(βn L) + cosh(β n L) φ n (x) = (cos(β n x) cosh(β n x)) (sin(β n x) sinh(β n x)). sin(β n L) + sinh(β n L) () The general solution of the partial differential equation can now be written in the form u(x, t) = φ n (x)t n (t) = φ n (x)(a n cos(cβnt) + b n sin(cβnt)). Two initial conditions on u(x, t) are needed to determine the coefficients a n and b n. These conditions will be given by specifying the initial deflection 3
4 u(x, ) = f(x) and the initial velocity u t (x, ) = g(x) at each point x along the beam. In order to use these functions to obtain the coefficients a n and b n, it must first be shown that the functions φ n (x) form an orthogonal set on the interval [, L]. Theorem 1 The set of functions {φ n (x)} 1 defined in equation () is an orthogonal family on [, L]; that is φ m(x)φ n (x)dx = if m n. Proof: We first use integration by parts to show that if u and v are any sufficiently differentiable functions on x L, then u vdx = v(x)u (x) L = v(x)u (x) L ( v (x)u (x) L u v dx ) u v dx and integrating by parts two more times = v(x)u (x) L v (x)u (x) L + v (x)u (x) L v (x)u(x) L + If the right-hand term is subtracted from both sides, u vdx uv dx, uv dx = (v(x)u (x) v (x)u (x)+v (x)u (x) v (x)u(x)) L. (3) Now assume u = φ m and v = φ n where φ m and φ n are eigenfunctions of X β 4 X = for two different eigenvalues β n and β m. Then φ m = βmφ 4 m and φ n = βnφ 4 n, and equation (3) shows that φ m φ n dx φ m φ n dx = = (β 4 m β 4 n) β 4 mφ m φ n dx φ m φ n dx = φ m β 4 nφ n dx (φ n (x)φ m(x) φ n(x)φ m(x) + φ n(x)φ m(x) φ n (x)φ m (x)) L When the right-hand side of the equation is evaluated at the two endpoints x = and x = L, the boundary conditions on φ m and φ n can be seen to imply that every product has one term equal to zero; therefore, (βm 4 βn) 4 L φ mφ n dx =, and since β m β n, this means that φ mφ n dx = as required. 4
5 The values of the coefficients a n and b n can now be found by using the initial conditions. Since u(x, ) f(x) = φ n (x)(a n cos() + b n sin()) = a n φ n (x), the a n are the coefficients in an orthogonal series for f(x); therefore, a n = f(x)φ n(x)dx (φ n(x)) dx. Similarly, u t (x, t) = φ n (x)cβn( a n sin(cβnt) + b n cos(cβnt)) and u t (x, ) g(x) = cβnb n φ n (x) b n = 1 g(x)φ n(x)dx cβn (φ n(x)) dx. Example 1 Assume that c = 1, and the initial displacement function is f(x) =.5x on x 1m. The initial velocity is g(x). Find the displacement u(x, t) and plot the position of the beam at time increments of 1 1 of the period of the function T 1(t)..6 u(x,t) Position at t=k*period/1 k=6.4 k=5...4 k=4 k= x k= k=1 k= 5
6 In the figure above, the position of the beam at time k period is labelled 1 by k =, 1,, 6. The period of T 1 (t) = a 1 cos(cβ1) + b 1 sin(cβ1), in this particular example, is π cβ 1. With c = 1 and Lβ , the period is π and the curves labelled k =, 1,, 6 correspond to times (.18751) t =, 14.9, 9.8,, In the figure on page 1, the initial beam position was set equal to the first eigenfunction φ 1 (x). In this case, a 1 is the only non-zero coefficient, and the beam will oscillate precisely with period π. For the straight-line initial cβ1 position in the example, the oscillation is no longer exactly periodic because the periods of the functions T, T 3, are not constant multiples of the period of T 1, and none of the coefficients a n are zero for this straight-line initial position function. Exercises: 1. * Determine the second eigenfunction φ (x) and plot a graph for x 1 (assume that the length of the beam is 1).. * Redo Example 1 and find u(x, t), assuming that the initial displacement function is f(x) = φ (x) and the initial velocity function g(x). Find the period of this function u(x, t) and plot a graph showing the position of the beam at several different times during its period. 6
Unit - 7 Vibration of Continuous System
Unit - 7 Vibration of Continuous System Dr. T. Jagadish. Professor for Post Graduation, Department of Mechanical Engineering, Bangalore Institute of Technology, Bangalore Continuous systems are tore which
More informationSAMPLE FINAL EXAM SOLUTIONS
LAST (family) NAME: FIRST (given) NAME: ID # : MATHEMATICS 3FF3 McMaster University Final Examination Day Class Duration of Examination: 3 hours Dr. J.-P. Gabardo THIS EXAMINATION PAPER INCLUDES 22 PAGES
More informationMath 220a - Fall 2002 Homework 6 Solutions
Math a - Fall Homework 6 Solutions. Use the method of reflection to solve the initial-boundary value problem on the interval < x < l, u tt c u xx = < x < l u(x, = < x < l u t (x, = x < x < l u(, t = =
More information2.4 Eigenvalue problems
2.4 Eigenvalue problems Associated with the boundary problem (2.1) (Poisson eq.), we call λ an eigenvalue if Lu = λu (2.36) for a nonzero function u C 2 0 ((0, 1)). Recall Lu = u. Then u is called an eigenfunction.
More informationWave Equation With Homogeneous Boundary Conditions
Wave Equation With Homogeneous Boundary Conditions MATH 467 Partial Differential Equations J. Robert Buchanan Department of Mathematics Fall 018 Objectives In this lesson we will learn: how to solve the
More informationu tt = a 2 u xx u tt = a 2 (u xx + u yy )
10.7 The wave equation 10.7 The wave equation O. Costin: 10.7 1 This equation describes the propagation of waves through a medium: in one dimension, such as a vibrating string u tt = a 2 u xx 1 This equation
More informationarxiv: v1 [math.gm] 16 Feb 2018
Differential Operator Method of Finding A Particular Solution to An Ordinary Nonhomogeneous Linear Differential Equation with Constant Coefficients Wenfeng Chen arxiv:802.09343v math.gm 6 Feb 208 Department
More informationFLUID STRUCTURE INTERACTIONS FLOW INDUCED VIBRATIONS OF STRUCTURES
FLUID STRUCTURE INTERACTIONS FLOW INDUCED VIBRATIONS OF STRUCTURES PREAMBLE There are two types of vibrations: resonance and instability. Resonance occurs when a structure is excited at a natural frequency.
More informationStrauss PDEs 2e: Section Exercise 4 Page 1 of 6
Strauss PDEs 2e: Section 5.3 - Exercise 4 Page of 6 Exercise 4 Consider the problem u t = ku xx for < x < l, with the boundary conditions u(, t) = U, u x (l, t) =, and the initial condition u(x, ) =, where
More informationMath 220B - Summer 2003 Homework 1 Solutions
Math 0B - Summer 003 Homework Soutions Consider the eigenvaue probem { X = λx 0 < x < X satisfies symmetric BCs x = 0, Suppose f(x)f (x) x=b x=a 0 for a rea-vaued functions f(x) which satisfy the boundary
More informationMa 221 Final Exam Solutions 5/14/13
Ma 221 Final Exam Solutions 5/14/13 1. Solve (a) (8 pts) Solution: The equation is separable. dy dx exy y 1 y0 0 y 1e y dy e x dx y 1e y dy e x dx ye y e y dy e x dx ye y e y e y e x c The last step comes
More informationFourier and Partial Differential Equations
Chapter 5 Fourier and Partial Differential Equations 5.1 Fourier MATH 294 SPRING 1982 FINAL # 5 5.1.1 Consider the function 2x, 0 x 1. a) Sketch the odd extension of this function on 1 x 1. b) Expand the
More information6 Non-homogeneous Heat Problems
6 Non-homogeneous Heat Problems Up to this point all the problems we have considered for the heat or wave equation we what we call homogeneous problems. This means that for an interval < x < l the problems
More informationThe Gram-Schmidt Process 1
The Gram-Schmidt Process In this section all vector spaces will be subspaces of some R m. Definition.. Let S = {v...v n } R m. The set S is said to be orthogonal if v v j = whenever i j. If in addition
More informationLECTURE 19: SEPARATION OF VARIABLES, HEAT CONDUCTION IN A ROD
ECTURE 19: SEPARATION OF VARIABES, HEAT CONDUCTION IN A ROD The idea of separation of variables is simple: in order to solve a partial differential equation in u(x, t), we ask, is it possible to find a
More informationMath 201 Assignment #11
Math 21 Assignment #11 Problem 1 (1.5 2) Find a formal solution to the given initial-boundary value problem. = 2 u x, < x < π, t > 2 u(, t) = u(π, t) =, t > u(x, ) = x 2, < x < π Problem 2 (1.5 5) Find
More informationFINAL EXAM, MATH 353 SUMMER I 2015
FINAL EXAM, MATH 353 SUMMER I 25 9:am-2:pm, Thursday, June 25 I have neither given nor received any unauthorized help on this exam and I have conducted myself within the guidelines of the Duke Community
More information1 Wave Equation on Finite Interval
1 Wave Equation on Finite Interval 1.1 Wave Equation Dirichlet Boundary Conditions u tt (x, t) = c u xx (x, t), < x < l, t > (1.1) u(, t) =, u(l, t) = u(x, ) = f(x) u t (x, ) = g(x) First we present the
More informationAn Introduction to Partial Differential Equations
An Introduction to Partial Differential Equations Ryan C. Trinity University Partial Differential Equations Lecture 1 Ordinary differential equations (ODEs) These are equations of the form where: F(x,y,y,y,y,...)
More informationMethod of Separation of Variables
MODUE 5: HEAT EQUATION 11 ecture 3 Method of Separation of Variables Separation of variables is one of the oldest technique for solving initial-boundary value problems (IBVP) and applies to problems, where
More informationName: Math Homework Set # 5. March 12, 2010
Name: Math 4567. Homework Set # 5 March 12, 2010 Chapter 3 (page 79, problems 1,2), (page 82, problems 1,2), (page 86, problems 2,3), Chapter 4 (page 93, problems 2,3), (page 98, problems 1,2), (page 102,
More informationSturm-Liouville Theory
More on Ryan C. Trinity University Partial Differential Equations April 19, 2012 Recall: A Sturm-Liouville (S-L) problem consists of A Sturm-Liouville equation on an interval: (p(x)y ) + (q(x) + λr(x))y
More informationCHAPTER 10 NOTES DAVID SEAL
CHAPTER 1 NOTES DAVID SEA 1. Two Point Boundary Value Problems All of the problems listed in 14 2 ask you to find eigenfunctions for the problem (1 y + λy = with some prescribed data on the boundary. To
More informationMA 262, Fall 2017, Final Version 01(Green)
INSTRUCTIONS MA 262, Fall 2017, Final Version 01(Green) (1) Switch off your phone upon entering the exam room. (2) Do not open the exam booklet until you are instructed to do so. (3) Before you open the
More informationMATH 251 Final Examination December 16, 2015 FORM A. Name: Student Number: Section:
MATH 5 Final Examination December 6, 5 FORM A Name: Student Number: Section: This exam has 7 questions for a total of 5 points. In order to obtain full credit for partial credit problems, all work must
More informationTHE WAVE EQUATION. F = T (x, t) j + T (x + x, t) j = T (sin(θ(x, t)) + sin(θ(x + x, t)))
THE WAVE EQUATION The aim is to derive a mathematical model that describes small vibrations of a tightly stretched flexible string for the one-dimensional case, or of a tightly stretched membrane for the
More informationMATH 251 Final Examination May 3, 2017 FORM A. Name: Student Number: Section:
MATH 5 Final Examination May 3, 07 FORM A Name: Student Number: Section: This exam has 6 questions for a total of 50 points. In order to obtain full credit for partial credit problems, all work must be
More informationMath 5587 Midterm II Solutions
Math 5587 Midterm II Solutions Prof. Jeff Calder November 3, 2016 Name: Instructions: 1. I recommend looking over the problems first and starting with those you feel most comfortable with. 2. Unless otherwise
More informationVIBRATION ANALYSIS OF WINGS WITH TIP-MOUNTED ENGINE
VIBRATION ANALYSIS OF WINGS WITH TIP-MOUNTED ENGINE by Sabrina Chowdhury Undergraduate Student, Widener University, Chester, Pennsylvania. AIAA Student Member Abstract Vibration analysis was being conducted
More informationLEGENDRE POLYNOMIALS AND APPLICATIONS. We construct Legendre polynomials and apply them to solve Dirichlet problems in spherical coordinates.
LEGENDRE POLYNOMIALS AND APPLICATIONS We construct Legendre polynomials and apply them to solve Dirichlet problems in spherical coordinates.. Legendre equation: series solutions The Legendre equation is
More informationMa 530 Power Series II
Ma 530 Power Series II Please note that there is material on power series at Visual Calculus. Some of this material was used as part of the presentation of the topics that follow. Operations on Power Series
More informationCIV-E1060 Engineering Computation and Simulation Examination, December 12, 2017 / Niiranen
CIV-E16 Engineering Computation and Simulation Examination, December 12, 217 / Niiranen This examination consists of 3 problems rated by the standard scale 1...6. Problem 1 Let us consider a long and tall
More informationDegree Master of Science in Mathematical Modelling and Scientific Computing Mathematical Methods I Thursday, 12th January 2012, 9:30 a.m.- 11:30 a.m.
Degree Master of Science in Mathematical Modelling and Scientific Computing Mathematical Methods I Thursday, 12th January 2012, 9:30 a.m.- 11:30 a.m. Candidates should submit answers to a maximum of four
More information8.7 Taylor s Inequality Math 2300 Section 005 Calculus II. f(x) = ln(1 + x) f(0) = 0
8.7 Taylor s Inequality Math 00 Section 005 Calculus II Name: ANSWER KEY Taylor s Inequality: If f (n+) is continuous and f (n+) < M between the center a and some point x, then f(x) T n (x) M x a n+ (n
More informationEssential Ordinary Differential Equations
MODULE 1: MATHEMATICAL PRELIMINARIES 10 Lecture 2 Essential Ordinary Differential Equations In this lecture, we recall some methods of solving first-order IVP in ODE (separable and linear) and homogeneous
More informationIntroduction to the Wave Equation
Introduction to the Ryan C. Trinity University Partial Differential Equations ecture 4 Modeling the Motion of an Ideal Elastic String Idealizing Assumptions: The only force acting on the string is (constant
More informationBoundary-value Problems in Rectangular Coordinates
Boundary-value Problems in Rectangular Coordinates 2009 Outline Separation of Variables: Heat Equation on a Slab Separation of Variables: Vibrating String Separation of Variables: Laplace Equation Review
More informationPhysics 250 Green s functions for ordinary differential equations
Physics 25 Green s functions for ordinary differential equations Peter Young November 25, 27 Homogeneous Equations We have already discussed second order linear homogeneous differential equations, which
More informationENGI 9420 Lecture Notes 8 - PDEs Page 8.01
ENGI 940 Lecture Notes 8 - PDEs Page 8.01 8. Partial Differential Equations Partial differential equations (PDEs) are equations involving functions of more than one variable and their partial derivatives
More informationConsider an ideal pendulum as shown below. l θ is the angular acceleration θ is the angular velocity
1 Second Order Ordinary Differential Equations 1.1 The harmonic oscillator Consider an ideal pendulum as shown below. θ l Fr mg l θ is the angular acceleration θ is the angular velocity A point mass m
More informationAP Calculus (BC) Chapter 9 Test No Calculator Section Name: Date: Period:
WORKSHEET: Series, Taylor Series AP Calculus (BC) Chapter 9 Test No Calculator Section Name: Date: Period: 1 Part I. Multiple-Choice Questions (5 points each; please circle the correct answer.) 1. The
More informationPractice Exercises on Differential Equations
Practice Exercises on Differential Equations What follows are some exerices to help with your studying for the part of the final exam on differential equations. In this regard, keep in mind that the exercises
More informationBessel s Equation. MATH 365 Ordinary Differential Equations. J. Robert Buchanan. Fall Department of Mathematics
Bessel s Equation MATH 365 Ordinary Differential Equations J. Robert Buchanan Department of Mathematics Fall 2018 Background Bessel s equation of order ν has the form where ν is a constant. x 2 y + xy
More informationand verify that it satisfies the differential equation:
MOTIVATION: Chapter One: Basic and Review Why study differential equations? Suppose we know how a certain quantity changes with time (for example, the temperature of coffee in a cup, the number of people
More informationThere are five problems. Solve four of the five problems. Each problem is worth 25 points. A sheet of convenient formulae is provided.
Preliminary Examination (Solutions): Partial Differential Equations, 1 AM - 1 PM, Jan. 18, 16, oom Discovery Learning Center (DLC) Bechtel Collaboratory. Student ID: There are five problems. Solve four
More informationReview For the Final: Problem 1 Find the general solutions of the following DEs. a) x 2 y xy y 2 = 0 solution: = 0 : homogeneous equation.
Review For the Final: Problem 1 Find the general solutions of the following DEs. a) x 2 y xy y 2 = 0 solution: y y x y2 = 0 : homogeneous equation. x2 v = y dy, y = vx, and x v + x dv dx = v + v2. dx =
More informationProblem (p.613) Determine all solutions, if any, to the boundary value problem. y + 9y = 0; 0 < x < π, y(0) = 0, y (π) = 6,
Problem 10.2.4 (p.613) Determine all solutions, if any, to the boundary value problem y + 9y = 0; 0 < x < π, y(0) = 0, y (π) = 6, by first finding a general solution to the differential equation. Solution.
More informationCIVL 7/8117 Chapter 4 - Development of Beam Equations - Part 2 1/34. Chapter 4b Development of Beam Equations. Learning Objectives
CIV 7/87 Chapter 4 - Development of Beam Equations - Part /4 Chapter 4b Development of Beam Equations earning Objectives To introduce the work-equivalence method for replacing distributed loading by a
More informationMATH 251 Final Examination December 16, 2014 FORM A. Name: Student Number: Section:
MATH 2 Final Examination December 6, 204 FORM A Name: Student Number: Section: This exam has 7 questions for a total of 0 points. In order to obtain full credit for partial credit problems, all work must
More informationReview Sol. of More Long Answer Questions
Review Sol. of More Long Answer Questions 1. Solve the integro-differential equation t y (t) e t v y(v)dv = t; y()=. (1) Solution. The key is to recognize the convolution: t e t v y(v) dv = e t y. () Now
More informationAnalysis III for D-BAUG, Fall 2017 Lecture 11
Analysis III for D-BAUG, Fall 2017 Lecture 11 Lecturer: Alex Sisto (sisto@math.ethz.ch) 1 Beams (Balken) Beams are basic mechanical systems that appear over and over in civil engineering. We consider a
More informationA NEW METHOD FOR VIBRATION MODE ANALYSIS
Proceedings of IDETC/CIE 25 25 ASME 25 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference Long Beach, California, USA, September 24-28, 25 DETC25-85138
More informationSolutions Serie 1 - preliminary exercises
D-MAVT D-MATL Prof. A. Iozzi ETH Zürich Analysis III Autumn 08 Solutions Serie - preliminary exercises. Compute the following primitive integrals using partial integration. a) cos(x) cos(x) dx cos(x) cos(x)
More informationPartial Differential Equations Separation of Variables. 1 Partial Differential Equations and Operators
PDE-SEP-HEAT-1 Partial Differential Equations Separation of Variables 1 Partial Differential Equations and Operators et C = C(R 2 ) be the collection of infinitely differentiable functions from the plane
More informationMAE 200B Homework #3 Solutions University of California, Irvine Winter 2005
Problem 1 (Haberman 5.3.2): Consider this equation: MAE 200B Homework #3 Solutions University of California, Irvine Winter 2005 a) ρ 2 u t = T 2 u u 2 0 + αu + β x2 t The term αu describes a force that
More informationMATH 251 Final Examination August 14, 2015 FORM A. Name: Student Number: Section:
MATH 251 Final Examination August 14, 2015 FORM A Name: Student Number: Section: This exam has 11 questions for a total of 150 points. Show all your work! In order to obtain full credit for partial credit
More informationLecture 32: Taylor Series and McLaurin series We saw last day that some functions are equal to a power series on part of their domain.
Lecture 32: Taylor Series and McLaurin series We saw last day that some functions are equal to a power series on part of their domain. For example f(x) = 1 1 x = 1 + x + x2 + x 3 + = ln(1 + x) = x x2 2
More informationMB4018 Differential equations
MB4018 Differential equations Part II http://www.staff.ul.ie/natalia/mb4018.html Prof. Natalia Kopteva Spring 2015 MB4018 (Spring 2015) Differential equations Part II 0 / 69 Section 1 Second-Order Linear
More informationMATH 118, LECTURES 27 & 28: TAYLOR SERIES
MATH 8, LECTURES 7 & 8: TAYLOR SERIES Taylor Series Suppose we know that the power series a n (x c) n converges on some interval c R < x < c + R to the function f(x). That is to say, we have f(x) = a 0
More informationCalculus I Announcements
Slide 1 Calculus I Announcements Read sections 3.9-3.10 Do all the homework for section 3.9 and problems 1,3,5,7 from section 3.10. The exam is in Thursday, October 22nd. The exam will cover sections 3.2-3.10,
More informationChapter 3 Second Order Linear Equations
Partial Differential Equations (Math 3303) A Ë@ Õæ Aë áöß @. X. @ 2015-2014 ú GA JË@ É Ë@ Chapter 3 Second Order Linear Equations Second-order partial differential equations for an known function u(x,
More informationMATH 412 Fourier Series and PDE- Spring 2010 SOLUTIONS to HOMEWORK 5
MATH 4 Fourier Series PDE- Spring SOLUTIONS to HOMEWORK 5 Problem (a: Solve the following Sturm-Liouville problem { (xu + λ x u = < x < e u( = u (e = (b: Show directly that the eigenfunctions are orthogonal
More informationTHE METHOD OF SEPARATION OF VARIABLES
THE METHOD OF SEPARATION OF VARIABES To solve the BVPs that we have encountered so far, we will use separation of variables on the homogeneous part of the BVP. This separation of variables leads to problems
More informationSeparation of variables in two dimensions. Overview of method: Consider linear, homogeneous equation for u(v 1, v 2 )
Separation of variables in two dimensions Overview of method: Consider linear, homogeneous equation for u(v 1, v 2 ) Separation of variables in two dimensions Overview of method: Consider linear, homogeneous
More informationMCE503: Modeling and Simulation of Mechatronic Systems. Modeling of Continuous Beams: Finite-Mode Approach
MCE503: Modeling and Simulation of Mechatronic Systems MCE503 p./23 Modeling of Continuous Beams: Finite-Mode Approach Reading: KMR Chapter 0 Cleveland State University Mechanical Engineering Hanz Richter,
More informationswapneel/207
Partial differential equations Swapneel Mahajan www.math.iitb.ac.in/ swapneel/207 1 1 Power series For a real number x 0 and a sequence (a n ) of real numbers, consider the expression a n (x x 0 ) n =
More informationExam 3 Solutions. Multiple Choice Questions
MA 4 Exam 3 Solutions Fall 26 Exam 3 Solutions Multiple Choice Questions. The average value of the function f (x) = x + sin(x) on the interval [, 2π] is: A. 2π 2 2π B. π 2π 2 + 2π 4π 2 2π 4π 2 + 2π 2.
More informationOR MSc Maths Revision Course
OR MSc Maths Revision Course Tom Byrne School of Mathematics University of Edinburgh t.m.byrne@sms.ed.ac.uk 15 September 2017 General Information Today JCMB Lecture Theatre A, 09:30-12:30 Mathematics revision
More informationOrdinary Differential Equations
Ordinary Differential Equations for Engineers and Scientists Gregg Waterman Oregon Institute of Technology c 2017 Gregg Waterman This work is licensed under the Creative Commons Attribution 4.0 International
More informationStudy # 1 11, 15, 19
Goals: 1. Recognize Taylor Series. 2. Recognize the Maclaurin Series. 3. Derive Taylor series and Maclaurin series representations for known functions. Study 11.10 # 1 11, 15, 19 f (n) (c)(x c) n f(c)+
More informationHeat Equation, Wave Equation, Properties, External Forcing
MATH348-Advanced Engineering Mathematics Homework Solutions: PDE Part II Heat Equation, Wave Equation, Properties, External Forcing Text: Chapter 1.3-1.5 ecture Notes : 14 and 15 ecture Slides: 6 Quote
More information17 Source Problems for Heat and Wave IB- VPs
17 Source Problems for Heat and Wave IB- VPs We have mostly dealt with homogeneous equations, homogeneous b.c.s in this course so far. Recall that if we have non-homogeneous b.c.s, then we want to first
More information2.2 Separable Equations
2.2 Separable Equations Definition A first-order differential equation that can be written in the form Is said to be separable. Note: the variables of a separable equation can be written as Examples Solve
More informationME 680- Spring Representation and Stability Concepts
ME 680- Spring 014 Representation and Stability Concepts 1 3. Representation and stability concepts 3.1 Continuous time systems: Consider systems of the form x F(x), x n (1) where F : U Vis a mapping U,V
More informationThe polar coordinates
The polar coordinates 1 2 3 4 Graphing in polar coordinates 5 6 7 8 Area and length in polar coordinates 9 10 11 Partial deravitive 12 13 14 15 16 17 18 19 20 Double Integral 21 22 23 24 25 26 27 Triple
More informationSection 9.7 and 9.10: Taylor Polynomials and Approximations/Taylor and Maclaurin Series
Section 9.7 and 9.10: Taylor Polynomials and Approximations/Taylor and Maclaurin Series Power Series for Functions We can create a Power Series (or polynomial series) that can approximate a function around
More informationTAYLOR AND MACLAURIN SERIES
TAYLOR AND MACLAURIN SERIES. Introduction Last time, we were able to represent a certain restricted class of functions as power series. This leads us to the question: can we represent more general functions
More informationCHAPTER 4. Introduction to the. Heat Conduction Model
A SERIES OF CLASS NOTES FOR 005-006 TO INTRODUCE LINEAR AND NONLINEAR PROBLEMS TO ENGINEERS, SCIENTISTS, AND APPLIED MATHEMATICIANS DE CLASS NOTES 4 A COLLECTION OF HANDOUTS ON PARTIAL DIFFERENTIAL EQUATIONS
More informationWave Equation Modelling Solutions
Wave Equation Modelling Solutions SEECS-NUST December 19, 2017 Wave Phenomenon Waves propagate in a pond when we gently touch water in it. Wave Phenomenon Our ear drums are very sensitive to small vibrations
More informationIntroduction to Sturm-Liouville Theory and the Theory of Generalized Fourier Series
CHAPTER 5 Introduction to Sturm-Liouville Theory and the Theory of Generalized Fourier Series We start with some introductory examples. 5.. Cauchy s equation The homogeneous Euler-Cauchy equation (Leonhard
More informationCLASSIFICATION AND PRINCIPLE OF SUPERPOSITION FOR SECOND ORDER LINEAR PDE
CLASSIFICATION AND PRINCIPLE OF SUPERPOSITION FOR SECOND ORDER LINEAR PDE 1. Linear Partial Differential Equations A partial differential equation (PDE) is an equation, for an unknown function u, that
More informationDifferential Equations
Differential Equations Problem Sheet 1 3 rd November 2011 First-Order Ordinary Differential Equations 1. Find the general solutions of the following separable differential equations. Which equations are
More informationJUST THE MATHS UNIT NUMBER DIFFERENTIATION APPLICATIONS 5 (Maclaurin s and Taylor s series) A.J.Hobson
JUST THE MATHS UNIT NUMBER.5 DIFFERENTIATION APPLICATIONS 5 (Maclaurin s and Taylor s series) by A.J.Hobson.5. Maclaurin s series.5. Standard series.5.3 Taylor s series.5.4 Exercises.5.5 Answers to exercises
More informationENGI 9420 Lecture Notes 8 - PDEs Page 8.01
ENGI 940 ecture Notes 8 - PDEs Page 8.0 8. Partial Differential Equations Partial differential equations (PDEs) are equations involving functions of more than one variable and their partial derivatives
More informationSeparation of variables
Separation of variables Idea: Transform a PDE of 2 variables into a pair of ODEs Example : Find the general solution of u x u y = 0 Step. Assume that u(x,y) = G(x)H(y), i.e., u can be written as the product
More informationMath Assignment 14
Math 2280 - Assignment 14 Dylan Zwick Spring 2014 Section 9.5-1, 3, 5, 7, 9 Section 9.6-1, 3, 5, 7, 14 Section 9.7-1, 2, 3, 4 1 Section 9.5 - Heat Conduction and Separation of Variables 9.5.1 - Solve the
More informationBoundary value problems for partial differential equations
Boundary value problems for partial differential equations Henrik Schlichtkrull March 11, 213 1 Boundary value problem 2 1 Introduction This note contains a brief introduction to linear partial differential
More information23 Elements of analytic ODE theory. Bessel s functions
23 Elements of analytic ODE theory. Bessel s functions Recall I am changing the variables) that we need to solve the so-called Bessel s equation 23. Elements of analytic ODE theory Let x 2 u + xu + x 2
More informationPartial differential equations (ACM30220)
(ACM3. A pot on a stove has a handle of length that can be modelled as a rod with diffusion constant D. The equation for the temperature in the rod is u t Du xx < x
More informationBOUNDARY-VALUE PROBLEMS IN RECTANGULAR COORDINATES
1 BOUNDARY-VALUE PROBLEMS IN RECTANGULAR COORDINATES 1.1 Separable Partial Differential Equations 1. Classical PDEs and Boundary-Value Problems 1.3 Heat Equation 1.4 Wave Equation 1.5 Laplace s Equation
More informationMATH 251 Final Examination May 4, 2015 FORM A. Name: Student Number: Section:
MATH 251 Final Examination May 4, 2015 FORM A Name: Student Number: Section: This exam has 16 questions for a total of 150 points. In order to obtain full credit for partial credit problems, all work must
More informationAppendix C. Modal Analysis of a Uniform Cantilever with a Tip Mass. C.1 Transverse Vibrations. Boundary-Value Problem
Appendix C Modal Analysis of a Uniform Cantilever with a Tip Mass C.1 Transverse Vibrations The following analytical modal analysis is given for the linear transverse vibrations of an undamped Euler Bernoulli
More informationIV Higher Order Linear ODEs
IV Higher Order Linear ODEs Boyce & DiPrima, Chapter 4 H.J.Eberl - MATH*2170 0 IV Higher Order Linear ODEs IV.1 General remarks Boyce & DiPrima, Section 4.1 H.J.Eberl - MATH*2170 1 Problem formulation
More informationFinite reductions of the two dimensional Toda chain
Journal of Nonlinear Mathematical Physics Volume 12, Supplement 2 (2005), 164 172 SIDE VI Finite reductions of the two dimensional Toda chain E V GUDKOVA Chernyshevsky str. 112, Institute of Mathematics,
More informationForced Vibration Analysis of Timoshenko Beam with Discontinuities by Means of Distributions Jiri Sobotka
21 st International Conference ENGINEERING MECHANICS 2015 Svratka, Czech Republic, May 11 14, 2015 Full Text Paper #018, pp. 45 51 Forced Vibration Analysis of Timoshenko Beam with Discontinuities by Means
More information22. Periodic Functions and Fourier Series
November 29, 2010 22-1 22. Periodic Functions and Fourier Series 1 Periodic Functions A real-valued function f(x) of a real variable is called periodic of period T > 0 if f(x + T ) = f(x) for all x R.
More informationChapter 3 Higher Order Linear ODEs
Chapter 3 Higher Order Linear ODEs Advanced Engineering Mathematics Wei-Ta Chu National Chung Cheng University wtchu@cs.ccu.edu.tw 1 2 3.1 Homogeneous Linear ODEs 3 Homogeneous Linear ODEs An ODE is of
More informationMath 4B Notes. Written by Victoria Kala SH 6432u Office Hours: T 12:45 1:45pm Last updated 7/24/2016
Math 4B Notes Written by Victoria Kala vtkala@math.ucsb.edu SH 6432u Office Hours: T 2:45 :45pm Last updated 7/24/206 Classification of Differential Equations The order of a differential equation is the
More informationChapter 11. Taylor Series. Josef Leydold Mathematical Methods WS 2018/19 11 Taylor Series 1 / 27
Chapter 11 Taylor Series Josef Leydold Mathematical Methods WS 2018/19 11 Taylor Series 1 / 27 First-Order Approximation We want to approximate function f by some simple function. Best possible approximation
More informationME 475 Modal Analysis of a Tapered Beam
ME 475 Modal Analysis of a Tapered Beam Objectives: 1. To find the natural frequencies and mode shapes of a tapered beam using FEA.. To compare the FE solution to analytical solutions of the vibratory
More information