# Solution of Matrix Eigenvalue Problem

Size: px
Start display at page:

Transcription

1 Outlines October 12, 2004

2 Outlines Part I: Review of Previous Lecture Part II: Review of Previous Lecture

3 Outlines Part I: Review of Previous Lecture Part II: Standard Matrix Eigenvalue Problem Other Forms of the Standard Eigenvalue Problem Solvability of the Standard Eigenvalue Problem Example Standard Eigenvalue Problems s

4 Part I Review of Previous Lecture

5 Review of Previous Lecture Jacobi Iteration Method Gauss-Seidel Iteration Method Use of Software Packages

6 Part II

7 Other Forms of the Standard Eigenvalue Problem Solvability of the Standard Eigenvalue Problem Standard Matrix Eigenvalue Problem: Consider a system of equations in algebraic form (a 11 λ)x 1 + a 12 x 2 + a 13 x a 1n x n =0 a 21 x 1 + (a 22 λ)x 2 + a 23 x a 2n x n =0. a n1 x 1 + a n2 x 2 + a n3 x (a nn λ)x n =0 This is not a normal system of linear algebraic equations we re used to. For one, there are n equations, but n + 1 unknowns (the x i values, and also λ). This particular system of equations is known as the standard eigenvalue problem.

8 Other Forms of the Standard Eigenvalue Problem Solvability of the Standard Eigenvalue Problem Form 1 The three forms shown are all algebraically equivalent. Any system of equations that can be expressed in these forms is a standard eigenvalue problem. a 11 λ a 12 a 1n a 21 a 22 λ a 2n a n1 a n2 a nn λ x 1 x 2. x n =

9 Other Forms of the Standard Eigenvalue Problem Solvability of the Standard Eigenvalue Problem Form 2 a 11 a 12 a 1n a 21 a 22 a 2n a n1 a n2 a nn λ x 1 x 2. x n = ([A] λ [I ]) {x} = {0}

10 Other Forms of the Standard Eigenvalue Problem Solvability of the Standard Eigenvalue Problem Form 3 a 11 a 12 a 1n a 21 a 22 a 2n a n1 a n2 a nn x 1 x 2. x n [A] {x} = λ {x} = λ x 1 x 2. x n

11 Other Forms of the Standard Eigenvalue Problem Solvability of the Standard Eigenvalue Problem Solvability of the Standard Eigenvalue Problem Recall form 2 of the standard eigenvalue problem: ([A] λ [I ]) {x} = {0} This system of equations has a solution for values of λ that cause the determinant of the coefficient matrix to equal 0, that is: [A] λ [I ] = 0

12 Other Forms of the Standard Eigenvalue Problem Solvability of the Standard Eigenvalue Problem Characteristic Equation Expanding out all the terms of the previous determinant a 11 λ a 12 a 1n a 21 a 22 λ a 2n..... = 0. a n1 a n2 a nn λ yields a long polynomial in terms of λ. This polynomial will be nth order, and will therefore have n roots, each of which may be real or complex.

13 Example : Many physical systems do not automatically present themselves as a standard eigenvalue problem, even though they can be reformatted as a standard eigenvalue problem. The form of a general eigenvalue problem is [A] {x} = λ [B] {x} where [A] and [B] are symmetric matrices of size n n.

14 Example Example A forging hammer of mass m 2 is mounted on a concrete foundation block of mass m 1. The stiffnesses of the springs underneath the forging hammer and the foundation block are given by k 2 and k 1, respectively.

15 Example Example The system undergoes simple harmonic motion at one of its natural frequencies ω. That is: x 1 (t) = cos(ωt + φ 1 ) x 2 (t) = cos(ωt + φ 2 ) a 1 (t) = ω 2 x 1 (t) a 2 (t) = ω 2 x 2 (t)

16 Example Example Each mass in the system obeys Newton s second law of motion, that is: ΣF = ma Forces on the foundation block: forces from the lower springs, which counteracts motion in the x direction at an amount k 1 x 1 forces from the upper springs, which act according to the amount of relative displacement of the masses m 1 and m 2 : k 2 (x 1 x 2 )

17 Example Example The equilibrium equation for the foundation mass is then ΣF = ma k 1 x 1 k 2 (x 1 x 2 ) = m 1 a ( k 2 k 1 )x 1 + k 2 x 2 = m 1 a ( k 2 k 1 )x 1 + k 2 x 2 = m 1 ω 2 x 1 (k 1 + k 2 )x 1 k 2 x 2 = m 1 ω 2 x 1

18 Example Example Similarly, the equilibrium equation for the forging hammer mass is k 2 x 1 + k 2 x 2 = m 2 ω 2 x 2

19 Example Example So the two equations of motion are (k 1 + k 2 )x 1 k 2 x 2 = m 1 ω 2 x 1 k 2 x 1 + k 2 x 2 = m 2 ω 2 x 2 or in matrix form [ k1 + k 2 k 2 k 2 k 2 ] { x1 x 2 } = ω 2 [ m1 0 0 m 2 ] { x1 x 2 } This is a general eigenvalue problem [A]{x} = λ[b]{x} where [A] is the spring matrix, {x} is the vector of x values, λ is ω 2, and [B] is the mass matrix.

20 Standard Eigenvalue Problems s : Standard Problems The design of a mechanical component requires that the maximum principal stress to be less than the material strength. For a component subjected to arbitrary loads, the principal stresses σ are given by the solution of the equation σ xx τ xy τ xz τ xy σ yy τ yz τ xz τ yz σ zz l x l y l z = σ where the σ values represent normal stresses in the x, y, and z directions, and the τ values represent shear stresses in the xy, xz, and yz planes. The l values represent direction cosines that define the principal planes on which the principal stress occurs. l x l y l z

21 Standard Eigenvalue Problems s : Standard Problems Determine the principal stresses and principal planes in a machine component for the following stress condition σ xx τ xy τ xz τ xy σ yy τ yz τ xz τ yz σ zz = MPa

22 Standard Eigenvalue Problems s MATLAB Solution clear all sigma =[ ]; [dirs, stresses ]= eig ( sigma ); % diag ( A) extracts the elements of the % [ A] matrix along the diagonal principalstresslist = diag ( stresses ) principaldirs = dirs

23 Standard Eigenvalue Problems s MATLAB Solution >> rao_ p431 principalstresslist = principaldirs =

24 Standard Eigenvalue Problems s : General Problems Solve the forging hammer problem for the following values: m 1 = kg m 2 = 5000 kg k 1 = N/m k 2 = N/m

25 Standard Eigenvalue Problems s Solving Eigenvalue Problems in MATLAB Solving this eigenvalue problem will yield 2 eigenvalues equal to the square of the system s natural frequencies, and 2 corresponding x vector values that show the relative displacements of the m 1 and m 2 masses at those frequencies.

26 Standard Eigenvalue Problems s MATLAB Solution (Part 1) clear all ; % Define spring constants and masses % for hammer and foundation block k1 =1 e7; k2 =5 e6; m1 =20000; m2 =5000; % Define system stiffness matrix K=[ k1+k2 -k2 -k2 k2 ]; % Define system mass matrix M=[ m1 0 0 m2 ];

27 Standard Eigenvalue Problems s MATLAB Solution (Part 2) % Solve general eigenvalue problem [X, Omega2 ]= eig (K,M); % diag ( A) extracts the elements of the % [ A] matrix along the diagonal Omega = diag ( sqrt ( Omega2 )); % Scale column 1 of the [ X] matrix by % the row 1, column 1 X value X (:,1)= X (:,1)/ X (1,1); % Scale column 2 of the [ X] matrix by % the row 1, column 2 X value X (:,2)= X (:,2)/ X (1,2); Omega X

28 Standard Eigenvalue Problems s MATLAB Solution (Results) >> rao_ ex42 Omega = X =

29 Continue working on the Gauss-Seidel and other homework problems already assigned.

### Review of Strain Energy Methods and Introduction to Stiffness Matrix Methods of Structural Analysis

uke University epartment of Civil and Environmental Engineering CEE 42L. Matrix Structural Analysis Henri P. Gavin Fall, 22 Review of Strain Energy Methods and Introduction to Stiffness Matrix Methods

### Applications of Eigenvalues & Eigenvectors

Applications of Eigenvalues & Eigenvectors Louie L. Yaw Walla Walla University Engineering Department For Linear Algebra Class November 17, 214 Outline 1 The eigenvalue/eigenvector problem 2 Principal

### 19. Principal Stresses

19. Principal Stresses I Main Topics A Cauchy s formula B Principal stresses (eigenvectors and eigenvalues) C Example 10/24/18 GG303 1 19. Principal Stresses hkp://hvo.wr.usgs.gov/kilauea/update/images.html

### Introduction to Seismology Spring 2008

MIT OpenCourseWare http://ocw.mit.edu 12.510 Introduction to Seismology Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. Stress and Strain

### UNIVERSITY OF SASKATCHEWAN ME MECHANICS OF MATERIALS I FINAL EXAM DECEMBER 13, 2008 Professor A. Dolovich

UNIVERSITY OF SASKATCHEWAN ME 313.3 MECHANICS OF MATERIALS I FINAL EXAM DECEMBER 13, 2008 Professor A. Dolovich A CLOSED BOOK EXAMINATION TIME: 3 HOURS For Marker s Use Only LAST NAME (printed): FIRST

### Math/Phys/Engr 428, Math 529/Phys 528 Numerical Methods - Summer Homework 3 Due: Tuesday, July 3, 2018

Math/Phys/Engr 428, Math 529/Phys 528 Numerical Methods - Summer 28. (Vector and Matrix Norms) Homework 3 Due: Tuesday, July 3, 28 Show that the l vector norm satisfies the three properties (a) x for x

### Gauss-Seidel method. Dr. Motilal Panigrahi. Dr. Motilal Panigrahi, Nirma University

Gauss-Seidel method Dr. Motilal Panigrahi Solving system of linear equations We discussed Gaussian elimination with partial pivoting Gaussian elimination was an exact method or closed method Now we will

### Finite Element Method in Geotechnical Engineering

Finite Element Method in Geotechnical Engineering Short Course on + Dynamics Boulder, Colorado January 5-8, 2004 Stein Sture Professor of Civil Engineering University of Colorado at Boulder Contents Steps

### Problem Set 2 Due Tuesday, September 27, ; p : 0. (b) Construct a representation using five d orbitals that sit on the origin as a basis: 1

Problem Set 2 Due Tuesday, September 27, 211 Problems from Carter: Chapter 2: 2a-d,g,h,j 2.6, 2.9; Chapter 3: 1a-d,f,g 3.3, 3.6, 3.7 Additional problems: (1) Consider the D 4 point group and use a coordinate

### Unit IV State of stress in Three Dimensions

Unit IV State of stress in Three Dimensions State of stress in Three Dimensions References Punmia B.C.,"Theory of Structures" (SMTS) Vol II, Laxmi Publishing Pvt Ltd, New Delhi 2004. Rattan.S.S., "Strength

### Strain Transformation equations

Strain Transformation equations R. Chandramouli Associate Dean-Research SASTRA University, Thanjavur-613 401 Joint Initiative of IITs and IISc Funded by MHRD Page 1 of 8 Table of Contents 1. Stress transformation

### Basic Equations of Elasticity

A Basic Equations of Elasticity A.1 STRESS The state of stress at any point in a loaded bo is defined completely in terms of the nine components of stress: σ xx,σ yy,σ zz,σ xy,σ yx,σ yz,σ zy,σ zx,andσ

### 10.2 ITERATIVE METHODS FOR SOLVING LINEAR SYSTEMS. The Jacobi Method

54 CHAPTER 10 NUMERICAL METHODS 10. ITERATIVE METHODS FOR SOLVING LINEAR SYSTEMS As a numerical technique, Gaussian elimination is rather unusual because it is direct. That is, a solution is obtained after

### The Finite Element Method for the Analysis of Non-Linear and Dynamic Systems

The Finite Element Method for the Analysis of Non-Linear and Dynamic Systems Prof. Dr. Eleni Chatzi Dr. Giuseppe Abbiati, Dr. Konstantinos Agathos Lecture 1-21 September, 2017 Institute of Structural Engineering

### Section 3.7: Mechanical and Electrical Vibrations

Section 3.7: Mechanical and Electrical Vibrations Second order linear equations with constant coefficients serve as mathematical models for mechanical and electrical oscillations. For example, the motion

### Properties of Linear Transformations from R n to R m

Properties of Linear Transformations from R n to R m MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Topic Overview Relationship between the properties of a matrix transformation

### Elementary Matrices. MATH 322, Linear Algebra I. J. Robert Buchanan. Spring Department of Mathematics

Elementary Matrices MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Outline Today s discussion will focus on: elementary matrices and their properties, using elementary

### Mechanics PhD Preliminary Spring 2017

Mechanics PhD Preliminary Spring 2017 1. (10 points) Consider a body Ω that is assembled by gluing together two separate bodies along a flat interface. The normal vector to the interface is given by n

### Module #3. Transformation of stresses in 3-D READING LIST. DIETER: Ch. 2, pp Ch. 3 in Roesler Ch. 2 in McClintock and Argon Ch.

HOMEWORK From Dieter -3, -4, 3-7 Module #3 Transformation of stresses in 3-D READING LIST DIETER: Ch., pp. 7-36 Ch. 3 in Roesler Ch. in McClintock and Argon Ch. 7 in Edelglass The Stress Tensor z z x O

### Two Posts to Fill On School Board

Y Y 9 86 4 4 qz 86 x : ( ) z 7 854 Y x 4 z z x x 4 87 88 Y 5 x q x 8 Y 8 x x : 6 ; : 5 x ; 4 ( z ; ( ) ) x ; z 94 ; x 3 3 3 5 94 ; ; ; ; 3 x : 5 89 q ; ; x ; x ; ; x : ; ; ; ; ; ; 87 47% : () : / : 83

### Seminar 6: COUPLED HARMONIC OSCILLATORS

Seminar 6: COUPLED HARMONIC OSCILLATORS 1. Lagrangian Equations of Motion Let consider a system consisting of two harmonic oscillators that are coupled together. As a model, we will use two particles attached

### 12. Stresses and Strains

12. Stresses and Strains Finite Element Method Differential Equation Weak Formulation Approximating Functions Weighted Residuals FEM - Formulation Classification of Problems Scalar Vector 1-D T(x) u(x)

### GG303 Lecture 6 8/27/09 1 SCALARS, VECTORS, AND TENSORS

GG303 Lecture 6 8/27/09 1 SCALARS, VECTORS, AND TENSORS I Main Topics A Why deal with tensors? B Order of scalars, vectors, and tensors C Linear transformation of scalars and vectors (and tensors) II Why

### MECH 5312 Solid Mechanics II. Dr. Calvin M. Stewart Department of Mechanical Engineering The University of Texas at El Paso

MECH 5312 Solid Mechanics II Dr. Calvin M. Stewart Department of Mechanical Engineering The University of Texas at El Paso Table of Contents Preliminary Math Concept of Stress Stress Components Equilibrium

### APPLICATIONS The eigenvalues are λ = 5, 5. An orthonormal basis of eigenvectors consists of

CHAPTER III APPLICATIONS The eigenvalues are λ =, An orthonormal basis of eigenvectors consists of, The eigenvalues are λ =, A basis of eigenvectors consists of, 4 which are not perpendicular However,

### Motivation: Sparse matrices and numerical PDE's

Lecture 20: Numerical Linear Algebra #4 Iterative methods and Eigenproblems Outline 1) Motivation: beyond LU for Ax=b A little PDE's and sparse matrices A) Temperature Equation B) Poisson Equation 2) Splitting

### Solving Linear Systems

Solving Linear Systems Iterative Solutions Methods Philippe B. Laval KSU Fall 207 Philippe B. Laval (KSU) Linear Systems Fall 207 / 2 Introduction We continue looking how to solve linear systems of the

### Numerical Modelling in Geosciences. Lecture 6 Deformation

Numerical Modelling in Geosciences Lecture 6 Deformation Tensor Second-rank tensor stress ), strain ), strain rate ) Invariants quantities independent of the coordinate system): - First invariant trace:!!

### AN ITERATION. In part as motivation, we consider an iteration method for solving a system of linear equations which has the form x Ax = b

AN ITERATION In part as motivation, we consider an iteration method for solving a system of linear equations which has the form x Ax = b In this, A is an n n matrix and b R n.systemsof this form arise

### GG612 Lecture 3. Outline

GG61 Lecture 3 Strain and Stress Should complete infinitesimal strain by adding rota>on. Outline Matrix Opera+ons Strain 1 General concepts Homogeneous strain 3 Matrix representa>ons 4 Squares of line

### Homework 1/Solutions. Graded Exercises

MTH 310-3 Abstract Algebra I and Number Theory S18 Homework 1/Solutions Graded Exercises Exercise 1. Below are parts of the addition table and parts of the multiplication table of a ring. Complete both

### Eigenvalue and Eigenvector Homework

Eigenvalue and Eigenvector Homework Olena Bormashenko November 4, 2 For each of the matrices A below, do the following:. Find the characteristic polynomial of A, and use it to find all the eigenvalues

### Problem Set 2 Due Thursday, October 1, & & & & # % (b) Construct a representation using five d orbitals that sit on the origin as a basis:

Problem Set 2 Due Thursday, October 1, 29 Problems from Cotton: Chapter 4: 4.6, 4.7; Chapter 6: 6.2, 6.4, 6.5 Additional problems: (1) Consider the D 3h point group and use a coordinate system wherein

### JACOBI S ITERATION METHOD

ITERATION METHODS These are methods which compute a sequence of progressively accurate iterates to approximate the solution of Ax = b. We need such methods for solving many large linear systems. Sometimes

### Classical Mechanics. Luis Anchordoqui

1 Rigid Body Motion Inertia Tensor Rotational Kinetic Energy Principal Axes of Rotation Steiner s Theorem Euler s Equations for a Rigid Body Eulerian Angles Review of Fundamental Equations 2 Rigid body

### TMA4125 Matematikk 4N Spring 2017

Norwegian University of Science and Technology Institutt for matematiske fag TMA15 Matematikk N Spring 17 Solutions to exercise set 1 1 We begin by writing the system as the augmented matrix.139.38.3 6.

«4 [< «

### MANY BILLS OF CONCERN TO PUBLIC

- 6 8 9-6 8 9 6 9 XXX 4 > -? - 8 9 x 4 z ) - -! x - x - - X - - - - - x 00 - - - - - x z - - - x x - x - - - - - ) x - - - - - - 0 > - 000-90 - - 4 0 x 00 - -? z 8 & x - - 8? > 9 - - - - 64 49 9 x - -

### MATH 423 Linear Algebra II Lecture 33: Diagonalization of normal operators.

MATH 423 Linear Algebra II Lecture 33: Diagonalization of normal operators. Adjoint operator and adjoint matrix Given a linear operator L on an inner product space V, the adjoint of L is a transformation

### Computer Applications in Engineering and Construction Programming Assignment #9 Principle Stresses and Flow Nets in Geotechnical Design

CVEN 302-501 Computer Applications in Engineering and Construction Programming Assignment #9 Principle Stresses and Flow Nets in Geotechnical Design Date distributed : 12/2/2015 Date due : 12/9/2015 at

### Stress transformation and Mohr s circle for stresses

Stress transformation and Mohr s circle for stresses 1.1 General State of stress Consider a certain body, subjected to external force. The force F is acting on the surface over an area da of the surface.

### A DARK GREY P O N T, with a Switch Tail, and a small Star on the Forehead. Any

Y Y Y X X «/ YY Y Y ««Y x ) & \ & & } # Y \#\$& / Y Y X» \\ / X X X x & Y Y X «q «z \x» = q Y # % \ & [ & Z \ & { + % ) / / «q zy» / & / / / & x x X / % % ) Y x X Y \$ Z % Y Y x x } / % «] «] # z» & Y X»

### Solution of System of Linear Equations & Eigen Values and Eigen Vectors

Solution of System of Linear Equations & Eigen Values and Eigen Vectors P. Sam Johnson March 30, 2015 P. Sam Johnson (NITK) Solution of System of Linear Equations & Eigen Values and Eigen Vectors March

### Linear Algebra: Matrix Eigenvalue Problems

CHAPTER8 Linear Algebra: Matrix Eigenvalue Problems Chapter 8 p1 A matrix eigenvalue problem considers the vector equation (1) Ax = λx. 8.0 Linear Algebra: Matrix Eigenvalue Problems Here A is a given

### Investigating Springs (Simple Harmonic Motion)

Investigating Springs (Simple Harmonic Motion) INTRODUCTION The purpose of this lab is to study the well-known force exerted by a spring The force, as given by Hooke s Law, is a function of the amount

### OWELL WEEKLY JOURNAL

Y \»< - } Y Y Y & #»»» q ] q»»»>) & - - - } ) x ( - { Y» & ( x - (» & )< - Y X - & Q Q» 3 - x Q Y 6 \Y > Y Y X 3 3-9 33 x - - / - -»- --

### Computational Linear Algebra

Computational Linear Algebra PD Dr. rer. nat. habil. Ralf Peter Mundani Computation in Engineering / BGU Scientific Computing in Computer Science / INF Winter Term 2017/18 Part 3: Iterative Methods PD

### NUMERICAL METHODS USING MATLAB

NUMERICAL METHODS USING MATLAB Dr John Penny George Lindfield Department of Mechanical Engineering, Aston University ELLIS HORWOOD NEW YORK LONDON TORONTO SYDNEY TOKYO SINGAPORE Preface 1 An introduction

### MATH Topics in Applied Mathematics Lecture 12: Evaluation of determinants. Cross product.

MATH 311-504 Topics in Applied Mathematics Lecture 12: Evaluation of determinants. Cross product. Determinant is a scalar assigned to each square matrix. Notation. The determinant of a matrix A = (a ij

### PEAT SEISMOLOGY Lecture 2: Continuum mechanics

PEAT8002 - SEISMOLOGY Lecture 2: Continuum mechanics Nick Rawlinson Research School of Earth Sciences Australian National University Strain Strain is the formal description of the change in shape of a

### Constitutive Equations

Constitutive quations David Roylance Department of Materials Science and ngineering Massachusetts Institute of Technology Cambridge, MA 0239 October 4, 2000 Introduction The modules on kinematics (Module

### Algebra C Numerical Linear Algebra Sample Exam Problems

Algebra C Numerical Linear Algebra Sample Exam Problems Notation. Denote by V a finite-dimensional Hilbert space with inner product (, ) and corresponding norm. The abbreviation SPD is used for symmetric

### Math 504 (Fall 2011) 1. (*) Consider the matrices

Math 504 (Fall 2011) Instructor: Emre Mengi Study Guide for Weeks 11-14 This homework concerns the following topics. Basic definitions and facts about eigenvalues and eigenvectors (Trefethen&Bau, Lecture

### review To find the coefficient of all the terms in 15ab + 60bc 17ca: Coefficient of ab = 15 Coefficient of bc = 60 Coefficient of ca = -17

1. Revision Recall basic terms of algebraic expressions like Variable, Constant, Term, Coefficient, Polynomial etc. The coefficients of the terms in 4x 2 5xy + 6y 2 are Coefficient of 4x 2 is 4 Coefficient

### In this section of notes, we look at the calculation of forces and torques for a manipulator in two settings:

Introduction Up to this point we have considered only the kinematics of a manipulator. That is, only the specification of motion without regard to the forces and torques required to cause motion In this

### Matrices and Deformation

ES 111 Mathematical Methods in the Earth Sciences Matrices and Deformation Lecture Outline 13 - Thurs 9th Nov 2017 Strain Ellipse and Eigenvectors One way of thinking about a matrix is that it operates

### Lecture 9 Approximations of Laplace s Equation, Finite Element Method. Mathématiques appliquées (MATH0504-1) B. Dewals, C.

Lecture 9 Approximations of Laplace s Equation, Finite Element Method Mathématiques appliquées (MATH54-1) B. Dewals, C. Geuzaine V1.2 23/11/218 1 Learning objectives of this lecture Apply the finite difference

### LINEAR ALGEBRA: NUMERICAL METHODS. Version: August 12,

LINEAR ALGEBRA: NUMERICAL METHODS. Version: August 12, 2000 74 6 Summary Here we summarize the most important information about theoretical and numerical linear algebra. MORALS OF THE STORY: I. Theoretically

### Structural Dynamics Lecture Eleven: Dynamic Response of MDOF Systems: (Chapter 11) By: H. Ahmadian

Structural Dynamics Lecture Eleven: Dynamic Response of MDOF Systems: (Chapter 11) By: H. Ahmadian ahmadian@iust.ac.ir Dynamic Response of MDOF Systems: Mode-Superposition Method Mode-Superposition Method:

### Jordan normal form notes (version date: 11/21/07)

Jordan normal form notes (version date: /2/7) If A has an eigenbasis {u,, u n }, ie a basis made up of eigenvectors, so that Au j = λ j u j, then A is diagonal with respect to that basis To see this, let

### Mechanics of materials Lecture 4 Strain and deformation

Mechanics of materials Lecture 4 Strain and deformation Reijo Kouhia Tampere University of Technology Department of Mechanical Engineering and Industrial Design Wednesday 3 rd February, 206 of a continuum

### MECH 5312 Solid Mechanics II. Dr. Calvin M. Stewart Department of Mechanical Engineering The University of Texas at El Paso

MECH 5312 Solid Mechanics II Dr. Calvin M. Stewart Department of Mechanical Engineering The University of Texas at El Paso Table of Contents Thermodynamics Derivation Hooke s Law: Anisotropic Elasticity

### Chapter 12: Iterative Methods

ES 40: Scientific and Engineering Computation. Uchechukwu Ofoegbu Temple University Chapter : Iterative Methods ES 40: Scientific and Engineering Computation. Gauss-Seidel Method The Gauss-Seidel method

### Variational principles in mechanics

CHAPTER Variational principles in mechanics.1 Linear Elasticity n D Figure.1: A domain and its boundary = D [. Consider a domain Ω R 3 with its boundary = D [ of normal n (see Figure.1). The problem of

### ME 243. Lecture 10: Combined stresses

ME 243 Mechanics of Solids Lecture 10: Combined stresses Ahmad Shahedi Shakil Lecturer, Dept. of Mechanical Engg, BUET E-mail: sshakil@me.buet.ac.bd, shakil6791@gmail.com Website: teacher.buet.ac.bd/sshakil

### 1 Last time: least-squares problems

MATH Linear algebra (Fall 07) Lecture Last time: least-squares problems Definition. If A is an m n matrix and b R m, then a least-squares solution to the linear system Ax = b is a vector x R n such that

### If the symmetry axes of a uniform symmetric body coincide with the coordinate axes, the products of inertia (Ixy etc.

Prof. O. B. Wright, Autumn 007 Mechanics Lecture 9 More on rigid bodies, coupled vibrations Principal axes of the inertia tensor If the symmetry axes of a uniform symmetric body coincide with the coordinate

### Department of Mathematics California State University, Los Angeles Master s Degree Comprehensive Examination in. NUMERICAL ANALYSIS Spring 2015

Department of Mathematics California State University, Los Angeles Master s Degree Comprehensive Examination in NUMERICAL ANALYSIS Spring 2015 Instructions: Do exactly two problems from Part A AND two

### Math 5630: Iterative Methods for Systems of Equations Hung Phan, UMass Lowell March 22, 2018

1 Linear Systems Math 5630: Iterative Methods for Systems of Equations Hung Phan, UMass Lowell March, 018 Consider the system 4x y + z = 7 4x 8y + z = 1 x + y + 5z = 15. We then obtain x = 1 4 (7 + y z)

### Tensor Transformations and the Maximum Shear Stress. (Draft 1, 1/28/07)

Tensor Transformations and the Maximum Shear Stress (Draft 1, 1/28/07) Introduction The order of a tensor is the number of subscripts it has. For each subscript it is multiplied by a direction cosine array

### Numerical Methods I Eigenvalue Problems

Numerical Methods I Eigenvalue Problems Aleksandar Donev Courant Institute, NYU 1 donev@courant.nyu.edu 1 MATH-GA 2011.003 / CSCI-GA 2945.003, Fall 2014 October 2nd, 2014 A. Donev (Courant Institute) Lecture

### Functions of Several Variables

Functions of Several Variables The Unconstrained Minimization Problem where In n dimensions the unconstrained problem is stated as f() x variables. minimize f()x x, is a scalar objective function of vector

### Iterative solution methods and their rate of convergence

Uppsala University Graduate School in Mathematics and Computing Institute for Information Technology Numerical Linear Algebra FMB and MN Fall 2007 Mandatory Assignment 3a: Iterative solution methods and

### CSE 167: Introduction to Computer Graphics Lecture #2: Linear Algebra Primer

CSE 167: Introduction to Computer Graphics Lecture #2: Linear Algebra Primer Jürgen P. Schulze, Ph.D. University of California, San Diego Fall Quarter 2016 Announcements Monday October 3: Discussion Assignment

### AA 242B / ME 242B: Mechanical Vibrations (Spring 2016)

AA 242B / ME 242B: Mechanical Vibrations (Spring 206) Solution of Homework #3 Control Tab Figure : Schematic for the control tab. Inadequacy of a static-test A static-test for measuring θ would ideally

### DEN: Linear algebra numerical view (GEM: Gauss elimination method for reducing a full rank matrix to upper-triangular

form) Given: matrix C = (c i,j ) n,m i,j=1 ODE and num math: Linear algebra (N) [lectures] c phabala 2016 DEN: Linear algebra numerical view (GEM: Gauss elimination method for reducing a full rank matrix

### AMS526: Numerical Analysis I (Numerical Linear Algebra for Computational and Data Sciences)

AMS526: Numerical Analysis I (Numerical Linear Algebra for Computational and Data Sciences) Lecture 19: Computing the SVD; Sparse Linear Systems Xiangmin Jiao Stony Brook University Xiangmin Jiao Numerical

### AIMS Exercise Set # 1

AIMS Exercise Set #. Determine the form of the single precision floating point arithmetic used in the computers at AIMS. What is the largest number that can be accurately represented? What is the smallest

### Stress, Strain, Mohr s Circle

Stress, Strain, Mohr s Circle The fundamental quantities in solid mechanics are stresses and strains. In accordance with the continuum mechanics assumption, the molecular structure of materials is neglected

### Lecture 8. Stress Strain in Multi-dimension

Lecture 8. Stress Strain in Multi-dimension Module. General Field Equations General Field Equations [] Equilibrium Equations in Elastic bodies xx x y z yx zx f x 0, etc [2] Kinematics xx u x x,etc. [3]

### MATH 425-Spring 2010 HOMEWORK ASSIGNMENTS

MATH 425-Spring 2010 HOMEWORK ASSIGNMENTS Instructor: Shmuel Friedland Department of Mathematics, Statistics and Computer Science email: friedlan@uic.edu Last update April 18, 2010 1 HOMEWORK ASSIGNMENT

### LOWELL WEEKLY JOURNAL

Y -» \$ 5 Y 7 Y Y -Y- Q x Q» 75»»/ q } # ]»\ - - \$ { Q» / X x»»- 3 q \$ 9 ) Y q - 5 5 3 3 3 7 Q q - - Q _»»/Q Y - 9 - - - )- [ X 7» -» - )»? / /? Q Y»» # X Q» - -?» Q ) Q \ Q - - - 3? 7» -? #»»» 7 - / Q

### Solving Linear Systems of Equations

November 6, 2013 Introduction The type of problems that we have to solve are: Solve the system: A x = B, where a 11 a 1N a 12 a 2N A =.. a 1N a NN x = x 1 x 2. x N B = b 1 b 2. b N To find A 1 (inverse

### Lectures 9-10: Polynomial and piecewise polynomial interpolation

Lectures 9-1: Polynomial and piecewise polynomial interpolation Let f be a function, which is only known at the nodes x 1, x,, x n, ie, all we know about the function f are its values y j = f(x j ), j

### Manipulator Dynamics 2. Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

Manipulator Dynamics 2 Forward Dynamics Problem Given: Joint torques and links geometry, mass, inertia, friction Compute: Angular acceleration of the links (solve differential equations) Solution Dynamic

### Name (Print) ME Mechanics of Materials Exam # 2 Date: March 29, 2016 Time: 8:00 10:00 PM - Location: PHYS 114

Name (Print) (Last) (First) Instructions: ME 323 - Mechanics of Materials Exam # 2 Date: March 29, 2016 Time: 8:00 10:00 PM - Location: PHYS 114 Circle your lecturer s name and your class meeting time.

### SECTION 5: POWER FLOW. ESE 470 Energy Distribution Systems

SECTION 5: POWER FLOW ESE 470 Energy Distribution Systems 2 Introduction Nodal Analysis 3 Consider the following circuit Three voltage sources VV sss, VV sss, VV sss Generic branch impedances Could be

### 6. 3D Kinematics DE2-EA 2.1: M4DE. Dr Connor Myant

DE2-EA 2.1: M4DE Dr Connor Myant 6. 3D Kinematics Comments and corrections to connor.myant@imperial.ac.uk Lecture resources may be found on Blackboard and at http://connormyant.com Contents Three-Dimensional

### Crystal Relaxation, Elasticity, and Lattice Dynamics

http://exciting-code.org Crystal Relaxation, Elasticity, and Lattice Dynamics Pasquale Pavone Humboldt-Universität zu Berlin http://exciting-code.org PART I: Structure Optimization Pasquale Pavone Humboldt-Universität

### Advanced Vibrations. Elements of Analytical Dynamics. By: H. Ahmadian Lecture One

Advanced Vibrations Lecture One Elements of Analytical Dynamics By: H. Ahmadian ahmadian@iust.ac.ir Elements of Analytical Dynamics Newton's laws were formulated for a single particle Can be extended to

### ECE 204 Numerical Methods for Computer Engineers MIDTERM EXAMINATION /8:00-9:30

ECE 204 Numerical Methods for Computer Engineers MIDTERM EXAMINATION 2007-10-23/8:00-9:30 The examination is out of 67 marks. Instructions: No aides. Write your name and student ID number on each booklet.

### CSE 167: Introduction to Computer Graphics Lecture #2: Linear Algebra Primer

CSE 167: Introduction to Computer Graphics Lecture #2: Linear Algebra Primer Jürgen P. Schulze, Ph.D. University of California, San Diego Spring Quarter 2016 Announcements Project 1 due next Friday at

### CHAPTER 4 Stress Transformation

CHAPTER 4 Stress Transformation ANALYSIS OF STRESS For this topic, the stresses to be considered are not on the perpendicular and parallel planes only but also on other inclined planes. A P a a b b P z

### ME 563 Mechanical Vibrations Lecture #15. Finite Element Approximations for Rods and Beams

ME 563 Mechanical Vibrations Lecture #15 Finite Element Approximations for Rods and Beams 1 Need for Finite Elements Continuous system vibration equations of motion are appropriate for applications where

### Diagonalization. P. Danziger. u B = A 1. B u S.

7., 8., 8.2 Diagonalization P. Danziger Change of Basis Given a basis of R n, B {v,..., v n }, we have seen that the matrix whose columns consist of these vectors can be thought of as a change of basis

### The Solution of Linear Systems AX = B

Chapter 2 The Solution of Linear Systems AX = B 21 Upper-triangular Linear Systems We will now develop the back-substitution algorithm, which is useful for solving a linear system of equations that has

### TMA Calculus 3. Lecture 21, April 3. Toke Meier Carlsen Norwegian University of Science and Technology Spring 2013

TMA4115 - Calculus 3 Lecture 21, April 3 Toke Meier Carlsen Norwegian University of Science and Technology Spring 2013 www.ntnu.no TMA4115 - Calculus 3, Lecture 21 Review of last week s lecture Last week

### HIGHER-ORDER THEORIES

HIGHER-ORDER THEORIES THIRD-ORDER SHEAR DEFORMATION PLATE THEORY LAYERWISE LAMINATE THEORY J.N. Reddy 1 Third-Order Shear Deformation Plate Theory Assumed Displacement Field µ u(x y z t) u 0 (x y t) +