Two mass-three spring system. Math 216 Differential Equations. Forces on mass m 1. Forces on mass m 2. Kenneth Harris
|
|
- Delilah Hood
- 7 years ago
- Views:
Transcription
1 Two mass-three spring system Math 6 Differential Equations Kenneth Harris kaharri@umich.edu m, m > 0, two masses k, k, k 3 > 0, spring elasticity t), t), displacement of m, m from equilibrium. Positive is right, negative left No friction or eternal forces Department of Mathematics University of Michigan November 3, 008 Kenneth Harris Math 6) Math 6 Differential Equations November 3, 008 / Kenneth Harris Math 6) Math 6 Differential Equations November 3, / Forces on mass m Apply Newton s law to mass m. m = k + k ) = k + k ) + k. Forces on mass m Apply Newton s law to mass m. m = k ) k 3 = k k + k 3 ). k k k k k 3 k k 3 m m m m k stretched k compressed k stretched k compressed Kenneth Harris Math 6) Math 6 Differential Equations November 3, / Kenneth Harris Math 6) Math 6 Differential Equations November 3, /
2 Two mass-three spring system: Equations First method for solving second order system Two mass, three spring system as a system of equations: m = k + k ) + k m = k k + k 3 ). As a matri equation dividing by m, m ): k +k ) = The equation has the form m k m k m k +k 3 ) m = A. First of three ways to solve the second-order system: m = k + k ) + k m = k k + k 3 ). Method. Rewrite using differential operators: m D + k + k )I ) k = 0 k + m D + k + k 3 )I ) = 0. Solve using Method of Elimination from Section 4.. Kenneth Harris Math 6) Math 6 Differential Equations November 3, / Kenneth Harris Math 6) Math 6 Differential Equations November 3, / Second method for solving second order system Second of three ways to solve the second-order system: m = k + k ) + k m = k k + k 3 ). Method. Rewrite as a system of four equations and four unknowns Section 4.): y = y y = k + k ) y + k y 3 m m y 3 = y 4 y 4 = k y k + k 3 ) y 3 m m Solve using Eigenvalue Method of Section 5.. Kenneth Harris Math 6) Math 6 Differential Equations November 3, / Third method for solving second order system Third of three ways to solve the second-order system: m = k + k ) + k m = k k + k 3 ). Method 3. Rewrite as a matri equation Section 5.): = k +k ) k m m k m k +k 3 ) m Solve using the etended) Eigenvalue Method of Section 5.3. Kenneth Harris Math 6) Math 6 Differential Equations November 3, /
3 Etended eigenvalue method to second-order systems Etended eigenvalue method Etended eigenvalue method to second-order systems Mechanical systems Solve. = A, where A has real constant components. Guess a solution: t) = ve αt. Substitute into = A: since ve αt) = α ve αt, Cancel e αt α ve αt = Ave αt α v = Av Answer. t) = ve αt is a solution when α is an eigenvalue and v an associated eigenvector of A. Solution. If λ = α is an eigenvalue and v is an eigenvector, ve αt is a solution to = A. Application. Mass-spring problems are of the form = A where where the eigenvalues are negative. Let α = ω and v an associated real-valued) eigenvector. Then is a solution to = A. t) = ve iωt = vcos ωt + i sin ωt) Real Solutions. The real and imaginary parts t) = v cos ωt t) = v sin ωt are linearly independent real-valued solutions to = A. Kenneth Harris Math 6) Math 6 Differential Equations November 3, 008 / Kenneth Harris Math 6) Math 6 Differential Equations November 3, 008 / Etended eigenvalue method to second-order systems Second-order homogeneous linear systems The following is useful for mechanical systems. Theorem Suppose an n n matri A has distinct negative eigenvalues ω, ω,..., ω n with associated real eigenvectors v, v,..., v n. Then a general solution to = A is t) = n a j cos ω j t + b j sin ω j t) j= where a i and b i are parameters. Problem. Find a general solution to the mass-spring system: m = m =, two masses k = k 3 = 4, k = 6, spring elasticity t), t), displacement of m, m from equilibrium. No friction or outside forces The second-order equations are = = 6 0. Equivalently, as a matri equation: [ [ [ 0 6 = 6 0 Kenneth Harris Math 6) Math 6 Differential Equations November 3, / Kenneth Harris Math 6) Math 6 Differential Equations November 3, /
4 continued continued Compute eigenvalues. 0 λ λ = λ + 0λ + 64 = λ + 6)λ + 4) So, λ = 6, 4. There are four linearly independent real solutions v cos 4t, v sin 4t, v cos t, v sin t where v is an eigenvector for λ = 6 and v is an eigenvector for λ = 4. Compute eigenvector for λ = 6. These are solutions to [ [ [ = This reduces to a single equation Eigenvector. v = = 0 or =. [. Kenneth Harris Math 6) Math 6 Differential Equations November 3, / Kenneth Harris Math 6) Math 6 Differential Equations November 3, / continued concluded Compute eigenvector for λ = 4. These are solutions to [ [ [ = This reduces to a single equation Eigenvector. v = = 0 or =. [. General Solution. The mass-spring system = = 6 0. has a general solution as a vector equation) t) = [ [ a cos 4t +a sin 4t ) + [ [ b cos t +b sin t ) and as a scalar system: t) = a cos 4t + a sin 4t) + b cos t + b sin t) t) = a cos 4t + a sin 4t) + b cos t + b sin t) Kenneth Harris Math 6) Math 6 Differential Equations November 3, / Kenneth Harris Math 6) Math 6 Differential Equations November 3, /
5 analysis analysis Analysis. The general solution t) = a cos 4t + a sin 4t) + b cos t + b sin t) t) = a cos 4t + a sin 4t) + b cos t + b sin t) can be simplified to t) = c cos4t α ) + c cost α ) t) = c cos4t α ) + c cost α ) where c = a + a, tan α = a c = b + b, tan α = b b a, The displacements of mass m and of mass m : t) = c cos4t α ) + c cost α ) t) = c cos4t α ) + c cost α ) A linear combination of two natural modes of oscillation: the natural frequencies ω = 4 and ω =. The natural mode ω = t) = c cost α ) t) = c cost α ) a free oscillation no damping) in which the masses move in synchrony in the same direction and with the same frequency ω = ) and equal amplitudes of oscillation c ). Kenneth Harris Math 6) Math 6 Differential Equations November 3, / Kenneth Harris Math 6) Math 6 Differential Equations November 3, 008 / analysis Forced oscillations Mass-spring systems with eternal forces The natural mode ω = 4 t) = c cos4t α ) t) = c cos4 ) a free oscillation no damping) in which the masses move in synchrony in the opposite directions and with the same frequency ω = 4) and equal amplitudes of oscillation c ). Suppose we apply eternal forces F to mass m and F to mass m in the mass-spring system. We now have a nonhomogeneous system m = k + k ) t) + k t) + F t) m = k t) k + k 3 ) t) + F t) t Kenneth Harris Math 6) Math 6 Differential Equations November 3, 008 / Kenneth Harris Math 6) Math 6 Differential Equations November 3, /
6 Forced oscillations Mechanical systems with forced oscillations Forced oscillations. We are interested in periodic eternal forces applied to the masses F 0 is a constant vector) [ F0 F 0 cos ωt = cos ωt F As a system of equations: As a matri equation = 6 4 m = k + k ) + k + F 0 cos ωt m = k k + k 3 ) + F cos ωt k +k ) k m m k k +k 3 ) m m Note the form = A + F 0 cos ωt F 0 F 3 5 cos ωt, Kenneth Harris Math 6) Math 6 Differential Equations November 3, / Forced oscillations First method for solving second order system First of two ways to solve the second-order system: m = k + k ) + k + F 0 cos ωt m = k k + k 3 ) + F cos ωt Method. Rewrite using differential operators: m D + k + k )I ) k = F 0 cos ωt k + m D + k + k 3 )I ) = F cos ωt Solve using Method of Elimination from Section 4.. Kenneth Harris Math 6) Math 6 Differential Equations November 3, / Forced oscillations Second method for solving second order system Second of two ways to solve the second-order system: m = k + k ) + k + F 0 cos ωt m = k k + k 3 ) + F cos ωt Method. Rewrite as a matri equation Section 5.): k +k ) k m m F 0 = + cos ωt k m k +k 3 ) m which is of the form = A + F 0 cos ωt. Guess a particular solution of the form p = c cos ωt, and determine the values of the parameter c. F Forced oscillations Second method for solving second order system Solve the second-order system = A + F 0 cos ωt Guess a particular solution of the form p = c cos ωt. Substitute p where p = ω c cos ωt and cancel the common cos ωt ω c cos ωt = Ac cos ωt + F 0 cos ωt ω c = Ac + F 0. New problem. We want a solution c to A + ω I ) c = F 0. Kenneth Harris Math 6) Math 6 Differential Equations November 3, / Kenneth Harris Math 6) Math 6 Differential Equations November 3, /
7 Forced oscillations Second method for solving second order system Eample Eample Solve for unknown c A + ω I ) c = F 0. Observation. As long as ω is not an eigenvalue for A, then A + ω I is invertible. Solution. If ω is not an eigenvalue for A, then A + ω I ) c = F 0. has a unique solution. Observation. If ω is an eigenvalue for A, then we have resonance. In this case, we can use techniques of Section 5.6 which generalize the Method of Undetermined Coefficients and Method of Variation of Parameters to vector systems. Problem. Find a general solution to the mass-spring system of when: Eternal forces: F = 30 cos t, F = 60 cos t The second-order equations are = cos t = cos t Equivalently, as a matri equation: [ [ 0 6 = 6 0 [ + [ 30 cos t 60 cos t Kenneth Harris Math 6) Math 6 Differential Equations November 3, / Kenneth Harris Math 6) Math 6 Differential Equations November 3, / Eample continued Eample Compute a particular solution for [ [ 0 6 = 6 0 [ + [ 30 cos t 60 cos t Guess. p = c cos t. Let ω =. Since ω = is not an eigenvalue for the system λ = 6, 4), there is no resonance. Substitute the guess p = c cos t into the equation. This reduces to [ c c cos t = [ [ c c cos t + [ [ [ [ c 30 = c 60 cos t Eample continued Solve the system of equations So, c = 4 and c = 6. Particular solution Eample 0 = 3c + c 0 = c 3c p = [ 4 cos t 6 General solution. c + p, where c is from : t) = a cos 4t + a sin 4t) + b cos t + b sin t) + 4 cos t t) = a cos 4t + a sin 4t) + b cos t + b sin t) + 6 cos t Kenneth Harris Math 6) Math 6 Differential Equations November 3, / Kenneth Harris Math 6) Math 6 Differential Equations November 3, /
8 Eample continued Eample Eample continued Eample Problem. Find the motion when the two masses begin at the equillibrium position at rest. The initial conditions are 0) = 0, 0) = 0, 0) = 0, 0) = 0 The general solution and derivatives are t) = a cos 4t + a sin 4t) + b cos t + b sin t) + 4 cos t t) = 4a sin 4t + 4a cos t) + b sin t + b cos t) 6 sin t t) = a cos 4t + a sin 4t) + b cos t + b sin t) + 6 cos t t) = 4a sin 4t + 4a cos 4t) + b sin t + b cos t) 4 sin t Substitute. 0 = a + b = 4a + b 0 = a + b = 4a + b Solve. The solutions are 0 = a + b = 4a + b 0 = a + b = 4a + b a = b = 5 a = b = 0 Solution. The general solution and derivatives are t) = cos 4t 5 cos t + 4 cos t t) = cos 4t 5 cos t + 6 cos t Kenneth Harris Math 6) Math 6 Differential Equations November 3, / Kenneth Harris Math 6) Math 6 Differential Equations November 3, /
Applications of Second-Order Differential Equations
Applications of Second-Order Differential Equations Second-order linear differential equations have a variety of applications in science and engineering. In this section we explore two of them: the vibration
More informationAPPLIED MATHEMATICS ADVANCED LEVEL
APPLIED MATHEMATICS ADVANCED LEVEL INTRODUCTION This syllabus serves to examine candidates knowledge and skills in introductory mathematical and statistical methods, and their applications. For applications
More informationThe Method of Partial Fractions Math 121 Calculus II Spring 2015
Rational functions. as The Method of Partial Fractions Math 11 Calculus II Spring 015 Recall that a rational function is a quotient of two polynomials such f(x) g(x) = 3x5 + x 3 + 16x x 60. The method
More informationEASTERN ARIZONA COLLEGE Differential Equations
EASTERN ARIZONA COLLEGE Differential Equations Course Design 2015-2016 Course Information Division Mathematics Course Number MAT 260 (SUN# MAT 2262) Title Differential Equations Credits 3 Developed by
More informationSecond Order Linear Differential Equations
CHAPTER 2 Second Order Linear Differential Equations 2.. Homogeneous Equations A differential equation is a relation involving variables x y y y. A solution is a function f x such that the substitution
More information3.2 Sources, Sinks, Saddles, and Spirals
3.2. Sources, Sinks, Saddles, and Spirals 6 3.2 Sources, Sinks, Saddles, and Spirals The pictures in this section show solutions to Ay 00 C By 0 C Cy D 0. These are linear equations with constant coefficients
More informationHow To Solve A Linear Dierential Equation
Dierential Equations (part 2): Linear Dierential Equations (by Evan Dummit, 2012, v. 1.00) Contents 4 Linear Dierential Equations 1 4.1 Terminology.................................................. 1 4.2
More informationLS.6 Solution Matrices
LS.6 Solution Matrices In the literature, solutions to linear systems often are expressed using square matrices rather than vectors. You need to get used to the terminology. As before, we state the definitions
More informationLinear Equations in Linear Algebra
1 Linear Equations in Linear Algebra 1.5 SOLUTION SETS OF LINEAR SYSTEMS HOMOGENEOUS LINEAR SYSTEMS A system of linear equations is said to be homogeneous if it can be written in the form A 0, where A
More informationCh 7 Kinetic Energy and Work. Question: 7 Problems: 3, 7, 11, 17, 23, 27, 35, 37, 41, 43
Ch 7 Kinetic Energy and Work Question: 7 Problems: 3, 7, 11, 17, 23, 27, 35, 37, 41, 43 Technical definition of energy a scalar quantity that is associated with that state of one or more objects The state
More informationboth double. A. T and v max B. T remains the same and v max doubles. both remain the same. C. T and v max
Q13.1 An object on the end of a spring is oscillating in simple harmonic motion. If the amplitude of oscillation is doubled, how does this affect the oscillation period T and the object s maximum speed
More informationASEN 3112 - Structures. MDOF Dynamic Systems. ASEN 3112 Lecture 1 Slide 1
19 MDOF Dynamic Systems ASEN 3112 Lecture 1 Slide 1 A Two-DOF Mass-Spring-Dashpot Dynamic System Consider the lumped-parameter, mass-spring-dashpot dynamic system shown in the Figure. It has two point
More informationPhysics 9e/Cutnell. correlated to the. College Board AP Physics 1 Course Objectives
Physics 9e/Cutnell correlated to the College Board AP Physics 1 Course Objectives Big Idea 1: Objects and systems have properties such as mass and charge. Systems may have internal structure. Enduring
More informationHW6 Solutions. MATH 20D Fall 2013 Prof: Sun Hui TA: Zezhou Zhang (David) November 14, 2013. Checklist: Section 7.8: 1c, 2, 7, 10, [16]
HW6 Solutions MATH D Fall 3 Prof: Sun Hui TA: Zezhou Zhang David November 4, 3 Checklist: Section 7.8: c,, 7,, [6] Section 7.9:, 3, 7, 9 Section 7.8 In Problems 7.8. thru 4: a Draw a direction field and
More informationSystem of First Order Differential Equations
CHAPTER System of First Order Differential Equations In this chapter, we will discuss system of first order differential equations. There are many applications that involving find several unknown functions
More informationA Brief Review of Elementary Ordinary Differential Equations
1 A Brief Review of Elementary Ordinary Differential Equations At various points in the material we will be covering, we will need to recall and use material normally covered in an elementary course on
More informationLinearly Independent Sets and Linearly Dependent Sets
These notes closely follow the presentation of the material given in David C. Lay s textbook Linear Algebra and its Applications (3rd edition). These notes are intended primarily for in-class presentation
More informationPhysics 41 HW Set 1 Chapter 15
Physics 4 HW Set Chapter 5 Serway 8 th OC:, 4, 7 CQ: 4, 8 P: 4, 5, 8, 8, 0, 9,, 4, 9, 4, 5, 5 Discussion Problems:, 57, 59, 67, 74 OC CQ P: 4, 5, 8, 8, 0, 9,, 4, 9, 4, 5, 5 Discussion Problems:, 57, 59,
More informationChapter 4. Forces and Newton s Laws of Motion. continued
Chapter 4 Forces and Newton s Laws of Motion continued 4.9 Static and Kinetic Frictional Forces When an object is in contact with a surface forces can act on the objects. The component of this force acting
More informationby the matrix A results in a vector which is a reflection of the given
Eigenvalues & Eigenvectors Example Suppose Then So, geometrically, multiplying a vector in by the matrix A results in a vector which is a reflection of the given vector about the y-axis We observe that
More informationAP Physics C. Oscillations/SHM Review Packet
AP Physics C Oscillations/SHM Review Packet 1. A 0.5 kg mass on a spring has a displacement as a function of time given by the equation x(t) = 0.8Cos(πt). Find the following: a. The time for one complete
More informationANALYTICAL METHODS FOR ENGINEERS
UNIT 1: Unit code: QCF Level: 4 Credit value: 15 ANALYTICAL METHODS FOR ENGINEERS A/601/1401 OUTCOME - TRIGONOMETRIC METHODS TUTORIAL 1 SINUSOIDAL FUNCTION Be able to analyse and model engineering situations
More informationr (t) = 2r(t) + sin t θ (t) = r(t) θ(t) + 1 = 1 1 θ(t) 1 9.4.4 Write the given system in matrix form x = Ax + f ( ) sin(t) x y 1 0 5 z = dy cos(t)
Solutions HW 9.4.2 Write the given system in matrix form x = Ax + f r (t) = 2r(t) + sin t θ (t) = r(t) θ(t) + We write this as ( ) r (t) θ (t) = ( ) ( ) 2 r(t) θ(t) + ( ) sin(t) 9.4.4 Write the given system
More informationLecture 8 : Dynamic Stability
Lecture 8 : Dynamic Stability Or what happens to small disturbances about a trim condition 1.0 : Dynamic Stability Static stability refers to the tendency of the aircraft to counter a disturbance. Dynamic
More informationCHAPTER 2. Eigenvalue Problems (EVP s) for ODE s
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 informationNonlinear Systems of Ordinary Differential Equations
Differential Equations Massoud Malek Nonlinear Systems of Ordinary Differential Equations Dynamical System. A dynamical system has a state determined by a collection of real numbers, or more generally
More informationElasticity Theory Basics
G22.3033-002: Topics in Computer Graphics: Lecture #7 Geometric Modeling New York University Elasticity Theory Basics Lecture #7: 20 October 2003 Lecturer: Denis Zorin Scribe: Adrian Secord, Yotam Gingold
More informationNonhomogeneous Linear Equations
Nonhomogeneous Linear Equations In this section we learn how to solve second-order nonhomogeneous linear differential equations with constant coefficients, that is, equations of the form ay by cy G x where
More informationEXAMPLE 8: An Electrical System (Mechanical-Electrical Analogy)
EXAMPLE 8: An Electrical System (Mechanical-Electrical Analogy) A completely analogous procedure can be used to find the state equations of electrical systems (and, ultimately, electro-mechanical systems
More informationChapter 3: Mathematical Models and Numerical Methods Involving First-Order Differential Equations
Massasoit Community College Instructor: Office: Email: Phone: Office Hours: Course: Differential Equations Course Number: MATH230-XX Semester: Classroom: Day and Time: Course Description: This course is
More informationUsing Microsoft Excel Built-in Functions and Matrix Operations. EGN 1006 Introduction to the Engineering Profession
Using Microsoft Ecel Built-in Functions and Matri Operations EGN 006 Introduction to the Engineering Profession Ecel Embedded Functions Ecel has a wide variety of Built-in Functions: Mathematical Financial
More informationMAT 242 Differential Equations Mathematics
MAT 242 Differential Equations Mathematics Catalog Course Description: This course includes the following topics: solution of linear and elementary non-linear differential equations by standard methods
More informationUnderstanding Poles and Zeros
MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING 2.14 Analysis and Design of Feedback Control Systems Understanding Poles and Zeros 1 System Poles and Zeros The transfer function
More information2.2 Magic with complex exponentials
2.2. MAGIC WITH COMPLEX EXPONENTIALS 97 2.2 Magic with complex exponentials We don t really know what aspects of complex variables you learned about in high school, so the goal here is to start more or
More informationSimple Harmonic Motion
Simple Harmonic Motion 1 Object To determine the period of motion of objects that are executing simple harmonic motion and to check the theoretical prediction of such periods. 2 Apparatus Assorted weights
More informationCode: MATH 274 Title: ELEMENTARY DIFFERENTIAL EQUATIONS
Code: MATH 274 Title: ELEMENTARY DIFFERENTIAL EQUATIONS Institute: STEM Department: MATHEMATICS Course Description: This is an introductory course in concepts and applications of differential equations.
More informationSimilarity and Diagonalization. Similar Matrices
MATH022 Linear Algebra Brief lecture notes 48 Similarity and Diagonalization Similar Matrices Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that
More informationChapter 20. Vector Spaces and Bases
Chapter 20. Vector Spaces and Bases In this course, we have proceeded step-by-step through low-dimensional Linear Algebra. We have looked at lines, planes, hyperplanes, and have seen that there is no limit
More informationSpring Simple Harmonic Oscillator. Spring constant. Potential Energy stored in a Spring. Understanding oscillations. Understanding oscillations
Spring Simple Harmonic Oscillator Simple Harmonic Oscillations and Resonance We have an object attached to a spring. The object is on a horizontal frictionless surface. We move the object so the spring
More informationDifferential Equations and Linear Algebra Lecture Notes. Simon J.A. Malham. Department of Mathematics, Heriot-Watt University
Differential Equations and Linear Algebra Lecture Notes Simon J.A. Malham Department of Mathematics, Heriot-Watt University Contents Chapter. Linear second order ODEs 5.. Newton s second law 5.2. Springs
More informationwww.mathsbox.org.uk Displacement (x) Velocity (v) Acceleration (a) x = f(t) differentiate v = dx Acceleration Velocity (v) Displacement x
Mechanics 2 : Revision Notes 1. Kinematics and variable acceleration Displacement (x) Velocity (v) Acceleration (a) x = f(t) differentiate v = dx differentiate a = dv = d2 x dt dt dt 2 Acceleration Velocity
More information12.4 UNDRIVEN, PARALLEL RLC CIRCUIT*
+ v C C R L - v i L FIGURE 12.24 The parallel second-order RLC circuit shown in Figure 2.14a. 12.4 UNDRIVEN, PARALLEL RLC CIRCUIT* We will now analyze the undriven parallel RLC circuit shown in Figure
More informationThese axioms must hold for all vectors ū, v, and w in V and all scalars c and d.
DEFINITION: A vector space is a nonempty set V of objects, called vectors, on which are defined two operations, called addition and multiplication by scalars (real numbers), subject to the following axioms
More informationOscillations. Vern Lindberg. June 10, 2010
Oscillations Vern Lindberg June 10, 2010 You have discussed oscillations in Vibs and Waves: we will therefore touch lightly on Chapter 3, mainly trying to refresh your memory and extend the concepts. 1
More informationChapter 6. Orthogonality
6.3 Orthogonal Matrices 1 Chapter 6. Orthogonality 6.3 Orthogonal Matrices Definition 6.4. An n n matrix A is orthogonal if A T A = I. Note. We will see that the columns of an orthogonal matrix must be
More informationPhysics Notes Class 11 CHAPTER 6 WORK, ENERGY AND POWER
1 P a g e Work Physics Notes Class 11 CHAPTER 6 WORK, ENERGY AND POWER When a force acts on an object and the object actually moves in the direction of force, then the work is said to be done by the force.
More informationUniversity of Lille I PC first year list of exercises n 7. Review
University of Lille I PC first year list of exercises n 7 Review Exercise Solve the following systems in 4 different ways (by substitution, by the Gauss method, by inverting the matrix of coefficients
More informationsin(θ) = opp hyp cos(θ) = adj hyp tan(θ) = opp adj
Math, Trigonometr and Vectors Geometr 33º What is the angle equal to? a) α = 7 b) α = 57 c) α = 33 d) α = 90 e) α cannot be determined α Trig Definitions Here's a familiar image. To make predictive models
More information1 Inner Products and Norms on Real Vector Spaces
Math 373: Principles Techniques of Applied Mathematics Spring 29 The 2 Inner Product 1 Inner Products Norms on Real Vector Spaces Recall that an inner product on a real vector space V is a function from
More informationMATH 423 Linear Algebra II Lecture 38: Generalized eigenvectors. Jordan canonical form (continued).
MATH 423 Linear Algebra II Lecture 38: Generalized eigenvectors Jordan canonical form (continued) Jordan canonical form A Jordan block is a square matrix of the form λ 1 0 0 0 0 λ 1 0 0 0 0 λ 0 0 J = 0
More information4.5 Linear Dependence and Linear Independence
4.5 Linear Dependence and Linear Independence 267 32. {v 1, v 2 }, where v 1, v 2 are collinear vectors in R 3. 33. Prove that if S and S are subsets of a vector space V such that S is a subset of S, then
More information1.5 SOLUTION SETS OF LINEAR SYSTEMS
1-2 CHAPTER 1 Linear Equations in Linear Algebra 1.5 SOLUTION SETS OF LINEAR SYSTEMS Many of the concepts and computations in linear algebra involve sets of vectors which are visualized geometrically as
More informationLecture L22-2D Rigid Body Dynamics: Work and Energy
J. Peraire, S. Widnall 6.07 Dynamics Fall 008 Version.0 Lecture L - D Rigid Body Dynamics: Work and Energy In this lecture, we will revisit the principle of work and energy introduced in lecture L-3 for
More information[1] Diagonal factorization
8.03 LA.6: Diagonalization and Orthogonal Matrices [ Diagonal factorization [2 Solving systems of first order differential equations [3 Symmetric and Orthonormal Matrices [ Diagonal factorization Recall:
More informationState of Stress at Point
State of Stress at Point Einstein Notation The basic idea of Einstein notation is that a covector and a vector can form a scalar: This is typically written as an explicit sum: According to this convention,
More informationNotice numbers may change randomly in your assignments and you may have to recalculate solutions for your specific case.
HW1 Possible Solutions Notice numbers may change randomly in your assignments and you may have to recalculate solutions for your specific case. Tipler 14.P.003 An object attached to a spring has simple
More information2.3. Finding polynomial functions. An Introduction:
2.3. Finding polynomial functions. An Introduction: As is usually the case when learning a new concept in mathematics, the new concept is the reverse of the previous one. Remember how you first learned
More informationUnit - 6 Vibrations of Two Degree of Freedom Systems
Unit - 6 Vibrations of Two Degree of Freedom Systems Dr. T. Jagadish. Professor for Post Graduation, Department of Mechanical Engineering, Bangalore Institute of Technology, Bangalore Introduction A two
More informationSecond Order Linear Nonhomogeneous Differential Equations; Method of Undetermined Coefficients. y + p(t) y + q(t) y = g(t), g(t) 0.
Second Order Linear Nonhomogeneous Differential Equations; Method of Undetermined Coefficients We will now turn our attention to nonhomogeneous second order linear equations, equations with the standard
More informationExamination paper for TMA4115 Matematikk 3
Department of Mathematical Sciences Examination paper for TMA45 Matematikk 3 Academic contact during examination: Antoine Julien a, Alexander Schmeding b, Gereon Quick c Phone: a 73 59 77 82, b 40 53 99
More information(Most of the material presented in this chapter is taken from Thornton and Marion, Chap. 7)
Chapter 4. Lagrangian Dynamics (Most of the material presented in this chapter is taken from Thornton and Marion, Chap. 7 4.1 Important Notes on Notation In this chapter, unless otherwise stated, the following
More informationCore Maths C3. Revision Notes
Core Maths C Revision Notes October 0 Core Maths C Algebraic fractions... Cancelling common factors... Multipling and dividing fractions... Adding and subtracting fractions... Equations... 4 Functions...
More informationMechanics 1: Conservation of Energy and Momentum
Mechanics : Conservation of Energy and Momentum If a certain quantity associated with a system does not change in time. We say that it is conserved, and the system possesses a conservation law. Conservation
More informationPractice Test SHM with Answers
Practice Test SHM with Answers MPC 1) If we double the frequency of a system undergoing simple harmonic motion, which of the following statements about that system are true? (There could be more than one
More informationMath, Trigonometry and Vectors. Geometry. Trig Definitions. sin(θ) = opp hyp. cos(θ) = adj hyp. tan(θ) = opp adj. Here's a familiar image.
Math, Trigonometr and Vectors Geometr Trig Definitions Here's a familiar image. To make predictive models of the phsical world, we'll need to make visualizations, which we can then turn into analtical
More informationAPPLICATIONS. are symmetric, but. are not.
CHAPTER III APPLICATIONS Real Symmetric Matrices The most common matrices we meet in applications are symmetric, that is, they are square matrices which are equal to their transposes In symbols, A t =
More informationORDINARY DIFFERENTIAL EQUATIONS
ORDINARY DIFFERENTIAL EQUATIONS GABRIEL NAGY Mathematics Department, Michigan State University, East Lansing, MI, 48824. SEPTEMBER 4, 25 Summary. This is an introduction to ordinary differential equations.
More informationCopyright 2011 Casa Software Ltd. www.casaxps.com
Table of Contents Variable Forces and Differential Equations... 2 Differential Equations... 3 Second Order Linear Differential Equations with Constant Coefficients... 6 Reduction of Differential Equations
More informationExample 4.1 (nonlinear pendulum dynamics with friction) Figure 4.1: Pendulum. asin. k, a, and b. We study stability of the origin x
Lecture 4. LaSalle s Invariance Principle We begin with a motivating eample. Eample 4.1 (nonlinear pendulum dynamics with friction) Figure 4.1: Pendulum Dynamics of a pendulum with friction can be written
More informationChapter 15 Collision Theory
Chapter 15 Collision Theory 151 Introduction 1 15 Reference Frames Relative and Velocities 1 151 Center of Mass Reference Frame 15 Relative Velocities 3 153 Characterizing Collisions 5 154 One-Dimensional
More informationFXA 2008. UNIT G484 Module 2 4.2.3 Simple Harmonic Oscillations 11. frequency of the applied = natural frequency of the
11 FORCED OSCILLATIONS AND RESONANCE POINTER INSTRUMENTS Analogue ammeter and voltmeters, have CRITICAL DAMPING so as to allow the needle pointer to reach its correct position on the scale after a single
More informationDYNAMICAL ANALYSIS OF SILO SURFACE CLEANING ROBOT USING FINITE ELEMENT METHOD
International Journal of Mechanical Engineering and Technology (IJMET) Volume 7, Issue 1, Jan-Feb 2016, pp. 190-202, Article ID: IJMET_07_01_020 Available online at http://www.iaeme.com/ijmet/issues.asp?jtype=ijmet&vtype=7&itype=1
More informationStanding Waves on a String
1 of 6 Standing Waves on a String Summer 2004 Standing Waves on a String If a string is tied between two fixed supports, pulled tightly and sharply plucked at one end, a pulse will travel from one end
More informationphysics 111N work & energy
physics 111N work & energy conservation of energy entirely gravitational potential energy kinetic energy turning into gravitational potential energy gravitational potential energy turning into kinetic
More informationMath 550 Notes. Chapter 7. Jesse Crawford. Department of Mathematics Tarleton State University. Fall 2010
Math 550 Notes Chapter 7 Jesse Crawford Department of Mathematics Tarleton State University Fall 2010 (Tarleton State University) Math 550 Chapter 7 Fall 2010 1 / 34 Outline 1 Self-Adjoint and Normal Operators
More informationOscillations: Mass on a Spring and Pendulums
Chapter 3 Oscillations: Mass on a Spring and Pendulums 3.1 Purpose 3.2 Introduction Galileo is said to have been sitting in church watching the large chandelier swinging to and fro when he decided that
More informationHOOKE S LAW AND OSCILLATIONS
9 HOOKE S LAW AND OSCILLATIONS OBJECTIVE To measure the effect of amplitude, mass, and spring constant on the period of a spring-mass oscillator. INTRODUCTION The force which restores a spring to its equilibrium
More informationAdequate Theory of Oscillator: A Prelude to Verification of Classical Mechanics Part 2
International Letters of Chemistry, Physics and Astronomy Online: 213-9-19 ISSN: 2299-3843, Vol. 3, pp 1-1 doi:1.1852/www.scipress.com/ilcpa.3.1 212 SciPress Ltd., Switzerland Adequate Theory of Oscillator:
More informationSECOND-ORDER LINEAR DIFFERENTIAL EQUATIONS
SECOND-ORDER LINEAR DIFFERENTIAL EQUATIONS A second-order linear differential equation has the form 1 Px d y dx dy Qx dx Rxy Gx where P, Q, R, and G are continuous functions. Equations of this type arise
More information1 of 79 Erik Eberhardt UBC Geological Engineering EOSC 433
Stress & Strain: A review xx yz zz zx zy xy xz yx yy xx yy zz 1 of 79 Erik Eberhardt UBC Geological Engineering EOSC 433 Disclaimer before beginning your problem assignment: Pick up and compare any set
More informationCore Maths C1. Revision Notes
Core Maths C Revision Notes November 0 Core Maths C Algebra... Indices... Rules of indices... Surds... 4 Simplifying surds... 4 Rationalising the denominator... 4 Quadratic functions... 4 Completing the
More information19.7. Applications of Differential Equations. Introduction. Prerequisites. Learning Outcomes. Learning Style
Applications of Differential Equations 19.7 Introduction Blocks 19.2 to 19.6 have introduced several techniques for solving commonly-occurring firstorder and second-order ordinary differential equations.
More informationcos Newington College HSC Mathematics Ext 1 Trial Examination 2011 QUESTION ONE (12 Marks) (b) Find the exact value of if. 2 . 3
1 QUESTION ONE (12 Marks) Marks (a) Find tan x e 1 2 cos dx x (b) Find the exact value of if. 2 (c) Solve 5 3 2x 1. 3 (d) If are the roots of the equation 2 find the value of. (e) Use the substitution
More information1 2 3 1 1 2 x = + x 2 + x 4 1 0 1
(d) If the vector b is the sum of the four columns of A, write down the complete solution to Ax = b. 1 2 3 1 1 2 x = + x 2 + x 4 1 0 0 1 0 1 2. (11 points) This problem finds the curve y = C + D 2 t which
More informationNotes on Determinant
ENGG2012B Advanced Engineering Mathematics Notes on Determinant Lecturer: Kenneth Shum Lecture 9-18/02/2013 The determinant of a system of linear equations determines whether the solution is unique, without
More informationKERN COMMUNITY COLLEGE DISTRICT CERRO COSO COLLEGE PHYS C111 COURSE OUTLINE OF RECORD
KERN COMMUNITY COLLEGE DISTRICT CERRO COSO COLLEGE PHYS C111 COURSE OUTLINE OF RECORD 1. DISCIPLINE AND COURSE NUMBER: PHYS C111 2. COURSE TITLE: Mechanics 3. SHORT BANWEB TITLE: Mechanics 4. COURSE AUTHOR:
More informationStudent name: Earlham College. Fall 2011 December 15, 2011
Student name: Earlham College MATH 320: Differential Equations Final exam - In class part Fall 2011 December 15, 2011 Instructions: This is a regular closed-book test, and is to be taken without the use
More informationLinear Equations in One Variable
Linear Equations in One Variable MATH 101 College Algebra J. Robert Buchanan Department of Mathematics Summer 2012 Objectives In this section we will learn how to: Recognize and combine like terms. Solve
More informationChapter 6 Work and Energy
Chapter 6 WORK AND ENERGY PREVIEW Work is the scalar product of the force acting on an object and the displacement through which it acts. When work is done on or by a system, the energy of that system
More informationAP1 Oscillations. 1. Which of the following statements about a spring-block oscillator in simple harmonic motion about its equilibrium point is false?
1. Which of the following statements about a spring-block oscillator in simple harmonic motion about its equilibrium point is false? (A) The displacement is directly related to the acceleration. (B) The
More informationSensor Performance Metrics
Sensor Performance Metrics Michael Todd Professor and Vice Chair Dept. of Structural Engineering University of California, San Diego mdtodd@ucsd.edu Email me if you want a copy. Outline Sensors as dynamic
More informationSlide 10.1. Basic system Models
Slide 10.1 Basic system Models Objectives: Devise Models from basic building blocks of mechanical, electrical, fluid and thermal systems Recognize analogies between mechanical, electrical, fluid and thermal
More informationLecture L2 - Degrees of Freedom and Constraints, Rectilinear Motion
S. Widnall 6.07 Dynamics Fall 009 Version.0 Lecture L - Degrees of Freedom and Constraints, Rectilinear Motion Degrees of Freedom Degrees of freedom refers to the number of independent spatial coordinates
More informationv v ax v a x a v a v = = = Since F = ma, it follows that a = F/m. The mass of the arrow is unchanged, and ( )
Week 3 homework IMPORTANT NOTE ABOUT WEBASSIGN: In the WebAssign versions of these problems, various details have been changed, so that the answers will come out differently. The method to find the solution
More informationElementary Differential Equations and Boundary Value Problems. 10th Edition International Student Version
Brochure More information from http://www.researchandmarkets.com/reports/3148843/ Elementary Differential Equations and Boundary Value Problems. 10th Edition International Student Version Description:
More informationBiggar High School Mathematics Department. National 5 Learning Intentions & Success Criteria: Assessing My Progress
Biggar High School Mathematics Department National 5 Learning Intentions & Success Criteria: Assessing My Progress Expressions & Formulae Topic Learning Intention Success Criteria I understand this Approximation
More information7. Beats. sin( + λ) + sin( λ) = 2 cos(λ) sin( )
34 7. Beats 7.1. What beats are. Musicians tune their instruments using beats. Beats occur when two very nearby pitches are sounded simultaneously. We ll make a mathematical study of this effect, using
More informationSystems of Linear Equations
Systems of Linear Equations Beifang Chen Systems of linear equations Linear systems A linear equation in variables x, x,, x n is an equation of the form a x + a x + + a n x n = b, where a, a,, a n and
More informationFigure 1.1 Vector A and Vector F
CHAPTER I VECTOR QUANTITIES Quantities are anything which can be measured, and stated with number. Quantities in physics are divided into two types; scalar and vector quantities. Scalar quantities have
More information