Oscillatory Motion. Simple pendulum: linear Hooke s Law restoring force for small angular deviations. Oscillatory solution

Size: px
Start display at page:

Download "Oscillatory Motion. Simple pendulum: linear Hooke s Law restoring force for small angular deviations. Oscillatory solution"

Transcription

1 Oscillatory Motion Simple pendulum: linear Hooke s Law restoring force for small angular deviations d 2 θ dt 2 = g l θ θ l Oscillatory solution θ(t) =θ 0 sin(ωt + φ) F with characteristic angular frequency Ω = g/l

2 Oscillatory Motion Simple pendulum: linear Hooke s Law restoring force for small angular deviations d 2 θ dt 2 = g l θ small angle approximation θ l Oscillatory solution θ(t) =θ 0 sin(ωt + φ) F with characteristic angular frequency Ω = g/l

3 Oscillatory Motion Important features: oscillations are perfectly regular in time and continue forever (no decay) θ l angular frequency is independent of the mass m and amplitude θ 0 F

4 Oscillatory Motion Usual trick for numerical solution: decompose the secondorder ODE into a system of first-order equations dω dt = g l θ dθ dt = ω Convert to a system of difference equations by discretizing the time variable, t t i = i t

5 Oscillatory Motion Simple pendulum is a Hamiltonian system described by an angular coordinate where I = ml 2 θ and angular momentum is the moment of inertia: L = Iω H = L2 2I IΩ2 θ 2 Hamilton s equations reproduce the first-order pair: L = H θ = IΩ2 θ θ = H L = L I dω dt = g l θ dθ dt = ω

6 Oscillatory Motion Hamiltonian system implies conservation of energy and preservation of the symplectic symmetry Neither are satisfied with Euler updates (small errors accumulate over each cycle) Important to use higher-order integration methods

7 Dissipation Simple pendulum is highly idealized More realistic models might include friction or damping terms proportional to the velocity d 2 θ dt 2 = g l θ q dθ dt Boring: Analytic solution shows that oscillations die out θ(t) e qt/2

8 Dissipation Depending on the strength of the parameter q, the behaviour may be under-, over-, or critically damped (Ω ] θ(t) =θ 0 e sin[ qt/ q2) 1/2 t + φ [ θ(t) =θ 0 exp q/2 ± ( 1 4 q2 Ω 2) ] 1/2 t θ(t) = ( θ 0 + Ct ) e qt/2

9 Dissipation + Driving Force We now add a driving force to the problem Assume a sinusoidal form with strength F D and angular frequency Ω D d 2 θ dt 2 = g l θ q dθ dt + F D sin Ω D t Pumps energy into or out of the system and competes with the natural frequency when Ω D Ω

10 Dissipation + Driving Force In this case, the steady state solution is sinusoidal in the driving frequency: θ(t) =θ 0 sin(ω D t + φ) But the amplitude has a resonance when the driving frequency is close to the natural frequency: θ 0 = F D (Ω2 Ω 2 D )2 +(qω D ) 2

11 Dissipation + Driving Force In this case, the steady state solution is sinusoidal in the driving frequency: θ(t) =θ 0 sin(ω D t + φ) But the amplitude has a resonance when the driving frequency is close to the natural frequency: natural frequency θ 0 = F D (Ω2 Ω 2 D )2 +(qω D ) 2 transient dies out

12 Non-linearity Small angular approximation is no longer appropriate Pendulum may swing completely around its pivot θ No analytical solution with a sinusoidal restoring force F d 2 θ dt 2 = g l sin θ q dθ dt + F D sin Ω D t

13 Chaotic Motion When F D is sufficiently large, the motion has no simple long-time behaviour The motion never repeats and is said to be chaotic But it is not random

14 Chaotic Motion What does it mean to be non-repeating, unpredictable, and yet still deterministic? Remember: the behaviour is unique and governed by the specification of the initial value problem

15 Chaotic Motion For chaotic systems, infitesimal variations in the initial conditions lead to different long-time behaviour For example, two identical chaotic pendulums with nearly identical initially conditions will show exponential growth in the angular distance θ θ 1 θ 2

16 Transition to Chaotic Motion Regular and chaotic regions distinguished by a change of sign of the Lyapunov exponent λ θ e λt λ < 0 λ > 0

17 Lyapunov Exponents In general, there is a Lyapunov exponent associated with each phase-space degree of freedom δ 1 (t) δ 2 (t). δ N (t) = M(t) δ 1 (0) δ 2 (0). δ N (0) M(t) =U T exp λ 1 t λ 2 t... U λn t For a conservative system: N λ k =0 k=1 For a dissipative system: N λ k < 0 k=1

18 Lyapunov Exponents In general, there is a Lyapunov exponent associated with each phase-space degree of freedom δ 1 (t) δ 2 (t). δ N (t) = M(t) δ 1 (0) δ 2 (0). δ N (0) For a conservative system: M(t) =U T exp N k=1 λ k =0 λ 1 t λ 2 t... U λn t Unitary transformation: For a dissipative system: N k=1 λ k < 0 matrix of Eigenvectors of M

19 The View from Phase Space Chaotic path through phase space still exhibits structure There are many orbits that are nearly closed and persist for one or two cycles

20 Strange Attractor Poincaré section: only plot points at times in phase with the driving force Ω D t =2nπ for integer n For a wide range of initial conditions, trajectories lie on this surface of points, known as a strange attractor

21 Strange Attractor Poincaré section: only plot points at times in phase with the driving force Ω D t =2nπ for integer n For a wide range of initial conditions, trajectories lie on this surface of points, known as a strange attractor fractal structure

22 Period Doubling For the pendulum, the route to chaos is via period doubling The system shows response at a subharmonic Ω D /2

23 Period Doubling Bifurcation diagram plots a Poincaré section versus driving force Regularity in windows of period 2 n Feigenbaum delta: δ n F n F n 1 F n+1 F n Universal value δ 4.669

Oscillatory Motion. Simple pendulum: linear Hooke s Law restoring force for small angular deviations. small angle approximation. Oscillatory solution

Oscillatory Motion. Simple pendulum: linear Hooke s Law restoring force for small angular deviations. small angle approximation. Oscillatory solution Oscillatory Motion Simple pendulum: linear Hooke s Law restoring force for small angular deviations d 2 θ dt 2 = g l θ small angle approximation θ l Oscillatory solution θ(t) =θ 0 sin(ωt + φ) F with characteristic

More information

Chaotic motion. Phys 420/580 Lecture 10

Chaotic motion. Phys 420/580 Lecture 10 Chaotic motion Phys 420/580 Lecture 10 Finite-difference equations Finite difference equation approximates a differential equation as an iterative map (x n+1,v n+1 )=M[(x n,v n )] Evolution from time t

More information

Chaotic motion. Phys 750 Lecture 9

Chaotic motion. Phys 750 Lecture 9 Chaotic motion Phys 750 Lecture 9 Finite-difference equations Finite difference equation approximates a differential equation as an iterative map (x n+1,v n+1 )=M[(x n,v n )] Evolution from time t =0to

More information

Lecture V : Oscillatory motion and spectral analysis

Lecture V : Oscillatory motion and spectral analysis Lecture V : Oscillatory motion and spectral analysis I. IDEAL PENDULUM AND STABILITY ANALYSIS Let us remind ourselves of the equation of motion for the pendulum. Remembering that the external torque applied

More information

Physics Mechanics. Lecture 32 Oscillations II

Physics Mechanics. Lecture 32 Oscillations II Physics 170 - Mechanics Lecture 32 Oscillations II Gravitational Potential Energy A plot of the gravitational potential energy U g looks like this: Energy Conservation Total mechanical energy of an object

More information

Midterm EXAM PHYS 401 (Spring 2012), 03/20/12

Midterm EXAM PHYS 401 (Spring 2012), 03/20/12 Midterm EXAM PHYS 401 (Spring 2012), 03/20/12 Name: Signature: Duration: 75 minutes Show all your work for full/partial credit! In taking this exam you confirm to adhere to the Aggie Honor Code: An Aggie

More information

MAS212 Assignment #2: The damped driven pendulum

MAS212 Assignment #2: The damped driven pendulum MAS Assignment #: The damped driven pendulum Sam Dolan (January 8 Introduction In this assignment we study the motion of a rigid pendulum of length l and mass m, shown in Fig., using both analytical and

More information

CHAOS -SOME BASIC CONCEPTS

CHAOS -SOME BASIC CONCEPTS CHAOS -SOME BASIC CONCEPTS Anders Ekberg INTRODUCTION This report is my exam of the "Chaos-part" of the course STOCHASTIC VIBRATIONS. I m by no means any expert in the area and may well have misunderstood

More information

Physics 106b: Lecture 7 25 January, 2018

Physics 106b: Lecture 7 25 January, 2018 Physics 106b: Lecture 7 25 January, 2018 Hamiltonian Chaos: Introduction Integrable Systems We start with systems that do not exhibit chaos, but instead have simple periodic motion (like the SHO) with

More information

The phenomenon: complex motion, unusual geometry

The phenomenon: complex motion, unusual geometry Part I The phenomenon: complex motion, unusual geometry Chapter 1 Chaotic motion 1.1 What is chaos? Certain long-lasting, sustained motion repeats itself exactly, periodically. Examples from everyday life

More information

Rotational Number Approach to a Damped Pendulum under Parametric Forcing

Rotational Number Approach to a Damped Pendulum under Parametric Forcing Journal of the Korean Physical Society, Vol. 44, No. 3, March 2004, pp. 518 522 Rotational Number Approach to a Damped Pendulum under Parametric Forcing Eun-Ah Kim and K.-C. Lee Department of Physics,

More information

Theoretical physics. Deterministic chaos in classical physics. Martin Scholtz

Theoretical physics. Deterministic chaos in classical physics. Martin Scholtz Theoretical physics Deterministic chaos in classical physics Martin Scholtz scholtzzz@gmail.com Fundamental physical theories and role of classical mechanics. Intuitive characteristics of chaos. Newton

More information

for changing independent variables. Most simply for a function f(x) the Legendre transformation f(x) B(s) takes the form B(s) = xs f(x) with s = df

for changing independent variables. Most simply for a function f(x) the Legendre transformation f(x) B(s) takes the form B(s) = xs f(x) with s = df Physics 106a, Caltech 1 November, 2018 Lecture 10: Hamiltonian Mechanics I The Hamiltonian In the Hamiltonian formulation of dynamics each second order ODE given by the Euler- Lagrange equation in terms

More information

PH 120 Project # 2: Pendulum and chaos

PH 120 Project # 2: Pendulum and chaos PH 120 Project # 2: Pendulum and chaos Due: Friday, January 16, 2004 In PH109, you studied a simple pendulum, which is an effectively massless rod of length l that is fixed at one end with a small mass

More information

ANALYTICAL MECHANICS. LOUIS N. HAND and JANET D. FINCH CAMBRIDGE UNIVERSITY PRESS

ANALYTICAL MECHANICS. LOUIS N. HAND and JANET D. FINCH CAMBRIDGE UNIVERSITY PRESS ANALYTICAL MECHANICS LOUIS N. HAND and JANET D. FINCH CAMBRIDGE UNIVERSITY PRESS Preface xi 1 LAGRANGIAN MECHANICS l 1.1 Example and Review of Newton's Mechanics: A Block Sliding on an Inclined Plane 1

More information

Linear and Nonlinear Oscillators (Lecture 2)

Linear and Nonlinear Oscillators (Lecture 2) Linear and Nonlinear Oscillators (Lecture 2) January 25, 2016 7/441 Lecture outline A simple model of a linear oscillator lies in the foundation of many physical phenomena in accelerator dynamics. A typical

More information

From Last Time. Gravitational forces are apparent at a wide range of scales. Obeys

From Last Time. Gravitational forces are apparent at a wide range of scales. Obeys From Last Time Gravitational forces are apparent at a wide range of scales. Obeys F gravity (Mass of object 1) (Mass of object 2) square of distance between them F = 6.7 10-11 m 1 m 2 d 2 Gravitational

More information

Project 3: Pendulum. Physics 2300 Spring 2018 Lab partner

Project 3: Pendulum. Physics 2300 Spring 2018 Lab partner Physics 2300 Spring 2018 Name Lab partner Project 3: Pendulum In this project you will explore the behavior of a pendulum. There is no better example of a system that seems simple at first but turns out

More information

CHALMERS, GÖTEBORGS UNIVERSITET. EXAM for DYNAMICAL SYSTEMS. COURSE CODES: TIF 155, FIM770GU, PhD

CHALMERS, GÖTEBORGS UNIVERSITET. EXAM for DYNAMICAL SYSTEMS. COURSE CODES: TIF 155, FIM770GU, PhD CHALMERS, GÖTEBORGS UNIVERSITET EXAM for DYNAMICAL SYSTEMS COURSE CODES: TIF 155, FIM770GU, PhD Time: Place: Teachers: Allowed material: Not allowed: April 06, 2018, at 14 00 18 00 Johanneberg Kristian

More information

LECTURE 8: DYNAMICAL SYSTEMS 7

LECTURE 8: DYNAMICAL SYSTEMS 7 15-382 COLLECTIVE INTELLIGENCE S18 LECTURE 8: DYNAMICAL SYSTEMS 7 INSTRUCTOR: GIANNI A. DI CARO GEOMETRIES IN THE PHASE SPACE Damped pendulum One cp in the region between two separatrix Separatrix Basin

More information

Physics 235 Chapter 4. Chapter 4 Non-Linear Oscillations and Chaos

Physics 235 Chapter 4. Chapter 4 Non-Linear Oscillations and Chaos Chapter 4 Non-Linear Oscillations and Chaos Non-Linear Differential Equations Up to now we have considered differential equations with terms that are proportional to the acceleration, the velocity, and

More information

Computational Physics (6810): Session 8

Computational Physics (6810): Session 8 Computational Physics (6810): Session 8 Dick Furnstahl Nuclear Theory Group OSU Physics Department February 24, 2014 Differential equation solving Session 7 Preview Session 8 Stuff Solving differential

More information

More Details Fixed point of mapping is point that maps into itself, i.e., x n+1 = x n.

More Details Fixed point of mapping is point that maps into itself, i.e., x n+1 = x n. More Details Fixed point of mapping is point that maps into itself, i.e., x n+1 = x n. If there are points which, after many iterations of map then fixed point called an attractor. fixed point, If λ

More information

Mechanical Resonance and Chaos

Mechanical Resonance and Chaos Mechanical Resonance and Chaos You will use the apparatus in Figure 1 to investigate regimes of increasing complexity. Figure 1. The rotary pendulum (from DeSerio, www.phys.ufl.edu/courses/phy483l/group_iv/chaos/chaos.pdf).

More information

Chapter 14. Oscillations. Oscillations Introductory Terminology Simple Harmonic Motion:

Chapter 14. Oscillations. Oscillations Introductory Terminology Simple Harmonic Motion: Chapter 14 Oscillations Oscillations Introductory Terminology Simple Harmonic Motion: Kinematics Energy Examples of Simple Harmonic Oscillators Damped and Forced Oscillations. Resonance. Periodic Motion

More information

Oscillatory Motion SHM

Oscillatory Motion SHM Chapter 15 Oscillatory Motion SHM Dr. Armen Kocharian Periodic Motion Periodic motion is motion of an object that regularly repeats The object returns to a given position after a fixed time interval A

More information

A plane autonomous system is a pair of simultaneous first-order differential equations,

A plane autonomous system is a pair of simultaneous first-order differential equations, Chapter 11 Phase-Plane Techniques 11.1 Plane Autonomous Systems A plane autonomous system is a pair of simultaneous first-order differential equations, ẋ = f(x, y), ẏ = g(x, y). This system has an equilibrium

More information

Chaos Theory. Namit Anand Y Integrated M.Sc.( ) Under the guidance of. Prof. S.C. Phatak. Center for Excellence in Basic Sciences

Chaos Theory. Namit Anand Y Integrated M.Sc.( ) Under the guidance of. Prof. S.C. Phatak. Center for Excellence in Basic Sciences Chaos Theory Namit Anand Y1111033 Integrated M.Sc.(2011-2016) Under the guidance of Prof. S.C. Phatak Center for Excellence in Basic Sciences University of Mumbai 1 Contents 1 Abstract 3 1.1 Basic Definitions

More information

Newton s laws. Chapter 1. Not: Quantum Mechanics / Relativistic Mechanics

Newton s laws. Chapter 1. Not: Quantum Mechanics / Relativistic Mechanics PHYB54 Revision Chapter 1 Newton s laws Not: Quantum Mechanics / Relativistic Mechanics Isaac Newton 1642-1727 Classical mechanics breaks down if: 1) high speed, v ~ c 2) microscopic/elementary particles

More information

OSCILLATIONS ABOUT EQUILIBRIUM

OSCILLATIONS ABOUT EQUILIBRIUM OSCILLATIONS ABOUT EQUILIBRIUM Chapter 13 Units of Chapter 13 Periodic Motion Simple Harmonic Motion Connections between Uniform Circular Motion and Simple Harmonic Motion The Period of a Mass on a Spring

More information

Contents Dynamical Systems Stability of Dynamical Systems: Linear Approach

Contents Dynamical Systems Stability of Dynamical Systems: Linear Approach Contents 1 Dynamical Systems... 1 1.1 Introduction... 1 1.2 DynamicalSystems andmathematical Models... 1 1.3 Kinematic Interpretation of a System of Differential Equations... 3 1.4 Definition of a Dynamical

More information

Oscillations. Simple Harmonic Motion of a Mass on a Spring The equation of motion for a mass m is attached to a spring of constant k is

Oscillations. Simple Harmonic Motion of a Mass on a Spring The equation of motion for a mass m is attached to a spring of constant k is Dr. Alain Brizard College Physics I (PY 10) Oscillations Textbook Reference: Chapter 14 sections 1-8. Simple Harmonic Motion of a Mass on a Spring The equation of motion for a mass m is attached to a spring

More information

Forced Oscillations in a Linear System Problems

Forced Oscillations in a Linear System Problems Forced Oscillations in a Linear System Problems Summary of the Principal Formulas The differential equation of forced oscillations for the kinematic excitation: ϕ + 2γ ϕ + ω 2 0ϕ = ω 2 0φ 0 sin ωt. Steady-state

More information

Chapter 14 Oscillations

Chapter 14 Oscillations Chapter 14 Oscillations Chapter Goal: To understand systems that oscillate with simple harmonic motion. Slide 14-2 Chapter 14 Preview Slide 14-3 Chapter 14 Preview Slide 14-4 Chapter 14 Preview Slide 14-5

More information

PHYSICS. Chapter 15 Lecture FOR SCIENTISTS AND ENGINEERS A STRATEGIC APPROACH 4/E RANDALL D. KNIGHT Pearson Education, Inc.

PHYSICS. Chapter 15 Lecture FOR SCIENTISTS AND ENGINEERS A STRATEGIC APPROACH 4/E RANDALL D. KNIGHT Pearson Education, Inc. PHYSICS FOR SCIENTISTS AND ENGINEERS A STRATEGIC APPROACH 4/E Chapter 15 Lecture RANDALL D. KNIGHT Chapter 15 Oscillations IN THIS CHAPTER, you will learn about systems that oscillate in simple harmonic

More information

Chapter 15 - Oscillations

Chapter 15 - Oscillations The pendulum of the mind oscillates between sense and nonsense, not between right and wrong. -Carl Gustav Jung David J. Starling Penn State Hazleton PHYS 211 Oscillatory motion is motion that is periodic

More information

Simple and Physical Pendulums Challenge Problem Solutions

Simple and Physical Pendulums Challenge Problem Solutions Simple and Physical Pendulums Challenge Problem Solutions Problem 1 Solutions: For this problem, the answers to parts a) through d) will rely on an analysis of the pendulum motion. There are two conventional

More information

NONLINEAR DYNAMICS AND CHAOS. Numerical integration. Stability analysis

NONLINEAR DYNAMICS AND CHAOS. Numerical integration. Stability analysis LECTURE 3: FLOWS NONLINEAR DYNAMICS AND CHAOS Patrick E McSharr Sstems Analsis, Modelling & Prediction Group www.eng.o.ac.uk/samp patrick@mcsharr.net Tel: +44 83 74 Numerical integration Stabilit analsis

More information

Selected Topics in Physics a lecture course for 1st year students by W.B. von Schlippe Spring Semester 2007

Selected Topics in Physics a lecture course for 1st year students by W.B. von Schlippe Spring Semester 2007 Selected Topics in Physics a lecture course for st year students by W.B. von Schlippe Spring Semester 7 Lecture : Oscillations simple harmonic oscillations; coupled oscillations; beats; damped oscillations;

More information

Dynamical Systems and Chaos Part I: Theoretical Techniques. Lecture 4: Discrete systems + Chaos. Ilya Potapov Mathematics Department, TUT Room TD325

Dynamical Systems and Chaos Part I: Theoretical Techniques. Lecture 4: Discrete systems + Chaos. Ilya Potapov Mathematics Department, TUT Room TD325 Dynamical Systems and Chaos Part I: Theoretical Techniques Lecture 4: Discrete systems + Chaos Ilya Potapov Mathematics Department, TUT Room TD325 Discrete maps x n+1 = f(x n ) Discrete time steps. x 0

More information

Chapter 14. PowerPoint Lectures for University Physics, Thirteenth Edition Hugh D. Young and Roger A. Freedman. Lectures by Wayne Anderson

Chapter 14. PowerPoint Lectures for University Physics, Thirteenth Edition Hugh D. Young and Roger A. Freedman. Lectures by Wayne Anderson Chapter 14 Periodic Motion PowerPoint Lectures for University Physics, Thirteenth Edition Hugh D. Young and Roger A. Freedman Lectures by Wayne Anderson Goals for Chapter 14 To describe oscillations in

More information

Creating and Analyzing Chaotic Attractors Using Mathematica Presented at the 2013 MAA MathFest

Creating and Analyzing Chaotic Attractors Using Mathematica Presented at the 2013 MAA MathFest Creating and Analyzing Chaotic Attractors Using Mathematica Presented at the 2013 MAA MathFest Ulrich Hoensch 1 Rocky Mountain College Billings, Montana hoenschu@rocky.edu Saturday, August 3, 2013 1 Travel

More information

2.4 Models of Oscillation

2.4 Models of Oscillation 2.4 Models of Oscillation In this section we give three examples of oscillating physical systems that can be modeled by the harmonic oscillator equation. Such models are ubiquitous in physics, but are

More information

Lecture 11 : Overview

Lecture 11 : Overview Lecture 11 : Overview Error in Assignment 3 : In Eq. 1, Hamiltonian should be H = p2 r 2m + p2 ϕ 2mr + (p z ea z ) 2 2 2m + eφ (1) Error in lecture 10, slide 7, Eq. (21). Should be S(q, α, t) m Q = β =

More information

2 Discrete growth models, logistic map (Murray, Chapter 2)

2 Discrete growth models, logistic map (Murray, Chapter 2) 2 Discrete growth models, logistic map (Murray, Chapter 2) As argued in Lecture 1 the population of non-overlapping generations can be modelled as a discrete dynamical system. This is an example of an

More information

Problem 1: Lagrangians and Conserved Quantities. Consider the following action for a particle of mass m moving in one dimension

Problem 1: Lagrangians and Conserved Quantities. Consider the following action for a particle of mass m moving in one dimension 105A Practice Final Solutions March 13, 01 William Kelly Problem 1: Lagrangians and Conserved Quantities Consider the following action for a particle of mass m moving in one dimension S = dtl = mc dt 1

More information

THREE DIMENSIONAL SYSTEMS. Lecture 6: The Lorenz Equations

THREE DIMENSIONAL SYSTEMS. Lecture 6: The Lorenz Equations THREE DIMENSIONAL SYSTEMS Lecture 6: The Lorenz Equations 6. The Lorenz (1963) Equations The Lorenz equations were originally derived by Saltzman (1962) as a minimalist model of thermal convection in a

More information

Chapter 14 Oscillations. Copyright 2009 Pearson Education, Inc.

Chapter 14 Oscillations. Copyright 2009 Pearson Education, Inc. Chapter 14 Oscillations Oscillations of a Spring Simple Harmonic Motion Energy in the Simple Harmonic Oscillator Simple Harmonic Motion Related to Uniform Circular Motion The Simple Pendulum The Physical

More information

Lecture 8 Phase Space, Part 2. 1 Surfaces of section. MATH-GA Mechanics

Lecture 8 Phase Space, Part 2. 1 Surfaces of section. MATH-GA Mechanics Lecture 8 Phase Space, Part 2 MATH-GA 2710.001 Mechanics 1 Surfaces of section Thus far, we have highlighted the value of phase portraits, and seen that valuable information can be extracted by looking

More information

Chaos in the Hénon-Heiles system

Chaos in the Hénon-Heiles system Chaos in the Hénon-Heiles system University of Karlstad Christian Emanuelsson Analytical Mechanics FYGC04 Abstract This paper briefly describes how the Hénon-Helies system exhibits chaos. First some subjects

More information

Chapter 15. Oscillatory Motion

Chapter 15. Oscillatory Motion Chapter 15 Oscillatory Motion Part 2 Oscillations and Mechanical Waves Periodic motion is the repeating motion of an object in which it continues to return to a given position after a fixed time interval.

More information

Damped & forced oscillators

Damped & forced oscillators SEISMOLOGY I Laurea Magistralis in Physics of the Earth and of the Environment Damped & forced oscillators Fabio ROMANELLI Dept. Earth Sciences Università degli studi di Trieste romanel@dst.units.it Damped

More information

Transitioning to Chaos in a Simple Mechanical Oscillator

Transitioning to Chaos in a Simple Mechanical Oscillator Transitioning to Chaos in a Simple Mechanical Oscillator Hwan Bae Physics Department, The College of Wooster, Wooster, Ohio 69, USA (Dated: May 9, 8) We vary the magnetic damping, driver frequency, and

More information

Lecture 1: A Preliminary to Nonlinear Dynamics and Chaos

Lecture 1: A Preliminary to Nonlinear Dynamics and Chaos Lecture 1: A Preliminary to Nonlinear Dynamics and Chaos Autonomous Systems A set of coupled autonomous 1st-order ODEs. Here "autonomous" means that the right hand side of the equations does not explicitly

More information

Vibrations: Second Order Systems with One Degree of Freedom, Free Response

Vibrations: Second Order Systems with One Degree of Freedom, Free Response Single Degree of Freedom System 1.003J/1.053J Dynamics and Control I, Spring 007 Professor Thomas Peacock 5//007 Lecture 0 Vibrations: Second Order Systems with One Degree of Freedom, Free Response Single

More information

11/17/10. Chapter 14. Oscillations. Chapter 14. Oscillations Topics: Simple Harmonic Motion. Simple Harmonic Motion

11/17/10. Chapter 14. Oscillations. Chapter 14. Oscillations Topics: Simple Harmonic Motion. Simple Harmonic Motion 11/17/10 Chapter 14. Oscillations This striking computergenerated image demonstrates an important type of motion: oscillatory motion. Examples of oscillatory motion include a car bouncing up and down,

More information

LAST TIME: Simple Pendulum:

LAST TIME: Simple Pendulum: LAST TIME: Simple Pendulum: The displacement from equilibrium, x is the arclength s = L. s / L x / L Accelerating & Restoring Force in the tangential direction, taking cw as positive initial displacement

More information

Chapter 13: Oscillatory Motions

Chapter 13: Oscillatory Motions Chapter 13: Oscillatory Motions Simple harmonic motion Spring and Hooe s law When a mass hanging from a spring and in equilibrium, the Newton s nd law says: Fy ma Fs Fg 0 Fs Fg This means the force due

More information

Chaos. Lendert Gelens. KU Leuven - Vrije Universiteit Brussel Nonlinear dynamics course - VUB

Chaos. Lendert Gelens. KU Leuven - Vrije Universiteit Brussel   Nonlinear dynamics course - VUB Chaos Lendert Gelens KU Leuven - Vrije Universiteit Brussel www.gelenslab.org Nonlinear dynamics course - VUB Examples of chaotic systems: the double pendulum? θ 1 θ θ 2 Examples of chaotic systems: the

More information

MAS212 Assignment #2: The damped driven pendulum

MAS212 Assignment #2: The damped driven pendulum MAS Assignment #: The damped driven pendulum Dr. Sam Dolan (s.dolan@sheffield.ac.uk) Introduction: In this assignment you will use Python to investigate a non-linear differential equation which models

More information

Notes on the Periodically Forced Harmonic Oscillator

Notes on the Periodically Forced Harmonic Oscillator Notes on the Periodically orced Harmonic Oscillator Warren Weckesser Math 38 - Differential Equations 1 The Periodically orced Harmonic Oscillator. By periodically forced harmonic oscillator, we mean the

More information

1 Simple Harmonic Oscillator

1 Simple Harmonic Oscillator Physics 1a Waves Lecture 3 Caltech, 10/09/18 1 Simple Harmonic Oscillator 1.4 General properties of Simple Harmonic Oscillator 1.4.4 Superposition of two independent SHO Suppose we have two SHOs described

More information

Lesson 4: Non-fading Memory Nonlinearities

Lesson 4: Non-fading Memory Nonlinearities Lesson 4: Non-fading Memory Nonlinearities Nonlinear Signal Processing SS 2017 Christian Knoll Signal Processing and Speech Communication Laboratory Graz University of Technology June 22, 2017 NLSP SS

More information

2.4 Harmonic Oscillator Models

2.4 Harmonic Oscillator Models 2.4 Harmonic Oscillator Models In this section we give three important examples from physics of harmonic oscillator models. Such models are ubiquitous in physics, but are also used in chemistry, biology,

More information

Nonlinear Oscillations and Chaos

Nonlinear Oscillations and Chaos CHAPTER 4 Nonlinear Oscillations and Chaos 4-. l l = l + d s d d l l = l + d m θ m (a) (b) (c) The unetended length of each spring is, as shown in (a). In order to attach the mass m, each spring must be

More information

Oscillations. Simple Harmonic Motion (SHM) Position, Velocity, Acceleration SHM Forces SHM Energy Period of oscillation Damping and Resonance

Oscillations. Simple Harmonic Motion (SHM) Position, Velocity, Acceleration SHM Forces SHM Energy Period of oscillation Damping and Resonance Oscillations Simple Harmonic Motion (SHM) Position, Velocity, Acceleration SHM Forces SHM Energy Period of oscillation Damping and Resonance 1 Revision problem Please try problem #31 on page 480 A pendulum

More information

Chapter 13 Lecture. Essential University Physics Richard Wolfson 2 nd Edition. Oscillatory Motion Pearson Education, Inc.

Chapter 13 Lecture. Essential University Physics Richard Wolfson 2 nd Edition. Oscillatory Motion Pearson Education, Inc. Chapter 13 Lecture Essential University Physics Richard Wolfson nd Edition Oscillatory Motion Slide 13-1 In this lecture you ll learn To describe the conditions under which oscillatory motion occurs To

More information

Lecture 6. Lorenz equations and Malkus' waterwheel Some properties of the Lorenz Eq.'s Lorenz Map Towards definitions of:

Lecture 6. Lorenz equations and Malkus' waterwheel Some properties of the Lorenz Eq.'s Lorenz Map Towards definitions of: Lecture 6 Chaos Lorenz equations and Malkus' waterwheel Some properties of the Lorenz Eq.'s Lorenz Map Towards definitions of: Chaos, Attractors and strange attractors Transient chaos Lorenz Equations

More information

Edward Lorenz. Professor of Meteorology at the Massachusetts Institute of Technology

Edward Lorenz. Professor of Meteorology at the Massachusetts Institute of Technology The Lorenz system Edward Lorenz Professor of Meteorology at the Massachusetts Institute of Technology In 1963 derived a three dimensional system in efforts to model long range predictions for the weather

More information

Homework 7: # 4.22, 5.15, 5.21, 5.23, Foucault pendulum

Homework 7: # 4.22, 5.15, 5.21, 5.23, Foucault pendulum Homework 7: # 4., 5.15, 5.1, 5.3, Foucault pendulum Michael Good Oct 9, 4 4. A projectile is fired horizontally along Earth s surface. Show that to a first approximation the angular deviation from the

More information

arxiv: v1 [physics.pop-ph] 13 May 2009

arxiv: v1 [physics.pop-ph] 13 May 2009 A computer controlled pendulum with position readout H. Hauptfleisch, T. Gasenzer, K. Meier, O. Nachtmann, and J. Schemmel Institut für Theoretische Physik, Universität Heidelberg, Philosophenweg 6, 69

More information

Mechanisms of Chaos: Stable Instability

Mechanisms of Chaos: Stable Instability Mechanisms of Chaos: Stable Instability Reading for this lecture: NDAC, Sec. 2.-2.3, 9.3, and.5. Unpredictability: Orbit complicated: difficult to follow Repeatedly convergent and divergent Net amplification

More information

Torque and Simple Harmonic Motion

Torque and Simple Harmonic Motion Torque and Simple Harmonic Motion Recall: Fixed Axis Rotation Angle variable Angular velocity Angular acceleration Mass element Radius of orbit Kinematics!! " d# / dt! " d 2 # / dt 2!m i Moment of inertia

More information

Vibrations and Waves Physics Year 1. Handout 1: Course Details

Vibrations and Waves Physics Year 1. Handout 1: Course Details Vibrations and Waves Jan-Feb 2011 Handout 1: Course Details Office Hours Vibrations and Waves Physics Year 1 Handout 1: Course Details Dr Carl Paterson (Blackett 621, carl.paterson@imperial.ac.uk Office

More information

Oscillations Simple Harmonic Motion

Oscillations Simple Harmonic Motion Oscillations Simple Harmonic Motion Lana Sheridan De Anza College Dec 1, 2017 Overview oscillations simple harmonic motion (SHM) spring systems energy in SHM pendula damped oscillations Oscillations and

More information

The Pendulum. The purpose of this tab is to predict the motion of various pendulums and compare these predictions with experimental observations.

The Pendulum. The purpose of this tab is to predict the motion of various pendulums and compare these predictions with experimental observations. The Pendulum Introduction: The purpose of this tab is to predict the motion of various pendulums and compare these predictions with experimental observations. Equipment: Simple pendulum made from string

More information

Classical Mechanics Comprehensive Exam Solution

Classical Mechanics Comprehensive Exam Solution Classical Mechanics Comprehensive Exam Solution January 31, 011, 1:00 pm 5:pm Solve the following six problems. In the following problems, e x, e y, and e z are unit vectors in the x, y, and z directions,

More information

Nonlinear dynamics & chaos BECS

Nonlinear dynamics & chaos BECS Nonlinear dynamics & chaos BECS-114.7151 Phase portraits Focus: nonlinear systems in two dimensions General form of a vector field on the phase plane: Vector notation: Phase portraits Solution x(t) describes

More information

Lab 11 - Free, Damped, and Forced Oscillations

Lab 11 - Free, Damped, and Forced Oscillations Lab 11 Free, Damped, and Forced Oscillations L11-1 Name Date Partners Lab 11 - Free, Damped, and Forced Oscillations OBJECTIVES To understand the free oscillations of a mass and spring. To understand how

More information

The Harmonic Oscillator

The Harmonic Oscillator The Harmonic Oscillator Math 4: Ordinary Differential Equations Chris Meyer May 3, 008 Introduction The harmonic oscillator is a common model used in physics because of the wide range of problems it can

More information

= 0. = q i., q i = E

= 0. = q i., q i = E Summary of the Above Newton s second law: d 2 r dt 2 = Φ( r) Complicated vector arithmetic & coordinate system dependence Lagrangian Formalism: L q i d dt ( L q i ) = 0 n second-order differential equations

More information

Physics 11b Lecture #15

Physics 11b Lecture #15 Physics 11b ecture #15 and ircuits A ircuits S&J hapter 3 & 33 Administravia Midterm # is Thursday If you can t take midterm, you MUST let us (me, arol and Shaun) know in writing before Wednesday noon

More information

Introduction to Dynamical Systems Basic Concepts of Dynamics

Introduction to Dynamical Systems Basic Concepts of Dynamics Introduction to Dynamical Systems Basic Concepts of Dynamics A dynamical system: Has a notion of state, which contains all the information upon which the dynamical system acts. A simple set of deterministic

More information

Robotics. Dynamics. Marc Toussaint U Stuttgart

Robotics. Dynamics. Marc Toussaint U Stuttgart Robotics Dynamics 1D point mass, damping & oscillation, PID, dynamics of mechanical systems, Euler-Lagrange equation, Newton-Euler recursion, general robot dynamics, joint space control, reference trajectory

More information

Periodic Skeletons of Nonlinear Dynamical Systems in the Problems of Global Bifurcation Analysis

Periodic Skeletons of Nonlinear Dynamical Systems in the Problems of Global Bifurcation Analysis Periodic Skeletons of Nonlinear Dynamical Systems in the Problems of Global Bifurcation Analysis M Zakrzhevsky, I Schukin, A Klokov and E Shilvan PERIODIC SKELETONS OF NONLINEAR DYNAMICAL SYSTEMS IN THE

More information

Mechanical Oscillations

Mechanical Oscillations Mechanical Oscillations Richard Spencer, Med Webster, Roy Albridge and Jim Waters September, 1988 Revised September 6, 010 1 Reading: Shamos, Great Experiments in Physics, pp. 4-58 Harmonic Motion.1 Free

More information

6 Parametric oscillator

6 Parametric oscillator 6 Parametric oscillator 6. Mathieu equation We now study a different kind of forced pendulum. Specifically, imagine subjecting the pivot of a simple frictionless pendulum to an alternating vertical motion:

More information

Entrainment Alex Bowie April 7, 2004

Entrainment Alex Bowie April 7, 2004 Entrainment Alex Bowie April 7, 2004 Abstract The driven Van der Pol oscillator displays entrainment, quasiperiodicity, and chaos. The characteristics of these different modes are discussed as well as

More information

Damped Harmonic Oscillator

Damped Harmonic Oscillator Damped Harmonic Oscillator Wednesday, 23 October 213 A simple harmonic oscillator subject to linear damping may oscillate with exponential decay, or it may decay biexponentially without oscillating, or

More information

PHY217: Vibrations and Waves

PHY217: Vibrations and Waves Assessed Problem set 1 Issued: 5 November 01 PHY17: Vibrations and Waves Deadline for submission: 5 pm Thursday 15th November, to the V&W pigeon hole in the Physics reception on the 1st floor of the GO

More information

The Nonlinear Pendulum

The Nonlinear Pendulum The Nonlinear Pendulum - Pádraig Ó Conbhuí - 08531749 TP Monday 1. Abstract This experiment was performed to examine the effects that linearizing equations has on the accuracy of results and to find ways

More information

6.2 Brief review of fundamental concepts about chaotic systems

6.2 Brief review of fundamental concepts about chaotic systems 6.2 Brief review of fundamental concepts about chaotic systems Lorenz (1963) introduced a 3-variable model that is a prototypical example of chaos theory. These equations were derived as a simplification

More information

Oscillatory Motion. Solutions of Selected Problems

Oscillatory Motion. Solutions of Selected Problems Chapter 15 Oscillatory Motion. Solutions of Selected Problems 15.1 Problem 15.18 (In the text book) A block-spring system oscillates with an amplitude of 3.50 cm. If the spring constant is 250 N/m and

More information

= w. These evolve with time yielding the

= w. These evolve with time yielding the 1 Analytical prediction and representation of chaos. Michail Zak a Jet Propulsion Laboratory California Institute of Technology, Pasadena, CA 91109, USA Abstract. 1. Introduction The concept of randomness

More information

Physics 207 Lecture 25. Lecture 25. HW11, Due Tuesday, May 6 th For Thursday, read through all of Chapter 18. Angular Momentum Exercise

Physics 207 Lecture 25. Lecture 25. HW11, Due Tuesday, May 6 th For Thursday, read through all of Chapter 18. Angular Momentum Exercise Lecture 5 Today Review: Exam covers Chapters 14-17 17 plus angular momentum, rolling motion & torque Assignment HW11, Due Tuesday, May 6 th For Thursday, read through all of Chapter 18 Physics 07: Lecture

More information

(a) Sections 3.7 through (b) Sections 8.1 through 8.3. and Sections 8.5 through 8.6. Problem Set 4 due Monday, 10/14/02

(a) Sections 3.7 through (b) Sections 8.1 through 8.3. and Sections 8.5 through 8.6. Problem Set 4 due Monday, 10/14/02 Physics 601 Dr. Dragt Fall 2002 Reading Assignment #4: 1. Dragt (a) Sections 1.5 and 1.6 of Chapter 1 (Introductory Concepts). (b) Notes VI, Hamilton s Equations of Motion (to be found right after the

More information

vii Contents 7.5 Mathematica Commands in Text Format 7.6 Exercises

vii Contents 7.5 Mathematica Commands in Text Format 7.6 Exercises Preface 0. A Tutorial Introduction to Mathematica 0.1 A Quick Tour of Mathematica 0.2 Tutorial 1: The Basics (One Hour) 0.3 Tutorial 2: Plots and Differential Equations (One Hour) 0.4 Mathematica Programs

More information

The Nonlinear Pendulum

The Nonlinear Pendulum The Nonlinear Pendulum Evan Sheridan 11367741 Feburary 18th 013 Abstract Both the non-linear linear pendulum are investigated compared using the pendulum.c program that utilizes the trapezoid method for

More information

Hamiltonian Dynamics In The Theory of Abstraction

Hamiltonian Dynamics In The Theory of Abstraction Hamiltonian Dynamics In The Theory of Abstraction Subhajit Ganguly. email: gangulysubhajit@indiatimes.com Abstract: This paper deals with fluid flow dynamics which may be Hamiltonian in nature and yet

More information

PHYS2330 Intermediate Mechanics Fall Final Exam Tuesday, 21 Dec 2010

PHYS2330 Intermediate Mechanics Fall Final Exam Tuesday, 21 Dec 2010 Name: PHYS2330 Intermediate Mechanics Fall 2010 Final Exam Tuesday, 21 Dec 2010 This exam has two parts. Part I has 20 multiple choice questions, worth two points each. Part II consists of six relatively

More information