Physics 401, Spring 2016 Eugene V. Colla
|
|
- Erika Long
- 5 years ago
- Views:
Transcription
1 Physics 41, Spring 16 Eugene V. Colla.8 (rad) time (s)
2 1.Driven torsional oscillator. Equations.Setup. Kinematics 3.Resonance 4.Beats 5.Nonlinear effects 6.Comments 3/7/16
3 3/7/16 3
4 Tacoma (WA) Narrows Bridge Disaster 3/7/16 4
5 Tacoma (WA) Narrows Bridge, 194 3/7/16 5
6 Tacoma (WA) Narrows Bridge, 194 3/7/16 6
7 Tacoma (WA) Narrows Bridge, 194 3/7/16 7
8 Egyptian Bridge disaster. January195, St. Petersburg, Russia. 3/7/16 8
9 Egyptian Bridge disaster. January195, St. Petersburg, Russia. 3/7/16 9
10 Egyptian Bridge disaster. January195, St. Petersburg, Russia. 3/7/16 1
11 Dancing Bridge in Volgograd (Russia) (record from st long). May miles 3/7/16 11
12 In autumn 11, 1 semi-active tuned mass dampers were installed in the bridge. Each one consists of a mass 5, kg (11,5 lb), a set of compression springs and a magnethoreological damper. 3/7/16 1
13 3/7/16 13
14 The goals: (i) analyze the response of the damped driven harmonic oscillator to a sinusoidal drive. (ii) transient response and (iii) steady-state solution. wire M L 1 L R disk motor Angular displacement: θ cos(ωt); torque: Kλθ cos(ωt) L 1 λ = L 1 + L I ሷ θ + Kθ + R ሶ θ = m = Kθ cos(ωt) I is momentum of inertia, [kgm ] R is a damping constant [N m s]. K is the total spring constant [Nm] Viscous damping Torque by motor 3/7/16 14
15 Motor Pendulum 3/7/16 15
16 I ሷ θ + Kθ + R ሶ θ = m = Kθ cos(ωt) Solutions: sum of (1) Transient solution + () steady solution due to torque m (1) Transient solution (1 st week experiment) Iθ+ Rθ+ Kθ = -at θ t = A e cos ω t- a = R I ω = o K I a 1 o 1 The homogeneous equation of motion Transient solution Attenuation constant Natural (angular) frequency Damped (angular) frequency 3/7/16 16
17 at t ( t) A e cos( t ) 1 1 a Transient solution Once this response dies away in time the system response only on the frequency of drive Initially the system responds on the characteristic frequency 1 ( ) Re ( ) i t ss t e So the steady-state solution must have the similar time dependence as the drive I ሷ θ + Kθ + R ሶ θ = m = Kθ cos(ωt) Substituting ss (t) in equation of motion we will find the equations for ( ) 4 a e i( ) and 1 a ( ) tan
18 I ሷ θ + Kθ + R ሶ θ = m = Kθ cos(ωt) () steady solution t B cos t s B tan R R a I I o o o o Steady state solution Amplitude function Phase function Damping constant 3/7/16 18
19 time domain form for steady-state solution will be ss( t) cos( t ( )) 4 a Phase Amplitude B() General solution for equation of motion consist of the sum of sum of two components: (t) = t (t) + ss (t) at ( t) ( t) ( t) Ae cos( t ) Bcos( t ( )) t ss 1 Coefficients A and could be determined from initial conditions
20 Fitting function: (rad) 1 f =.5Hz (fitting) ( f ) A f f f f =pf; =a To create a new fitting function go Tools Fitting Function Builder or press F8 Model Equation Reduced Chi-Sqr.1 1 f d (Hz) Resonance1 (User) Adj. R-Square y=a*f^/sqrt((f^-x^)^+x^*gamma^) 3.E-4 Value Standard Error pend A pend f e-4 pend gamma E-4 Count Model Equation Reduced Chi-S qr Regular Residual of Sheet1 pend Gauss Fit Counts Gauss y=y + (A/(w*sqrt(PI/)))*exp(-*((x-xc)/w)^) Adj. R-Square Value Standard Error Counts y Counts xc 6.543E Counts w Counts A Counts sigma.1199 Counts FWHM.84 Counts Height Regular Residual of Sheet1 pend
21 (rad) 3 1 Phase.1 1 f d (Hz) Scanning the driving frequency we can measure the amplitude of the pendulum oscillating and the phase shift Both parameters Amplitude and phase can be defined by DAQ program or using Origin
22 Amplitude () t ss 4 a At resonance = ss() t Q a Combination of high initial amplitude, and high quality Q or low damping factor a could be result of the destruction of the mechanical system 3/7/16
23 For correct representation of the resonance curve take care about choosing of the step size in frequency. 3/7/16 3
24 There are two parameters used to measure the rate at which the oscillations of a system are damped out. One parameter is the logarithmic decrement d, and the other is the quality factor, Q. d, is defined by 6 (t max ) (t max +T 1 ) 8.49 d ln (rad) time (s) Q ~ 1.8
25 It can be shown that Q can f 1 =.496Hz be calculated as 1 / or Amplitude 1 f=.66hz f 1 /f. is bandwidth of the resonance curve on the θ half power level or max for amplitude graph f d (Hz) Here Q~7.9
26 Consider sum of two harmonic signals of frequencies 1 and y 1 =Asin( 1 t+ 1 ); y =Bsin( t+ ) In case A=B y=y 1 +y =Asin ω 1+ω β 1 = φ 1+φ ; β = φ 1 φ If 1 ω 1+ω = and y= Acos Wt + β sin t + β 1 ω 1 ω t + β 1 =W cos ω 1 ω t + β ; 1 =.78 =.94 1 y time (s) f (Hz)
27 More general case A B 1 and y 1 =Asin( t); y =Bsin(( +)t) y=y 1 +y =Csin( + )t where C = A + B + ABcos(at) β = tan 1 B sin(αt) A + B cos αt if A + Bcos( αt) + ቊ π if A + Bcos αt < A + B + AB A + B AB 1 1 = = 1 y time (s).8.3 f (Hz)
28 pend (rad) Amplitude 6 4 Two peaks corresponding and t (sec) Time domain trace Frequency (Hz) Beating spectrum Use Origin to analyze the frequency spectrum!
29 at ( t) ( t) ( t) Ae cos( t ) Bcos( t ( )) t ss 1 t (t) 4 Beats dying in time. How fast it depends on damping. When you will work on resonance data wait until you will see the steady state oscillations. (rad) -4 4 time (s) 4 (rad) (rad) time(s) time(s)
30 at ( t) ( t) ( t) Ae cos( t ) Bcos( t ( )) t ss (rad) (rad) time (s) time (s) t (t) This can be seen well from envelope plot Origin 8.6: Analysis Signal Processing Envelope
31 at ( t) ( t) ( t) Ae cos( t ) B cos( t ( )) C t ss 1 First let we apply FFT to find 1 and Result: =3.14rad -1 and =.898 rad -1
32 t ss t t ( t) ( t) ( t) Ae cos( t ) Bcos( t ( )) C 1 8 fitting parameters From fitting A.651 t B C Result from FFT: =3.14rad -1 and =.898 rad -1
33 FFT Compare with original pendulum spectrum Pendulum Possible origin of extra peaks: (i) Nonlinear behavior of pendulum (ii) Not a single frequency driving force provided by motor (iii) Not ideal fitting function Residuals
34 at ( t) ( t) ( t) Ae cos( t ) Bcos( t ( )) t ss 1 4 t (t) (rad) We also can analyze the decrease of the amplitude of the 1 component by analyzing the spectrum as a function of time -4 4 time (s) motor.4 First 55 sec 1 s1 1 R (V) First 55 sec Last 55 sec Origin 9.: Analysis Signal Processing FFT f (Hz) Last 55 sec
35 .1 C VC. -.1 L 5 time ( s) 1 5 V (V) V(t) VC (V) R -5-1 : Graph 5 time ( s) 1 1
36 Find peaks.1 R V(t) C L V C VC (V) time (s) Envelope FFT.1 31kHz 1 34kHz VC (V). Magnitude time (s) f (khz)
37 .4 d~ /3 1-1 f =.4891Hz f d =.163 Hz f d =.163 f -f d f (rad). Amplitude f +f d time (s) Frequency (Hz)
38 In the case of driving frequency f d =f1/n where N is integer we can observe more complicated motion of the pendulum f d f =.4891Hz f 1-1 f d =.44 Hz (rad).3. d~ Amplitude f d +f time (s) Frequency (Hz)
39 1 7 Magnitude Frequency (Hz) Detailed analyzes* shows that even if = sin ωt the driving torque contains several harmonics of *P. Debevec (UIUC, Department of Physics)
Physics 401. Fall 2018 Eugene V. Colla. 10/8/2018 Physics 401 1
Physics 41. Fall 18 Eugene V. Colla 1/8/18 Physics 41 1 Electrical RLC circuits Torsional Oscillator Damping Data Analysis 1/8/18 Physics 41 V R +V L +V C =V(t) If V(t)= R d d q(t) q(t) dt dt C C L q(t)
More informationPhysics 401. Fall 2017 Eugene V. Colla. 10/9/2017 Physics 401 1
Physics 41. Fall 17 Eugene V. Colla 1/9/17 Physics 41 1 Electrical RLC circuits Torsional Oscillator Damping Data Analysis 1/9/17 Physics 41 V R +V L +V C =V(t) If V(t)= R d d q(t) q(t) dt dt C C L q(t)
More informationChapter 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 informationChapter 14 Periodic Motion
Chapter 14 Periodic Motion 1 Describing Oscillation First, we want to describe the kinematical and dynamical quantities associated with Simple Harmonic Motion (SHM), for example, x, v x, a x, and F x.
More informationOscillatory 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 informationFaculty of Computers and Information. Basic Science Department
18--018 FCI 1 Faculty of Computers and Information Basic Science Department 017-018 Prof. Nabila.M.Hassan 18--018 FCI Aims of Course: The graduates have to know the nature of vibration wave motions with
More informationChapter 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 informationPREMED COURSE, 14/08/2015 OSCILLATIONS
PREMED COURSE, 14/08/2015 OSCILLATIONS PERIODIC MOTIONS Mechanical Metronom Laser Optical Bunjee jumping Electrical Astronomical Pulsar Biological ECG AC 50 Hz Another biological exampe PERIODIC MOTIONS
More informationDEVIL PHYSICS THE BADDEST CLASS ON CAMPUS IB PHYSICS
DEVIL PHYSICS THE BADDEST CLASS ON CAMPUS IB PHYSICS TSOKOS LESSON 4-1 SIMPLE HARMONIC MOTION Introductory Video: Simple Harmonic Motion IB Assessment Statements Topic 4.1, Kinematics of Simple Harmonic
More informationChapter 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 informationVibrations and Waves MP205, Assignment 4 Solutions
Vibrations and Waves MP205, Assignment Solutions 1. Verify that x = Ae αt cos ωt is a possible solution of the equation and find α and ω in terms of γ and ω 0. [20] dt 2 + γ dx dt + ω2 0x = 0, Given x
More informationLAST 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 informationPeriodic Motion. Periodic motion is motion of an object that. regularly repeats
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 special kind of periodic motion occurs in mechanical systems
More informationChapter 13. Hooke s Law: F = - kx Periodic & Simple Harmonic Motion Springs & Pendula Waves Superposition. Next Week!
Chapter 13 Hooke s Law: F = - kx Periodic & Simple Harmonic Motion Springs & Pendula Waves Superposition Next Week! Review Physics 2A: Springs, Pendula & Circular Motion Elastic Systems F = kx Small Vibrations
More informationChapter 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 informationPhysics 141, Lecture 7. Outline. Course Information. Course information: Homework set # 3 Exam # 1. Quiz. Continuation of the discussion of Chapter 4.
Physics 141, Lecture 7. Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 07, Page 1 Outline. Course information: Homework set # 3 Exam # 1 Quiz. Continuation of the
More information16 SUPERPOSITION & STANDING WAVES
Chapter 6 SUPERPOSITION & STANDING WAVES 6. Superposition of waves Principle of superposition: When two or more waves overlap, the resultant wave is the algebraic sum of the individual waves. Illustration:
More informationChapter 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 Exam 3 results Class Average - 57 (Approximate grade
More informationThe distance of the object from the equilibrium position is m.
Answers, Even-Numbered Problems, Chapter..4.6.8.0..4.6.8 (a) A = 0.0 m (b).60 s (c) 0.65 Hz Whenever the object is released from rest, its initial displacement equals the amplitude of its SHM. (a) so 0.065
More informationLaboratory notes. Torsional Vibration Absorber
Titurus, Marsico & Wagg Torsional Vibration Absorber UoB/1-11, v1. Laboratory notes Torsional Vibration Absorber Contents 1 Objectives... Apparatus... 3 Theory... 3 3.1 Background information... 3 3. Undamped
More informationOscillations. Phys101 Lectures 28, 29. Key points: Simple Harmonic Motion (SHM) SHM Related to Uniform Circular Motion The Simple Pendulum
Phys101 Lectures 8, 9 Oscillations Key points: Simple Harmonic Motion (SHM) SHM Related to Uniform Circular Motion The Simple Pendulum Ref: 11-1,,3,4. Page 1 Oscillations of a Spring If an object oscillates
More informationTorque 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 informationOscillations. 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 informationChapter 2: Complex numbers
Chapter 2: Complex numbers Complex numbers are commonplace in physics and engineering. In particular, complex numbers enable us to simplify equations and/or more easily find solutions to equations. We
More information本教材僅供教學使用, 勿做其他用途, 以維護智慧財產權
本教材內容主要取自課本 Physics for Scientists and Engineers with Modern Physics 7th Edition. Jewett & Serway. 注意 本教材僅供教學使用, 勿做其他用途, 以維護智慧財產權 教材網址 : https://sites.google.com/site/ndhugp1 1 Chapter 15 Oscillatory Motion
More informationChapter 16: Oscillatory Motion and Waves. Simple Harmonic Motion (SHM)
Chapter 6: Oscillatory Motion and Waves Hooke s Law (revisited) F = - k x Tthe elastic potential energy of a stretched or compressed spring is PE elastic = kx / Spring-block Note: To consider the potential
More informationPhysics 201, Lecture 28
Physics 01, Lecture 8 Today s Topics n Oscillations (Ch 15) n n n More Simple Harmonic Oscillation n Review: Mathematical Representation n Eamples: Simple Pendulum, Physical pendulum Damped Oscillation
More informationFundamentals Physics. Chapter 15 Oscillations
Fundamentals Physics Tenth Edition Halliday Chapter 15 Oscillations 15-1 Simple Harmonic Motion (1 of 20) Learning Objectives 15.01 Distinguish simple harmonic motion from other types of periodic motion.
More informationOscillations. 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 informationChapter 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 informationPHYSICS. 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 informationPhysics 41: Waves, Optics, Thermo
Physics 41: Waves, Optics, Thermo Particles & Waves Localized in Space: LOCAL Have Mass & Momentum No Superposition: Two particles cannot occupy the same space at the same time! Particles have energy.
More informationSection 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
More informationFaculty of Computers and Information Fayoum University 2017/ 2018 Physics 2 (Waves)
Faculty of Computers and Information Fayoum University 2017/ 2018 Physics 2 (Waves) 3/10/2018 1 Using these definitions, we see that Example : A sinusoidal wave traveling in the positive x direction has
More informationEDEXCEL NATIONAL CERTIFICATE UNIT 28 FURTHER MATHEMATICS FOR TECHNICIANS OUTCOME 3 TUTORIAL 1 - TRIGONOMETRICAL GRAPHS
EDEXCEL NATIONAL CERTIFICATE UNIT 28 FURTHER MATHEMATICS FOR TECHNICIANS OUTCOME 3 TUTORIAL 1 - TRIGONOMETRICAL GRAPHS CONTENTS 3 Be able to understand how to manipulate trigonometric expressions and apply
More informationHarmonic Oscillator. Mass-Spring Oscillator Resonance The Pendulum. Physics 109 Experiment Number 12
Harmonic Oscillator Mass-Spring Oscillator Resonance The Pendulum Physics 109 Experiment Number 12 Outline Simple harmonic motion The vertical mass-spring system Driven oscillations and resonance The pendulum
More informationMechanical 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 informationMATH 251 Week 6 Not collected, however you are encouraged to approach all problems to prepare for exam
MATH 51 Week 6 Not collected, however you are encouraged to approach all problems to prepare for exam A collection of previous exams could be found at the coordinator s web: http://www.math.psu.edu/tseng/class/m51samples.html
More informationCHAPTER 12 OSCILLATORY MOTION
CHAPTER 1 OSCILLATORY MOTION Before starting the discussion of the chapter s concepts it is worth to define some terms we will use frequently in this chapter: 1. The period of the motion, T, is the time
More informationSimple Harmonic Motion
Chapter 9 Simple Harmonic Motion In This Chapter: Restoring Force Elastic Potential Energy Simple Harmonic Motion Period and Frequency Displacement, Velocity, and Acceleration Pendulums Restoring Force
More information4.1 KINEMATICS OF SIMPLE HARMONIC MOTION 4.2 ENERGY CHANGES DURING SIMPLE HARMONIC MOTION 4.3 FORCED OSCILLATIONS AND RESONANCE Notes
4.1 KINEMATICS OF SIMPLE HARMONIC MOTION 4.2 ENERGY CHANGES DURING SIMPLE HARMONIC MOTION 4.3 FORCED OSCILLATIONS AND RESONANCE Notes I. DEFINING TERMS A. HOW ARE OSCILLATIONS RELATED TO WAVES? II. EQUATIONS
More informationKEELE UNIVERSITY PHYSICS/ASTROPHYSICS MODULE PHY OSCILLATIONS AND WAVES PRACTICE EXAM
KEELE UNIVERSITY PHYSICS/ASTROPHYSICS MODULE PHY-10012 OSCILLATIONS AND WAVES PRACTICE EXAM Candidates should attempt ALL of PARTS A and B, and TWO questions from PART C. PARTS A and B should be answered
More informationChapter 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 informationKEY SOLUTION. 05/07/01 PHYSICS 223 Exam #1 NAME M 1 M 1. Fig. 1a Fig. 1b Fig. 1c
KEY SOLUTION 05/07/01 PHYSICS 223 Exam #1 NAME Use g = 10 m/s 2 in your calculations. Wherever appropriate answers must include units. 1. Fig. 1a shows a spring, 20 cm long. The spring gets compressed
More informationNotes 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 informationLinear 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 informationPhysics 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 informationOSCILLATIONS 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 informationDriven RLC Circuits Challenge Problem Solutions
Driven LC Circuits Challenge Problem Solutions Problem : Using the same circuit as in problem 6, only this time leaving the function generator on and driving below resonance, which in the following pairs
More informationChapter 14: Periodic motion
Chapter 14: Periodic motion Describing oscillations Simple harmonic motion Energy of simple harmonic motion Applications of simple harmonic motion Simple pendulum & physical pendulum Damped oscillations
More informationCHAPTER 11 VIBRATIONS AND WAVES
CHAPTER 11 VIBRATIONS AND WAVES http://www.physicsclassroom.com/class/waves/u10l1a.html UNITS Simple Harmonic Motion Energy in the Simple Harmonic Oscillator The Period and Sinusoidal Nature of SHM The
More informationME 563 HOMEWORK # 7 SOLUTIONS Fall 2010
ME 563 HOMEWORK # 7 SOLUTIONS Fall 2010 PROBLEM 1: Given the mass matrix and two undamped natural frequencies for a general two degree-of-freedom system with a symmetric stiffness matrix, find the stiffness
More informationOscillations. Tacoma Narrow Bridge: Example of Torsional Oscillation
Oscillations Mechanical Mass-spring system nd order differential eq. Energy tossing between mass (kinetic energy) and spring (potential energy) Effect of friction, critical damping (shock absorber) Simple
More informationChapter 16: Oscillations
Chapter 16: Oscillations Brent Royuk Phys-111 Concordia University Periodic Motion Periodic Motion is any motion that repeats itself. The Period (T) is the time it takes for one complete cycle of motion.
More information11/17/10. 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 informationIn-Class Problems 30-32: Moment of Inertia, Torque, and Pendulum: Solutions
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics Physics 8.01 TEAL Fall Term 004 In-Class Problems 30-3: Moment of Inertia, Torque, and Pendulum: Solutions Problem 30 Moment of Inertia of a
More informationMathematical Modeling and response analysis of mechanical systems are the subjects of this chapter.
Chapter 3 Mechanical Systems A. Bazoune 3.1 INRODUCION Mathematical Modeling and response analysis of mechanical systems are the subjects of this chapter. 3. MECHANICAL ELEMENS Any mechanical system consists
More informationDate: 1 April (1) The only reference material you may use is one 8½x11 crib sheet and a calculator.
PH1140: Oscillations and Waves Name: Solutions Conference: Date: 1 April 2005 EXAM #1: D2005 INSTRUCTIONS: (1) The only reference material you may use is one 8½x11 crib sheet and a calculator. (2) Show
More informationChapter 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 informationMechanical 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 informationChapter 15. Oscillations
Chapter 15 Oscillations 15.1 Simple Harmonic Motion Oscillatory Motion: Motion which is periodic in time; motion that repeats itself in time. Examples: SHM: Power line oscillates when the wind blows past.
More informationChapter 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 informationA4. Free and Forced Oscillations
A4. Free and Forced Oscillations I. OBJECTIVE OF THE EXPERIMENT Physics classes are generally divided into chapters: mechanics, Thermodynamics, electrodynamics, etc. One interesting aspect can be noted:
More informationDate: 31 March (1) The only reference material you may use is one 8½x11 crib sheet and a calculator.
PH1140: Oscillations and Waves Name: SOLUTIONS AT END Conference: Date: 31 March 2005 EXAM #1: D2006 INSTRUCTIONS: (1) The only reference material you may use is one 8½x11 crib sheet and a calculator.
More informationOscillatory 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 informationIntroductory Physics. Week 2015/05/29
2015/05/29 Part I Summary of week 6 Summary of week 6 We studied the motion of a projectile under uniform gravity, and constrained rectilinear motion, introducing the concept of constraint force. Then
More informationChapter 5 Oscillatory Motion
Chapter 5 Oscillatory Motion Simple Harmonic Motion An object moves with simple harmonic motion whenever its acceleration is proportional to its displacement from some equilibrium position and is oppositely
More information1 Oscillations MEI Conference 2009
1 Oscillations MEI Conference 2009 Some Background Information There is a film clip you can get from Youtube of the Tacoma Narrows Bridge called Galloping Gertie. This shows vibrations in the bridge increasing
More informationEnergy in a Simple Harmonic Oscillator. Class 30. Simple Harmonic Motion
Simple Harmonic Motion Class 30 Here is a simulation of a mass hanging from a spring. This is a case of stable equilibrium in which there is a large extension in which the restoring force is linear in
More informationTOPIC E: OSCILLATIONS SPRING 2019
TOPIC E: OSCILLATIONS SPRING 2019 1. Introduction 1.1 Overview 1.2 Degrees of freedom 1.3 Simple harmonic motion 2. Undamped free oscillation 2.1 Generalised mass-spring system: simple harmonic motion
More informationLaboratory handouts, ME 340
Laboratory handouts, ME 340 This document contains summary theory, solved exercises, prelab assignments, lab instructions, and report assignments for Lab 4. 2014-2016 Harry Dankowicz, unless otherwise
More informationMATHEMATICS FOR ENGINEERING TRIGONOMETRY TUTORIAL 3 PERIODIC FUNCTIONS
MATHEMATICS FOR ENGINEERING TRIGONOMETRY TUTORIAL 3 PERIODIC FUNCTIONS This is the one of a series of basic tutorials in mathematics aimed at beginners or anyone wanting to refresh themselves on fundamentals.
More informationDifferential equation of wave motion
Differential equation of wave motion Lecture-10 A plane progressive wave is one which travels onward through the medium in a given direction without attenuation, i.e., with its amplitude constant. y asin
More informationLab 1: Damped, Driven Harmonic Oscillator
1 Introduction Lab 1: Damped, Driven Harmonic Oscillator The purpose of this experiment is to study the resonant properties of a driven, damped harmonic oscillator. This type of motion is characteristic
More informationWEEKS 8-9 Dynamics of Machinery
WEEKS 8-9 Dynamics of Machinery References Theory of Machines and Mechanisms, J.J.Uicker, G.R.Pennock ve J.E. Shigley, 2011 Mechanical Vibrations, Singiresu S. Rao, 2010 Mechanical Vibrations: Theory and
More informationSimple Harmonic Motion
3/5/07 Simple Harmonic Motion 0. The Ideal Spring and Simple Harmonic Motion HOOKE S AW: RESTORING FORCE OF AN IDEA SPRING The restoring force on an ideal spring is F x k x spring constant Units: N/m 3/5/07
More informationChapter 14 Oscillations. Copyright 2009 Pearson Education, Inc.
Chapter 14 Oscillations 14-1 Oscillations of a Spring If an object vibrates or oscillates back and forth over the same path, each cycle taking the same amount of time, the motion is called periodic. The
More informationGeneral Physics I. Lecture 12: Applications of Oscillatory Motion. Prof. WAN, Xin ( 万歆 )
General Physics I Lecture 1: Applications of Oscillatory Motion Prof. WAN, Xin ( 万歆 ) inwan@zju.edu.cn http://zimp.zju.edu.cn/~inwan/ Outline The pendulum Comparing simple harmonic motion and uniform circular
More informationDON T PANIC! If you get stuck, take a deep breath and go on to the next question. Come back to the question you left if you have time at the end.
Math 307, Midterm 2 Winter 2013 Name: Instructions. DON T PANIC! If you get stuck, take a deep breath and go on to the next question. Come back to the question you left if you have time at the end. There
More informationPHY217: 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 informationLab 1: damped, driven harmonic oscillator
Lab 1: damped, driven harmonic oscillator 1 Introduction The purpose of this experiment is to study the resonant properties of a driven, damped harmonic oscillator. This type of motion is characteristic
More informationCh 3.7: Mechanical & Electrical Vibrations
Ch 3.7: Mechanical & Electrical Vibrations Two important areas of application for second order linear equations with constant coefficients are in modeling mechanical and electrical oscillations. We will
More informationEXAMPLE 2: CLASSICAL MECHANICS: Worked examples. b) Position and velocity as integrals. Michaelmas Term Lectures Prof M.
CLASSICAL MECHANICS: Worked examples Michaelmas Term 2006 4 Lectures Prof M. Brouard EXAMPLE 2: b) Position and velocity as integrals Calculate the position of a particle given its time dependent acceleration:
More informationLecture XXVI. Morris Swartz Dept. of Physics and Astronomy Johns Hopkins University November 5, 2003
Lecture XXVI Morris Swartz Dept. of Physics and Astronomy Johns Hopins University morris@jhu.edu November 5, 2003 Lecture XXVI: Oscillations Oscillations are periodic motions. There are many examples of
More information3.1 Centrifugal Pendulum Vibration Absorbers: Centrifugal pendulum vibration absorbers are a type of tuned dynamic absorber used for the reduction of
3.1 Centrifugal Pendulum Vibration Absorbers: Centrifugal pendulum vibration absorbers are a type of tuned dynamic absorber used for the reduction of torsional vibrations in rotating and reciprocating
More informationExperiment IV. To find the velocity of waves on a string by measuring the wavelength and frequency of standing waves.
Experiment IV The Vibrating String I. Purpose: To find the velocity of waves on a string by measuring the wavelength and frequency of standing waves. II. References: Serway and Jewett, 6th Ed., Vol., Chap.
More informationPhysics 5B PRACTICE MIDTERM EXAM I-B Winter 2009
Physics 5B PRACTICE MIDTERM EXAM I-B Winter 2009 PART I: Multiple choice questions Only one of the choices given is the correct answer. No explanation for your choice is required. Each multiple choice
More informationPhysics 231 Lecture 18
Physics 31 ecture 18 τ = Fd;d is the lever arm Main points of today s lecture: Energy Pendulum T = π g ( ) θ = θmax cos πft + ϑ0 Damped Oscillations x x equibrium = Ae bt/(m) cos(ω damped t) ω damped =
More informationChapter 15 Oscillations
Chapter 15 Oscillations Summary Simple harmonic motion Hook s Law Energy F = kx Pendulums: Simple. Physical, Meter stick Simple Picture of an Oscillation x Frictionless surface F = -kx x SHM in vertical
More informationOscillatory Motion and Wave Motion
Oscillatory Motion and Wave Motion Oscillatory Motion Simple Harmonic Motion Wave Motion Waves Motion of an Object Attached to a Spring The Pendulum Transverse and Longitudinal Waves Sinusoidal Wave Function
More informationRutgers University Department of Physics & Astronomy. 01:750:271 Honors Physics I Fall Lecture 20 JJ II. Home Page. Title Page.
Rutgers University Department of Physics & Astronomy 01:750:271 Honors Physics Fall 2015 Lecture 20 Page 1 of 31 1. No quizzes during Thanksgiving week. There will be recitation according to the regular
More information2. Determine whether the following pair of functions are linearly dependent, or linearly independent:
Topics to be covered on the exam include: Recognizing, and verifying solutions to homogeneous second-order linear differential equations, and their corresponding Initial Value Problems Recognizing and
More informationFigure XP3.1 (a) Mass in equilibrium, (b) Freebody diagram, (c) Kinematic constraint relation Example Problem 3.1 Figure XP3.1 illustrates a mass m
LECTURE 7. MORE VIBRATIONS ` Figure XP3.1 (a) Mass in equilibrium, (b) Freebody diagram, (c) Kinematic constraint relation Example Problem 3.1 Figure XP3.1 illustrates a mass m that is in equilibrium and
More informationChapter 14 Oscillations
Chapter 14 Oscillations If an object vibrates or oscillates back and forth over the same path, each cycle taking the same amount of time, the motion is called periodic. The mass and spring system is a
More informationChapter 15+ Revisit Oscillations and Simple Harmonic Motion
Chapter 15+ Revisit Oscillations and Simple Harmonic Motion Revisit: Oscillations Simple harmonic motion To-Do: Pendulum oscillations Derive the parallel axis theorem for moments of inertia and apply it
More informationOscillations. Oscillations and Simple Harmonic Motion
Oscillations AP Physics C Oscillations and Simple Harmonic Motion 1 Equilibrium and Oscillations A marble that is free to roll inside a spherical bowl has an equilibrium position at the bottom of the bowl
More informationChapter 15 Periodic Motion
Chapter 15 Periodic Motion Slide 1-1 Chapter 15 Periodic Motion Concepts Slide 1-2 Section 15.1: Periodic motion and energy Section Goals You will learn to Define the concepts of periodic motion, vibration,
More informationEngineering Science OUTCOME 2 - TUTORIAL 3 FREE VIBRATIONS
Unit 2: Unit code: QCF Level: 4 Credit value: 5 Engineering Science L/60/404 OUTCOME 2 - TUTORIAL 3 FREE VIBRATIONS UNIT CONTENT OUTCOME 2 Be able to determine the behavioural characteristics of elements
More informationWORK SHEET FOR MEP311
EXPERIMENT II-1A STUDY OF PRESSURE DISTRIBUTIONS IN LUBRICATING OIL FILMS USING MICHELL TILTING PAD APPARATUS OBJECTIVE To study generation of pressure profile along and across the thick fluid film (converging,
More informationIntroduction to Vibration. Professor Mike Brennan
Introduction to Vibration Professor Mie Brennan Introduction to Vibration Nature of vibration of mechanical systems Free and forced vibrations Frequency response functions Fundamentals For free vibration
More information