# Oscillations. Vern Lindberg. June 10, 2010

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 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 Undamped Simple Harmonic Motion If a mass m is attached to an ideal spring of constant k, and the spring force is the only net force, then one dimensional simple harmonic motion (SHM) results. The equation of motion is The solution can be written is several forms. ẍ = k m x (1) x = A 0 sin(ωt + φ 0 ) (2) = A 1 cos ωt + A 2 sin ωt (3) = A 3 exp(iωt) + A 4 exp( iωt) (4) = A 5 exp(i(ωt + φ 5 )) (5) The angular frequency, ω = 2πf, is independent of amplitude and is k ω = m Since this is a second order differential equation, all solutions have two integration constants (such as amplitude and phase, A 0 and φ 0, or two amplitudes A 1 and A 2 ) that are determined by the initial conditions. Picking Eq. 2, the first solution, If at t = 0 we have x = x 0 and v = v 0 then (6) x(0) = A sin φ 0 = x 0 (7) ẋ(0) = ωa cos φ 0 = v 0 (8) 1

2 Solving these we get tan φ 0 = ωx 0 (9) v 0 A = x v2 0 ω 2 (10) We will choose the positive square root value for amplitude, but there remains an ambiguity in the inverse tangent that is resolved by deciding what quadrant for φ 0 is needed to produce the proper signs for initial position and velocity. We will spend some time on this in class. Example: Suppose the spring constant is 18 N/m and the mass is 2 kg. Then the angular frequency is ω = 3 rad/s. If x 0 = cm and v 0 = cm/s, then the amplitude is 5.0 cm and the inverse tangent gives phase angles of either or 2.80 radians. With a bit of thought you can decide that the correct phase is 2.80 radians. Example Using Eq. 3, find the values of A 1, A 2 Use the same values as in the previous example. The potential energy associated with this simple model SHM is V (x) = 1 2 kx2 (11) This is a parabolic potential energy, with the oscillations occurring around the minimum in the potential energy. The spring force is a restoring force, linear in displacement. 1.1 The Morse Potential for Diatomic Molecules: Approximations Any potential energy that has a minimum can have oscillations around that minimum. A natural question is to ask whether the oscillations are simple harmonic. If the system oscillates about x = x 0 with force F = k(x x 0 ) and potential energy V = 1 2 k(x x 0) 2 then the motion is SHM for any amplitude. What if we don t have forces of that form? For small oscillations about the minimum we can make suitable approximations to see if the small amplitude motion is SHM. Taylor Series and Approximations A vital skill in physics is making approximations. The Taylor series is a polynomial expansion of a function about some point. If we expand f(x) about x = a, the Taylor series is f(x) = f(a) + ( f x ) x=a (x a) + 1 2! ( 2 ) f x 2 (x x=a a)2 + (12) The terms are called the zeroeth order, first order, second order,.... We frequently want to find the first higher order term (first and above) that is non-zero. 2

3 Consider the Morse potential energy, a model of the potential energy for a vibrating diatomic molecule like O 2. [ V (x) = V 0 1 exp( x x 2 0 )] V 0 (13) δ First we find the location of the minimum by solving dv/dx = 0 and find that the minimum occurs at x = x 0. This is done as an example in Chapter 2 of the text. At this minimum we find V = V 0. Computational Hint Make appropriate substitutions of variables to make the calculus easier. For the Morse potential define z = (x x 0 )/δ. Then V = V 0 (1 exp z) 2 V 0, and we can do derivatives as dv/dx = dv/dz dz/dx. For small oscillations we expand the exponential in a Taylor series, V = V 0 [1 (1 z + z 2 /2 z 3 /6 + )] 2 V 0 = V 0 [z z 2 /2 + z 3 /6 + ] 2 V 0 = V 0 [z 2 z 3 + 7z 4 /12 + ] V 0 ( ) x 2 = V 0 z 2 x0 V 0 = V 0 V 0 (14) δ This is parabolic and so we expect SHM. The restoring force for small oscillations is linear, F (x) = dv dx = 2V 0 δ 2 (x x 0) (15) Thus for small oscillation we expect SHM with an angular frequency of 2V0 ω = mδ 2 (16) Example The following system is set up on a horizontal, frictionless table. A small block of mass m is attached to two identical springs of constant k. When the springs are unstretched they have identical lengths L 0. The equilibrium position is a straight line consisting of a wall, a spring, the block, the other spring, and a second wall. The block is pulled down a small distance y L 0 perpendicular to the equilibrium line. Determine whether or not the subsequent motion is simple harmonic. 1.2 Angular SHM Consider a simple pendulum of length l with point mass m attached at one end, in a uniform gravitational field g. The motion is along an arc of a circle, so it is natural to use 3

4 an angular variable θ for the displacement. Using the angular form of Newton s Second Law, in scalar form since the angular motion is along a fixed direction, N = I θ (17) mgl sin θ = (ml 2 ) θ (18) Providing the angle (in radians) is small, we can use the Taylor Series expansion sin θ = θ θ 3 /3! + and keep it to first order to get the equation of motion. θ + g l θ = 0 (19) This is the same form as the SHM equation of motion, Equation 1, so the solution is SHM with angular frequency ω = g/l. We will solve the general equation of motion (large angles) in Mechanics 2 using Elliptic Integrals. 1.3 Energy and SHM For ideal SHM the energy is a constant of the motion and is E = 1 2 mẋ kx2 (20) 2 Damped Simple Harmonic Motion Linear Damping Suppose we add a linear damping term to the equation of motion to get ẍ = c mẋ k m x (21) It is convenient to define a damping factor γ = so that the equation of motion becomes ẍ + 2γẋ + ω0 2 = 0 (22) Next let s introduce a differential operator, D = d/dt. The operator has no independent meaning, but once it is applied to a function its meaning is clear. The equation of motion can be written [ D 2 + 2γD + ω0 2 ] x = 0 (23) [ ] [ ] D + γ γ 2 ω0 2 D + γ + γ 2 ω0 2 x = 0 (24) There are two solutions, one for the first bracket operating on x equalling 0 and one for the second bracket operating on x being zero. c 2m 4

5 These are first order differential equations and are easy to solve resulting in a general solution x(t) = A 1 exp [ (γ q)t] + A 2 exp [ (γ + q)t] (25) where q = γ 2 ω 2 0. Case 1 When γ 2 ω0 2 > 0, q is real and positive and we have the overdamped case, where a displaced mass slowly returns to equilibrium with no oscillations. Note that (γ q) > 0. Case 3 When γ 2 ω0 2 < 0, q is imaginary and we have the underdamped case where the mass oscillates about equilibrium with a decreasing amplitude. Defining = ω0 2 γ2 (26) and doing some work (shown in the text) we can write the solution can as where A and θ 0 are determined by initial conditions. x(t) = e γt A cos( t + θ 0 ) (27) Case 2 When γ 2 ω0 2 = 0 we have critical damping. The mass quickly returns to equilibrium without overshooting. If we start with Equation 25, we run into a problem: we have only one arbitrary integration constant. To find a solution with two integration constants we must return to the differential equation, Equation 24, that becomes [D + γ] [D + γ] x = 0 (28) This is a second order differential equation and we need two arbitrary constants. Here is the trick needed to get the general solution. Define u = (D + γ)x. Then (D + γ)u = 0 and this has the solution u = Ae γt. The solution for x is found from Ae γt = (D + γ)x A = e γt (D + γ)x = D(xe γt ) xe γt = At + B x = Ate γt + Be γt (29) Example Suppose we have a mass of 3.0 kg attached to a spring of constant 12.0 N/m, and with initial conditions that at t = 0, position is 1.0 m and velocity is 0.0 m/s. (a) For the underdamped case with c = 4.0 kg/s find the values of the integration constants. [A = 1.06 m, θ 0 = rad] (b) For the critically damped case with c = 12 kg/s find the values of the integration constants. [A = 2.0 m/s, B = 1.0 m] (c) For the overdamped case of c = 20 kg/s, find the values of the integration constants. [A 1 = m, A 2 = m] 5

6 2.1 Energy and Linear Damping The instantaneous energy is still E = mẋ 2 /2 + kx 2 /2 so we can look at how this changes with time. de = mẋẍ + kẋx = (mẍ + kx)ẋ (30) dt But from Equation 21, mẍ + kx = cẋ, hence de dt = cẋ2 (31) 2.2 Quality Factor for Damped SHM For the underdamped case we can characterize the system by its Q-factor, Q = 2π E E (32) where E is the energy stored in the oscillator (think in terms of the average energy) and E is the energy lost in one period of the oscillation. Q is calculated usually for weakly damped systems where the energy does not change quickly. In this case the energy at time t can be written Using Equation 27 for x we differentiate to get Then, changing variables to z = t + θ 0 E = E = 1 2 ka2 e 2γt = 1 2 mω2 0A 2 e 2γt (33) ẋ = Ae γt (γ cos( t + θ 0 ) + sin( t + θ 0 )) (34) Td 0 = ca2 cẋ 2 dt (35) 2π+θ0 θ 0 e 2γt [ γ 2 cos 2 z + 2γ cos z sin z + ω 2 d sin2 z ] dz (36) But for weakly decaying systems the exponential changes very little during the integration, so we remove it from the integral. The integration limits can start anywhere as long as they extend over 2π. So 2π E = ca2 e 2γt 0 [ γ 2 cos 2 z + 2γ cos z sin z + ω 2 d sin2 z ] dz (37) 6

7 The integrals for cos 2 and sin 2 each yield π, and the cross term becomes 0, giving Then E = πca2 e 2γt [ γ 2 + ω 2 ] πca 2 d = e 2γt ω0 2 (38) Q = 2π 1 2 mω2 0 A2 e 2γt πca 2 e 2γt ω 2 0 = 2γ Table in the text lists some Q s for underdamped systems. The earth (responding to earthquakes) has Q = 250 to 1400 while a piano string has Q = (39) 2.3 An Example Suppose a mass of 500 kg is attached to an ideal spring of force constant 2000 N/m and a linear damping shock absorber of constant 20 N s/m. Find (a) The natural (undamped) angular frequency and frequency in Hz. (b) The frequency with damping applied. (c) The Q of the system (d) The time for the amplitude to reduce to 1/10 of its original value. (e) The energy when the amplitude has reduced to 1/10 its original value. We have m = 500 kg, k = 2000 N/m, and c = 20 N s/m. (a) Use ω 0 = k/m = rad/s. This gives a frequency of f = ω 0 /2π = Hz (and a period of s). (b) Compute γ = c/2m = 0.02 rad/s. Then = ω 2 0 γ2 = rad/s. (c) The quality factor is Q = /2γ = (d) The envelope of the oscillation varies as A(t) = Ae γt, so solve 0.1 = e 0.02t to get t = s. (37 oscillations during this time.) (e) Since E A 2, when the amplitude is reduced to 1/10 of its original value, the energy is reduced to 1/100 of its original value. 3 Phase Space Plots of position versus time and velocity versus time are useful, but often a plot of velocity (or momentum) versus position is drawn. A plot such as this is called a phase plot. 7

8 For undamped SHM, x = A sin(ω 0 t + φ 0 ) and ẋ = Aω 0 cos(ω 0 t + φ 0 ). We combine these so as to eliminate time and get the parametric equation x 2 A 2 + ẋ2 A 2 ω 2 0 = 1 (40) which is the equation of an ellipse. Figure shows several trajectories in phase space corresponding to different initial conditions. All trajectories in this situation form closed paths. For underdamped oscillations the phase plot spirals in toward a final value of v = 0. The text does some very clever changes of variables to make the phase plot more obvious: I will just demonstrate using Easy JAVA Simulation, and extend to the case of critical damping and over damping. Non linear oscillators can lead to chaotic motion, a topic left for other courses. 4 Forced Damped Harmonic Motion Consider the damped harmonic oscillator subject to a periodic driving force F 0 cos ωt. The equation of motion is mẍ = kx cẋ + F 0 cos ωt (41) First consider undamped motion and guess the solution to be x = A cos(ωt φ) with two variables, A, φ to be determined. The equation of motion becomes maω 2 cos(ωt φ) + ka cos(ωt φ) = F 0 cos ωt (42) This only works for the two values φ = 0, π, implying A = F 0/m ω 2 0 ω2 φ = 0 Low frequency (43) = F 0/m ω 2 ω 2 0 This gives a resonance at ω = ω 0. φ = π High frequency (44) When we add damping and the velocity term we have difficulty when starting with a cosine solution. Instead we assume a driving force as F 0 e iωt and consider the real part of 8

9 the complex quantity to describe the physical situation. We guess a solution x = Ae iωt φ so that the equation of motion becomes mω 2 Ae iωt φ + kae iωt φ + icωae iωt φ = F 0 e iωt (45) mω 2 A + ka + icωa = F 0 e iφ = F 0 (cos φ + i sin φ) (46) The real parts must be equal, and the imaginary parts must be equal giving These can be solved simultaneously and give A(k mω 2 ) = F 0 cos φ (47) c ω A = F 0 sin φ (48) tan φ = A = 2γω ω0 2 ω2 (49) F 0 /m (ω 2 0 ω 2 ) 2 + 4γ 2 ω 2 (50) Further details should have been discussed in Vibrations and Waves. You will probably see resonance in an electrical circuit when you take Electronic Measurements. When the resonance is fairly sharp, meaning relatively small damping, the resonant frequency is very close to the natural frequency. For these cases we define the sharpness of the resonance with another Q factor same symbol and definition as before, but a different meaning. If the resonance has an amplitude A max at resonance, ω 0, then there are frequencies above and below the resonant frequency at which A = A max / 2. These two frequencies are called the half-energy (or half-power) frequencies, and the difference in frequencies is ω = 2γ = c (51) m and Q resonance = ω 0 ω ω 0 2γ (52) 9

### Spring 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

### Mechanical Vibrations

Mechanical Vibrations A mass m is suspended at the end of a spring, its weight stretches the spring by a length L to reach a static state (the equilibrium position of the system). Let u(t) denote the displacement,

### Periodic Motion or Oscillations. Physics 232 Lecture 01 1

Periodic Motion or Oscillations Physics 3 Lecture 01 1 Periodic Motion Periodic Motion is motion that repeats about a point of stable equilibrium Stable Equilibrium Unstable Equilibrium A necessary requirement

### Simple Harmonic Motion

Simple Harmonic Motion Restating Hooke s law The equation of motion Phase, frequency, amplitude Simple Pendulum Damped and Forced oscillations Resonance Harmonic Motion A lot of motion in the real world

### Physics 1120: Simple Harmonic Motion Solutions

Questions: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Physics 1120: Simple Harmonic Motion Solutions 1. A 1.75 kg particle moves as function of time as follows: x = 4cos(1.33t+π/5) where distance is measured

### Homework #7 Solutions

MAT 0 Spring 201 Problems Homework #7 Solutions Section.: 4, 18, 22, 24, 4, 40 Section.4: 4, abc, 16, 18, 22. Omit the graphing part on problems 16 and 18...4. Find the general solution to the differential

### 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

### 226 Chapter 15: OSCILLATIONS

Chapter 15: OSCILLATIONS 1. In simple harmonic motion, the restoring force must be proportional to the: A. amplitude B. frequency C. velocity D. displacement E. displacement squared 2. An oscillatory motion

### 1.10 Using Figure 1.6, verify that equation (1.10) satisfies the initial velocity condition. t + ") # x (t) = A! n. t + ") # v(0) = A!

1.1 Using Figure 1.6, verify that equation (1.1) satisfies the initial velocity condition. Solution: Following the lead given in Example 1.1., write down the general expression of the velocity by differentiating

### Physics 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,

### Oscillations. Chapter 1. 1.1 Simple harmonic motion. 1.1.1 Hooke s law and small oscillations

Chapter 1 Oscillations David Morin, morin@physics.harvard.edu A wave is a correlated collection of oscillations. For example, in a transverse wave traveling along a string, each point in the string oscillates

### ANALYTICAL 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

### Physics 207 Lecture 25. Lecture 25. For Thursday, read through all of Chapter 18. Angular Momentum Exercise

Lecture 5 Today Review: Exam covers Chapters 14-17 17 plus angular momentum, statics Assignment For Thursday, read through all of Chapter 18 Physics 07: Lecture 5, Pg 1 Angular Momentum Exercise A mass

### Hooke s Law. Spring. Simple Harmonic Motion. Energy. 12/9/09 Physics 201, UW-Madison 1

Hooke s Law Spring Simple Harmonic Motion Energy 12/9/09 Physics 201, UW-Madison 1 relaxed position F X = -kx > 0 F X = 0 x apple 0 x=0 x > 0 x=0 F X = - kx < 0 x 12/9/09 Physics 201, UW-Madison 2 We know

### HOOKE S LAW AND SIMPLE HARMONIC MOTION

HOOKE S LAW AND SIMPLE HARMONIC MOTION Alexander Sapozhnikov, Brooklyn College CUNY, New York, alexs@brooklyn.cuny.edu Objectives Study Hooke s Law and measure the spring constant. Study Simple Harmonic

### Simple Harmonic Motion(SHM) Period and Frequency. Period and Frequency. Cosines and Sines

Simple Harmonic Motion(SHM) Vibration (oscillation) Equilibrium position position of the natural length of a spring Amplitude maximum displacement Period and Frequency Period (T) Time for one complete

### 2.2 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

### Physics 53. Oscillations. You've got to be very careful if you don't know where you're going, because you might not get there.

Physics 53 Oscillations You've got to be very careful if you don't know where you're going, because you might not get there. Yogi Berra Overview Many natural phenomena exhibit motion in which particles

### Solution: F = kx is Hooke s law for a mass and spring system. Angular frequency of this system is: k m therefore, k

Physics 1C Midterm 1 Summer Session II, 2011 Solutions 1. If F = kx, then k m is (a) A (b) ω (c) ω 2 (d) Aω (e) A 2 ω Solution: F = kx is Hooke s law for a mass and spring system. Angular frequency of

### Chapter 13, example problems: x (cm) 10.0

Chapter 13, example problems: (13.04) Reading Fig. 13-30 (reproduced on the right): (a) Frequency f = 1/ T = 1/ (16s) = 0.0625 Hz. (since the figure shows that T/2 is 8 s.) (b) The amplitude is 10 cm.

### AP 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

### Introduction to Complex Numbers in Physics/Engineering

Introduction to Complex Numbers in Physics/Engineering ference: Mary L. Boas, Mathematical Methods in the Physical Sciences Chapter 2 & 14 George Arfken, Mathematical Methods for Physicists Chapter 6 The

### Simple Harmonic Motion Concepts

Simple Harmonic Motion Concepts INTRODUCTION Have you ever wondered why a grandfather clock keeps accurate time? The motion of the pendulum is a particular kind of repetitive or periodic motion called

### A C O U S T I C S of W O O D Lecture 3

Jan Tippner, Dep. of Wood Science, FFWT MU Brno jan. tippner@mendelu. cz Content of lecture 3: 1. Damping 2. Internal friction in the wood Content of lecture 3: 1. Damping 2. Internal friction in the wood

### 2.6 The driven oscillator

2.6. THE DRIVEN OSCILLATOR 131 2.6 The driven oscillator We would like to understand what happens when we apply forces to the harmonic oscillator. That is, we want to solve the equation M d2 x(t) 2 + γ

### Physics 231 Lecture 15

Physics 31 ecture 15 Main points of today s lecture: Simple harmonic motion Mass and Spring Pendulum Circular motion T 1/f; f 1/ T; ω πf for mass and spring ω x Acos( ωt) v ωasin( ωt) x ax ω Acos( ωt)

### Practice 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

### Unit - 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

### Updated 2013 (Mathematica Version) M1.1. Lab M1: The Simple Pendulum

Updated 2013 (Mathematica Version) M1.1 Introduction. Lab M1: The Simple Pendulum The simple pendulum is a favorite introductory exercise because Galileo's experiments on pendulums in the early 1600s are

### Wave topics 1. Waves - multiple choice

Wave topics 1 Waves - multiple choice When an object is oscillating in simple harmonic motion in the vertical direction, its maximum speed occurs when the object (a) is at its highest point. (b) is at

### LABORATORY 9. Simple Harmonic Motion

LABORATORY 9 Simple Harmonic Motion Purpose In this experiment we will investigate two examples of simple harmonic motion: the mass-spring system and the simple pendulum. For the mass-spring system we

### turn-table in terms of SHM and UCM: be plotted as a sine wave. n Think about spinning a ball on a string or a ball on a

RECALL: Angular Displacement & Angular Velocity Think about spinning a ball on a string or a ball on a turn-table in terms of SHM and UCM: If you look at the ball from the side, its motion could be plotted

### Simple harmonic motion

PH-122- Dynamics Page 1 Simple harmonic motion 02 February 2011 10:10 Force opposes the displacement in A We assume the spring is linear k is the spring constant. Sometimes called stiffness constant Newton's

### Simple Harmonic Motion

5 Simple Harmonic Motion Note: this section is not part of the syllabus for PHYS26. You should already be familiar with simple harmonic motion from your first year course PH115 Oscillations and Waves.

### Seminar 4: CHARGED PARTICLE IN ELECTROMAGNETIC FIELD. q j

Seminar 4: CHARGED PARTICLE IN ELECTROMAGNETIC FIELD Introduction Let take Lagrange s equations in the form that follows from D Alembert s principle, ) d T T = Q j, 1) dt q j q j suppose that the generalized

### Second 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

### AP1 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

### Torsion Pendulum. Life swings like a pendulum backward and forward between pain and boredom. Arthur Schopenhauer

Torsion Pendulum Life swings like a pendulum backward and forward between pain and boredom. Arthur Schopenhauer 1 Introduction Oscillations show up throughout physics. From simple spring systems in mechanics

### MECHANICS IV - SIMPLE HARMONIC MOTION

M-IV-p.1 A. OSCILLATIONS B. SIMPLE PENDULUM C. KINEMATICS OF SIMPLE HARMONIC MOTION D. SPRING-AND-MASS SYSTEM E. ENERGY OF SHM F. DAMPED HARMONIC MOTION G. FORCED VIBRATION A. OSCILLATIONS A to-and-fro

### Understanding 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

### Columbia University Department of Physics QUALIFYING EXAMINATION

Columbia University Department of Physics QUALIFYING EXAMINATION Monday, January 13, 2014 1:00PM to 3:00PM Classical Physics Section 1. Classical Mechanics Two hours are permitted for the completion of

### Solving the Harmonic Oscillator Equation. Morgan Root NCSU Department of Math

Solving the Harmonic Oscillator Equation Morgan Root NCSU Department of Math Spring-Mass System Consider a mass attached to a wall by means of a spring. Define y to be the equilibrium position of the block.

### Experiment Type: Open-Ended

Simple Harmonic Oscillation Overview Experiment Type: Open-Ended In this experiment, students will look at three kinds of oscillators and determine whether or not they can be approximated as simple harmonic

### Mechanics 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

### Lesson 5 Rotational and Projectile Motion

Lesson 5 Rotational and Projectile Motion Introduction: Connecting Your Learning The previous lesson discussed momentum and energy. This lesson explores rotational and circular motion as well as the particular

### Chapter 15, example problems:

Chapter, example problems: (.0) Ultrasound imaging. (Frequenc > 0,000 Hz) v = 00 m/s. λ 00 m/s /.0 mm =.0 0 6 Hz. (Smaller wave length implies larger frequenc, since their product,

### Experiment 4: Harmonic Motion Analysis

Experiment 4: Harmonic Motion Analysis Background In this experiment you will investigate the influence of damping on a driven harmonic oscillator and study resonant conditions. The following theoretical

### Module P5.3 Forced vibrations and resonance

F L E X I B L E L E A R N I N G A P P R O A C H T O P H Y S I C S Module P5.3 Forced vibrations and resonance 1 Opening items 1.1 Module introduction 1.2 Fast track questions 1.3 Ready to study? 2 Driven

### Lecture L19 - Vibration, Normal Modes, Natural Frequencies, Instability

S. Widnall 16.07 Dynamics Fall 2009 Version 1.0 Lecture L19 - Vibration, Normal Modes, Natural Frequencies, Instability Vibration, Instability An important class of problems in dynamics concerns the free

### Simple 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

### SECOND-ORDER LINEAR HOMOGENEOUS DIFFERENTIAL EQUATIONS

L SECOND-ORDER LINEAR HOOGENEOUS DIFFERENTIAL EQUATIONS SECOND-ORDER LINEAR HOOGENEOUS DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS A second-order linear differential equation is one of the form d

### How do we obtain the solution, if we are given F (t)? First we note that suppose someone did give us one solution of this equation

1 Green s functions The harmonic oscillator equation is This has the solution mẍ + kx = 0 (1) x = A sin(ωt) + B cos(ωt), ω = k m where A, B are arbitrary constants reflecting the fact that we have two

### Chapter 1. Oscillations. Oscillations

Oscillations 1. A mass m hanging on a spring with a spring constant k has simple harmonic motion with a period T. If the mass is doubled to 2m, the period of oscillation A) increases by a factor of 2.

### www.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

### Physics 2305 Lab 11: Torsion Pendulum

Name ID number Date Lab CRN Lab partner Lab instructor Physics 2305 Lab 11: Torsion Pendulum Objective 1. To demonstrate that the motion of the torsion pendulum satisfies the simple harmonic form in equation

### Simple Harmonic Motion Experiment. 1 f

Simple Harmonic Motion Experiment In this experiment, a motion sensor is used to measure the position of an oscillating mass as a function of time. The frequency of oscillations will be obtained by measuring

### 19.7. 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.

### Response to Harmonic Excitation

Response to Harmonic Excitation Part 1 : Undamped Systems Harmonic excitation refers to a sinusoidal external force of a certain frequency applied to a system. The response of a system to harmonic excitation

### ORDINARY DIFFERENTIAL EQUATIONS

Page 1 of 21 ORDINARY DIFFERENTIAL EQUATIONS Lecture 15 Physical Systems Modelled by Linear Homogeneous Second- Order ODEs (Revised 22 March, 2009 @ 18:00) Professor Stephen H Saperstone Department of

### Determination of Acceleration due to Gravity

Experiment 2 24 Kuwait University Physics 105 Physics Department Determination of Acceleration due to Gravity Introduction In this experiment the acceleration due to gravity (g) is determined using two

### 1 of 10 11/23/2009 6:37 PM

hapter 14 Homework Due: 9:00am on Thursday November 19 2009 Note: To understand how points are awarded read your instructor's Grading Policy. [Return to Standard Assignment View] Good Vibes: Introduction

### Lesson 11. Luis Anchordoqui. Physics 168. Tuesday, December 8, 15

Lesson 11 Physics 168 1 Oscillations and Waves 2 Simple harmonic motion If an object vibrates or oscillates back and forth over same path each cycle taking same amount of time motion is called periodic

### y = a sin ωt or y = a cos ωt then the object is said to be in simple harmonic motion. In this case, Amplitude = a (maximum displacement)

5.5 Modelling Harmonic Motion Periodic behaviour happens a lot in nature. Examples of things that oscillate periodically are daytime temperature, the position of a weight on a spring, and tide level. If

### EDEXCEL NATIONAL CERTIFICATE/DIPLOMA UNIT 5 - ELECTRICAL AND ELECTRONIC PRINCIPLES NQF LEVEL 3 OUTCOME 4 - ALTERNATING CURRENT

EDEXCEL NATIONAL CERTIFICATE/DIPLOMA UNIT 5 - ELECTRICAL AND ELECTRONIC PRINCIPLES NQF LEVEL 3 OUTCOME 4 - ALTERNATING CURRENT 4 Understand single-phase alternating current (ac) theory Single phase AC

### People s Physics book 3e Ch 25-1

The Big Idea: In most realistic situations forces and accelerations are not fixed quantities but vary with time or displacement. In these situations algebraic formulas cannot do better than approximate

### SIMPLE HARMONIC MOTION: SHIFTED ORIGIN AND PHASE

MISN-0-26 SIMPLE HARMONIC MOTION: SHIFTED ORIGIN AND PHASE SIMPLE HARMONIC MOTION: SHIFTED ORIGIN AND PHASE by Kirby Morgan 1. Dynamics of Harmonic Motion a. Force Varies in Magnitude and Direction................

### Midterm Solutions. mvr = ω f (I wheel + I bullet ) = ω f 2 MR2 + mr 2 ) ω f = v R. 1 + M 2m

Midterm Solutions I) A bullet of mass m moving at horizontal velocity v strikes and sticks to the rim of a wheel a solid disc) of mass M, radius R, anchored at its center but free to rotate i) Which of

### PHYS 101-4M, Fall 2005 Exam #3. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

PHYS 101-4M, Fall 2005 Exam #3 Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) A bicycle wheel rotates uniformly through 2.0 revolutions in

### Hooke s Law and Simple Harmonic Motion

Hooke s Law and Simple Harmonic Motion OBJECTIVE to measure the spring constant of the springs using Hooke s Law to explore the static properties of springy objects and springs, connected in series and

### Chapter 24 Physical Pendulum

Chapter 4 Physical Pendulum 4.1 Introduction... 1 4.1.1 Simple Pendulum: Torque Approach... 1 4. Physical Pendulum... 4.3 Worked Examples... 4 Example 4.1 Oscillating Rod... 4 Example 4.3 Torsional Oscillator...

### Advanced Higher Physics: MECHANICS. Simple Harmonic Motion

Advanced Higher Physics: MECHANICS Simple Harmonic Motion At the end of this section, you should be able to: Describe examples of simple harmonic motion (SHM). State that in SHM the unbalanced force is

### Solutions 2.4-Page 140

Solutions.4-Page 4 Problem 3 A mass of 3 kg is attached to the end of a spring that is stretched cm by a force of 5N. It is set in motion with initial position = and initial velocity v = m/s. Find the

### Lecture 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

### Lecture L13 - Conservative Internal Forces and Potential Energy

S. Widnall, J. Peraire 16.07 Dynamics Fall 008 Version.0 Lecture L13 - Conservative Internal Forces and Potential Energy The forces internal to a system are of two types. Conservative forces, such as gravity;

### Let s first see how precession works in quantitative detail. The system is illustrated below: ...

lecture 20 Topics: Precession of tops Nutation Vectors in the body frame The free symmetric top in the body frame Euler s equations The free symmetric top ala Euler s The tennis racket theorem As you know,

### Dynamics. Figure 1: Dynamics used to generate an exemplar of the letter A. To generate

Dynamics Any physical system, such as neurons or muscles, will not respond instantaneously in time but will have a time-varying response termed the dynamics. The dynamics of neurons are an inevitable constraint

### Lecture L5 - Other Coordinate Systems

S. Widnall, J. Peraire 16.07 Dynamics Fall 008 Version.0 Lecture L5 - Other Coordinate Systems In this lecture, we will look at some other common systems of coordinates. We will present polar coordinates

### SOLUTIONS TO CONCEPTS CHAPTER 15

SOLUTIONS TO CONCEPTS CHAPTER 15 1. v = 40 cm/sec As velocity of a wave is constant location of maximum after 5 sec = 40 5 = 00 cm along negative x-axis. [(x / a) (t / T)]. Given y = Ae a) [A] = [M 0 L

### Discussion Session 1

Physics 102 Fall 2016 NAME: Discussion Session 1 Math Review and Temperature The goal of Physics is to explain the Universe in terms of equations, and so the ideas of mathematics are central to your success

### 1.2 Second-order systems

1.2. SECOND-ORDER SYSTEMS 25 if the initial fluid height is defined as h() = h, then the fluid height as a function of time varies as h(t) = h e tρg/ra [m]. (1.31) 1.2 Second-order systems In the previous

### Mechanics Lecture Notes. 1 Notes for lectures 12 and 13: Motion in a circle

Mechanics Lecture Notes Notes for lectures 2 and 3: Motion in a circle. Introduction The important result in this lecture concerns the force required to keep a particle moving on a circular path: if the

### SIMPLE HARMONIC MOTION

SIMPLE HARMONIC MOTION PURPOSE The purpose of this experiment is to investigate one of the fundamental types of motion that exists in nature - simple harmonic motion. The importance of this kind of motion

### Lab M1: The Simple Pendulum

Lab M1: The Simple Pendulum Introduction. The simple pendulum is a favorite introductory exercise because Galileo's experiments on pendulums in the early 1600s are usually regarded as the beginning of

### Experiment 9. The Pendulum

Experiment 9 The Pendulum 9.1 Objectives Investigate the functional dependence of the period (τ) 1 of a pendulum on its length (L), the mass of its bob (m), and the starting angle (θ 0 ). Use a pendulum

### Response to Harmonic Excitation Part 2: Damped Systems

Response to Harmonic Excitation Part 2: Damped Systems Part 1 covered the response of a single degree of freedom system to harmonic excitation without considering the effects of damping. However, almost

### 11. Rotation Translational Motion: Rotational Motion:

11. Rotation Translational Motion: Motion of the center of mass of an object from one position to another. All the motion discussed so far belongs to this category, except uniform circular motion. Rotational

### Chapter 6 Circular Motion

Chapter 6 Circular Motion 6.1 Introduction... 1 6.2 Cylindrical Coordinate System... 2 6.2.1 Unit Vectors... 3 6.2.2 Infinitesimal Line, Area, and Volume Elements in Cylindrical Coordinates... 4 Example

### Notice 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

### Physics 1022: Chapter 14 Waves

Phys 10: Introduction, Pg 1 Physics 10: Chapter 14 Waves Anatomy of a wave Simple harmonic motion Energy and simple harmonic motion Phys 10: Introduction, Pg Page 1 1 Waves New Topic Phys 10: Introduction,

### Design for Vibration Suppression

Design for Vibration Suppression Outlines: 1. Vibration Design Process. Design of Vibration Isolation Moving base Fixed base 3. Design of Vibration Absorbers Vibration Design Process Actual system Modeling

### SOLID MECHANICS TUTORIAL MECHANISMS KINEMATICS - VELOCITY AND ACCELERATION DIAGRAMS

SOLID MECHANICS TUTORIAL MECHANISMS KINEMATICS - VELOCITY AND ACCELERATION DIAGRAMS This work covers elements of the syllabus for the Engineering Council exams C105 Mechanical and Structural Engineering

### Center of Mass/Momentum

Center of Mass/Momentum 1. 2. An L-shaped piece, represented by the shaded area on the figure, is cut from a metal plate of uniform thickness. The point that corresponds to the center of mass of the L-shaped

### Aim : To study how the time period of a simple pendulum changes when its amplitude is changed.

Aim : To study how the time period of a simple pendulum changes when its amplitude is changed. Teacher s Signature Name: Suvrat Raju Class: XIID Board Roll No.: Table of Contents Aim..................................................1

### Second Order Systems

Second Order Systems Second Order Equations Standard Form G () s = τ s K + ζτs + 1 K = Gain τ = Natural Period of Oscillation ζ = Damping Factor (zeta) Note: this has to be 1.0!!! Corresponding Differential

### Differential 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

### State Newton's second law of motion for a particle, defining carefully each term used.

5 Question 1. [Marks 20] An unmarked police car P is, travelling at the legal speed limit, v P, on a straight section of highway. At time t = 0, the police car is overtaken by a car C, which is speeding

### Chapter 2 Second Order Differential Equations

Chapter 2 Second Order Differential Equations Either mathematics is too big for the human mind or the human mind is more than a machine. - Kurt Gödel (1906-1978) 2.1 The Simple Harmonic Oscillator The

### From Figure 5.1, an rms displacement of 1 mm (1000 µm) would not cause wall damage at frequencies below 3.2 Hz.

5-1 Problems and Solutions Section 5.1 (5.1 through 5.5) 5.1 Using the nomograph of Figure 5.1, determine the frequency range of vibration for which a machine oscillation remains at a satisfactory level