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

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

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

Transcription

1 Jan Tippner, Dep. of Wood Science, FFWT MU Brno jan. cz

2 Content of lecture 3: 1. Damping 2. Internal friction in the wood

3 Content of lecture 3: 1. Damping 2. Internal friction in the wood

4 Damping sine wave is a waveform generated by a system that is characterised by simple harmonic motion ideal system which exhibits simple harmonic motion is a system that loses no energy (or has its energy replenished from outside the system) such a waveform can also be called a continuous waveform as it continues forever without eventually reducing to zero intensity

5 Damping real systems are never ideal; all naturally occuring systems loose energy (eg. as heat due to friction) system loses energy as heat (both internally as a consequence of heat loss during physical deformation and externally as a consequence of friction with air) this loss of energy in an oscillating system is know as damping; a damped waveform is also know as a noncontinuous waveform

6 Damping damped waveform can die out quickly or slowly; waveform that dies out quickly is said to be strongly damped as it loses energy quickly; waveform that dies out slowly is said to be weakly damped as it loses energy slowly damping is not just a characteristic of systems that generate non continuous sine wave like patterns, damping is a characteristic of systems that produce sounds with very complex spectral patterns in physics, damping is any effect that tends to reduce the amplitude of oscillations in an oscillatory system, particularly the harmonic oscillator in mechanics, friction is one such damping effect. For many purposes the frictional force F f can be modeled as being proportional to the velocity v of the object: F f = c v where: c is the viscous damping coefficient, given in units of newton seconds per meter

7 Damping damped harmonic oscillators satisfy the second order differential equation: where: ω 0 is the undamped angular frequency of the oscillator and ζ is a constant called the damping ratio for a mass on a spring having a spring constant k and a damping coefficient c: ω 0 = (k/m) ζ = c/2mω 0.

8 Damping value of the damping ratio ζ determines the behavior of the system. A damped harmonic oscillator can be: 1. Overdamped (ζ > 1) system returns (exponentially decays) to equilibrium without oscillating; larger values of the damping ratio ζ return to equilibrium slower 2. Critically damped ( ζ = 1) system returns to equilibrium as quickly as possible without oscillating (often desired) 3. Underdamped (ζ < 1) system oscillates (with a slightly different frequency than the undamped case) with the amplitude gradually decreasing to zero frequency of the underdamped harmonic oscillator is given by:

9 Example of Spring Mass System A mass m attached to a spring and damper. The damping coefficient is represented by B, F denotes an external force.

10 Example of Spring Mass System ideal mass spring damper system with mass m (in kilograms), spring constant k (in newtons per meter) and viscous damper of damping coefficient c (in newton seconds per meter or kilograms per second) is subject to an oscillatory force... and a damping force... treating the mass as a free body and applying Newton's second law (F=ma), the total force F tot on the body is where: a is the acceleration (in meters per second squared) of the mass and x is the displacement (in meters) of the mass relative to a fixed point of reference F tot = F s + F d >>>>>>

11 Example of Spring Mass System rearr. to: where: (undamped) natural frequency of the system: the damping ratio: the natural frequency represents an angular frequency, expressed in radians per second the damping ratio is a dimensionless quantity

12 Example of Spring Mass System the differential equation now becomes we can solve the equation by assuming a solution x such that: where: the parameter γ (gamma) is, in general, a complex number. substituting this assumed solution back into the differential equation gives which is the characteristic equation. solving the characteristic equation will give two roots, γ + and γ ; and the solution to the differential equation is: where: A and B are determined by the initial conditions of the system:

13 Example of Spring Mass System rearr. to: where: (undamped) natural frequency of the system: the damping ratio: the natural frequency represents an angular frequency, expressed in radians per second the damping ratio is a dimensionless quantity

14 A harmonic oscillator can be: 1. Overdamped (ζ > 1) system returns (exponentially decays) to equilibrium without oscillating; larger values of the damping ratio ζ return to equilibrium slower 2. Critically damped ( ζ = 1) system returns to equilibrium as quickly as possible without oscillating (often desired) 3. Underdamped (ζ < 1) system oscillates (with a slightly different frequency than the undamped case) with the amplitude gradually decreasing to zero 4. Undamped (ζ = 0)

15 Dependence of the system behavior on the value of the damping ratio, for under damped, ζ critically damped, over damped, and undamped cases, for zero velocity initial condition.

16 the behavior of the system depends on the relative values of the two fundamental parameters, the natural frequency ω 0 and the damping ratio ζ in particular, the qualitative behavior of the system depends crucially on whether the quadratic equation for γ has one real solution, two real solutions, or two complex conjugate solutions. 1. Critical damping ( ζ = 1) When ζ = 1, there is a double root γ (defined above), which is real. The system is said to be critically damped. A critically damped system converges to zero faster than any other, and without oscillating. t In this case, with only one root γ, there is in addition to the solution x(t) = e γ a solution x(t) = te t γ : where A and B are determined by the initial conditions of the system (usually the initial position and velocity of the mass):

17 2. Over damping ( ζ > 1) When ζ > 1, the system is over damped and there are two different real roots. The solution to the motion equation is: where A and B are determined by the initial conditions of the system:

18 3. Under damping (0 ζ < 1) Finally, when 0 ζ < 1, γ is complex, and the system is under damped. In this situation, the system will oscillate at the natural damped frequency ω d, which is a function of the natural frequency and the damping ratio. In this case, the solution can be generally written as: where: represents the natural damped frequency of the system, and A and B are again determined by the initial conditions of the system: for an under damped system, the value of can be found by examining the logarithm of the ratio of ζ succeeding amplitudes of a system this is called the logarithmic decrement

19 Logarithmic Decrement of Damping Logarithmic decrement, δ, is used to find the damping ratio of an underdamped system in the time domain. The logarithmic decrement is the natural log of the amplitudes of any two successive peaks: where x 0 is the greater of the two amplitudes and x n is the amplitude of a peak n periods away; the damping ratio is then found from the logarithmic decrement:

20 Q factor the quality factor or Q factor is a dimensionless parameter that describes how under damped an oscillator or resonator is, or equivalently, characterizes a resonator's bandwidth relative to its center frequency higher Q indicates a lower rate of energy loss relative to the stored energy of the oscillator; the oscillations die out more slowly (a pendulum suspended from a high quality bearing, oscillating in air, has a high Q, while a pendulum immersed in oil has a low one) for a single damped mass spring system, the Q factor represents the effect of simplified viscous damping or drag, where the damping force or drag force is proportional to velocity. The formula for the Q factor is: where M is the mass, k is the spring constant, and D is the damping coefficient, defined by the equation F damping = D v, where v is the velocity bandwidth, Δf, of a damped oscillator is shown on a graph of energy versus frequency. The Q factor of the damped oscillator, or filter, is f0 / Δf. The higher the Q, the narrower and 'sharper' the peak is.

21 Simulation by finite element method Ansys software unsteady: full transient solution 1D (beam), 2D (shell), ev. 3D (solid) 1D solution: preprocessing (geometry, physics), solution, postprocessing (time history) modal analysis >>>>> full transient analysis adaptation for wood (changes of material model)

22 Content of lecture 3: 1. Damping 2. Internal friction in the wood

Periodic Motion or Oscillations. Physics 232 Lecture 01 1

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

More information

Mechanical Vibrations

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,

More information

VIBRATION DUE TO ROTATION UNBALANCE

VIBRATION DUE TO ROTATION UNBALANCE Fall 08 Prepared by: Keivan Anbarani Abstract In this experiment four eccentric masses are used in conjunction with four springs and one damper to simulate the vibration. Masses are aligned in different

More information

Experiment 08: RLC Circuits and Resonance Dr. Pezzaglia

Experiment 08: RLC Circuits and Resonance Dr. Pezzaglia Mar9 RLC Circuit Page Experiment 8: RLC Circuits and Resonance Dr. Pezzaglia Theory When a system at a stable equilibrium is displaced, it will tend to oscillate. An Inductor combined with Capacitor will

More information

Simple Harmonic Motion

Simple Harmonic Motion Simple Harmonic Motion -Theory Simple harmonic motion refers to the periodic sinusoidal oscillation of an object or quantity. Simple harmonic motion is eecuted by any quantity obeying the Differential

More information

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. 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

More information

Advanced Higher Physics: MECHANICS. Simple Harmonic Motion

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

More information

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 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

More information

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

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

More information

Lab 5: Harmonic Oscillations and Damping

Lab 5: Harmonic Oscillations and Damping 3 Lab 5: Harmonic Oscillations and Damping I. Introduction A. Objectives for this lab: 1. Learn how to quantitatively model a real harmonic oscillator 2. Learn how damping affects simple harmonic motion

More information

2.3 Cantilever linear oscillations

2.3 Cantilever linear oscillations .3 Cantilever linear oscillations Study of a cantilever oscillation is a rather science - intensive problem. In many cases the general solution to the cantilever equation of motion can not be obtained

More information

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

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.

More information

both double. A. T and v max B. T remains the same and v max doubles. both remain the same. C. T and v max

both double. A. T and v max B. T remains the same and v max doubles. both remain the same. C. T and v max Q13.1 An object on the end of a spring is oscillating in simple harmonic motion. If the amplitude of oscillation is doubled, how does this affect the oscillation period T and the object s maximum speed

More information

Oscillations. Vern Lindberg. June 10, 2010

Oscillations. Vern Lindberg. June 10, 2010 Oscillations Vern Lindberg June 10, 2010 You have discussed oscillations in Vibs and Waves: we will therefore touch lightly on Chapter 3, mainly trying to refresh your memory and extend the concepts. 1

More information

Simple Harmonic Motion

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.

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

Dynamic Response of Measurement Systems

Dynamic Response of Measurement Systems ME231 Measurements Laboratory Spring 1999 Dynamic Response of Measurement Systems Edmundo Corona c The handout Getting Ready to Measure presented a couple of examples where the results of a static calibration

More information

p = F net t (2) But, what is the net force acting on the object? Here s a little help in identifying the net force on an object:

p = F net t (2) But, what is the net force acting on the object? Here s a little help in identifying the net force on an object: Harmonic Oscillator Objective: Describe the position as a function of time of a harmonic oscillator. Apply the momentum principle to a harmonic oscillator. Sketch (and interpret) a graph of position as

More information

1: (ta initials) 2: first name (print) last name (print) brock id (ab13cd) (lab date)

1: (ta initials) 2: first name (print) last name (print) brock id (ab13cd) (lab date) 1: (ta initials) 2: first name (print) last name (print) brock id (ab13cd) (lab date) Experiment 5 Harmonic motion In this Experiment you will learn that Hooke s Law F = kx can be used to model the interaction

More information

Applications of Second-Order Differential Equations

Applications of Second-Order Differential Equations Applications of Second-Order Differential Equations Second-order linear differential equations have a variety of applications in science and engineering. In this section we explore two of them: the vibration

More information

Coupled Electrical Oscillators Physics 3600 Advanced Physics Lab Summer 2010 Don Heiman, Northeastern University, 5/10/10

Coupled Electrical Oscillators Physics 3600 Advanced Physics Lab Summer 2010 Don Heiman, Northeastern University, 5/10/10 Coupled Electrical Oscillators Physics 3600 Advanced Physics Lab Summer 00 Don Heiman, Northeastern University, 5/0/0 I. Introduction The objectives of this experiment are: () explore the properties of

More information

Response to Harmonic Excitation Part 2: Damped Systems

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

More information

Chapter 1. Oscillations. Oscillations

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.

More information

Simple Harmonic Motion Concepts

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

More information

Chapter 13, example problems: x (cm) 10.0

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.

More information

Materials Design: Vibration Isolation and Damping, the Basics

Materials Design: Vibration Isolation and Damping, the Basics Materials Design: Vibration Isolation and Damping, the Basics Vibration management should always be considered in any engineering design. Applications that have effectively incorporated vibration management

More information

Soil Dynamics Prof. Deepankar Choudhury Department of Civil Engineering Indian Institute of Technology, Bombay

Soil Dynamics Prof. Deepankar Choudhury Department of Civil Engineering Indian Institute of Technology, Bombay Soil Dynamics Prof. Deepankar Choudhury Department of Civil Engineering Indian Institute of Technology, Bombay Module - 2 Vibration Theory Lecture - 8 Forced Vibrations, Dynamic Magnification Factor Let

More information

ORDINARY DIFFERENTIAL EQUATIONS

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

More information

1 CHAPTER 12 FORCED OSCILLATIONS. In Section 11.4 we argued that the most general solution of the differential equation. + cy =

1 CHAPTER 12 FORCED OSCILLATIONS. In Section 11.4 we argued that the most general solution of the differential equation. + cy = CHAPTER FORCED OSCILLATIONS More on Differential Equations In Section 4 we argued that the most general solution of the differential equation is of the form In this chapter we shall be looking at equations

More information

Second Order Systems

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

More information

Simple Harmonic Motion

Simple Harmonic Motion Simple Harmonic Motion Objective: In this exercise you will investigate the simple harmonic motion of mass suspended from a helical (coiled) spring. Apparatus: Spring 1 Table Post 1 Short Rod 1 Right-angled

More information

Homework #7 Solutions

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

More information

Simple harmonic motion

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

More information

LABORATORY 9. Simple Harmonic Motion

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

More information

Lab M1: The Simple Pendulum

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

More information

Simple Harmonic Motion

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

More information

Spring Simple Harmonic Oscillator. Spring constant. Potential Energy stored in a Spring. Understanding oscillations. Understanding oscillations

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

More information

Forced Oscillations in a Linear System

Forced Oscillations in a Linear System Forced Oscillations in a Linear System Manual Eugene Butikov Annotation. The manual includes a description of the simulated physical system and a summary of the relevant theoretical material for students

More information

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!

1.10 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

More information

Overdamped system response

Overdamped system response Second order system response. Im(s) Underdamped Unstable Overdamped or Critically damped Undamped Re(s) Underdamped Overdamped system response System transfer function : Impulse response : Step response

More information

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

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

More information

SIMPLE HARMONIC MOTION

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

More information

226 Chapter 15: OSCILLATIONS

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

More information

FXA 2008. UNIT G484 Module 2 4.2.3 Simple Harmonic Oscillations 11. frequency of the applied = natural frequency of the

FXA 2008. UNIT G484 Module 2 4.2.3 Simple Harmonic Oscillations 11. frequency of the applied = natural frequency of the 11 FORCED OSCILLATIONS AND RESONANCE POINTER INSTRUMENTS Analogue ammeter and voltmeters, have CRITICAL DAMPING so as to allow the needle pointer to reach its correct position on the scale after a single

More information

MECHANICS IV - SIMPLE HARMONIC MOTION

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

More information

AR-9161 B.Tech. VI Sem. Chemical Engineering Process Dynamics &Control Model Answer

AR-9161 B.Tech. VI Sem. Chemical Engineering Process Dynamics &Control Model Answer AR-9161 B.Tech. VI Sem. Chemical Engineering Process Dynamics &Control Model Answer Ans (1) Section A i. (A) ii. iii. iv. (B) (B) (B) v. (D) vi. vii. viii. ix. (C) (B) (B) (C) x. (A) Section B (1) (i)

More information

EXPERIMENT 5: SERIES AND PARALLEL RLC RESONATOR CIRCUITS

EXPERIMENT 5: SERIES AND PARALLEL RLC RESONATOR CIRCUITS EXPERIMENT 5: SERIES AND PARALLEL RLC RESONATOR CIRCUITS Equipment List S 1 BK Precision 4011 or 4011A 5 MHz Function Generator OS BK 2120B Dual Channel Oscilloscope V 1 BK 388B Multimeter L 1 Leeds &

More information

LAB #1: TIME AND FREQUENCY RESPONSES OF SERIES RLC CIRCUITS Updated July 19, 2003

LAB #1: TIME AND FREQUENCY RESPONSES OF SERIES RLC CIRCUITS Updated July 19, 2003 SFSU - ENGR 3 ELECTRONICS LAB LAB #: TIME AND FREQUENCY RESPONSES OF SERIES RLC CIRCUITS Updated July 9, 3 Objective: To investigate the step, impulse, and frequency responses of series RLC circuits. To

More information

Experiment 4: Harmonic Motion Analysis

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

More information

PHYS 130 Laboratory Experiment 11 Hooke s Law & Simple Harmonic Motion

PHYS 130 Laboratory Experiment 11 Hooke s Law & Simple Harmonic Motion PHYS 130 Laboratory Experiment 11 Hooke s Law & Simple Harmonic Motion NAME: DATE: SECTION: PARTNERS: OBJECTIVES 1. Verify Hooke s Law and use it to measure the force constant of a spring. 2. Investigate

More information

BASIC VIBRATION THEORY

BASIC VIBRATION THEORY CHAPTER BASIC VIBRATION THEORY Ralph E. Blae INTRODUCTION This chapter presents the theory of free and forced steady-state vibration of single degree-of-freedom systems. Undamped systems and systems having

More information

Engineering Feasibility Study: Vehicle Shock Absorption System

Engineering Feasibility Study: Vehicle Shock Absorption System Engineering Feasibility Study: Vehicle Shock Absorption System Neil R. Kennedy AME40463 Senior Design February 28, 2008 1 Abstract The purpose of this study is to explore the possibilities for the springs

More information

Experiment Type: Open-Ended

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

More information

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

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

More information

Damping in a variable mass on a spring pendulum

Damping in a variable mass on a spring pendulum Damping in a variable mass on a spring pendulum Rafael M. Digilov, a M. Reiner, and Z. Weizman Department of Education in Technology and Science, Technion-Israel Institute of Technology, Haifa 32000, Israel

More information

(Most of the material presented in this chapter is taken from Thornton and Marion, Chap. 3)

(Most of the material presented in this chapter is taken from Thornton and Marion, Chap. 3) Chapter 2. Small Oscillations Most of the material presented in this chapter is taken from Thornton and Marion, Chap. 3) 2.1 Introduction If a particle, originally in a position of equilibrium we limit

More information

Hooke s Law and Simple Harmonic Motion

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

More information

Runge-Kutta - Numerical Solutions of Differential Equations

Runge-Kutta - Numerical Solutions of Differential Equations Runge-Kutta - Numerical Solutions of Differential Equations Kamalu J. Beamer March 21, 2013 1 Introduction As with all dynamical systems, it is interesting to observe the position and velocity of an object

More information

11 Vibration Analysis

11 Vibration Analysis 11 Vibration Analysis 11.1 Introduction A spring and a mass interact with one another to form a system that resonates at their characteristic natural frequency. If energy is applied to a spring mass system,

More information

2.6 The driven oscillator

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

More information

Second Order Linear Differential Equations

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

More information

Structural Dynamics, Dynamic Force and Dynamic System

Structural Dynamics, Dynamic Force and Dynamic System Structural Dynamics, Dynamic Force and Dynamic System Structural Dynamics Conventional structural analysis is based on the concept of statics, which can be derived from Newton s 1 st law of motion. This

More information

physics 111N oscillations & waves

physics 111N oscillations & waves physics 111N oscillations & waves periodic motion! often a physical system will repeat the same motion over and over! we call this periodic motion, or an oscillation the time it takes for the motion to

More information

Resonance. The purpose of this experiment is to observe and evaluate the phenomenon of resonance.

Resonance. The purpose of this experiment is to observe and evaluate the phenomenon of resonance. Resonance Objective: The purpose of this experiment is to observe and evaluate the phenomenon of resonance. Background: Resonance is the tendency of a system to oscillate with greater amplitude at some

More information

A B = AB sin(θ) = A B = AB (2) For two vectors A and B the cross product A B is a vector. The magnitude of the cross product

A B = AB sin(θ) = A B = AB (2) For two vectors A and B the cross product A B is a vector. The magnitude of the cross product 1 Dot Product and Cross Products For two vectors, the dot product is a number A B = AB cos(θ) = A B = AB (1) For two vectors A and B the cross product A B is a vector. The magnitude of the cross product

More information

Prelab Exercises: Hooke's Law and the Behavior of Springs

Prelab Exercises: Hooke's Law and the Behavior of Springs 59 Prelab Exercises: Hooke's Law and the Behavior of Springs Study the description of the experiment that follows and answer the following questions.. (3 marks) Explain why a mass suspended vertically

More information

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

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

More information

Up and Down: Damped Harmonic Motion

Up and Down: Damped Harmonic Motion Up and Down: Damped Harmonic Motion Activity 27 An object hanging from a spring can bounce up and down in a simple way. The vertical position of the object can be described mathematically in terms of a

More information

Oscillations. The Force. The Motion

Oscillations. The Force. The Motion Team Oscillations Oscillatory motion is motion that repeats itself. An object oscillates if it moves back and forth along a fixed path between two extreme positions. Oscillations are everywhere in the

More information

Response to Harmonic Excitation

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

More information

Electrical Resonance RLC circuits

Electrical Resonance RLC circuits Purpose: To investigate resonance phenomena that result from forced motion near a system's natural frequency. In this case the system will be a variety of RLC circuits. Theory: You are already familiar

More information

1.2. Mathematical Models: A Catalog of Essential Functions

1.2. Mathematical Models: A Catalog of Essential Functions 1.2. Mathematical Models: A Catalog of Essential Functions Mathematical model A mathematical model is a mathematical description (often by means of a function or an equation) of a real-world phenomenon

More information

SIMPLE HARMONIC MOTION

SIMPLE HARMONIC MOTION PERIODIC MOTION SIMPLE HARMONIC MOTION If a particle moves such that it repeats its path regularly after equal intervals of time, its motion is said to be periodic. The interval of time required to complete

More information

Advanced Loads. Patrick Cunningham. CAE Associates Inc. and ANSYS Inc. Proprietary 2013 CAE Associates Inc. and ANSYS Inc. All rights reserved.

Advanced Loads. Patrick Cunningham. CAE Associates Inc. and ANSYS Inc. Proprietary 2013 CAE Associates Inc. and ANSYS Inc. All rights reserved. Advanced Loads Webinar Patrick Cunningham January 2013 CAE Associates Inc. and ANSYS Inc. Proprietary 2013 CAE Associates Inc. and ANSYS Inc. All rights reserved. E-Learning Webinar Series This presentation

More information

Simple Harmonic Motion

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

More information

Modeling Mechanical Systems

Modeling Mechanical Systems chp3 1 Modeling Mechanical Systems Dr. Nhut Ho ME584 chp3 2 Agenda Idealized Modeling Elements Modeling Method and Examples Lagrange s Equation Case study: Feasibility Study of a Mobile Robot Design Matlab

More information

Chapter 24 Physical Pendulum

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...

More information

Practice Test SHM with Answers

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

More information

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 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

More information

Chapter 4 HW Solution

Chapter 4 HW Solution Chapter 4 HW Solution Review Questions. 1. Name the performance specification for first order systems. Time constant τ. 2. What does the performance specification for a first order system tell us? How

More information

Acceleration levels of dropped objects

Acceleration levels of dropped objects Acceleration levels of dropped objects cmyk Acceleration levels of dropped objects Introduction his paper is intended to provide an overview of drop shock testing, which is defined as the acceleration

More information

2. The graph shows how the displacement varies with time for an object undergoing simple harmonic motion.

2. The graph shows how the displacement varies with time for an object undergoing simple harmonic motion. Practice Test: 29 marks (37 minutes) Additional Problem: 31 marks (45 minutes) 1. A transverse wave travels from left to right. The diagram on the right shows how, at a particular instant of time, the

More information

Simple Harmonic Motion Experiment. 1 f

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

More information

AP1 Oscillations. 1. Which of the following statements about a spring-block oscillator in simple harmonic motion about its equilibrium point is false?

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

More information

Lab #4 examines inductors and capacitors and their influence on DC circuits.

Lab #4 examines inductors and capacitors and their influence on DC circuits. Transient DC Circuits 1 Lab #4 examines inductors and capacitors and their influence on DC circuits. As R is the symbol for a resistor, C and L are the symbols for capacitors and inductors. Capacitors

More information

Unit G484: The Newtonian World

Unit G484: The Newtonian World Define linear momentum (and appreciate the vector nature of momentum) net force on a body impulse of a force a perfectly elastic collision an inelastic collision the radian gravitational field strength

More information

Lab 5: Conservation of Energy

Lab 5: Conservation of Energy Lab 5: Conservation of Energy Equipment SWS, 1-meter stick, 2-meter stick, heavy duty bench clamp, 90-cm rod, 40-cm rod, 2 double clamps, brass spring, 100-g mass, 500-g mass with 5-cm cardboard square

More information

AP Physics C. Oscillations/SHM Review Packet

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

More information

a) Determine the equation for v(t), including numerical values for all constants.

a) Determine the equation for v(t), including numerical values for all constants. a) Determine the equation for v(t), including numerical values for all constants. 1 pt for correct trig eqn with ( ) sign 1 pt. for using eqn. to solve for omega 1 pt for correct period of 0.70 s 1 pt

More information

SIMPLE HARMONIC MOTION Ken Cheney

SIMPLE HARMONIC MOTION Ken Cheney SIMPLE HARMONIC MOTION Ken Cheney INTRODUCTION GENERAL Probably no tools that you will learn in Physics are more widely used than those that deal with simple harmonic motion. Here we will be investigating

More information

School of Biotechnology

School of Biotechnology Physics reference slides Donatello Dolce Università di Camerino a.y. 2014/2015 mail: donatello.dolce@unicam.it School of Biotechnology Program and Aim Introduction to Physics Kinematics and Dynamics; Position

More information

Acoustics Analysis of Speaker

Acoustics Analysis of Speaker Acoustics Analysis of Speaker 1 Introduction ANSYS 14.0 offers many enhancements in the area of acoustics. In this presentation, an example speaker analysis will be shown to highlight some of the acoustics

More information

Physics 53. Wave Motion 1

Physics 53. Wave Motion 1 Physics 53 Wave Motion 1 It's just a job. Grass grows, waves pound the sand, I beat people up. Muhammad Ali Overview To transport energy, momentum or angular momentum from one place to another, one can

More information

HOOKE S LAW AND SIMPLE HARMONIC MOTION

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

More information

Unit 6 Practice Test: Sound

Unit 6 Practice Test: Sound Unit 6 Practice Test: Sound Name: Multiple Guess Identify the letter of the choice that best completes the statement or answers the question. 1. A mass attached to a spring vibrates back and forth. At

More information

Frequency Response Method

Frequency Response Method Frequency Response Method Transfer function: For a stable system, the real parts of s i lie in the left half of the complex plane. The response of the system to a sinusoidal input of amplitude X, is: Where

More information

Solutions 2.4-Page 140

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

More information

ANALYTICAL METHODS FOR ENGINEERS

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

More information

A2 Physics Notes OCR Unit 4: The Newtonian World

A2 Physics Notes OCR Unit 4: The Newtonian World A2 Physics Notes OCR Unit 4: The Newtonian World Momentum: - An object s linear momentum is defined as the product of its mass and its velocity. Linear momentum is a vector quantity, measured in kgms -1

More information

A simple, low-cost, data-logging pendulum built from a computer mouse

A simple, low-cost, data-logging pendulum built from a computer mouse A simple, low-cost, data-logging pendulum built from a computer mouse Vadas Gintautas, Alfred Hübler Center for Complex Systems Research, Department of Physics, University of Illinois at Urbana-Champaign

More information