The student will experimentally determine the parameters to represent the behavior of a damped oscillatory system of one degree of freedom.

Size: px
Start display at page:

Download "The student will experimentally determine the parameters to represent the behavior of a damped oscillatory system of one degree of freedom."

Transcription

1 Practice 3 NAME STUDENT ID LAB GROUP PROFESSOR INSTRUCTOR Vibrations of systems of one degree of freedom with damping QUIZ 10% PARTICIPATION & PRESENTATION 5% INVESTIGATION 10% DESIGN PROBLEM 15% CALCULATIONS 25% RESULTS 15% DISCUSSION OF RESULTS 10% CONCLUSIONS 10% TOTAL 100% OBJECTIVE The student will experimentally determine the parameters to represent the behavior of a damped oscillatory system of one degree of freedom. FUNDAMENTALS In real life, it is possible to appreciate that free vibrations are not infinite. This phenomenon can be easily verified by observing that the free vibrations decrease their amplitude of oscillation with respect to time. The differential equation used to represent a mass-spring system such as that shown in Figure 1. F = ma mx + kx = 0 (1) (2) Fig. 1. (a) System mass-spring of one degree of freedom. (b) Free body diagram of said system. The solution of Eq. (1), as discussed in the previous practice, has the form. x(t) = C 1 e iω nt + C 2 e iω nt (3) Where C1 and C2 are constants that are defined from the initial conditions of the oscillatory system and through the use of the identity. e ±iαt = cos αt + sin αt (4) 1

2 It can be seen from Eq. (4) that the displacement is periodic (given the nature of the sine and cosine functions) in time and infinity. However, this does not agree with most of the vibration systems within our reach, which oscillate freely and stop as time goes on. This suggests then the existence of forms of energy dissipation (also known as mechanisms of damping) in the vibratory systems which produce the end of the oscillatory movements of these systems. During damping the energy of the vibratory system is dissipated as friction, heat or sound. Damping mechanisms exist in several ways, for example: Coulomb damping or dry friction. - In this case the damping force is constant. Solid damping or hysteresis. - This is caused by the internal friction of a solid when opposing to vibration. Turbulent damping. - In this case the damping force is proportional to the square of the average velocity. Damping in viscous fluid. - In this case the damping force is proportional to the speed. The most commonly used damping mechanism is viscous damping, in which the damping force is proportional to the speed of movement. These types of damping phenomena occur strictly when a laminar flow of a viscous fluid through a groove is present, for example: an impact absorber, the damping occurring around a piston in a cylinder, the damping mechanism of Automatic closing doors, automobile dampers, etc. Cushioning A basic vibratory system such as that shown in Figure 2 essentially counts with: Mass (m), which is directly related to the kinetic energy of the system. Spring (k), whose function is to store potential energy. Damper (c). - whose function is to dissipate the vibrational energy of the system. From an analysis of the free-body diagram shown in figure 2 it is possible to obtain the equation of motion. mx + cx + kx = f(t) Eq. (5) is a non-homogeneous second-order differential equation, so its solution contains two parts. The first one, which refers to the case of free vibration, is when f(t) = 0 is obtained to solve a second-order homogeneous differential equation, which corresponds physically to the case of a vibration damped system (Figure 2) Which can better represent the oscillations we traditionally appreciate in real life. (5) 2

3 Fig. 2.- (a) System mass spring damping of one degree of freedom. (b) Free body diagram of said system. The homogeneous differential equation that we must solve for this practice is given by: mx + cx + kx = 0 (6) A traditional approach to solve the differential equation (6) is to assume that the solution has the form. x = e rt (7) With, x = re rt x = r 2 e rt (8) (9) Gets Where r is a constant. When replacing Eqs. (7) and (8) in the differential equation (6) e rt (mr 2 + cr + k) = 0 (10) Eq. (9) is satisfied for all values of t when the characteristic equation is zero, that is r 2 + c m r + k m = 0 (11) Eq. (10) has the roots r 1,2 = c 2m ± ( c 2m ) 2 k m (12) In such a way that it is possible to find the following general solution for the displacement. c ( X = e 2m )t (C 1 e t ( c 2m )2 k m + C 2 e t ( c 2m )2 m) k (13) Where C 1 and C 2 are constants that depend on the initial conditions of the movement, while the terms within the radical of Eq. (12) will indicate three different damping cases with their respective characteristic curves, which will be discussed below. 3

4 OVERCURRENT CASE: ( c 2m )2 > k m In this case the radical is real and the roots r 1,2 expressed in Eq. (11) will be real and distinct. The movement of the system is dominated by damping. This means that the system will approach its equilibrium position exponentially without any oscillation occurring and will never return to its original position (from which the movement occurred). Examples of this type of damping can be observed in the mechanisms that serve for the automatic closing of doors. This motion is expressed by Eq. (12) and can be seen graphically in Figure 3 (a). CRITICALLY DAMAGED CASE: ( c 2m )2 = k m In this case the radical is equal to zero and the roots r 1,2 will be real and equal. In this case it is said that the system is critically damped. This value of the damping constant is known as the critical damping constant c cr and its value depends exclusively on k and m. c cr 2 = 4mk c cr 2 = 4mk The relation between the damping of the system and its constant damping criticism is known as damping ratio and is called by the Greek letter ζ ζ = c c cr Which is a dimensionless parameter. In a critically damped oscillation, the damped system is brought to its equilibrium position exponentially in a minimum time without oscillation occurring. In this case, the displacement will also be represented by Eq. (12), which can be written as Eq. (15) by remembering that the radical is equal to zero. (14) (15) (16) c ( X = (C 1 + C 2 )e 2m )2 (17) The movement of a critically damped oscillatory system can be seen in Figure 3 (b). SUB DAMPING CASE: ( c 2m )2 < k m In this case a harmonic oscillatory motion will be shown around an equilibrium position in which the amplitude will decrease over time in each oscillation. In this case, since the radical is negative, the roots r 1,2 are complex and conjugate having the form. r 1,2 = α ± iβ (18) 4

5 Where, α = c 2m β = k m r2 4m 2 (19) (20) And the solution in this case will have the form. X = e αt [C 1 e iβt + C 2 e iβt ] (21) Which can also be written - using Eq. (3) as c ( X = e 2m )t [A cos(ωt) + B sin(ωt)] (22) Where, ω d = β = k m c2 4m 2 = ω n 1 ζ2 (23) Torsional Pendulum If a rigid body oscillates around a specific reference axis, the resulting motion is said to be a torsional vibration. In this case, the displacement of the body is measured in terms of an angular coordinate denoted by </s>. In this type of vibration, the restorative moment can be due to the twisting of an elastic body or the imbalance of a force. Fig. 3.- (a) Torsional pendulum not damped. (b) Free body diagram of said system [1]. In figure 3 is shown a disk with mass moment of inertia J0 mounted at the end of a solid circular arrow, which is recessed at the other end. If the movement of the disk is angular and is described by the coordinate θ, from the theory of torsion of circular arrows it is possible to obtain the relation. M t = GJ bar l (24) 5

6 Where M t is a torque producing the rotation θ, G is the shear modulus, l is the length of the arrow and J bar is the polar moment of inertia of the cross section of the bar, which is given by J bar = πd4 32 (25) d is the diameter of the arrow. If the disk is displaced by an angle from its equilibrium position, the arrow has a restoring torque (resembling a spring restoring force) of magnitude M t. Consequently, the stiffness constant for a torsional spring such as Bar of figure 3 is. k t = M t θ = GJ bar = πgd4 l 32 (26) The equation of motion of a torsional pendulum is obtained through Newton's second law or through some energetic method. Considering the free-body diagram shown in Figure 3(b) for a torsional pendulum with damping you get. J 0 θ + c t θ + k t θ = 0 (27) And writing Eq. (24) as, θ + c t J 0 θ + k t J 0 θ = 0 (28) Where the moment of inertia of the disk J 0 is J 0 = ρhπd4 32 = WD2 32 (29) From Eq. (25) the natural oscillation frequency. ω n = k t J 0 (30) And finally, ζ = c t = c t = c t c ct 2J 0 ω n 2 k t J 0 (31) 6

7 MATERIAL AND EQUIPMENT TO BE USED Mass-Spring system Torsional pendulum Vibration analyzer Accelerometer Vernier caliper Oil REPORT Mass-Spring system Determine the natural frequency using the cycles method, then determine the value of the constant spring stiffness using the two methods seen in Practice 1. Using the vibration analyzer, determine the value of the natural oscillation frequency (ω n ). Using the vibration analyzer, determine the value of the damped oscillation frequency (ω d ). For this use an oil vessel in which the fins of the mass are fully in contact with the oil. Determine the value of the damping ratio (ξ). Determine the value of the critical damping constant (c cr ). Determine the value of the damping constant provided by the oil (c). Discuss the obtain results, under which conditions the percentage of error decrease and which conditions affects the experimental analysis. Conclude according to the objectives of the practice and add a personal conclusion of why is important what we learn? Torsional Pendulum Determine the value of the torsional stiffness constant (k t ) using the equations (22) and (23). Using the vibration analyzer, determine the value of the natural oscillation frequency (ω n ). Using the vibration analyzer, determine the value of the damped oscillation frequency (ω d ). To do this use a container with oil in which the disc is immersed in oil. Determine the value of the damping ratio (ξ). Determine the value of the critical torsional damping constant (c ct ). Determine the value of the torsional damping constant provided by the oil (c t ). Discuss the obtain results, under which conditions the percentage of error decrease and which conditions affects the experimental analysis. Conclude according to the objectives of the practice and add a personal conclusion of why is important what we learn? Note: Extension of the discussion of results and conclusions of the two systems should be 1 page. 7

8 RESULTS Complete the results table below and evaluate the reliability of the techniques applied in the spaces designated for this propose. Table 1. Cycles Method Mass-Spring Time (s) Cycles Average Mass (kg) Period (s) Natural Frequency (rad/s) Spring Constant (N/m) Table 2. Deformation Method Mass-Spring m 1 (kg) δ 1 (m) m 2 (kg) δ 2 (m) m δ Spring constant (N/m) Table 3. Vibration Analyzer Mass-Spring Frequency Amplitude (Hz) (G-S) No damped Damped ω n (rad/s) ω d (rad/s) Table 4. Vibration Analyzer Torsional Pendulum Frequency Amplitude (Hz) (G-S) No damped Damped ω n (rad/s) ω d (rad/s) INVESTIGATION Explain how accelerometers work (Extension 1 page) Search for types of accelerometers and their definition (Extension Half page) Note: All should be referenced following the IEEE format. 8

9 DESIGN PROBLEM Although we find a large number of examples of damping daily, one of the most illustrative is the automobile. In our case, the car we are going to study has a mass of 1520 kg and is supported by its four springs and four shock absorbers. If the static elongation of the springs due to the car's own weight is 0.20 m, how would you determine the damping constant required in each damper to obtain the critical damping case? Assume the car has only one degree of freedom. Before determining the critical damping constant (c cr ), mention what the assumptions you took into account to arrive at your result. REFERENCES [1] Rao, Singiresu S. Mechanical Vibrations, Fourth Edition, Pearson. USA (2003). [2] Steidel, Robert F. An introduction to mechanical vibrations. Third Edition, John Wiley, USA (1989). [3] Thomson, William T. Theory of vibrations: applications. Second Edition, Prentice Hall, USA (1982). [4] Kelly, Graham S. Fundamentals of mechanical vibrations. Second Edition. McGraw Hill. USA (2000). [5] Stile, Hidgon. Ingeniería Mecánica, tomo II: Dinámica Vectorial. Prentice Hall, (1982). 9

WEEKS 8-9 Dynamics of Machinery

WEEKS 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 information

Step 1: Mathematical Modeling

Step 1: Mathematical Modeling 083 Mechanical Vibrations Lesson Vibration Analysis Procedure The analysis of a vibrating system usually involves four steps: mathematical modeling derivation of the governing uations solution of the uations

More information

LANMARK UNIVERSITY OMU-ARAN, KWARA STATE DEPARTMENT OF MECHANICAL ENGINEERING COURSE: MECHANICS OF MACHINE (MCE 322). LECTURER: ENGR.

LANMARK UNIVERSITY OMU-ARAN, KWARA STATE DEPARTMENT OF MECHANICAL ENGINEERING COURSE: MECHANICS OF MACHINE (MCE 322). LECTURER: ENGR. LANMARK UNIVERSITY OMU-ARAN, KWARA STATE DEPARTMENT OF MECHANICAL ENGINEERING COURSE: MECHANICS OF MACHINE (MCE 322). LECTURER: ENGR. IBIKUNLE ROTIMI ADEDAYO SIMPLE HARMONIC MOTION. Introduction Consider

More information

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

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 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 information

Chapter 14 Oscillations. Copyright 2009 Pearson Education, Inc.

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

More information

Engineering Science OUTCOME 2 - TUTORIAL 3 FREE VIBRATIONS

Engineering 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 information

Dynamic Modelling of Mechanical Systems

Dynamic Modelling of Mechanical Systems Dynamic Modelling of Mechanical Systems Dr. Bishakh Bhattacharya Professor, Department of Mechanical Engineering g IIT Kanpur Joint Initiative of IITs and IISc - Funded by MHRD Hints of the Last Assignment

More information

TOPIC E: OSCILLATIONS SPRING 2019

TOPIC 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 information

Periodic Motion. Periodic motion is motion of an object that. regularly repeats

Periodic 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 information

EQUIVALENT SINGLE-DEGREE-OF-FREEDOM SYSTEM AND FREE VIBRATION

EQUIVALENT SINGLE-DEGREE-OF-FREEDOM SYSTEM AND FREE VIBRATION 1 EQUIVALENT SINGLE-DEGREE-OF-FREEDOM SYSTEM AND FREE VIBRATION The course on Mechanical Vibration is an important part of the Mechanical Engineering undergraduate curriculum. It is necessary for the development

More information

Mathematical Modeling and response analysis of mechanical systems are the subjects of this chapter.

Mathematical 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 information

Introduction to Vibration. Professor Mike Brennan

Introduction 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

Vibrations Qualifying Exam Study Material

Vibrations Qualifying Exam Study Material Vibrations Qualifying Exam Study Material The candidate is expected to have a thorough understanding of engineering vibrations topics. These topics are listed below for clarification. Not all instructors

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

CE 6701 Structural Dynamics and Earthquake Engineering Dr. P. Venkateswara Rao

CE 6701 Structural Dynamics and Earthquake Engineering Dr. P. Venkateswara Rao CE 6701 Structural Dynamics and Earthquake Engineering Dr. P. Venkateswara Rao Associate Professor Dept. of Civil Engineering SVCE, Sriperumbudur Difference between static loading and dynamic loading Degree

More information

Chapter 15. Oscillatory Motion

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

More information

Lectures Chapter 10 (Cutnell & Johnson, Physics 7 th edition)

Lectures Chapter 10 (Cutnell & Johnson, Physics 7 th edition) PH 201-4A spring 2007 Simple Harmonic Motion Lectures 24-25 Chapter 10 (Cutnell & Johnson, Physics 7 th edition) 1 The Ideal Spring Springs are objects that exhibit elastic behavior. It will return back

More information

VTU-NPTEL-NMEICT Project

VTU-NPTEL-NMEICT Project MODULE-II --- SINGLE DOF FREE S VTU-NPTEL-NMEICT Project Progress Report The Project on Development of Remaining Three Quadrants to NPTEL Phase-I under grant in aid NMEICT, MHRD, New Delhi SME Name : Course

More information

Vibrations and Waves MP205, Assignment 4 Solutions

Vibrations 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 information

Dynamics of structures

Dynamics of structures Dynamics of structures 2.Vibrations: single degree of freedom system Arnaud Deraemaeker (aderaema@ulb.ac.be) 1 Outline of the chapter *One degree of freedom systems in real life Hypothesis Examples *Response

More information

Chapter 14 Oscillations. Copyright 2009 Pearson Education, Inc.

Chapter 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 information

Structural Dynamics. Spring mass system. The spring force is given by and F(t) is the driving force. Start by applying Newton s second law (F=ma).

Structural Dynamics. Spring mass system. The spring force is given by and F(t) is the driving force. Start by applying Newton s second law (F=ma). Structural Dynamics Spring mass system. The spring force is given by and F(t) is the driving force. Start by applying Newton s second law (F=ma). We will now look at free vibrations. Considering the free

More information

In the presence of viscous damping, a more generalized form of the Lagrange s equation of motion can be written as

In the presence of viscous damping, a more generalized form of the Lagrange s equation of motion can be written as 2 MODELING Once the control target is identified, which includes the state variable to be controlled (ex. speed, position, temperature, flow rate, etc), and once the system drives are identified (ex. force,

More information

2007 Problem Topic Comment 1 Kinematics Position-time equation Kinematics 7 2 Kinematics Velocity-time graph Dynamics 6 3 Kinematics Average velocity

2007 Problem Topic Comment 1 Kinematics Position-time equation Kinematics 7 2 Kinematics Velocity-time graph Dynamics 6 3 Kinematics Average velocity 2007 Problem Topic Comment 1 Kinematics Position-time equation Kinematics 7 2 Kinematics Velocity-time graph Dynamics 6 3 Kinematics Average velocity Energy 7 4 Kinematics Free fall Collisions 3 5 Dynamics

More information

Fundamentals Physics. Chapter 15 Oscillations

Fundamentals 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 information

Oscillatory Motion SHM

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

More information

F = ma, F R + F S = mx.

F = ma, F R + F S = mx. Mechanical Vibrations As we mentioned in Section 3.1, linear equations with constant coefficients come up in many applications; in this section, we will specifically study spring and shock absorber systems

More information

Dynamics of Ocean Structures Prof. Dr. Srinivasan Chandrasekaran Department of Ocean Engineering Indian Institute of Technology, Madras

Dynamics of Ocean Structures Prof. Dr. Srinivasan Chandrasekaran Department of Ocean Engineering Indian Institute of Technology, Madras Dynamics of Ocean Structures Prof. Dr. Srinivasan Chandrasekaran Department of Ocean Engineering Indian Institute of Technology, Madras Module - 01 Lecture - 09 Characteristics of Single Degree - of -

More information

Chapter 14 Oscillations

Chapter 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 information

The Torsion Pendulum (One or two weights)

The Torsion Pendulum (One or two weights) The Torsion Pendulum (One or two weights) Exercises I through V form the one-weight experiment. Exercises VI and VII, completed after Exercises I -V, add one weight more. Preparatory Questions: 1. The

More information

Oscillations. Oscillations and Simple Harmonic Motion

Oscillations. 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 information

Structural Dynamics Lecture 4. Outline of Lecture 4. Multi-Degree-of-Freedom Systems. Formulation of Equations of Motions. Undamped Eigenvibrations.

Structural Dynamics Lecture 4. Outline of Lecture 4. Multi-Degree-of-Freedom Systems. Formulation of Equations of Motions. Undamped Eigenvibrations. Outline of Multi-Degree-of-Freedom Systems Formulation of Equations of Motions. Newton s 2 nd Law Applied to Free Masses. D Alembert s Principle. Basic Equations of Motion for Forced Vibrations of Linear

More information

PREMED COURSE, 14/08/2015 OSCILLATIONS

PREMED 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 information

Chapter 15 Periodic Motion

Chapter 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 information

Chapter 5 Design. D. J. Inman 1/51 Mechanical Engineering at Virginia Tech

Chapter 5 Design. D. J. Inman 1/51 Mechanical Engineering at Virginia Tech Chapter 5 Design Acceptable vibration levels (ISO) Vibration isolation Vibration absorbers Effects of damping in absorbers Optimization Viscoelastic damping treatments Critical Speeds Design for vibration

More information

Chapter 14 Periodic Motion

Chapter 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 information

IJSRD - International Journal for Scientific Research & Development Vol. 4, Issue 07, 2016 ISSN (online):

IJSRD - International Journal for Scientific Research & Development Vol. 4, Issue 07, 2016 ISSN (online): IJSRD - International Journal for Scientific Research & Development Vol. 4, Issue 07, 2016 ISSN (online): 2321-0613 Analysis of Vibration Transmissibility on a System using Finite Element Analysis Rajesh

More information

Experiment 12 Damped Harmonic Motion

Experiment 12 Damped Harmonic Motion Physics Department LAB A - 120 Experiment 12 Damped Harmonic Motion References: Daniel Kleppner and Robert Kolenkow, An Introduction to Mechanics, McGraw -Hill 1973 pp. 414-418. Equipment: Air track, glider,

More information

Chapter 12. Recall that when a spring is stretched a distance x, it will pull back with a force given by: F = -kx

Chapter 12. Recall that when a spring is stretched a distance x, it will pull back with a force given by: F = -kx Chapter 1 Lecture Notes Chapter 1 Oscillatory Motion Recall that when a spring is stretched a distance x, it will pull back with a force given by: F = -kx When the mass is released, the spring will pull

More information

Forced Damped Vibrations

Forced Damped Vibrations Forced Damped Vibrations Forced Damped Motion Definitions Visualization Cafe door Pet door Damped Free Oscillation Model Tuning a Dampener Bicycle trailer Forced Damped Motion Real systems do not exhibit

More information

Introduction to Mechanical Vibration

Introduction to Mechanical Vibration 2103433 Introduction to Mechanical Vibration Nopdanai Ajavakom (NAV) 1 Course Topics Introduction to Vibration What is vibration? Basic concepts of vibration Modeling Linearization Single-Degree-of-Freedom

More information

Unforced Oscillations

Unforced Oscillations Unforced Oscillations Simple Harmonic Motion Hooke s Law Newton s Second Law Method of Force Competition Visualization of Harmonic Motion Phase-Amplitude Conversion The Simple Pendulum and The Linearized

More information

Section 3.7: Mechanical and Electrical Vibrations

Section 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 information

OSCILLATIONS ABOUT EQUILIBRIUM

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

More information

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

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

More information

Energy Considerations

Energy Considerations Physics 42200 Waves & Oscillations Lecture 4 French, Chapter 3 Spring 2016 Semester Matthew Jones Energy Considerations The force in Hooke s law is = Potential energy can be used to describe conservative

More information

7. Vibrations DE2-EA 2.1: M4DE. Dr Connor Myant

7. Vibrations DE2-EA 2.1: M4DE. Dr Connor Myant DE2-EA 2.1: M4DE Dr Connor Myant 7. Vibrations Comments and corrections to connor.myant@imperial.ac.uk Lecture resources may be found on Blackboard and at http://connormyant.com Contents Introduction...

More information

Oscillations. PHYS 101 Previous Exam Problems CHAPTER. Simple harmonic motion Mass-spring system Energy in SHM Pendulums

Oscillations. PHYS 101 Previous Exam Problems CHAPTER. Simple harmonic motion Mass-spring system Energy in SHM Pendulums PHYS 101 Previous Exam Problems CHAPTER 15 Oscillations Simple harmonic motion Mass-spring system Energy in SHM Pendulums 1. The displacement of a particle oscillating along the x axis is given as a function

More information

Chap. 15: Simple Harmonic Motion

Chap. 15: Simple Harmonic Motion Chap. 15: Simple Harmonic Motion Announcements: CAPA is due next Tuesday and next Friday. Web page: http://www.colorado.edu/physics/phys1110/phys1110_sp12/ Examples of periodic motion vibrating guitar

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 information

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

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

More information

OSCILLATIONS OF A SPRING-MASS SYSTEM AND A TORSIONAL PENDULUM

OSCILLATIONS OF A SPRING-MASS SYSTEM AND A TORSIONAL PENDULUM EXPERIMENT Spring-Mass System and a Torsional Pendulum OSCILLATIONS OF A SPRING-MASS SYSTEM AND A TORSIONAL PENDULUM Structure.1 Introduction Objectives. Determination of Spring Constant Static Method

More information

Physics Mechanics. Lecture 32 Oscillations II

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

More information

SHAKING TABLE DEMONSTRATION OF DYNAMIC RESPONSE OF BASE-ISOLATED BUILDINGS ***** Instructor Manual *****

SHAKING TABLE DEMONSTRATION OF DYNAMIC RESPONSE OF BASE-ISOLATED BUILDINGS ***** Instructor Manual ***** SHAKING TABLE DEMONSTRATION OF DYNAMIC RESPONSE OF BASE-ISOLATED BUILDINGS ***** Instructor Manual ***** A PROJECT DEVELOPED FOR THE UNIVERSITY CONSORTIUM ON INSTRUCTIONAL SHAKE TABLES http://wusceel.cive.wustl.edu/ucist/

More information

LAB 11: FREE, DAMPED, AND FORCED OSCILLATIONS

LAB 11: FREE, DAMPED, AND FORCED OSCILLATIONS Lab 11 ree, amped, and orced Oscillations 135 Name ate Partners OBJECTIVES LAB 11: REE, AMPE, AN ORCE OSCILLATIONS To understand the free oscillations of a mass and spring. To understand how energy is

More information

Sound Propagation through Media. Nachiketa Tiwari Indian Institute of Technology Kanpur

Sound Propagation through Media. Nachiketa Tiwari Indian Institute of Technology Kanpur Sound Propagation through Media Nachiketa Tiwari Indian Institute of Technology Kanpur LECTURE-13 WAVE PROPAGATION IN SOLIDS Longitudinal Vibrations In Thin Plates Unlike 3-D solids, thin plates have surfaces

More information

1792. On the conditions for synchronous harmonic free vibrations

1792. On the conditions for synchronous harmonic free vibrations 1792. On the conditions for synchronous harmonic free vibrations César A. Morales Departamento de Mecánica, Universidad Simón Bolívar, Apdo. 89000, Caracas 1080A, Venezuela E-mail: cmorales@usb.ve (Received

More information

Lab 11. Spring-Mass Oscillations

Lab 11. Spring-Mass Oscillations Lab 11. Spring-Mass Oscillations Goals To determine experimentally whether the supplied spring obeys Hooke s law, and if so, to calculate its spring constant. To find a solution to the differential equation

More information

Chapter 7 Hooke s Force law and Simple Harmonic Oscillations

Chapter 7 Hooke s Force law and Simple Harmonic Oscillations Chapter 7 Hooke s Force law and Simple Harmonic Oscillations Hooke s Law An empirically derived relationship that approximately works for many materials over a limited range. Exactly true for a massless,

More information

T1 T e c h n i c a l S e c t i o n

T1 T e c h n i c a l S e c t i o n 1.5 Principles of Noise Reduction A good vibration isolation system is reducing vibration transmission through structures and thus, radiation of these vibration into air, thereby reducing noise. There

More information

Dynamic Analysis on Vibration Isolation of Hypersonic Vehicle Internal Systems

Dynamic Analysis on Vibration Isolation of Hypersonic Vehicle Internal Systems International Journal of Engineering Research and Technology. ISSN 0974-3154 Volume 6, Number 1 (2013), pp. 55-60 International Research Publication House http://www.irphouse.com Dynamic Analysis on Vibration

More information

Thursday, August 4, 2011

Thursday, August 4, 2011 Chapter 16 Thursday, August 4, 2011 16.1 Springs in Motion: Hooke s Law and the Second-Order ODE We have seen alrealdy that differential equations are powerful tools for understanding mechanics and electro-magnetism.

More information

Exponential decay. The deviations in amplitude over 30 periods rise to more than ±20%. Fig 1 a rod and ball pendulum

Exponential decay. The deviations in amplitude over 30 periods rise to more than ±20%. Fig 1 a rod and ball pendulum Exponential decay A counter example There is a common belief that the damping of the motion of a pendulum in air is exponential, or nearly so, in all situations. To explore the limits of that approximation

More information

Forced Oscillation and Resonance

Forced Oscillation and Resonance Chapter Forced Oscillation and Resonance The forced oscillation problem will be crucial to our understanding of wave phenomena Complex exponentials are even more useful for the discussion of damping and

More information

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

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

More information

Undetermined Coefficents, Resonance, Applications

Undetermined Coefficents, Resonance, Applications Undetermined Coefficents, Resonance, Applications An Undetermined Coefficients Illustration Phase-amplitude conversion I Phase-amplitude conversion II Cafe door Pet door Cafe Door Model Pet Door Model

More information

Introduction to structural dynamics

Introduction to structural dynamics Introduction to structural dynamics p n m n u n p n-1 p 3... m n-1 m 3... u n-1 u 3 k 1 c 1 u 1 u 2 k 2 m p 1 1 c 2 m2 p 2 k n c n m n u n p n m 2 p 2 u 2 m 1 p 1 u 1 Static vs dynamic analysis Static

More information

Contents. Dynamics and control of mechanical systems. Focus on

Contents. Dynamics and control of mechanical systems. Focus on Dynamics and control of mechanical systems Date Day 1 (01/08) Day 2 (03/08) Day 3 (05/08) Day 4 (07/08) Day 5 (09/08) Day 6 (11/08) Content Review of the basics of mechanics. Kinematics of rigid bodies

More information

4.9 Free Mechanical Vibrations

4.9 Free Mechanical Vibrations 4.9 Free Mechanical Vibrations Spring-Mass Oscillator When the spring is not stretched and the mass m is at rest, the system is at equilibrium. Forces Acting in the System When the mass m is displaced

More information

Dynamics of structures

Dynamics of structures Dynamics of structures 1.2 Viscous damping Luc St-Pierre October 30, 2017 1 / 22 Summary so far We analysed the spring-mass system and found that its motion is governed by: mẍ(t) + kx(t) = 0 k y m x x

More information

Chapter 14 Oscillations

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

More information

Dynamics of Structures: Theory and Analysis

Dynamics of Structures: Theory and Analysis 1. Free vibrations 2. Forced vibrations 3. Transient response 4. Damping mechanisms Dynamics of Structures: Theory and Analysis Steen Krenk Technical University of Denmark 5. Modal analysis I: Basic idea

More information

Simple Harmonic Motion

Simple 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 information

Aircraft Dynamics First order and Second order system

Aircraft Dynamics First order and Second order system Aircraft Dynamics First order and Second order system Prepared by A.Kaviyarasu Assistant Professor Department of Aerospace Engineering Madras Institute Of Technology Chromepet, Chennai Aircraft dynamic

More information

PLANAR KINETICS OF A RIGID BODY: WORK AND ENERGY Today s Objectives: Students will be able to: 1. Define the various ways a force and couple do work.

PLANAR KINETICS OF A RIGID BODY: WORK AND ENERGY Today s Objectives: Students will be able to: 1. Define the various ways a force and couple do work. PLANAR KINETICS OF A RIGID BODY: WORK AND ENERGY Today s Objectives: Students will be able to: 1. Define the various ways a force and couple do work. In-Class Activities: 2. Apply the principle of work

More information

TOPIC E: OSCILLATIONS EXAMPLES SPRING Q1. Find general solutions for the following differential equations:

TOPIC E: OSCILLATIONS EXAMPLES SPRING Q1. Find general solutions for the following differential equations: TOPIC E: OSCILLATIONS EXAMPLES SPRING 2019 Mathematics of Oscillating Systems Q1. Find general solutions for the following differential equations: Undamped Free Vibration Q2. A 4 g mass is suspended by

More information

STRUCTURAL DYNAMICS BASICS:

STRUCTURAL DYNAMICS BASICS: BASICS: STRUCTURAL DYNAMICS Real-life structures are subjected to loads which vary with time Except self weight of the structure, all other loads vary with time In many cases, this variation of the load

More information

Physical and Biological Properties of Agricultural Products Acoustic, Electrical and Optical Properties and Biochemical Property

Physical and Biological Properties of Agricultural Products Acoustic, Electrical and Optical Properties and Biochemical Property Physical and Biological Properties of Agricultural Products Acoustic, Electrical and Optical Properties and Biochemical Property 1. Acoustic and Vibrational Properties 1.1 Acoustics and Vibration Engineering

More information

A SHORT INTRODUCTION TO ADAMS

A SHORT INTRODUCTION TO ADAMS A. AHADI, P. LIDSTRÖM, K. NILSSON A SHORT INTRODUCTION TO ADAMS FOR MECHANICAL ENGINEERS DIVISION OF MECHANICS DEPARTMENT OF MECHANICAL ENGINEERING LUND INSTITUTE OF TECHNOLOGY 2017 1 FOREWORD THESE EXERCISES

More information

3.1 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 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 information

Mechatronics. MANE 4490 Fall 2002 Assignment # 1

Mechatronics. MANE 4490 Fall 2002 Assignment # 1 Mechatronics MANE 4490 Fall 2002 Assignment # 1 1. For each of the physical models shown in Figure 1, derive the mathematical model (equation of motion). All displacements are measured from the static

More information

18.12 FORCED-DAMPED VIBRATIONS

18.12 FORCED-DAMPED VIBRATIONS 8. ORCED-DAMPED VIBRATIONS Vibrations A mass m is attached to a helical spring and is suspended from a fixed support as before. Damping is also provided in the system ith a dashpot (ig. 8.). Before the

More information

Chapter 5 Oscillatory Motion

Chapter 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 information

You may use your books and notes. Moreover, you are encouraged to freely discuss the questions..which doesn't mean copying answers.

You may use your books and notes. Moreover, you are encouraged to freely discuss the questions..which doesn't mean copying answers. Section: Oscillations Take-Home Test You may use your books and notes. Moreover, you are encouraged to freely discuss the questions..which doesn't mean copying answers. 1. In simple harmonic motion, the

More information

CEE 271: Applied Mechanics II, Dynamics Lecture 27: Ch.18, Sec.1 5

CEE 271: Applied Mechanics II, Dynamics Lecture 27: Ch.18, Sec.1 5 1 / 42 CEE 271: Applied Mechanics II, Dynamics Lecture 27: Ch.18, Sec.1 5 Prof. Albert S. Kim Civil and Environmental Engineering, University of Hawaii at Manoa Tuesday, November 27, 2012 2 / 42 KINETIC

More information

CHAPTER 12 OSCILLATORY MOTION

CHAPTER 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 information

Chapter 10: Simple Harmonic Motion

Chapter 10: Simple Harmonic Motion Chapter 10: Simple Harmonic Motion Oscillations are back-and forth motions. Sometimes the word vibration is used in place of oscillation; for our purposes, we can consider the two words to represent the

More information

Vibration Control Prof. Dr. S. P. Harsha Department of Mechanical & Industrial Engineering Indian Institute of Technology, Roorkee

Vibration Control Prof. Dr. S. P. Harsha Department of Mechanical & Industrial Engineering Indian Institute of Technology, Roorkee Vibration Control Prof. Dr. S. P. Harsha Department of Mechanical & Industrial Engineering Indian Institute of Technology, Roorkee Module - 1 Review of Basics of Mechanical Vibrations Lecture - 2 Introduction

More information

Phase-Amplitude Conversion, Cafe Door, Pet Door, Damping Classifications

Phase-Amplitude Conversion, Cafe Door, Pet Door, Damping Classifications Phase-Amplitude Conversion, Cafe Door, Pet Door, Damping Classifications Phase-amplitude conversion Cafe door Pet door Cafe Door Model Pet Door Model Classifying Damped Models Phase-amplitude conversion

More information

Chapter 15. Oscillations

Chapter 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 information

Lab 11 - Free, Damped, and Forced Oscillations

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

More information

Wilberforce Pendulum (One or two weights)

Wilberforce Pendulum (One or two weights) Wilberforce Pendulum (One or two weights) For a 1 weight experiment do Part 1 (a) and (b). For a weight experiment do Part1 and Part Recommended readings: 1. PHY15 University of Toronto. Selected Material

More information

Engineering Mechanics Prof. U. S. Dixit Department of Mechanical Engineering Indian Institute of Technology, Guwahati Introduction to vibration

Engineering Mechanics Prof. U. S. Dixit Department of Mechanical Engineering Indian Institute of Technology, Guwahati Introduction to vibration Engineering Mechanics Prof. U. S. Dixit Department of Mechanical Engineering Indian Institute of Technology, Guwahati Introduction to vibration Module 15 Lecture 38 Vibration of Rigid Bodies Part-1 Today,

More information

2.4 Models of Oscillation

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

More information

Simple Harmonic Motion

Simple Harmonic Motion Simple Harmonic Motion (FIZ 101E - Summer 2018) July 29, 2018 Contents 1 Introduction 2 2 The Spring-Mass System 2 3 The Energy in SHM 5 4 The Simple Pendulum 6 5 The Physical Pendulum 8 6 The Damped Oscillations

More information

Mechanical Oscillations

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

More information

LECTURE 19: Simple harmonic oscillators

LECTURE 19: Simple harmonic oscillators Lectures Page 1 Select LEARNING OBJECTIVES: LECTURE 19: Simple harmonic oscillators Be able to identify the features of a system that oscillates - i.e. systems with a restoring force and a potential energy

More information

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

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

More information

5.6 Unforced Mechanical Vibrations

5.6 Unforced Mechanical Vibrations 5.6 Unforced Mechanical Vibrations 215 5.6 Unforced Mechanical Vibrations The study of vibrating mechanical systems begins here with examples for unforced systems with one degree of freedom. The main example

More information

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Exam Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) A 4.8-kg block attached to a spring executes simple harmonic motion on a frictionless

More information