# Project: Vibration Damping

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Mekanik GK för I (FMEA10) 2015 Project: Vibration Damping Project team: Name: Personal id-number: Mekanik 1

2 Project: Vibration damping Project Specification 1. Introduction In this project we will study the vibrations of a simple mechanical system by using our knowledge in mechanics and the Multi Body Simulation package ADAMS. We will investigate the motion of a machine-housing due to an internal rotating mass unbalance. The machine housing is mounted on a foundation via an elastic spring system. The motion of the housing creates a force on the foundation. This force is unwanted and we will try to reduce the motion by adding a specific mass-spring-damper system to the machine housing. The purpose of this project will be to investigate some of the properties of this vibration damping concept. 2. Problem formulation A machine-housing contains a rotor with a mass unbalance according to Figure 1 below. The unbalance is given by the offset of the centre of mass for the rotor relative to the rotor axle. The rotor is spinning with a constant angular velocity and this produces an acceleration on the housing which is transmitted to the foundation via the supporting spring system. By adding a specific mass-spring-damper system on top of the machine housing the motion of the housing and the forces exerted on the foundation may be reduced. We want to find out how this damping mechanism works and how it could be designed. See Figure Theoretical model In our theoretical model we assume that the machine housing is a rigid body with mass m h and with the possibility of a translational motion in the vertical direction x= xt () (1) where x is a coordinate measuring the vertical position of the housing relative to the foundation (see Figure 1). The rotor unbalance is modelled as a particle with mass m r performing a circular motion with radius e and constant angular velocity ω relative to the housing. The total mass of the machine is denoted by m and m= m + m (2) h r 2

3 The housing is supported by a spring-damper system connected to the foundation (ground). This system is modelled by using a linear spring and linear viscous damper. The force on the housing from the spring-damper system may then be written F = k( x x ) cx (3) 0 where k is the spring constant, x 0 is the position of the housing when the spring is unloaded. and c is the viscous damping coefficient. The machine model is visualized in Figure 1 and will be referred to as the un-damped machine. m r ϕ Motion: ϕ = ωt+ϕ 0 m h e x k c Figure 1: Un-damped machine The damping device ( vibration absorber ) mounted on top of the housing is modelled as a particle with mass m a connected to the housing with a spring-damper system according to Figure 2. The elastic spring constant is denoted κ and the viscous damping coefficient η. If the position of the damping mass relative to the machine housing is given by y= yt () (4) the damping force D, acting on the housing from the damping device, may be written D= κ( y y ) ηy (5) 0 where y 0 is the position of the absorber mass (relative to the housing) when the spring is unloaded. The machine model is visualized in Figure 2 and will be referred to as the damped machine. 3

4 m a κ η y Figure 2: The damped machine. 4. Basic data The following basic data should be used: Table 1: Basic data Parameter Symbol Unit Numerical value Machine mass m = m h +m r kg 100 Foundation spring constant k N/m -- (Team specific)* Foundation damping constant c Ns/m 3000 Rotor mass m r kg 10 Rotor mass eccentricity e m 0.2 Rotor velocity N Rpm** -- (Team specific)* Absorber mass m a kg - Absorber spring constant κ N/m - Absorber damping constant η Ns/m - *See Appendix 2 at the back of this specification. ** Revolutions per minute. 5. Theoretical investigations and damper design We want to examine the motion of the un-damped machine and then design a vibration absorber which reduces the motion of the machine housing as much as possible. The design problem may be formulated as follows: Choose the mass m a, the elastic stiffness κ and the viscous damping η of the vibration absorber so that the amplitude of the housing motion becomes as small as possible. We call this the optimal design. 4

5 5.1 The motion of the un-damped machine Tasks and questions: 1. Calculate analytically the amplitude of the steady state motion of the un-damped machine housing. Draw a free-body diagram including external forces. Formulate the force equation and solve the obtained differential equation. Use data in table 1. Consult chapter 8 in DYNAMICS and the Lecture Notes for the theory. 2. Construct a model of the problem using the ADAMS software. Simulate the motion of the machine. Plot a diagram of the vertical position of the machine housing as a function of time. Compare the ADAMS results of the machine housing motion with the analytical solution. 3. Reduce the viscous damping c (c 0) of the foundation spring-damper system and observe the motion of the housing. Try to explain what is happening! What is this phenomenon usually called. (Hint: In the simulation choose End time 2, Steps 1000.) 5.2 The motion of the damped machine The damping mechanism cannot be allowed to be too heavy. We will design a damper with a mass of maximum 10% of the total machine mass, i.e. we have the design restriction: ma 0.10m (6) where m is the total machine mass, according to (2). Tasks and questions: 4. Restore the damping coefficient c to the default value according to Table 1! 5. Construct a model of the damped machine using the ADAMS software. 6. Simulate the motion of the damped machine for different values of mass, stiffness and damping coefficients (m a, κ and η) of the vibration absorber. Do you find any damping of the housing motion? If not change the values and try again! Compare the amplitudes of the damped machine for different values of absorber mass m a, absorber spring constant κ and absorber damping constant η. Include plots of these simulations in the report. 7. What mechanical effect may be causing the damping of the housing motion? Try to formulate, in words, a simple explanation (a theory )! 5.3 The optimal design Tasks and questions: 8. Formulate a strategy for the selection of mass and stiffness coefficients (m a, κ) for the vibration absorber in order to obtain an optimal design. 5

6 9. Try to find mass, stiffness and damping coefficients (m a, κ and η) for an optimal design. Use the strategy and your simulation model! 10. Compare the motions of the optimally damped and the un-damped housings by plotting a diagram of the vertical position of the machine housing as a function of time. What is the amplitude reduction ratio R, defined by: amplitude of damped machine housing R = amplitude of undamped machine housing 11. Can you think of any practical problems with the design of a vibration absorber of this type? 12. Produce a written report on your findings! See Appendix 3! 6

7 Appendix 1 Short ADAMS manual for Project: Vibration Damping 5.1 The motion of the un-damped machine See Computer Exercise 2 ( Datorlabben ). If we choose to build a machine housing as a cube with side 100cm then choose the following Working Grid X Y Size (2m) (2m) Spacing (10cm) (10cm) Zoom in the entire grid of the working area. We will not need the gravity in this problem so: Select No Gavity. As the machine housing choose for instance Rigid body: Box. Select Length = Height = Depth = 100cm! Place the box on the Working Space. Change the mass properties of the box! Connect the Box to Ground with a Translational Spring-damper. Make the Spring-Damper as long as possible! Use a Rigid body: Link (Length = 20cm) to create the eccentricity. Connect the Link to the Box. Use a Joint: Revolute. Change the mass properties of the Link. Set mass = 0. Use a Rigid body: Sphere to create the rotor mass. Connect the Sphere to the Link Change the mass properties of the Sphere. Create a Rotational Joint Motion for the Link+Sphere. Create a Measure of the y-displacement of the centre of mass of the Box. Test the model by running a simulation. Select an appropriate End Time so that steady state of the housing motion is obtained. Calculate (by hand) an estimate of the amplitude of the machine housing. Did the model work as expected? 5.2 The motion of the damped machine Create the vibration absorber, in the same way as you built the machine housing, and place it on top of the machine housing. Assign specific values to the Design Variables m a, κ and η. Test the model by running a simulation. 5.3 The optimal design How should the mass of the vibration absorber be selected? How should the natural frequency of the vibration absorber be designed? 7

8 Appendix 2 Team specific parameters Set up Spring constant, k (N/m) Rotor velocity, N (rpm) S S S S S S S S Project Team Set up Project Team A1 S1 B1 A2 S2 B2 A3 S3 B3 A4 S4 B4 A5 S5 B5 A6 S6 B6 A7 S7 B7 A8 S8 B8 A9 S1 B9 A10 S2 B10 A11 S3 B11 A12 S4 B12 A13 S5 B13 A14 S6 B14 A15 S7 B15 A16 S8 B16 A17 S1 B17 A18 S2 B18 A19 S3 B19 A20 S4 B20 A21 S5 B21 A22 S6 B22 A23 S7 B23 A24 S8 B24 A25 S1 B25 A26 S2 B26 A27 S3 B27 8

9 Appendix 3: Anvisningar för skriftlig rapport Rapport ska skrivas på ett sådant sätt att en kurskamrat som inte har utfört samma projektuppgift har möjlighet att sätta sig in i frågeställningen, utförandet och resultatet för att därefter själv kunna bilda sig en uppfattning om projektuppgiftens utfall. Rapporten kan skrivas på svenska eller engelska. Den skriftliga rapporten ska granskas av en lärare vid avdelningen för Mekanik. För att underlätta diskussion av rapporten ska både ekvationerna och sidorna numreras. Följande rubriker skall ingå i rapporten: 1. The motion of the un-damped machine. 2. The motion of the damped machine. 3. The optimal design. 4. Conclusions and discussion. Utformningen av rapporten i detalj bestäms av projektgruppen men där skall ingå Problembeskrivning Beskrivning av vad uppgiften gick ut på. Metod Beskriv kortfattat vilka hjälpmedel som använts vid analysen. Vilka antaganden och vilka approximationer har gjorts? Vilken teori har använts? Utförande Berätta hur undersökningen har genomförts. Organisera eventuellt resultatet i tabellform. Diagram ritas med hjälp av dator. Varje diagram ska numreras och förses med beskrivande text. Diagram placeras sist i redogörelsen och refereras till med hjälp av diagramnumret. Inga orefererade diagram får infogas i rapporten. Resultat Ange vilka resultat som erhölls i form av beräknade värden och kvalitativa resultat. Resultatpresentation i form av tabeller och diagram. Diskussion Stämmer resultatet med den teori som har antagits? Hur kan eventuella avvikelser förklaras? Finns det skäl att tro att den verkliga problemställningen inte är så idealiserad som teorin förutsätter? 9

### 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

### 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

### 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

### 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

### Spinning Stuff Review

Spinning Stuff Review 1. A wheel (radius = 0.20 m) is mounted on a frictionless, horizontal axis. A light cord wrapped around the wheel supports a 0.50-kg object, as shown in the figure below. When released

### 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

### 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

### 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

### Manufacturing Equipment Modeling

QUESTION 1 For a linear axis actuated by an electric motor complete the following: a. Derive a differential equation for the linear axis velocity assuming viscous friction acts on the DC motor shaft, leadscrew,

### Equivalent Spring Stiffness

Module 7 : Free Undamped Vibration of Single Degree of Freedom Systems; Determination of Natural Frequency ; Equivalent Inertia and Stiffness; Energy Method; Phase Plane Representation. Lecture 13 : Equivalent

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

Chapter 3 Mechanical Systems Dr. A.Aziz. Bazoune 3.1 INTRODUCTION Mathematical Modeling and response analysis of mechanical systems are the subjects of this chapter. 3. MECHANICAL ELEMENTS Any mechanical

### 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

### Chapter 8- Rotational Motion

Chapter 8- Rotational Motion Textbook (Giancoli, 6 th edition): Assignment 9 Due on Thursday, November 26. 1. On page 131 of Giancoli, problem 18. 2. On page 220 of Giancoli, problem 24. 3. On page 221

### Rotational Dynamics. Luis Anchordoqui

Rotational Dynamics Angular Quantities In purely rotational motion, all points on the object move in circles around the axis of rotation ( O ). The radius of the circle is r. All points on a straight line

### 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

### 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

### Rotational Mechanics - 1

Rotational Mechanics - 1 The Radian The radian is a unit of angular measure. The radian can be defined as the arc length s along a circle divided by the radius r. s r Comparing degrees and radians 360

### KINAR13h, KINLO13h. The marking period is, for the most part, 15 working days, otherwise it s the following date:

Materialplanering och Styrning 7.5 ECTS Ladokcode: The exam is given to: 41I29M KININ13h, KININ13h1, KINAF13h-pgrp3, KINAF13h-pgrp4, KINAR13h, KINLO13h ExamCode: Date of exam: 2016-01-12 Time: 14:00 18:00

### 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

### 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

### DECISION/BESLUT 2003-09-30

DECISION/BESLUT 2003-09-30 Ärendenr. 23. Sökande/Complainant IT Butikken A/S, Danmark Motpart/Respondent Shoppingsajterna Svenska AB, 556577-2547, Önsvala Gård, 245 93 Staffanstorp Saken/The Matter Alternativt

### 5.2 Rotational Kinematics, Moment of Inertia

5 ANGULAR MOTION 5.2 Rotational Kinematics, Moment of Inertia Name: 5.2 Rotational Kinematics, Moment of Inertia 5.2.1 Rotational Kinematics In (translational) kinematics, we started out with the position

### Homework 4. problems: 5.61, 5.67, 6.63, 13.21

Homework 4 problems: 5.6, 5.67, 6.6,. Problem 5.6 An object of mass M is held in place by an applied force F. and a pulley system as shown in the figure. he pulleys are massless and frictionless. Find

### 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

### SAS Data Integration SAS Business Intelligence

Kursöversikt 2010 SAS Education Providing knowledge through global training and certification SAS Data Integration SAS Business Intelligence Specialkurser SAS Forum 2010 Kontaktinformation Stora Frösunda

### 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,

### = mg [down] =!mg [up]; F! x

Section 4.6: Elastic Potential Energy and Simple Harmonic Motion Mini Investigation: Spring Force, page 193 Answers may vary. Sample answers: A. The relationship between F g and x is linear. B. The slope

### Rotational Motion. So far, you have studied translational motion. Here you will explore the physics of rotational motion.

Team: Rotational Motion Rotational motion is everywhere. When you push a door, it rotates. When you pedal a bike, the wheel rotates. When you start an engine, many parts rotate. Electrons rotate in an

### physics 111N rotational motion

physics 111N rotational motion rotations of a rigid body! suppose we have a body which rotates about some axis! we can define its orientation at any moment by an angle, θ (any point P will do) θ P physics

### 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

### 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

### Slide 10.1. Basic system Models

Slide 10.1 Basic system Models Objectives: Devise Models from basic building blocks of mechanical, electrical, fluid and thermal systems Recognize analogies between mechanical, electrical, fluid and thermal

Tillämpad Transformteori TNG032 Zhuangwei Liu Linköpings universitet January 18, 2016 Lärare Examinator: Zhuangwei Liu Språklärare: Pamela Vang Övningsledare: Zhuangwei Liu (ED2, MT2a) Olof Svensson (KTS2,

### Name: Lab Partner: Section:

Chapter 10 Simple Harmonic Motion Name: Lab Partner: Section: 10.1 Purpose Simple harmonic motion will be examined in this experiment. 10.2 Introduction A periodic motion is one that repeats itself in

### SAS Education Providing knowledge through global training and certification. SAS Foundation. Kursöversikt 2010

Kursöversikt 2010 SAS Education Providing knowledge through global training and certification SAS Foundation SAS Forum 2010 och specialkurser SAS Master Class Kontaktinformation Stora Frösunda Gård Box

### EQUATIONS OF MOTION: ROTATION ABOUT A FIXED AXIS

EQUATIONS OF MOTION: ROTATION ABOUT A FIXED AXIS Today s Objectives: Students will be able to: 1. Analyze the planar kinetics of a rigid body undergoing rotational motion. In-Class Activities: Applications

### Angular Momentum Problems Challenge Problems

Angular Momentum Problems Challenge Problems Problem 1: Toy Locomotive A toy locomotive of mass m L runs on a horizontal circular track of radius R and total mass m T. The track forms the rim of an otherwise

### 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

### End-of-Chapter Exercises

End-of-Chapter Exercises Exercises 1 12 are conceptual questions that are designed to see if you have understood the main concepts of the chapter. 1. Figure 11.20 shows four different cases involving a

### 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

### Laplace Transforms. (This command loads the functions required for computing Laplace and Inverse Laplace transforms)

Laplace Transforms For the design of a control system, it is important to know how the system of interest behaves and how it responds to different controller designs. To do this, the dynamic equations

### 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

### Repetition inför tentan. Kommunikation. Infrastruktur. ÖP 13-17 Inga hjälpmedel. v v v

Repetition inför tentan v v v ÖP 13-17 Inga hjälpmedel Kommunikation Infrastruktur 1 Skalbarhet När det inte går bra Organisation 2 Illvilja Repetition inför tentan Tentan hur kommer den att bli? 13-17

### A ball, attached to a cord of length 1.20 m, is set in motion so that it is swinging backwards and forwards like a pendulum.

MECHANICS: SIMPLE HARMONIC MOTION QUESTIONS THE PENDULUM (2014;2) A pendulum is set up, as shown in the diagram. The length of the cord attached to the bob is 1.55 m. The bob has a mass of 1.80 kg. The

### 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

### Rotation: Kinematics

Rotation: Kinematics Rotation refers to the turning of an object about a fixed axis and is very commonly encountered in day to day life. The motion of a fan, the turning of a door knob and the opening

### F mg (10.1 kg)(9.80 m/s ) m

Week 9 homework IMPORTANT NOTE ABOUT WEBASSIGN: In the WebAssign versions of these problems, various details have been changed, so that the answers will come out differently. The method to find the solution

### 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

### 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...

### Analytical Mechanics with Computer Algebra

Analytical Mechanics with Computer Algebra Bachelor s Degree Martin Lindberg mli6@kth.se Anders Pärlstrand apar@kth.se Department of mechanics KTH 2014-04-28 Supervisor: Nicholas Apazidis Abstract In this

### Calculate the centripetal acceleration of the boot just before impact

(ii) alculate the centripetal acceleration of the boot just before impact....... (iii) iscuss briefly the radial force on the knee joint before impact and during the impact................. (4) (Total

### ANGULAR POSITION. 4. Rotational Kinematics and Dynamics

ANGULAR POSITION To describe rotational motion, we define angular quantities that are analogous to linear quantities Consider a bicycle wheel that is free to rotate about its axle The axle is the axis

### Chapter 10 Rotational Motion. Copyright 2009 Pearson Education, Inc.

Chapter 10 Rotational Motion Angular Quantities Units of Chapter 10 Vector Nature of Angular Quantities Constant Angular Acceleration Torque Rotational Dynamics; Torque and Rotational Inertia Solving Problems

### SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

Exam Name SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 1) A lawn roller in the form of a uniform solid cylinder is being pulled horizontally by a horizontal

### No Brain Too Small PHYSICS. 2 kg

MECHANICS: ANGULAR MECHANICS QUESTIONS ROTATIONAL MOTION (2014;1) Universal gravitational constant = 6.67 10 11 N m 2 kg 2 (a) The radius of the Sun is 6.96 10 8 m. The equator of the Sun rotates at a

### 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

### 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

### Vatten(byggnad) VVR145 Vatten. 2. Vätskors egenskaper (1.1, 4.1 och 2.8) (Föreläsningsanteckningar)

Vatten(byggnad) Vätskors egenskaper (1) Hydrostatik (3) Grundläggande ekvationer (5) Rörströmning (4) 2. Vätskors egenskaper (1.1, 4.1 och 2.8) (Föreläsningsanteckningar) Vätska som kontinuerligt medium

### Exemplar Problems Physics

Chapter Eight GRAVITATION MCQ I 8.1 The earth is an approximate sphere. If the interior contained matter which is not of the same density everywhere, then on the surface of the earth, the acceleration

### Seismic Design of Shallow Foundations

Shallow Foundations Page 1 Seismic Design of Shallow Foundations Reading Assignment Lecture Notes Other Materials Ch. 9 FHWA manual Foundations_vibrations.pdf Homework Assignment 10 1. The factored forces

### MODELING RAIL WHEEL-FLAT DYNAMICS

MODELING RAIL WHEEL-FLAT DYNAMICS NS Vyas & AK Gupta Indian Institute of Technology, Kanpur 208 016, India Abstract: Development of flats on the wheels of railway engines and wagons is one of the major

### SVSFEM s.r.o. Brno, Czech Republic. Ing. László IVÁN, Technical support for Rotordynamics. Rotordynamics for.

SVSFEM s.r.o. Brno, Czech Republic Technical support for Rotordynamics Ing. László IVÁN, rotor@svsfem.cz The aim of the project ANSYS has been created as a problem oriented macro based tool which is integrated

### Hjälpmedel: Physics Handbook samt räknedosa. Tabell över vissa kärndata bifogas.

Tentamensskrivning i Kärnfysik (FK7010), 7,5hp Fredag den 28 mars 2008 kl 9-15 Hjälpmedel: Physics Handbook samt räknedosa. Tabell över vissa kärndata bifogas. Denna tentamen består av två delar. Den första

### Rotation, Angular Momentum

This test covers rotational motion, rotational kinematics, rotational energy, moments of inertia, torque, cross-products, angular momentum and conservation of angular momentum, with some problems requiring

### 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

### 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

### 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

### 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

### 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

### KINAFF12, KINAR12, KININ12, KINLO12. The results are posted no longer than three weeks after the exam

Materialplanering och styrning på grundnivå 7.5 ECTS Ladokcode: The exam is given to: (TI6612) 41I29M KINAFF12, KINAR12, KININ12, KINLO12 Name: (Ifylles av student) Personal number: (Ifylles av student)

### 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

### 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

### 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

### Fric-3. force F k and the equation (4.2) may be used. The sense of F k is opposite

4. FRICTION 4.1 Laws of friction. We know from experience that when two bodies tend to slide on each other a resisting force appears at their surface of contact which opposes their relative motion. The

### 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 Balance of Fans and Blowers

AHRI Guideline G (SI) 2011 Guideline for Mechanical Balance of Fans and Blowers IMPORTANT SAFETY DISCLAIMER AHRI does not set safety standards and does not certify or guarantee the safety of any products,

### Linear and Rotational Kinematics

Linear and Rotational Kinematics Starting from rest, a disk takes 10 revolutions to reach an angular velocity. If the angular acceleration is constant throughout, how many additional revolutions are required

### Simple Harmonic Motion

Simple Harmonic Motion 9M Object: Apparatus: To determine the force constant of a spring and then study the harmonic motion of that spring when it is loaded with a mass m. Force sensor, motion sensor,

### Chapter 7 - Rotational Motion w./ QuickCheck Questions

Chapter 7 - Rotational Motion w./ QuickCheck Questions 2015 Pearson Education, Inc. Anastasia Ierides Department of Physics and Astronomy University of New Mexico October 1, 2015 Review of Last Time Uniform

### Physics 131: Test/Exam Problems: Rotations

Physics 131: Test/Exam Problems: Rotations 1) A uniform one-meter bar has a mass of 10 kg. One end of the bar is hinged to a building, and the other end is suspended by a cable that makes an angle of 45

### Physics 53. Rotational Motion 1. We're going to turn this team around 360 degrees. Jason Kidd

Physics 53 Rotational Motion 1 We're going to turn this team around 360 degrees. Jason Kidd Rigid bodies To a good approximation, a solid object behaves like a perfectly rigid body, in which each particle

### Chapter 07: Kinetic Energy and Work

Chapter 07: Kinetic Energy and Work Conservation of Energy is one of Nature s fundamental laws that is not violated. Energy can take on different forms in a given system. This chapter we will discuss work

### Lecture 24 - Surface tension, viscous flow, thermodynamics

Lecture 24 - Surface tension, viscous flow, thermodynamics Surface tension, surface energy The atoms at the surface of a solid or liquid are not happy. Their bonding is less ideal than the bonding of atoms

### 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.

### Chapter 5: Circular Motion

Page 1 Chapter 5: Circular Motion Rotating Objects: Wheels, moon, earth, CDs, DVDs etc. Rigid bodies. Description of circular motion. Angular Position, Angular Displacement θ r s Angle (in radians) θ =

### Assignment Work (Physics) Class :Xi Chapter :04: Motion In PLANE

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. Assignment Work (Physics) Class :Xi Chapter :04: Motion In PLANE State law of parallelogram of vector addition and derive expression for resultant of two vectors

### Interactive Animation: A new approach to simulate parametric studies

Interactive Animation: A new approach to simulate parametric studies Darwin Sebayang and Ignatius Agung Wibowo Kolej Universiti Teknologi Tun Hussein Onn (KUiTTHO) Abstract Animation is the one of novel

### Readme10_054.doc page 1 of 7

Readme10_054.doc page 1 of 7 Date of production: 2007-12-03 News in 10_054 Hardware 1) New thyristor module DASD 145, 600 V - 500 A 2) Improved speed control with incremental encoder at very low speed

### Solutions to Homework Set #10 Phys2414 Fall 2005

Solution Set #0 Solutions to Homework Set #0 Phys244 Fall 2005 Note: The numbers in the boxes correspond to those that are generated by WebAssign. The numbers on your individual assignment will vary. Any

### Rotational inertia (moment of inertia)

Rotational inertia (moment of inertia) Define rotational inertia (moment of inertia) to be I = Σ m i r i 2 or r i : the perpendicular distance between m i and the given rotation axis m 1 m 2 x 1 x 2 Moment

### Question on Class IX» Science» Gravitation» The Universal Law Of Gravitation.

Question on Class IX» Science» Gravitation» The Universal Law Of Gravitation. Q.1. When we move from the poles to the equator. Hence, the value of g decreases. Why Ans: The shape of earth is an ellipse

### Physics 1A Lecture 10C

Physics 1A Lecture 10C "If you neglect to recharge a battery, it dies. And if you run full speed ahead without stopping for water, you lose momentum to finish the race. --Oprah Winfrey Static Equilibrium

### Measurement of Equivalent Stiffness and Damping of Shock Absorbers

Measurement of Equivalent Stiffness and Damping of Shock Absorbers Mohan D. Rao a and Scott Gruenberg b a Mechanical Engineering-Engineering Mechanics Department b Keweenaw Research Center Michigan Technological

### Computer animations in physical chemistry. Kim Bolton School of Engineering, University College of Borås, SE-501 90, Borås

Computer animations in physical chemistry Kim Bolton School of Engineering, University College of Borås, SE-501 90, Borås Introduction The immense increase in computational power, variety of computer-based

### FALL 2015 Pre-Test Solution for Exam 3 10/29/15 Time Limit: 75 Minutes

PHYC 151-002 FALL 2015 Pre-Test Solution for Exam 3 10/29/15 Time Limit: 75 Minutes Name (Print): This exam contains 10 pages (including this cover page) and 8 problems. Check to see if any pages are missing.

### 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

### Mechanical Principles

Unit 4: Mechanical Principles Unit code: F/601/1450 QCF level: 5 Credit value: 15 OUTCOME 4 POWER TRANSMISSION TUTORIAL 2 BALANCING 4. Dynamics of rotating systems Single and multi-link mechanisms: slider

### 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