THEORY OF VIBRATIONS Page 1
|
|
- Dayna Heath
- 7 years ago
- Views:
Transcription
1 THEORY OF VIBRATIONS THEORY OF VIBRATIONS Compiled by: K. Prabakaran aeroguys.tnj@gmail.com Page 1
2 THEORY OF VIBRATIONS Theory of vibrations Introduction There are two general classes of vibrations - free and forced. Free vibration takes place when a system oscillates under the action of forces inherent in the system itself, and when external impressed forces are absent. The system under free vibration will vibrate at one or more of its natural frequencies, which are properties of the dynamic system established by its mass and stiffness distribution. Vibration that takes place under the excitation of external forces is called forced vibration. When the excitation is oscillatory, the system is forced to vibrate at the excitation frequency. If the frequency of excitation coincides with one of the natural frequencies of the system, a condition of resonance is encountered, and dangerously large oscillations may result. The failure of major structures such as bridges, buildings, or airplane wings is an awesome possibility under resonance. Thus, the calculation of the natural frequencies of major importance in the study of vibrations. Vibrating systems are all subject to damping to some degree because energy is dissipated by friction and other resistances. If the damping is small, it has very little influence on the natural frequencies of the system, and hence the calculation for the natural frequencies is generally made on the basis of no damping. On the other hand, damping is of great importance in limiting the amplitude of oscillation at resonance. The number of independent coordinates required to describe the motion of a system is called degrees of freedom of the system. Thus, a free particle undergoing general motion in space will have three degrees of freedom, and a rigid body will have six degrees of freedom, i.e., three components of position and three angles Page 2
3 defining its orientation. Furthermore, a continuous elastic body will require an infinite number of coordinates (three for each point on the body) to describe its motion; hence, its degrees of freedom must be infinite. However, in many cases, parts of such bodies may be assumed to be rigid, and the system may be considered to be dynamically equivalent to one having finite degrees of freedom. In fact, a surprisingly large number of vibration problems can be treated with sufficient accuracy by reducing the system to one having a few degrees of freedom. Harmonic Motion Oscillatory motion may repeat itself regularly, as in the balance wheel of a watch, or display considerable irregularity, as in earthquakes. When the motion is repeated in equal intervals of time T, it is called period motion. The repetition time t is called the period of the oscillation, and its reciprocal,,is called the frequency. If the motion is designated by the time function x(t), then any periodic motion must satisfy the relationship. Harmonic motion is often represented as the projection on a straight line of a point that is moving on a circle at constant speed, as shown in Fig. 1. With the angular speed of the line o-p designated by w, the displacement x can be written as (1) Page 3
4 Figure 1 Harmonic Motion as a Projection of a Point Moving on a Circle The quantity w is generally measured in radians per second, and is referred to as the angular frequency. Because the motion repeats itself in 2p radians, we have the relationship (2) where t and f are the period and frequency of the harmonic motion, usually measured in seconds and cycles per second, respectively. The velocity and acceleration of harmonic motion can be simply determined by differentiation of Eq. 1. Using the dot notation for the derivative, we obtain (3) (4) Vibration Model The basic vibration model of a simple oscillatory system consists of a mass, a massless spring, and a damper. The spring supporting the mass is assumed to be of negligible mass. Its force-deflection relationship is considered to be linear, Page 4
5 following Hooke's law, newtons/meter., where the stiffness k is measured in The viscous damping, generally represented by a dashpot, is described by a force proportional to the velocity, or newtons/meter/second.. The damping coefficient c is measured in Figure 2 Spring-Mass System and Free-Body Diagram Equation of Motion : Natural Frequency Figure 2 shows a simple undamped spring-mass system, which is assumed to move only along the vertical direction. It has one degree of freedom (DOF), because its motion is described by a single coordinate x. When placed into motion, oscillation will take place at the natural frequency fn which is a property of the system. We now examine some of the basic concepts associated with the free vibration of systems with one degree of freedom. Page 5
6 Figure 2 Spring-Mass System and Free-Body Diagram Newton's second law is the first basis for examining the motion of the system. As shown in Fig. 2 the deformation of the spring in the static equilibrium position is D, and the spring force kd is equal to the gravitational force w acting on mass m (5) By measuring the displacement x from the static equilibrium position, the forces acting on m are and w. With x chosen to be positive in the downward direction, all quantities - force, velocity, and acceleration are also positive in the downward direction. We now apply Newton's second law of motion to the mass m : and because kd = w, we obtain : (6) It is evident that the choice of the static equilibrium position as reference for x has eliminated w, the force due to gravity, and the static spring force kd from the equation of motion, and the resultant force on m is simply the Page 6
7 spring force due to the displacement x. By defining the circular frequency w n by the equation (7) Eq. 6 can be written as (8) and we conclude that the motion is harmonic. Equation (8), a homogeneous second order linear differential equation, has the following general solution : where A and B are the two necessary constants. These constants are evaluated (9) from initial conditions, and Eq. (9) can be shown to reduce to (10) The natural period of the oscillation is established from, or (11) and the natural frequency is Page 7
8 (12) These quantities can be expressed in terms of the static deflection D by observing Eq. (5),. Thus, Eq. (12) can be expressed in terms of the static deflection D as (13) Note that, depend only on the mass and stiffness of the system, which are properties of the system. Viscously Damped Free Vibration Viscous damping force is expressed by the equation where c is a constant of proportionality. (14) Page 8
9 Symbolically. it is designated by a dashpot, as shown in Fig. 3. From the free body diagram, the equation of motion is.seen to be (15) The solution of this equation has two parts. If F(t) = 0, we have the homogeneous differential equation whose solution corresponds physically to that of freedamped vibration. With F(t) ¹ 0, we obtain the particular solution that is due to the excitation irrespective of the homogeneous solution. We will first examine the homogeneous equation that will give us some understanding of the role of damping. Page 9
10 Figure 3 Viscously Damped Free Vibration With the homogeneous equation : (16) the traditional approach is to assume a solution of the form : (17) where s is a constant. Upon substitution into the differential equation, we obtain : which is satisfied for all values of t when (18) Equation (18), which is known as the characteristic equation, has two roots : (19) Page 10
11 Hence, the general solution is given by the equation: (20) where A and B are constants to be evaluated from the initial conditions and. Equation (19) substituted into (20) gives : (21) The first term,, is simply an exponentially decaying function of time. The behavior of the terms in the parentheses, however, depends on whether the numerical value within the radical is positive, zero, or negative. When the damping term (c/2m)2 is larger than k/m, the exponents in the previous equation are real numbers and no oscillations are possible. We refer to this case as overdamped. When the damping term (c/2m)2 is less than k/m, the exponent becomes an imaginary number,. Because the terms of Eq. (21) within the parentheses are oscillatory. We refer to this case as underdamped. Page 11
12 In the limiting case between the oscillatory and non oscillatory motion, and the radical is zero. The damping corresponding to this case is called critical damping, cc. (22) Any damping can then be expressed in terms of the critical damping by a non dimensional number z, called the damping ratio: and (23) (24) (i) Oscillatory Motion (z < 1.0) Underdamped Case : The frequency of damped oscillation is equal to : (25) (26) Figure 4 shows the general nature of the oscillatory motion. Page 12
13 Figure 4 Damped Oscillation z < 1 (ii) Non oscillatory Motion (z > 1.0) Overdamped Case : (27) The motion is an exponentially decreasing function of time as shown in Fig. 5. Figure 5 Aperiodic Motion z > 1 Page 13
14 (iii) Critically Damped Motion (z = 1.0) : (28) Figure 6 shows three types of response with initial displacement x(0). Figure 6 Critically Damped Motion z = 1 Forced Harmonic Vibration Harmonic excitation is often encountered in engineering systems. It is commonly produced by the unbalance in rotating machinery. Although pure harmonic excitation is less likely to occur than periodic or other types of excitation, understanding the behavior of a system undergoing harmonic excitation is essential in order to comprehend how the system will respond to more general types of excitation. Harmonic excitation may be in the form of a force or displacement of some point in the system. We will first consider a single DOF system with viscous damping, excited by a harmonic force, as shown in Fig. 7. Its differential equation of motion is found from the free-body diagram. Page 14
15 (29) Figure 7 Viscously Damped System with Harmonic Excitation The solution to this equation consists of two parts, the complementary function, which is the solution of the homogeneous equation, and the particular integral. The complementary function. in this case, is a damped free vibration. The particular solution to the preceding equation is a steady-state oscillation of the same frequency w as that of the excitation. We can assume the particular solution to be of the form : (30) where X is the amplitude of oscillation and f is the phase of the displacement with respect to the exciting force. The amplitude and phase in the previous equation are found by substituting Eqn. (30) into the differential equation (29). Remembering that in harmonic motion the phases of the velocity and acceleration are ahead of the displacement by 90 Page 15
16 and 180, respectively, the terms of the differential equation can also be displayed graphically, as in Fig. 8. Figure 8 Vector Relationship for Forced Vibration with Damping It is easily seen from this diagram that and (31) (32) We now express Eqs (31) and (32) in non-dimensional term that enables a concise graphical presentation of these results. Dividing the numerator and denominator of Eqs. (31) and (32) by k, we obtain : (33) Page 16
17 and (34) These equations can be further expressed in terms of the following quantities: The non-dimensional expressions for the amplitude and phase then become (35) Compiled by: K. Prabakaran aeroguys.tnj@gmail.com Page 17
18 THEORY OF VIBRATIONS and (36) Page 18
19 These equations indicate that the non dimensional amplitude, and the phase f are functions only of the frequency ratio and can be plotted as shown in Fig 9., and the damping factor z Figure 9 Plot of Eqs. (35) and (36) Page 19
20 Rotating Unbalance Unbalance in rotating machines is a common source of vibration excitation. We consider here a spring-mass system constrained to move in the vertical direction and excited by a rotating machine that is unbalanced, as shown in Fig. 10. The unbalance is represented by an eccentric mass m with eccentricity e that is rotating with angular velocity w. By letting x be the displacement of the non rotating mass (M - m) from the static equilibrium position, the displacement of m is : Page 20
21 Figure 10 Harmonic Disturbing Force Resulting from Rotating Unbalance The equation of motion is then : which can be rearranged to : (37) It is evident that this equation is identical to Eq. (29), where is replaced by, and hence the steady-state solution of the previous section can be replaced by : and (38) (39) These can be further reduced to non dimensional form : (40) Page 21
22 and (41) Page 22
23 Example A counter rotating eccentric weight exciter is used to produce the forced oscillation of a spring-supported mass as shown in Fig. 11. By varying the speed of rotation, a resonant amplitude of 0.60 cm was recorded. When the speed of rotation was increase considerably beyond the resonant frequency, the amplitude appeared to approach a fixed value of 0.08 cm. Determine the damping factor of the system. Figure 11 Solution : From Eqn. (40), the resonant amplitude is : When w is very much greater than, the same equation becomes Page 23
24 By solving the two equations simultaneously, the damping factor of the system is REFRENCE: Theory of vibrations - V.P. SINGH Compiled by: K. Prabakaran aeroguys.tnj@gmail.com Page 24
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 informationStructural 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 informationSoil 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 informationOscillations. 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 informationSpring 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 informationUnderstanding Poles and Zeros
MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING 2.14 Analysis and Design of Feedback Control Systems Understanding Poles and Zeros 1 System Poles and Zeros The transfer function
More informationSimple 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 informationPhysics 41 HW Set 1 Chapter 15
Physics 4 HW Set Chapter 5 Serway 8 th OC:, 4, 7 CQ: 4, 8 P: 4, 5, 8, 8, 0, 9,, 4, 9, 4, 5, 5 Discussion Problems:, 57, 59, 67, 74 OC CQ P: 4, 5, 8, 8, 0, 9,, 4, 9, 4, 5, 5 Discussion Problems:, 57, 59,
More informationAP1 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 informationUnit - 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
More informationCopyright 2011 Casa Software Ltd. www.casaxps.com
Table of Contents Variable Forces and Differential Equations... 2 Differential Equations... 3 Second Order Linear Differential Equations with Constant Coefficients... 6 Reduction of Differential Equations
More informationModeling 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 informationEquivalent 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
More informationState 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
More informationPrelab 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 informationDetermination of Acceleration due to Gravity
Experiment 2 24 Kuwait University Physics 105 Physics Department Determination of Acceleration due to Gravity Introduction In this experiment the acceleration due to gravity (g) is determined using two
More informationHOOKE S LAW AND OSCILLATIONS
9 HOOKE S LAW AND OSCILLATIONS OBJECTIVE To measure the effect of amplitude, mass, and spring constant on the period of a spring-mass oscillator. INTRODUCTION The force which restores a spring to its equilibrium
More informationPhysics 1120: Simple Harmonic Motion Solutions
Questions: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Physics 1120: Simple Harmonic Motion Solutions 1. A 1.75 kg particle moves as function of time as follows: x = 4cos(1.33t+π/5) where distance is measured
More informationBASIC 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 informationState Newton's second law of motion for a particle, defining carefully each term used.
5 Question 1. [Marks 28] 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
More informationPhysics 9e/Cutnell. correlated to the. College Board AP Physics 1 Course Objectives
Physics 9e/Cutnell correlated to the College Board AP Physics 1 Course Objectives Big Idea 1: Objects and systems have properties such as mass and charge. Systems may have internal structure. Enduring
More informationEngineering 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 informationSlide 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
More information226 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 information19.7. Applications of Differential Equations. Introduction. Prerequisites. Learning Outcomes. Learning Style
Applications of Differential Equations 19.7 Introduction Blocks 19.2 to 19.6 have introduced several techniques for solving commonly-occurring firstorder and second-order ordinary differential equations.
More informationASEN 3112 - Structures. MDOF Dynamic Systems. ASEN 3112 Lecture 1 Slide 1
19 MDOF Dynamic Systems ASEN 3112 Lecture 1 Slide 1 A Two-DOF Mass-Spring-Dashpot Dynamic System Consider the lumped-parameter, mass-spring-dashpot dynamic system shown in the Figure. It has two point
More informationANALYTICAL 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 informationInteractive 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
More informationSample Questions for the AP Physics 1 Exam
Sample Questions for the AP Physics 1 Exam Sample Questions for the AP Physics 1 Exam Multiple-choice Questions Note: To simplify calculations, you may use g 5 10 m/s 2 in all problems. Directions: Each
More informationPrecise Modelling of a Gantry Crane System Including Friction, 3D Angular Swing and Hoisting Cable Flexibility
Precise Modelling of a Gantry Crane System Including Friction, 3D Angular Swing and Hoisting Cable Flexibility Renuka V. S. & Abraham T Mathew Electrical Engineering Department, NIT Calicut E-mail : renuka_mee@nitc.ac.in,
More informationFXA 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 informationAPPLIED MATHEMATICS ADVANCED LEVEL
APPLIED MATHEMATICS ADVANCED LEVEL INTRODUCTION This syllabus serves to examine candidates knowledge and skills in introductory mathematical and statistical methods, and their applications. For applications
More informationwww.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
More informationThe simulation of machine tools can be divided into two stages. In the first stage the mechanical behavior of a machine tool is simulated with FEM
1 The simulation of machine tools can be divided into two stages. In the first stage the mechanical behavior of a machine tool is simulated with FEM tools. The approach to this simulation is different
More informationExperiment 9. The Pendulum
Experiment 9 The Pendulum 9.1 Objectives Investigate the functional dependence of the period (τ) 1 of a pendulum on its length (L), the mass of its bob (m), and the starting angle (θ 0 ). Use a pendulum
More informationPhysics 231 Lecture 15
Physics 31 ecture 15 Main points of today s lecture: Simple harmonic motion Mass and Spring Pendulum Circular motion T 1/f; f 1/ T; ω πf for mass and spring ω x Acos( ωt) v ωasin( ωt) x ax ω Acos( ωt)
More informationLecture L2 - Degrees of Freedom and Constraints, Rectilinear Motion
S. Widnall 6.07 Dynamics Fall 009 Version.0 Lecture L - Degrees of Freedom and Constraints, Rectilinear Motion Degrees of Freedom Degrees of freedom refers to the number of independent spatial coordinates
More informationboth 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 information2.6 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 informationLecture L22-2D Rigid Body Dynamics: Work and Energy
J. Peraire, S. Widnall 6.07 Dynamics Fall 008 Version.0 Lecture L - D Rigid Body Dynamics: Work and Energy In this lecture, we will revisit the principle of work and energy introduced in lecture L-3 for
More informationHOOKE 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 informationMounting instructions. Acceleration Transducer B12. B 26.B12.10 en
Mounting instructions Acceleration Transducer B12 B 26.B12.10 en B12 3 Contents Page Safety instructions.............................................. 4 1 Scope of supply..............................................
More informationSTATICS. Introduction VECTOR MECHANICS FOR ENGINEERS: Eighth Edition CHAPTER. Ferdinand P. Beer E. Russell Johnston, Jr.
Eighth E CHAPTER VECTOR MECHANICS FOR ENGINEERS: STATICS Ferdinand P. Beer E. Russell Johnston, Jr. Introduction Lecture Notes: J. Walt Oler Texas Tech University Contents What is Mechanics? Fundamental
More informationThis page intentionally left blank
This page intentionally left blank Vibration of Mechanical Systems This is a textbook for a first course in mechanical vibrations. There are many books in this area that try to include everything, thus
More informationELASTIC FORCES and HOOKE S LAW
PHYS-101 LAB-03 ELASTIC FORCES and HOOKE S LAW 1. Objective The objective of this lab is to show that the response of a spring when an external agent changes its equilibrium length by x can be described
More informationSecond 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 informationChapter 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
More informationSHAKING TABLE DEMONSTRATION OF DYNAMIC RESPONSE OF BASE-ISOLATED BUILDINGS ***** Student Manual *****
SHAKING TABLE DEMONSTRATION OF DYNAMIC RESPONSE OF BASE-ISOLATED BUILDINGS ***** Student Manual ***** A PROJECT DEVELOPED FOR THE UNIVERSITY CONSORTIUM ON INSTRUCTIONAL SHAKE TABLES http://wusceel.cive.wustl.edu/ucist/
More informationv v ax v a x a v a v = = = Since F = ma, it follows that a = F/m. The mass of the arrow is unchanged, and ( )
Week 3 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
More informationPENDULUM PERIODS. First Last. Partners: student1, student2, and student3
PENDULUM PERIODS First Last Partners: student1, student2, and student3 Governor s School for Science and Technology 520 Butler Farm Road, Hampton, VA 23666 April 13, 2011 ABSTRACT The effect of amplitude,
More informationDynamic Analysis of the Dortmund University Campus Sky Train
Dynamic Analysis of the Dortmund University Campus Sky Train Reinhold Meisinger Mechanical Engineering Department Nuremberg University of Applied Sciences Kesslerplatz 12, 90121 Nuremberg, Germany Abstract
More information11. Rotation Translational Motion: Rotational Motion:
11. Rotation Translational Motion: Motion of the center of mass of an object from one position to another. All the motion discussed so far belongs to this category, except uniform circular motion. Rotational
More informationDifferential Relations for Fluid Flow. Acceleration field of a fluid. The differential equation of mass conservation
Differential Relations for Fluid Flow In this approach, we apply our four basic conservation laws to an infinitesimally small control volume. The differential approach provides point by point details of
More informationSOLID MECHANICS TUTORIAL MECHANISMS KINEMATICS - VELOCITY AND ACCELERATION DIAGRAMS
SOLID MECHANICS TUTORIAL MECHANISMS KINEMATICS - VELOCITY AND ACCELERATION DIAGRAMS This work covers elements of the syllabus for the Engineering Council exams C105 Mechanical and Structural Engineering
More information1.2 Second-order systems
1.2. SECOND-ORDER SYSTEMS 25 if the initial fluid height is defined as h() = h, then the fluid height as a function of time varies as h(t) = h e tρg/ra [m]. (1.31) 1.2 Second-order systems In the previous
More informationStructural Dynamics of Linear Elastic Single-Degree-of-Freedom (SDOF) Systems
Structural Dynamics of Linear Elastic Single-Degree-of-Freedom (SDOF) Systems SDOF Dynamics 3-1 This set of slides covers the fundamental concepts of structural dynamics of linear elastic single-degree-of-freedom
More informationSimple Vibration Problems with MATLAB (and Some Help from MAPLE) Original Version by Stephen Kuchnicki
Simple Vibration Problems with MATLAB (and Some Help from MAPLE) Original Version by Stephen Kuchnicki December 7, 29 Contents Preface ix 1 Introduction 1 2 SDOF Undamped Oscillation 3 3 A Damped SDOF
More informationMechanics 1: Conservation of Energy and Momentum
Mechanics : Conservation of Energy and Momentum If a certain quantity associated with a system does not change in time. We say that it is conserved, and the system possesses a conservation law. Conservation
More informationForce on Moving Charges in a Magnetic Field
[ Assignment View ] [ Eðlisfræði 2, vor 2007 27. Magnetic Field and Magnetic Forces Assignment is due at 2:00am on Wednesday, February 28, 2007 Credit for problems submitted late will decrease to 0% after
More information2.2 Magic with complex exponentials
2.2. MAGIC WITH COMPLEX EXPONENTIALS 97 2.2 Magic with complex exponentials We don t really know what aspects of complex variables you learned about in high school, so the goal here is to start more or
More informationPhysics 2A, Sec B00: Mechanics -- Winter 2011 Instructor: B. Grinstein Final Exam
Physics 2A, Sec B00: Mechanics -- Winter 2011 Instructor: B. Grinstein Final Exam INSTRUCTIONS: Use a pencil #2 to fill your scantron. Write your code number and bubble it in under "EXAM NUMBER;" an entry
More information3.2 Sources, Sinks, Saddles, and Spirals
3.2. Sources, Sinks, Saddles, and Spirals 6 3.2 Sources, Sinks, Saddles, and Spirals The pictures in this section show solutions to Ay 00 C By 0 C Cy D 0. These are linear equations with constant coefficients
More informationAdequate Theory of Oscillator: A Prelude to Verification of Classical Mechanics Part 2
International Letters of Chemistry, Physics and Astronomy Online: 213-9-19 ISSN: 2299-3843, Vol. 3, pp 1-1 doi:1.1852/www.scipress.com/ilcpa.3.1 212 SciPress Ltd., Switzerland Adequate Theory of Oscillator:
More informationRotation: Moment of Inertia and Torque
Rotation: Moment of Inertia and Torque Every time we push a door open or tighten a bolt using a wrench, we apply a force that results in a rotational motion about a fixed axis. Through experience we learn
More informationEXPERIMENT 2 Measurement of g: Use of a simple pendulum
EXPERIMENT 2 Measurement of g: Use of a simple pendulum OBJECTIVE: To measure the acceleration due to gravity using a simple pendulum. Textbook reference: pp10-15 INTRODUCTION: Many things in nature wiggle
More informationPHYSICAL QUANTITIES AND UNITS
1 PHYSICAL QUANTITIES AND UNITS Introduction Physics is the study of matter, its motion and the interaction between matter. Physics involves analysis of physical quantities, the interaction between them
More informationVibration Course Enhancement through a Dynamic MATLAB Graphic User Interface
Vibration Course Enhancement through a Dynamic MATLAB Graphic User Interface Elizabeth K. Ervin 1 [Weiping Xu 2 ] Abstract From the string of a guitar to the radio wave, vibration occurs all the time and
More informationPHY121 #8 Midterm I 3.06.2013
PHY11 #8 Midterm I 3.06.013 AP Physics- Newton s Laws AP Exam Multiple Choice Questions #1 #4 1. When the frictionless system shown above is accelerated by an applied force of magnitude F, the tension
More informationCentripetal Force. This result is independent of the size of r. A full circle has 2π rad, and 360 deg = 2π rad.
Centripetal Force 1 Introduction In classical mechanics, the dynamics of a point particle are described by Newton s 2nd law, F = m a, where F is the net force, m is the mass, and a is the acceleration.
More informationKERN COMMUNITY COLLEGE DISTRICT CERRO COSO COLLEGE PHYS C111 COURSE OUTLINE OF RECORD
KERN COMMUNITY COLLEGE DISTRICT CERRO COSO COLLEGE PHYS C111 COURSE OUTLINE OF RECORD 1. DISCIPLINE AND COURSE NUMBER: PHYS C111 2. COURSE TITLE: Mechanics 3. SHORT BANWEB TITLE: Mechanics 4. COURSE AUTHOR:
More informationF = ma. F = G m 1m 2 R 2
Newton s Laws The ideal models of a particle or point mass constrained to move along the x-axis, or the motion of a projectile or satellite, have been studied from Newton s second law (1) F = ma. In the
More informationVELOCITY, ACCELERATION, FORCE
VELOCITY, ACCELERATION, FORCE velocity Velocity v is a vector, with units of meters per second ( m s ). Velocity indicates the rate of change of the object s position ( r ); i.e., velocity tells you how
More informationAP 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 informationUniversal Law of Gravitation
Universal Law of Gravitation Law: Every body exerts a force of attraction on every other body. This force called, gravity, is relatively weak and decreases rapidly with the distance separating the bodies
More informationRotational Motion: Moment of Inertia
Experiment 8 Rotational Motion: Moment of Inertia 8.1 Objectives Familiarize yourself with the concept of moment of inertia, I, which plays the same role in the description of the rotation of a rigid body
More information2.5 Physically-based Animation
2.5 Physically-based Animation 320491: Advanced Graphics - Chapter 2 74 Physically-based animation Morphing allowed us to animate between two known states. Typically, only one state of an object is known.
More informationOrbital Mechanics. Angular Momentum
Orbital Mechanics The objects that orbit earth have only a few forces acting on them, the largest being the gravitational pull from the earth. The trajectories that satellites or rockets follow are largely
More informationPractice 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 informationActive Vibration Isolation of an Unbalanced Machine Spindle
UCRL-CONF-206108 Active Vibration Isolation of an Unbalanced Machine Spindle D. J. Hopkins, P. Geraghty August 18, 2004 American Society of Precision Engineering Annual Conference Orlando, FL, United States
More informationRANDOM VIBRATION AN OVERVIEW by Barry Controls, Hopkinton, MA
RANDOM VIBRATION AN OVERVIEW by Barry Controls, Hopkinton, MA ABSTRACT Random vibration is becoming increasingly recognized as the most realistic method of simulating the dynamic environment of military
More informationME 563 MECHANICAL VIBRATIONS
ME 563 MECHANICAL VIBRATIONS Fall 2010 Potter MWF 4:30 p.m.-5:20 p.m. Instructor: Prof. D. E. Adams Room: ME 361 Email: deadams@purdue.edu Phone: 496-6033 1-1 1 Introduction to Mechanical Vibrations 1.1
More information! n. Problems and Solutions Section 1.5 (1.66 through 1.74)
Problems and Solutions Section.5 (.66 through.74).66 A helicopter landing gear consists of a metal framework rather than the coil spring based suspension system used in a fixed-wing aircraft. The vibration
More informationPHYS 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
More informationDEVELOPMENT AND APPLICATIONS OF TUNED/HYBRID MASS DAMPERS USING MULTI-STAGE RUBBER BEARINGS FOR VIBRATION CONTROL OF STRUCTURES
13 th World Conference on Earthquake Engineering Vancouver, B.C., Canada August 1-6, 2004 Paper No. 2243 DEVELOPMENT AND APPLICATIONS OF TUNED/HYBRID MASS DAMPERS USING MULTI-STAGE RUBBER BEARINGS FOR
More informationCHAPTER 6 WORK AND ENERGY
CHAPTER 6 WORK AND ENERGY CONCEPTUAL QUESTIONS. REASONING AND SOLUTION The work done by F in moving the box through a displacement s is W = ( F cos 0 ) s= Fs. The work done by F is W = ( F cos θ). s From
More informationWeight The weight of an object is defined as the gravitational force acting on the object. Unit: Newton (N)
Gravitational Field A gravitational field as a region in which an object experiences a force due to gravitational attraction Gravitational Field Strength The gravitational field strength at a point in
More informationSection 5.0 : Horn Physics. By Martin J. King, 6/29/08 Copyright 2008 by Martin J. King. All Rights Reserved.
Section 5. : Horn Physics Section 5. : Horn Physics By Martin J. King, 6/29/8 Copyright 28 by Martin J. King. All Rights Reserved. Before discussing the design of a horn loaded loudspeaker system, it is
More informationJournal of Engineering Science and Technology Review 2 (1) (2009) 76-81. Lecture Note
Journal of Engineering Science and Technology Review 2 (1) (2009) 76-81 Lecture Note JOURNAL OF Engineering Science and Technology Review www.jestr.org Time of flight and range of the motion of a projectile
More informationC B A T 3 T 2 T 1. 1. What is the magnitude of the force T 1? A) 37.5 N B) 75.0 N C) 113 N D) 157 N E) 192 N
Three boxes are connected by massless strings and are resting on a frictionless table. Each box has a mass of 15 kg, and the tension T 1 in the right string is accelerating the boxes to the right at a
More information8.2 Elastic Strain Energy
Section 8. 8. Elastic Strain Energy The strain energy stored in an elastic material upon deformation is calculated below for a number of different geometries and loading conditions. These expressions for
More informationPrecession of spin and Precession of a top
6. Classical Precession of the Angular Momentum Vector A classical bar magnet (Figure 11) may lie motionless at a certain orientation in a magnetic field. However, if the bar magnet possesses angular momentum,
More informationPhysics Midterm Review Packet January 2010
Physics Midterm Review Packet January 2010 This Packet is a Study Guide, not a replacement for studying from your notes, tests, quizzes, and textbook. Midterm Date: Thursday, January 28 th 8:15-10:15 Room:
More informationReview of First- and Second-Order System Response 1
MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING 2.151 Advanced System Dynamics and Control Review of First- and Second-Order System Response 1 1 First-Order Linear System Transient
More informationDYNAMIC ANALYSIS ON STEEL FIBRE
International Journal of Civil Engineering and Technology (IJCIET) Volume 7, Issue 2, March-April 2016, pp. 179 184, Article ID: IJCIET_07_02_015 Available online at http://www.iaeme.com/ijciet/issues.asp?jtype=ijciet&vtype=7&itype=2
More informationLinear Motion vs. Rotational Motion
Linear Motion vs. Rotational Motion Linear motion involves an object moving from one point to another in a straight line. Rotational motion involves an object rotating about an axis. Examples include a
More informationNotice numbers may change randomly in your assignments and you may have to recalculate solutions for your specific case.
HW1 Possible Solutions Notice numbers may change randomly in your assignments and you may have to recalculate solutions for your specific case. Tipler 14.P.003 An object attached to a spring has simple
More informationPhysics Notes Class 11 CHAPTER 3 MOTION IN A STRAIGHT LINE
1 P a g e Motion Physics Notes Class 11 CHAPTER 3 MOTION IN A STRAIGHT LINE If an object changes its position with respect to its surroundings with time, then it is called in motion. Rest If an object
More informationHow To Solve A Linear Dierential Equation
Dierential Equations (part 2): Linear Dierential Equations (by Evan Dummit, 2012, v. 1.00) Contents 4 Linear Dierential Equations 1 4.1 Terminology.................................................. 1 4.2
More informationAim : To study how the time period of a simple pendulum changes when its amplitude is changed.
Aim : To study how the time period of a simple pendulum changes when its amplitude is changed. Teacher s Signature Name: Suvrat Raju Class: XIID Board Roll No.: Table of Contents Aim..................................................1
More information