Dynamics and control of mechanical systems
|
|
- Emmeline Webster
- 5 years ago
- Views:
Transcription
1 Dynamics and control of mechanical systems Date Day 1 (03/05) - 05/05 Day 2 (07/05) Day 3 (09/05) Day 4 (11/05) Day 5 (14/05) Day 6 (16/05) Content Review of the basics of mechanics. Kinematics of rigid bodies - coordinate transformation, angular velocity vector, description of velocity and acceleration in relatively moving frames. Euler angles, Review of methods of momentum and angular momentum of system of particles, inertia tensor of rigid body. Dynamics of rigid bodies - Euler's equation, application to motion of symmetric tops and gyroscopes and problems of system of bodies. Kinetic energy of a rigid body, virtual displacement and classification of constraints. D Alembert s principle. Introduction to generalized coordinates, derivation of Lagrange's equation from D Alembert s principle. Small oscillations, matrix formulation, Eigen value problem and numerical solutions. Modelling mechanical systems, Introduction to MATLAB, computer generation and solution of equations of motion. Introduction to complex analytic functions, Laplace and Fourier transform. PID controllers, Phase lag and Phase lead compensation. Analysis of Control systems in state space, pole placement, computer simulation through MATLAB. 1
2 Contents Focus on Introduction to modeling of mechanical systems Modeling fundamentals of elements of mechanical systems Inertia element (mass) Energy storage elements (spring) Energy dissipation elements (dampers) Examples and exercises 2
3 Introduction A model is an abstraction of the physical world Used for analysis and design, commonly before realization of the physical system Can be obtained from first principles or experimentally The modeling purpose determines level of abstraction Can be complex enough, but no more than the physical object Model derivation from first principles Using physical laws to derive the models Models based on first principles provides better undertanding Such models can also use empirical data to determine parameters, and validate the model Input System Output Give known input (empirical data) and observe output fit model to your data - Applicable for complex systems and blackbox systems 3
4 Introduction Derivation of Differential Equations (Des) is an important part of modeling of mechanical systems. Sources of such DEs Newton s 2 nd Law FF = ma Euler s 2 nd Law MM = Iαα Hooke s Law F = kx Lagrangian Hamiltonian dd dddd LL qq jj - LL qq jj = 0 dddd dddd - HH qq jj = 0 4
5 Introduction Modeling of static and dynamic systems Static systems Relationship is static (not changing with time) Dynamic systems Relationship changes with time, past inputs and initial conditions influence current output ( it is dynamic) Only current input determines the output, which reacts immediately Relationship is represented by algebraic equations Output takes time to respond to inputs Relationship is represented by differential equations Input Output (system response) System Volt, 5v (Ang. Speed = 200 rpm) El-motor Force and motion sources cause elements to respond 5
6 1) Energy storage elements Inertia elements Stiffness (spring) elements 2) Energy dissipation elements Dampers Friction Mechanical systems can be either translational or rotational, for instance, for energy storage elements. Translational system - Inertia element: mass, m - Stiffness element: k For rotational system: - Inertia element: Mass moment of inertia, I - Stiffness element: k t 6 These are passive (non-energy producing) devices
7 1) Inertia elements Inertia elements store energy in the form of KE Commonly components are not added to a mech. system for the purpose of adding inertia Since all materials have mass, however, the mass or inertia element often may represents an undesirable effect in the system. There are some applications in which mass itself serves a useful function, e.g., flywheels. Response of engine with flywheel Flywheels are used as energy-storage devices or as a means of smoothing out speed fluctuations in in mechanical systems such as IC engines or other machines 7 Response of engine without flywheel
8 1) Inertia elements... modelling Inertia elements are modelled as follows Particle (point mass), m Rid body of mass, m Symbol for translational element T, θθ F, x Torque θθ = TT II F, x m Moment of inertia xx = FF mm Mathematical model from Newton s 2 nd Law Note that, for a rigid body in translation, every particle in the body has the same velocity and acceleration, while in flexible bodies the velocities and/or accelerations can vary Symbol for rotational element Mathematical model from Newton s 2 nd Law Note: No real rigid body exists when being accelerated, and the inertia element is a model, not a real/physical object. 8
9 1) Inertia elements... Mass moment of inertia for typical geometries Particle of mass m, rotating at end of a massless rod of length l: I = ml 2 I = mm rr 22 dddd A cylinder (r, m), rotating about center axis: I = 11 mr 2 22 Solid sphere (r, m), rotating about center axis: I = 22 mr 2 33 Uniform rod (l, m), rotating about its center: I = 11 ml Uniform rod (l, m), rotating about its end: I = 11 ml 2 33 Note: moment of inertia of rigid bodies with complex geometry are determined experimentally 9
10 Inertia elements... Stiffness elements (springs) Real springs - Are neither pure nor ideal - Can have inertia and friction Pure spring is a mathematical model, - It is not a real device - It is ideal: massless, no damping and linear For ideal spring element Note: The inverse of stiffness is called compliance (c) / flexibility F k x Force: F = k(x 1 x 2 ) = kx Torque: T = k t (θθ 11 θθ 22 ) = k t θθ Energy: E = kkkk22 10 x c F
11 Inertia elements... Stiffness elements (springs) Real springs are not purely linear - Linearization of a nonlinear spring - Taylor s expansion FF(xx) = FF(xx 0 ) + (xx xx 0 ). dddd dddd 0 + xx xx 0 2! 2 dd 2 FF dddd ! dd 3 FF dddd FF(xx) Ignoring higher order derivatives FF(xx) - FF(xx 0 ) = (xx xx 0 ). F = kx dddd dddd 0 FF(xx 0 ) Nonlinear 11 xx 0
12 Inertia elements... Stiffness elements (springs) Real springs are not purely linear Coil spring Cantilever beam spring Tension rod spring Torsion bar spring clamped-end beam spring ring spring rubber spring (shock mount) hydraulic (oil) spring pneumatic (air) spring kk = GGdd DD 33 kk = LL 33 kk tt = 12 kk = AAAA LL EEEE LL
13 Inertia elements... Stiffness elements (springs) Key steps to modelling of mechanical systems 1) Establish an inertial (fixed) coordinate system 2) Identify discrete mechanical system elements (masses, springs, dampers) and isolate them 3) Determine min. number of variables needed to uniquely define the configuration of system and subtract constraints 4) Draw the free body diagram for each inertia element 5) Write equations that relate each element s loading to deformation 6) Apply Newton s 2nd Law to each element: Translational motion: Newton s 2 nd Law: F = ma Rotational motion: Euler s 2 nd Law: T = Iα 13
14 Inertia elements... Stiffness elements (springs) Example 1: 2 mass and 3 spring system* 14
15 Energy dissipation elements: Damping elements Dampers converts the mechanical energy to thermal energy. Two main forms Friction damping (coulomb dry friction damping) Viscoelastic damping or viscous damping (most common type) F Static friction F Kinetic friction F xx xx xx Linear damper (Assuming small rel. velocity) F = Cxx Coulomb friction F = C Sin( xx) 15 Drag damper F = Cxx 22
16 Energy dissipation elements: Damping elements Characteristics of damping elements Viscoelastic damper The damping force is proportional to the velocity across the damper, The force acts in the direction opposite to the velocity. Friction damping (coulomb damping). Takes place between two surfaces in relative motion Magnitude of damping force is assumed constant not function of the relative velocity at the interface The force is opposite to that of the relative velocity 16
17 Applications: A damper element is used to model a device designed into a system (e.g., automotive shock absorbers) Sprang mass B Unsprang mass 17
18 Energy dissipation elements: Damping elements... Damper force or torque is directly proportional to the relative velocity of its two ends. FF DD = CC dd dddd xx 11 xx 22 = CC ddxx TT DD = CC dd dddd (θθ 1 θθ 2) dddd = CC ddθθ dddd Where C is the damping coefficient = CC xx for translational motion = C θθ - for rotational motion In some textbooks B is used as a damping coeff. C Power dissipated FFxx = CC( xx) 2 Symbol of a damper (translational motion) 18
19 Example 2*: mass spring and damper system C 1 C 2 C 3 19
20 Exercise: Modelling the suspension system of a vehicle k 2 m 2 x 2 m 1 C x 1 m 1 = Unsprung mass m 2 = Spring mass, a quarter of the car s mass C = damping coeff. of shock absorber k 1 = Tire stiffness (damping in tire is neglected) k 2 = Stiffness of suspension coil spring x 0 = motion of the ground stiffness x 1 = coordinate for motion of unsprung mass x 2 = coordinate for motion of sprung mass k 1 x 0 Derive the DEs that describe the motion of the suspension system and formulate it in matrix form 20
21 This lecture has focused on Summary Explanation of the building blocks of modeling mechanical systems: - Inertia elements, - energy storage (spring) elements and - energy dissipation (damper) elements Use of Newton s 2 nd Law and Euler s 2 nd Law to develop mathematical models of the modeling elements Next: Brief Introduction about multibody dynamics. 21
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 informationFundamental principles
Dynamics and control of mechanical systems Date Day 1 (03/05) - 05/05 Day (07/05) Day 3 (09/05) Day 4 (11/05) Day 5 (14/05) Day 6 (16/05) Content Review of the basics of mechanics. Kinematics of rigid
More informationDynamics and control of mechanical systems
Dynamics and control of mechanical systems Date Day 1 (03/05) - 05/05 Day 2 (07/05) Day 3 (09/05) Day 4 (11/05) Day 5 (14/05) Day 6 (16/05) Content Review of the basics of mechanics. Kinematics of rigid
More informationIntroduction 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 informationMechanical System Elements
Mechanical System Elements Three basic mechanical elements: Spring (elastic) element Damper (frictional) element Mass (inertia) element Translational and rotational versions These are passive (non-energy
More informationLecture 6 mechanical system modeling equivalent mass gears
M2794.25 Mechanical System Analysis 기계시스템해석 lecture 6,7,8 Dongjun Lee ( 이동준 ) Department of Mechanical & Aerospace Engineering Seoul National University Dongjun Lee Lecture 6 mechanical system modeling
More informationTranslational Mechanical Systems
Translational Mechanical Systems Basic (Idealized) Modeling Elements Interconnection Relationships -Physical Laws Derive Equation of Motion (EOM) - SDOF Energy Transfer Series and Parallel Connections
More informationStructural 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 informationEQUIVALENT 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 informationStep 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 informationIn 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 informationIndex. Index. More information. in this web service Cambridge University Press
A-type elements, 4 7, 18, 31, 168, 198, 202, 219, 220, 222, 225 A-type variables. See Across variable ac current, 172, 251 ac induction motor, 251 Acceleration rotational, 30 translational, 16 Accumulator,
More informationDynamic 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 informationDEPARTMENT OF MECHANICAL ENGINEERING Dynamics of Machinery. Submitted
DEPARTMENT OF MECHANICAL ENGINEERING Dynamics of Machinery Submitted 1 UNIT I - Force Analysis INDEX (1) Introduction (2) Newton s Law (3) Types of force Analysis (4) Principle of Super Position (5) Free
More informationCHAPTER 8: ROTATIONAL OF RIGID BODY PHYSICS. 1. Define Torque
7 1. Define Torque 2. State the conditions for equilibrium of rigid body (Hint: 2 conditions) 3. Define angular displacement 4. Define average angular velocity 5. Define instantaneous angular velocity
More informationTorque and Rotation Lecture 7
Torque and Rotation Lecture 7 ˆ In this lecture we finally move beyond a simple particle in our mechanical analysis of motion. ˆ Now we consider the so-called rigid body. Essentially, a particle with extension
More informationRotational Motion. Chapter 10 of Essential University Physics, Richard Wolfson, 3 rd Edition
Rotational Motion Chapter 10 of Essential University Physics, Richard Wolfson, 3 rd Edition 1 We ll look for a way to describe the combined (rotational) motion 2 Angle Measurements θθ ss rr rrrrrrrrrrrrrr
More informationMathematical 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 informationIntroduction to Continuous Systems. Continuous Systems. Strings, Torsional Rods and Beams.
Outline of Continuous Systems. Introduction to Continuous Systems. Continuous Systems. Strings, Torsional Rods and Beams. Vibrations of Flexible Strings. Torsional Vibration of Rods. Bernoulli-Euler Beams.
More information3. Mathematical Modelling
3. Mathematical Modelling 3.1 Modelling principles 3.1.1 Model types 3.1.2 Model construction 3.1.3 Modelling from first principles 3.2 Models for technical systems 3.2.1 Electrical systems 3.2.2 Mechanical
More informationStructural 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 informationModeling of Dynamic Systems: Notes on Bond Graphs Version 1.0 Copyright Diane L. Peters, Ph.D., P.E.
Modeling of Dynamic Systems: Notes on Bond Graphs Version 1.0 Copyright 2015 Diane L. Peters, Ph.D., P.E. Spring 2015 2 Contents 1 Overview of Dynamic Modeling 5 2 Bond Graph Basics 7 2.1 Causality.............................
More informationEXAMPLE: MODELING THE PT326 PROCESS TRAINER
CHAPTER 1 By Radu Muresan University of Guelph Page 1 EXAMPLE: MODELING THE PT326 PROCESS TRAINER The PT326 apparatus models common industrial situations in which temperature control is required in the
More informationMembers Subjected to Torsional Loads
Members Subjected to Torsional Loads Torsion of circular shafts Definition of Torsion: Consider a shaft rigidly clamped at one end and twisted at the other end by a torque T = F.d applied in a plane perpendicular
More informationMechanical System Modeling
Mechanical Engineer Modeling & Simulation Electro- Mechanics Electrical- Electronics Engineer Sensors Actuators Computer Systems Engineer Embedded Control Controls Engineer Mechatronic System Design K.
More information41514 Dynamics of Machinery
41514 Dynamics of Machinery Theory, Experiment, Phenomenology and Industrial Applications Ilmar Ferreira Santos 1. Recapitulation Mathematical Modeling & Steps 2. Example System of Particle 3. Example
More informationVideo 2.1a Vijay Kumar and Ani Hsieh
Video 2.1a Vijay Kumar and Ani Hsieh Robo3x-1.3 1 Introduction to Lagrangian Mechanics Vijay Kumar and Ani Hsieh University of Pennsylvania Robo3x-1.3 2 Analytical Mechanics Aristotle Galileo Bernoulli
More informationLANMARK 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 informationChapter 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 informationT1 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 informationMechatronics. 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 informationfor any object. Note that we use letter, m g, meaning gravitational
Lecture 4. orces, Newton's Second Law Last time we have started our discussion of Newtonian Mechanics and formulated Newton s laws. Today we shall closely look at the statement of the second law and consider
More information2007 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 informationPhysical Pendulum, Torsion Pendulum
[International Campus Lab] Physical Pendulum, Torsion Pendulum Objective Investigate the motions of physical pendulums and torsion pendulums. Theory ----------------------------- Reference --------------------------
More information557. Radial correction controllers of gyroscopic stabilizer
557. Radial correction controllers of gyroscopic stabilizer M. Sivčák 1, J. Škoda, Technical University in Liberec, Studentská, Liberec, Czech Republic e-mail: 1 michal.sivcak@tul.cz; jan.skoda@pevnosti.cz
More informationMechanics II. Which of the following relations among the forces W, k, N, and F must be true?
Mechanics II 1. By applying a force F on a block, a person pulls a block along a rough surface at constant velocity v (see Figure below; directions, but not necessarily magnitudes, are indicated). Which
More informationLecture Module 5: Introduction to Attitude Stabilization and Control
1 Lecture Module 5: Introduction to Attitude Stabilization and Control Lectures 1-3 Stability is referred to as a system s behaviour to external/internal disturbances (small) in/from equilibrium states.
More informationIDENTIFICATION OF FRICTION ENERGY DISSIPATION USING FREE VIBRATION VELOCITY: MEASUREMENT AND MODELING
IDENTIFICATION OF FRICTION ENERGY DISSIPATION USING FREE VIBRATION VELOCITY: MEASUREMENT AND MODELING Christoph A. Kossack, Tony L. Schmitz, and John C. Ziegert Department of Mechanical Engineering and
More information7. FORCE ANALYSIS. Fundamentals F C
ME 352 ORE NLYSIS 7. ORE NLYSIS his chapter discusses some of the methodologies used to perform force analysis on mechanisms. he chapter begins with a review of some fundamentals of force analysis using
More informationContents. Dynamics and control of mechanical systems. Focuses on
Dnamics and control of mechanical sstems Date Da (/8) Da (3/8) Da 3 (5/8) Da 4 (7/8) Da 5 (9/8) Da 6 (/8) Content Review of the basics of mechanics. Kinematics of rigid bodies - coordinate transformation,
More informationThe student will experimentally determine the parameters to represent the behavior of a damped oscillatory system of one degree of freedom.
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
More informationCEE 271: Applied Mechanics II, Dynamics Lecture 25: Ch.17, Sec.4-5
1 / 36 CEE 271: Applied Mechanics II, Dynamics Lecture 25: Ch.17, Sec.4-5 Prof. Albert S. Kim Civil and Environmental Engineering, University of Hawaii at Manoa Date: 2 / 36 EQUATIONS OF MOTION: ROTATION
More information1820. Selection of torsional vibration damper based on the results of simulation
8. Selection of torsional vibration damper based on the results of simulation Tomasz Matyja, Bogusław Łazarz Silesian University of Technology, Faculty of Transport, Gliwice, Poland Corresponding author
More informationChapter 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 informationModeling Mechanical Systems
Modeling Mechanical Systems Mechanical systems can be either translational or rotational. Although the fundamental relationships for both types are derived from Newton s law, they are different enough
More informationTable of Contents. Preface... 13
Table of Contents Preface... 13 Chapter 1. Vibrations of Continuous Elastic Solid Media... 17 1.1. Objective of the chapter... 17 1.2. Equations of motion and boundary conditions of continuous media...
More informationCE 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 informationRead textbook CHAPTER 1.4, Apps B&D
Lecture 2 Read textbook CHAPTER 1.4, Apps B&D Today: Derive EOMs & Linearization undamental equation of motion for mass-springdamper system (1DO). Linear and nonlinear system. Examples of derivation of
More informationAS3010: Introduction to Space Technology
AS3010: Introduction to Space Technology L E C T U R E 22 Part B, Lecture 22 19 April, 2017 C O N T E N T S Attitude stabilization passive and active. Actuators for three axis or active stabilization.
More informationGame Physics. Game and Media Technology Master Program - Utrecht University. Dr. Nicolas Pronost
Game and Media Technology Master Program - Utrecht University Dr. Nicolas Pronost Essential physics for game developers Introduction The primary issues Let s move virtual objects Kinematics: description
More informationTextbook Reference: Wilson, Buffa, Lou: Chapter 8 Glencoe Physics: Chapter 8
AP Physics Rotational Motion Introduction: Which moves with greater speed on a merry-go-round - a horse near the center or one near the outside? Your answer probably depends on whether you are considering
More informationEngineering 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 informationSimple Harmonic Motion
1. Object Simple Harmonic Motion 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 informationReview: control, feedback, etc. Today s topic: state-space models of systems; linearization
Plan of the Lecture Review: control, feedback, etc Today s topic: state-space models of systems; linearization Goal: a general framework that encompasses all examples of interest Once we have mastered
More informationMET 327 APPLIED ENGINEERING II (DYNAMICS) 1-D Dynamic System Equation of Motion (EOM)
Handout #1 by Hejie Lin MET 327 APPLIED ENGINEERING II (DYNAMICS) 1. Introduction to Statics and Dynamics 1.1 Statics vs. Dynamics 1 Ch 9 Moment of Inertia A dynamic system is characterized with mass (M),
More informationUnit 7: Oscillations
Text: Chapter 15 Unit 7: Oscillations NAME: Problems (p. 405-412) #1: 1, 7, 13, 17, 24, 26, 28, 32, 35 (simple harmonic motion, springs) #2: 45, 46, 49, 51, 75 (pendulums) Vocabulary: simple harmonic motion,
More informationAA 242B/ ME 242B: Mechanical Vibrations (Spring 2016)
AA 242B/ ME 242B: Mechanical Vibrations (Spring 2016) Homework #2 Due April 17, 2016 This homework focuses on developing a simplified analytical model of the longitudinal dynamics of an aircraft during
More informationUnit WorkBook 4 Level 4 ENG U8 Mechanical Principles 2018 UniCourse Ltd. All Rights Reserved. Sample
2018 UniCourse Ltd. A Rights Reserved. Pearson BTEC Levels 4 Higher Nationals in Engineering (RQF) Unit 8: Mechanical Principles Unit Workbook 4 in a series of 4 for this unit Learning Outcome 4 Translational
More informationAutomatic Control Systems. -Lecture Note 15-
-Lecture Note 15- Modeling of Physical Systems 5 1/52 AC Motors AC Motors Classification i) Induction Motor (Asynchronous Motor) ii) Synchronous Motor 2/52 Advantages of AC Motors i) Cost-effective ii)
More informationSTATICS & DYNAMICS. Engineering Mechanics. Gary L. Gray. Francesco Costanzo. Michael E. Plesha. University of Wisconsin-Madison
Engineering Mechanics STATICS & DYNAMICS SECOND EDITION Francesco Costanzo Department of Engineering Science and Mechanics Penn State University Michael E. Plesha Department of Engineering Physics University
More informationPhysical Dynamics (PHY-304)
Physical Dynamics (PHY-304) Gabriele Travaglini March 31, 2012 1 Review of Newtonian Mechanics 1.1 One particle Lectures 1-2. Frame, velocity, acceleration, number of degrees of freedom, generalised coordinates.
More informationINSTITUTE OF AERONAUTICAL ENGINEERING (Autonomous) Dundigal, Hyderabad
INSTITUTE OF AERONAUTICAL ENGINEERING (Autonomous) Dundigal, Hyderabad - 500 043 AERONAUTICAL ENGINEERING DEFINITIONS AND TERMINOLOGY Course Name : ENGINEERING MECHANICS Course Code : AAEB01 Program :
More information1.053J/2.003J Dynamics and Control I Fall Final Exam 18 th December, 2007
1.053J/2.003J Dynamics and Control I Fall 2007 Final Exam 18 th December, 2007 Important Notes: 1. You are allowed to use three letter-size sheets (two-sides each) of notes. 2. There are five (5) problems
More informationChapter 22 : Electric potential
Chapter 22 : Electric potential What is electric potential? How does it relate to potential energy? How does it relate to electric field? Some simple applications What does it mean when it says 1.5 Volts
More informationDynamics 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 informationVideo 3.1 Vijay Kumar and Ani Hsieh
Video 3.1 Vijay Kumar and Ani Hsieh Robo3x-1.3 1 Dynamics of Robot Arms Vijay Kumar and Ani Hsieh University of Pennsylvania Robo3x-1.3 2 Lagrange s Equation of Motion Lagrangian Kinetic Energy Potential
More informationUNIT-I (FORCE ANALYSIS)
DHANALAKSHMI SRINIVASAN INSTITUTE OF RESEACH AND TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING QUESTION BANK ME2302 DYNAMICS OF MACHINERY III YEAR/ V SEMESTER UNIT-I (FORCE ANALYSIS) PART-A (2 marks)
More informationEngineering 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 informationAP PHYSICS 1 Learning Objectives Arranged Topically
AP PHYSICS 1 Learning Objectives Arranged Topically with o Big Ideas o Enduring Understandings o Essential Knowledges o Learning Objectives o Science Practices o Correlation to Knight Textbook Chapters
More informationRoad Vehicle Dynamics
Road Vehicle Dynamics Table of Contents: Foreword Preface Chapter 1 Introduction 1.1 General 1.2 Vehicle System Classification 1.3 Dynamic System 1.4 Classification of Dynamic System Models 1.5 Constraints,
More informationPhysics for Scientists and Engineers 4th Edition, 2017
A Correlation of Physics for Scientists and Engineers 4th Edition, 2017 To the AP Physics C: Mechanics Course Descriptions AP is a trademark registered and/or owned by the College Board, which was not
More information7. 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 informationAP Physics C: Rotation II. (Torque and Rotational Dynamics, Rolling Motion) Problems
AP Physics C: Rotation II (Torque and Rotational Dynamics, Rolling Motion) Problems 1980M3. A billiard ball has mass M, radius R, and moment of inertia about the center of mass I c = 2 MR²/5 The ball is
More informationThe Modeling of Single-dof Mechanical Systems
The Modeling of Single-dof Mechanical Systems Lagrange equation for a single-dof system: where: q: is the generalized coordinate; T: is the total kinetic energy of the system; V: is the total potential
More informationAP Physics C Mechanics Objectives
AP Physics C Mechanics Objectives I. KINEMATICS A. Motion in One Dimension 1. The relationships among position, velocity and acceleration a. Given a graph of position vs. time, identify or sketch a graph
More informationCEE 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 information8.012 Physics I: Classical Mechanics Fall 2008
IT OpenCourseWare http://ocw.mit.edu 8.012 Physics I: Classical echanics Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. ASSACHUSETTS INSTITUTE
More information1 2 Models, Theories, and Laws 1.5 Distinguish between models, theories, and laws 2.1 State the origin of significant figures in measurement
Textbook Correlation Textbook Correlation Physics 1115/2015 Chapter 1 Introduction, Measurement, Estimating 1.1 Describe thoughts of Aristotle vs. Galileo in describing motion 1 1 Nature of Science 1.2
More informationQuantitative Skills in AP Physics 1
This chapter focuses on some of the quantitative skills that are important in your AP Physics 1 course. These are not all of the skills that you will learn, practice, and apply during the year, but these
More information202 Index. failure, 26 field equation, 122 force, 1
Index acceleration, 12, 161 admissible function, 155 admissible stress, 32 Airy's stress function, 122, 124 d'alembert's principle, 165, 167, 177 amplitude, 171 analogy, 76 anisotropic material, 20 aperiodic
More information8. More about calculus in physics
8. More about calculus in physics This section is about physical quantities that change with time or change when a different quantity changes. Calculus is about the mathematics of rates of change (differentiation)
More informationChapter 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 informationCHAPTER 3 QUARTER AIRCRAFT MODELING
30 CHAPTER 3 QUARTER AIRCRAFT MODELING 3.1 GENERAL In this chapter, the quarter aircraft model is developed and the dynamic equations are derived. The quarter aircraft model is two degrees of freedom model
More informationMechanical Principles
Unit 8: Unit code Mechanical Principles F/615/1482 Unit level 4 Credit value 15 Introduction Mechanical principles have been crucial for engineers to convert the energy produced by burning oil and gas
More informationChapter 9 TORQUE & Rotational Kinematics
Chapter 9 TORQUE & Rotational Kinematics This motionless person is in static equilibrium. The forces acting on him add up to zero. Both forces are vertical in this case. This car is in dynamic equilibrium
More informationLecture 19. Measurement of Solid-Mechanical Quantities (Chapter 8) Measuring Strain Measuring Displacement Measuring Linear Velocity
MECH 373 Instrumentation and Measurements Lecture 19 Measurement of Solid-Mechanical Quantities (Chapter 8) Measuring Strain Measuring Displacement Measuring Linear Velocity Measuring Accepleration and
More information28. Pendulum phase portrait Draw the phase portrait for the pendulum (supported by an inextensible rod)
28. Pendulum phase portrait Draw the phase portrait for the pendulum (supported by an inextensible rod) θ + ω 2 sin θ = 0. Indicate the stable equilibrium points as well as the unstable equilibrium points.
More informationPeriodic 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 informationWebreview Torque and Rotation Practice Test
Please do not write on test. ID A Webreview - 8.2 Torque and Rotation Practice Test Multiple Choice Identify the choice that best completes the statement or answers the question. 1. A 0.30-m-radius automobile
More informationChapter 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 informationTranslational Motion Rotational Motion Equations Sheet
PHYSICS 01 Translational Motion Rotational Motion Equations Sheet LINEAR ANGULAR Time t t Displacement x; (x = rθ) θ Velocity v = Δx/Δt; (v = rω) ω = Δθ/Δt Acceleration a = Δv/Δt; (a = rα) α = Δω/Δt (
More informationPLANAR 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 informationUnforced 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 informationModeling and Experimentation: Compound Pendulum
Modeling and Experimentation: Compound Pendulum Prof. R.G. Longoria Department of Mechanical Engineering The University of Texas at Austin Fall 2014 Overview This lab focuses on developing a mathematical
More informationPLANAR KINETIC EQUATIONS OF MOTION (Section 17.2)
PLANAR KINETIC EQUATIONS OF MOTION (Section 17.2) We will limit our study of planar kinetics to rigid bodies that are symmetric with respect to a fixed reference plane. As discussed in Chapter 16, when
More informationEE Homework 3 Due Date: 03 / 30 / Spring 2015
EE 476 - Homework 3 Due Date: 03 / 30 / 2015 Spring 2015 Exercise 1 (10 points). Consider the problem of two pulleys and a mass discussed in class. We solved a version of the problem where the mass was
More information16. Rotational Dynamics
6. Rotational Dynamics A Overview In this unit we will address examples that combine both translational and rotational motion. We will find that we will need both Newton s second law and the rotational
More informationSimple Harmonic Motion
[International Campus Lab] Objective Investigate simple harmonic motion using an oscillating spring and a simple pendulum. Theory ----------------------------- Reference -------------------------- Young
More informationDynamics. describe the relationship between the joint actuator torques and the motion of the structure important role for
Dynamics describe the relationship between the joint actuator torques and the motion of the structure important role for simulation of motion (test control strategies) analysis of manipulator structures
More informationDynamic Systems. Modeling and Analysis. Hung V. Vu. Ramin S. Esfandiari. THE McGRAW-HILL COMPANIES, INC. California State University, Long Beach
Dynamic Systems Modeling and Analysis Hung V. Vu California State University, Long Beach Ramin S. Esfandiari California State University, Long Beach THE McGRAW-HILL COMPANIES, INC. New York St. Louis San
More information