Multibody and coordinates of slider crank mechanism. • B A C y C x C ¸ • B Slider...

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Transcript of Multibody and coordinates of slider crank mechanism. • B A C y C x C ¸ • B Slider...

  • Multibody Dynamics Spring 2018

    www.mek.lth.se Education/Advanced Courses

    MULTIBODY DYNAMICS (FLERKROPPSDYNAMIK)

    (FMEN02, 7.5p) 1

    http://www.mek.lth.se/

  • Course coordinator

    Prof. Aylin Ahadi, Mechanics, LTH aylin.ahadi@mek.lth.se 046-2223039

    Prof. Emeritus Per Lidstrm

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    mailto:aylin.ahadi@mek.lth.se

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    COURSE REGISTRATION Multibody Dynamics (FMEN02) Spring 2018 pnr efternamn frnamn email

    Abramowitz Ryan ryan.abramowitz.3016@student.lu.se Adibi Pardis pardis.adibi.257@student.lu.se Backstam Alexander alexander.backstam.533@student.lu.se Berggren Jenny jenny.berggren.2632@student.lu.se Berglund Darell Olivia olivia.berglund_darell.362@student.lu.se Dalklint Anna anna.dalklint.079@student.lu.se Eek Hofmann Ludvig ludvig.eek_hofmann.580@student.lu.se Foster John Stewart john_stewart.foster.7656@student.lu.se Hellholm Linnea linnea.hellholm.122@student.lu.se Hodali Cabrera Felipe Javier felipe_javier.hodali_cabrera.8755@student.lu.se Hultgren Viktor viktor.hultgren.839@student.lu.se Iteka Rita rita.iteka.781@student.lu.se Jnsson Gustaf gustaf.jonsson.723@student.lu.se Kingsley Albert albert.kingsley.5612@student.lu.se Lelorieux Virginie virginie.lelorieux.0750@student.lu.se Lindqvist Max max.lindqvist.762@student.lu.se Lindstrm Malin malin.lindstrom.757@student.lu.se Nyberg Johannes johannes.nyberg.429@student.lu.se Ong Samuel samuel.ong.4336@student.lu.se Persson Gustav gustav.persson.961@student.lu.se Sandell Karolina karolina.sandell.182@student.lu.se Sarajrvi Marko marko.sarajarvi.041@student.lu.se Schultheis Robin robin.schultheis.7336@student.lu.se Stenson Dennis dennis.stenson.172@student.lu.se Svanberg Gustaf gustaf.svanberg.412@student.lu.se

  • FUNDAMENTALS OF

    MULTIBODY DYNAMICS

    LECTURE NOTES

    Per Lidstrm Kristina Nilsson

    Division of Mechanics, Lund University

    Multibody system (MSC ADAMS)

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  • Password: euler

    Leonard Euler 1707-1783 5

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    Name Signature

    Abramowitz Ryan Adibi Pardis Backstam Alexander

    Berggren Jenny

    Berglund Darell Olivia

    Dalklint Anna

    Eek Hofmann Ludvig

    Foster John Stewart

    Hellholm Linnea Hodali Cabrera Felipe Javier Hultgren Viktor

    Iteka Rita

    Jnsson Gustaf

    Kingsley Albert

    Lelorieux Virginie

    Lindqvist Max

    Lindstrm Malin

    Nyberg Johannes

    Ong Samuel

    Persson Gustav

    Sandell Karolina

    Sarajrvi Marko

    Schultheis Robin

    Stenson Dennis

    Svanberg Gustaf

    COURSE BOOK RESERVATION (HARD COPY) Fundamentals of Multibody Dynamics, Lecture Notes SEK 350

  • Course program

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    Course literature: Lidstrm P., Nilsson K.: Fundamentals of Multibody Dynamics, Lecture Notes (LN). Div. of Mechanics, LTH, 2017. The Lecture Notes will be on the Course web site. Hand out material: Solutions to selected Exercises, Examination tasks (Assignments), Project specification. Examples of previous written tests. Teacher: Prof. Aylin Ahadi (Lectures, Exercises, Course coordination) Phone: 046-222 3039, email: aylin.ahadi@mek.lth.se Schedule (w. 1-7): Lectures: Monday 13 - 15 Tuesday 13 - 15 Thursday 10 - 12, room M:IP2 Exercises: Wednesday 8 - 10, room M:IP2

    mailto:aylin.ahadi@mek.lth.se

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  • Course objectives and contents

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    Course objectives: The objective of this course is to present the basic theoretical knowledge of the Foundations of Multibody Dynamics with applications to machine and structural dynamics. The course gives a mechanical background for applications in, for instance, control theory and vehicle dynamics. Course contents: 1) Topics presented at lectures and in the Lecture Notes. 2) Exercises. 3) Examination tasks 4) Application project The scope of the course is defined by the curriculum above and the lecture notes (Fundamentals of Multibody Dynamics, Lecture Notes). The teaching consists of Lectures and Exercises: Lectures: Lectures will present the topics of the course in accordance with the curriculum presented above. Exercises: Recommended exercises worked out by the student will serve as a preparation for the Examination tasks. Solutions to selected problems will be distributed.

  • Examination

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  • CONTENTS

    1. INTRODUCTION 1 2. BASIC NOTATIONS 12 3. PARTICLE DYNAMICS 13

    3.1 One particle system 13 3.2 Many particle system 27

    4. RIGID BODY KINEMATICS 36 4.1 The rigid body transplacement 36 4.2 The Euler theorem 47 4.3 Mathematical representations of rotations 59 4.4 Velocity and acceleration 72

    5. THE EQUATIONS OF MOTION 89 5.1 The Newton and Euler equations of motion 89 5.2 Balance laws of momentum and moment of momentum 94 5.3 The relative moment of momentum 99 5.4 Power and energy 100

    6. RIGID BODY DYNAMICS 104 6.1 The equations of motion for the rigid body 104 6.2 The inertia tensor 109 6.3 Fixed axis rotation and bearing reactions 138 6.4 The Euler equations for a rigid body 143 6.5 Power, kinetic energy and stability 158 6.6 Solutions to the equations of motion: An introduction to the case of

    7. THE DEFORMABLE BODY 172 7.1 Kinematics 172 7.2 Equations of motion 196 7.3 Power and energy 205 7.4 The elastic body 209

    8. THE PRINCIPLE OF VIRTUAL POWER 217 8.1 The principle of virtual power in continuum mechanics 219

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  • 1. THE MULTIBODY 222 9.1 Multibody systems 222 9.2 Degrees of freedom and coordinates 224 9.3 Multibody kinematics 229 9.4 Lagranges equations 232 9.5 Constitutive assumptions and generalized internal forces 252

    9.6 External forces 261 9.7 The interaction between parts 267 9.8 The Lagrangian and the Power theorem 295

    10. CONSTRAINTS 300 10.1 Constraint conditions 300 10.2 Lagranges equations with constraint conditions 305 10.3 The Power Theorem 323

    11. COORDINATE REPRESENTATIONS 327 11.1 Coordinate representations 328 11.2 Linear coordinates 331 11.3 Floating frame of reference 359 APPENDICES A1 A.1 Matrices A1 A.2 Vector spaces A6 A.3 Linear mappings A12 A.4 The Euclidean point space A37 A.5 Derivatives of matrices A45 A.6 The Frobenius theorem A54 A.7 Analysis on Euclidean space A56 SOME FORMULAS IN VECTOR AND MATRIX ALGEBRA EXERCISES

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  • Mechanics in perspective

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    Quantum Nano Macro-Continuum Micro-meso -

    nm m mm

    lengthscale

  • Prerequicites

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    Multibody Dynamics

    Linear algebra

    3D Analysis

    Mechanics (Basic course

    Continuum mechanics

    Matlab

    Finite element method

    Mathematics Mechanics Numerical analysis

    Synthesis

  • Content

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    Multibody Dynamics

    Flexible bodies

    Rigid bodies

    Coordinates

    Lagranges equations

    Constraints

    A multibody system is a mechanical system consisting of a number of interconnected components, or parts, performing motions that may involve large translations and rotations as well as small displacements such as vibrations.

    Interconnections between the components are of vital importance. They will introduce constraints on the relative motion between components and in this way limit the possible motions which a multibody system may undertake.

  • Course objectives

    Primary objectives are to obtain a thorough understanding of

    rigid body dynamics Lagrangian techniques for multibody systems

    containing constraint conditions a good understanding of

    the dynamics multibodies consisting of coupled rigid bodies

    some understanding of

    the dynamics of multibodies containing flexible structures 17

  • an ability to

    perform an analysis of a multibody system and to present the result in a written report

    read technical and scientific reports and articles

    on multibody dynamics some knowledge of

    industrial applications of multibody dynamics

    computer softwares for multibody problems

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  • Industrial applications

    Aerospace

    Automotive

    Mechanism Robotics 19

    Multibody system dynamics (MBS dynamics) is motivated by an increasing need for analysis, simulations and assessments of the behaviour of machine systems during the product design process.

  • Commercial MBS softwares

    Adams Dads Simpac Ansys Dymola/Modelinc Simulink/Matlab RecurDyn

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    Adams Student Edition Multibody Dynamics Simulation

    https://www.youtube.com/watch?v=hd-cFu7yMGw

    https://www.youtube.com/watch?v=hd-cFu7yMGw

  • Multibody system

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    Classical steam engine equipped with a slider crank mechanism

    Slider crank mechanism converts reciprocating translational motion of the slider into rotational motion of the crank or the other way around.

  • Multibody system

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    Figure 1.2 Steam engine with slider crank mechanism.

    Slider

    Connecting rod

    Flywheel

    Crank Revolute joint

    Revolute joint

    Fundament

    Translatonal joint

    The crank is connected to the slider via a rod which performs a translational as well as rotational motion. The connection between the slider and the rod is maintained through a so-called revolute joint and a similar arrangement connects the rod and the crank.

  • Multibody system

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    A schematic (topological) picture of the slider crank mechanism. The mechanism consists of three bodies; the slider, the crank and the rod. These are connected by revolute joints which