ME451 Kinematics and Dynamics

of Machine Systems

IntroductionIntroductionSeptember 6, 2011

Dan NegrutUniversity of Wisconsin, Madison

© Dan Negrut, 2011ME451, UW-Madison

Overview, Today’s Lecture…

� Discuss Syllabus

Discuss schedule related issues� Discuss schedule related issues

� Quick overview of ME451 is going to be about

� Start a review of linear algebra (vectors and matrices)

2

Instructor: Dan Negrut

� Bucharest Polytechnic Institute, Romania� B.S. – Aerospace Engineering (1992)

� The University of Iowa� Ph.D. – Mechanical Engineering (1998)

� MSC.Software � MSC.Software � Product Development Engineer 1998-2004

� The University of Michigan� Adjunct Assistant Professor, Dept. of Mathematics (2004)

� Division of Mathematics and Computer Science, Argonne National Laboratory� Visiting Scientist (2005, 2006)

� The University of Wisconsin-Madison, Joined in Nov. 2005� Research Focus: Computational Dynamics

� Leading the Simulation-Based Engineering Lab - http://sbel.wisc.edu/

3

Good to know…

� Time 11:00 – 12:15 PM [ Tu, Th ]

� Room 1152ME

� Office 2035ME

� Phone 608 890-0914

� E-Mail negrut@engr.wisc.edu

� Course Webpage:

� https://learnuw.wisc.edu – solution to HW problems and grades� https://learnuw.wisc.edu – solution to HW problems and grades

� http://sbel.wisc.edu/Courses/ME451/2011/index.htm - for slides, audio files, examples covered in class, etc.

� Forum Page:

� http://sbel.wisc.edu/Forum/

� Teaching Assistant: Toby Heyn (heyn@wisc.edu)

� Office Hours:

� Monday 2 – 3:30 PM

� Wednesday 2 – 3:30 PM

� Stop by my office anytime in the PM if you have quick ME451 questions

4

Text

� Edward J. Haug: Computer Aided Kinematics and Dynamics of Mechanical Systems: Basic Methods (1989)

� Allyn and Bacon series in Engineering

� Book is out of print� Book is out of print

� Author provided PDF copy of the book, available for download at course website

� On a couple of occasions, the material in the book will be supplemented with notes

� We’ll cover Chapters 1 through 6 (a bit of 7 too)

5

Information Dissemination

� Handouts will be printed out and provided before each lecture

� PPT slides for each lecture made available online at lab website� I intend to also provide MP3 audio files

� Homework solutions will be posted at Learn@UW

� Grades will be maintained online at Learn@UW

� Syllabus available at lab website� Updated as we go, will change to reflect progress made in covering material

� Topics we cover

� Homework assignments and due dates

� Exam dates

6

Grading

� Homework 40%

� Exam 1 15%

� Exam 2 15%

� Final Exam 20%

� Final Project 10%� Final Project 10%

� Total 100%

NOTE:• Score related questions (homework/exams) must be raised prior to next

class after the homework/exam is returned.

7

Homework & Final Project

� I’m planning for weekly homework assignments

� Assigned at the end of each class

� Typically due one week later at beginning of class, unless stated otherwise

� No late homework accepted

� We’ll probably end up with 11 assignments

� There will be a Final Project, you’ll choose one of two options:

� ADAMS option: you’ll choose the project topic, I decide if it’s good enough

� MATLAB option: you implement a dynamics engine, simEngine2D

� HW Grading Approach

� 50% - One random problem graded thoroughly

� 50% - For completing the other problems

� Solutions will be posted on at Learn@UW

8

A Word on simEngine2D

� A code that you put together and by the end of the semester should be capable of running basic 2D Kinematics and Dynamics analysis

� Each assignment will add a little bit to the core functionality of the simulation engine

� You will:Setup a procedure to input (describe) your model� Setup a procedure to input (describe) your model

� Example Model: 2D model of truck, wrecker boom, etc.

� Implement a numerical solution sequence

� Example: use Newton-Raphson to determine the position of your system as a function of time

� Plot results of interest

� Example: plot of reaction forces, of peak acceleration, etc.

� Link to past simEngine2D (from Fall 2010):

� http://sbel.wisc.edu/Courses/ME451/2010/SimEngine2D/index.htm

9

Exams

� Two midterm exams, as indicated in syllabus� Tuesday, 11/03

� Thursday, 12/01

� Review sessions in 1152ME at 7:15PM the evening before the exam

� They’ll have take-home components related to simEngine2D� They’ll have take-home components related to simEngine2D

� Final Exam

� Saturday, Dec. 17, at 2:45 PM

� Comprehensive

� Room: 1255ME (computer room)

� It’ll require you to use your simEngine2D to solve a simple problem

10

Scores and Grades

Score Grade

94-100 A � Grading will not be done on a curve

87-93 AB

80-86 B

73-79 BC

66-72 C

55-65 D

11

� Final score will be rounded to the

nearest integer prior to having a letter assigned

� Example:� 86.59 becomes AB

� 86.47 becomes B

MATLAB and Simulink

� MATLAB will be used extensively for HW� It’ll be the vehicle used to formulate and solve the equations

governing the time evolution of mechanical systems

� You are responsible for brushing up your MATLAB skills

� Simulink might be used for ADAMS co-simulation

� If you feel comfortable with using C or C++ that is also ok

12

Quick Suggestions

� Be active, pay attention, ask questions

� Reading the text is good

� Doing your homework is critical

� Provide feedback� Both during and at end of the semester

� I can change small things that that could make a difference in the learning process

13

Goals of ME451

� Goals of the class

� Given a general mechanical system, understand how to generate in a systematic and general fashion the equations that govern the time evolution of the mechanical system

� These equations are called the equations of motion (EOM)

� Have a basic understanding of the techniques (called numerical methods) used to solve the EOM� We’ll rely on MATLAB to implement/illustrate some of the numerical methods used to

solve EOM

� Be able to use commercial software to simulate and interpret the dynamics associated with complex mechanical systems� We’ll used the commercial package ADAMS, available at CAE

14

Why/How Do Bodies Move?

� Why? � The configuration of a mechanism changes in time based on forces and motions

applied to its components� Forces

� Internal (reaction forces)� External, or applied forces (gravity, compliant forces, etc.)

� Prescribed motion� Prescribed motion� Somebody prescribes the motion of a component of the mechanical system

� Recall Finite Element Analysis, boundary conditions are of two types:� Neumann, when the force is prescribed� Dirichlet, when the displacement is prescribed

� How?� They move in a way that obeys Newton’s second law

� Caveat: there are additional conditions (constraints) that need to be satisfies by the time evolution of these bodies, and these constraints come from the joints that connect the bodies (to be covered in detail later…)

15

Putting it all together…

MECHANICAL SYSTEM =

BODIES + JOINTS + FORCES

16

THE SYSTEM CHANGES ITS CONFIGURATION IN TIME

WE WANT TO BE ABLE TO PREDICT & CHANGE/CONTROL

HOW SYSTEM EVOLVES

Examples, Multibody Dynamics

17

Vehicle Suspension

Vehicle Simulation

Examples, Multibody Dynamics

Examples, Multibody Dynamics

Examples, Multibody Dynamics

Examples of Mechanisms

� Examples below are considered 2D

21

Windshield wiper mechanism

Quick-return shaper mechanism

Nomenclature

� Mechanical System, definition:

� A collection of interconnected rigid bodies that can move relative to one another, consistent with mechanical joints that limit relative motions of pairs of bodies

� Why type of analysis can one speak of in conjunction with a mechanical system?

� Kinematics analysis

� Dynamics analysis

� Inverse Dynamics analysis

� Equilibrium analysis

22

Kinematics Analysis

� Concerns the motion of the system independent of the forces that produce the motion

� Typically, the time history of one body in the system is one body in the system is prescribed

� We are interested in how the rest of the bodies in the system move

� Requires the solution linear and nonlinear systems of equations

23

Windshield wiper mechanism

Dynamics Analysis

� Concerns the motion of the system that is due to the action of applied forces/torques

� Typically, a set of forces acting on the system is provided. Motions can also be specified on some can also be specified on some bodies

� We are interested in how each body in the mechanism moves

� Requires the solution of a combined system of differential and algebraic equations (DAEs)

24

Cross Section of Engine

Inverse Dynamics Analysis

� It is a hybrid between Kinematics and Dynamics

� Basically, one wants to find the set of forces that lead to a certain desirable

motion of the mechanism

� Your bread and butter in Controls…

25Windshield wiper mechanism

Robotic Manipulator

What is the Slant of This Course?

� When it comes to dynamics, there are several ways to approach the solution of the

problem, that is, to find the time evolution of the mechanical system

� The ME240 way, on a case-by-case fashion

� In many circumstances, this required following a recipe, not always clear where it came from

� Typically works for small problems, not clear how to go beyond textbook cases

� Use a graphical approach

� This was the methodology that used to be emphasized in ME451 (Prof. Uicker)

� Intuitive but doesn’t scale particularly well

� Use a computational approach

� This is methodology emphasized in this class

� Leverages the power of the computer

� Relies on a unitary approach to finding the time evolution of any mechanical system� Sometimes the approach might seem to be an overkill, but it’s general, and remember, it’s the computer that does

the work and not you

� In other words, we hit it with a heavy hammer that takes care of all jobs, although at times it seems like killing a mosquito with a cannon…

26

Modeling & Simulation

� Computer modeling and simulation: what does it mean?

� The state of a system (in physics, economics, biology, etc.) changes due to a set of inputs

� Write a set of equations that capture how the universal law[s] apply to the *specific* problem you’re dealing with

� Solve this equation to understand the behavior of the system

� Applies to what we do in ME451 but also to many other disciplines

27

More on the Computational Perspective…

� Everything that we do in ME451 is governed by Newton’s Second Law

� We pose the problem so that it is suited for being solved using a computer

A) Identify in a simple and general way the data that is needed to formulate the � A) Identify in a simple and general way the data that is needed to formulate the equations of motion

� B) Automatically solve the set of nonlinear equations of motion using appropriate numerical solution algorithms: Newton Raphson, Newmark Numerical Integration Method, etc.

� C) Consider providing some means for post-processing required for analysis of results. Usually it boils down to having a GUI that enables one to plot results and animate the mechanism

28

Overview of the Class[Chapter numbers according to Haug’s book]

� Chapter 1 – general considerations regarding the scope and goal of Kinematics and Dynamics (with a computational slant)

� Chapter 2 – review of basic Linear Algebra and Calculus� Linear Algebra: Focus on geometric vectors and matrix-vector operations� Calculus: Focus on taking partial derivatives (a lot of this), handling time derivatives, chain rule (a lot of this too)

� Chapter 3 – introduces the concept of kinematic constraint as the mathematical building block used to represent joints in mechanical systemsto represent joints in mechanical systems� This is the hardest part of the material covered� Basically poses the Kinematics problem

� Chapter 4 – quick discussion of the numerical algorithms used to solve kinematics problem formulated in Chapter 3

� Chapter 5 – applications, will draw on the simulation facilities provided by the commercial package ADAMS� Only tangentially touching it

� Chapter 6 – states the dynamics problem

� Chapter 7 – only tangentially touching it, in order to get an idea of how to solve the set of DAEs obtained in Chapter 6

29Haug’s book is available online at the class website

ADAMS

� Automatic Dynamic Analysis of Mechanical Systems

� It says Dynamics in name, but it does a whole lot more� Kinematics, Statics, Quasi-Statics, etc.

� Philosophy behind software package� Offer a pre-processor (ADAMS/View) for people to be able to generate models� Offer a pre-processor (ADAMS/View) for people to be able to generate models

� Offer a solution engine (ADAMS/Solver) for people to be able to find the time evolution of their models

� Offer a post-processor (ADAMS/PPT) for people to be able to animate and plot results

� It now has a variety of so-called vertical products, which all draw on the ADAMS/Solver, but address applications from a specific field:� ADAMS/Car, ADAMS/Rail, ADAMS/Controls, ADAMS/Linear, ADAMS/Hydraulics,

ADAMS/Flex, ADAMS/Engine, etc.

� I used to work for six years in the ADAMS/Solver group

30

End: Chapter 1 (Introduction)End: Chapter 1 (Introduction)Begin: Review of Linear Algebra

31

ME451 Kinematics and Dynamics

of Machine Systems

Review of Linear AlgebraReview of Linear Algebra2.1 through 2.4

Th, Sept. 08

© Dan Negrut, 2011ME451, UW-Madison

Before we get started…

� Last time:

� Syllabus

� Quick overview of course

� Starting discussion about vectors, their geometric representation

� HW Assigned:

� ADAMS assignment, will be emailed to you today

� Problems: 2.2.5, 2.2.8. 2.2.10

� Due in one week

33

Geometric Entities: Their Relevance

� Kinematics & Dynamics of systems of rigid bodies:

� Requires the ability to describe the position, velocity, and acceleration of each rigid body in the system as functions of time

� In the Euclidian 2D space, geometric vectors and 2X2 orthonormal matrices are extensively used to this end

� Geometric vectors - used to locate points on a body or the center of mass of a rigid body

� 2X2 orthonormal matrices - used to describe the orientation of a body

34

Geometric Vectors

� What is a “Geometric Vector”?

� A quantity that has three attributes:� A support line (given by the blue line)

� A direction along this line (from O to P)

� A magnitude, ||OP||

Note that all geometric vectors are defined in relation to an origin O

O

P

� Note that all geometric vectors are defined in relation to an origin O

� IMPORTANT:

� Geometric vectors are entities that are independent of any reference frame

� ME451 deals planar kinematics and dynamics

� We assume that all the vectors are defined in the 2D Euclidian space

� A basis for the Euclidian space is any collection of two independent vectors35

Geometric Vectors: Operations

� What geometric vectors operations are defined out there?

� Scaling by a scalar ®

� Addition of geometric vectors (the parallelogram rule)Addition of geometric vectors (the parallelogram rule)

� Multiplication of two geometric vectors

� The inner product rule (leads to a number)

� The outer product rule (leads to a vector)

� One can measure the angle θ between two geometric vectors

� A review these definitions follows over the next couple of slides

36

G. Vector Operation : Scaling by ®

37

G. Vector Operation: Addition of Two G. Vectors

� Sum of two vectors (definition)

� Obtained by the parallelogram rule

� Operation is commutative

38

� Easy to visualize, pretty messy to

summarize in an analytical fashion:

G. Vector Operation: Inner Product of Two G. Vectors

� The product between the magnitude of the first geometric vector and the projection of the second vector onto the first vector

� Note that operation is commutative

� Don’t call this the “dot product” of the two vectors

� This name is saved for algebraic vectors

39

G. Vector Operation: Angle Between Two G. Vectors

� Regarding the angle between two vectors, note that

40

� Important: Angles are positive counterclockwise

� This is why when measuring the angle between two vectors it’s important which one is the first (start) vector

Combining Basic G. Vector Operations

� P1 – The sum of geometric vectors is associative

� P2 – Multiplication with a scalar is distributive with respect to the sum:

( ) ( )+ + = + +a b c a b c� �� � � �

· · ·( )k k k+ = +a b a b� �� �

� P3 – The inner product is distributive with respect to sum:

� P4:

41

· · ·( )+ = +a b c a b a c� �� � � � �

· · ·( )k k k+ = +a b a b� �

· ·( )α β α β+ = +b b b� � �

[AO]

Exercise, P3:

� Prove that inner product is distributive with respect to sum:

· · ·( )+ = +a b c a b a c� �� � � � �

42

Geometric Vectors: Reference Frames ! Making Things Simpler

� Geometric vectors:

� Easy to visualize but cumbersome to work with

� The major drawback: hard to manipulate� Was very hard to carry out simple operations (recall proving the distributive

property on previous slide)

� When it comes to computers, which are good at storing matrices and vectors, having to deal with a geometric entity is cumbersome

� We are about to address these drawbacks by first introducing a Reference Frame (RF) in which we’ll express all our vectors

43

Basis (Unit Coordinate) Vectors

� Basis (Unit Coordinate) Vectors: a set of unit vectors used to express all other vectors of the 2D Euclidian space

� In this class, to simplify our life, we use a set of two orthonormal unit vectors� These two vectors, and , define the x and y directions of the RF

A vector a can then be resolved into components and , along the axes � A vector a can then be resolved into components and , along the axes x and y :

� Nomenclature: and are called the Cartesian components of the vector

� We’re going to exclusively work with right hand mutually orthogonal RFs

44x

y

O

x

y

O

~j

~i

~j

~i

Geometric Vectors: Operations

� Recall the distributive property of the dot product

� Based on the relation above, the following holds (expression for inner product when working in a reference frame):product when working in a reference frame):

45

� Used to prove identity above (recall angle between basis vectors is π/2):

� Also, it’s easy to see that the projections ax and ay on the two axes are

� Given a vector , the orthogonal vector is obtained as

Geometric Vectors: Loose Ends

� Length of a vector expressed using Cartesian coordinates:

� Notation used:

� Notation convention: throughout this class, vectors/matrices are in bold font, scalars are not (most often they are in italics)

46

New Concept: Algebraic Vectors

� Given a RF, each vector can be represented by a triplet

( , )x y x ya a a a+ ⇔=a i j a� �� �

�

� It doesn’t take too much imagination to associate to each geometricvector a two-dimensional algebraic vector:

� Note that I dropped the arrow on a to indicate that we are talking about an algebraic vector

47

x

x y

y

aa a

a

= + =

⇔a i j a� ��

Putting Things in Perspective…

� Step 1: We started with geometric vectors

� Step 2: We introduced a reference frame

� Step 3: Relative to that reference frame each geometric vector is uniquely represented as a pair of scalars (the Cartesian coordinates)uniquely represented as a pair of scalars (the Cartesian coordinates)

� Step 4: We generated an algebraic vector whose two entries are provided by the pair above

� This vector is the algebraic representation of the geometric vector

� Note that the algebraic representations of the basis vectors are

48

1 0

0 1

ji��

� �

Fundamental Question:How do G. Vector Ops. Translate into A. Vector Ops.?

� There is a straight correspondence between the operations

� Just a different representation of an old concept

� Scaling a G. Vector , Scaling of corresponding A. Vector� Scaling a G. Vector , Scaling of corresponding A. Vector

� Adding two G. Vectors , Adding the corresponding two A. Vectors

� Inner product of two G. Vectors , Dot Product of the two A. Vectors

� We’ll talk about outer product later

� Measure the angle θ between two G. Vectors ! uses inner product, so it is based on the dot product of the corresponding A. Vectors

49

Algebraic Vector and

Reference Frames

� Recall that an algebraic vector is just a representation of a

geometric vector in a particular reference frame (RF)

� Question: What if I now want to represent the same geometric

vector in a different RF?

50

Algebraic Vector and

Reference Frames

� Representing the same geometric vector in a different RF leads

to the concept of Rotation Matrix A:

� Getting the new coordinates, that is, representation of the samegeometric vector in the new RF is as simple as multiplying the geometric vector in the new RF is as simple as multiplying the coordinates by the rotation matrix A:

� NOTE 1: what is changed is the RF used for representing the

vector, and not the underlying geometric vector

� NOTE 2: rotation matrix A is sometimes called “orientation matrix”51

The Rotation Matrix A

� Very important observation ! the matrix A is orthonormal:

52

Important Relation

� Expressing a given vector in one reference frame (local) in a different reference frame (global)

53Also called a change of base.

Example 1

y’ x’ θ

B

Y

� Express the geometric vector in the local reference frame O’X’Y’.

� Express the same geometric vector in the global reference frame OXY

54

O’

E

L

XO

Y frame OXY

� Do the same for the geometric vector

Example 2

L

θ

G

O

Y

X

� Express the geometric vector

in the local reference frame O’X’Y’.� Express the same geometric

vector in the global reference frame OXY

55

y’

x’

O’

P

frame OXY� Do the same for the geometric

vector