1. ME451 Kinematics and Dynamics of Machine Systems
Introduction
September 6, 2011
© Dan Negrut, 2011 ME451, UW-Madison
Dan Negrut
University of Wisconsin, Madison

2. Overview, Today’s Lecture…
Discuss Syllabus
Discuss schedule related issues
Quick overview of ME451 is going to be about
Start a review of linear algebra (vectors and matrices) 2

3. Instructor: Dan Negrut
Bucharest Polytechnic Institute, Romania
B.S. – Aerospace Engineering (1992)
The University of Iowa
Ph.D. – Mechanical Engineering (1998)
MSC.Software
Product Development Engineer
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 - 3

4. Good to know… 4 Time 11:00 – 12:15 PM [ Tu, Th ] Room 1152ME
Office 2035ME
Phone
Course Webpage:
– solution to HW problems and grades
- for slides, audio files, examples covered in class, etc.
Forum Page:
Teaching Assistant: Toby Heyn
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

5. 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
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

6. 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
Grades will be maintained online at
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

7. Grading Homework 40% Exam 1 15% Exam 2 15% Final Exam 20%
Final Project %
Total 100%
NOTE:
Score related questions (homework/exams) must be raised prior to next class after the homework/exam is returned. 7

8. 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 8

9. 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
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): 9

10. Exams Two midterm exams, as indicated in syllabus Final Exam
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
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

11. Scores and Grades Score Grade 94-100 A 87-93 AB 80-86 B 73-79 BC
Grading will not be done on a curve
Final score will be rounded to the nearest integer prior to having a letter assigned
Example:
86.59 becomes AB
86.47 becomes B 11

12. 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

13. 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

14. 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

15. Why/How Do Bodies Move? Why? How? 15
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
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

16. Putting it all together…
MECHANICAL SYSTEM = BODIES + JOINTS + FORCES
THE SYSTEM CHANGES ITS CONFIGURATION IN TIME
WE WANT TO BE ABLE TO PREDICT & CHANGE/CONTROL HOW SYSTEM EVOLVES 16

17. Examples, Multibody Dynamics
Vehicle Suspension
Vehicle Simulation 17

18. Examples, Multibody Dynamics

19. Examples, Multibody Dynamics

20. Examples, Multibody Dynamics

21. Examples of Mechanisms
Examples below are considered 2D
Windshield wiper mechanism
Quick-return shaper mechanism 21

22. 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

23. 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 prescribed
We are interested in how the rest of the bodies in the system move
Requires the solution linear and nonlinear systems of equations
Windshield wiper mechanism 23

24. 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 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)
Cross Section of Engine 24

25. 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…
Windshield wiper mechanism 25
Robotic Manipulator

26. 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

27. 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

28. 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 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

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

30. 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 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

31. End: Chapter 1 (Introduction) Begin: Review of Linear Algebra31

32. ME451 Kinematics and Dynamics of Machine Systems
Review of Linear Algebra
2.1 through 2.4
Th, Sept. 08
© Dan Negrut, 2011 ME451, UW-Madison
TexPoint fonts used in EMF.
Read the TexPoint manual before you delete this box.: AAAAAAA

33. Before we get started… Last time: HW Assigned: Syllabus
Quick overview of course
Starting discussion about vectors, their geometric representation
HW Assigned:
ADAMS assignment, will be ed to you today
Problems: 2.2.5,
Due in one week 33

34. 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

35. Geometric Vectors What is a “Geometric Vector”? IMPORTANT:
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
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 vectors

36. Geometric Vectors: Operations
What geometric vectors operations are defined out there?
Scaling by a scalar ®
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  between two geometric vectors
A review these definitions follows over the next couple of slides

37. G. Vector Operation : Scaling by ®

38. G. Vector Operation: Addition of Two G. Vectors
Sum of two vectors (definition)
Obtained by the parallelogram rule
Operation is commutative
Easy to visualize, pretty messy to summarize in an analytical fashion:

39. 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

40. G. Vector Operation: Angle Between Two G. Vectors
Regarding the angle between two vectors, note that
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

41. 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:
P3 – The inner product is distributive with respect to sum:
P4:

42. Prove that inner product is distributive with respect to sum:
[AO] Exercise, P3:
Prove that inner product is distributive with respect to sum:

43. Geometric Vectors: Reference Frames ! Making Things Simpler
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

44. 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 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 44

45. 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):
Used to prove identity above (recall angle between basis vectors is /2):
Also, it’s easy to see that the projections ax and ay on the two axes are 45

46. Geometric Vectors: Loose Ends
Given a vector , the orthogonal vector is obtained as
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

47. New Concept: Algebraic Vectors
Given a RF, each vector can be represented by a triplet
It doesn’t take too much imagination to associate to each geometric vector a two-dimensional algebraic vector:
Note that I dropped the arrow on a to indicate that we are talking about an algebraic vector

48. 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)
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

49. Fundamental Question: How do G. Vector Ops. Translate into A
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
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  between two G. Vectors ! uses inner product, so it is based on the dot product of the corresponding A. Vectors 49

50. 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

51. 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 same 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

52. The Rotation Matrix A
Very important observation ! the matrix A is orthonormal: 52

53. Important Relation
Expressing a given vector in one reference frame (local) in a different reference frame (global)
Also called a change of base. 53

54. Example 1
Express the geometric vector in the local reference frame O’X’Y’.
Express the same geometric vector in the global reference frame OXY
Do the same for the geometric vector 54

55. Example 2 Express the geometric vector
in the local reference frame O’X’Y’.
Express the same geometric vector in the global reference frame OXY
Do the same for the geometric vector 55