lecture 0906
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ME451Kinematics and Dynamics
of Machine Systems
September 6, 2011
Dan NegrutUniversity of Wisconsin, Madison
Dan Negrut, 2011ME451, UW-Madison
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Overview, Todays 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)
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Instructor: Dan Negrut
Bucharest Polytechnic Institute, Romania B.S. Aerospace Engineering (1992)
The University of Iowa Ph.D. Mechanical Engineering (1998)
. 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/
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Good to know
Time 11:00 12:15 PM [ Tu, Th ]
Room 1152ME
Office 2035ME
Phone 608 890-0914
E-Mail [email protected]
Course Webpage:
. .
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 ([email protected])
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
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Text
Edward J. Haug: Computer Aided Kinematics and
Dynamics of Mechanical Systems: Basic Methods (1989)
Allyn and Bacon series in Engineering
Author provided PDF copy of the book, available fordownload at course website
On a couple of occasions, the material in the bookwill be supplemented with notes
Well cover Chapters 1 through 6 (a bit of 7 too)
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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
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Grading
Homework 40%
Exam 1 15% Exam 2 15%
Final Exam 20%
Total 100%
NOTE: Score related questions (homework/exams) must be raised prior to next
class after the homework/exam is returned.
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Homework & Final Project Im 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
Well probably end up with 11 assignments
There will be a Final Project, youll choose one of two options: ADAMS option: youll choose the project topic, I decide if its 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
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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):
http://sbel.wisc.edu/Courses/ME451/2010/SimEngine2D/index.htm9
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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
ey ave a e- ome componen s re a e o s m ng ne
Final Exam
Saturday, Dec. 17, at 2:45 PM Comprehensive
Room: 1255ME (computer room)
Itll require you to use your simEngine2D to solve a simple problem
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Scores and Grades
Score Grade
94-100 A Grading will not be done on a curve
87-93 AB80-86 B
73-79 BC
66-72 C
55-65 D
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Final score will be rounded to thenearest integer prior to having aletter assigned
Example: 86.59 becomes AB
86.47 becomes B
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MATLAB and Simulink
MATLAB will be used extensively for HW Itll 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
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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 thelearning process
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Goals of ME451
Goals of the class
Given a general mechanical system, understand how to generate in asystematic and general fashion the equations that govern the time evolutionof 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 Well 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 dynamicsassociated with complex mechanical systems Well used the commercial package ADAMS, available at CAE
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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.)
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 Newtons second law
Caveat: there are additionalconditions (constraints) that need to be satisfies by thetime evolution of these bodies, and these constraints come from the joints thatconnect the bodies (to be covered in detail later)
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Putting it all together
MECHANICAL SYSTEM
=BODIES + JOINTS + FORCES
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THE SYSTEM CHANGES ITSCONFIGURATION IN TIME
WE WANT TO BE ABLE TOPREDICT & CHANGE/CONTROL
HOW SYSTEM EVOLVES
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Examples, Multibody Dynamics
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Vehicle Suspension
Vehicle Simulation
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Examples, Multibody Dynamics
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Examples, Multibody Dynamics
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Examples, Multibody Dynamics
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Examples of Mechanisms
Examples below are considered 2D
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Windshield wiper mechanism
Quick-return shaper mechanism
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Nomenclature
Mechanical System, definition: A collection of interconnected rigid bodies that can move relative to
one another, consistent with mechanical joints that limit relativemotions of pairs of bodies
Why type of analysis can one speak of in conjunction with amechanical system?
Kinematics analysis
Dynamics analysis Inverse Dynamics analysis
Equilibrium analysis
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Kinematics Analysis
Concerns the motion of thesystem independent of theforces that produce the motion
Typically, the time history ofone bod in the s stem is
prescribed
We are interested in how therest of the bodies in thesystem move
Requires the solution linearand nonlinear systems ofequations
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Windshield wiper mechanism
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Dynamics Analysis
Concerns the motion of the systemthat is due to the action of applied
forces/torques
Typically, a set of forces acting onthe system is provided. Motions
bodies
We are interested in how eachbody in the mechanism moves
Requires the solution of acombined system of differentialand algebraic equations (DAEs)
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Cross Section of Engine
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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 desirablemotion of the mechanism
Your bread and butter in Controls
25Windshield wiper mechanism Robotic Manipulator
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What is the Slant of This Course?
When it comes to dynamics, there are several ways to approach the solution of theproblem, 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 doesnt 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 its general, and remember, its 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 amosquito with a cannon
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Modeling & Simulation
Computer modeling and simulation: what does it mean?
The state of a system (in physics, economics, biology, etc.) changes due to aset of inputs
Write a set of equations that capture how the universal law[s] apply to the*specific* problem youre 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
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More on the Computational Perspective
Everything that we do in ME451 is governed by Newtons Second Law
We pose the problem so that it is suited for being solved using a computer
en y n a s mp e an genera way e a a a s nee e o ormu a e e
equations of motion
B) Automatically solve the set of nonlinear equations of motion usingappropriate numerical solution algorithms: Newton Raphson, NewmarkNumerical Integration Method, etc.
C) Consider providing some means for post-processing required for analysis ofresults. Usually it boils down to having a GUI that enables one to plot resultsand animate the mechanism
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Overview of the Class[Chapter numbers according to Haugs book]
Chapter 1 general considerations regarding the scope and goal of Kinematics and Dynamics (witha 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 usedo represen o n s n mec an ca sys ems
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 problemformulated 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 DAEsobtained in Chapter 6
29Haugs book is available online at the class website
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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 solution engine (ADAMS/Solver) for people to be able to find the timeevolution 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
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End: Cha ter 1 Introduction
Begin: Review of Linear Algebra
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ME451Kinematics and Dynamics
of Machine Systems
2.1 through 2.4
Th, Sept. 08
Dan Negrut, 2011ME451, UW-Madison
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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
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Geometric Entities: Their Relevance
Kinematics & Dynamics of systems of rigid bodies:
Requires the ability to describe the position, velocity, and accelerationof each rigid body in the system as functions of time
In the Euclidian 2D space, geometric vectors and 2X2 orthonormalmatrices are extensively used to this end
Geometric vectors - used to locate points on a body or thecenter of mass of a rigid body
2X2 orthonormal matrices - used to describe the orientation of a body
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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||
O
P
o e a a geome r c vec ors are e ne n re a on o an or g n
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
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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
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G. Vector Operation :
Scaling by
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G. Vector Operation:
Addition of Two G. Vectors
Sum of two vectors (definition)
Obtained by the parallelogram rule
Operation is commutative
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Easy to visualize, pretty messy tosummarize in an analytical fashion:
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G. Vector Operation:
Inner Product of Two G. Vectors
The product between the magnitude of the first geometric vector andthe projection of the second vector onto the first vector
Note that operation is commutative
Dont call this the dot product of the two vectors
This name is saved for algebraic vectors
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G. Vector Operation:
Angle Between Two G. Vectors
Regarding the angle between two vectors, note that
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Important: Angles are positive counterclockwise
This is why when measuring the angle between two vectors itsimportant which one is the first (start) vector
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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
P3 The inner product is distributive with respect to sum:
P4:
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( )+ = +a b c a b a c
( ) + = +b b b
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[AO]
Exercise, P3:
Prove that inner product is distributive with respect to sum:
( )+ = +a b c a b a c
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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 aReference Frame (RF) in which well express all our vectors
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Basis (Unit Coordinate) Vectors
Basis (Unit Coordinate) Vectors: a set of unit vectors used to expressall 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
xand y :
Nomenclature: and are called the Cartesian components of the vector
Were going to exclusively work with right hand mutually orthogonal RFs
44x
y
O
x
y
O
~j
~i
~j
~i
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Geometric Vectors: Operations
Recall the distributive property of the dot product
Based on the relation above, the following holds (expression for inner
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Used to prove identity above (recall angle between basis vectors is /2):
Also, its easy to see that the projections ax and ay on the two axes are
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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)
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New Concept: Algebraic Vectors
Given a RF, each vector can be represented by a triplet
( , )x y x y
a a a a + =a i j a
It doesnt 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 talkingabout an algebraic vector
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x
x y
y
aa a
a
= + =
a i j a
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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
Step 4: We generated an algebraic vector whose two entries areprovided by the pair above
This vector is the algebraic representation of the geometric vector
Note that the algebraic representations of the basis vectors are
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1 0
0 1
ji
Fundamental Question:
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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
Adding two G. Vectors , Adding the corresponding two A. Vectors
Inner product of two G. Vectors , Dot Product of the two A. Vectors
Well talk about outer product later
Measure the angle between two G. Vectors ! uses inner product, so itis based on the dot product of the corresponding A. Vectors
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Algebraic Vector
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Algebraic Vector
andReference 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?
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Algebraic Vector
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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
coordinates by the rotation matrixA
:
NOTE 1: what is changed is the RF used for representing thevector, and not the underlying geometric vector
NOTE 2: rotation matrix A is sometimes called orientation matrix51
Th R t ti M t i A
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The Rotation Matrix A
Very important observation ! the matrix A is orthonormal:
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I t t R l ti
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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
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Example 1
y x
B Express the geometric vector
in the local reference frameOXY.
Express the same geometricvector in the global reference
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O
E
L
XO
Do the same for the geometricvector
Example 2
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Example 2
L
G
O
Y
X
Express the geometric vector
in the local reference frame OXY. Express the same geometric
vector in the global reference
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y
x
O
P
Do the same for the geometricvector