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Transcript
ME451
Kinematics and Dynamics
of Machine Systems
Introduction
September 6, 2011
© Dan Negrut, 2011
ME451, UW-Madison
Dan Negrut
University of Wisconsin, Madison
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
Instructor: Dan Negrut

Bucharest Polytechnic Institute, Romania


The University of Iowa


Adjunct Assistant Professor, Dept. of Mathematics (2004)
Division of Mathematics and Computer Science, Argonne National Laboratory


Product Development Engineer 1998-2004
The University of Michigan


Ph.D. – Mechanical Engineering (1998)
MSC.Software


B.S. – Aerospace Engineering (1992)
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 [email protected]
Course Webpage:



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 ([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
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

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 [email protected]

Grades will be maintained online at [email protected]

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
Exam 1
Exam 2
Final Exam
Final Project
40%
15%
15%
20%
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





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



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
50% - One random problem graded thoroughly
50% - For completing the other problems
Solutions will be posted on at [email protected]
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


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 Model: 2D model of truck, wrecker boom, etc.
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
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
94-100
87-93
80-86
73-79
66-72
55-65
Grade
A
AB
B
BC
C
D

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
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


Have a basic understanding of the techniques (called numerical methods)
used to solve the EOM


These equations are called the equations of motion (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



Prescribed motion


Somebody prescribes the motion of a component of the mechanical system
Recall Finite Element Analysis, boundary conditions are of two types:



Internal (reaction forces)
External, or applied forces (gravity, compliant forces, etc.)
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
THE SYSTEM CHANGES ITS
CONFIGURATION IN TIME
WE WANT TO BE ABLE TO
PREDICT & CHANGE/CONTROL
HOW SYSTEM EVOLVES
16
Examples, Multibody Dynamics
Vehicle Suspension
Vehicle Simulation
17
Examples, Multibody Dynamics
Examples, Multibody Dynamics
Examples, Multibody Dynamics
Examples of Mechanisms

Examples below are considered 2D
Windshield wiper mechanism
Quick-return shaper mechanism
21
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
prescribed

We are interested in how the
rest of the bodies in the
system move

Windshield wiper mechanism
Requires the solution linear
and nonlinear systems of
equations
23
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
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
Robotic Manipulator
25
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
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 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
ADAMS

Automatic Dynamic Analysis of Mechanical Systems

It says Dynamics in name, but it does a whole lot more


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:


Kinematics, Statics, Quasi-Statics, etc.
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)
Begin: Review of Linear Algebra
31
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
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
center of mass
used to locate points on a body or the
of a rigid body

2X2 orthonormal matrices - used to describe the orientation of a body
34
Geometric Vectors
P

What is a “Geometric Vector”?

A quantity that has three attributes:





O
Note that all geometric vectors are defined in relation to an origin O
IMPORTANT:


A support line (given by the blue line)
A direction along this line (from O to P)
A magnitude, ||OP||
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
35
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
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

Easy to visualize, pretty messy to
summarize in an analytical fashion:
38
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

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
40
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:
r r r
r r r
a + (b + c) = (a + b) + c
r
r r
r
k ·(a + b) = k ·a + k ·b
r r r
rr rr
a·(b + c) = a·b + a·c

P4:
r
r
r
b(a + b ) = a ·b + b ·b
41
[AO]
Exercise, P3:

Prove that inner product is distributive with respect to sum:
r r r
rr rr
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
x and y :

Nomenclature:

We’re going to exclusively work with right hand mutually orthogonal RFs
and
, along the axes
are called the Cartesian components of the vector
y
x
~j
~i
~i
O
and
y
x
O
~j
44
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
Geometric Vectors: Loose Ends

Given a vector

Length of a vector expressed using Cartesian coordinates:

Notation used:

46
, the orthogonal vector
is obtained as
Notation convention: throughout this class, vectors/matrices are in
bold font, scalars are not (most often they are in italics)
New Concept: Algebraic Vectors

Given a RF, each vector can be represented by a triplet
r
r
r
a = a x i + ay j

It doesn’t take too much imagination to associate to each geometric
vector a two-dimensional algebraic vector:
r
r
r
a = a x i + ay j

r
a a (a x , ay )
Û
Û
éa ù
a = êê x ú
ú
êëay ú
û
Note that I dropped the arrow on a to indicate that we are talking
about an algebraic vector
47
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
é1ù
r
i a êê ú
ú
êë0ú
û
é0ù
r
j a êê ú
ú
êë1ú
û
48
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
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 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
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)
Also called a change of base.
53
Example 1
B
x’
y’
2222
5555
O’%
%
%
%
Y
O


θ

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
X
L
E
54
Example 2
Y
G
θ

O
X

L
2
22
2
5
55
5 O’
%
%
%
%
y’

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
x’
P
55