ME451 Kinematics and Dynamics

of Machine Systems

IntroductionSeptember 6, 2011

Dan NegrutUniversity of Wisconsin, Madison

© Dan Negrut, 2011ME451, UW-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)

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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)

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/

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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 negrut@engr.wisc.edu 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 (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

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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

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)

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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% Final Project 10%

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

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@UW8

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.htm

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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

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

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Scores and Grades

Score Grade94-100 A

87-93 AB

80-86 B

73-79 BC

66-72 C

55-65 D

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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

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

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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 the

learning process

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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

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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.)

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…)

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Putting it all together…

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MECHANICAL SYSTEM =

BODIES + JOINTS + FORCES

THE SYSTEM CHANGES ITS CONFIGURATION IN TIME

WE WANT TO BE ABLE TO PREDICT & CHANGE/CONTROL

HOW SYSTEM EVOLVES

Examples, Multibody Dynamics

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Vehicle Suspension

Vehicle Simulation

Examples, Multibody Dynamics

Examples, Multibody Dynamics

Examples, Multibody Dynamics

Examples of Mechanisms

Examples below are considered 2D

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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

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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

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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 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)

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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…

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Windshield wiper mechanismRobotic 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…

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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 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

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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

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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

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Haug’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 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

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

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ME451 Kinematics and Dynamics

of Machine Systems

Review 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

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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

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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||

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

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O

P

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

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Sum of two vectors (definition) Obtained by the parallelogram rule

Operation is commutative

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

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G. Vector Operation: Angle Between Two G. Vectors

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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

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:

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( ) ( )+ + = + +a b c a b cr rr r r r

· · ·( )+ = +a b c ab acr rr r r rr

· · ·( )k k k+ = +a b a br rr r

· ·( )a b a b+ = +b b br r r

[AO]

Exercise, P3:

Prove that inner product is distributive with respect to sum:

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· · ·( )+ = +a b c ab acr rr r r rr

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

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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

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):

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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

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)

Geometric Vectors: Loose Ends

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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

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( , )x y x ya a a a+ Û=a i j ar rr r

a

xx y

y

aa a

a

é ùê ú= + = ê úê úë û

Ûa i j ar rr

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

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1 0

0 1

éù éùêú êúêú êúêú êúëû ëû

jirr

a 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

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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?

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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:

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Important Relation

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

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Also called a change of base.

Example 1

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y’ x’ θ

O’

E

B

L

XO

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

Example 2

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L

y’

x’

θ

O’

P

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

Do the same for the geometric vector