2. Presented By : 140023119089 NARENDRA LUHAR 140023119091 ABHISHEKH MEHATA 140023119092 ABHIJEET PANCHAL

3. APPLY FUNDAMENTAL OF DYNAMIC ANALYSIS TO VARIOUS MECHANICAL SYSTEM Introduction September 6, 2011 Dan Negrut University of Wisconsin, Madison © Dan Negrut, 2011 ME451, UW-Madison

4. 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…) 4

5. Putting it all together… 5 MECHANICAL SYSTEM = BODIES + JOINTS + FORCES THE SYSTEM CHANGES ITS CONFIGURATION IN TIME WE WANT TO BE ABLE TO PREDICT & CHANGE/CONTROL HOW SYSTEM EVOLVES

6. Examples, Multibody Dynamics 6 Vehicle Suspension Vehicle Simulation

7. Examples, Multibody Dynamics

10. Examples of Mechanisms Examples below are considered 2D 10 Windshield wiper mechanism Quick-return shaper mechanism

11. 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 11

12. 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 12 Windshield wiper mechanism

13. 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) 13 Cross Section of Engine

14. 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… 14 Windshield wiper mechanism Robotic Manipulator

15. 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… 15

16. 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 16

17. 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 17

18. 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 18 Haug’s book is available online at the class website

19. 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 19

20. End: Chapter 1 (Introduction) Begin: Review of Linear Algebra 20

21. 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

22. 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 22

23. 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 23

24. 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 24

25. 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 25

26. G. Vector Operation : Scaling by ® 26

27. 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: 27

28. 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 28

29. G. Vector Operation: Angle Between Two G. Vectors 29 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

30. 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: 30

31. [AO] Exercise, P3: Prove that inner product is distributive with respect to sum: 31

32. 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 32

33. 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 33

34. 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): 34 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

35. 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) 35

36. 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 36

37. 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 37

38. 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 38

39. 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? 39

40. 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” 40

41. The Rotation Matrix A Very important observation ! the matrix A is orthonormal: 41

42. Important Relation Expressing a given vector in one reference frame (local) in a different reference frame (global) 42 Also called a change of base.

43. 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 43

44. 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 44