1. ME451 Kinematics and Dynamics of Machine Systems Review of Linear Algebra 2.1 through 2.4 Th, Jan. 22 © Dan Negrut, 2009 ME451, UW-Madison

2. 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: 2.1.2, 2.1.3 Due in one week 2

3. Basis (Unit Coordinate) Vectors Basis (Unit Coordinate) Vectors: a set of unit vectors used to express all other vectors In this class, to simplify our life, we use a set of two orthogonal unit vectors 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 Notation convention: throughout this class, vectors/matrices are in bold font, scalars are not (most often they are in italics) 3

4. Geometric Vectors: Operations Dot product of two vectors, definition: The dot-product of two vectors is commutative 4 Since the angle between coordinate unit vectors is  /2: Therefore, the projection a x and a y on the two axes are Regarding the angle between two vectors, note that Important: Angles are positive counterclockwise

5. Geometric Vectors: Operations The distributive property of the dot product (take it at face value) If you buy the relation above (and you should), the following holds: Given a vector, the orthogonal vector is obtained as 5

6. Geometric Vectors: Moving away… Geometric vectors, concluding remarks: Easy to visualize, but cumbersome to work with The major drawback: hard to manipulate, especially when it comes to computers, which are good at storing matrices and vectors… …and this motivates the algebraic representation of a vector, which is now married to a Reference Frame Reference Frame (RF): as soon as you define the coordinate unit vectors and, you defined a RF We’re going to exclusively work with right hand mutually orthogonal RFs 6

7. Algebraic representation of vectors It all hinges upon the following representation: Then we can write this: In the algebraic notation, drop the arrow… 7

8. What can you do with algebraic vectors? Everything that you could with the geometric vectors This is just a different notation (representation) of the same concept Scale them Add them (this is very simple now) Multiply two of them Inner product (leads to a number) Measure the angle  between two of them 8

9. 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? 9

10. 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” 10

11. The Rotation Matrix A Very important observation: the matrix A is orthogonal: 11

12. Important Relation Expressing a given vector in one reference frame (local) in a different reference frame (global) 12 Also called a change of base.

13. Example 1 13 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

14. Example 2 14 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

15. Translation + Rotation What we just discussed dealt with the case when you are interested in finding the representing the location of a point P when you change the reference frame, but yet the new and old reference frames share the same origin 15 What if they don’t share the same origin? How would you represent the position of the point P in this new reference frame?

16. Example The location of point O’ in the OXY global RF is [x,y] T. The orientation of the bar is described by the angle  1. Find the location of C and D expressed in the global reference frame as functions of x, y, and  1. 16

17. Example 2.4.3: Slider Crank Based on information provided in figure (b), derive the position vector associated with point P (that is, find position of point P in the global reference frame OXY) 17 O

18. Example (RGC) Use the array q of generalized coordinates to locate the point B in the GRF 18