1. Lesson 2 – Vector manipulation

2. Plan In this lesson we shall Discuss grading scheme Discuss homework
Talk about Inkscape – they need to get it
Discuss features of vector manipulation
Components
Unit vectors
Rotational vectors
Right-hand rule
See how vector manipulation applies to machines
Discuss relationships between position, velocity, acceleration

3. Plan
Since homework is due Sunday, we’ll talk about homework problems from Chapter 1 too

4. Homework Want to track your progress as we go along
Homework is important; assigned weekly
Turn in homework every Sunday by noon EDT

5. Grading scheme Homework: 20% of grade
Midterms: 22.5% of grade each = 45%
Class participation: 5% of grade
Exam: 30% of grade

6. Inkscape I think I’ve figured out screen sharing now…let’s see
I prepare many figures in Inkscape
I’ll make these available to you in material for course
Inkscape is a useful tool for analyzing a machine’s kinematics
More precise and modifiable than pencil and paper
Inkscape produces .svg files (Scalable Vector Graphics)
You can write SVG programs yourself that then work in web pages

7. Inkscape
Inkscape is open-source, so you can download and use it for no cost
Inkscape is only 2D
It is not a solid modeler like SolidWorks or Fusion
Many, many problems involving machines are 2D
We want simplicity so that we can focus on the kinematics
We shall make stick figures (see figures in book)

8. Inkscape Analyzing geometry (kinematics) for problem 1-5
We know the length and angle of T,
we know the lengths of S and R, components of T. So we draw T at its length and angle. Then, adding S and R tail-to-tip, we must get T as a result. Since S’s length is known, we draw a circle of radius S from the tip of T. Since R’s length is known, we draw a circle of radius R from the tail of T. Where these circles cross is the tip of S. Note that there are actually two solutions, since the circles cross in two different places.

9. Some notations on Martin’s vector terminology
We use arrows over variable names to indicate vectors
Not boldface type
His funny notation for adding and subtracting vectors is non-standard; we won’t use it:
𝑯 = 𝑨 + 𝑩
𝑰 = 𝑨 − 𝑩
etc.

10. Vector manipulation
Let’s go through a few vector manipulations:

11. Vector manipulation – cross product
Once we study rotation, we shall often be calculating the cross product
𝑣 = 𝜔 × 𝑟
Let’s talk about how we perform this calculation using the right-hand rule

12. How vector manipulation applies to machines
Example, rolling wheel: v C
𝒗 = 𝝎 × 𝒓

13. How vector manipulation applies to machines
Example, rolling wheel:
𝒗 = 𝝎 × 𝒓 v B

14. How vector manipulation applies to machines
Example, rolling wheel:
𝒗 = 𝝎 × 𝒓 v D

15. How vector manipulation applies to machines
Example, rolling wheel:

16. Relationship between x, v, a
We shall do this for rectilinear motion first. We can then extend it to 2D and 3D cases.
Derivative relationships
Definition of velocity:
Definition of acceleration:

17. Relationship between x, v, a
Integral relationships
Start with the definition of acceleration:
You can do the same with velocity:

18. Relationship between x, v, a
See further discussion of this in “Rectilinear motion” article
It turns out that we encounter rectilinear motion often in working with machines:
The machine rotates
The rotation is the rotational version of 1D motion
x becomes q
v becomes w
a becomes a

19. Homework for Chapter 1
Problems due Sunday:

20. End Lesson 2