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Elements of Mechanical Design

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2.72 Elements of Mechanical Design
Spring 2009

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2.72

Elements of 

Mechanical Design

Lecture 05: Structures


Schedule and reading assignment
Quizzes
‰

Quiz – None

Topics
‰
‰

Finish fatigue
Finish HTMs in structures

Reading assignment
‰
‰

None
Quiz next time on HTMs

© Martin Culpepper, All rights reserved

2

Matrix Review


What is a Matrix?
A matrix is an easy way to
represent a system of linear
equations
Linear algebra is the set of
rules that governs matrix
and vector operations

⎡ b1 ⎤
⎢b ⎥
⎣ 2⎦
“Vector”

⎡ a1
⎢a
⎣ 3

a2
⎤
⎥
a4
⎦

“Matrix”

© Martin Culpepper, All rights reserved

4

3D HTMs
For x-axis
rotation

For y-axis
rotation

For z-axis
rotation

0
⎡1
⎢0 cos θ
x
⎢
⎢0 − sin θ x
⎢
0
⎣0
⎡cos θ y
⎢ 0
⎢
⎢ sin θ y
⎢
⎣ 0

⎡ cos θ z
⎢− sin θ
z
⎢
⎢ 0
⎢
⎣ 0

© Martin Culpepper, All rights reserved

0
sin θ x
cos θ x
0

0 − sin θ y
1

0

0
0

cos θ y
0

sin θ z
cos θ z
0
0

X
⎤
Y ⎥⎥
Z
⎥
⎥
1⎦
X⎤
Y ⎥⎥
Z⎥
⎥
1⎦

For small Ө:
cos(Ө)~1 & sin(Ө)~ Ө

0 X⎤
0 Y ⎥⎥
1 Z⎥
⎥
0 1⎦
42

HTM Rotation
•Remember order of multiplication matters:


180
deg
flip


90
deg
right

90
deg
right

180
deg
flip

•To combine a translation and rotation, again multiply the HTM matrices
together
•Note that the order of the rotation and translation matrices does matter, so
makes sure the answer makes sense!!!
© Martin Culpepper, All rights reserved

43

...
MIT OpenCourseWare
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2.72 Elements of Mechanical Design
Spring 2009
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms
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