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Two-Dimensional Coordinate Transformations Solved Exercises, Study notes of Computer-Aided Analysis of Machine Dynamics

Dr. Bhim Rao Ambedkar UniversityComputer-Aided Analysis of Machine Dynamics

Main objectives of this course are: 1. Recognize constrained kinematic chains embedded in larger engineering systems 2. Identify forward and inverse dynamic problems 3. Use numerical integration methods and other numerical solution techniques 4. Communicate well using verbal, written and electronic methods. Key points for this lecture are: Two-Dimensional Coordinate Transformations, Haug Notation, Coordinate Transformation, Attitude Angle, Numerical Values, Unit Vectors

Typology: Study notes

2012/2013

Uploaded on 10/02/2013

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ajaey 🇮🇳

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Two-Dimensional Coordinate Transformations – Haug Notation
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Download Two-Dimensional Coordinate Transformations Solved Exercises and more Study notes Computer-Aided Analysis of Machine Dynamics in PDF only on Docsity!

Two-Dimensional Coordinate Transformations – Haug Notation

P i i i

P ri  r A s '

P

2

2

2

2

P

2

y

x

__________ __________

y

x

y

x

__________

__________

__________ __________

__________ __________

b

a

__________

__________

__________

__________ __________

__________ __________

b

a

__________

i i

i i i S C

C S

A

A i     fˆi g ˆi fˆ ^ andg ˆare unit vectors

     fˆ  A fˆ ' gˆ  A g ˆ'

gˆ ' 0

fˆ^ i ' i i  i i i  i i 

P i

1 i

P i

P i i

P s (^) i A s ' s ' A s

  

[A] matrices are orthonormal [A]

  • = [A]

T

● all columns are unit vectors

● all columns are mutually orthogonal

● all rows are unit vectors

● all rows are mutually orthogonal

● det( [A] ) = +

x 1

y 1

x 2 ’

y 2 ’



P

O (^2)

a

b

c

d

Provide numerical values for the three coordinate transformations shown below.

      

____________

____________

s '

__________

s ' 6

r

P 7

P 4

P

       

P 4 4 4

P r 4  r A s '

    (^)  

__________ __________

__________ __________

A

__________

__________

r (^44)

       

P 7 7 7

P r 7  r A s '

    (^)  

__________ __________

__________ __________

A

__________

__________

r (^77)

 (^) ij attitude angle for body j with respect to body i ij jj

A (^) ij attitude matrix for body j with respect to body i

  ^   (^) j

T i ij ij

ij ij ij A A sin cos

cos sin A (^)  

  (^)  

__________ __________

__________ __________

A 47

check using MATLAB

x 1

y 1

Ground

15 deg

x 7 ’

y 7 ’

Body 7

45 deg

y 4 ’ (^) x 4 ’

Body 4

units = cm

P

1.4142 cm

1.4142 cm

3 cm

4 cm

7 cm

2 cm

C(45°)

S(45°)

- S(45°)

C(45°)

C(15°)

S(15°)

- S(15°)

C(15°)

C(-30°)

S(-30°)

- S(-30°)

C(-30°)

~ -2.8 cm

~ +5 cm