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Two-Dimensional Vector and Matrix Notation: Haug Notation and Vector Transformations, Study notes of Computer-Aided Analysis of Machine Dynamics

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

An overview of the haug notation for two-dimensional vector and matrix transformations. It covers the representation of global and local positions, velocities, and accelerations of points attached to bodies. The document also explains the concept of relative locations between points, attitude angles, and the use of rotation matrices to convert information between local and global directions.

Typology: Study notes

2012/2013

Uploaded on 10/02/2013

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Two-Dimensional Vector and Matrix Notation – Haug Notation

i
r global position of the origin of reference frame attached to body i

P
i
r global position of point P attached to body i
example

B
4
B
4
B
4y
x
r global position of point B attached to body 4

i
r
global velocity of the origin of the reference attached to body i

P
i
r
global velocity of point P attached to body i

i
r
global acceleration of the origin of the reference attached to body i

P
i
r
global acceleration of point P attached to body i

i
r global jerk of the origin of the reference attached to body i

P
i
r global jerk of point P attached to body i

P
i's position of point P on body i relative to the reference frame for body i measured in local
body-fixed directions
example

B
4
B
4
B
4'y
'x
's location of point B on body 4 relative to the reference frame for
body 4 measured in local body-fixed directions for body 4

P
i
s position of point P on body i relative to the reference frame for body i but measured in
global directions
ij
d relative location between two points on bodies i and j measured in global directions
example


P
i
P
jij rrd relative location of point P on body j with respect to point P on
body i measured in global directions
i
attitude angle for reference frame attached to body i
ij
attitude angle of body j with respect to reference frame attached to body i
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Two-Dimensional Vector and Matrix Notation – Haug Notation

 ri global position of the origin of reference frame attached to body i

 

P ri global position of point P attached to body i

example  

 B

4

B B 4 4 y

x r global position of point B attached to body 4

 ri global velocity of the origin of the reference attached to body i

 

P ri global velocity of point P attached to body i

 r (^) i global acceleration of the origin of the reference attached to body i

 

P r (^) i global acceleration of point P attached to body i

 r^ i global jerk of the origin of the reference attached to body i

 

P r (^) i global jerk of point P attached to body i

 

P s (^) i ' position of point P on body i relative to the reference frame for body i measured in local

body-fixed directions

example  

B 4

B B 4 4 y '

x ' s ' location of point B on body 4 relative to the reference frame for

body 4 measured in local body-fixed directions for body 4

 

P s (^) i position of point P on body i relative to the reference frame for body i but measured in

global directions

d (^) ij relative location between two points on bodies i and j measured in global directions

example      

P i

P d (^) ij  rj  r relative location of point P on body j with respect to point P on

body i measured in global directions

i attitude angle for reference frame attached to body i

ij attitude angle of body j with respect to reference frame attached to body i

example ij jj

 i  (^) angular velocity of body i

i  (^) angular acceleration of body i

i   angular jerk of body i

P Fi (^) /j force from body i on body j acting through point P measured in global directions

P Fi (^) /j' force from body i on body j acting through point P measured in body-fixed directions

local to body j

Ti torque on body i

 A i orthonormal rotation matrix that describes attitude of body i

example   

i i

i i i sin cos

cos sin A

example     

P i i

P s (^) i  A s ' rotation matrix converts information in local body-fixed

directions into global directions

fˆ i^  global direction of unit vector along local x axis attached to body i

gˆ i global direction of unit vector along local y axis attached to body i

example A i    ^ fˆi^ gˆ^ i unit directions of local axes for body i

 Bi  second rotation matrix

example      

cos sin

sin cos B (^) i R Ai

A ij rotation matrix that describes attitude of body j with respect to body i

example      j

T i ij ij

ij ij ij A A sin cos

cos sin A (^)  