This document presents the components of an angular velocity vector along the body set of axes. It discusses angular velocity, finding the components of angular velocity, and the components of an angular velocity vector along the body set of axes. Equations are provided to calculate the components of angular velocity vectors ωφ, ωθ, and ωψ along the x', y', and z' axes of the body frame. The components are added vectorially to obtain the full angular velocity vector ω with respect to the body axes.

1. Presentation on Components
of angular velocity vector along
the body set of axes
Presented by : Tamim Bin Shafique
Student ID:111232

2. Discussion summary
Angular velocity
Components of angular velocity
Components of angular velocity vector
along the body set of axes

3. Angular Velocity
v = ω x r
Note: ω, r, v vectors
This is a spinning top having angular velocity
Here ω is constant

4. Finding the components of angular
velocity
Consider any axis
Transformation of axis
Adding the components

5. Components of angular velocity vector
along the body set of axes
In consequence of the vector property of ω ,we can combine the
components of angular velocity vectorially to form ω .we shall
obtain the components of ω along the body set of axes.
Now ωϕ is along the space z-axis, Therefore its components
along the body axes are found by applying complete orthogonal
transformation A=BCD , since there orthogonal transformations
are required to come to body axex.

6. Components of angular velocity vector
along the body set of axes





















.
'
'
'
0
0
)(
)(
)(






A
z
y
x

7. Components of angular velocity vector
along the body set of axes
Giving






cos)(
)1(..........cossin)(
sinsin)(
.
.
.
'
'
'



z
y
x

8. Components of angular velocity vector
along the body set of axes
lies along the line of nodes. We perform
orthogonal transformation B to come to
body axes after rotation has been
preformed so the components of along
axes are obtained upon applying the final
transformation B.


9. Components of angular velocity vector
along the body set of axes























0
0
)(
)(
)(
.
'
'
'







B
z
y
x

10. Components of angular velocity vector
along the body set of axes
Giving
0)(
)2(..............................sin)(
sincos)(
'
'
'
.
.



z
y
x







11. Components of angular velocity vector
along the body set of axes
Since ωψ is already parallel to 𝑧′axis, no transformation is
necessary .Its only component is 𝑧′ component equal to Ψ























.
'
'
'
0
0
)(
)(
)(






z
y
x

12. Components of angular velocity vector
along the body set of axes
Giving
.
'
'
'
)(
)3(..............................0)(
0)(









z
y
x

13. Components of angular velocity vector
along the body set of axes
From equation 1,2,3 we write the components of ωϕ,
ωѳ, ωѰ along 𝑥′, 𝑦′, 𝑧′ axes as follows :
0)(
cos)(
sinsin)(
'
'
'
.
.



x
x
x







14. Components of angular velocity vector
along the body set of axes
0)(
sin)(
cossin)(
'
'
'
.
.



y
y
y







15. Components of angular velocity vector
along the body set of axes
.
.
'
'
'
)(
0)(
cos)(









z
z
z

16. Components of angular velocity vector
along the body set of axes
Adding the components of the three velocities along
individual axes we have the components of
w.r.t. the body axes :

  
'''
)()()(' xxxx   
 cossinsin
..


17. Components of angular velocity vector
along the body set of axes
'''
)()()(' yyyy   
 cossinsin
..

  cos
.
'''
)()()(' zzzz   

18. Thanks from :
Tamim Bin Shafique
Student ID:111232
Mathematics Discipline,
Khulna University.