presentation on components of angular velocity vector along the body set of axes
TRANSCRIPT
Presentation on Components of angular velocity vector along the body set of axes
Presented by : Tamim Bin Shafique
Student ID:111232
Discussion summary
Angular velocity
Components of angular velocity
Components of angular velocity vector along the body set of axes
Angular Velocity
v = ω x r
Note: ω, r, v vectors
This is a spinning top having angular velocity
Here ω is constant
Finding the components of angular velocity
Consider any axis
Transformation of axis
Adding the components
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.
Components of angular velocity vector along the body set of axes
.'
'
'
0
0
)(
)(
)(
A
z
y
x
Components of angular velocity vector along the body set of axes
Giving
cos)(
)1(..........cossin)(
sinsin)(
.
.
.
'
'
'
z
y
x
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.
Components of angular velocity vector along the body set of axes
0
0
)(
)(
)(.
'
'
'
B
z
y
x
Components of angular velocity vector along the body set of axes
Giving
0)(
)2(..............................sin)(
sincos)(
'
'
'
.
.
z
y
x
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
Components of angular velocity vector along the body set of axes
Giving
.
'
'
'
)(
)3(..............................0)(
0)(
z
y
x
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
Components of angular velocity vector along the body set of axes
0)(
sin)(
cossin)(
'
'
'
.
.
y
y
y
Components of angular velocity vector along the body set of axes
.
.
'
'
'
)(
0)(
cos)(
z
z
z
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..
Components of angular velocity vector along the body set of axes
''' )()()(' yyyy
cossinsin..
cos.
''' )()()(' zzzz
Thanks from :
Tamim Bin Shafique
Student ID:111232
Mathematics Discipline,
Khulna University.