presentation on components of angular velocity vector along the body set of axes

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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.

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