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Thesis - Leigh Moody.pdf - Bad Request - Cranfield University

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Appendix D / Point Mass Dynamics<br />

_ _<br />

17.1 Transformation from Angular to Euler Rates<br />

The transform from Euler to angular rates is given by the vector sum of the<br />

angular velocity components about the intermediate Euler axes.<br />

ω<br />

R<br />

I, R<br />

⎛<br />

R<br />

− sin Θ ⎞ I<br />

⎛ 0 ⎞ ⎛ 1 ⎞<br />

⎜<br />

⎟ ⎜ ⎟ ⎜ ⎟<br />

⎜<br />

⎟ ⎜ ⎟ ⎜ ⎟<br />

R R R<br />

R R<br />

: =<br />

⎜<br />

⎟<br />

sin I cos<br />

⎜<br />

I I cos ⎟ ⎜<br />

I I 0 ⎟<br />

⎜ Φ ⋅ Θ ⋅ Ψ&<br />

+ Φ ⋅ Θ&<br />

+ ⋅ Φ&<br />

⎟ ⎜ ⎟ ⎜ ⎟<br />

⎜<br />

⎟ ⎜ ⎟ ⎜ ⎟<br />

⎜ R R ⎟ ⎜ R<br />

cos<br />

sin<br />

⎟ ⎜<br />

I cos I<br />

I<br />

0<br />

⎟<br />

⎝ Φ ⋅ Θ ⎠ ⎝ − Φ ⎠ ⎝ ⎠<br />

ω<br />

R<br />

I, R<br />

: =<br />

⎛<br />

⎜<br />

⎜<br />

⎜<br />

⎜<br />

⎜<br />

⎜<br />

⎝<br />

I<br />

Z<br />

a<br />

1<br />

0<br />

0<br />

,<br />

,<br />

,<br />

0<br />

cosΦ<br />

17-3<br />

R<br />

I<br />

− sin Φ<br />

P XR<br />

ab<br />

Y I<br />

R<br />

I<br />

R<br />

Ψ<br />

I<br />

Y R<br />

,<br />

,<br />

,<br />

R<br />

−θ I<br />

X I<br />

R<br />

Z<br />

sin Φ<br />

cosΦ<br />

− sin Θ<br />

R<br />

I<br />

R<br />

I<br />

b<br />

R<br />

I<br />

⋅ cosΘ<br />

⋅ cosΘ<br />

X R<br />

R<br />

I<br />

R<br />

I<br />

R<br />

I<br />

Equation 17.1-1<br />

⎞<br />

⎟<br />

⎟<br />

⎟<br />

⎟<br />

⎟<br />

⎟<br />

⎠<br />

⋅ E&<br />

R<br />

I<br />

Equation 17.1-2<br />

Figure 17-1 : Inertial and Rotating Frames of Reference

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