F. Profito Ângulos de Euler

Prévia do material em texto

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Francisco J. Profito – fprofito@hotmail.com March 2017 1
Francisco Profito
Surface Phenomena Laboratory (LFS)
Department of Mechanical Engineering
University of Sao Paulo (USP)
Sao Paulo – Brazil
Euler Angles 
& 
Axisymmetric Bodies
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Francisco J. Profito – fprofito@hotmail.com March 2017 2
Euler Angles
 Angular velocity ( 𝑅𝑅𝜔𝜔𝑆𝑆 ) and acceleration ( 𝑅𝑅�⃗�𝛼𝑆𝑆 ) are used to calculate the translational 
velocities and accelerations of the rigid body’s particles.
How the body’s orientation and 
rotation can be measured?
Euler Angles
Rotation of a body-fixed 
reference frame
OXYZ (inertial, R)
Oxyz (body-fixed, S)
𝝓𝝓,𝜽𝜽,𝝍𝝍
 In general, rigid body orientation cannot be fully determined by integrating the angular 
velocity, since 𝑅𝑅𝜔𝜔𝑆𝑆 is derived by assuming infinitesimal rotations (i.e. 𝑅𝑅𝜔𝜔𝑆𝑆 ≢ 𝑑𝑑𝜃𝜃
𝑑𝑑𝑑𝑑
)
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Francisco J. Profito – fprofito@hotmail.com March 2017 3
𝑌𝑌
𝑍𝑍
𝑋𝑋 𝑋𝑋
𝑌𝑌
𝑍𝑍 𝒚𝒚
𝒛𝒛
𝒙𝒙
𝑍𝑍
𝑌𝑌
𝑋𝑋
𝑍𝑍
𝑋𝑋
𝑌𝑌
Euler Angles
 Transform OXYZ Oxyz in 3 steps (3-1-3 sequence):
𝑂𝑂𝑋𝑋𝑌𝑌𝑍𝑍
𝑂𝑂𝜉𝜉𝜉𝜉𝜉𝜉
𝑂𝑂𝜉𝜉′𝜉𝜉′𝜉𝜉′
𝑂𝑂𝑥𝑥𝑦𝑦𝑦𝑦
Rotate 𝝓𝝓 about 𝑍𝑍 axis
Rotate 𝜽𝜽 about 𝜉𝜉 axis
Rotate 𝝍𝝍 about 𝜉𝜉′ axis
𝑫𝑫 𝑪𝑪 𝑩𝑩
𝑢𝑢𝑋𝑋𝑋𝑋𝑋𝑋
𝑢𝑢𝜉𝜉𝜉𝜉𝜉𝜉 = 𝑫𝑫 𝑢𝑢𝑋𝑋𝑋𝑋𝑋𝑋
𝑢𝑢𝜉𝜉𝜉𝜉𝜉𝜉𝜉𝜉𝜉 = 𝑪𝑪 𝑫𝑫 𝑢𝑢𝑋𝑋𝑋𝑋𝑋𝑋
𝑢𝑢𝑥𝑥𝑥𝑥𝑥𝑥 = 𝑩𝑩 𝑪𝑪 𝑫𝑫 𝑢𝑢𝑋𝑋𝑋𝑋𝑋𝑋 = 𝑨𝑨 𝑢𝑢𝑋𝑋𝑋𝑋𝑋𝑋
Inertial
(R)
Body-Fixed
(S)
ϕ: precession
θ: nutation
ψ: spin
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Francisco J. Profito – fprofito@hotmail.com March 2017 4
Euler Angles
 Definition of Euler angles is somewhat arbitrary
 May rotate around different axes in different order
 Many conventions exist
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Francisco J. Profito – fprofito@hotmail.com March 2017 5
Euler Angles
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Francisco J. Profito – fprofito@hotmail.com March 2017 6
Euler Angles
 Body’s angular velocity in terms of time-derivatives of Euler angles:
𝑅𝑅𝜔𝜔𝑆𝑆 = �̇�𝜙 𝒏𝒏𝒁𝒁 + �̇�𝜃 𝒏𝒏𝜉𝜉 + �̇�𝜓 𝒏𝒏𝒛𝒛
 Rewriting the above equation in the body-fixed system Oxyz:
𝑌𝑌
𝑍𝑍
𝑋𝑋 𝑋𝑋
𝑌𝑌
𝑍𝑍 𝒚𝒚
𝒛𝒛
𝒙𝒙
𝑍𝑍
𝑌𝑌
𝑋𝑋
𝑪𝑪 𝑩𝑩
𝒏𝒏𝒁𝒁 = 𝑨𝑨 𝒏𝒏𝒛𝒛
𝒏𝒏𝜉𝜉 = 𝑩𝑩 𝒏𝒏𝜉𝜉𝜉
𝒏𝒏𝑥𝑥 = 𝑶𝑶𝑶𝑶
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Francisco J. Profito – fprofito@hotmail.com March 2017 7
Euler Angles
𝑅𝑅𝜔𝜔𝑆𝑆 = �̇�𝜙𝑠𝑠𝑠𝑠𝑠𝑠𝜓𝜓𝑠𝑠𝑠𝑠𝑠𝑠𝜃𝜃 + �̇�𝜃𝑐𝑐𝑐𝑐𝑠𝑠𝜓𝜓 ⃗+�̇�𝜙𝑐𝑐𝑐𝑐𝑠𝑠𝜓𝜓𝑠𝑠𝑠𝑠𝑠𝑠𝜃𝜃 − �̇�𝜃𝑠𝑠𝑠𝑠𝑠𝑠𝜓𝜓 ⃗+
�̇�𝜙𝑐𝑐𝑐𝑐𝑠𝑠𝜃𝜃 + �̇�𝜓 𝒌𝒌
 Remember that this expression is written in the body-fixed system Oxyz
 Now we can have the equations of motion in terms of 6 independent coordinates
𝑥𝑥,𝑦𝑦, 𝑦𝑦,𝜙𝜙, 𝜃𝜃,𝜓𝜓
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Francisco J. Profito – fprofito@hotmail.com March 2017 8
Axisymmetric Bodies
 For axisymmetric bodies, the last Euler rotation is NOT needed
 The principal reference frame is now 𝑶𝑶𝒙𝒙𝒚𝒚𝒛𝒛 ≡ 𝑶𝑶𝝃𝝃𝜉𝜼𝜼𝜉𝜻𝜻𝜉, denoted as F, which is NOT 
fully attached to the body
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Francisco J. Profito – fprofito@hotmail.com March 2017 9
Axisymmetric Bodies
 The angular velocity of the Oxyz system (F 
reference frame) can now be written as:
𝑅𝑅𝜔𝜔𝐹𝐹 = �̇�𝜙 𝒏𝒏𝒁𝒁 + �̇�𝜃 𝒏𝒏𝜉𝜉
𝑅𝑅𝜔𝜔𝐹𝐹 = �̇�𝜃 ⃗+ �̇�𝜙𝑠𝑠𝑠𝑠𝑠𝑠𝜃𝜃 ⃗+ �̇�𝜙𝑐𝑐𝑐𝑐𝑠𝑠𝜃𝜃 𝒌𝒌Remember: only 2 Euler rotations
Much simpler than the body-fixed frame!
 The absolute angular velocity of the body can 
be obtained by composition of motion:
𝑅𝑅𝜔𝜔𝑆𝑆 = 𝑅𝑅𝜔𝜔𝐹𝐹+ 𝐹𝐹𝜔𝜔𝑆𝑆
𝑅𝑅𝜔𝜔𝑆𝑆 = �̇�𝜃 ⃗+ �̇�𝜙𝑠𝑠𝑠𝑠𝑠𝑠𝜃𝜃 ⃗+ �̇�𝜙𝑐𝑐𝑐𝑐𝑠𝑠𝜃𝜃 + �̇�𝜓 𝒌𝒌
Relative rotation (spin)
𝐹𝐹𝜔𝜔𝑆𝑆 = �̇�𝜓𝒌𝒌
‘Arrastamento’
ϕ: precession
θ: nutation
ψ: spin
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Francisco J. Profito – fprofito@hotmail.com March 2017 10
Axisymmetric Bodies
𝒙𝒙
𝒚𝒚
𝒛𝒛
ϕ: precession
θ: nutation
ψ: spin
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Francisco J. Profito – fprofito@hotmail.com March 2017 11
Other Applications
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