Attitude dynamics 1 – Main aspects of the rigid body dynamics
Transcript of Attitude dynamics 1 – Main aspects of the rigid body dynamics
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Attitude dynamics
1 – Main aspects of the rigid body dynamics
Doroshin A.V.
Samara State Aerospace University
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The main dynamical parameters of rigid body motion about fixed point
The angular momentum and the kinetic energy of the rigid body with fixed point:
The (linear) momentum | quantity of motion:
The angular momentum at the change of the point-pole (A to B):
The kinetic energy:
The angular momentum:
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The dynamical theorems
The change of the angular momentum:
The change of the kinetic energy:
the elementary mechanical work of all EXTERNAL and INTERNAL forces
the vector of EXTERNAL torques
The potential (the partial case of forces):
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The physical pendulum angular motion as an important example of rigid body motion
Point A fixed point O
0
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The physical pendulum angular motion as an important example of rigid body motion
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The physical pendulum angular motion as an important example of rigid body motion
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The physical pendulum angular motion as an important example of rigid body motion
Some aspects of elliptic functions
The Jacobi elliptical integral (the first kind elliptic integral)
Then:
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The physical pendulum angular motion as an important example of rigid body motion
Some aspects of elliptic functions
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The physical pendulum angular motion as an important example of rigid body motion
β – the maximum of ϕ value
Changing variables
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The physical pendulum angular motion as an important example of rigid body motion
If the angle is small then we can linearize the equation; and as the result t we obtain the harmonic solutions with ordinal SINUS the practical seminar
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Equations of the angular motion of rigid body about the fixed point
Rotations: the precession ψ-> the nutation θ -> the intrinsic rotation ϕ
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Equations of the angular motion in the Euler case
0
Leonhard Euler
Born: April 15, 1707,
Basel, Switzerland
Died: September 18, 1783,
Saint Petersburg, Russian Empire
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Equations of the angular motion in the Euler case
Poinsot's construction. Polhodes
https://www.youtube.com/watch?v=BwYFT3T5uIw
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Equations of the angular motion in the Euler case
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Equations of the angular motion in the Euler case
Areas I and II
Changing variables
Case 1
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Equations of the angular motion in the Euler case
Areas I and II
By back substitutions
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Equations of the angular motion in the Euler case
Areas III and IV
Changing of variablesCase 2
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Equations of the angular motion in the Euler case
Trajectories 1 and 2
Case 3
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Equations of the angular motion in the Euler case
OZ ≡ K
Solutions for Euler angles
Let us to direct the axes with fulfilment of the condition
Integrating elliptic function…
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Equations of the angular motion in the Lagrange case
Joseph-Louis Lagrange
Born: 25 January 1736,
Turin, Piedmont-Sardinia
Died: 10 April 1813 (aged 77)Paris, France
(e)
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Equations of the angular motion in the Lagrange case
The first integrals:
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Equations of the angular motion in the Lagrange case
First integrals:
- The energy conservation
- The “vertical” component of K conservation
- The “logitudinal” component of K conservation
In the “canonical” designation
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Equations of the angular motion in the Lagrange case
The energy conservation, and
The “vertical” componentof K conservation
- The “logitudinal” component of K conservation
1
2
3
b - const
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Equations of the angular motion in the Lagrange case
By reducing:
The change of variables:
The polynomial of the fourth power has the 3 real roots
A real motion area
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Equations of the angular motion in the Lagrange case
A real motion area
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Equations of the angular motion in the Lagrange case
A real motion area
Integrating in elliptic integrals:
The detailed consideration can be found in books and articles…
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Elementary gyroscopic effect
Let us to find conditions of the motion with constant velocities of precession (ψ) and intrinsic rotation (ϕ) at constant nutation:
(DYN.SYS)
(DYN.SYS)
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This can be reduced to the form:
Vector-part Scalar-part
The main formula of the gyroscopic effect
The velocity Va of the end of the Ko vector is
1) equal to Mo value2) directed along Mo vector
Va =
Ko
Va
Let us now consider The next slide
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The main formula of the gyroscopic effect
Ko VaLet us now consider
0 If assume
A backward consideration: if we intend to rotate the spun rigid body, the REACTION-torque will arise
This is the gyroscopic torque
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The main formula of the gyroscopic effect
The gyroscopic torque
a rotating cause
The obtained force-effect
The obtained force-effect
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Sofia Vasilyevna Kovalevskaya
Born: 15 January 1850Moscow, Russian Empire
Died: 10 February 1891Stockholm, Sweden
S.V. KOVALEVSKAYA’S CASE OF RIGID BODY MOTION
b=c=0; a≠0
The first integrals:
where
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THE S.V. KOVALEVSKAYA TOP
In 1888 she won the Prix Bordin of the French Academy of Science, for
her work on the question: "Mémoire sur un cas particulier du problème de le
rotation d'un corps pesant autour d'un point fixe, où l'intégration s'effectue à
l'aide des fonctions ultraelliptiques du temps"
Kovalevskaya’s first intrgral:
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THE S.V. KOVALEVSKAYA TOP
Kovalevskaya’s variables:
Where:
The motion equations in Kovalevskaya’s variables:
Polynomial of 5th power
Then the hyperelliptic integrals/functions follow… The end of classical cases of the rigid body motion…
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Special aspects of the rigid body dynamics
The Andoyer–Deprit variables
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Special aspects of the rigid body dynamics
Hamiltonian form of equations:
The potential and kinetic energy in the A-D variables:
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Directional cosines can be expressed in A-D variables:
1 2 3
1 2 3
1 2 3
X
Y
Z
e i j k
e i j k
e i j k
𝐞𝑋
𝐞𝑌
𝐞𝑍
𝐢
𝐣
𝐤
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The Euler parameters / Euler–Rodrigues formula
Let we have the unit-vector
Then the E-R parameters are:
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The Euler parameters / Euler–Rodrigues formula
Connections with the Euler angles and the angular velocity components:
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The Euler parameters / Euler–Rodrigues formula
Connections with the directional cosines:
Then we have the connections of the angles and parameters… -we can use it for rewriting of torques expressions and
building of dynamical equations… :𝐞𝑋
𝐞𝑌
𝐞𝑍
𝐢
𝐣
𝐤
2 2 2 2
0 1 2 3 0 1 2 32P c b
B…
C…
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The Euler parameters / Euler–Rodrigues formula
Kinematical equations: