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Study on the dynamics, relative equilibria, and stability for liquid-filled spacecraft with flexible appendage

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Abstract

From the Lagrangian side of geometric mechanics, the dynamics, relative equilibria, and stability analysis for a rigid spacecraft coupled with liquid propellant, flexible appendage, and momentum wheel are studied. The liquid propellant nonlinear sloshing is equivalently modeled by a 3D rigid pendulum, which can effectively imitate the large-amplitude lateral sloshing accompanied by liquid rotation-starting, rotary sloshing, and spin motion of partially filled liquid in a closed cylindrical or spherical tank. The flexible attachment is modeled as a geometrically exact rod, which can accommodate arbitrarily large deformations in three dimensions, including extension, shear, bending, and twist. The global and coordinate-free dynamical equations are derived from the perspective of Lagrangian geometry. According to the principle of symmetric criticality, the relative equilibria and associated properties are investigated. Finally, in terms of the energy–momentum method and block diagonalization technique, the formal stability conditions for particular relative equilibria of the coupled spacecraft system are obtained.

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Abbreviations

h :

virtual hanging point of 3D rigid pendulum

l :

length of pendulum rod

d :

fixed position of frozen fuel respective to the centroid of fuel tank

B 1 :

attitude matrix of spacecraft platform

B 2 :

attitude matrix of 3D rigid pendulum

B r(s):

attitude matrix of cross-section at s position of geometrically exact rod

B r(s):

attitude matrix of cross-section s of geometrically exact rod

\(\varphi_0\) :

position vector of center of mass of system

\(\varphi_1\) :

position vector of spacecraft platform

\(\varphi_2\) :

position vector of 3D rigid pendulum

\(\varphi_f\) :

position vector of frozen fuel

\(\varphi_r\) :

position vector of cross-section s of geometrically exact rod

l 0 :

installation position of geometrically exact rod

\({L}\) :

length of geometrically exact rod

\(\rho_r\) :

density of of geometrically exact rod

\(\phi\) :

rotation angle of the rotor

\(\varvec{\varOmega}\) :

body angular velocity

\({\varvec{\omega_{\phi}}},{\varvec{\varPhi}}\) :

angular velocity of rotor

V r :

potential energy of geometrically exact rod

y(·):

stored energy function of s cross-section of geometrically exact rod

n(s, t):

force of s cross-section of geometrically exact rod

m(s, t):

moment of s cross-section of geometrically exact rod

Q :

configuration manifold

TQ :

tangent bundle of Q

SO(3):

special orthogonal group

TSO(3):

tangent bundle of special orthogonal group

TTSO(3):

second tangent bundle of special orthogonal group

T*TQ :

cotangent bundle of TSO(3)

G :

Lie group

\(\mathfrak{g}\) :

Lie algebra

\(\mathfrak{g}^*\) :

dual Lie algebra

\(\varvec{\xi}_0\) :

infinitesimal generator

I lock(q):

locked inertia tensor

\(\ll\cdot,\cdot\gg\) :

Riemannian metric

\(\langle \cdot,\cdot \rangle\) :

duality pairing between \(\mathfrak{g}^*\) and \(\mathfrak{g}\)

Y(q):

gyroscopic field

I Y(q):

gyro-momentum

\({V_\xi}(q)\) :

amended potential energy

I ° lock(q):

locked inertia dyadic

I°Y(q):

gyro-momentum dyadic

\({\varvec{\mu}}_{\text e}={\varvec{\tilde{J}}}(q_{\text e},\xi)\) :

pre-momentum mapping at relative equilibria

\(\mathfrak{g}_\mu\) :

isotropy subalgebra

\({{\mathfrak{g}^\bot_{\mu}}_{\text {e}}}\) :

orthogonal complement of isotropy subalgebra

V :

space of admissible configuration variations modulo variations

V RIG :

space of “rigid” configuration variations

V INT :

space of “internal” configuration variations

\({\varvec{\omega}}_s\) :

spatial rotating angular velocity of system

\(\varXi\) :

filled ratio of propellant

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Acknowledgements

We are particularly grateful to Associated Professor Donghua Shi for valuable discussions, especially for the energy-momentum method. This work was supported by the National Natural Science Foundation of China (Grant Nos: 12132002, 11772049).

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  1. School of Aerospace Engineering, Beijing Institute of Technology, Beijing, 100081, People’s Republic of China

    Zhonggui Yi & Baozeng Yue

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  1. Zhonggui Yi
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Yi, Z., Yue, B. Study on the dynamics, relative equilibria, and stability for liquid-filled spacecraft with flexible appendage. Acta Mech 233, 3557–3578 (2022). https://doi.org/10.1007/s00707-022-03269-5

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  • DOI: https://doi.org/10.1007/s00707-022-03269-5

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