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The attitude stability of a spacecraft with two flexible solar arrays in the gravitational field
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Chaos, Solitons and Fractals 37 (2008) 108–112 www.elsevier.com/locate/chaos
The attitude stability of a spacecraft with two flexible solar arrays in the gravitational field Xin-Sheng Ge b
a,*
, Yan-Zhu Liu
b
a Basic Science Courses Department, Beijing Institute of Machinery, Beijing 100085, China Department of Engineering Mechanics, Shanghai Jiaotong University, Shanghai 200030, China
Accepted 24 July 2006
Communicated by Prof. Ji-Huan He
Abstract The attitude stability of flexible spacecraft is discussed in this paper. The Euler’s equations of spacecraft and the equations of forced vibration of two flexible solar arrays are derived. The governing equation is discretized via the Galerkin method, and the stability of relative equilibrium of spacecraft in the orbit frame is determined based on the Kelvin–Tait–Chetayev theorem. The sufficient condition of stability in the analytical form for the n-term truncated modal is established based on the governing equation. Ó 2006 Elsevier Ltd. All rights reserved.
1. Introduction After the US-launched satellite Explorer-I lost its equilibrium, the influence of the elastic deformation of flexible appendages on the attitude stability of spacecraft has attracted a great deal of attention in the field. Since large flexible solar arrays are extensively used in aerospace engineering, the importance of attitude dynamics and control are widely recognised. By using analytical modes and numerical methods, Cherchas and Houghes [1] and Pfeiffer and Pohl [2] analysed the vibration and stability of a dual-spin satellite with flexible solar arrays. Ibrahim and Misra [3] investigated the effects of the spacecraft libration and the solar array vibration on the stability of the system by the theory and experiments. Liu [4] discussed the stability of relative equilibrium of an unsymmetrical spacecraft with a flexible plate in its orbital frame, and obtain a stability criterion in the analytical form. The attitude stability of a spacecraft with flexible solar arrays in both sides and moving in a circle orbit under the action of the gravitational torque will be discussed in this paper. 2. The governing equation of flexible spacecraft The spacecraft consists of a main rigid body B0 and two rectangular flexible plates B1 and B2 that are symmetrical along the principal axes. The mass of main body B0 is much larger than those of plate Bi (i = 1, 2). Neglecting the influence of small elastic deformation due to the displacement of mass center, we establish an orbital frame (O–XYZ) with *
Corresponding author. E-mail address: [email protected] (X.-S. Ge).
0960-0779/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2006.07.040
X.-S. Ge, Y.-Z. Liu / Chaos, Solitons and Fractals 37 (2008) 108–112
109
its origin at O, axis X directed from the earth center O0 to the point O, axis Y along the tangent of orbital, and axis Z along the normal of orbital plane. The principal axis of inertia of main body are selected as the floating reference frame (O–xyz) with axis z paralleled to the clamped edge of plate Bi, and axis x along the direction of the elastic deformation of plate. Denote Cardan’s angles of the principal frame (O–xyz) relative to the orbital frame (O–XYZ) as h, w, u. The orbital frame rotates about axis Z at the angular velocity xc. Euler’s equation of flexible spacecraft are obtained form the moment of momentum theorem Z Z Z Z ou1 ou2 A€h ðA þ B CÞxc w_ þ ðC BÞx2c h 2xc hc z dy dz þ z dy dz ¼ 0; S 1 ot S 2 ot Z Z Z Z 2 o u1 o2 u2 2 _ € z dy dz þ z dy dz Bw þ ðA þ B CÞxc h þ ðC BÞxc w þ hc 2 ot2 S S 2 ot Z Z 1 Z Z u1 z dy dz þ u2 z dy dz ¼ 0; 4x2c hc ð1Þ S1
S2
S1
S2
Z Z Z Z o2 u1 o2 u2 € þ 3x2c ðB AÞu hc y dy dz þ y dy dz Cu 2 2 S 1 ot S 2 ot Z Z Z Z u1 y dy dz þ u2 y dy dz ¼ 0; þ 3x2c hc where A, B, C are the principle moments of inertia of the system, ui (i = 1, 2) are the elastic displacement of plate, r and h are the density and the thickness of plate respectively. The equation of transverse vibrations of plate Bi can be written as o2 u2 ¼ pi ðy; z; tÞ; ð2Þ ot2 where D is the flexural rigidity of the plate, pi ðy; z; tÞ is the distributing load produced by the inertia and the gravitational force along axis x, defined by € þ 2xc h_ 4x2 wÞz þ ð€ pi ðy; z; tÞ ¼ hc½ðw u 3x2c uÞy: ð3Þ c Dr4 ui þ hc
Substituting the distributing force Eq. (3) into the forced vibration Eq. (2) of plate, one obtains o2 u1 € þ 2xc h_ 4x2 wÞz ð€ þ hc½ðw u 3x2c uÞy ¼ 0; c ot2 o2 u2 € þ 2xc h_ 4x2 wÞz ð€ Dr4 u2 þ hc 2 þ hc½ðw u 3x2c uÞy ¼ 0: c ot
Dr4 u1 þ hc
ð4Þ
3. The discretization of the governing equation If the plate B1 andB2 have the same mode function, the special solution ui(y, z, t) of plate Bi under the inertia and the gravitational force can be written as a linear combination of the modes of free vibration Uj(y, z) of an one-side clamped rectangular plate, which is based on the Galerkin method [5] ui ðy; z; tÞ ¼
N X
Uj ðy; zÞqij ðtÞ;
ð5Þ
j¼1
where qij is modal coordinates of plate Bi. Substituting Eq. (5) into Eqs. (1) and (4) yields A€h ðA þ B CÞxc w_ þ ðC BÞx2c h 2xc
N X
nj ðq1j þ q2j Þ ¼ 0;
j¼1
€ þ ðA þ B CÞxc h_ þ 4ðC BÞx2 w þ Bw c
N X
nj ð€q1j þ € q2j Þ 4x2c
j¼1
€ þ 3x2c ðB AÞu Cu
N X
mj ð€q1j þ €q2j Þ þ 3x2c
j¼1
N X j¼1
N X
nj ðq1j þ q2j Þ ¼ 0; ð6Þ
mj ðq1j þ q2j Þ ¼ 0;
j¼1
€ þ 2xc h_ 4x2 wÞ mj ð€ u 3x2c uÞ ¼ 0; djk ð€q1j þ r2j q1j Þ þ nj ðw c € þ 2xc h_ 4x2 wÞ mj ð€ djk ð€q2j þ r2 q Þ þ nj ðw u 3x2 uÞ ¼ 0; j 2j
c
c
where djk is the Kronecher symbol, rj is the jth order eigen-frequencies of the plate, and parameter mj, nj is defined as
110
X.-S. Ge, Y.-Z. Liu / Chaos, Solitons and Fractals 37 (2008) 108–112
Z Z
Z Z
mj ¼ hc Uj ðy; zÞy dy dz ¼ hc Uj ðy; zÞy dy dz; S S Z Z 1 Z Z 2 nj ¼ hc Uj ðy; zÞz dy dz ¼ hc Uj ðy; zÞz dy dz S1
ð7Þ
S2
in which mj and nj are constituted by the integral of symmetrical and skew-symmetrical modes respectively. They are caused by the bending and torsion deformation of plate.
4. Condition of attitude stability Eq. (6) may be written compactly as a single matrix equation, in which djk is replaced by 1 or 0. Eq. (6) are rewritten in the form M€q þ Gq_ þ Kq ¼ 0;
ð8Þ
where q is a column matrix composed of (2N+3) generalized coordinates qi, M is a (2N+3)-order symmetrical inertia matrix, G is a (2N+3)-order skew-symmetric gyroscope matrix, K is a (2M+3)-order symmetrical stiffness matrix. Each matrix is defined respectively as q ¼ ½h; w; u; q11 ; . . . ; q1N ; q21 ; . . . ; q2N T ; 2 A 0 0 0 0 6 B 0 n1 nN 6 0 6 6 0 0 C m1 mN 6 6 1 6 0 n1 m1 6 6 .. . M ¼6 6 6 1 6 0 nN mN 6 6 0 n1 mN 6 6 6 0 4
3 0 7 nN 7 7 mN 7 7 7 7 7 7 7; 0 7 7 7 7 7 1 7 7 .. 7 . 5 1 2n1 2nN 0 0 0 0 0 .. . 0
0 n1 m1
mN 0 ðA þ B CÞ 0 6 0 0 6 ðA þ B CÞ 6 6 0 0 0 6 6 2n1 0 0 6 6 6 G ¼ xc 6 6 6 2n 0 0 6 N 6 6 2n1 0 0 6 6 6 4 2nN 0 0 2 ðC BÞ 0 0 0 6 0 4ðC AÞ 0 4n 1 6 6 6 0 0 3ðB AÞ 3m1 6 6 0 4n1 3m1 ðr1 =xc Þ2 6 6 6 K ¼ x2c 6 6 6 0 4nN 3mN 6 6 6 0 4n 3mN 1 6 6 6 4 0 2
nN
0
4nN
3mN
0
0
3 2nN 7 0 7 7 0 7 7 7 7 7 7 7; 7 7 7 7 7 7 7 7 5 0
4nN 3mN
4n1 3m1
2n1 0 0
0 0 ..
0
..
.
ð9Þ
0 4nN 3mN
0
. ðrN =xc Þ
2
ðr1 =xc Þ2 0
..
. ðrN =xc Þ2
3 7 7 7 7 7 7 7 7 7 7: 7 7 7 7 7 7 7 7 5
X.-S. Ge, Y.-Z. Liu / Chaos, Solitons and Fractals 37 (2008) 108–112
111
According to the Kelvin–Tait–Chetayev theorem [6], the stability of satellite relative the orbital frame is determined. The system is stable if the stiffness matrix K is positive definite. From the positive definite of the stiffness matrix K in Eq. (9), the stability criterion can be derived as C > B; C >Aþ8
N X
ðxc =rj Þ2 n2j ;
j¼1
hP i2 N N 48 ðxc =rj Þ2 mj nj X j¼1 : ðxc =rj Þ2 m2j þ B>Aþ6 P C A 8 Nj¼1 ðxc =rj Þ2 n2j j¼1
ð10Þ
Consider the following special cases: (1) If nj = 0 in the foregoing condition, the flexible plate becomes the flexible bar that has bending deformation along direction of axis x, and condition (10) can be converted into the stability criterion of spacecraft with a couple of flexible bars [7] C > B;
C > A; N X B>Aþ6 ðxc =rj Þ2 mj :
ð11Þ
j¼1
(2) If mj = nj = 0, condition (10) leads to the well-known attitude stability criterion of rigid body satellite in the gravitational field
ð12Þ
C > B > A: Define the following dimensionless parameters k, b, d and e as C k¼ ; A
B b¼ ; A
PN d¼
2 2 j¼1 mj ðxc =rj Þ
A
PN ;
e¼
2 2 j¼1 nj ðxc =rj Þ
A
ð13Þ
where d and e are small parameters. Retaining the 1st order of d and e only, condition (10) can be simplified into the form k > b;
k > 1 þ 8e;
b > 1 þ 6d:
ð14Þ
Introducing parameter plane with k and b as the axes of frame, one can shows the stable region with variable d in parameter (k, b) plane, and shows the stable region with variable e in parameter (k, b) plane. According to the discussions in this paper, it is shown that the reduction of Lagrange’s triangle region of stability in the parameters (k, b) plane with the increasing of the flexibility of plate. It reflects the influences of the bending and the torsion deformation of plate on the attitude stability of the spacecraft respectively.
Acknowledgment This work is supported by the National Natural Science Foundation of PR China (No. 10372014).
References [1] Cherchas DB, Houghes PC. Attitude stability of a dual-spin satellite with a large flexible solar array. J Spacecraft Rockets 1973;10(2):126–32. [2] Pfeiffer F, Pohl A. Dynamics of a satellite with a flywheel and flexible solar array. Raumfahrtforschung 1974;6:277–85. [3] Ibrahim AE, Misra AK. Attitude dynamics of satellite during deployment of large plate-type structures. J Guidance Control Dyn 1982;5(5):442–7.
112
X.-S. Ge, Y.-Z. Liu / Chaos, Solitons and Fractals 37 (2008) 108–112
[4] Liu YZ. Attitude stability of an unsymmetrical spacecraft with flexible plate in gravitational field. Acta Mech Solida Sinica 1994;15(4):296–302 [in Chinese]. [5] Meirovitch L. Elements of vibration analysis. McGraw-Hill; 1975. [6] Zajac EE. The Kelvin–Tait–Chetayev theorem and extensions. J Astronaut Sci 1964;11(2):46–9. [7] Liu YZ. The attitude dynamics of spacecraft. Beijing: National Defense Industry Press; 1995 [in Chinese].
Chaos, Solitons and Fractals 37 (2008) 108–112 www.elsevier.com/locate/chaos
The attitude stability of a spacecraft with two flexible solar arrays in the gravitational field Xin-Sheng Ge b
a,*
, Yan-Zhu Liu
b
a Basic Science Courses Department, Beijing Institute of Machinery, Beijing 100085, China Department of Engineering Mechanics, Shanghai Jiaotong University, Shanghai 200030, China
Accepted 24 July 2006
Communicated by Prof. Ji-Huan He
Abstract The attitude stability of flexible spacecraft is discussed in this paper. The Euler’s equations of spacecraft and the equations of forced vibration of two flexible solar arrays are derived. The governing equation is discretized via the Galerkin method, and the stability of relative equilibrium of spacecraft in the orbit frame is determined based on the Kelvin–Tait–Chetayev theorem. The sufficient condition of stability in the analytical form for the n-term truncated modal is established based on the governing equation. Ó 2006 Elsevier Ltd. All rights reserved.
1. Introduction After the US-launched satellite Explorer-I lost its equilibrium, the influence of the elastic deformation of flexible appendages on the attitude stability of spacecraft has attracted a great deal of attention in the field. Since large flexible solar arrays are extensively used in aerospace engineering, the importance of attitude dynamics and control are widely recognised. By using analytical modes and numerical methods, Cherchas and Houghes [1] and Pfeiffer and Pohl [2] analysed the vibration and stability of a dual-spin satellite with flexible solar arrays. Ibrahim and Misra [3] investigated the effects of the spacecraft libration and the solar array vibration on the stability of the system by the theory and experiments. Liu [4] discussed the stability of relative equilibrium of an unsymmetrical spacecraft with a flexible plate in its orbital frame, and obtain a stability criterion in the analytical form. The attitude stability of a spacecraft with flexible solar arrays in both sides and moving in a circle orbit under the action of the gravitational torque will be discussed in this paper. 2. The governing equation of flexible spacecraft The spacecraft consists of a main rigid body B0 and two rectangular flexible plates B1 and B2 that are symmetrical along the principal axes. The mass of main body B0 is much larger than those of plate Bi (i = 1, 2). Neglecting the influence of small elastic deformation due to the displacement of mass center, we establish an orbital frame (O–XYZ) with *
Corresponding author. E-mail address: [email protected] (X.-S. Ge).
0960-0779/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2006.07.040
X.-S. Ge, Y.-Z. Liu / Chaos, Solitons and Fractals 37 (2008) 108–112
109
its origin at O, axis X directed from the earth center O0 to the point O, axis Y along the tangent of orbital, and axis Z along the normal of orbital plane. The principal axis of inertia of main body are selected as the floating reference frame (O–xyz) with axis z paralleled to the clamped edge of plate Bi, and axis x along the direction of the elastic deformation of plate. Denote Cardan’s angles of the principal frame (O–xyz) relative to the orbital frame (O–XYZ) as h, w, u. The orbital frame rotates about axis Z at the angular velocity xc. Euler’s equation of flexible spacecraft are obtained form the moment of momentum theorem Z Z Z Z ou1 ou2 A€h ðA þ B CÞxc w_ þ ðC BÞx2c h 2xc hc z dy dz þ z dy dz ¼ 0; S 1 ot S 2 ot Z Z Z Z 2 o u1 o2 u2 2 _ € z dy dz þ z dy dz Bw þ ðA þ B CÞxc h þ ðC BÞxc w þ hc 2 ot2 S S 2 ot Z Z 1 Z Z u1 z dy dz þ u2 z dy dz ¼ 0; 4x2c hc ð1Þ S1
S2
S1
S2
Z Z Z Z o2 u1 o2 u2 € þ 3x2c ðB AÞu hc y dy dz þ y dy dz Cu 2 2 S 1 ot S 2 ot Z Z Z Z u1 y dy dz þ u2 y dy dz ¼ 0; þ 3x2c hc where A, B, C are the principle moments of inertia of the system, ui (i = 1, 2) are the elastic displacement of plate, r and h are the density and the thickness of plate respectively. The equation of transverse vibrations of plate Bi can be written as o2 u2 ¼ pi ðy; z; tÞ; ð2Þ ot2 where D is the flexural rigidity of the plate, pi ðy; z; tÞ is the distributing load produced by the inertia and the gravitational force along axis x, defined by € þ 2xc h_ 4x2 wÞz þ ð€ pi ðy; z; tÞ ¼ hc½ðw u 3x2c uÞy: ð3Þ c Dr4 ui þ hc
Substituting the distributing force Eq. (3) into the forced vibration Eq. (2) of plate, one obtains o2 u1 € þ 2xc h_ 4x2 wÞz ð€ þ hc½ðw u 3x2c uÞy ¼ 0; c ot2 o2 u2 € þ 2xc h_ 4x2 wÞz ð€ Dr4 u2 þ hc 2 þ hc½ðw u 3x2c uÞy ¼ 0: c ot
Dr4 u1 þ hc
ð4Þ
3. The discretization of the governing equation If the plate B1 andB2 have the same mode function, the special solution ui(y, z, t) of plate Bi under the inertia and the gravitational force can be written as a linear combination of the modes of free vibration Uj(y, z) of an one-side clamped rectangular plate, which is based on the Galerkin method [5] ui ðy; z; tÞ ¼
N X
Uj ðy; zÞqij ðtÞ;
ð5Þ
j¼1
where qij is modal coordinates of plate Bi. Substituting Eq. (5) into Eqs. (1) and (4) yields A€h ðA þ B CÞxc w_ þ ðC BÞx2c h 2xc
N X
nj ðq1j þ q2j Þ ¼ 0;
j¼1
€ þ ðA þ B CÞxc h_ þ 4ðC BÞx2 w þ Bw c
N X
nj ð€q1j þ € q2j Þ 4x2c
j¼1
€ þ 3x2c ðB AÞu Cu
N X
mj ð€q1j þ €q2j Þ þ 3x2c
j¼1
N X j¼1
N X
nj ðq1j þ q2j Þ ¼ 0; ð6Þ
mj ðq1j þ q2j Þ ¼ 0;
j¼1
€ þ 2xc h_ 4x2 wÞ mj ð€ u 3x2c uÞ ¼ 0; djk ð€q1j þ r2j q1j Þ þ nj ðw c € þ 2xc h_ 4x2 wÞ mj ð€ djk ð€q2j þ r2 q Þ þ nj ðw u 3x2 uÞ ¼ 0; j 2j
c
c
where djk is the Kronecher symbol, rj is the jth order eigen-frequencies of the plate, and parameter mj, nj is defined as
110
X.-S. Ge, Y.-Z. Liu / Chaos, Solitons and Fractals 37 (2008) 108–112
Z Z
Z Z
mj ¼ hc Uj ðy; zÞy dy dz ¼ hc Uj ðy; zÞy dy dz; S S Z Z 1 Z Z 2 nj ¼ hc Uj ðy; zÞz dy dz ¼ hc Uj ðy; zÞz dy dz S1
ð7Þ
S2
in which mj and nj are constituted by the integral of symmetrical and skew-symmetrical modes respectively. They are caused by the bending and torsion deformation of plate.
4. Condition of attitude stability Eq. (6) may be written compactly as a single matrix equation, in which djk is replaced by 1 or 0. Eq. (6) are rewritten in the form M€q þ Gq_ þ Kq ¼ 0;
ð8Þ
where q is a column matrix composed of (2N+3) generalized coordinates qi, M is a (2N+3)-order symmetrical inertia matrix, G is a (2N+3)-order skew-symmetric gyroscope matrix, K is a (2M+3)-order symmetrical stiffness matrix. Each matrix is defined respectively as q ¼ ½h; w; u; q11 ; . . . ; q1N ; q21 ; . . . ; q2N T ; 2 A 0 0 0 0 6 B 0 n1 nN 6 0 6 6 0 0 C m1 mN 6 6 1 6 0 n1 m1 6 6 .. . M ¼6 6 6 1 6 0 nN mN 6 6 0 n1 mN 6 6 6 0 4
3 0 7 nN 7 7 mN 7 7 7 7 7 7 7; 0 7 7 7 7 7 1 7 7 .. 7 . 5 1 2n1 2nN 0 0 0 0 0 .. . 0
0 n1 m1
mN 0 ðA þ B CÞ 0 6 0 0 6 ðA þ B CÞ 6 6 0 0 0 6 6 2n1 0 0 6 6 6 G ¼ xc 6 6 6 2n 0 0 6 N 6 6 2n1 0 0 6 6 6 4 2nN 0 0 2 ðC BÞ 0 0 0 6 0 4ðC AÞ 0 4n 1 6 6 6 0 0 3ðB AÞ 3m1 6 6 0 4n1 3m1 ðr1 =xc Þ2 6 6 6 K ¼ x2c 6 6 6 0 4nN 3mN 6 6 6 0 4n 3mN 1 6 6 6 4 0 2
nN
0
4nN
3mN
0
0
3 2nN 7 0 7 7 0 7 7 7 7 7 7 7; 7 7 7 7 7 7 7 7 5 0
4nN 3mN
4n1 3m1
2n1 0 0
0 0 ..
0
..
.
ð9Þ
0 4nN 3mN
0
. ðrN =xc Þ
2
ðr1 =xc Þ2 0
..
. ðrN =xc Þ2
3 7 7 7 7 7 7 7 7 7 7: 7 7 7 7 7 7 7 7 5
X.-S. Ge, Y.-Z. Liu / Chaos, Solitons and Fractals 37 (2008) 108–112
111
According to the Kelvin–Tait–Chetayev theorem [6], the stability of satellite relative the orbital frame is determined. The system is stable if the stiffness matrix K is positive definite. From the positive definite of the stiffness matrix K in Eq. (9), the stability criterion can be derived as C > B; C >Aþ8
N X
ðxc =rj Þ2 n2j ;
j¼1
hP i2 N N 48 ðxc =rj Þ2 mj nj X j¼1 : ðxc =rj Þ2 m2j þ B>Aþ6 P C A 8 Nj¼1 ðxc =rj Þ2 n2j j¼1
ð10Þ
Consider the following special cases: (1) If nj = 0 in the foregoing condition, the flexible plate becomes the flexible bar that has bending deformation along direction of axis x, and condition (10) can be converted into the stability criterion of spacecraft with a couple of flexible bars [7] C > B;
C > A; N X B>Aþ6 ðxc =rj Þ2 mj :
ð11Þ
j¼1
(2) If mj = nj = 0, condition (10) leads to the well-known attitude stability criterion of rigid body satellite in the gravitational field
ð12Þ
C > B > A: Define the following dimensionless parameters k, b, d and e as C k¼ ; A
B b¼ ; A
PN d¼
2 2 j¼1 mj ðxc =rj Þ
A
PN ;
e¼
2 2 j¼1 nj ðxc =rj Þ
A
ð13Þ
where d and e are small parameters. Retaining the 1st order of d and e only, condition (10) can be simplified into the form k > b;
k > 1 þ 8e;
b > 1 þ 6d:
ð14Þ
Introducing parameter plane with k and b as the axes of frame, one can shows the stable region with variable d in parameter (k, b) plane, and shows the stable region with variable e in parameter (k, b) plane. According to the discussions in this paper, it is shown that the reduction of Lagrange’s triangle region of stability in the parameters (k, b) plane with the increasing of the flexibility of plate. It reflects the influences of the bending and the torsion deformation of plate on the attitude stability of the spacecraft respectively.
Acknowledgment This work is supported by the National Natural Science Foundation of PR China (No. 10372014).
References [1] Cherchas DB, Houghes PC. Attitude stability of a dual-spin satellite with a large flexible solar array. J Spacecraft Rockets 1973;10(2):126–32. [2] Pfeiffer F, Pohl A. Dynamics of a satellite with a flywheel and flexible solar array. Raumfahrtforschung 1974;6:277–85. [3] Ibrahim AE, Misra AK. Attitude dynamics of satellite during deployment of large plate-type structures. J Guidance Control Dyn 1982;5(5):442–7.
112
X.-S. Ge, Y.-Z. Liu / Chaos, Solitons and Fractals 37 (2008) 108–112
[4] Liu YZ. Attitude stability of an unsymmetrical spacecraft with flexible plate in gravitational field. Acta Mech Solida Sinica 1994;15(4):296–302 [in Chinese]. [5] Meirovitch L. Elements of vibration analysis. McGraw-Hill; 1975. [6] Zajac EE. The Kelvin–Tait–Chetayev theorem and extensions. J Astronaut Sci 1964;11(2):46–9. [7] Liu YZ. The attitude dynamics of spacecraft. Beijing: National Defense Industry Press; 1995 [in Chinese].