ON THE DYNAMICS OF DUAL-SPIN SPACECRAFT CONTAINING A NUTATION DAMPER

The dynamics of dual-spin spacecraft which containing a proposed nutation damper which consisting of a ring totally filled with a viscous liquid with offset center, to improve damping, is investigated. The equations of motion were developed using Newton-Lagrange approach resulting equations in terms of spacecraft’s and damper’s parameters , which are given in dimensionless form. The expression of the nutation angle and time constant in both modes are developed using zero-order approximation technique. The equilibria states and stability condition, and the analytical expression for residual nutation angle were derived. The analytical results were compared with those found numerically using computer simulation program named MATLAB, ver. 7. Also the effect of various spacecraft’s and damper’s parameters on the dynamic and damping characteristics are discussed. The three dimensional graphical representation of the first and the second relative equilibria states are introduced. The numerical results are compared with the analytical for both modes of motion, where the percentage error of the time constant for nutation mode is less than (  3.6%)


INTRODUCTION
The attitude control system of the spacecraft is to control the attitude and position of the spacecraft as it performs its mission.The techniques that provide attitude stabilization and control of spacecraft are; passive control system, semi-passive and active control system (15) .
The type of the system adopted in the present study is the passive type system.Passive system does not require any external power source, once they are in place, they use gravity or momentum to create the necessary control forces and moments (19) .
Dual-spin stabilization type is the method of attitude stabilization adopted in the present study.A spin and Dual-spin stabilized spacecraft, or spinners, utilizes its own spinning motion to keep it's self aligned in a certain inertial direction.The spinning motion creates stiff angular momentum vector, which tend to resist external disturbance torques.A spinner is stable if it is spun about the axis of largest principal moment of inertia, if it is spun about a different axis, any disturbance could cause the spin axis to shift to the major axis.
In single spin stabilization the whole body rotates about the axis of maximum principal moment of inertia.Early communication satellites, such Syncom I, ATS I, II and Inelsat I, II were single spin stabilized.Its advantages are simple, reliable, and long life time but the main limitations of these satellites are that they could not use earth oriented antennas.These limitations are overcome in a dual spin spacecraft.Whereas dual spin spacecraft consisting of spinning rotor producing gyroscopic stiffness and a platform rotating at a much slower rate in accordance with the desired attitude of the spacecraft.There are two types which are commonly known as the external rotor and body stabilized spacecraft, each type employs a different method of attitude stabilization.The external rotor type or "Gyrostat" uses spin stabilization where the rotor of relatively large moment of inertia rotates to provide gyroscopic stiffness, while the platform usually containing communication equipment and antennas are despun.
Chang, and Liu (4) , studied the dynamic and stability of an inertially symmetric, spinning, rigid body with a partial filled viscous-ring damper mounted normal to the spin axis.They used the nonlinear equations directly by using center manifold theory; they generated the stability criteria and the decay time constant.Then Alfriend (2) , studied the attitude stability of  (31) , studied the ball in ring nutation damper, he utilized neon gas and many percentage of glycerin water mixture as damping fluid in his study.
The present work represents an attempt to study the full filled viscous ring damper mounted normally to the spin axis with offset centre (d) from the spin axis.

-EQUATION OF MOTION
The total angular momentum can be written in terms of angular velocity component: When the external torque components are zero, the system referred to as a freely precessing system, then the principle of conservation of angular momentum can be applied, such that (21) :  the equation which describes the motion of the fluid inside the ring, can be obtained by using Lagrange's equation expressed in terms of quasi-coordinate.
where  Q is the generalized moment associated with the generalized coordinate  , and it is given by: The kinetic energy (T) of the system is in the form: Using Eqs.1, 4 and 5, then Eqs. 2 and 3 yields: where, ( )′ = (d/dt), D 1 & D 2 are given in the appendix

-SOLUTION OF THE PROBLEM
Before developing the solution for the attitude equations of the spacecraft, the equation which is describing the nutation angle in terms of dimensionless parameters and variables, will be developed.
differentiate Eq. 10a and substitute in Eq. 10b gives: from Eq. 2, the time derivative of the spin axis angular momentum vector h z , may be given by (in dimensionless angular velocity variables (p, q)): Substitute Eq. 11 into Eq.12, yields the following equation: Substitute h u , h v , h t and h uz into Eq.13 yields (details in appendix): 14 apply zero-order approximation procedure to Eqs. 6, 7, 8, and 9 to get: The solution of the angular velocity component r is obtained from Eq. 17, with the fact that the initial value of the r-components equal to 1, then the solution of Eq. 17 is: r=1 19 substitute the solution of the r-component Eq. 19 in Eqs. 15 and 16, then one get: where, the solution of p and q are given by: 22, 23 substitute Eqs.22 and 23 into Eq.14, then Eq. 14 becomes: Now, it is required to express for t  in terms of the nutation angle .Divide Eq. 10b by Eq. 10a and substitute t h and z h then apply zero-order approximation to get: where, p given by Eq. 15 into Eq.18 results:

DAMPER MOTION
The symmetric rigid body, which is spinning about its axis of symmetry, has a constant nutation angle when no damping is present.The transverse angular velocity and the body rotates relative to when no damping is present, a plane containing the angular momentum vector Eventually the component of the centrifugal force is not large enough to balance the damping and friction forces, and the fluid begins to decelerate and oscillate until becomes at rest.This type of motion is called "spin-synchronous" motion.

Nutation-Synchronous Mode
Let  measure the position of the center of a portion of the fluid with respect to the nutation plane.Assuming that at Substituting for  in Eq. 26 and using Eqs. 22, 23then Eq. 26 becomes: where, n  refer to the damping constant in the nutation-synchronous mode.
Since that, the motion of the fluid in this mode is constant, then the solution of Eq.28 is, , substituting in Eq.28 and take into account that into Eq.24, then 24 becomes: where, n  refer to the nutation angle in the nutation-synchronous mode.
Substituting for the left hand side of Eq.29 in Eq.30, and using Eq.25 then, Carrying out the integration, then the nutation angle n  is given by: 0 cos cos where, 0 n  is the initial value of n  , n  is the time constant of the system which it is given by: At the end of the nutation-synchronous mode, the system goes into the spin-synchronous mode and the nutation angle n  gain its minimum value.Referring to Eq.29, to satisfy the condition of minimum value of the nutation angle in this mode, the angle s  should be equal to 2   substituting for this value and for t  from Eq.25, then;

Spin-Synchronous Mode
In the Spin-Synchronous mode, the spacecraft becomes more closely to the state of pure spin about the spin axis (z-axis).Accordingly, the relative speed of the fluid, about the spin axis with respect to the spinning rotor, will be decreased.Eventually the relative speed between the fluid and the spinning rotor becomes zero, then the spacecraft spin axis is aligned with the initial direction of the total angular momentum vector.In the spin-synchronous mode, it can be shown that the fluid is moving with a small variation in its speed with respect to the damper ring therefore it is necessary to find the solution for the motion of the fluid ( ) as a function of dimensionless time ( ), substituting for p and q from Eqs.22 and 23 into Eq.26,yields: As mentioned above that the fluid moving with a small variation in its speed, then the following equation can be assumed: where, 0  is the initial value of  and  ~ represent the small variation of the speed of the fluid such that   0  , which gives the following expressions: The basis of the above assumptions is that the change in  is small compared with   n .Using the above assumption, then Eq.35 becomes: assume the forced oscillating solution of Eq.38 is given by to find the constants A and B, substituting Eq.39 in Eq.38, and after some mathematical manipulations we get: 1 substitute A and B in Eq. 39, and substitute for t  , then the expression of  ~ becomes: where s  referred to the nutation angle in the spin-synchronous mode and the constant F is given by: The differential equation of the nutation angle rate Eq.24 is given by: it was mentioned that  ~represents small variation in  such that , where   0  .using the approximation of small angle, then , 42 using the above mentioned relations, then the expression of may be given by 43 Substituting  ~ from Eq.40 in Eq.43 ( s  is small in the spin-synchronous mode ), then (by expanding the trigonometric terms) Eq.41 becomes: Eq.45 can be written in terms of spin-synchronous time constant as: where, one can see that the nutation angle time history consists of dominant exponential decay super imposed on it an oscillation of small amplitude represents the effect of the trigonometric terms.So that, the general solution of Eq.46 is given by: where, sc  : is the complementary part of the, ..I sP  : is the particular part of the solution.These solutions may be given by: cos sin 50 substitute Eq.50 in Eq.46 to get: substituting the solution of Eq.50 in Eq.48 and using the initial condition ( 0 s s    at time 0    ) then the constant c is given by:   (2) .One can see that the trend of results of the present work is acceptable in comparison with Alfriend (2) .Figure (3) shows the comparison of the nutation angle time history of this work compared with that predicted by Alfriend (2) , for nutation-synchronous mode.This figure shows that the time constant in this work is decreased compared with the time constant Alfriend (2) .In figure (4), the comparison of the nutation angle time history for both nutation-synchronous mode and spin-synchronous mode is shown.It is seen that analytical solution very well agrees with the numerical solution.The time constant for nutation-synchronous mode obtained analytically is (99981) and numerically is (96428.5)which means that the percentage error is less than (  3.6%), and this means that the analytical solution predicts the time constant very well, and for spin-synchronous mode, it could be seen the numerical is (866.66)and the analytical is (802.17),i.e. the percentage error is (  8%).
Figures (5, 6, 7, 8, 9 and 10)   15).It is shown that even the system being at a point in neighborhood of the second relative equilibrium state, it will converge to the first relative equilibrium state.This is because that the system parameters satisfy the stability condition of the first relative equilibrium state.

CONCLUSIONS
From the results shown, it is concluded that a good agreement was obtained between the analytical and numerical solutions.The proposed damper overcomes the problems of the spreading and sloshing which occur in the partially filled nutation dampers.Utilizing fluids with high damping coefficient will decrease the time constant in both modes of motion.

Appendix
, and

A
Principal transverse moment of inertia of the spacecraft kg.m 2 b Ratio of the ring height to the ring mean radius -C, C p Moment of inertia of the rotor and the platform along z -axis kg.the fluid to the transverse inertia of the spacecraft η Damping constant of the damping fluid θ Nutation angle of the spacecraft rad θ n Nutation angle in the nutation-synchronous mode rad

Fig
Fig. A: Body fixed coordinate system.

Fig. C :
Fig.C: Spacecraft model with viscous ring damper.
viscous friction between fluid and ring wall.(Ns/m) axis, called the nutation plane, is formed.the fluid is then moving at a constant rate of ), with respect to the body.At the same time, the fluid subjected to centrifugal force due to the relative rotation (spin) about the z-axis.This type of motion is called "nutation-synchronous" motion.In this mode the fluid is moving at a constant rate with respect to the spacecraft (damper ring); hence the energy dissipation rate is constant.If  >1the nutation angle decreases which cause a decrease in the centrifugal force.
.48 the complete solution of the nutation angle s  is given by: and (2) shows the nutation angle time history prepared in this work and that presented by Alfriend (2) respectively, where figure (2) represent the experimental work of Alfriend Figures(5, 6, 7, 8, 9 and 10) show the variation of the time constant with the inertia ratio  , ring mean radius (R), and damping constant, respectively.The variation of the time constant with the ratio of the ring height to the ring mean radius (b) for spin-synchronous mode is shown in Fig. (11).In Fig. (12), the variation of the time constant with the distance of offset center (d) is shown.Figure (13) shows the degradation of p component while, the time history of the r component of the spacecraft angular velocity for the first relative equilibrium state is shown in Fig. (14).A three dimensional visualization of the first relative equilibrium state is shown in Fig. (15).It is shown that even the system being at a point in neighborhood

Fig
Fig. (1).Nutation angle time history of present work.

Fig. ( 3
Fig. (3).Comparison of the nutation angle time history of present work with ref.(2) for the nutation-synchronous mode.

Fig. ( 8 )
Fig. (8) Influence of the ring mean radius on the time constant for spin-synchronous mode.

Fig. ( 9 )
Fig. (9) Influence of the damping constant on the time constant for nutation-synchronous mode.

Fig. ( 11 )
Fig. (11) Influence of the ratio of the ring height to the ring mean radius, b on the time constant for spinsynchronous.mode. .

Fig. (
Fig. (15) Three dimensional visualization shows that the spacecraft diverge from second relative stability and reaches to