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We study the spin-up dynamics of a dual-spin spacecraft containing one axisymmetric rotor which is parallel to one of the principal axes of the spacecraft. It will be supposed that one of the moments of inertia of the platform is a periodic function of time and that the center of mass of the spacecraft is not modified. Under these assumptions, it i...
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... relative orientation between these two reference frames results from three consecutive rotations involving the Euler angles (ψ, θ, φ) [Goldstein, 1992, pp. 183-188] (see Fig. ...
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... Fig. 9 indicates, for a fixed frequency, that the bigger the amplitude ε the greater the pa- rameter q, suggesting a possible linear dependence of ε. On the other hand, when ε is fixed, Fig. 10 indicates that for low frequencies the parameter q is close to zero increasing its value to reach a max- imum and to decrease asymptotically to zero, ac- cording to the two integrable limits for ν = 0 and ν → ∞. This behavior is observed for the width of the stochastic layer when the rotors are at rest. In fact, the width of the ...
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... order to show the connection between these two measures of the amount of apparent chaotic mo- tion, we proceed to plot the parameter q as a func- tion of the amplitude ε and the frequency ν. Fig- ure 11(a) indicates, for a fixed frequency ν = 0.3, that q behaves as a linear function, as we expected. On the other hand, Fig. 11(b) shows, for a fixed amplitude ε = 0.01, that the parameter q increases from zero until it reaches a maximum to decrease asymptotically to zero. ...
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... order to show the connection between these two measures of the amount of apparent chaotic mo- tion, we proceed to plot the parameter q as a func- tion of the amplitude ε and the frequency ν. Fig- ure 11(a) indicates, for a fixed frequency ν = 0.3, that q behaves as a linear function, as we expected. On the other hand, Fig. 11(b) shows, for a fixed amplitude ε = 0.01, that the parameter q increases from zero until it reaches a maximum to decrease asymptotically to ...
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... behavior observed in Fig. 11 is the same observed in Fig. 4 corresponding to the graph of the width of the stochastic layer as a function of the amplitude ε and the frequency ν. To emphasize this point we plot in Fig. 12 both the graph of pa- rameter q and 100 ∆H, a hundred times the width of the stochastic ...
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... behavior observed in Fig. 11 is the same observed in Fig. 4 corresponding to the graph of the width of the stochastic layer as a function of the amplitude ε and the frequency ν. To emphasize this point we plot in Fig. 12 both the graph of pa- rameter q and 100 ∆H, a hundred times the width of the stochastic ...
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... = 0.01, only good agreement is observed for low and high frequencies, that is, near the two integrable limits and it is in an intermediate range of frequencies where we observe strong devia- tions. Moreover, these deviations are also observed plotting q as a function of the amplitude ε in the intermediate range of frequencies as it is depicted in Fig. 13 for ν = ...
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... begin with, we show a graphical description of this phenomenon where we notice the sudden increment of the stochastic layer when a resonant orbit is en- gulfed. Figure 14(a) shows a 10:1 resonant orbit near the border of the stochastic layer, just before it is absorbed. In Fig. 14(b) we appreciate the ab- sorption of the resonant orbit and the sudden incre- ment of the width of the stochastic layer due to the width of the resonant configuration, the resonant bandwidth. ...
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... be explained taking into account the effect of nonlinear resonances. To begin with, we show a graphical description of this phenomenon where we notice the sudden increment of the stochastic layer when a resonant orbit is en- gulfed. Figure 14(a) shows a 10:1 resonant orbit near the border of the stochastic layer, just before it is absorbed. In Fig. 14(b) we appreciate the ab- sorption of the resonant orbit and the sudden incre- ment of the width of the stochastic layer due to the width of the resonant configuration, the resonant bandwidth. This abrupt increment for the values of Fig. 14 matches with the increment detected nu- merically and depicted in Fig. ...
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... 14(a) shows a 10:1 resonant orbit near the border of the stochastic layer, just before it is absorbed. In Fig. 14(b) we appreciate the ab- sorption of the resonant orbit and the sudden incre- ment of the width of the stochastic layer due to the width of the resonant configuration, the resonant bandwidth. This abrupt increment for the values of Fig. 14 matches with the increment detected nu- merically and depicted in Fig. ...
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... this case we can calculate the values of the fre- quency ν for which the stochastic layer and the res- onant band overlap. Figure 15 shows the overlap for a gyrostat with a 10 = 0.1, a 2 = 0.2, a 3 = 0.3 and ε = 0.01. We can appreciate that the over- lapping begins for ν ≈ 0.1 (for smaller values the resonant band is entirely inside the stochastic layer) and it finishes for ν ≈ 0.15 (for greater values the resonant band is outside the stochastic layer disap- pearing for ν great enough). ...
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... 10 = 0.1, a 2 = 0.2, a 3 = 0.3 and ε = 0.01. We can appreciate that the over- lapping begins for ν ≈ 0.1 (for smaller values the resonant band is entirely inside the stochastic layer) and it finishes for ν ≈ 0.15 (for greater values the resonant band is outside the stochastic layer disap- pearing for ν great enough). Note that the forming pick in Fig. 15 explains the abrupt increment for both the numerical estimation of the width of the stochastic layer and the parameter q in the range of frequencies between 0.1 and 0.15 [compare Fig. 15 with Figs. 6 and ...
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... it finishes for ν ≈ 0.15 (for greater values the resonant band is outside the stochastic layer disap- pearing for ν great enough). Note that the forming pick in Fig. 15 explains the abrupt increment for both the numerical estimation of the width of the stochastic layer and the parameter q in the range of frequencies between 0.1 and 0.15 [compare Fig. 15 with Figs. 6 and ...
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Citations
... It is a known fact, that a rigid body under the action of external perturbations or in the presence of internal asymmetric rotators or movable elements can have chaotic regimes of the angular motion [5,13,15,20,21,[25][26][27][30][31][32][33]35,39,40,[48][49][50]. ...
... As we see from (49), the sliding parameter t 0 now ''shifts'' along the τ -axis the graph of the factor f (τ − t 0 ), which is analytically defined by the expression (31). The factor ϕ 1 (τ ) corresponds to the time-dependence of the small perturbation (in the present case it is a small angle α(τ )). ...
... Here we should give some comments on the improper integration during the calculation of the Melnikov integral in the form (48) with the integrand (49). ...
... A gyrostat G is a mechanical system made up of a rigid body P, called the platform, and other bodies R, called the rotors, connected to the platform in such a way that the motion of the rotors does not modify the distribution of mass of the gyrostat G. Due to this double spinning, the platform, on the one hand, and the rotors, on the other, the gyrostat is also known under the name of a dual-spin body, especially in astrodynamics, where these artifacts are widely used in spacecraft dynamics in order to stabilize their rotations; see, e.g., [1][2][3][4][5]. ...
... In this way, most of the works are devoted to the consideration of external torques or inner perturbations. For instance, some authors assume elasticity or periodic time dependence of the moments of inertia [4,5], while other authors focus on the attitude dynamics of a gyrostat rotating and moving on a circular orbit [15,19,20], or under the action of a uniform gravity field [21][22][23][24][25][26][27][28][29]. ...
In this paper, we consider the motion of an asymmetric heavy gyrostat, when its center of mass lies along one of the principal axes of inertia. We determine the possible permanent rotations and, by means of the Energy-Casimir method, we give sufficient stability conditions. We prove that there exist permanent stable rotations when the gyrostat is oriented in any direction of the space, by the action of two spinning rotors, one of them aligned along the principal axis, where the center of mass lies. We also derive necessary stability conditions that, in some cases, are the same as the sufficient ones.
... Volterra is the first one who announces the concepts of a gyrostat to investigate Earth's polar axis' motion and illustrate variations in the Earth's latitude by means of the internal motion that does not alter the planet's mass distribution [2]. Also, it has multiple implementations in distinct categories of science such as Astrodynamics (see, e.g., [3][4][5][6][7]). The majority of studies related to a rigid body-gyrostat is sorted into three categories. ...
This work is devoted to investigating the stability of motion of a rigid body-gyrostat in an incompressible ideal fluid. This motion is assumed to happen under the action of neutral forces (buoyancy and gravity) whose centers are not coincide. The equations of motion are introduced, and they are expressed by means of the Hamiltonian function in the framework of the Lie-Poisson system. We prove that the problem of motion of a gyrostat in an incompressible ideal fluid is equivalent to the general problem of the motion of a rigid body under the action of a combination of potential and gyroscopic forces. We endeavor for studying the stability of two possible types of stationary solutions which describe the spin motion and non-spin motion of a gyrostat. For the non-spin motion, we consider two stationary solutions for which the translation motion is either in direction of the gravity or in the perpendicular direction on it. For the spin motion, we consider an stationary solution describing physically translation along and rotation about the same axis, say, the third one. The linear approximation method is applied to get the sufficient conditions for instability or in other words, the necessary conditions for stability. The energy-Casimir method is utilized to provide sufficient conditions for stability.
... A fairly wide class of problems concerns the control of the orientation of satellites by using moving internal masses and rotors. In this case, the dynamics of the system is described by the Euler -Poisson equations with a gyrostat and variable moments of inertia in the general case (Liouville equations) [19,23,24]. The properties of the Euler -Poisson equations were studied, for example, in [8,13,17]. ...
... Gray et al. [20,21] have modeled a flexible satellite with a small flexible appendage and a dissipation that drives the satellite from minor to major axis spin. There are other important studies, such as Meehan et al. [22], who investigate the onset of chaotic instability in a rotating satellite with a small spring-mass-damper, Baozeng [23,24], who studies chaos in a liquid-filled flexible spacecraft with a small appendage, and Iñarrea et al. [25][26][27], who study the effects of a time-periodic moments of inertia in a satellite. ...
... In this section, we analyze chaos in the satellite using Melnikov-Wiggins method [31][32][33]. First, we divide the Hamiltonian in Eq. (27) into two parts, namely the integrable and perturbation parts. Then, we reduce the perturbation part of the Hamiltonian by assuming the relative angles of the panels θ to be small. ...
... To divide the Hamiltonian associated with the integrable and perturbation, first, we divide the matrix I −1 t in Eq. (27) which is a function of the variables θ. This matrix can be divided into two parts as follows: ...
In this paper, we analytically and numerically investigate chaos in attitude dynamics of a flexible satellite composed of a rigid body and two identical rigid panels attached to the main body with springs. Flexibility, viewed as a perturbation, can cause chaos in the satellite. To show this, first, we use a novel approach to define this perturbation. Then, we employ canonical transformation to transform the Hamiltonian of the system from five to three degrees of freedom. Next, we approximate the system by a second-order differential equation with a time quasiperiodic perturbation. Finally, we apply Melnikov–Wiggins’ method near the heteroclinic orbits to prove the existence of chaos. Using the maximum value of Melnikov–Wiggins function and the small perturbation parameter, we find a tool to predict the size of the chaotic layers. Results show that this approach is useful even if the panels are not small. In addition, it is observed that though the satellite attitude dynamics is chaotic, in many cases the width of chaotic layers is very small and therefore negligible.
... This is signified by virtue of two important facts, namely, the simplifying assumptions made in the equations of motion, and ignoring the effect of the nonlinear resonance phenomena, which may occur in different sets of parameters and initial conditions and can change the width of chaotic layers. 35 Case study II: Effect of the size of the rotational angular velocity ...
In this paper, chaos in spatial attitude dynamics of a triaxial rigid satellite in an elliptic orbit is investigated analytically and numerically. The goal in the analytical part is to prove the existence of chaos and then to find a relation for the width of chaotic layers (i.e. the initial values needed to have a chaotic attitude motion) based on the parameters of the system. The numerical part is aimed at validating the analytical method using the Poincaré maps and the maximum value of the Lyapunov exponents. The rotational–translational Hamiltonian of the system is first derived. This Hamiltonian has six degrees of freedom. Choosing a proper set of coordinates and given the fact that the total angular momentum is constant, the Hamiltonian is then reduced to a four-degree-of-freedom system. Assuming the effect of attitude on the orbital dynamics to be negligible, and assuming a nearly symmetric and fast-spinning satellite, the system is approximated by a second-order differential equation with a time quasi-periodic perturbation. Next, the Melnikov–Wiggins’s method is used to prove the existence of a chaotic behavior followed by the determination of an analytical relation for the width of chaotic layers. Although in the analytical method some restrictive assumptions are enforced, the results show that the analytical relation gives a good estimate for the width of chaotic layers even if these assumptions are not entirely satisfied. The results also show that this method is useful for finding the effects of all the parameters (the orbit and the satellite) and the initial values on the existence of a regular behavior.
... This is an integrable problem and the stability and bifurcations of the permanent rotations have been studied in detail by many authors from different points of view, especially in the context of spacecraft attitude dynamics [5,13,15,17,18,21]. These studies help in the analysis of the motion of the gyrostat under small perturbations, giving rise to chaotic behaviors which can be controlled by the action of the rotors [4,24,28,29]. ...
... It is worth noting that, in this case, as z 0 = 0, the center of mass G of the gyrostat coincides with the fixed point O. Therefore, although the gyrostat is under the action of a uniform gravitational field, the weight of the system is applied in the fixed point O. Thus, the gravitational torque N about O is zero, and then we have the case of an asymmetric gyrostat in free rotational motion about O [16,24,29]. Moreover, for l 1 = 0 we have the well-known problem of the free triaxial rigid body. ...
We consider the motion of an asymmetric gyrostat under the attraction of a uniform Newtonian field. It is supposed that the center of mass lies along one of the principal axes of inertia, while a rotor spins around a different axis of inertia. For this problem, we obtain the possible permanent rotations, that is, the equilibria of the system. The Lyapunov stability of these permanent rotations is analyzed by means of the Energy–Casimir method and necessary and sufficient conditions are derived, proving that there exist permanent stable rotations when the gyrostat is oriented in any direction of the space. The geometry of the gyrostat and the value of the gyrostatic momentum are relevant in order to get stable permanent rotations. Moreover, it seems that the necessary conditions are also sufficient, but this fact can only be proved partially.
... This type of chaos in fact was fundamentally predicted in the work of Poincaré [51], and also were described in applied tasks of spacecraft (SC) attitude dynamics in many works, e.g. [12,[17][18][19][20][21][27][28][29]. ...
In this work the chaos in dynamical systems is considered as a positive aspect of dynamical behavior which can be applied to change systems dynamical parameters and, moreover, to change systems qualitative properties. From this point of view, the chaos can be characterized as a hub for the system dynamical regimes, because it allows to interconnect separated zones of the phase space of the system, and to fulfill the jump into the desirable phase space zone. The concretized aim of this part of the research is to focus on developing the attitude control method for magnetized gyrostat-satellites, which uses the passage through the intentionally generated heteroclinic chaos. The attitude dynamics of the satellite/spacecraft in this case represents the series of transitions from the initial dynamical regime into the chaotic heteroclinic regime with the subsequent exit to the final target dynamical regime with desirable parameters of the attitude dynamics.
... Also, it has various applications in diverse branches of science such as Astrodynamics. For instance, it is used as a control device in spacecrafts for stabilizing their rotations (see, e.g., [17][18][19][20][21]). Moreover, the majority of the problems concerning the rigid body-gyrostat can be summarized in the following: ...
... On the stability of the permanent rotations of a gyrostat 3951 Equation (20) holds in four possible cases. They are (γ 1 = 0, γ 2 = 0) or (γ 1 = 0, γ 2 = 0) or (γ 1 = 0, γ 2 = 0) and (γ 1 = γ 2 = 0). ...
... • For γ 1 = 0 and γ 2 = 0, the two Eqs. (19) and (20) are identically satisfied. Taking into account expression (1) and Casimir (13), we obtain γ 2 = sin θ and γ 3 = cos θ . ...
We consider the motion of a charged rigid body about a fixed point carrying a rotor that is attached along one of the principal axes of the body. This motion occurs under the action of the resultant of the uniform gravity field and the homogeneous magnetic field. The equations of motion are formulated, and they are presented by means of the Hamiltonian function in the framework of the Lie–Poisson system. These equations of motion have six equilibrium solutions. The sufficient conditions for instability for these equilibria are studied by utilizing the linear approximation method, while the sufficient conditions for stability are presented by means of the energy-Casimir method. For certain configuration of the body, the regions of Lyapunov stability and instability are determined in the plane of some parameters. Furthermore, we clarify that the regions of Lyapunov stability are a portion of the regions of linear stability.
... In Astrodynamics, gyrostats play an essential role, since they are used for controlling the attitude dynamics of a spacecraft and for stabilizing their rotations. See, for instance, Cochran [8] , Hall [15][16][17] , Elipe and coworkers [10][11][12][13]20,26] , Vera [37] , Aslanov [4,5] and also Hughes [19] for further references. Besides its practical interest, the rotational motion of a gyrostat is very interesting from a mathematical point of view. ...
The stability of the permanent rotations of a heavy gyrostat is analyzed by means of the Energy-Casimir method. Sufficient and necessary conditions are established for some of the permanent rotations. The geometry of the gyrostat and the value of the gyrostatic moment are relevant in order to get stable permanent rotations. Moreover, the necessary conditions are also sufficient, for some configurations of the gyrostat.
