Resonance Effects when Moving a Small Spacecraft around the Center of Mass in a Deployable Tether System

Abstract—

The resonance motions of a small spacecraft relative to the center of mass when deploying a tether system are analyzed. The tether system is deployed from a base spacecraft moving in near-Earth orbit. The small spacecraft makes angular motions relative to the direction of the tether; in this case, the tension force of the tether varies according to a given program. The small spacecraft is a revolving body and is characterized by small static and dynamic asymmetries. The lowest-order resonance is analyzed, during implementing of which there is a “lunar” motion of the spacecraft relative to the tether direction. To study the resonance effects, we use the approximate nonlinear equations of motion of the spacecraft obtained by the method of integral manifolds using the asymptotic approach. The necessary conditions for the “capture” of the system into a resonance, i.e., the conditions under which the implementation of long-term resonance modes of the spacecraft motion is possible, have been obtained. The value of the probability of “capture” into a resonance is estimated and the conditions, under which the probability of “capture” is close to unity, are determined. A typical tether release program is considered that is analyzed from the point of view of the implementation of possible resonance effects. The results obtained using the approximate equations of motion are confirmed by numerical modeling using the initial equations of the angular spacecraft motion.

APPENDIX

Terms included in the initial equations of SS motion (1)–(5).

$$\begin{gathered} \Delta M_{x}^{d} = \frac{{\Delta J}}{J}\left\{ {2{{\omega }_{{zn}}}\frac{{{{K}_{{xt}}} - {{K}_{x}}\cos \alpha }}{{\sin \alpha }}\cos 2\varphi } \right. \\ + \,\,\sin 2\varphi \left. {\left[ {{{{\left( {\frac{{{{K}_{{xt}}} - {{K}_{x}}\cos \alpha }}{{J\sin \alpha }}} \right)}}^{2}} - \omega _{{zn}}^{2}} \right]J} \right\} \\ + \,\,\frac{{{{J}_{{yz}}}}}{J}\left\{ { - 2{{\omega }_{{zn}}}\frac{{{{K}_{{xt}}} - {{K}_{x}}\cos \alpha }}{{\sin \alpha }}\sin 2\varphi } \right. \\ + \,\,\cos 2\varphi \left. {\left[ {{{{\left( {\frac{{{{K}_{{xt}}} - {{K}_{x}}\cos \alpha }}{{J\sin \alpha }}} \right)}}^{2}} - \omega _{{zn}}^{2}} \right]J} \right\} \\ - \,\,{{J}_{{xzn}}}\frac{{({{K}_{{xt}}} - {{K}_{x}}\cos \alpha )}}{{J\sin \alpha }}\frac{{{{K}_{x}}}}{{{{J}_{x}}}} - {{J}_{{xyn}}}\frac{{{{K}_{x}}}}{{{{J}_{x}}}}{{\omega }_{{zn}}}, \\ \end{gathered} $$

$$\begin{gathered} \Delta M_{{zn}}^{d} = {{J}_{{xyn}}}\frac{{{{K}_{x}}}}{{{{J}_{x}}}}\left[ {\frac{{{{K}_{x}}}}{{{{J}_{x}}}} + \frac{{{{K}_{x}}\left( {1 + {{{\cos }}^{2}}\alpha } \right) - 2{{K}_{{xt}}}\cos \alpha }}{{J{{{\sin }}^{2}}\alpha }}} \right] \\ + \,\,2\Delta J\left[ {\left( {\frac{{{{K}_{x}}}}{{{{J}_{x}}}} + \frac{{{{K}_{x}}\cos \alpha - {{K}_{{xt}}}}}{{J{{{\sin }}^{2}}\alpha }}\cos \alpha } \right)} \right. \\ \times \,\,\left( {{{\omega }_{{zn}}}\sin 2\varphi + \frac{{{{K}_{x}}\cos \alpha - {{K}_{{xt}}}}}{{J\sin \alpha }}\cos 2\varphi } \right) \\ \left. { + \,\,\frac{{\left( {{{K}_{x}}\cos \alpha \, - \,{{K}_{{xt}}}} \right)\left( {{{K}_{x}}\, - \,{{K}_{{xt}}}\cos \alpha } \right)}}{{{{J}^{2}}{{{\sin }}^{3}}\alpha }}\, + \,\frac{{{{M}_{{zn}}}}}{J}{{{\sin }}^{2}}\varphi } \right] \\ - \,\,{{J}_{{yz}}}\left[ {2\left( {\frac{{{{K}_{x}}}}{{{{J}_{x}}}} + \frac{{{{K}_{x}}\cos \alpha - {{K}_{{xt}}}}}{{J{{{\sin }}^{2}}\alpha }}\cos \alpha } \right)} \right. \\ \times \,\,\left. {\left( { - {{\omega }_{{zn}}}\cos 2\varphi \, + \,\frac{{{{K}_{x}}\cos \alpha \, - \,{{K}_{{xt}}}}}{{J\sin \alpha }}\sin 2\varphi } \right)\, - \,\frac{{{{M}_{{zn}}}}}{J}\sin 2\varphi } \right], \\ \end{gathered} $$

$$\begin{gathered} \Delta M_{x}^{s} = T\sin \alpha \left( {\Delta y\sin \varphi + \Delta z\cos \varphi } \right), \\ \Delta M_{{zn}}^{s} = T\cos \alpha \left( { - \Delta y\cos \varphi + \Delta z\sin \varphi } \right), \\ \Delta M_{{xt}}^{{}} = \frac{{{{K}_{x}} - {{K}_{{xt}}}\cos {{\alpha }}}}{{\sin {{\alpha }}}}\Delta {{{{\omega }}}_{{zt}}} - J{{{{\omega }}}_{{zn}}}\Delta {{{{\omega }}}_{{yt}}}, \\ \end{gathered} $$

$$\begin{gathered} \Delta {{{\dot {\varphi }}}_{d}} = \frac{{{{J}_{{xyn}}}}}{{{{J}_{x}}}}\frac{{2{{K}_{x}}\cos \alpha - {{K}_{{xt}}}}}{{J\sin \alpha }} + \frac{{{{J}_{{xzn}}}}}{{{{J}_{x}}}}{{\omega }_{{zn}}} \\ + \,\,\frac{{{{J}_{{yzn}}}}}{{{{J}_{x}}}}{{\omega }_{{zn}}}{\text{cot}}\alpha + \frac{{\Delta {{J}_{n}}}}{J}\frac{{{{K}_{x}}\cos \alpha - {{K}_{{xt}}}}}{{J\sin \alpha }}{\text{cot}}\alpha , \\ \end{gathered} $$

$$\begin{gathered} \Delta {{{\dot {\psi }}}_{d}} = - \frac{{{{J}_{{xyn}}}}}{J}\frac{{{{K}_{x}}}}{{{{J}_{x}}\sin \alpha }} - \frac{{{{J}_{{yzn}}}}}{{J\sin \alpha }}{{\omega }_{{zn}}} \\ - \,\,\frac{{\Delta {{J}_{n}}}}{J}\frac{{{{K}_{x}}\cos \alpha - {{K}_{{xt}}}}}{{J{{{\sin }}^{2}}\alpha }}, \\ \end{gathered} $$

where

$$\begin{gathered} {{J}_{{yn}}} = J - \Delta {{J}_{n}},\,\,\,\,{{J}_{{zn}}} = J + \Delta {{J}_{n}},\,\,\,\,\,J = {{\left( {{{J}_{y}} + {{J}_{z}}} \right)} \mathord{\left/ {\vphantom {{\left( {{{J}_{y}} + {{J}_{z}}} \right)} 2}} \right. \kern-0em} 2}, \\ \Delta J = {{\left( {{{J}_{z}} - {{J}_{y}}} \right)} \mathord{\left/ {\vphantom {{\left( {{{J}_{z}} - {{J}_{y}}} \right)} 2}} \right. \kern-0em} 2}, \\ \end{gathered} $$

$$\begin{gathered} {{J}_{{xyn}}} = {{J}_{{xy}}}\cos \varphi - {{J}_{{xz}}}\sin \varphi , \\ {{J}_{{xzn}}} = {{J}_{{xz}}}\cos \varphi + {{J}_{{xy}}}\sin \varphi , \\ \end{gathered} $$

$$\begin{gathered} {{J}_{{yzn}}} = {{J}_{{yz}}}\cos 2\varphi + \Delta J\sin 2\varphi , \\ \Delta {{J}_{n}} = \Delta J\cos 2\varphi - {{J}_{{yz}}}\sin 2\varphi , \\ \end{gathered} $$

$\Delta {{{{\omega }}}_{{yt}}},\Delta {{{{\omega }}}_{{zt}}}$ are angular velocities of rotation of the coordinate system $c{{x}_{t}}{{y}_{t}}{{z}_{t}}$ [5].