Chaotic behavior of a body in a resistant medium

• Chaotic attitude motion of a rigid body in a resistant medium is studied.
• A biharmonic torque and small periodic perturbation act on the body.
• The biharmonic torque is an odd function of a nutation angle.
• Separatrix orbits and Melnikov function are determined in an analytical form.

We study the pitch motion dynamics of a rigid body in a resistant medium under the influence of a biharmonic torque asinθ+bsin2θ. Such nutation angle dependence of the biharmonic aerodynamic torque is typical for uncontrolled re-entry vehicles of segmentally conical, blunted conical, and other shapes (Soyuz, Mars, Apollo, Viking, Galileo Probe, Dragon). The presence of the second harmonic in the biharmonic torque is the cause of additional unstable equilibrium. In case of spatial motion a small perturbation is a small difference of the transverse inertia moments of the body. In this case, two Euler angles θθand ψψ are the positional coordinates, and we can observe a chaos. In case of the planar motion the body is perturbed by a small aerodynamic damping torque and a small periodic torque of time. We show by means of the Melnikov method that the system exhibits a transient chaotic behavior. This method gives us an analytical criterion for heteroclinic chaos in the planar motion and an integral criterion for the spatial motion. The results of the study can be useful for studying the chaotic behavior of a spacecraft in the atmosphere.