Chaotic Behavior of the Biharmonic Dynamics System
Motion of a biharmonic system under action of small periodic force and small damped force is studied. The biharmonic oscillator is a physical system acting under a biharmonic force like . The article contains biharmonic oscillator analysis, phase space research, and analytic solutions for separatrixes. The biharmonic oscillator performs chaotic motion near separatrixes under small perturbations. Melnikov method gives analytical criterion for heteroclinic chaos in terms of system parameters. A transition from chaotic to regular motion of the biharmonic oscillator was found as the heteroclinic chaos can be removed by increasing the coefficient of a damping force. The analytical results obtained using Melnikov method have been confirmed by a good match with numeric research.
Chaotic behavior of a body in a resistant medium
Abstract We study the pitch motion dynamics of a rigid body in a resistant medium under the influence of a biharmonic torque a sin θ + b sin 2 θ . 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 θ and ψ 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.
Realizability of Reynolds-Stress Turbulence Models
It is shown that certain existing phenomenological models of turbulence in terms of differential equations for the Reynolds stresses Rαβ=〈u′α u′β〉 do not guarantee realizable solutions. The known realizability conditions are Rαβ⩾0 for α=β and Rαβ2⩾Rαα Rββ for α≠β. A stronger requirement is that the matrix Rαβ be positive semi‐definite. This implies three conditions like non‐negative eigenvalues or non‐negative principal invariants. Conditions are given which must be satisfied by the model itself in order to guarantee realizable solutions for any realizable initial and boundary conditions. Some means are proposed which can be used to change the existing models into realizable ones.