Arun K. Misra

Dynamics and stability of coaxial cylindrical shells containing flowing fluid

Abstract This paper presents an analytical model for the dynamics and stability of coaxial cylindrical shells conveying incompressible or compressible fluid in the inner shell and in the annulus between the two shells. Shell motions are described by Flugges thin-shell equations and the fluid forces are determined by means of linearized potential flow theory and formulated with the aid of generalized force Fourier transform techniques. Calculations have been conducted with two flexible shells or with one replaced by an identical rigid cylinder, with inner or annular flow, or both, mainly with incompressible flows but also with compressible ones; with steel or rubber shells conveying either water or air. It is found that, for the systems studied, annular flow destabilizes the system at lower flow velocities than flow in the inner shell and that the stability threshold is lower when both shells are flexible. The critical flow velocities are in the range of interest for industrial systems. The effect of compressibility is found to be small.

General formulation for N-body tethered satellite system dynamics

General three-dimensional motion of tethered TV-body systems is considered. Equations of motion are derived using the Lagrangian formulation. The derived equations are valid for large librational motion, variable lengths, and an arbitrary orbit. The elasticity and mass of the tethers are taken into account. Longitudinal and transverse displacements of the tethers are assumed to be small compared to the lengths and are discretized using the assumed-mode method. The nonlinear equations of motion are used for simulation of the system dynamics. For eigenvalue analysis and control applications, however, the equations need to be linearized; this is done analytically. Several examples are considered. Librational as well as longitudinal and transverse elastic frequencies of several multibody systems are obtained. A segmented-tether model is used to obtain the higher frequencies of a system. It is observed that the frequencies associated with the in-plane and out-of-plane motion of the system, a;/ and u>o, are related by (u;0/J7c)? w (w//fl c)? +1> where Hc is the orbital frequency. Typical transient responses of three-body and four-body tethered systems are obtained.

On the deployment dynamics of tether connected two-body systems☆

Abstract A general formulation of the deployment dynamics of tether connected two-body systems taking into account the three dimensional librational motion, longitudinal and transverse vibrations of the continuous tether, eccentricity of the orbit and aerodynamic drag in a rotating, oblate atmosphere, is presented. Three simple deployment procedures (uniform, exponential and their combination) are considered. In the beginning analytical solutions are obtained for the degenerate case of negligible vibrations in a circular orbit, which help in establishing trends for the more general situation. This is followed by the general dynamics of the system investigated through integration of the linearized autonomous equations of motion numerically. Typical plots are presented which describe the effects of various system parameters on the tether dynamics during deployment of a subsatellite from the Space Shuttle.

On the dynamics of curved pipes transporting fluid. Part I: Inextensible theory

This paper treats the dynamics and stability of curved pipes conveying fluid. The cases of pipes supported at both ends, as well as cantilevered pipes, are considered. A “modified” inextensible theory is developed, which retains the assumption of inextensibility of the centreline of the pipe as in the “conventional” inextensible theory, but the steady-state initial forces due to the centrifugal and pressure forces are nevertheless taken into account. The problem is solved via a finite-element formulation. The behaviour of curved pipes with supported ends according to the new theory is similar to that predicted by the extensible theories, i.e., the system does not lose stability as the flow velocity is increased, for either in-plane or out-of-plane motions; however, the new theory involves much less computational effort. For cantilevered pipes, the theory predicts a smaller critical flow velocity, as compared to the “conventional” inextensible theory. It is also observed that viscosity of the fluid has no effect on the dynamics, similarly to the case of straight pipes conveying fluid.

Attitude dynamics of three-body tethered systems

Abstract The dynamics of a tether-connected three-body system is investigated. The system is treated as a double-pendulum and the motion in the orbital plane is considered for the two cases of fixed-length tethers and variable-length tethers. For fixed-length tethers, it is noted that there are four possible equilibrium configurations: both tethers aligned along the local vertical; both tethers horizontal; and for certain combinations of parameters, two other configurations where one tether is along the local vertical while the other is inclined to the local vertical. Only the vertical equilibrium configuration is stable. Frequencies of oscillations around the stable configuration and corresponding modes are given in the paper. The dynamic response of the system during deployment of the three-body constellation is obtained. Dynamical behaviour during transportation of a cargo from one end-body to the other is also studied.

Dynamics of a 3-DOF spatial parallel manipulator with flexible links

In this paper, the dynamics of a three-degree-of-freedom (3-dof) spatial parallel manipulator with flexible links is studied. Finite elements are used for discretization of the flexible links, while the Euler-Lagrange formulation is used to derive the equations of motion of the uncoupled links. The equations of motion of all the links are then assembled to obtain the governing equations of the manipulator. The method of the natural orthogonal complement is used to eliminate the constraint forces and to derive the minimum number of equations of motion. To highlight the link flexibility effect, the governing equations of motion are used in a simulation of the manipulator at hand for both rigid and flexible-link models.

Effect of electrodynamic forces on the orbital dynamics of tethered satellites

Tethered satellites have a vast potential for space applications. If the tether is conductive, electrodynamic forces result from the motion of the conductive tether relative to the magnetic field of the Earth. The long-term effect of electrodynamic forces on the equinoctial orbital elements of a tethered system is studied with use of a detailed model of the Earth’s magnetic field as well as of the tether dynamics. A detailed gravity and aerodynamic model is used. In all cases studied, bare tethers induce larger electrodynamic forces than insulated tether systems. The results show that Lorentz forces can remove a satellite from orbit more effectively than air drag if a conductive tether is attached to it. A simple control law is used to stabilize tether librations caused by electrodynamic forces. Such an approach for junk removal is practical if the satellite mass is greater than 100 kg or so.

On the control of the space shuttle based tethered systems

Abstract Spatial dynamics of the Space Shuttle based tethered satellite system is investigated using a nonlinear model that accounts for the aerodynamic drag in a rotating oblate atmosphere. Results show that the normally unstable retrieval manoeuver can be stabilized satisfactorily using a nonlinear tether tension control strategy which depends on the tether length, its variation with time and pitch rate. Effectiveness of the control is illustrated through an example involving a 100 km tether supporting a proposed satellite for charting the earths magnetic field.

Thruster-augmented active control of a tethered subsatellite system during its retrieval

The paper considers control of the rotational motion as well as longitudinal and transverse vibrations of a tethered subsatellite system during its retrieval to the shuttle. Control using a set of thrusters alone is studied first; then, a length rate control law augmented by thrusters is examined. The functional forms of the thrust components applicable to both cases are determined by analyzing simplified equations of motion. These are validated by computer simulation of the original equations. The schemes considered appear to be fairly effective in arresting the growth of rotations as well as vibrations while maintaining a nonzero tension in the tether. It is recommended that the exponential retrieval be replaced by uniform retrieval rate toward the end of retrieval.

Three-dimensional dynamics and control of tether-connected N-body systems☆

Abstract General three-dimensional motion of tethered N-body systems is considered. The equations of motion derived are valid for large motion, variable length and arbitrary orbit, but the tethers are assumed to be massless and straight. It is shown that the out-of-plane libration frequencies are always related to the in-plane librational frequencies by the relation ( ω o j Ω 2 =( ω i j Ω ) 2 + 1 , j = 1, 2, …, N − 1, where Ω is the orbital frequency. Typical transient response of three-body tethered systems is shown. Possible reel rate laws to control both in-plane and out-of-plane motions are presented.