Computational dynamics of a 3D elastic string pendulum attached to a rigid body and an inertially fixed reel mechanism

Abstract

A high fidelity model is developed for an elastic string pendulum, one end of which is attached to a rigid body while the other end is attached to an inertially fixed reel mechanism which allows the unstretched length of the string to be dynamically varied. The string is assumed to have distributed mass and elasticity that permits axial deformations. The rigid body is attached to the string at an arbitrary point, and the resulting string pendulum system exhibits nontrivial coupling between the elastic wave propagation in the string and the rigid body dynamics. Variational methods are used to develop coupled ordinary and partial differential equations of motion. Computational methods, referred to as Lie group variational integrators, are then developed, based on a finite element approximation and the use of variational methods in a discrete-time setting to obtain discrete-time equations of motion. This approach preserves the geometry of the configurations, and leads to accurate and efficient algorithms that have guaranteed accuracy properties that make them suitable for many dynamic simulations, especially over long simulation times. Numerical results are presented for typical examples involving a constant length string, string deployment, and string retrieval. These demonstrate the complicated dynamics that arise in a string pendulum from the interaction of the rigid body motion, elastic wave dynamics in the string, and the disturbances introduced by the reeling mechanism. Such interactions are dynamically important in many engineering problems, but tend be obscured in lower fidelity models.

High-fidelity numerical simulation of complex dynamics of tethered spacecraft

“…This is an extension of preliminary work that studies a string pendulum model with a reeling mechanism [15,16]. The first part of this paper provides a realistic and accurate analytical tethered spacecraft model including tether deformations, attitude dynamics of rigid bodies, and a reeling mechanism.…”

“…Now, we focus on the first term of (15). We cannot apply integration by parts with respect to s, since the order of the integrals cannot be interchanged as s p (t) depends on t. Instead, we use Green's theorem [15]:…”

Geometric numerical integration for complex dynamics of tethered spacecraft

“…Here, we cannot apply integration by parts with respect to s, since the order of the integrals cannot be interchanged due to the time dependency in the variable s p (t). Instead, we use Green's theorem [7],…”

“…pendulum model with a reeling mechanism [7], [8]. The first part of this paper provides a realistic and accurate analytical tethered spacecraft model including tether deformations, attitude dynamics of rigid bodies, and a reeling mechanism.…”

Dynamics and control of a chain pendulum on a cart

“…For example, the dynamics of a double spherical pendulum and a Lagrange top have been studied in [3], [4], [5]. Generalized models, such as a 3D pendulum [6] or a 3D pendulum attached to an elastic string [7], have been considered. A variety of control techniques have been applied, such as passivitybased approaches [8], [9], swing-up strategies [10], [11], Lyapunov-based method [12], controlled-Lagrangian [13], and hybrid control systems [14].…”