Two Specialized Structural Elements for Airdrop System Modeling

Abstract

Two new structural elements are presented that were developed specifically for modeling airdrop systems. The first element is used to model Pneumatic Muscle Actuators, which have recently been used as control devices in several new airdrop systems. The second element is called a Sliding Cable Element, which is used in the current paper to model a parachute with slider reefing. Both elements are based on a threedimensional, geometrically nonlinear, transient finite element formulation and can easily be implemented in any general-purpose finite element code. In the current paper, the formulation of these two new elements is briefly discussed. Several examples are given to verify the behavior of the elements. Finally, several simulations are presented to demonstrate the application of the elements to modeling airdrop systems. Introduction Pneumatic Muscle Actuator Element Pneumatic Muscle Actuators (PMAs) consist of a braided fiber cylinder with an interior pressure bladder. When pressurized, the radial and axial dimensions of the PMA change and the braided fibers rotate to a new equilibrium configuration. The fiber rotation and dimensional changes can be quite large and, therefore, the relation between these variables is generally nonlinear. The ability to change the PMA length and corresponding axial force by changing the internal pressure makes PMAs suitable for control devices. PMAs have been used extensively for robotics applications (see, for example, Tondu et al 1 ) and have ____________________________________________ Copyright © 2003 The American Institute of Aeronautics and Astronautics Inc. All Rights Reserved. recently been used for airdrop applications, such as soft landing (see, for example, Brown et al 2 ) and for steering control (see, for example, Brown et al 3 ). Tondu et al developed an analytical PMA model based on the assumption that the PMA fibers are inextensible. This model has been shown to be reasonably accurate for predicting the nonlinear behavior of PMAs. In the current paper, this analytical model is generalized to develop a PMA element that is implemented in a three-dimensional geometrically nonlinear finite element program. The new PMA element has a two node linear geometry and therefore is easy to include in a general finite element model. Analytical expressions for the PMA internal force vector and tangent stiffness matrix are determined. Sliding Cable Elements Sliding Cable Elements (SCEs) address the general mechanics problem of constraining a string of cable elements to pass through a predefined moving point, called the slider point. This general simulation capability has numerous applications in airdrop system modeling. In the current paper, SCEs are used to model slider reefing of a round canopy during inflation. Slider reefing is a commonly used method to slow down the rate of inflation of a canopy and thereby reduce the opening shock. A slider consists of a piece of fabric with multiple grommets. Groups of suspension lines are threaded through the grommets. The slider is initially positioned at the top of the suspension lines near the canopy skirt. During inflation, the opening of the canopy causes the slider to move down the suspension lines. The resistance of the slider, however, slows down the opening. In our simulation, the suspension lines are modeled as SCEs, the slider is modeled as a membrane element, and the grommets are the slider points that coincide with nodes on the slider membrane element. American Institute of Aeronautics and Astronautics 1 17th AIAA Aerodynamic Decelerator Systems Technology Conference and Seminar 19-22 May 2003, Monterey, California AIAA 2003-2152 Copyright © 2003 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. Methodology Pneumatic Muscle Actuator Element A brief summary of the geometrically nonlinear, total Lagrangian formulation for the PMA element is presented. A detailed derivation is given by Zhou . A circular cylindrical PMA is shown in Figure 1a with internal pressure P and axial force F. The corresponding unrolled PMA is shown in Figure 1b with radius r, element length L, and fiber angle α . The fiber unit length, l, is defined as the axial distance corresponding to one loop of the fiber around the circumference, which is generally different than the element length, L. The arc length of the fiber for one circumferential loop is S. Figure 1: (a) PMA Cylinder, and (b) Unrolled PMA The virtual work of the pressure acting on the cylindrical surface and two end caps is given by 2 2 PMA W r P L rLP δ π δ π = + r δ