Multibody Parafoil Model

Abstract

The advantages of different approaches to multibody analysis of a parafoil system are discussed. The usual approach to this subject is to have two bodies, representing the canopy and the payload, connected at a common pivot. If the pivot is free to rotate in three axes, the resulting system has nine degrees of freedom. Some researchers model this as two separate bodies, each with six DOF, and then enforce the condition that the hinge point is the same on both bodies by modeling the interbody forces at the pivot point. This adds three extra states to the problem. In order to avoid this and to obtain the maximum numerical efficiency, a solution is pursued for a minimal set of nine equations. In order to highlight the effects of apparent mass, a derivation is first presented of a system in which only the real mass properties are included. The real mass solution closely follows a derivation presented in a textbook by P. Hughes [1]. Results are obtained that are equivalent his previous results, with certain simplifications that further improve the numerical efficiency. A symmetrical mass matrix is produced. This is followed by a derivation showing explicitly how the effects of apparent mass should be added. An apparent mass matrix that incorporates spanwise camber is used [2]. The overall mass matrix remains symmetrical when the apparent mass effects are included. Very little information has been presented in the literature describing the orientation of the principal axes of the apparent inertia matrix of a parafoil. It is shown that a good approximation of the Z axis of this matrix can be obtained by taking a line running from the confluence point to the center of pressure of the central airfoil section. This choice of orientation results in some simplification of the expressions in the system mass matrix.