Modeling and Simulation of Variable Mass, Flexible Structures

Abstract

The advent of the new Ares I launch vehicle has highlighted the need for advanced dynamic analysis tools for variable mass, flexible structures. This system is composed of interconnected flexible stages or components undergoing rapid mass depletion through the consumption of solid or liquid propellant. In addition to large rigid body configuration changes, the system simultaneously experiences elastic deformations. In most applications, the elastic deformations are compatible with linear straindisplacement relationships and are typically modeled using the assumed modes technique. The deformation of the system is approximated through the linear combination of the products of spatial shape functions and generalized time coordinates. Spatial shape functions are traditionally composed of normal mode shapes of the system or even constraint modes and static deformations derived from finite element models of the system. Equations of motion for systems undergoing coupled large rigid body motion and elastic deformation have previously been derived through a number of techniques [1]. However, in these derivations, the mode shapes or spatial shape functions of the system components were considered constant. But with the Ares I vehicle, the structural characteristics of the system are changing with the mass of the system. Previous approaches to solving this problem involve periodic updates to the spatial shape functions or interpolation between shape functions based on system mass or elapsed mission time. These solutions often introduce misleading or even unstable numerical transients into the system. Plus, interpolation on a shape function is not intuitive. This paper presents an approach in which the shape functions are held constant and operate on the changing mass and stiffness matrices of the vehicle components. Each vehicle stage or component finite element model is broken into dry structure and propellant models. A library of propellant models is used to describe the distribution of mass in the fuel tank or Solid Rocket Booster (SRB) case for various propellant levels. Based on the mass consumed by the liquid engine or SRB, the appropriate propellant model is coupled with the dry structure model for the stage. Then using vehicle configuration data, the integrated vehicle model is assembled and operated on by the constant system shape functions. The system mode shapes and frequencies can then be computed from the resulting generalized mass and stiffness matrices for that mass configuration. The rigid body mass properties of the vehicle are derived from the integrated vehicle model. The coupling terms between the vehicle rigid body motion and elastic deformation are also updated from the constant system shape functions and the integrated vehicle model. This approach was first used to analyze variable mass spinning beams and then prototyped into a generic dynamics simulation engine. The resulting code was tested against Crew Launch Vehicle (CLV-)class problems worked in the TREETOPS simulation package and by Wilson [2]....

Moving Kriging shape function modeling of vector TARMA models for modal identification of linear time-varying structural systems

“…17, the selected order of the AR model is 2 (N a ¼ 2) and that of the MA order is 1 (N b ¼ 1). In addition, the initial identified model is FS-VTARMA (2,1,11) [20,10] . The functional space is also selected by the BIC and the GA.…”

Elastic Model Transitions Using Quadratic Inequality Constrained Least Squares

“…2, the physical configuration after the transition is in no way related to the physical configuration before. A standard pseudoinverse initialization using the minimum square error of the displacement and velocity is shown in Figures (3) and (4). Due to the ability of the new basis to accurately model the preceding physical configuration, the model fit in physical coordinates is very good.…”

“…The method of assumed modes 3 fits elasticity to a linear combination of a set of basis functions that are not necessarily orthogonal to the rigid body, so the coupling between the elastic response and the rigid body motion has to be explicitly applied 4 . Determination of the time-varying terms requires at least partial knowledge of the mass and stiffness properties of the original FEM, and increases the computational burden significantly.…”

Mode Selection Techniques in Variable Mass Flexible Body Modeling

“…Banerjee, 7 which lists a few other works on dynamics of flexible bodies losing mass, provides a new formulation for a variable mass flexible body system, which is suitable for finite element-based model of a rocket. Tobbe et al 4 also address variable mass flexible structures in support of the propellant depletion for the Ares I launch vehicle, another element of the NASA Constellation Program.…”

“…We call it Method 1, and it is essentially the standard Assumed-modes method, 3 except that the mass matrix and other parameters in the EOMs are now time-dependent. This approach was used by Tobbe et al 4 Given that the eigenmodes change with mass depletion, we had sought to characterize the accuracy of using a set of fixed assumed modes as the basis for integrating the EOMs. This characterization was pursued on two fronts: (1) To develop stand-alone tests of how accurately one reproduces the evolving eigenmodes by solving the eigenvalue problem in this fixed basis as the mass depletes; and (2) To develop two alternative techniques where the modes used in integrating the EOMs accurately reflect the evolving eigenmodes at all times.…”