Flexible Bodies with COTS Control Design and Physical Modeling Tools

A common problem in aerodynamic and astrodynamic engineering is accurate modeling of systems that combine flexible and rigid bodies with a controller. Mechanical and control simulators typically represent only rigid bodies and require preliminary transformation of flexible body problems into a suitable approximation. Analytical methods and commercial off-the-shelf (COTS) flexible-body analysis tools can fill this gap. Easily available COTS multibody simulation packages can efficiently simulate rigid-body machines. They can also simulate mixed rigid-flexible systems if and when they can make dynamic use of flexible-body response data from the variety of available COTS finite-element analysis programs. I. Introduction HE ability to construct high quality mechanical models often plays an important role in the design of control systems. Frequently the mechanics being modeled include flexible parts. Because of its computational expense and conceptual difficulty, many mechanical simulators do not offer native support for flexible-body modeling. In such tools, none of the simulated machine parts is assumed to change its shape or mass distribution. Modeling mixed flexible and rigid body motion is an important requirement in aerospace, automotive, and civil engineering, among other applications. It is especially important in flight simulation where aircraft structures are bent by aerodynamic loads. In this paper, we explore two distinct mathematical approaches to modeling flexible bodies, each of which has direct application to COTS modeling tools: the lumped-parameter method and the finite-element analysis (FEA)/state-space method. These methods break the flexible-body modeling down into pieces that are less computationally demanding and that can be easily incorporated into rigid-body simulations. Both methods assume that beam deflection is small and in the linear regime. Both methods can simulate flexibility with fidelity sufficient for many applications, such as plant analysis and control design. The two methods differ in their computational expense; how open they are to greater mathematical rigor; and, in the FEA case, the need for additional COTS FEA software. The one-dimensional bending beam is the starting point for modeling flexible bodies in physics and engineering. The lumped-parameter method, best suited for modeling beam-like geometries, discretizes the flexible body into a series of constituent or beam elements, each pair of which is linked by a spring-damper that models the flexibility between them. In its simplest version, the lumped-parameter method makes the spring moment between neighboring elements a function of local beam deflection only. While not a correct representation of the bending moment, this type of lumped-parameter approximation is useful and easy to implement, allowing a quick determination of gross flexible behavior. The method can be improved for greater accuracy and turned into a controlled approximation that converges on the correct continuum behavior. However, such improvements require more complex modeling and greater effort. For a more rigorous approximation that can be progressively refined and made mathematically convergent, it is better to turn to the FEA/state-space method. The finite-element analysis approach, in contrast to the lumped-parameter method, makes use of the vibration analysis output of FEA programs to model flexible bodies in motion relative to a rigid-body machine. The FEA real-space discrete mesh is transformed by such analysis into a modal, frequency-response state-space that can be embedded in a larger