Discrete Time Finite Element Transfer Matrix Method Development for Modeling and Decentralized Control

Abstract

The current emphasis on increasing aeronautical efficiency is leading the way to a new class of lighter more flexible airplane materials and structures, which unfortunately can result in aeroelastic instabilities.To effectively control the wings deformation and shape, appropriate modeling is necessary. Wings are often modeled as cantilever beams using finite element analysis. The drawback of this approach is that large aeroelastic models cannot be used for embedded controllers. Therefore, to effectively control wings shape, a simple, stable and fast equivalent predictive model that can capture the physical problem and could be used for in-flight control is required.The current paper proposes a Discrete Time Finite Element Transfer Matrix (DT-FETMM) model beam deformation and use it to design a regulator. The advantage of the proposed approach over existing methods is that the proposed controller could be designed to suppress a larger number of vibration modes within the fidelity of the selected time step. We will extend the discrete * Address all correspondence to this author time transfer matrix method to finite element models and present the decentralized models and controllers for structural control. NomenclatureA n = Acceleration integration scaling value for the n th node B n = Acceleration integration constant for the n th node C i j = Galerkin finite element damping matrix sub-block D n = Velocity integration constant for the n th node E n = Velocity integration scaling value for the n th node f n = Control input force at n th node F n = Forward propagation matrix of the right force for the n th node H n = Reverse propagation matrix of the left force for the n th node J n = Reverse propagation matrix for the n th node K i j = Galerkin finite element elastic matrix sub-block M i j = Galerkin finite element mass matrix sub-block P n = Forward propagation matrix for the n th node Q n = Transfer matrix from left boundary condition to n th node T n = Transfer matrix from right boundary condition to n th node v n = Propagation vector 1Copyright c 2015 by ASME x n = n th node position stateṡ x n = n th node velocity states x n = n th node acceleration states τ n = Internal forces at the n th node for either left or right side 7