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

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. Nomenclature An = Acceleration integration scaling value for the nth node Bn = Acceleration integration constant for the nth node Ci j = Galerkin finite element damping matrix sub-block Dn = Velocity integration constant for the nth node En = Velocity integration scaling value for the nth node fn = Control input force at nth node Fn = Forward propagation matrix of the right force for the nth node Hn = Reverse propagation matrix of the left force for the nth node Jn = Reverse propagation matrix for the nth node Ki j = Galerkin finite element elastic matrix sub-block Mi j = Galerkin finite element mass matrix sub-block Pn = Forward propagation matrix for the nth node Qn = Transfer matrix from left boundary condition to nth node Tn = Transfer matrix from right boundary condition to nth node vn = Propagation vector 1 Copyright c © 2015 by ASME xn = nth node position states ẋn = nth node velocity states ẍn = nth node acceleration states τn = Internal forces at the nth node for either left or right side