The Application Research of Inverse Finite Element Method for Frame Deformation Estimation

Abstract

A frame deformation estimation algorithm is investigated for the purpose of real-time control and health monitoring of flexible lightweight aerospace structures. The inverse finite element method (iFEM) for beam deformation estimation was recently proposed by Gherlone and his collaborators. The methodology uses a least squares principle involving section strains of Timoshenko theory for stretching, torsion, bending, and transverse shearing. The proposed methodology is based on staindisplacement relations only, without invoking force equilibrium. Thus, the displacement fields can be reconstructed without the knowledge of structural mode shapes, material properties, and applied loading. In this paper, the number of the locations where the section strains are evaluated in the iFEM is discussed firstly, and the algorithm is subsequently investigated through a simple supplied beam and an experimental aluminum wing-like frame model in the loading case of end-node force. The estimation results from the iFEM are compared with reference displacements from optical measurement and computational analysis, and the accuracy of the algorithm estimation is quantified by the root-mean-square error and percentage difference error.

Optimal Sensor Placement Based on Eigenvalues Analysis for Sensing Deformation of Wing Frame Using iFEM

“… is the length of the beam element; and n are the axial coordinate of the locations where the section strains are evaluated and the number of locations, respectively. For the end-node loads, and for the uniformly distributed loads, [ 27 , 28 , 29 , 41 ].…”

“…The two key steps in iFEM are: (1) the selection of suitable shape functions for the beam deformation sensing with iFEM; (2) the calculation of section strains from the measured surface strain data. For step (1), literatures [ 27 , 28 , 29 , 41 , 42 ] provide the detailed derivation process and results. For step (2), the section strains are computed through the following equation [ 27 , 28 ]: where, is the in-situ section strains vector at location , along the x -axis.…”

“…In the case of end-node loads, 12 strain measurements are required for the calculation of Section 2 strain vectors; in the case of uniformly distributed loads, 18 strain measurements are required for the calculation of Section 3 strain vectors. As aforementioned in the literatures [ 27 , 28 , 29 , 41 ], the number of strain measurements is reduced through the constitutive equations of Equation (14) (refer to Figure 3 ) and the equilibrium equations of Equation (15). where, the section forces ( ) and moments ( ) are related to the section strains .…”

“…For the uniformly distributed transverse loads, the section strains are constant, are linear and are parabolic. [ 28 , 41 ]. Through the aforementioned results, the section strains are calculated as:…”

Real-Time Monitoring of the Position and Orientation of a Radio Telescope Sub-Reflector with Fiber Bragg Grating Sensors

“…When only Equation (8) is used, six strain sensors need to be placed in one section for the calculation of the section strains vector eε; thus, the total number of the strain sensors used to capture the surface strain measurements is 6 × n in one inverse finite beam element. Herein, n is the minimum number of the sections where the section strains are evaluated, which is different under different loading cases [21]. For the end-node load, n equals 2, whereas for the uniformly distributed load, n equals 3.…”