Uncertainty quantification for the modal shape sensing of structures undergoing geometrically non-linear deformation

Abstract

Shape sensing techniques allow for the time-efficient reconstruction of displacements based on measured strain data. There are technical applications, where the structure of interest is deformed in the geometrically non-linear domain. In aeronautics, this is the case for high-aspect-ratio wings, which are more frequently found in future designs. Only shape sensing methods that specifically take the non-linearity into account, can deliver appropriate displacement estimates for such application. A shape sensing method based on the linear modal approach can be utilised incrementally to capture the geometric non-linearity; it has therefore been denoted incremental modal method (IMM). This paper presents analytical relations for the uncertainty propagation for the various input quantities of the method, specifically strain mode shapes, displacement mode shapes, and measured strain. Deterministic shape sensing and uncertainty propagation are demonstrated using data obtained with a finite element model of a high-aspect-ratio wing experiencing geometric non-linear deflections in flapwise bending. Virtual strain and acceleration sensors are assumed for this setup, imitating the instrumentation conceivable for experimental work. The results obtained by analytical propagation are compared to Monte Carlo simulations for the purpose of validation. The derived propagation formulas make it possible to follow the evolution of the uncertainties over the number of increments. Given that the variability of the input quantities is known, the number of increments that minimise uncertainties can be determined for a model-free application of the shape sensing. Together with the deterministic estimates provided by an FE model, it is possible to determine the ideal number of increments for a specific shape sensing application in the geometrically non-linear domain.