Sensitivity-Based Structural Model Updating Based on Incomplete and Unnormalized Mode Shapes

Abstract

Abstract

Vibration-based techniques are widely used for structural parameter estimation and finite-element model updating. In large-scale structures, however, practical limitations on sensor deployment result in incomplete modal data, which severely restricts the applicability of conventional sensitivity-based updating methods. Existing sensitivity formulations often rely on mass-normalized mode shapes. In full-scale applications, mass normalization of incomplete measured modes is inherently unreliable because accurate mass matrices of undamaged or damaged structures are typically unavailable. Moreover, mass-normalized approaches require complete response measurements, including translational and rotational degrees of freedom, which are rarely attainable in practice. Although various data expansion and model-reduction techniques have been proposed to address data incompleteness, their effectiveness for accurate and robust model calibration remains limited. Hence, structural model updating based on the unnormalized mode shapes is of great interest. This research introduces a novel sensitivity formulation that completely avoids the use of mass-normalized mode shapes. The proposed approach is based on arbitrary scaling of unnormalized incomplete mode shapes. It establishes a direct sensitivity relationship between measured modal data and structural parameter variations by selecting a single measured degree of freedom in each mode shape as a reference. This formulation enables finite-element model updating using incomplete modal data without requiring mass-normalized mode shapes, rotational measurements, modal expansion, or model reduction procedures. Additional data relationships are created by varying the selected degree of freedom for mode shape scaling. It increases the number of equations and mitigates the adverse effects of an ill-conditioned system, an issue in inverse methods. The method is validated through model updating of a frame structure. It demonstrates high accuracy and robustness even under substantial measurement and modeling errors, highlighting its suitability for real-world structural health monitoring and large-scale engineering applications. An example of the proposed model calibration, using incomplete and unnormalized mode shape data, produced accurate estimates of stiffness parameters, even in the presence of measurement and mass modeling errors, for structural model updating and damage assessment. Keywords: Vibration measurements, modal data, unnormalized incomplete mode shapes, structural parameter estimation, finite element, model updating, innovative sensitivity equation