A source detection problem for thin plate vibration with L-curve regularization based on the Menger curvature

Abstract The dynamic vibrations from the shipboard engine are transmitted to hull plates through the metal pedestal, yielding the structure fatigue. To detect these source loads from extra measurable data, we propose a thin plate vibration model for which the frequency response function (FRF) is expressed in terms of the Green function of this governed system explicitly. Then we describe this load identification problem in frequency domain by a linear system with the FRF matrix as a linear map, transforming the amplitudes of the vibration excitations to the measurement data at specified points. To deal with the ill-conditioned FRF matrix due to either the resonance frequencies or the inappropriate distributions of measurement points, which leads to the instability of solving this linear system, we solve this system under the Tikhonov regularizing framework. To determine the regularization parameters under the L-curve scheme, we propose an adaptive shrinkage strategy by approximating the standard curvature as the discrete Menger curvature, with rigorous convergence analysis. Numerical tests using our scheme are presented to solve this vibration source recovery problem, as well as an imaging deblurring problem in higher-dimensional space, showing the general applicability and validity of our proposed scheme with reduced computational costs.