A comparative study on the suitability of orthogonal basis functions in the full-field response reconstruction of beams via spatial compressive sensing

Spatial compressive sensing (CS) is a promising data-driven technique for reconstructing full-field structural responses using measurements recorded by few sensors. A prerequisite for its successful implementation is the construction of a dictionary capable of sparsely representing the full-field response. As structural vibrations are typically dominated by the first few vibration modes, mode shapes can be used for dictionary construction. However, analytical mode shapes of variable-section and continuous beams are generally difficult to derive. To address this difficulty, the mode shape approximation strategy has been proposed. In this study, the efficacy of four commonly used orthogonal basis functions (OBFs) in quickly approximating mode shapes is first evaluated. It is found that sine functions can quickly approximate mode shapes of beams with fixed or pinned ends; however, they are not effective for beams with free ends. Two sets of OBFs are therefore derived to rapidly approximate the mode shapes of beams with one and two free ends, respectively. An analytical dictionary composed of sine functions and the derived OBFs is then proposed. Subsequently, a numerical example of a three-span, variable cross-section beam (length of 120 m) is presented to illustrate the procedure of spatial CS. Using measurements from ten sensors, the full-field response is reconstructed with a spatial resolution of 0.5 m and a relative error of 3.68 %. Finally, the effectiveness of the constructed dictionary is validated through laboratory beam vibration tests under six kinds of boundary configurations. This study expands the scope of spatial CS applications and provides a paradigm for dictionary construction that may be applied to other types of structures.