Real-Time Response Estimation of Structural Vibration with Inverse Force Identification

Virtual sensing is an effective method to identify the inaccessible state of the structural systems by compensating the limitations of the conventional physical sensing techniques. Recently, it becomes popular in structural vibration field and their relevant physical domains such as civil, mechanical, aerospace engineering, thermal dynamics, and acoustics. This study aimed to develop a virtual sensing algorithm of structural vibration for the real-time identification of unmeasured information. First, certain local point vibration responses (such as displacement and acceleration) are measured using physical sensors, and the data sets are extended using a numerical model to cover the unmeasured quantities through the entire spatial domain in the real-time computation process. A modified time integrator is then proposed to synchronize the physical sensors and the numerical model using inverse dynamics. In particular, an efficient inverse force identification method is derived using implicit time integration. The second-order ordinary differential formulation and its projection-based reduced-order modeling are used to avoid two times larger degrees of freedom within the state-space form. Then, the Tikhonov regularization noise-filtering algorithm is employed instead of Kalman filtering. The performance of the proposed method is investigated on both numerical and experimental testbeds under sinusoidal and random excitation loading conditions. In the numerical test, the system could identify the status of the motor housing structure in the speed of 16,181 $\left(\mathrm{S}\mathrm{a}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{e}\mathrm{s}/\mathrm{s}\right)$ . The FDE and RMSE values are bounded under 0.1816 and 1503.2. In the case of the experimental test, the algorithm is implemented to the beam structure using a single-board computer, including inverse force identification and unmeasured response prediction. Even in the limited computational environment, the system could identify the applied forces in real time in the speed of 2,173 $\left(\mathrm{S}\mathrm{a}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{e}\mathrm{s}/\mathrm{s}\right)$ . Through all experimental cases, FDE and RMSE values are bounded under 0.2925 and 0.1660. The results show that the virtual sensing algorithm can accurately identify unmeasured information, forces, and displacements throughout the vibration model in real time in a very limited computing environment.