*Article* **Theoretical and Numerical Examination of a Novel Method for Identifying Bridge Moving Force Using an Instrumented Vehicle**

**Dexin Liu 1, Bo Liu 1, Xingui Li 1,2,\* and Kang Shi <sup>3</sup>**


**Abstract:** This study presents a novel and highly efficient technique to identify moving forces by utilising the acceleration response of an instrumented moving vehicle. The complex task of determining moving forces is transformed into one of solving linear equations, made easier through the Newmark-β method. To overcome the ill-conditioned problem and improve the recognition accuracy, Tikhonov regularisation is employed. Uniquely, this approach necessitates the placement sensors on the vehicle alone to record the responses of the vehicle–bridge coupled system as it traverses the bridge. Once the coupled responses are acquired, the moving force is quickly identified using the proposed method. The reliability of this method is numerically verified using a case study of a single degree-of-freedom vehicle crossing a simply supported beam. The performance of the proposed method is further demonstrated by examining several typical external factors. The results indicate that the method presents a high recognition accuracy, demonstrating good robustness and reliability even amidst substantial environmental noise interference. This proposed method offers a new perspective for identifying the moving force of small- to medium-span bridges.

**Keywords:** moving force identification; vehicle–bridge interaction; Newmark-β method; Tikhonov regularisation

**Citation:** Liu, D.; Liu, B.; Li, X.; Shi, K. Theoretical and Numerical Examination of a Novel Method for Identifying Bridge Moving Force Using an Instrumented Vehicle. *Buildings* **2023**, *13*, 1481. https:// doi.org/10.3390/buildings13061481

Academic Editors: Humberto Varum and Rajai Zuheir Al-Rousan

Received: 7 April 2023 Revised: 28 May 2023 Accepted: 1 June 2023 Published: 7 June 2023

**Copyright:** © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).
