Research Article Adaptive Reconstruction of a Dynamic Force Using Multiscale Wavelet Shape Functions Wen-Yu He , 1 Yang Wang, 1 and Songye Zhu 2 1 Department of Civil Engineering, Hefei University of Technology, Hefei, Anhui, China 2 Department of Civil and Environmental Engineering, Te Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong Correspondence should be addressed to Songye Zhu; songye.zhu@polyu.edu.hk Received 25 July 2017; Revised 13 December 2017; Accepted 31 December 2017; Published 31 January 2018 Academic Editor: Jussi Sopanen Copyright © 2018 Wen-Yu He et al. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Te shape function-based method is one of the very promising time-domain methods for dynamic force reconstruction, because it can signifcantly reduce the number of unknowns and shorten the reconstruction time. However, it is challenging to determine the optimum time unit length that can balance the tradeof between reconstruction accuracy and efciency in advance. To address this challenge, this paper develops an adaptive dynamic force reconstruction method based on multiscale wavelet shape functions and time-domain deconvolution. A concentrated dynamic force is discretized into units in time domain and the local force in each unit is approximated by wavelet scale functions at an initial scale. Subsequently, the whole response matrix is formulated by assembling the responses induced by the wavelet shape function forces of all time units which are calculated by the structural fnite element model (FEM). Ten, the wavelet shape function-based force-response equation is established for force reconstruction. Finally, the scale of the force-response equation is lifed by refning the wavelet shape function with high-scale wavelets and dynamic responses with more point data to improve the reconstruction accuracy gradually. Numerical examples of diferent structural types are analyzed to verify the feasibility and efectiveness of the proposed method. 1. Introduction Forward and inverse dynamic analyses are two typical types of structural dynamic problems, containing three basic com- ponents, namely, excitations, structures, and responses. Te forward analysis refers to structural response calculation with the knowledge of excitation and structure parameters. Te inverse analysis can be classifed into two types: struc- tural parameters identifcation using the known excitation and response and force reconstruction using the known structure parameters and response. Dynamic force recon- struction methods based on structural dynamic response have attracted great interest in the feld of structural health monitoring (SHM), as it is ofen difcult or impractical to measure a dynamic force directly. Dynamic force reconstruc- tion methods can generally be categorized into two groups: frequency-domain [1–4] and time-domain [5–8] methods. Te frequency- and time-domain methods estimate dynamic forces by establishing the relationship between dynamic forces and structural responses based on frequency response functions [9, 10] and impulse response functions [11, 12], respectively. Compared to the frequency-domain methods, the time-domain methods have received increasing attention in the past years because of their distinct physical meaning and relatively higher accuracy [13]. Most time-domain methods are based on discretization in the time domain. Te sampling time interval size is one main factor that afects the reconstruction accuracy and efciency. In general, a smaller size of sampling time interval can achieve higher reconstruction accuracy. However, a very small size would result in too many undetermined coefcients, which not only increases the computation cost but also tends to make the inverse problem ill-conditioned [13]. Terefore, it is challenging to set an optimal sampling time interval to balance the tradeof between reconstruc- tion accuracy and efciency. Approaches based on various basis functions were proposed to address this challenge, in which unknown dynamic forces were approximated by Hindawi Shock and Vibration Volume 2018, Article ID 8213105, 11 pages https://doi.org/10.1155/2018/8213105