Chapter 8 Investigation of Using Log-Spectrum Averaging (Cepstral Averaging) for Blind Reconstruction of an Unknown Impact Input Force Sa’ed Alajlouni, Vijaya V. N. Sriram Malladi, and Murat Ambarkutuk Abstract Consider the case of a mechanical structure being impacted at an arbitrary location by an unknown loading profile (i.e., force–time curve). Then, estimating the unknown impact loading profile (ILP) based on response vibrations is a challenging problem. If the impact location is also unknown, traditional inverse problem approaches (i.e., deconvolution) cannot reconstruct the ILP. This problem is particularly complex when inferring someone’s footstep loading profile by monitoring floor vibrations. Therefore, this preliminary study attempts to overcome the missing input location issue by producing a blind estimate (without knowledge of excitation location point) of the unknown ILP. Producing a blind ILP estimate is appealing since there is no need to know the location of the input force. Additionally, knowledge of the ILP can potentially uncover important information about the excitation source, such as, for example, identifying individuals from their footfall-induced floor vibration. The blind input reconstruction is done using log-spectrum averaging of the structural response at several locations. Our approach investigation is done via a MATLAB simulation, utilizing a Timoshenko finite element (FE) beam model as the virtual mechanical structure. Simulation results encourage further refinement of the approach. Keywords Deconvolution · Input reconstruction · Cepstrum 8.1 Introduction Estimating the input loading profile (ILP) exciting a mechanical structure (i.e., force-time function f(t)) can potentially uncover important information about the event that caused the vibration (i.e., the source). For example, when considering a floor excited by a footstep impact, estimating the ILP can uncover pathological gait [1] and identify individuals [2, 3]. Given a mechanical structure, in particular, a beamimpacted at a single input excitation point and instrumented with more than one accelerometer. Then, all what is needed to solve for (estimate) the ILP is knowledge of the frequency response functions (FRFs) from the input point to each of the sensor points. This is a classical inverse problem known as “deconvolution” [4]. However, estimating the ILP (deconvolution) requires knowledge of the excitationlocation point,which is unknown. Unfortunately, state of the art localization methods are either (1) not accurate enough (error is on the order of 1 m [5, 6]) for estimating the ILP through classical deconvolution or (2) require an analytic model of the mechanical structure (e.g., [7]), which in many cases is not feasible (e.g., a concrete floor section in a building will have material inhomogeneity, uncertainty in material properties/thickness, and uncertainty in boundary conditions). In this chapter, we provide an alternative approach to classical deconvolution, which does not require knowledge of the input excitation source location. The approach is based on finding an averaged cepstrum, or averaged log-spectrum, of the sensor response signals. S. Alajlouni ( ) Department of Mechatronics Engineering, The Hashemite University, Zarqa, Jordan e-mail: saed@hu.edu.jo V. V. N. S. Malladi Department of Mechanical Engineering-Engineering Mechanics, Michigan Tech, Houghton, MI, USA e-mail: smalladi@mtu.edu M. Ambarkutuk Department of Mechanical Engineering, Virginia Tech, Blacksburg, VA, USA e-mail: murata@vt.edu © The Society for Experimental Mechanics, Inc. 2023 M. Allen et al. (eds.), Special Topics in Structural Dynamics & Experimental Techniques, Volume 5, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-05405-1_8 63
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