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Defect Characterization in a 1D Spring Mass System Using the Laplace and Z-Transforms

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Abstract

Purpose

We present a scheme to characterize the defects within a one-dimensional spring–mass system comprised of an arbitrary number of bodies with otherwise uniform masses connected in series by springs using only a discrete set of vibrational data of the first body.

Methods

The system of ordinary differential equations modeling spring–mass systems was analyzed using the Laplace transform with the unknown mass and location of the defects as parameters. We propose a two-phase strategy to determine these unknown parameters using a set of discrete measurements of the longitudinal displacements of the first mass after the system is excited by a Dirac \(\delta\) impulse on the first mass. The Z-transform of the discrete time-measurements is used to obtain an approximation for the Laplace-domain solution curve of the vibration of the first body. First, we show how the poles of this simulated data can be used to determine the masses of the defects. Then the location of these defects were calculated using an optimization routine.

Results

We also show several simulations with two defects highlighting the instances when the scheme is highly accurate as well as its limitations. In these cases, the algorithm was able to predict the mass and locations accurately.

Conclusions

In this paper, we were able to design a stable numerical scheme that can characterize the defects, i.e., estimate their masses and locations, using only a discrete set of vibrational data of the first mass.

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Funding

Larry Guan was supported by the University of Houston through an Undergraduate Research Fellowship.

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Authors and Affiliations

  1. Institute of Mathematical Sciences and Physics, University of the Philippines Los Baños, Los Baños, Laguna, Philippines

    Neil Jerome A. Egarguin

  2. Department of Mathematics, University of Houston, Houston, TX, USA

    Larry Guan & Daniel Onofrei

Authors
  1. Neil Jerome A. Egarguin
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  2. Larry Guan
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  3. Daniel Onofrei
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Correspondence to Neil Jerome A. Egarguin.

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Egarguin, N.J.A., Guan, L. & Onofrei, D. Defect Characterization in a 1D Spring Mass System Using the Laplace and Z-Transforms. J. Vib. Eng. Technol. 10, 1121–1134 (2022). https://doi.org/10.1007/s42417-022-00433-y

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  • DOI: https://doi.org/10.1007/s42417-022-00433-y

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