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Purpose <p>In the vibration-based method, the natural frequency and mode shape are the two most important signals for crack identification, which is an inverse problem and thus huge computation is required. An efficient algorithm for the natural frequency computation and analytical solution form for the mode shape are in high demand.</p> Method <p>Based on the generalized functions, the exact solution for natural frequency and the analytical solution form for the mode shape of a cracked bar are derived.</p> Results <p>For a bar with <i>N</i> cracks, the eigenvalue problem formulated by the classical method is the determinant of a <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(2(N+1)\times 2(N+1)\)</EquationSource> </InlineEquation> matrix and there are <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(2(N+1)\)</EquationSource> </InlineEquation> coefficients to be determined for an eigenvector. In comparison, the eigenvalue problem formulated with the generalized functions is the determinant of a <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(2\times 2\)</EquationSource> </InlineEquation> matrix, which is <i>de facto</i> <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(1\times 1\)</EquationSource> </InlineEquation> matrix for the bar free vibration. And there is no need to determine any coefficient for an eigenvector.</p> Conclusions <p>The analytical solutions based on the the generalized functions are with the advantage of satisfying all the transition conditions and only boundary conditions are needed to formulate the eigenvalue problem. A new and efficient method for the eigenvalue and eigenvector computations of a bar with an arbitrary number of cracks is thus provided.</p>

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An Efficient Solution to Free Vibration of a Bar with an Arbitrary Number of Cracks

  • 期刊论文

  • 发表日期：

    2026年1月16日

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  • Yin Zhang,
  • Han Wu,
  • Yunlin Xing

摘要

Purpose

In the vibration-based method, the natural frequency and mode shape are the two most important signals for crack identification, which is an inverse problem and thus huge computation is required. An efficient algorithm for the natural frequency computation and analytical solution form for the mode shape are in high demand.

Method

Based on the generalized functions, the exact solution for natural frequency and the analytical solution form for the mode shape of a cracked bar are derived.

Results

For a bar with N cracks, the eigenvalue problem formulated by the classical method is the determinant of a \(2(N+1)\times 2(N+1)\) matrix and there are \(2(N+1)\) coefficients to be determined for an eigenvector. In comparison, the eigenvalue problem formulated with the generalized functions is the determinant of a \(2\times 2\) matrix, which is de facto \(1\times 1\) matrix for the bar free vibration. And there is no need to determine any coefficient for an eigenvector.

Conclusions

The analytical solutions based on the the generalized functions are with the advantage of satisfying all the transition conditions and only boundary conditions are needed to formulate the eigenvalue problem. A new and efficient method for the eigenvalue and eigenvector computations of a bar with an arbitrary number of cracks is thus provided.