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<p>The present study analyzes the transverse vibrations of slender and flexible structures, modelled as uniform strings, moving in axial direction, by considering three different mathematical formulations, namely: (i) a uniform string model subjected to an axial tension load and moving with constant axial velocity (here referred to as model 1), (ii) the same string model 1 that includes the nonlinear effect of axial deformation (model 2), and (iii) the same string model 1 moving with a time-dependent axial velocity (model 3). In all analyzed cases, we make use of the Generalized Integral Transformed Technique (GITT) to handle the related Partial Differential Equations (PDEs) of such flexible string models and to obtain the correspondent coupled systems of Ordinary Differential Equations (ODEs), which are numerically solved with user-prescribed controlled error estimate through appropriate subroutines such as the DIVPAG from the IMSL Library. Firstly, we perform an extensive convergence behavior to guarantee the completely converged GITT results with a determined truncation order depending on the governing parameters of each physical flexible string model. Secondly, verifications of the GITT results for typical cases are also realized to demonstrate the numerical results' consistency and accuracy. Parametric evaluations are analyzed for the three models adopted to illustrate the governing parameters' influence on the string displacements' dynamics. Finally, because model 1 is the only linear one and with time-independent axial velocity, a modal analysis is also performed for this case.</p>

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Integral transform solution for transverse vibrations of axially moving slender structures

  • 期刊论文

  • 发表日期：

    2025年9月2日

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  • Rogilson N. S. Porfírio,
  • Emanuel N. Macêdo,
  • Carlos F. T. Matt,
  • João N. N. Quaresma

摘要

The present study analyzes the transverse vibrations of slender and flexible structures, modelled as uniform strings, moving in axial direction, by considering three different mathematical formulations, namely: (i) a uniform string model subjected to an axial tension load and moving with constant axial velocity (here referred to as model 1), (ii) the same string model 1 that includes the nonlinear effect of axial deformation (model 2), and (iii) the same string model 1 moving with a time-dependent axial velocity (model 3). In all analyzed cases, we make use of the Generalized Integral Transformed Technique (GITT) to handle the related Partial Differential Equations (PDEs) of such flexible string models and to obtain the correspondent coupled systems of Ordinary Differential Equations (ODEs), which are numerically solved with user-prescribed controlled error estimate through appropriate subroutines such as the DIVPAG from the IMSL Library. Firstly, we perform an extensive convergence behavior to guarantee the completely converged GITT results with a determined truncation order depending on the governing parameters of each physical flexible string model. Secondly, verifications of the GITT results for typical cases are also realized to demonstrate the numerical results' consistency and accuracy. Parametric evaluations are analyzed for the three models adopted to illustrate the governing parameters' influence on the string displacements' dynamics. Finally, because model 1 is the only linear one and with time-independent axial velocity, a modal analysis is also performed for this case.