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<p>The dynamic behaviour of non-uniform beams is of critical interest in a wide range of engineering applications, particularly where geometric or material gradation plays a significant role. In this study, we develop a semi-analytical inverse approach for obtaining closed-form solutions to the free vibration problem of non-uniform Rayleigh beams with free-free boundary conditions. To the best of the authors’ knowledge, this is the first work to present exact closed-form inverse solutions for Rayleigh free-free beams with internal nodes—a significantly more challenging case than Euler–Bernoulli beams due to the explicit appearance of the natural frequency in the boundary conditions and the need to satisfy multiple internal node constraints. Unlike conventional forward methods that compute frequencies and mode shapes from known distributions, the inverse method begins with a prescribed polynomial mode shape and fundamental frequency and seeks the corresponding spatial variations in mass and stiffness. Specifically, we demonstrate that sixth-order polynomial mode shapes—satisfying all boundary and internal node conditions—admit exact analytical solutions for Rayleigh beams with cubic and quartic mass variations, and we further propose a general determinant-based framework applicable to any polynomial order. The formulation also provides a direct mapping from analytically obtained mass and stiffness distributions to physically realizable beam geometries, enabling practical structural design. The governing differential equation is reduced to a linear system whose solvability condition leads to a constraint involving internal node locations and frequency. The resulting mass and stiffness functions are demonstrated through both direct substitution and finite element analysis using the h-version FEM. The analytical solutions not only serve as benchmark functions for numerical validation—addressing a known scarcity of such cases for Rayleigh beams—but also offer a structured methodology for designing beams with tailored dynamic properties. The approach is further extended to accommodate higher-order polynomial mass distributions, making the framework broadly applicable across structural dynamics, vibration-sensitive design, and optimization scenarios.</p>

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Closed-form solutions for non-uniform Rayleigh free-free beams

  • 期刊论文

  • 发表日期：

    2026年4月2日

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  • Nikhil Kirodiwal,
  • Vishal Saha Chowdhury,
  • Korak Sarkar,
  • Ranjan Ganguli,
  • Isaac Elishakoff

摘要

The dynamic behaviour of non-uniform beams is of critical interest in a wide range of engineering applications, particularly where geometric or material gradation plays a significant role. In this study, we develop a semi-analytical inverse approach for obtaining closed-form solutions to the free vibration problem of non-uniform Rayleigh beams with free-free boundary conditions. To the best of the authors’ knowledge, this is the first work to present exact closed-form inverse solutions for Rayleigh free-free beams with internal nodes—a significantly more challenging case than Euler–Bernoulli beams due to the explicit appearance of the natural frequency in the boundary conditions and the need to satisfy multiple internal node constraints. Unlike conventional forward methods that compute frequencies and mode shapes from known distributions, the inverse method begins with a prescribed polynomial mode shape and fundamental frequency and seeks the corresponding spatial variations in mass and stiffness. Specifically, we demonstrate that sixth-order polynomial mode shapes—satisfying all boundary and internal node conditions—admit exact analytical solutions for Rayleigh beams with cubic and quartic mass variations, and we further propose a general determinant-based framework applicable to any polynomial order. The formulation also provides a direct mapping from analytically obtained mass and stiffness distributions to physically realizable beam geometries, enabling practical structural design. The governing differential equation is reduced to a linear system whose solvability condition leads to a constraint involving internal node locations and frequency. The resulting mass and stiffness functions are demonstrated through both direct substitution and finite element analysis using the h-version FEM. The analytical solutions not only serve as benchmark functions for numerical validation—addressing a known scarcity of such cases for Rayleigh beams—but also offer a structured methodology for designing beams with tailored dynamic properties. The approach is further extended to accommodate higher-order polynomial mass distributions, making the framework broadly applicable across structural dynamics, vibration-sensitive design, and optimization scenarios.