Application of Chebyshev-tau method to the free vibration analysis of stepped beams

• Chebyshev-tau method is applied to the free vibration analysis of beams.
• Present method is based on Timoshenko theory and Euler–Bernoulli theory.
• Orthogonality of Chebyshev polynomials is made use of.
• Present method is accurate when compare to exact solutions and experimental results.

A Chebyshev-tau method based on Euler–Bernoulli beam theory and Timoshenko beam theory is applied to the free vibration analyses of stepped beams. The lateral deflection and the rotation of each segment of stepped beam are approximated by partial sums of Chebyshev expansions. The number of expansions per segment is larger by four than the number of intended degrees of freedom, and the surplus expansions accommodate the continuity conditions and the boundary conditions. The governing equation and test function are integrated to result in inner products and the orthogonality property of Chebyshev polynomials is utilized. Numerical examples are provided for a various number of steps and boundary conditions. The results of the present method coincide with those of theoretical results for both of lower modes and high-order modes. It is demonstrated that the present method computes the natural frequencies of stepped beams accurately when compared with experimental measurements.