A Non‐Classical Algorithm for Vibration Analysis of Non‐Uniform Beams

ABSTRACT A novel numerical computation technique called the Chebyshev‐matrix collocation method is proposed. The method is based on the Chebyshev polynomials of the second kind and collocation points. The new approach is first applied to vibration models of non‐uniform (i.e., tapered) Euler–Bernoulli beams. The non‐dimensional equation of motion for transverse vibration of a non‐uniform Euler–Bernoulli beam is presented in a general form. With the general expression, different types of non‐uniform beams can be modeled by changing the parameters related to non‐uniform geometry. The Chebyshev‐matrix collocation algorithm is described using a detailed flow chart. The algorithm makes it possible to obtain solutions with alternative support combinations. The solution algorithm is applied to three different non‐uniformity cases of Euler–Bernoulli beams: tapered height and constant width; tapered width and height; and constant thickness with decreasing width. Transverse vibration responses are presented for simple‐simple, clamped‐clamped, and clamped‐free support conditions. The natural frequencies are obtained, and mode shapes are plotted for all cases. The residual error analyses are proposed to demonstrate the accuracy of the new approach. The results are compared with exact solutions to prove the validity of the proposed method.