Natural frequency analyses of segmented Timoshenko–Euler beams using the Rayleigh–Ritz method

Abstract

This paper proposes part by part usage of Timoshenko and Euler–Bernoulli beam theories for obtaining natural frequencies of the non-uniform beam that has partially thick and thin beam vibration characteristics. The paper also presents convergence tests to determine proper function among the simple admissible shape functions taken into consideration. By doing so, closer approximation of the Rayleigh–Ritz method is achieved. The method is applied using a simple computation technique. In the analyses of the Timoshenko beams, an additional function is employed to identify shear deformation and rotational inertia effects. Modified angular displacement functions are defined to improve convergence capability of the method. Furthermore, optimal numbers of lateral and angular displacement terms are investigated for suggested function couple of the Timoshenko beams. Efficiencies of part by part modelling and advantages of novel approaches suggested for the Rayleigh–Ritz approximation are introduced by the comparative studies performed on tapered, stepped and continuously segmented beams under some classical end conditions. All of the computational outcomes for the beams with rectangular cross-section are validated by the results given in current literature and also those obtained by the finite element analyses.