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.

A Modified Couple Stress Elasticity for Non-Uniform Composite Laminated Beams Based on the Ritz Formulation

“…Among different numerical approaches, in the present work we apply a Ritz-type solution with harmonic trial functions to solve the problem, whose stability and accuracy is verified through a systematic investigation. In line with predictions from the literature [65][66][67][68][69][70][71][72][73], the Rayleigh-Ritz method, represents an efficient tool for the analysis of the structural behavior of beams, whose accuracy and stability are well known to be related to the selected trial functions. The trial functions must satisfy the enforced boundary conditions.…”

“…where ω is the natural frequency, i 2 = −1 refers to the imaginary unit, u j , w j are the unknowns of the problem, and ψ j are the trial functions which depend on the selected boundary conditions. In the present study we consider two different types of boundary conditions, namely simply supports (S-S) and clamped-free (C-F) supports, such that the following trial functions are assumed [71]:…”

Modelling of Equivalent Mass and Rigidity of Continual Segment of the Inter-Resonance Vibration Machine

“…The Rayleigh-Ritz method ensures sufficient accuracy of the results (Yuan et al [23], Kumar [24], Rahbar-Ranji et al [25], Mazanoglu [26], Babakov [27]). This method can be applied to a wide range of plates, bars, rods, beams, etc., with any aspect ratio.…”

Frequency equation using new set of fundamental solutions with application on the free vibration of Timoshenko beams with intermediate rigid or elastic span

“…These methods include the traditional numerical assembly method (NAM) (Farghaly, 1994; Lin, 2010; El-Sayed and Farghaly, 2016; Farghaly and El-Sayed, 2016), dynamic stiffness matrix (DSM) (Banerjee and Williams, 1985; Banerjee and Williams, 1992; Banerjee, 2012; Calio and Greco, 2013), transfer matrix method (TMM) (Attar, 2012; Wu and Chang, 2013; Boiangiu et al., 2016) and inverse Laplace transformation (Magrab, 2007; Wang and Qiao, 2007). Approximate methods include the formation of approximate eigenvalue problem such as finite element methods (FEM) (Balasubramanian and Subramanian, 1985; Laura et al., 1991; Eltaher et al., 2013), finite difference (Popplewell and Chang, 1996), differential quadrature (Liu and Wu, 2001; Wang and Wang, 2013; Mohandes and Ghasemi, 2016), Rayleigh-Ritz method (Lee and Ng, 1994; Mohamed et al., 2016; Mazanoglu, 2017), Adomian decomposition method (ADM) (Mao and Pietrzko, 2010; Mao, 2011; Wattanasakulpong and Chaikittiratana, 2016) and discrete singular convolution (DSC) (Duan and Wang, 2013).…”