General closed-form expressions for the three-dimensional vibrations of elastic bodies using the Ritz method

• The proposed novel Ritz- β approach benefits from a low computational cost. • Vibrational frequencies for 3D structures not discussed previously have been found. • Correctness of proposed method has been proved through convergence studies. • Superiority of closed-form expressions supports effectiveness of proposed method. A novel Ritz- β approach is proposed for accurately determining the vibrational frequencies of a body based on its three-dimensional elasticity. In the conventional Ritz method, when admissible 3D displacement fields are expressed as either simple algebraic, Chebyshev, or Legendre polynomials, the volumetric integrals are transformed into beta integrals through a change of variables and then furthered simplified to beta functions and gamma functions; the resulting stiffness and mass matrices are expressed in a closed-form and the computational time is significantly decreased. The proposed approach does not involve the intensive computational procedures employed in the classical finite element, finite difference, and spectral approximation theories. Furthermore, it does not involve the kinematic constraints employed in classical beam, plate, and shell theories. The correctness and accuracy of the proposed approach were confirmed through comprehensive convergence studies and comparisons with existing results.