Elzaki Transform Method for Natural Frequency Analysis of Euler-Bernoulli Beams

Abstract

Free vibration analysis of thin beams under dynamic excitation is important in design to prevent resonance failures that occur when the excitation frequency coincides with the natural vibration frequency. In this paper, the Elzaki transform method (ETM) is used, for the first time, to solve the free vibration problem of thin beams. The beam is assumed to be homogeneous and prismatic, and the vibration is assumed to be harmonic. As a result, the field equation becomes a fourth-order homogeneous ordinary differential equation (ODE). The Elzaki transform method is chosen in this study due to its proven ability to solve ODEs, systems of ODEs, integro-differential equations, integral equations, and fractional differential equations. The Elzaki transformation simplifies the field equation into an algebraic equation in terms of the unknown deflection in the Elzaki space. By inverting the transformation, the general solution for the deflection is obtained in the physical domain, considering the initial conditions. The enforcement of boundary conditions for each case of end supports is utilized to determine the eigenequations, which are then solved for their roots using Symbolic Algebra methods. The eigenvalues are used to determine the exact natural frequencies of flexural vibration for each considered classical boundary condition. The eigenequations obtained are exact and identical to the ones previously derived by other scholars.

Keywords

• characteristic frequency equation
• eigenequation
• elzaki transform method
• euler-bernoulli beam
• natural frequencies