VIBRATIONS OF MULTIPLE INTERCONNECTED BEAMS BY DIRAC-LAPLACE-HEAVISIDE METHOD

Abstract

The Multiple Inter-Connected Beams (MICB) system is widely used in civil and mechanical engineering. Adopting Euler-Bernoulli beam theory, this thesis presents novel exact and closed-form solutions for free and forced vibrations of the MICB system with intermediate connections and masses using a continuum approach.

Recently, Roncevic et al. (2019) adopted Green's function method to study the free vibrations of a beam system with arbitrary intermediate supports. A total of 49 cases of Green's functions were derived and tabulated for each combination of the boundary conditions, i.e., fixed, pinned, sliding, free, translational spring-supported, rotational spring-supported, and combined translational-rotational spring-supported. Han et al. (2021) adopted the dynamic stiffness matrix method to study the free vibrations of a double-beam system with intermediate supports. With this method, one must discretize the beams at the connections and then apply the continuity conditions.

This thesis is intended to treat MICB with arbitrary boundary conditions in a unified manner and avoid the need to discretize the beams. In this work, the Dirac-Laplace-Heaviside (DLH) method is proposed to investigate vibrations of the MICB with arbitrary intermediate connections and concentrated masses under the arbitrary boundary conditions subjected to arbitrary external excitation. The Dirac's delta function is adopted to formulate the mathematical model; the Laplace transform is utilized to solve the model; and the Heaviside function is used to implement the solution. Euler-Bernoulli beam is assumed to have a uniform cross-section. Axial loads, arbitrary external exciting forces, and arbitrary boundary conditions are also incorporated in the studied model.

The exact mode shape solutions are developed by the proposed DLH method. The arbitrary boundary conditions are handled in a unified form. The solutions are validated by numerical results using a Finite Element Method (FEM). They are also compared to the results of specific cases and Green's function method.

This study contributes to a continuum approach to handle beams with interconnections and concentrated masses. It is a general model that could be reduced to the specific models investigated in the literature. The method and model reported in this thesis may be useful for vibration analysis, dynamic control, vibration attenuation, design optimization of the MICB.