Abstract Summary
This paper provides explicit closed-form analytical expressions for eigenfrequencies, and mode shapes of a uniform Euler-Bernoulli beam subjected to free undamped transverse vibration. The beam is having uniformly distributed mass with an elastic translational spring support at an intermediate location and carrying a concentrated mass at the location of the translational spring. One end of the beam has a an in-plane elastic rotational restraint while the other end is freely supported. The exact mathematical expressions for bending eigenfrequencies and explicit mathematical equations of eigenfunctions have been derived from the fundamental principle of vibration of a continuous system under the condition of undamped free vibrations. Special mathematical functions such as infinitely peaked Dirac-delta function and Heaviside-step function coupled with methods such as Laplace transformations are employed to deal with the fourth-order partial differential equation of motion. An example calculation with numerical values of natural frequencies for few significant eigen-modes has been presented in a tabular form considering various ranges of translational and rotational spring stiffnesses. The numerical results are then subsequently compared with the output of the Modal analysis of Finite element models using ANSYS and SESAM GeniE software. The general shapes of few meaningful eigenmodes are also compared with the Modal analysis output of Finite element program of ANSYS and SESAM GeniE software. The output of this comparative study reveals an excellent agreement among the closed-form solutions with that of output of ANSYS and SESAM GeniE. The analytical expression of this paper can be further deployed to study the free vibration characteristics, and derive the natural frequencies, and mode shape functions of several cases of Euler-Bernoulli beams having classical and non-classical boundary conditions. It is worth noting that all these cases, both classical and non-classical have significant real-world applications in Industry, especially in Mechanical and Structural Engineering to study the vibration characteristics of dynamical systems and to investigate the sensitivity when subjected to external forced excitation. While there are literatures and references that address similar problems related to eigenfrequencies however, the Author is not aware of any such article or technical paper which specifically provides an explicit mathematical expression of the eigenfunction for such a problem. The solution method that has been adopted can further be extended and generically applied to establish classical mathematical expressions for similar problems of varying complexities. As for illustration, be it a tall guyed mast or a free-standing steel chimney (including the effect of soil flexibility) etc., transcendental equations derived in this paper, when properly implemented to matching scenarios, will allow engineers to obtain a reasonable perception of dynamic characteristics of the system prior to conducting a rigorous free and forced vibration analysis using finite element software packages.
