Resonant Green's function for Euler-Bernoulli beams by means of the Fredholm Alternative Theorem

This paper presents the Green's function for a uniform thin beam which is assumed to obey the Euler-Bernoulli theory at resonant condition. The beam under study has a simple support at one end and a sliding support at the other. First, the differential equation governing the free vibration of the beam is obtained in the frequency domain using the Fourier transform. Then, we try to find the corresponding Green's function of the problem. But a contradiction occurs due to the special properties of resonance. In order to overcome this hurdle, the Fredholm Alternative Theorem is utilized. Remarkably, it is shown that this theorem, by adding a particular term to the Green's function, can remedy this problem and the modified Green's function is consequently established. Moreover, the deformation function of the beam is found in an integral equation form. Some diagrams and tables conclude this study.