Green's function for uniform Euler-Bernoulli beams at resonant condition: Introduction of Fredholm Alternative Theorem

This paper deals with the dynamic analysis of Euler-Bernoulli beams at the resonant condition. The governing partial differential equation of the problem is converted into an ordinary differential equation by applying the well-known Fourier transform. The solution develops a Green's function method which involves establishing the Green's function of the problem, applying the pertinent boundary conditions of the beam. Due to the special conditions of the resonant situation, a significant obstacle arises during the derivation of the Green's function. In order to overcome this hurdle, however, the Fredholm Alternative Theorem is employed; and it is shown that the modified Green's function of the beam may still be achievable. Furthermore, the necessary requirement so that the resonant response will be found is introduced. A special case which refers to a case in the absence of resonance is also included, for some verification purposes. (C) 2014 Elsevier Inc. All rights reserved.