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

Hozhabrossadati, S. M ; Sharif University of Technology | 2015

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  1. Type of Document: Article
  2. DOI: 10.1016/j.apm.2014.11.038
  3. Publisher: Elsevier Inc , 2015
  4. Abstract:
  5. 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
  6. Keywords:
  7. Euler-Bernoulli beam ; Fredholm Alternative Theorem ; Modified Green's function ; Boundary conditions ; Ordinary differential equations ; Resonance ; Alternative theorem ; Euler Bernoulli beams ; Fredholm ; Resonant condition ; Resonant response ; Uniform Euler-Bernoulli beams ; Green's function
  8. Source: Applied Mathematical Modelling ; Volume 39, Issue 12 , 2015 , Pages 3366-3379 ; 0307904X (ISSN)
  9. URL: http://www.sciencedirect.com/science/article/pii/S0307904X14006222