ارایه یک معادله یکتا جهت بررسی ارتعاشات آزاد تیرهای اویلر-برنولی غیریکنواخت و مدرج تابعی محوری

Beams are significant due to their wide application in engineering structures. Therefore, examining the frequencies and mode shapes to understand their vibrational behavior is the first priority of beam-related research works. In this thesis, the free vibrations of non-uniform and non-homogeneous beams with general non-classical boundary conditions, including end objects with the mass moment of inertia, eccentricity, and rotational and translational flexible constraints, have been studied. The literature review indicates that, unlike many studies and various methods used to investigate the vibrations of non-uniform and non-homogeneous beams, the presented methods are not fully able to evaluate the vibrations of beams carrying end objects with the mass moment of inertia, eccentricity, and flexible rotational and translational constraints. Hence, the purpose of this review is to present a new process for iterated integration based on the application of Cauchy's equation, which according to the author's investigation, it has not been done so far. By utilizing this process and the related equations, it is possible to analyze the vibrations of different sorts of functionally graded non-uniform, homogeneous nonuniform, functionally graded uniform, and homogeneous uniform beams. The validation results obtained for some of the considered configurations have been reviewed by comparing existing and reported data in related studies. Also, parametric studies were carried out and provided to investigate the effect of alteration in end objects, the stiffness of rotational and translational springs, the
power law index (n), the geometric shape of end objects, and the aspect ratio on the frequency parameters and mode shapes.