On an alternative approach for the derivation of the frequency equation of a bending beam carrying several axially vibrating rods with tip mass

It is witnessed in technical literature, often encountered in continuous systems of applications of mechanical, constructional and aviation engineering that is modelled as a Bernoulli-Euler beam carrying several axially vibrating rods with tip mass. Generally, however, mass of helical spring is neglected. On the other hand, it is observed in some studies in recent years that mass of helical spring has been taken into account. These studies have showed that it revealed to significant errors in results, if mass of helical spring is neglected. One of the methods of taking into account approximate mass of helical spring is to model spring as a axially vibrating rod. In the studies of Philip D. Cha and others [J.S.V. vol. 130(061008) p. 1-9 (2008)], Assumed Modes Method provides basic and approximately solution to analyze the free vibration of a unified system consisting of a Bernoulli-Euler beam carrying several axially vibrating rods with tip mass hinged independently of particular conditions. They developed in their studies an approximately solution method, Philip D. Cha and Masanori Honda [J.S.V. vol.132(051011) p.1-9 (2010)], as they called Characteristic Force Method in order to obtain approximately natural frequencies and regarding mode shapes of a unified system consisting of a Bernoulli-Euler beam carrying various discrete elements. This method is essentially based on fact that discrete elements on the beam are represented by effect of their forces and moments at the connection point to the beam. Thus, a free vibration problem becomes a forced vibration problem, and obtaining equation of motion in the second condition has been quite simplified than in the first condition. However, Metin Gürgöze and Serkan Zeren, in their studies [S.E.M. vol.53(6) p.1105-1126 (2015)], have reached to the solution through different way to get formulation defining in [J.S.V. vol.130(061008) p.1-9 (2008)] by using "Characteristic Force Method" for the condition one axially vibrating rod with tip mass. It is automatically taken into account the system's "dynamic boundary conditions" in kinetic and potential energy expressions, because "Assumed Modes Method" uses the kinetic and potential energy. Otherwise, if "Characteristic Force Method" is used, then system's "dynamic boundary conditions" must be considered. Because of this purpose, in the article [S.E.M. vol.53 (6) p. 1105-1126 (2015)], system's "dynamic boundary conditions" are considered by using Galerkin Procedure. Another point to be considered when using the approximate solution methods is that it should provide the boundary conditions of trial functions, bending beam and axially vibrating rods, so it should be 'comparison functions' type. In order to ensure these conditions mentioned in the article above, beam eigenfunctions are used with different hinged formats as trial functions. It is used as eigenfunctions of fixed-free axially vibrating rods trial functions, because axially vibrating rods on beam behaves like fixed-free rod. First of all, formulation defining in [J.S.V. vol.130(061008) p.1-9 (2008)] is obtained, in this study, for the condition one axially vibrating rod with tip mass. Then, this formulation has been generalized to the case where number of axially vibrating rods is n. In the second part of this thesis, the formulation in the article [S.E.M. vol.53(6) s.s.1105-1126 (2015)], obtained by using "Characteristic Force Method" for the condition two axially vibrating rods with tip mass. The main contribution of this thesis is that earlier developed formulation for two axially vibrating rods has been generalized to the case where number of axially vibrating rods is n. In the conclusion of this thesis, formulation, obtained by two approximately methods, is considered as numerically. Approximately nondimensional natural frequencies of a unified system are calculated according to the various numbers of axially vibrating rods and the bending beam hinged in various forms. Approximate values, then, are compared with real values if they are available, if not compared with FEM results. Eventually, it was observed that very good results are obtained in both alternative methods. There are many advantages of using the approximate solution methods, described above, for the solution of the unified system of free vibration consisting of Bernoulli-Euler beam carrying one or more helical spring-mass system. Primarily, solution of N-DOF of a finite system is obtained, instead of a continuous system solution by using "Assumed Modes Method" or "Characteristic Force Method". Secondly, it is easy to encode 'characteristic equation' or "eigenvalue problem' with software (MATLAB, Mathematica, Maple etc.). Moreover, obtained formulations are applicable to beams any hinged format.