4/9/2023 0 Comments Geometric boundary![]() ![]() Where and are independent of time and position, respectively. ![]() Assume that the displacement can be separated into two parts: one is depending on the position and the other is depending on time, as follows: (2) is sought by separation of variables. Where is the linear mass density of the beam. Where is the area moment of inertia of the beam cross section, is the transverse displacement, and t is time. Using Euler-Bernoulli beam theory, one can obtain the equation of motion of a beam with homogeneous material properties and constant cross section as follows : Basic Equations Consider an elastic beam of length, Young's modulus, and mass density with uniform cross section, as shown in Figure 1. Present analysis can be used as a comparative study or data for the different solution methods of future works in the related field. Furthermore, to confirm the reliability of the vibration analysis carried out in the present paper as well, all the analytical results are checked with the corresponding numerical results obtained from Finite Element Method (FEM)-based software called ANSYS, where the method is established on the idea of building a complicated object with simple blocks, or, dividing a complicated object into smaller and manageable pieces. Solutions including the effects of the geometric characteristics, i.e., length and cross sectional area, and boundary conditions are obtained and discussed for the natural frequencies of the first three modes. An example is the calculation of natural frequencies of continuous structures. Such equations occur in vibration analysis. It is particularly useful for transcendental equations, composed of mixed trigonometric and hyperbolic terms. This method is based on the simple idea of linear approximation, and used for finding the roots of equations. Analytical solution is carried out using Euler-Bernoulli beam theory, in which material is assumed to be linear-isotropic, and Newton Raphson Method. In this study, the free vibration of square cross-sectioned aluminum beams is investigated analytically and numerically under four different boundary conditions. Then an increasing interest has been observed regarding the vibration of beams, and several studies have appeared in the general literature, some of which are provided in the Refs. Early investigations of the theory of vibration were given in Refs. Due to beams are important structural elements, vibration analysis has been a vital task in their design for engineers and researchers for more than a century. In addition, if the vibration exceeds certain limits, there is the danger of beam breakage or failure. In many engineering applications, beams are subjected to dynamic loads, which can excite beam structural vibrations and cause durability concerns or discomfort because of the resulting noise and vibration. Introduction A beam is a slender horizontal structural member that resists lateral loads by bending, and this important element of engineering structures appears in various forms and comprises various artifacts, such as supporting members in high-rise buildings, railways, long-span bridges, flexible satellites, gun barrels, robot arms, airplane wings, etc. ![]()
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