Calculation of Natural Frequencies of Two-Dimensional Prismatic Bending Beams with Distributed Mass and Elasticity Using an Innovative Method

Usually, by modeling the structures using the finite element method, their undamped free vibration frequencies are calculated analytically. In addition, the issue of accurate calculation of natural frequencies and the shape of vibration modes corresponding to them for bending systems that have distributed mass and elasticity and possibly a combination of several bending beams, sometimes requires solving complex mathematical equations and requires a relatively heavy mathematical work demands. Bending beams are beams whose axial deformations is insignificant compared to their bending deformations, and as a result, these members are assumed to be axially rigid. By using the conventional finite element method, the natural vibration frequencies of these beams can be obtained approximately. By increasing the number of finite elements used in the model, the calculation error of natural frequencies of vibration decreases. When the consistent-mass matrix is used, the frequency values obtained from the finite element method converge to the exact frequency values with larger values, while if the lumped-mass matrix is used, the frequency values obtained from the finite element method converge to the exact frequency values with smaller values. It should be noted that the consistent-mass matrix is non-diagonal, but the lumped-mass matrix is diagonal. The interpolation functions (shape functions) used for bending finite elements (beam elements) are polynomial functions of the 3rd degree. This bending finite element has two nodes, each node has one translational degree of freedom and one rotational degree of freedom. The new idea that came to the authors of this article is that instead of using polynomial functions, trigonometric and exponential interpolation functions are used to calculate the stiffness matrix and mass matrix of the finite element. In fact, these trigonometric and exponential functions are the solutions of the differential equation governing the free vibration of bending beams with distributed mass and elasticity. The argument of these trigonometric and exponential functions includes a parameter called beta, which is proportional to the square root of angular frequency of the bending beam. By changing this parameter in a suitable range and with a certain step, it is possible to plot the changes in the frequencies of the different modes of the studied prismatic beam in terms of beta. In this paper, three models were studied, which included a uniform cantilever beam, a uniform beam clamped at left side and simply supported at right side, and a uniform beam free at both ends. Using the conventional finite element method and using the consistent-mass matrix, these three models were analyzed and the approximate frequencies of the first few modes of these beams were calculated, which were greater than their corresponding exact values. In the innovative method presented in this article, a uniform beam was modeled with a finite element model with one translational degree of freedom and one rotational degree of freedom. The stiffness matrix and the mass matrix of this beam were calculated for different betas and having these two matrices, the first and second frequency values of this model were calculated for different beta values and its graph was drawn for different betas. The values of the maximum frequency of the first frequency are the same as the values of the minimum frequency of the second for certain betas, and by specifying these betas, the frequencies of different vibration modes can be accurately determined. The detected frequencies of different modes with this method had a very good match with their exact corresponding frequencies. For the second model investigated in this paper, one rotational degree of freedom was considered. Considering that this beam had only one rotational degree of freedom, therefore, by plotting the first frequency of this model for different betas and finding its minimum, the frequency values of different modes of this beam were obtained, which matched the exact values like the previous model very well. The third model was the same as the previous two models. The diagram of the first to fourth natural frequencies of this model was drawn for different betas. By having the approximate values of the frequencies of different modes obtained from the conventional finite element method and these diagrams, the frequencies of different modes of the model were identified, which were in good agreement with their corresponding exact values.