Mass and Stiffness Matrices and Frequencies of Simple Beam Elements Based on Real Shape Functions

In this research, we investigate and compare the natural frequencies of simple beams and their mass and stiffness matrices of the two methods: classic shape functions and real shape functions. To this end, we solve the beam motion Equation and apply boundary conditions. This article shows that the coefficients of the real shape functions, and consequently, the real shape functions, become harmonic and hyperbolic and also, they are dependent on the natural frequency value of the element. As a result, the real mass and the real stiffness matrix of each element are also dependent on the element frequency.  The frequency values obtained from these two methods are compared with the exact frequency values of two simple beam types with different support conditions. In this way, we determine which method leads to more accurate and acceptable frequencies for these beams. Based on the obtained results, the percentage of frequency error obtained by the classical method is relatively high in the sample beams. Hence, the natural frequency value of the beams studied using exact shape functions shows a small error compared to the classical method in terms of the exact frequency value of these beams. It is of note that the frequency error obtained from the classical method is greater in the elements with a higher natural frequency. Overall, obtaining the exact natural frequency of an element will result in accurate dynamic responses and more appropriate analyses and designs.

Keywords:

Beam Element , high natural frequency , mass and stiffness matrices , Real and classic shape functions