A complete mathematical support of the 3D finite beam element for modeling of the micromechanical inertial measurement sensors and their components has been developed. The mathematical support includes mass matrix, stiffness matrix, Coriolis matrix, and centrifugal matrix. The mathematical support takes full account of the gyroscopic effect and the theory of Timoshenko. Employing of the Variational principles of mechanics and Lagrange's equation makes the process of derivation of the mathematical support clear, accurate and well-founded. The developed software was verified by a numerical simulation of the influence of the gyroscopic effect on the dynamics of the simplest model of the vibrating gyro. The results obtained due to the numerical simulation by using the developed mathematical support were compared with the results obtained in ANSYS, well-known engineering simulation software. The difference between these results was less than 5 %. This difference can be explained by the dissimilarity of the elements used in ANSYS and in the developed software. This paper shows that the developed mathematical support can be used for development of special software, which ensures, in contrast to the universal proprietary closed-source software such as ANSYS, a transparent implementation of the algorithms, a complete control of the progress of computing and significantly lower cost. Thus, the developed mathematical support for the three-dimensional finite element based on the theory of Timoshenko can be used to solve a wide range of problems of statics and dynamics, including the gyroscopic effect, e.g. in the area of research and development of the microelectromechanical sensors of the inertial information.