Development of a Mass Matrix of the 3D Finite Element for Modeling of the Dynamics of Micromechanical Inertial Sensor Data and their Components

Abstract

Mass matrix of the 3D finite element with twelve degrees of freedom was constructed taking fully into account Timoshenko theory - the stiffness of the beam section bending and shear deformation of the cross section. The created mass matrix generalizes the mass matrices of the beam element, obtained in compliance with Eu-ler-Bernoulli theory and Rayleigh theory previously constructed by the other authors. These matrices can be obtained by zeroing the specified coefficients in the mass matrix proposed in the paper. In order to verify the constructed mass matrix several numerical experiments were performed. The results were compared with the results of the numerical simulation in ANSYS. Numerical modeling demonstrated that the difference between the values of the natural frequencies in the translational motion obtained with the constructed mass matrix and in ANSYS, is less than 1 %. The difference of the maximum displacement under the harmonic loads is less than 5 %. The beats take place, when the frequency of the driving force is close to the value of the natural frequency. The effect of the resonance was obtained in the conditions of coincidence of the values of frequency of the driving force and the natural frequency. Thus the feasibility of using the proposed finite element mass matrix for the numerical simulation of the vibrating processes and loads of micromechanical inertial sensors was proved. The main advantages of the proposed 3D finite element with twelve degrees of freedom for modeling are: full respect of the inertia and shear deformation of the cross area; full control over the process of computing at any stage; low requirements for a computer power in comparison with the universal programs for the finite element modeling; possibility to do a study of high-frequency oscillations with a step of calculation equal to or less than 10^-7 s.