Abstract
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.
Highlights
Уpавнение движения тpехмеpного конечного элементаPассìотpиì коне÷ный эëеìент (КЭ) äëины L (pис. 1). В äанной pаботе pассìатpивается тоëüко эëеìент с постоянныì пpяìоуãоëüныì попеpе÷ныì се÷ениеì, хотя боëüøинство выкëаäок ìожет бытü pаспpостpанено на баëки с пpоизвоëüныì постоянныì попеpе÷ныì се÷ениеì
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
This paper shows that the developed mathematical support can be used for development of special software, which ensures, in contrast to the universal proprietary closedsource software such as ANSYS, a transparent implementation of the algorithms, a complete control of the progress of computing and significantly lower cost.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
Summary
Pассìотpиì коне÷ный эëеìент (КЭ) äëины L (pис. 1). В äанной pаботе pассìатpивается тоëüко эëеìент с постоянныì пpяìоуãоëüныì попеpе÷ныì се÷ениеì, хотя боëüøинство выкëаäок ìожет бытü pаспpостpанено на баëки с пpоизвоëüныì постоянныì попеpе÷ныì се÷ениеì. Известно [4], ÷то потенöиаëüнуþ энеpãиþ Π систеìы без äеìпфиpования ìожно пpеäставитü в виäе кваäpати÷ной фоpìы по обобщенныì кооpäинатаì: Π. Кинети÷ескуþ энеpãиþ pассìатpиваеìой ìехани÷еской систеìы также ìожно пpеäставитü в виäе pазëожения по обобщенныì скоpостяì и кооpäинатаì. Uz = uz(x, t) + φ(x, t)y, ãäе x, y, z — кооpäинаты то÷ки в ëокаëüной систеìе кооpäинат (xyz); ux(x, t) опpеäеëяет пpоäоëüное пеpеìещение öентpа се÷ения; uy(x, t), uz(x, t) — попеpе÷ные пеpеìещения öентpа се÷ения в напpавëении осей у и z соответственно; φ(x, t) — уãоë кpу÷ения вокpуã оси х; ψ(x, t), θ(x, t) — уãëы изãиба в пëоскости (xz) и (xy) соответственно. Поäставëяя соотноøения (7) в фоpìуëы (10), у÷итывая (8) и интеãpиpуя, поëу÷аеì, ÷то T (0), T (1), T (2) ìоãут бытü пpеäставëены в виäе сëеäуþщих pазëожений по обобщенныì кооpäинатаì и скоpостяì (N = 12 — ÷исëо степеней свобоäы коне÷ноãо эëеìента): 12-. Поäставëяя поëу÷енное выpажение (12) ëанãpажиана в уpавнения Лаãpанжа 2-ãо pоäа (2), поëу÷иì сëеäуþщуþ систеìу уpавнений: mi, j q··j ( c~i, j c~j, i ) q· j j=1 j=1
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