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Using a Combination of FEM and Perturbation Method in Frequency Split Calculation of a Nearly Axisymmetric Shell with Middle Surface Shape Defect
# 05, May 2016
DOI: 10.7463/0516.0839190
authors: D.S. Vakhlyarskiy^{1,*}, A.M. Guskov^{1}, M.A. Basarab^{1}, V.A. Matveev^{1}
 ^{1} Bauman Moscow State Technical University, Moscow, Russia 
This paper proposes a method to calculate the splitting of natural frequency of the shell of hemispherical resonator gyro. (HRG). The paper considers splitting that arises from the small defect of the middle surface, which makes the resonator different from the rotary shell. The presented method is a combination of the perturbation method and the finite element method. The method allows us to find the frequency splitting caused by defects in shape, arbitrary distributed in the circumferential direction. This is achieved by calculating the perturbations of multiple natural frequencies of the second and higher orders. The proposed method allows us to calculate the splitting of multiple frequencies for the shell with the meridian of arbitrary shape. A developed finite element is an annular element of the shell and has two nodes. Projections of movements are used on the axis of the global cylindrical system of coordinates, as the unknown. To approximate the movements are used polynomials of the second degree. Within the finite element the geometric characteristics are arranged in a series according to the small parameter of perturbations of the middle surface geometry. Movements on the final element are arranged in series according to the small parameter, and in a series according to circumferential angle. With computer used to implement the method, threedimensional arrays are used to store the perturbed quantities. This allows the use of regular expressions for the mass and stiffness matrices, when building the finite element, instead of analytic dependencies for each perturbation of these matrices of the required order with desirable mathematical operations redefined in accordance with the perturbation method. As a test task, is calculated frequency splitting of noncircular cylindrical resonator with Navier boundary conditions. The discrepancy between the results and semianalytic solution to this problem is less than 1%. For a cylindrical shell is made a comparison of results with solution in ANSYS commercial complex  a difference is less than 1%. For a hemispherical shell was found the frequency splitting. The comparison has shown that a discrepancy between the results and ANSYS solution is less than 1%. The solution of this problem allows us to estimate further a mutual influence of defects of different nature (shape, thickness, density, modulus of elasticity, etc.) on splitting the frequency of the HRG. This is an urgent problem in terms of balancing the HRG resonators. References
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