Free Vibrations of an Open Elliptical Cylindrical Shell*
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International Applied Mechanics, Vol. 56, No. 4, July, 2020
FREE VIBRATIONS OF AN OPEN ELLIPTICAL CYLINDRICAL SHELL* A. Ya. Grigorenko1, M. Yu. Borisenko1, and E. V. Boichuk2
The dynamic characteristics of an open elliptical cylindrical shell clamped at one and two ends are numerically determined using FEMAP computer-aided design and engineering analysis software system with NX NASTRAN solver. The dependences of the natural frequencies on the opening angle relative to the major and minor semi-axes are compared. The behavior of the first natural modes is analyzed. Keywords: natural frequencies, mode shapes, open elliptical cylindrical shell, finite-element method, clamping Introduction. Different industries widely use noncircular cylindrical shells, either closed or open, subject to different boundary conditions. In this connection, there is a need to know their dynamic characteristics, such as natural frequencies and modes since in actual operating conditions, it is necessary to avoid resonance that can cause collapse of structures. The natural frequencies can be determined for objects of simple geometry using the theory of thin shells [5, 6, 10, 13–15, 18, 19, 21, 22, 24]. It is also possible to use stroboscopic holographic interferometry [3, 9, 11, 16, 17] for which it is necessary to make a real shell and qualitatively implement boundary conditions and experiment itself, which require much time and money. Note that this experimental method can be applied to not only two-dimensional, but also three-dimensional dynamic problems [11, 12]. In the cases of complex geometries such as elliptic shells with differently varying thickness, it is expedient to use the finite-element method (FEM) [1, 2, 7, 8, 12, 20, 23], which shows good agreement with experiments [3, 12]. This confirms the validity of the method. Moreover, the method does not require making the real object and saves much time and money. In the present paper, we will numerically determine the dynamic characteristics of an elliptic cylindrical shell open along the semiminor axis or along the semimajor axis and having two types of clamping at the ends. The natural frequencies and modes are calculated using FEMAP finite-element software. The dynamics of nonclosed (open) shells has been studied inadequately in the world’s literature. Let us address one of such publications [4]. We used two finite-element models to study the natural vibrations of thin-walled open circular shells with different boundary conditions and different opening angles. In the range of opening angle considered, all the vibration frequencies increase. This dependence is nonmonotonic. It is characterized by a substantial increase in the frequencies at opening angle j Î(60°; 90°). We will use one of the problems solved in [4] as a test one. 1. Basic FEM Equations for Problems of Free Vibrations. The dynamic FEM equations can be derived from a system of Lagrange equations of the second kind with n degrees of freedom:
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S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of
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