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Numerical Methods
Published in Ansel C. Ugural, Plates and Shells, 2017
Difference operators in Cartesian coordinates x and y are well adapted to the solution of problems involving rectangular domains. When the plate has a curved or irregular boundary, special operators must be used at nodal points adjacent to the boundary (Section 7.4). One of the non-Cartesian meshes most commonly employed to cover the polygonal and irregular boundaries is the triangular mesh (Section 7.6). If the plate shape is a parallelogram, it may often be more accurate and convenient to use coordinates parallel to the edges of the plate, skew coordinates. The polar mesh is used in connection with shapes having some degree of axisymmetry (Section 7.5). The finite difference operators in any coordinate set are developed through transformation of the equations that relate the x and y coordinates to that set. In all cases, the procedure for determining the deflections and the moments is the same as described in the following section.
Effect of temperature on the performance of active constrained layer damping of skew sandwich plate with CNT reinforced composite core
Published in Mechanics of Advanced Materials and Structures, 2022
Vinayak Kallannavar, Subhaschandra Kattimani
The three-point and two-point Gaussian integration rules are used to obtain the element matrices corresponding to the bending and the transverse shear deformation, respectively. The elemental matrices such as the initial stress stiffness matrix [], the stiffness matrix [Ke], and the mass matrix [Me] are computed separately. The computed element matrices are then assembled to obtain the global stiffness matrices [K], [Kσ], and [M]. The global matrices can then be used to calculate the natural frequency of the system as where ω is the natural frequency of the system. While dealing with skew plates, the skew coordinates () are considered in place of rectangular coordinates The generalized displacement vectors of an arbitrary point lying on the skew edge are transformed as follows: where [Lt] & [Lr] are transformation matrices, and and are generalized displacement vectors of the point concerning the new () coordinate system.