with: 2 2 22 12 12 22 22 11, 21 12, 11 12, 12 11, 2 2 21 11 11 21 22 21, 21 22, 11 22, 12 21, 2 21 22 11 12 11 22 12 21 22 11, 21 22, 11 22, 12 11, J J 2J J J J J J J J J J 1 1 ( , ) J J 2J J , ( , ) J J J J J J J J det[ ] det [ ] J J J J J J J J J J J J J J J J [ [ [ [ K K K K K [ [ [ K [ K ª º « » « » « » ¬ ¼ A B J J K ª º « » « » « » ¬ ¼ (4) where the subscripts [ and K denote differentiation with respect to the natural variables. The elements in the Jacobian matrix J as well as those in matrices A and B depend on the mapping of the Cartesian coordinate system. 3 Analysis technique An isotropic homogeneous Kirchhoff rectangular plate in free flexural vibration is considered [12]. The functional of the total potential energy can be written as the sum of a term U due to the strain energy of the system plus a term V representing the potential of all applied loads (including the inertial forces), and a term 'V taking into account lumped and distributed elastic constraints: U V V 3 ' (5) The potential of the strain energy can be written in terms of second order derivatives of the out-of-plane displacement w: 2 2 2 1 2 2(1 ) , ( , ) det[ ] 2 xx yy xx yy xy S U D w w w w w dS dS d d Q Q [ K [ K ª º ¬ ¼ ³ J (6) In Eq. 6 the subscripts denote differentiation with respect to the spatial variables, S is the spatial domain and D is the flexural stiffness of the plate, which can be expressed as a function of Young’s modulus E, Poisson’s ratio Q and the thickness of the plate h [12]: 3 2 12(1 ) Eh D Q (7) In the present formulation the inertial forces are included in the potential of applied loads: S h V wwdS U ³ (8) where Uh denotes the mass per unit area of the plate. The additional term 'V in Eq. 5 is considered to take into account lumped and distributed elastic constraints. In the case of translational constraints it can be expressed in the form: 2 constraint domain 1 , ( , ) distributed stiffness 2 V w d x y J J N J N ' ® ¯ ³ (9) In Eq. 9 J can refer to surfaces, lines and separate points. Terms 'V with rotational stiffness may be included as well, for modelling rotational elastic constraints. The out-of-plane displacement w is expressed by means of a linear combination of shape functions [8-9]. If external constraints do not explicitly appear in the functional of Eq. 5, i.e. 'V = 0, then each of these functions must respect the essential conditions at the boundary of the plate (also known as principal or kinematic conditions). In the present study, the shape functions are selected as products of homogeneous uniform prismatic beam eigenfunctions I. In natural coordinates they can be expressed as: 1 1 ( ) ( ) ( ) N N ij i j i j w q t [ K I [ I K ¦¦ (10) 79
RkJQdWJsaXNoZXIy MTMzNzEzMQ==