Turbo Machinery Dynamics Part 6

Tham khảo tài liệu 'turbo machinery dynamics part 6', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | FAN AND COMPRESSOR AIRFOILS 183 and where n 0 N 2 1 I 1 - n2 Using a thickness vector and midsurface coordinates then x xi 8 z . Jy1 t N Ỉ n JyiJ t N z V z i 1 z. i 1 LztZimid where xi y and zi are the average values of the coordinates at the two surfaces. Then u 1 v 1 w ui 8 t N1 v ii i 1 wi J t N z Vi - vj i 1 I i where V and V2i are unit orthogonal vectors with displacements in global axes x y z. ui vi wi are displacements at the midsurface nodes and b a are rotations about V1i and V2i providing a total of 5 of freedom at each node. In matrix notation U N ữ where ữ is a column vector and N is obtained by expanding Eq. . With the displacements available element properties strains and stresses need to be defined. From Fig. the strain components are ey u x v J 1 Yxy 1 1 v y 1 L U YxV w x u x x J Yyx Wy vx J where x 0 0 0 y 0 L y x 0 and z 0 x _ 0 z y _ U 1 v u w Hence U N a and e B ữ where B L N . Corresponding stresses in matrix form are Ơ 0 e where elasticity matrix O is 1 0 0 0 0 O 1Z- 0 1 0 0 0 0 0 1-v 2 0 0 0 0 0 2k 0 l_0 0 0 0 1-v 2k 184 COMPONENT DESIGN where E and V are Young s modulus and Poisson s ratio. Factor K approximates displacement due to shear. Properties of elements call for integration over their volume and take the form of J 5 X dx X dy X dz where 5 is a function of the global coordinates x y z and 5 B r D B with strain defined by ae B a a a2 1 a 8 Thus B is defined in terms of displacement derivatives in the local cartesian coordinates x y z and a e is the displacement field. Integration of the element in the curvilinear coordinates can be performed after transformations from the local to the global system and then to the curvilinear x h z coordinates. Equation relates global displacements u v w to the curvilinear coordinates. Derivatives of these displacements with respect to x h z may be obtained by the jacobian matrix so derivatives of displacements in global coordinates are given by d .