XXI ICTAM, 15–21 August 2004, Warsaw, Poland MINDLIN CYLINDRICAL PANELS WITH TWIST AND DOUBLE CURVATURE T. Sakiyama a , C. W. Lim b , X. M. Yuan c , X. X. Hu b,c ,H. Matsuda a , C. Morita a a Nagasaki University,1-14 Bunkyomachi, Nagasaki 852-8521, Japan b City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong c Zhejiang University of Technology, Hangzhou, Zhejiang 310032, PRC Summary An open cylindrical panel, which has a twist around the lengthways direction and double curvature in the radial and lengthways directions, is a better shell model of turbine blades. In order to analyse vibration characteristics accurately, a precise relationship between strains and displacement components of the shell model is derived on the general shell theory and the first order shear deformation theory. By using the principle of virtual work and the Rayleigh Ritz method, the governing equation for free vibration of the model is presented. The effects of parameters such as curvature and twist on vibration are studied. INTRODUCTION A comprehensive summary about the research of blades before 1980 was made by Rao [1-3]. It is known that three models such as beam, plate and shell were adopted for studying the dynamics of blades, where the beam was the most common model and it is also used for the research of blades now. Although the plate and the shell models are more approximate to the geometric configurations of blades, and there is a little related work reported. Since 80s more complicated and precise shell models have been presented. For example, Leissa, Lee et al. used twisted plate, shallow cylindrical shell and doubly curved shallow shell models for the vibrations of turbo machinery blades and studied their vibration performance by Ritz method [4-7]. Introducing a pb-2 Ritz method, Lim et al. studied the problem using several models such as the trapezoidal plate, the shallow cylindrical shell and the shallow conical shell [8-11]. Based on an exact strain-deformation relationship on the general shell theory, Tsuiji et al. presented a pretwisted thin plate and thin cylindrical models [12,13] and Hu et al. presented a curved and twisted thin cylindrical model [14]. Sakiyama et al. further studied the vibration of the cylindrical panel with non-uniform thickness [15]. The purpose of this work is to extend the research of thin shell model and to introduce an accurate strain- deformation relationship of the model under considering the influence of shear deformation and rotary inertia. By the use of the principle of virtual work the energy equation of the panel is given out, and then the governing equations are achieved by the Rayleigh-Ritz method with orthonormal polynomials as admissible displacement components. The vibration characteristics and the effects on it are studied briefly. THEORETICAL FORMULATION The profile, geometric parameters and the coordinate system of the new blade model is shown in Figure 1 where the cylindrical panel rotates around the curvilinear axis x at a twist rate k, and has two curvature (1/R and 1/r ). Based on the accurate relation between strains and displacement components for the 3D panel [14] and the first order shear deformation theory, a precise relationship for the Mindlin cylindrical panel is achieved. R l / 2 = W R ' z y y ' z z h l x kl e r 1 W b q f Figure 1. A schematic diagram of a Mindlin cylindrical panel  ε ξξ ε ηη γ ξη γ ξζ γ ηζ  T = Z 1 G (1) R 1 + Z 2 G (2) R 2 , (1) R T 1 =  ∂u ∂x ∂u ∂y u ∂v ∂x ∂v ∂y v ∂w ∂x ∂w ∂y w  , R T 2 =  ∂ Θ 1 ∂x ∂ Θ 1 ∂y Θ 1 ∂ Θ 2 ∂x ∂ Θ 2 ∂y Θ 2  , (2)