134 L. Carassale et al. 13.2 1-D Modelling of a Turbine Blade This section discusses a strategy for the realization of a ROM of a turbine blade. Since the ROM is intended to be predictive, its parameters should be obtained from the sole blade geometry and material properties, without relying on an existing 3-D reference solution. A possible choice consists of a model composed by 1-D finite elements, whose properties are evaluated from the blade geometry. In practice the blade geometry is provided through a series of closed lines representing a number of cross-sections. These lines are not necessarily plane and are described by sets of points in the 3-D space. The definition of a 1-D ROM based on the classic beam theory requires (1) the identification of the beam axis and of a family of plane cross-sections orthogonal to the axis, (2) the definition of the FEs in terms of stiffness and mass matrices, (3) the assemblage of the FEs into a blade model, (4) the calculation of the static and dynamic response with pertinent load and constraint conditions. The use of the mentioned modeling approach in the preliminary stage of a blade design in very inefficient as a relatively large ensemble. 13.3 Cross-Section The definition of a blade cross section starts from a closed curve defined by the positionX(s), s being a curvilinear abscissa measuring the curve starting from an arbitrary point. Since the input geometry is given as a set of points, the function X(s) is conveniently parametrized through cubic splines. The position Xis referred to a blade reference system having origin somewhere at the clamping point. The orientation of the blade reference system is defined by the unit vectors e(0) x , e (0) y and e(0) z withe (0) x having radial direction ande (0) y being parallel to the machine axis. Let XG be the Center of Gravity (CG) of the cross section andabe the unit vector defining its normal direction. Both this quantities are unknown, however they can be used to define, at least in principle, four orthogonal reference systems that are proper of the cross section. The reference system SYS0 is parallel to the blade reference system, but centered in the CG of the cross section. In SYS0, the border of the cross-section has coordinates x0.s/ DX.s/ XG (13.1) The reference system SYS1 is obtained rotating SYS0 about e(0) z of anangle˛ (0) z . The orientation of SYS1 is defined through the unit vectors e(1) x , e (1) y ande .1/ z De .0/ z . The reference system SYS2 is obtained by rotating SYS1 around the direction e(1) y of an angle ˛ (1) y . The orientation of SYS2 is defined through the unit vectors e(2) x , e .2/ y De .1/ y ande (2) z . The reference system SYS3 is the is obtained by rotating SYS2 about e(2) x of an angle ˛ (2) 2 . The orientation of SYS2 is defined through the unit vectors e.3/ x De .2/ x , e (3) y and e (3) z , with e (3) y and e (3) z being inertial principal axes of the cross section. The angles ˛(0) z , ˛ (1) y and˛ (2) x are usually referred to as Tait-Bryan angles and define completely the orientation of the crosssection. The angles ˛(0) z and˛ (1) y can be calculated on the basis of the normal vector a as tan˛z0 D ay ax I tan˛y1 D az axy (13.2) where ax, ay andaz are the components of ainSYS0 andaxy 2 Dax 2 Cay 2. Theplane (e(2) y , e (2) z ) contains the orthogonal cross-section of the blade, but does not contain, necessarily, the given profile x0(s). A plane profile contained in (e (2) y , e (2) z ) can be obtained by orthogonal projection as (Fig. 13.1): x0 Dx0 ET 2x0E2 (13.3) where E2 Dhe .2/ x ; e .2/ y ; e .2/ z i D 2 64 ax ay axy axaz axy ay ax axy ayaz axy az 0 axy 3 75 (13.4)
RkJQdWJsaXNoZXIy MTMzNzEzMQ==