tailieunhanh - UNSTEADY AERODYNAMICS, AEROACOUSTICS AND AEROELASTICITY OF TURBOMACHINES Episode 2

Tham khảo tài liệu 'unsteady aerodynamics, aeroacoustics and aeroelasticity of turbomachines episode 2', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Flutter Design of Low Pressure Turbine Blades with Cyclic Symmetric Modes 43 in the Panovsky-Kielb method baseline unsteady aerodynamic analyses must be performed for the 3 fundamental motions two translations and a rotation. Unlike the current method work matrices must be saved for a range of reduced frequencies and interblade phase angles. These work matrices are used to generate the total work for the complex mode shape. Since it only requires knowledge of the reduced frequency and mode shape complex this new method is still very quick and easy to use. This paper describes the complete theory for extension of the Panovsky-Kielb method gives example results discusses the importance of the interaction effects of the cosine and sine modes and discusses the contribution to work associated with the steady pressure. 2. Work for General Complex Mode Shapes Work per cycle can be calculated by integrating the dot product of the local velocity vector with the local force vector over one cycle of vibration and the entire airfoil surface. Airfoil Surface Figure 2. Coordinate Systems Wcyc ff - Í NP dtdA A 0 where ị velocity N normal vector P surface pressure 44 For harmonic motion the displacement normal vector and surface pressure can be written as the bar over the variables indicates the complex conjugate 1 N ỉ ne iiVt nc p p ps i pe- pe The work per cycle then becomes wcyc - J p ộ n - A nj Ns ộp - ộp dA A For real mode shapes the work per cycle reduces to the familiar expression NCyC IS real 7T Ị Ws ỘPimag dA A Thus the steady pressure term does not contribute to the work when the mode shape is real. As will be shown this is not the case when the mode shape is complex. 3. Work for Cyclic Symmetry Mode Shapes For cyclic symmetry eigensolutions the mode shapes are complex. For a forward traveling wave ỹ ac - ias n nc ins Where ac and as are commonly referred to as the cosine and sine modes. The work per cycle becomes AẠjc I Ns facPimag H IsPreal Ps C Tls S c dA A L J Thus .