Natural vibrations of hydroturbine cascades having small geometric nonuniformity

Abstract

In the theory of turbomachinery elasticity, it is welt known that the introduction of even a small nonuniformity in cascades can affect the stability of blade vibrations in a flow and increase considerably the dynamic stresses in individual blades with forced vibrations of the cascades. For gas turbomachines, for which the natural frequencies of blade vibrations depend only slightly on the ambient medium, this problem has been adequately studied. For hydraulic turbines, there have been few studies in this direction. In the present work, we examine the effect of small geometric nonuniformity of hydraulic-turbine cascades on the natural frequencies and modes of their vibrations in a quiescent liquid.1. Formulation of the Problem. We study free vibrations of a hydraulic-turbine cascade of sufficiently thin blades. The cascade is located between two coaxial cylinders in a quiescent, ideal, incompressible liquid. In contrast to the model studied in [1], we shall assume that the cascade has small geometric nonuniformity because of inaccurate manufacture and assembly.The geometric parameters of the cascade are assumed to be specified. We introduce a homogeneous cascade with blades of identical thickness h(r0m), where r0m are the radius vectors of the centroidal surfaces S~ ) of the blades. The geometric nonuniformity of the cascade considered is then given by the functions 6rm(r0m) = rm -r0m, ~hn~(r0m) = hm(rm) -h(r0m). where w~ ) is the natural frequency of first-mode vibrations of the ruth blade the and w(D is the natural frequency of the corresponding uniform cascade. The introduction of the second small parameter r is due to the fact the order of smallness of the quantities in relations (1.2) and (1.3) can be different. Assuming that the interaction between the blades is realized only via the liquid, we obtain a matrix system of equations that describe the free vibrations of the blade cascade considered: N-1

Mathematical model of aeroelastic vibrations of cascades of axial turbomachine blades, caused by circumferential nonuniformity of the flow

“…Small geometric irregularity of the cascade, caused by technological inaccuracy of blade manufacturing and wheel assembling, is admitted in the general case, because the associated inhomogeneity of dynamic characteristics of the blades exerts a significant effect on their aeroelastic vibrations [7][8][9]. We assume that the effect of this inhomogeneity on the unsteady component of aerodynamic forces can be neglected.…”

“…Many publications deal with solving the third subproblem within the model of aeroelastic vibrations of cascades, which are described by linear differential equations with constant coefficients [3,4,[7][8][9]. In reality, the coefficients of the aerodynamic forces in the corresponding differential equations are time-dependent because of circumferential nonuniformity of the flow in the turbomachine duct.…”