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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

533.6 During operation, hydraulic-turbine blades are inevitably subjected to unsteady flow action, which induces forced vibrations of the blades. If the frequencies of the initiating forces are close to the natural frequencies of the blades, resonance can occur. This frequently causes blade failure in the operation of turbines. The above circumstances lead to the necessity of increasing the accuracy of calculation of eigenfrequencies and eigenmodes of blade vibrations. The difficulties in achieving this goal are due to a large number of factors that influence the solution of the corresponding hydroelastic problem, in which elastic deformations of blades and fluid motion should be determined simultaneously.In most of the former papers devoted to this problem, these factors were not taken into account. Thus. Gorelov and Guseva [1] did not take into account the curvature of the blade surface, and also determined approximately (in a two-dimensional formulation) the hydrodynamic interaction. Naumenko et al.[2] studied a blade model in the form of a plate of variable thickness. Sundqvist [3] considered this problem for a blade model in the form of a variable-thickness shell in the most complete approximation and with allowance for spatial flow. However, the finite element method (FEM) employed in [3] to describe motion of both solid and fluid particles requires a large computer memory and time, and this limits the possibilities of the method in engineering practice.The present work seeks to develop a method for calculating the natural vibrations of blades using a sufficiently general model similar to [3], which can be realized on personal computers. For this, to determine the hydrodynamic loads acting on the blades, we used the integral equation method, which decreased by unity the dimension of the corresponding hydrodynamic problem.1. Basic Assumptions and Formulation of the Problem. We consider the problem of small free linear vibrations of rotor blades (Fig. 1) of an axial hydraulic turbine in a fluid. The rotor is simulated by an annular cascade consisting of N identical, fairly thin blades located between two infinite circular cylinders with radii R1 and R2. We assume that the interaction between the blades occurs only via the fluid, which is ideal and incompressible and whose motion is caused only by blade vibrations.By virtue of the assumption of small blade thickness, we write the problem for the displacements of the centroidal surfaces. According to the condition of cascade uniformity [4], the normal vibrations of the centroidal surfaces can be written as wJm=uJm(x)ei(~j+uJ m) (j,m=O, 1,2,...,N-1). (1.1) Here #5 = 27rj/N is the phase shift between vibrations of neighboring blades, wj is the eigenfrequency of blade vibrations with a phase shift pj, UJm is an amplitude function of the displacement of the centroidal surface Sm of the ruth blade, and x is the radius vector of the points on the surface Sin.Taking into account (1.1), we can reduce the equation of the normal vibrations of the blade cascade wit...

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