Perpustakaan Digital ITB

The present work deals with an analytical study of aeroelastic behavior of two - dimensional cascades in incompressible flow. The results are intended to give better knowledge on the flutter characteristics of turbomachine bladings necessary for safe designs and operations. In this regard, the effects of the most essential design parameters on the flutter occurence should be investigated. Those parameters are : (a) The cascade geometry, i.e. the stagger angle, pitch to chord ratio, and angle of attack. (b) The elastomechanic parameters, e.i. the position of rotation (or elastic) axis, natural frequency ratio, mass ratio, and structural damping. (c) The unsteady aerodynamic parameters, e.i. the reduced frequency, and interblade phase angle. Flutter behavior in the coupled two degree of freedom motions in bending and pitching as well as possible occurence of instabilities in the uncoupled one degree of freedom modes are the main interest. Such a systematic study to reveal the influences of those parameters mentioned above has never been done before for cascades. The development of aerodynamic load computation method is started from Martensen's theory [121 for the steady condition followed by Carsten's approaches [4,5] to formulate the unsteady solution. In this regard, the analysis is based on the following assumptions. The two dimensional cascade consists of a straight row of infinite number of identical blades. The blades are profiled with finite camber and thickness. They are arranged parallel to one another with constant distances in staggered position. Each of the blades are assumed to be supported elastically to enable independent oscillations in a purely vertical translation (bending) and rotation (torsion) about the elastic axis of the blade. During vibration, the phase angle differences between the adjacent blades are assumed to be identical for the entire cascade. As the blades oscillate, the flow around the profiles fluctuates as well. However, the amplitudes of oscillations are assumed to be small that the flow never separates from the profile wall. The treatment of the flow problem to solve the unsteady aerodynamic forces is based on the conservation form of vorticity transport equation, where the singularity method is applied to obtain numerical solutions. Here, the blade surfaces are replaced by vortex layers. As the blades oscillate, the circulation around each of them changes periodically. This in turn creates counter directional vortices at the trailing edges which are convected down stream along the wake by the outlet flow. The mathematical formulation of the aeroelastic stability problem is based on the Lagrange's equations of motion. It involves the elastomechanic (inertial, elastic, and friction) and aerodynamic forces inducing two degree of freedom motion in the translation and rotational modes. To obtain the flutter characteristics, it is sufficient to solve the stability boundaries where the motions are in the neutrally stable conditions. In this case the oscillations can be assummed as purely time harmonic. Linearization of the problem is based on the assumption that the amplitudes of the vibrations are small. In order to verify the analysis, isolated flat plate flutter is simulated with the cascade method by introducing flat plate profile with large interblade distances. Comparissons with analytic solution as well as Theodorsen's results (15] show sufficienly good agreements. The results of the computations made show in general that each parameter considered plays a certain role in the aeroelastic behavior of the cascade. Qualitatively their individual effects in the two degree of freedom flutter strongly influences the flutter characteristic of the cascade ; highly cambered and thick profiles tend to be more succeptible to flutter. In the two degree of freedom motion, the natural frequency ratio which is a measure of relative stiffnes in bending and torsion, apparently determine the critical velocity and can be taken as a controling variable in the structural design of the blade. The location of elastic axis relative to the center of mass clearly effects the stability. For common turbomachine blades where the frequency ratios are small, positioning the elastic axis behind the profile center of mass tends to improve stability. Enlargement of the interblade distance gives better stability in general. Increasing the stagger angle, tends to destabilize the cascade in the lower range of frequency ratio. Regarding the influence of mass ratio, it is found that heavier blades produce better stability. Stabilizing effect is shown as well by introduction of structural dampings ; appreciable improvement is ensured even with very small value of damping existing in the solid blades. The values of interblade phase angle where flutter starts are influenced by all considered parameters. However, there are indications that blade camber and thickness strongly affect the location of the critical phase angle where flutter velocities take the lowest values. Computations for single degree of freedom motions show that flutter can occur in the uncoupled bending and torsion modes. The critical velocities in bending are considerably higher than those in torsion. Comparison of results from the two degree with one degree of freedom calculations suggests that there are certain parallelity, so that the mechanism of flutter can be better understood by studying both asppects. The validity checks of the present computation with those obtained by Th. Theodorsen as well as by analytical methods for flat plates, show good agreements for low mass ratios and low frequency ratios. The analysis carried out in the present work to systematically investigate the aeroelastic behavior of two dimensional oscillating cascades with large cambers and thicknesses in incompressible flow, has indicated the occurence of flutter in the two as well as one degree of freedom modes. The individual as well as common influences of the steady loading, cascade geometry and elastomechanic parameters to the flutter characteristics are presented and discussed.