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tations has been carried out (105). Lang and Nemat-Nasser (1061 deve!oped approximate step functions to inve$tigate nonuniform blade vibrations; they discussed two example$ and verified them with other results. A semj-empirical method has been used to measure the dynamie stresses of groups of blades affected by synphase and intra-packet vibrations (1071. Bending and torsional frequencies for lower modes have been calculated accurately (1081. Nu-merlcal methods have been formulated to predict the vibrational behavior of blades subjected to eon-tinuous erosion (109]. Nied (1101 has determined the values of a set of frequency response functions from impulse excitation and used it to calculate modal parameters and hence vibrational aspeets. A stationary functionai method with normal modes has been utilized to approximate a nonlinear analysis of blade vibration; the effects of shear deformation. rotary inertia, and coriolis forces were included (1111.

Dickerson (1121 has summarized developments in turbinę blade analysis and compared results from a test model and a theoretical analysis. Gudmundson (1131 used a perturbation technique to investigate variations of natural frequencies as a function of blade geometry. The damping decay ratę has been predicted accurately by using standing wave flutter analysis for tuned and mistuned blades (1141. The effects of interactions between bending and torsional motions of nonuniform blades were included (1151. Bendiksen and Friedmann (1161 discussed blade flutter boundaries with bending-torsional coupling. Dynamie stresses and eigenvalues have been determined from the potential admission condition (117-1191. Wake-induced vibrations have been analyzed to determine dynamie stresses (120. 1211. The se-quential optimization technique has been used to tune the natural frequencies of blades away from resonant conditions (1221. An optimum method for controlling the cross sections of blade to reduce weight has been outlined (1231. Life prediction theories have been used to evaluate blade tip dura-bility from heat transfer and stress analysis (1241. Blade composite materials have been discussed for various influencing factors (125*1271.

BLADE-DISC INTERACTIONS Stress Analysis

Frlctlonal forces between lock ing pieces significantly alter the contact stress distribution at the blade root (1281. Stress distribution for an orthotropic rotating disc of varying thickness has been investi-gated and compared with FE Solutions (129. 1301. The centrifugal stresses in vanes and disc have been determined by treating unshrouded radial impellers as co-axial rings. and using the matrix method (1311. A shoulder at the root joint has been found to decrease dynamie loading and inerease fatigue life (1321. Gamner (1331 showed the invalidity of Tresca's yield criterion for a rotating disc from its discontinuity at the elasto-plastic interface. Rotating dises with linearly, exponentially. and hyperbolically varying thicknesses have been studied for strength analysis (1341 using the matrix method. The matrix method has also been used to determine rhenomic and scleronomic components of inelastic deformation in an annular section of a disc (1351. Singh and Rawtani (136,1371 researched the stress distribution at the fir tree root fastening of turbinę blades by simplifying the root as an assembly of steps. They incorporated the effects of centrifugal body force of the root. the force due to friction between the blade and disc steps. and the dissimilarities in the materiał properties of the blade and the disc. Kopecki and Walczak (1381 have reported some important conclusions about the axisymmetry of the stress distribution of rotating dises madę up of nonlinear materia). A detailed study of stresses in rotating dises of variable thickness has been done using the finite different method (1391; included were thermal gradient and shrinkage effects.

Vibr«tion Analysis

Lalanne (1401 has presented references for the vibra-tion control of blades; blade-disc interactions. axi-symmetric structures, and rotors were considered. Ewins (141J has summarized fluid-structure inter-action. vibration modes of bladed disc assemblies. tuned and mistuned assemblies, and flutter.

Numerous articles in which FEM was used to analyze blade disc interactions are available. Partington (1421 extended the FEM to study the vibrating stresses and frequencies of pinned root blades and confirmed the results. The dynamics of compressor impellers have been studied (1431. Nagraj and Sahu (1441 compared the Rayleigh Ritz method with the Galerkin technique in a study of torsional yibration of nonuniform turbinę blades; also included was root flexi-bility. FEM and the holographic technique have been used to investigate the static. dynamie, and thermal

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