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loadings of a turbinę impeller (145]. Kuo [146] modeled a blade-disc segment asa rotating symmetri-cal structure and studied the effects of coupled blade-disc resonance. the dissimilar behavior of blade groups. and the uncoupled blade or disc natural fre-quencies on the vibrational characteristics of the blade disc assembly. Efficient NASTRAN programs have been developed to anafyze the aeroelastic and structural properties of bladed shrouded discs [ 147-155]. These programs also incorporate the effects of mistuning and tuning, forced vibration analysis, coriolis and centripedal accelerations, and flutter.

A special bladed disc element and a special shrouded blade element have been used to simplify the static and dynamie deflection analyses of blade [156]. Mindlin's thick-plate theory was used in a disc analysis with both annular and sector elements [157]; theoretical results from an eight noded, superpara-metric thick shell element analysis were compared with experimental results for a shrouded. bladed disk. Nigh and Olson [158] have reportea the effects of damped and undamped transverse point loads in space-fixed coordinates; they also determined critical speeds in space fixed coordinates. Singh (159] dis-cussed the torsional yibrations of blades with root flexibi(ity; he presented a correction factor for the fixed root frequency to account for root flexibility. Impeller blade frequencies, modę shapes. and stress distribution have been analyzed and verified with experimental results obtained by using semiconductor strain gauges and a mechanical telemetry system [160], A model havinga five-degrees-of-freedom sub-system has been used to study the dynamics of a bladed disc assembly; dry friction, structural damp-ing, mistuning. and type of excitation were eon-sidered (161 ].

Loewy (162] used a classical structural dynamie analysis with Lagrange equations to discuss the response of bladed disc assemblies for a wide rangę of rotor flexibilities. Hunt (163] has suggested a transfer matrix method to study the in-plane yibrations of blades with known boundary conditions. Free and forced vibration characteristics of blades have been studied by applying the concept of rota-tionally periodic structure and the transfer matrix method (164). An inertia and elastic hinge model has been used to consider flexibility of impellers [165].

Experimental Methods

Experimental results obtained from an image dero-tated holographic interferometry method have been checked with FEM Solutions (166J. Modę splitting has been studied for mistuned bladed disc with the help of the holographic interferometry technigueand strain gauges [167]. Speeds up to 13.000 RPM have been measured by utilizing eight-channel telemetry systems with holographic interferometry [168]. Experimental results of bench resonance are in good agreement with predicted frequencie$ [169]; in-cluded are the effects of in-plane inertial couplings in tuned and mistuned bladed disc assemblies. The fast Fourier transform technique has been used to investigate the yibrations of periodically symmetric structure such as bladed disc assemblies [170].

Jones (171] has formulated a simple discrete analyti-cał model to predict the fundamental frequency of a blade with slip at the root; he verified the results experimentaliy. It has been shown that the fundamental frequency is a function of spin frequency for a rotating blade with an off-set root (172]. Singh and Rawtani [ 173] have modeled a blade with spring hinged-free end boundary conditions. Sollmann [174] has demonstrated a method for calculating the centrifugal force coefficient for rigidly mounted as wed as f/exibły mounted blades. Khader (175] has incorporated the effects of disc flexibility and shaft and bearing flexibiłities in a bladed disc vibration analysis. A traveling wave solution has been used to obtain the dynamie characteristics of a shrouded bladed disc assembly [176].

Jendryschik used a co-rotating accelerometer to analyze the yibrations of rotating disc; he found that the mistuning of a bladed disc is caused by the irregularities of friction forces and by harmonie excitations [178]. Wildheim [179, 180] used the principle of dynamie substructures with free modes of a disc and the clamped free modes of a blade to predict the natural frequencies of a rotating bladed di9C with lacing wire. Irretier [181] used the effects of variable thickness. centrifugal force. and the location of blades on the disc to determine the bend-ing and torsional yibrations of a blade; direct integra-tion techniques and the shooting method were used to solve the problem [182]. It has been shown that the effects of blade twist, shear deformation. rotary inertia for bending. centrifugal force. and stagger angle are significant [183]. Eigenyalues and modę