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by treating arbitrary forces as a series of rotating forces 125].

Kinetostatic strains have been considered in deriva-tions of the equation$ of motion for obtaining blade displacements, vibrational frequencies, and modę shapes [26]. The dynamics of packets of blades have been analyzed by incorporating responses due to the number of blades in the packet, stiffness and mass ratios. size and position of bracing wires, and rotating speed [27, 28]. A new FEM program called PVAST was used; it gives graphic results of dynamie stresses. displacements. frequencie$, and modę shapes [29].

Many authors use different energy principles to deal with blade vibration problems. The accuracy of the energy method and transmissibility method have been compared [30]. Forced vibrational aspects of rotating blades have been studied using Lagrange's equations [31J. The effects of various parameters on blade dynamics have been studied using the Ritz method. Reissner's variational principle, Trefftz method, and modified lower bound method [32]. Results from the finite difference energy method have been examined with previously available results 133).

Rotating blade vibration has been considered using shallow shell theory and the Ritz method [34]; shal dw shell theory has been compared with deep Shell theory [35]. The potential energy method has been extended to blade vibration analysis [36-38]. The superiority of the Reissner method was pointed out; shear deformation, rotary inertia, speed of rotation, and stagger angle were accounted for. Nondimensional frequency parameters have been obt8ined for different aspect ratios, shallowness ratios. and thickness ratios [39]. Leissa addressed the problem of shallow shells by using the Ritz method with algebraic polynomial troi functions [40]. Vibration analyses for blades with and without camber are available [41 ].

A transfer matrix has been used to investigate the vibrational behavior of turbinę blades [42]. Experi-mental and analytical results were also obtained for blades with shrouds by varying the number of blades in a group, the method of connection, geo-metrical tolerances, and centrifugal forces. Chiatti and Sestieri [43] used FEM in combination with transfer matrix method to study the free vibrations of turbinę blades. Free and forced vibration analyses of closed periodic structures have been presented [44]. Non-massive blades of axial vents have been studied for different blade designs using a simple numerical procedurę [45].

Flexural vibrations have been analyzed by treating the blade as a cantilever beam [46]. Subrahmanyam [47] has described the superiority of the Reissner method. Downs [48] has formulated the discretiza-tion technique for blade vibration and compared it with other techniques. Borishanski [49] stressed the vibration effects of fastening connectors on the blade. The beam analysis of blade vibration has been im-proved by accounting for earlier discrepancies [50].

Wood [51 ] brought out a new mathematical tool -the boundary integral technique - to investigate blade vibration. He incorporated torsional stiffness, warping stiffness, and shear center coordinates. The causesof gyroscopic forces and damping have been reported from beam analysis [52. 53]. It was found that coriolis acceleration due to gyroscopic motion of spinning rotors is creating new regions of instabihty [54]; also shown was the importance of second order resonance.

Hodges [55] presented a simple but efficient method for obtaining the fundamental frequency of blade vibration from the composite expansion technique. Leissa and Lee [56] used a numerical integration technique to solve the shallow shell analysis of blade. They also summarized the effects of rotational inertia, coriolis acceleration, variable curvature and thickness. and arbitrary quadrilateral planform. Free and forced vibrations and resonant stresses of rotating blades have been examined [57].

Nonlinear analysis. A number of articles have been published in the field of nonlinear vibration analysis of blades. Murthy and Hammond [58] used trans-mission matrices to linearize the blade equations of motion. Nonlinear differential equations have been converted into nonlinear algebraic equations and simplified by the method of iteration [59]. Ven-katesan and Nagaraj [60] demonstrated the softening effect of large amplitudę on flapwise frequency. Nonlinear blade vibration analyses have been discussed with the use of FEM [61-63].