  Mechanics of Solids A Journal of Russian Academy of Sciences   Founded
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M. A. Belova, V. I. Gulyaev, and I. L. Solov'ev, "Static and dynamic critical states of parabolic shells subjected to simple or compound rotations," Mech. Solids. 39 (3), 124132 (2004) 
Year 
2004 
Volume 
39 
Number 
3 
Pages 
124132 
Title 
Static and dynamic critical states of parabolic shells subjected to simple or compound rotations 
Author(s) 
M. A. Belova (Kiev)
V. I. Gulyaev (Kiev)
I. L. Solov'ev (Kiev) 
Abstract 
Bifurcation problems for thin truncated parabolic shells
subjected to simple or compound rotation are considered. The analogies and differences in statements and techniques
of solution of these problems are discussed. The buckling of the shell in a state of simple rotation is described in the reference frame that is spinning together with the shell. The equations of precession vibrations induced by the compound rotation are related to a rotating reference frame with respect to which the shell is spinning. In both cases, the buckling shape is described in terms of the first harmonic with respect to the hoop coordinate. Numerical analysis shows that for parabolic shells extended along the axis, the critical angular velocities are lower than the resonant angular velocities in the compound rotation. For shallow parabolic shells, the inverse tendency is observed.
As a rule, the static and dynamic buckling analysis of turbines is based on simplified models in which rotors are considered to be rigid bodies and the critical state is assumed to be able to occur due to the elastic compliance of the shaft [14]. For example, on the basis of such a model it has been established that an elastic spinning shaft to which rigid disks are attached can buckle under the action of centrifugal forces of inertia. As a result, the axis of the shaft deforms and became a plane curve in the reference frame spinning with the shaft. The buckling effect of the spinning shaft can be explained by the fact that the centrifugal forces of inertia acting on the shaft are positional (i.e., depend only on the generalized coordinates), since they depend on the deflection of the shaft axis. It turns out in this case that the equation of the critical equilibrium of the shaft coincides in form with the equation of natural vibrations of this shaft and that the critical angular velocity coincides with the fundamental frequency of natural vibrations of the elastic system.
The mechanical behavior of a rotor on an elastic shaft becomes considerably more complicated in the case of compound rotation, for example, when the rotor is installed on a flying vehicle that is performing attitude motions. In this case, the axis of the rotor performs an additional forced rotation in some plane. The superposition and interaction of various kinds of rotation excites the precession vibrations of the rotor on the elastic shaft. In an inertial reference frame, these vibrations are represented as a steady deformed state in the plane that passes through the spinning axis of the system and the axis of its plane rotation. The appearance of the critical states of the rotating system in this case can be due to precession resonances that occur at certain (critical) values of the angular velocity of the system.
The critical states similar to those appearing in the shaft in the case of simple or compound rotations can appear also in the rotor if it is thinwalled and has a sufficient elastic compliance [57]. In view of this, it is important to establish the conditions for the appearance of static and dynamic critical states for simple shell rotors in the case of simple or compound rotation and to determine which of these types of instability occurs earlier. Although being apparently different, these problems have much in common and, therefore, joint analysis and comparison of these problems seems to be reasonable. In what follows, this analysis is conducted for shallow and extended parabolic shells.
A number of related issues of mechanics of thin rotating shells was considered in [811]. 
References 
1.  H. Ziegler, Principles of Structural Stability
[Russian translation], Mir, Moscow, 1971. 
2.  A. S. Kel'zon, Yu. P. Tsimanskii, and V. I. Yakovlev,
Dynamics of Rotors in Elastic Bearings [in Russian], Nauka, Moscow, 1982. 
3.  G. S. Maslov, Vibration Analysis of Shafts
[in Russian], Mashinostroenie, Moscow, 1980. 
4.  A. Tondl, Dynamics of Rotors of Turbine Generators
[Russian translation], Energiya, Leningrad, 1971. 
5.  V. I. Gulyaev, A. A. Grom, and N. A. Snezhko, "Precession vibrations of conical shapes in the case of compound rotation," Izv. AN. MTT [Mechanics of Solids], No. 2, pp. 156163, 1999. 
6.  V. I. Gulyaev and I. L. Solov'ev, "Precession vibrations and resonances of composite shells in the case of compound rotation," Prikladnaya Mekhanika, Vol. 35, No. 6, pp. 7481, 1999. 
7.  V. I. Gulyaev, I. L. Solov'ev, and P. Z. Lugovoi, "Analysis of precession vibrations of thinwall elastic shells in compound rotation,"
J.Sound and Vibrat., Vol. 246, No. 3, pp. 491504, 2001. 
8.  N. E. Egarmin, "The precession of standing waves of vibrations of a rotating axially symmetric shell," Izv. AN SSSR. MTT [Mechanics of Solids],
No. 1, pp. 142148, 1986. 
9.  A. L. Smirnov and P. E. Tovstik, "Qualitative analysis of the dynamics of rotating shells of revolution," in Modern Issues of Mechanics and Aviation [in Russian], pp. 280290, Mashinostroenie, Moscow, 1982. 
10.  Y. Padovan, "Natural frequencies in rotating prestressed cylinders,"
J. Sound and Vibrat., Vol. 31, No. 4, pp. 469482, 1973. 
11.  K. R. Sivadas, Vibration analysis of prestressed rotating thick circular conical shells," J. Sound and Vibrat., Vol. 186, No. 1, pp. 99109, 1995. 
12.  V. I. Gulyaev, V. A. Bazhenov, E. A. Gotsulyak, et al.,
Stability of Periodic Processes in Nonlinear Mechanical Systems [in Russian], Vyshcha Shkola, L'vov, 1983. 
13.  A. I. Lur'e, Analytical Mechanics
[in Russian], Fizmatgiz, Moscow, 1961. 

Received 
06 September 2001 
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