Natural frequencies of rotating functionally graded cylindrical shells

Love’s first approximation theory is used to analyze the natural frequencies of rotating functionally graded cylindrical shells.To verify the validity of the present method,the natural frequencies of the simply supported non-rotating isotropic cylindrical shell and the functionally graded cylindrical shell are compared with available published results.Good agreement is obtained.The effects of the power law index,the wave numbers along the x-and θ-directions,and the thickness-to-radius ratio on the natural frequencies of the simply supported rotating functionally graded cylindrical shell are investigated by several numerical examples.It is found that the fundamental frequencies of the backward waves increase with the increasing rotating speed,the fundamental frequencies of the forward waves decrease with the increasing rotating speed,and the forward and backward waves frequencies increase with the increasing thickness-to-radius ratio.     

Natural frequencies of circular arcs with varying cross section

This paper presents a transter matrix method for conveniently predicting the natural frequencies of circular arcs with varying cross section. The equations of in-plane vibration of an arc are written as a coupled set of first-order differential equations by using the transfer matrix of the arc. Once the transfer matrix has been determined, the natural frequencies and the mode shapes of the vibration are calculated in terms of the elements of the matrix for a given set of boundary conditions at the ends of the arc. The numerical results obtained by the method are compared with the experimental results for free-clamped arcs with tapered breadth, which are in good agreement with each other.     

On the spectrum of natural frequencies of circular cylindrical shells completely filled with a fluid

The natural frequencies of cylindrical shells filled with a fluid and having the ends either simply supported or clamped are determined. Conditions are studied under which the natural frequencies of the shell are close or multiple __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 5, pp. 42–49, May 2006.     

Natural frequencies of thick, complete, circular rings with an elliptical or circular cross-section from a three-dimensional theory

A three-dimensional (3D) method of analysis is presented for determining the free vibration frequencies and mode shapes of thick, complete (circumferentially closed), circular rings with an elliptical or circular cross-section. Displacement components u_r, u_θ, and u_z in the radial, circumferential, and axial directions, respectively, are taken to be periodic in θ and in time, and algebraic polynomials in the r and z directions. Potential (strain) and kinetic energies of the circular rings are formulated, and upper-bound values of the frequencies are obtained by minimizing the frequencies. As the degree of the polynomials is increased, frequencies converge to the exact values. Convergence to four-digit exactitude is demonstrated for the first five frequencies of the rings. Novel numerical results are presented for the circular rings having an elliptical cross-section based upon 3D theory. Comparisons are also made between the frequencies from the present 3D Ritz method and ones obtained from thin and thick ring theories, experiments, and other 3D methods.     

Natural dynamic characteristics of a circular cylindrical Timoshenko tube made of three-directional functionally graded material

The natural dynamic characteristics of a circular cylindrical tube made of three-directional(3 D) functional graded material(FGM) based on the Timoshenko beam theory are investigated. Hamilton’s principle is utilized to derive the novel motion equations of the tube, considering the interactions among the longitudinal, transverse,and rotation deformations. By dint of the differential quadrature method(DQM), the governing equations are discretized to conduct the analysis of natural dynamic charact...     

Electric forces on a circular cylindrical inclusion at the surface separating two dielectrics

Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, No. 4, pp. 14–24, July–August, 1993.     

Natural frequencies of a hemispherical shell

An experimental study of the natural frequencies of a hemispherical thin shell with a clamped edge is described. For the particular shell model studied, the four lowest symmetric modes were identified conclusively by establishing the node-line patterns. The associated natural frequencies were found to be in excellent agreement with theoretical predictions. A great number of higher modes were also excited but could not be conclusively identified due to difficulties in establishing the node-line patterns. The symmetric frequency spectrum up to and including the 23rd mode was, however, established by means of a qualitative argument and showed good correlation with the theoretical spectrum. A number of natural frequencies were found to be present in a particular frequency band for the test shell resulting in a severe crowding in this region. This was found to be in agreement with the more general results for hemispherical shells which indicate that the crowding takes place around a frequency parameterp=1.0.     

Instability analysis of nonlinear surface waves in a circular cylindrical container subjected to a vertical excitation

Singular perturbation theory of two-time scale expansions was developed both in inviscid and weak viscous fluids to investigate the motion of single surface standing wave in a liquid-filled circular cylindrical vessel, which is subject to a vertical periodical oscillation. Firstly, it is assumed that the fluid in the circular cylindrical vessel is inviscid, incompressible and the motion is irrotational, a nonlinear evolution equation of slowly varying complex amplitude, which incorporates cubic nonlinear term, external excitation and the influence of surface tension, was derived from solvability condition of high-order approximation. It shows that when forced frequency is low, the effect of surface tension on mode selection of surface wave is not important. However, when forced frequency is high, the influence of surface tension is significant, and can not be neglected. This proved that the surface tension has the function, which causes free surface returning to equilibrium location. Theoretical results much close to experimental results when the surface tension is considered. In fact, the damping will appear in actual physical system due to dissipation of viscosity of fluid. Based upon weakly viscous fluids assumption, the fluid field was divided into an outer potential flow region and an inner boundary layer region. A linear amplitude equation of slowly varying complex amplitude, which incorporates damping term and external excitation, was derived from linearized Navier–Stokes equation. The analytical expression of damping coefficient was determined and the relation between damping and other related parameters (such as viscosity, forced amplitude and depth of fluid) was presented. The nonlinear amplitude equation and a dispersion, which had been derived from the inviscid fluid approximation, were modified by adding linear damping. It was found that the modified results much reasonably close to experimental results. Moreover, the influence both of the surface tension and the weak viscosity on the mode formation was described by comparing theoretical and experimental results. The results show that when the forcing frequency is low, the viscosity of the fluid is prominent for the mode selection. However, when the forcing frequency is high, the surface tension of the fluid is prominent. Finally, instability of the surface wave is analyzed and properties of the solutions of the modified amplitude equation are determined together with phase-plane trajectories. A necessary condition of forming stable surface wave is obtained and unstable regions are illustrated.     

CAPILLARY EFFECT ON VERTICALLY EXCITED SURFACE WAVE IN CIRCULAR CYLINDRICAL VESSEL

In a vertically oscillating circular cylindrical container, singular perturbation theory of two-time scale expansions was developed in inviscid fluids to investigate the motion of single free surface standing wave including the effect of surface tension. A nonlinear slowly varying amplitude equation, which incorporates cubic nonlinear term, external excitation and the influence of surface tension, was derived from potential flow equation. The results show that, when forced frequency is lower, the effect of surface tension on mode selection of surface wave is not important. However, when forced frequency is higher, the surface tension can not be neglected. This proved that the surface tension causes free surface returning to equilibrium location. In addition, due to considering the effect of surface tension, the theoretical result approaches to experimental results much more than that of no surface tension.     

Dynamics of an elastic cylindrical shell with a gas–liquid medium subject to two-frequency vibrational excitation

The paper presents experimental results on nonlinear dynamic processes such as the deformation of the elastic wall of a cylindrical shell filled with a fluid and the formation and clustering of gas bubbles, which interact under two-frequency excitation Translated from Prikladnaya Mekhanika, Vol. 44, No. 11, pp. 112–122, November 2008.     

Small twist of a circular cylindrical tube exhibiting cylindrical aeolotropy of a nonorthogonal type

Summary At each point of an elastic cylindrical tube the aeolotropy is defined with respect to a triad of nonorthogonal curves. When the tube undergoes a small twist the effect of the nonorthogonality appears in the form of extra terms in the expression for the couple on the tube. Restrictions on the elastic properties of the material are obtained for zero warping of the cylinder sections, and a solution is given when warping is present.     

The plane problem on vertical impact of a rotating circular cylindrical shell against the surface of a liquid

The plane nonsymmetric problem on impact against and immersion into a compressible fluid of a thin electic cylindrical shell is considered. The shell rotates about its axis with a given angular velocity. The boundary-value problem is reduced to an infinite system of integral Volterra equations of the second kind. Translated from Prikladnaya Mekhanika, Vol. 36, No. 5, pp. 103–113, May, 2000.     

Torsion of a circular cylinder weakened by a cylindrical crack

Leningrad Wood Technology Academy. Translated from Prikladnaya Mekhanika, Vol. 25, No. 3, pp. 11–14, March, 1989.