Nonlinear stability of circular vortex patches

This paper proves that circular vortex patches in the plane are stable for the nonlinear dynamical system generated by the Euler equations of incompressible fluids. This is achieved by establishing a relative variational principle in terms of either energy or angular momentum. Thus, we exploit and extend Arnold's idea in (1965, 1969) to a nonsmooth setting as well.Research partially supported by DOE contract DE-AT03-82ER12097 to the University of California, Berkeley     

The vibration modes of concentrically supported free circular plates

Vibration of a circular plate internally concentrically supported is basic in structural mechanics. The complete first six frequencies and mode shapes are determined for clamped, simply-supported, sliding, and ring supports using exact characteristic frequency equations. Complex mode switches occur as the support radius is varied. Asymptotic formulas for large support radii are derived. Different types of singularities as the support radius shrinks to zero are distinguished and classified.     

Theoretical investigation of frequency characteristics of free oscillation and injection-locked magnetrons

The frequency characteristics of free oscillation magnetron(FOM) and injection-locked magnetron(ILM) are theoretically investigated.By using the equal power voltage obtained from the experiment data,expressions of the frequency and radio frequency(RF) voltage of FOM and ILM,as well as the locking bandwidth,on the anode voltage and magnetic field are derived.With the increase of the anode voltage and the decrease of the magnetic field,the power and its growth rate increase,while the frequency increases and its growth rate decreases.The theoretical frequency and power of FOM agree with the particle-in-cell(PIC) simulation results.Besides,the theoretical trends of the power and frequency with the anode voltage and magnetic field are consistent with the experimental results,which verifies the accuracy of the theory.The theory provides a novel calculation method of frequency characteristics.It can approximately analyze the power and frequency of both FOM and ILM,which promotes the industrial applications of magnetron and microwave energy.     

Nonlinear free vibration behavior of functionally graded plates

In this paper, an analytical solution is provided for the nonlinear free vibration behavior of plates made of functionally graded materials. The material properties of the functionally graded plates are assumed to vary continuously through the thickness, according to a power-law distribution of the volume fraction of the constituents. The fundamental equations for thin rectangular plates of functionally graded materials are obtained using the von Karman theory for large transverse deflection, and the solution is obtained in terms of mixed Fourier series. The effect of material properties, boundary conditions and thermal loading on the dynamic behavior of the plates is determined and discussed. The results reveal that nonlinear coupling effects play a major role in dictating the fundamental frequency of functionally graded plates.     

A semi-analytic solution for free vibration of annular sector plates

The paper describes a semi-analytical method in which the basic function in the circumferential direction satisfying the boundary conditions of the radial edges is substituted into the free vibration equation of the curved plate. By a suitable transformation, an ordinary differential equation is obtained. The resulting equation is solved by a finite difference technique. Tabulated results have been presented for annular sector plates possessing different boundary conditions. Excellent accuracy has been obtained wherever comparisons have been possible.     

Correlation between free oscillation frequency and stiffness in high temperature superconducting bearings

A superconducting magnetic bearing is a dynamic system, which undergoes vibrations at various frequencies during its operation. In this study, we investigated the free vibration frequency modes of a permanent magnet (PM) levitated over a high temperature superconductor (HTS) where the vibration was provided by the seismic activities of the earth. The amplitude of the vibration was less than 1 μm as measured by a vibrometer. A disk shaped PM was levitated over a melt-textured HTS YBCO (yttrium barium copper oxide). The experimental setup was adopted to do the fast Fourier transform analysis of the vibration characteristics of the levitated PM. A cross-coupling between the vibration frequency modes of vertical, lateral and angular is observed in all respective directions for any particular vibration frequency measurement. The results indicate that all the vibration modes are actually the combination of the pure vibration frequency modes. The theoretical predictions based on the frozen-image concept show that the ratio of the vertical to lateral stiffness should be higher than 2 in the dynamic case, which is observed experimentally.     

Nonlinear standing waves,resonance phenomena,and frequency characteristics of distributed systems

This review is dedicated to resonator oscillations under conditions of a strongly expressed nonlinearity under which steep shock fronts emerge in the wave profiles. Models and approximated methods for their analysis for quadratic and cubic nonlinear media are examined, as well as for nonlinearity when taking into account the mobility of boundaries. The forms of the profiles are calculated both for a steady-state oscillation regime and during the establishment of the profiles. Dissipative losses and selective losses at specially chosen frequencies are considered. An analysis of nonlinear Q-factor is given. The possibility of increasing the acoustic energy accumulated in the cavity of the resonator is discussed. Special attention is given to various physical phenomena that are exhibited only in nonlinear acoustic fields.     

The frequency stability of single-tuned parametric systems

A general fluctuational analysis of a single-tuned parametric system has been carried out in an operating mode with multiplication and division of the frequency by two. Expressions are derived for the short-term frequency instability which is associated with natural and engineering noises of the system.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 15, No. 4, pp. 562–572, April, 1972.The authors thank A. N. Malakhov for discussing the work and for his valuable comments.     

Approximate closed form solutions for free vibration of polar orthotropic circular plates

For free vibrations of polar orthotropic plate, simple approximate closed form solutions for mode shapes and its natural frequencies were obtained using the Rayleigh-Ritz method. Coordinate function satisfying the natural boundary conditions and the predetermined coefficients was adapted, which results in compact expressions and enables to readily calculate symmetric and nonsymmetric natural frequencies for arbitrary values of the elastic constants. The derived formulation can be used in designing of circular plates such as wood disk, which are naturally endowed with material orthotropy as well as fiber reinforced composite materials. The model can easily be used for the evaluation of parametric studies on dynamic behaviors and nondestructive methods during the initial design process.     

Buckling and free vibration analyses of stiffened plates using the FSDT mesh-free method

This paper presents a mesh-free Galerkin method for the free vibration and stability analyses of stiffened plates via the first-order shear deformable theory (FSDT). The model of a stiffened plate is formed by (1) regarding the plate and the stiffener separately, (2) imposing displacement compatible conditions between the plate and the stiffener so that displacement fields of the stiffener can be expressed in terms of the mid-surface displacement of the plate, and (3) superimposing the strain energy of plate and stiffener. Because there are no meshes used in this method, the stiffeners can be placed anywhere on the plate and need not be placed along the mesh lines. Several numerical examples are computed by this method to show its accuracy and convergence. The present results demonstrate good agreement with the existing solutions given by other researchers and the ANSYS. Influences of support size and order of the complete basis functions on the numerical accuracy are also investigated.     

Vibration and stability of a free circular plate subjected to non-conservative loading

This paper is concerned with the stability and vibration of a completely free circular plate subjected to a non-conservative edge loading. The eigencurves and mode shapes of the plate are obtained for various values of the non-conservativeness parameter. Numerical results are presented for the asymmetrical mode shapes of the plate. Interesting conclusions are drawn from these results some of which are verified analytically.     

Nonlinear vibration of functionally graded circular cylindrical shells based on improved Donnell equations

In the present work, the study of the nonlinear vibration of a functionally graded cylindrical shell subjected to axial and transverse mechanical loads is presented. Material properties are graded in the thickness direction of the shell according to a simple power law distribution in terms of volume fractions of the material constituents. Governing equations are derived using improved Donnell shell theory ignoring the shallowness of cylindrical shells and kinematic nonlinearity is taken into consideration. One-term approximate solution is assumed to satisfy simply supported boundary conditions. The Galerkin method, the Volmir's assumption and fourth-order Runge–Kutta method are used for dynamical analysis of shells to give explicit expressions of natural frequencies, nonlinear frequency–amplitude relation and nonlinear dynamic responses. Numerical results show the effects of characteristics of functionally graded materials, pre-loaded axial compression and dimensional ratios on the dynamical behavior of shells. The proposed results are validated by comparing with those in the literature.     

The decoupling of damped linear systems in free or forced vibration

The purpose of this paper is to extend classical modal analysis to decouple any viscously damped linear system in non-oscillatory free vibration or in forced vibration. Based upon an exposition of how exponential decay in a system can be regarded as imaginary oscillations, the concept of damped modes of imaginary vibration is introduced. By phase synchronization of these real and physically excitable modes, a time-varying transformation is constructed to decouple non-oscillatory free vibration. When time drifts caused by viscous damping and by external excitation are both accounted for, a time-varying decoupling transformation for forced vibration is derived. The decoupling procedure devised herein reduces to classical modal analysis for systems that are undamped or classically damped. This paper constitutes the second and final part of a solution to the “classical decoupling problem.” Together with an earlier paper, a general methodology that requires only the solution of a quadratic eigenvalue problem is developed to decouple any damped linear system.     

Dynamic analysis of circular and sector thick,layered plates

The finite element method is extended to the free vibration analysis of laminated thick plates with curved boundaries. Two elements are developed on the basis of Mindlin's thick plate theory in which the effects of thickness-shear deformation and rotary inertia are included. Both elements are derived in polar co-ordinates and can be joined together to handle annular as well as circular laminated anisotropic plate problems. Since axisymmetry has not been assumed, variations in material properties in the tangential direction can be dealt with. Numerical results are presented to demonstrate the influence of geometrical shape as well as that of thickness-shear deformation on the free vibrations of both homogeneous and layered plates. Comparisons between the numerical results obtained and those presented by other investigators confirm the accuracy of the new elements. The elements also can be used in the analysis of rectangular plates by assuming very large radii and very small subtended angle values.     

Accurate free vibration analysis of thick laminated circular plates with attached rigid core

This paper deals with the free vibration behavior of laminated transversely isotropic circular plates with axisymmetric rigid core attached at the center. The governing equations of motion are obtained based on Mindlin's first-order shear deformation plate theory. Two possible categories of vibration modes related to up-down translation of the core and wobbly rotation of the core about a diameter are studied. Accurate natural frequencies hitherto not reported in the literature are presented for a wide range of thickness-to-radius ratio, inner-to-outer radius ratio, mass and moment of inertia ratios of the core and various boundary conditions at the outer edge of the plate. Numerical results are compared with those of a three-dimensional finite element method (3-D FEM) to demonstrate the high accuracy and reliability of the current analysis.