Aeroelastic flutter of a disk rotating in an unbounded acoustic medium

The linear aeroelastic stability of an unbaffled flexible disk rotating in an unbounded fluid is investigated by modeling the disk-fluid system as a rotating Kirchhoff plate coupled to the irrotational motions of a compressible inviscid fluid. A perturbed eigenvalue formulation is used to compute systematically the coupled system eigenvalues. Both a semi-analytical and a numerical method are employed to solve the fluid boundary value problem. The semi-analytical approach involves a perturbation series solution of the dual integral equations arising from the fluid boundary value problem. The numerical approach is a boundary element method based on the Hadamard finite part. Unlike previous works, it is found that a disk with zero material damping destabilizes immediately beyond its lowest critical speed. Upon the inclusion of small disk material damping, the flutter speeds become supercritical and increase with decreasing fluid density. The competing effects of radiation damping into the surrounding fluid and disk material damping control the onset of flutter at supercritical speed. The results are expected to be relevant for the design of rotating disk systems in data storage, turbomachinery and manufacturing applications.     

Effects of internal viscous damping on the stability of a rotating shaft driven through a universal joint

A rotating flexible shaft, with both external and internal viscous damping, driven through a universal joint is considered. The mathematical model consists of a set of coupled, linear partial differential equations with time-dependent coefficients. Use of Galerkin's technique leads to a set of coupled linear differential equations with time-dependent coefficients. Using these differential equations some effects of internal viscous damping on parametric and flutter instability zones are investigated by the monodromy matrix technique. The flutter zones are also obtained on discarding the time-dependent coefficients in the differential equations which leads to an eigenvalue analysis. A one-term Galerkin approximation aided this analysis. Two different shafts (“automotive” and “lab”) were considered. Increasing internal damping is always stabilizing as regards to parametric instabilities. For flutter type instabilities it was found that increasing internal damping is always stabilizing for rotational speeds v below the first critical speed, v₁. For v>v₁, there is a value of the internal viscous damping coefficient, C_iv, which depends on the rotational speed and torque, above which destabilization occurs.The value of C_iv (“critical value”) at which the unstable zone first enters the practical range of operation was determined. The dependence of C_iv critical on the external damping was investigated. It was found for the automotive case that a four-fold increase in external damping led to an increase of about 20% of the critical value. For the lab model an increase of two orders of magnitude of the external damping led to an increase of critical value of only 10%.For the automotive shaft it was found that this critical value also removed the parametric instabilities out of the practical range. For the lab model it is not always possible to completely stabilize the system by increasing the internal damping. For this model using C_iv critical, parametric instabilities are still found in the practical range of operation.     

RELIABILITY ANALYSIS OF SUSPENSION BRIDGES AGAINST FLUTTER

A reliability analysis of suspension bridges against flutter failure is presented using the basic theory of reliability. For the purpose of analysis, uncertainties considered are those arising from the variations in geometric and mechanical properties of bridge, modelling, damping and experimentally obtained flutter derivatives. These uncertainties are incorporated by multiplying the computed flutter wind speed with a number of independent factors, which are considered as random variables. Each factor is assumed to follow log-normal distribution. The wind environment at the site, which may cause flutter failure, is considered as the other uncertainty necessary for computing the reliability against flutter failure. The flutter wind speed for the bridge is determined using a finite element approach and a multimode analysis. The effect of some important parameters such as the mean wind speed at the site, coefficients of variation of the multiplying factors associated with damping, modelling and flutter derivatives on the reliability estimates is investigated. The results of the study show that the reliability against flutter failure is sensitive to the variation of the above parameters.     

Shell-structure inertia for slow collective motion

The modified shell correction method is suggested for the calculation of transport coefficients in a slow nuclear collective dynamics. For the multipole low-lying vibrations near the spherical shape of a nucleus, the smooth transport coefficients corresponding to the extended Thomas-Fermi approach are used as a macroscopic background. The time-dependent mean field is approximated through the infinitely deep square-well potential for the calculation of the shell corrections. Significant shell effects in stiffness and inertia are found at small temperatures. These effects disappear approximately at the same large enough temperature as in the free energy. It is shown that the collective inertia is substantially larger than that of irrotational flow owing to the consistency condition of particle density and potential variations. The collective vibration energies and reduced friction and effective damping coefficients with accounting for the shell effects are in better agreement with allowance data than that found from the hydrodynamic model. The text was submitted by the authors in English.     

VIBRATION ATTENUATION VIA CONSTRAINED LAYER DAMPING AND DASHPOT ON A SPRING-MASS-PLATE SYSTEM

The vibrations and damping characteristics of an annular plate with constrained layer damping (CLD) treatment subject to a traveling spring-mass-damper (SMD) are investigated. The equations of the CLD-treated plate are first derived from the energy principle. These equations are simplified via the Donnell-Mushtari-Vlasov assumptions. The response equations are eventually uncoupled for each mode and are in terms of a single-degree-of-freedom (s.d.o.f.) linear oscillator with hysteretic damping. The receptance method follows to joint the plate and the SMD, and the resulting change of natural frequencies and damping ratios are investigated. Individual effects due to the inertia and the stiffness are illustrated as well. The results shows that the damping ratios resulted from the viscoelastic core are more significant than that from the viscous damper. In addition, there exists a best design on the thickness of the viscoelastic material core to have the maximum damping ratios. The results also show that the attachment of SMD bifurcated the plate's natural frequencies for every mode but n = 0. The bifurcation becomes more obvious with the rotational speed. These results provide useful information for vibration suppression in engineering design.     

Microscopic theory of He II-He3 mixtures including dissipative processes

The conclusions following from the solutions of the linearized hydrodynamic equations are discussed. By using some of symmetry properties of the Green functions the kinetic coefficients coupled to the external fields are determined and some interesting informations concerning the thermodynamic derivatives of the condensate density arise. A number of the Green functions is calculated in the hydrodynamic approximation. The examination of their denominators leads to the expressions for damping of the acoustic modes in He II-He³. A set of Kubo-type formulae for the kinetic coefficients and certain sum rules are obtained.     

Estimating the nonlinear resonant frequency of a single pile in nonlinear soil

Analytical expressions are determined for the nonlinear resonant frequency (or natural frequency) of the fundamental lateral mode of a pile. A pile with a floating toe, with and without pile cap is considered in this paper. The influence of a nonlinear soil spring model that varies with depth and a nonlinear damping model that is strain amplitude dependent is considered. A non-dimensional equation of motion for the system dynamics is derived from an energy based formulation. This equation is a Duffing's type nonlinear differential system that has nonlinear damping. Harmonic balance with numerical continuation is employed to determine the nonlinear resonance curves of the system. Comparison with some experimental results is made.     

Amplitude equation and pattern selection in Faraday waves

A nonlinear theory of pattern selection in parametric surface waves (Faraday waves) is presented that is not restricted to small viscous dissipation. By using a multiple scale asymptotic expansion near threshold, a standing wave amplitude equation is derived from the governing equations. The amplitude equation is of gradient form, and the coefficients of the associated Lyapunov function are computed for regular patterns of various symmetries as a function of a viscous damping parameter gamma. For gamma approximately 1, the selected wave pattern comprises a single standing wave (stripe pattern). For gamma<1, patterns of square symmetry are obtained in the capillary regime (large frequencies). At lower frequencies (the mixed gravity-capillary regime), a sequence of sixfold (hexagonal), eightfold, ...patterns are predicted. For even lower frequencies (gravity waves) a stripe pattern is again selected. Our predictions of the stability regions of the various patterns are in quantitative agreement with recent experiments conducted in large aspect ratio systems.     

Extended symmetry transformation of (3+1)-dimensional generalized nonlinear Schrdinger equation with variable coefficients

In this paper, the extended symmetry transformation of (3+1)-dimensional (3D) generalized nonlinear Schrdinger (NLS) equations with variable coefficients is investigated by using the extended symmetry approach and symbolic computation. Then based on the extended symmetry, some 3D variable coefficient NLS equations are reduced to other variable coefficient NLS equations or the constant coefficient 3D NLS equation. By using these symmetry transformations, abundant exact solutions of some 3D NLS equations with distributed dispersion, nonlinearity, and gain or loss are obtained from the constant coefficient 3D NLS equation.     

A GROUP THEORETIC APPROACH TO THE LINEAR FREE VIBRATION ANALYSIS OF SHELLS WITH DIHEDRAL SYMMETRY

This paper deals with a group theoretic approach to the finite element analysis of linear free vibrations of shells with dihedral symmetry. Examples of such shell structures are cylindrical shells, conical shells, shells with circumferential stiffeners, corrugated shells, spherical shells, etc. The group theoretic approach is used to exploit the inherent symmetry in the problem. For vibration analysis, the group theoretic results give the correct symmetry-adapted basis for the displacement field. The stiffness matrix K and the mass matrix M are identically block diagonalized in this basis. The generalized linear eigenvalue problem of free vibration gets split into independent subproblems due to this block diagonalization. The Simo element is used in the finite element formulation of the shell equilibrium equations. Numerical results for natural frequencies and natural modes of vibration of several dihedral shell structures are presented. The results are shown to be in very good agreement with those reported in the literature. The computational advantages and physical insights due to the group theoretic approach are also discussed.     

High-frequency vibration analysis of thin elastic plates under heavy fluid loading by an energy finite element formulation

An energy finite element analysis (EFEA) formulation for computing the high frequency behavior of plate structures in contact with a dense fluid is presented. The heavy fluid loading effect is incorporated in the derivation of the EFEA governing differential equations and in the computation of the power transfer coefficients between plate members. The new formulation is validated through comparison of EFEA results to classical techniques such as statistical energy analysis (SEA) method and the modal decomposition method for bodies of revolution. Good correlations are observed and the advantages of the EFEA formulation are identified.     

Universe as a representation of affine and conformal symmetries

The evolution of the Universe is considered by means of a nonlinear realization of affine and conformal symmetries via Maurer-Cartan forms. Conformal symmetry is realized by the geometry of similarity with the Dirac scalar dilaton. We provide preliminary quantitative evidence that the zeroth harmonic of the Dirac scalar dilaton may lead to observationally viable cosmology, where the type la supernova luminosity distances-redshift relation can be explained by vacuum dilaton dark energy. The diffeo-invariance of spin connection coefficients of the affine formulation leaves only one degree of freedom of strong gravitation waves. We discuss that the dark matter effect in spiral galaxies can be described by the gravitation waves expressed through the spin connection coefficients of the affine formulation. We show that, the evolution equations of the affine gravitons with respect to the dilaton zeroth mode coincide with the equations of “squeezed oscillator”. The list of theoretical and observational arguments is given in favor of that the origin of the Universe can be described by quantum vacuum creation of these squeezed oscillators.     

Effects of the symmetry energy slope on the axial oscillations of neutron stars

The impact of symmetry energy slope L on the axial w-mode oscillations is explored, where the range of the con- strained slope L of symmetry energy at saturation density is adopted from 25 MeV to 115 MeV while keeping the equation of state （EOS） of symmetric nuclear matter fixed. Based on the range of the symmetry energy slope, a constraint on the frequency and damping time of the wi-mode of the neutron star is given. It is found that there is a perfect linear relation between the frequency and the stellar mass for a fixed slope L, and the softer symmetry energy corresponds to a higher frequency. Moreover, it is confirmed that both the frequencies and damping times have a perfect universal scaling behavior for the EOSs with different symmetry energy slopes at saturation density.     

Laser short-pulse heating of metallic surface: Consideration of Seebeck effect

In the present study, laser short-pulse heating is formulated using an electron kinetic theory approach. Temperature predictions are compared with that obtained from two-equation model. Seebeck effect is considered during the heating process. The predicted Seebeck coefficients are compared with the results based on the early formulation. Electron excess energy loss due to Seebeck effect is compared with electron mean energy. It is found that Seebeck coefficient decays sharply in the surface region due to sharp decay of electron temperature in this region. Seebeck coefficient obtained from the present study is in agreement with the predictions based on the early formulation, provided electron temperature is used in the previous formulation. However, Seebeck coefficient differs significantly once the lattice site temperature is used in the previous formulation. Electron excess energy loss due to Seebeck effect is considerably less than electron mean energy, i.e. the ratio is in the order of 10⁻⁵.     

Sparse identification method of extracting hybrid energy harvesting system from observed data

Hybrid energy harvesters under external excitation have complex dynamical behavior and the superiority of promoting energy harvesting efficiency. Sometimes, it is difficult to model the governing equations of the hybrid energy harvesting system precisely, especially under external excitation. Accompanied with machine learning, data-driven methods play an important role in discovering the governing equations from massive datasets. Recently, there are many studies of data-driven models done in aspect of ordinary differential equations and stochastic differential equations (SDEs). However, few studies discover the governing equations for the hybrid energy harvesting system under harmonic excitation and Gaussian white noise (GWN). Thus, in this paper, a data-driven approach, with least square and sparse constraint, is devised to discover the governing equations of the systems from observed data. Firstly, the algorithm processing and pseudo code are given. Then, the effectiveness and accuracy of the method are verified by taking two examples with harmonic excitation and GWN, respectively. For harmonic excitation, all coefficients of the system can be simultaneously learned. For GWN, we approximate the drift term and diffusion term by using the Kramers-Moyal formulas, and separately learn the coefficients of the drift term and diffusion term. Cross-validation (CV) and mean-square error (MSE) are utilized to obtain the optimal number of iterations. Finally, the comparisons between true values and learned values are depicted to demonstrate that the approach is well utilized to obtain the governing equations for the hybrid energy harvester under harmonic excitation and GWN.     

On geometric approach to Lie symmetries of differential-difference equations

Based upon Cartan's geometric formulation of differential equations, Harrison and Estabrook proposed a geometric approach for the symmetries of differential equations. In this Letter, we extend Harrison and Estabrook's approach to analyze the symmetries of differential-difference equations. The discrete exterior differential technique is applied in our approach. The Lie symmetry of (2+1)-dimensional Toda equation is investigated by means of our approach.     

Brownian motion in periodic potentials; nonlinear response to an external force

The Brownian motion of particles in a periodic potential in response to a constant external force is investigated. By expanding the distribution function into Hermite-functions and into a Fourier-series, the Fokker-Planck-equation is transformed into a set of coupled equations for the expansion coefficients. These equations are solved by a continued fraction method for matrices. This continued fraction for the matrices converges for large, intermediate and even for very small damping constants. The mobility, the kinetic and potential energy for various damping constants and external forces are given for a cos-potential. The current-voltagecharacteristic of the Josephson tunneling junction is also shown.     

Steady-state vectorial self-diffraction on a non-local photorefractive grating in a crystal of symmetry \overline{4}3m at symmetrical transmitting geometry

The steady-state two-wave interaction in a cubic crystal of the symmetry group 3m with the non-local photorefractive response in the absence of an external electric field is considered for the case of arbitrary interaction orientation with respect to the crystallographic coordinate system and for arbitrary intensities and polarization states of incident light waves. The self-diffraction problem is described on the basis of four coupled-wave equations in terms of the complex scalar amplitudes of components of the light waves with orthogonal linear polarization. The derived conservation laws are valid for the non-linear dependency of the photorefractive-grating amplitude on the modulation coefficient of the interference light pattern. It follows from these laws that the two non-unidirectional energy fluxes can form the total energy exchange between the two interacting light waves. A set of independent conservation laws allows us to decouple the coupled-wave equations and to obtain their analytical solution, at least, in the form of quadrature formulae. For example, such a solution is derived for the case of linearly polarized incident light waves and for the linearized dependency of the photorefractive-grating amplitude on the modulation coefficient. The explicit analytical expressions for the scalar amplitudes are obtained for the transversal electro-optic configuration of interaction. The possibility of polarization-state transformation of light waves without energy exchange between them is shown. Received: 30 July 2002 / Published online: 11 December 2002 RID="*" ID="*"Corresponding author. Fax: +7-3822/414321, E-mail: litvinov@ed.rk.tusur.ru     

A finite element method for transmission in non-uniform ducts without flow: Comparison with the method of weighted residuals

A finite element method is developed for the study of transmission of sound in non-uniform ducts without flow. The formulation is based on a weighted residual approach and eight noded isoparametric elements are used. Two computational schemes are described, one based on the Helmholtz equation obtained by combining the basic conservation equations and one based on the conservation equations themselves. The latter case is considered because in future extensions to problems involving mean flow a single governing equation is not readily obtainable except for irrotational flows. Both two-dimensional and circular duct geometries are considered. Comparisons are made with a Method of Weighted Residuals in the form of a Modified Galerkin Method in the two-dimensional case to assess both accuracy and computational efficiency. It is found that the finite element method produces results for transmission and reflection coefficients nearly identical to those from the Galerkin approach. Used to its best advantage the finite element method is of comparable efficiency.