The coupling vibration characteristics of a flexible shaft-disk-blades system with mistuned features

The purpose of this paper is to investigate the coupling vibration characteristics of a flexible shaft-disk-blades system with mistuned features. There are some new phenomena due to the coupling effects of shaft-bending, shaft-torsion, disk-transverse and blade-bending. In this investigation, this paper mainly focuses on the influence of mistuned features of the blade's length and the stagger angle. It is found that there are four types of coupling modes: the coupling mode of shaft bending, disk transverse and blade bending (SDB), the coupling mode of shaft torsion disk transverse-blade bending (TDB), the coupling mode of disk transverse and blade bending (DB), the repeated mode of blade bending-blade bending (BB). With the effect of mistuned features, the natural frequencies and the coupling mode type will change correspondingly. With the mistuning value of blade length employed in this study, the TDB mode in the tuned system will disappear and shift into TSDB mode instead, and one of the repeated SDB modes will be replaced by STDB modes. Due to this mistuned features, the blades and disk experience a certain degree of vibration localization phenomenon. Different from the length feature, the influence of mistuning values of blade's stagger angle mainly take effect on the coupling modes. At last, by inspection on the Campbell diagrams, the influence of rotational speed on the transformation of natural frequencies is illustrated on the tuned/mistuned flexible shaft-disk-blades coupling structure.     

FROM ENERGY IMPROVEMENT TO ACCURACYENHANCEMENT： IMPROVEMENT OF PLATE BENDING ELEMENTS BY THE COMBINED HYBRID METHOD

By following the geometric point of view in mechanics, a novel expression of the combined hybrid method for plate bending problems is introduced to clarify its intrinsic mechanism of enhancing coarse-mesh accuracy of conforming or nonconforming plate elements. By adjusting the combination parameter α∈(0, 1) and adopting appropriate bending moments modes, reduction of energy error for the discretized displacement model leads to enhanced numerical accuracy. As an application, improvement of Adini‘s rectangle is discussed. Numerical experiments show that the combined hybrid counterpart of Adini‘s element is capable of attaining high accuracy at coarse meshes.     

Dynamic modeling and analysis of rotating FG beams for capturing steady bending deformation

A dynamic analysis of rotating functionally gradient (FG) beams is presented for capturing the steady bending deformation by using a novel floating frame reference (FFR) formulation. Usually, the cross section of bending beams should rotate round the point at the neutral axis while centrifugal inertial forces are supposed to act on centroid axis. Due to material inhomogeneity of FG beams, centroid and neutral axes may be in different positions, which leads to the eccentricity of centrifugal forces. Thus, centrifugal forces can be divided into three componets: transverse component, axial component and force moment acting on the points of the neutral axis, in which transverse component and force moment can make the beam produce the steady bending deformation. However, this speculation has not been presented and discussed in existing literatures. To this end, a novel FFR formulation of rotating FG beams is especially developed considering centroid and neutral axes. The FFR and its nodal coordinates are used to determine the displacement field, in which kinetic and elastic energies can be accurately formulated according to centroid and neutral axes, respectively. By using the Lagrange's equations of the second kind, the nonlinear dynamic equations are derived for transient dynamics problems of rotating FG beams. Simplifying the nonlinear dynamic equations obtains the equilibrium equations about inertial and elastic forces. The equilibrium equations can be solved to capture the steady bending deformation. Based on the steady bending state, the nonlinear dynamic equations are linearized to obtain eigen-frequency equations. Transient responses obtained from the nonlinear dynamic equations and frequencies obtained from the eigen-frequency equations are compared with available results in existing literatures. Finally, effects of material gradient index and angular speed on the steady bending deformation and vibration characteristics are investigated in detail.     

Simplified equations and solutions for the free vibration of an orthotropic oval cylindrical shell with variable thickness

Based on the framework of the Flügge's shell theory, the transfer matrix approach and the Romberg integration method, this paper presents the vibration behavior of an isotropic and orthotropic oval cylindrical shell with parabolically varying thickness along its circumference. The governing equations of motion of the shell, which have variable coefficients are formulated and solved. The analysis is formulated to overcome the mathematical difficulties related to mode coupling, which comes from variable curvature and thickness of shell. The vibration equations of the shell are reduced to eight first‐order differential equations in the circumferential coordinate and by using the transfer matrix of the shell, these equations can be written in a matrix differential equation. The proposed model is adopted to get the vibration frequencies and the corresponding mode shapes for the symmetrical and antisymmetrical modes of vibration. The sensitivity of the frequency parameters and the bending deformations to the shell geometry, ovality parameter, thickness ratio, and orthotropic parameters corresponding to different type modes of vibration is investigated. Copyright © 2011 John Wiley & Sons, Ltd.     

The Generalized Patch Test for Zienkiewicz's Triangles

It is proved that Zienkiewicz's triangles for plate bending problems pass Stummel's generalized patch test — a necessary and sufficient condition for convergence of nonconforming finite elements — for mesh (a) generated by three sets of parallel lines, but do not pass it when "union jack" mesh (b) or when another mesh (c) is used. In the latter two cases the approximations are divergent.     

Investigation,Modelling and Analysis of stiffened GFRP-Samples

Glasfibre structures feature high potentials for optimization and substitution of conventional materials like steel and aluminum and their alloys. The paper deals with the insertion of glasfibre trusses into thin glasfibre structures to reinforce them. The effective material properties of the glasfibre structures were estimated by experiments and simulations. Furthermore the Young's modulus of the trusses was obtained by bending tests and tension tests. A comparison between bending experiments and bending simulations is given. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Natural damped vibrations of anisotropic box beams of polymer composite materials. 2. Numerical experiments

The influence of the orientation of reinforcing fibers on the natural frequencies and mechanical loss coefficient of coupled vibrations of unsupported symmetric and asymmetric box beams, as evaluated in numerical experiments, is discussed. The calculations were performed under the assumption that the real parts of the complex moduli and mechanical loss coefficient are frequency-independent. Vibration modes were identified by their surface shapes. The boundaries of the regions of mutual transformation of interacting vibration modes were determined by the joint analysis of the dependences of the coupled and partial eigenfrequencies and the mechanical loss coefficients on the orientation angle of reinforcing fibers. It is established that vibrations of a symmetric box beam give rise to two primary interactions: bending–torsional and longitudinal–shear ones, which are united into a unique longitudinal–bending–torsional–shear interaction by the secondary interaction caused by transverse shear strains. Vibrations of an asymmetric box beam give rise to longitudinal–torsional and bending–bending (in two mutually orthogonal planes) interactions. It is shown that in a number of cases variation in the orientation angle of reinforcing fibers is accompanied with a mutual transformation of coupled vibration modes. If the differential equations for natural vibrations involve odd-order derivatives with respect to the spatial variable (a symmetric beam and the bending–bending interaction of an asymmetric beam), then, with variation in the orientation angle of reinforcing fibers, the mutual transformation of coupled vibration modes proceeds. If the differential equations for natural vibrations involve only even-order derivatives (the longitudinal–torsional interaction of an asymmetric beam), no mutual transformation of coupled vibration modes occurs.     

Green's functions for forced vibration analysis of bending-torsion coupled Timoshenko beam

A method based on Green's functions is proposed for the analysis of the steady-state dynamic response of bending-torsion coupled Timoshenko beam subjected to distributed and/or concentrated loadings. Damping effects on the bending and torsional directions are taken into account in the vibration equations. The elastic boundary conditions with bending-torsion coupling and damping effects are derived and the classical boundary conditions can be obtained by setting the values of specific stiffness parameters of the artificial springs. The Laplace transform technology is employed to work out the Green's functions for the beam with arbitrary boundary conditions. The Green's functions are obtained for the beam subject to external lateral force and external torque, respectively. Coupling effects between bending and torsional vibrations of the beam can be studied conveniently through these analytical Green's functions. The direct expressions of the steady-state responses with various loadings are obtained by using the superposition principle. The present Green's functions for the Timoshenko beam can be reduced to those for Euler–Bernoulli beam by setting the values of shear rigidity and rotational inertia. In order to demonstrate the validity of the Green's functions proposed, results obtained for special cases are given for a comparison with those given in the literature and they agree with each other exactly. The influences of external loading frequency and eccentricity on Green's functions of bending-torsion coupled Timoshenko beam are investigated in terms of the numerical results for both simply supported and cantilever beams. Moreover, the symmetric property of the Green's functions and the damping effects on the amplitude of Green's functions of the beam are discussed particularly.     

The exact and approximate method in mechanical system's analysis

The main aim of this work is to present results of the mechanical system's analysis based on the exact and approximate Galerkin's methods. The considered system is the flexural vibrating one-dimension bending beam. The exact and approximate method were used to assign the dynamic flexibility of the considered system and results of this work were juxtaposed to verify the approximate method's accuracy. The correction coefficients were introduced into the approximate method to unify results of both methods. The aim of this work was to check accuracy of the approximate method and to verify if this method may be used to mechatronic system's analysis, where it is impossible to use the exact method. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An Experimental Study of Inertial Waves in a Closed Cone

An experimental study of inertial waves in a closed cone is presented in which the inertial waves are excited by a slight periodic oscillation superimposed on the cone's basic rotation rate. The dynamic pressure field is measured along the cone's rotation axis; no standing modal structure is observed, confirming Greenspan's (1969) argument that the closed cone appears open to inertial oscillations and the inertial wave spectrum is continuous. Similar pressure measurements made in the frustum of a right circular cone show that removal of the singular apex of the cone leads to standing wave modes.     

Complex behavior in biosystems: an information-theoretic approach

The particular question of transport of vibrational energy in biosystems is considered within the scope of Fröhlich–Davydov's model. It is shown that Davydov's solitary waves, strongly damped in near equilibrium conditions, can display long-range propagation when travelling in Fröhlich's condensate. The latter consists in the emergence of a self-organized dissipative structure (in Prigogine's sense), resembling a nonequilibrium Bose–Einstein-like condensation in the low-lying in frequency modes of vibration, once a critical level of pumping of metabolic energy is achieved.     

The Mode Tree: A Tool for Visualization of Nonparametric Density Features

Abstract

Recognition and extraction of features in a nonparametric density estimate are highly dependent on correct calibration. The data-driven choice of bandwidth h in kernel density estimation is a difficult one that is compounded by the fact that the globally optimal h is not generally optimal for all values of x. In recognition of this fact a new type of graphical tool, the mode tree, is proposed. The basic mode tree plot relates the locations of modes in density estimates with the bandwidths of those estimates. Additional information can be included on the plot indicating factors such as the size of modes, how modes split, and the locations of antimodes and bumps. The use of a mode tree in adaptive multimodality investigations is proposed, and an example is given to show the value in using a normal kernel, as opposed to the biweight or other kernels, in such investigations. Examples of such investigations are provided for Ahrens's chondrite data and van Winkle's Hidalgo stamp data. Finally, the bivariate mode tree is introduced, together with an example using Scott's lipid data.     

Modified Shape Functions for Specht's Plate Bending Element

1.IntroductionThesolutionoftheC'-continuityrequlrementofKirchhoffbendingwithfiniteele-mentmodelsresult8incomplicatedhigher.l...nt.I2J'[4'I7].Besidesthelargenumberofunknowns,difficultiesmayalsoarisefrommiredsecondderiVativesattheverticestakenasnodalvari.bl.I8l.Toovercomesuchdifficulties,asplittingsplineelemelltmethodisintrod.cedl5j'l9],butthisalwayscausescomplicatedcomputation-nomthepracticalpoilitofviewlower-degreepolynomialfiniteelemelitsaremoredesirable.Unfortunately,thesimpleelementsbase…     

Shaking table tests of a model-scale building with 2DOF pendulum mass damper

In this paper, the numerical and experimental studies of a tall building's model with 2DOF pendulum mass damper (PMD) are considered. It is assumed that the model excitation is in the form of horizontal and/or torsional motion of the ground caused by earthquake. The construction consists of the main system (tall building's model) and a double pendulum mass damper, which is attuned to the first (bending) and the second (torsional) eigenfrequencies of the main structure. The analysis focuses on reduction of structure vibration caused by horizontal or torsional component of ground motions. Therefore, results presented in this work show efficiency of 2DOF PMD for vibration reduction. The numerical analysis of the problem is performed with using COSMOS/M system (a FEM numerical model is defined), while experimental analysis is carried out on a physical model-scale building with 2DOF PMD. Model consists of twenty five recurrent storeys (height 2.5m) with a PMD located on the highest one. Shaking table device is used to simulate an earthquake excitation in horizontal and torsional component, independently. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Eigenmode Asymptotics in Thin Elastic Plates

In this paper, we investigate the behavior of the vibration modes (eigenvalues) of an isotropic homogeneous plate as its thickness tends to zero. As lateral boundary conditions, we consider clamped or free edge. We prove distinct asymptotics for bending and membrane modes: the smallest bending eigenvalues behave as the square of the thickness whereas the membrane eigenvalues tend to non-zero limits. Moreover, we prove that all these eigenvalues have an expansion in power series with respect to the thickness regardless of their multiplicities or of the multiplicities of the limit in-plane problems.     

The nonlocal frequency behavior of nanomechanical mass sensors based on the multi-directional vibrations of a buckled nanoribbon

Considering the potential applications of the buckled structures as nanomechanical mass sensors, this paper presents an analytical method to solve the frequency shifts of the first-order transverse and longitudinal vibration modes when a mass attaches on the surface of a buckled nanoribbon based on the nonlocal elastic theory and the Lagrange's motion equation. The first-order transverse and longitudinal vibration modes of the buckled nanoribbon are introduced. A comparison between the analytic solution and the finite element analysis (FEA) is presented. Then, the effects of the compressive strain, the magnitude and location of attached mass, the nonlocal parameter, and attached mass numbers on the frequency shifts are presented. From example calculations, it is seen that the magnitude of attached mass increases the frequency shifts of the first-order modes, except the first-order transverse vibration mode at the location Z1=0. The frequency shifts for the first-order transverse and longitudinal vibration modes are different, and could be used as important principles in mass sensing. What's more, the compressive strain and the nonlocal parameter play significant roles on the sensing process of the buckled nanoribbon. The results could serve as useful references for the fabrications and applications of buckled structures based nanomechanical mass sensors.     

Optimal parameter choice for the pole condition

The pole condition is a framework for the derivation of transparent boundary conditions that identifies non-physical modes by the location of the corresponding singularities in the complex plane of the solution's spatial Laplace transform. A complex half-plane is then defined that contains all poles corresponding to non-physical modes. A key parameter in the pole condition arises in the Möbius transformation that maps this half-plane onto the complex unit circle. The effect of variations in this parameter on the quality of the approximate TBC realized by the pole condition is explored here for the two-dimensional drift-diffusion equation with inhomogeneous coefficients. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Stability of Navier-Stokes Equations and the Estimation of Its Attractor Dimension

In this paper, for a class of exterior force term 2s²W^'_{s,s} we analyse the existence of unstable modes of linearized Navier-Stokes Equations (NSE), and associate them with integer points in plane. Furthermore we give the lower boundary dimension estimation of the attractor of NSE. Liu discussed the condition where the exterior force term is W^'_{0,s} in (1, 2), but his method can't be extended to the condition where the exterior force term is W^'_{s_1,s_2} (s_1 ≠ 0, s_2 ≠ 0). So this paper may look as the extention of [1, 2]. The method which we give in this paper has direct application for further study of other properties of NSE (such as Hopf bifurcation). See [3].