Nonlinear Transversal Vibration Analysis of an Axially Moving Viscoelastic Yarn in Tufting Process

The nonlinear transversal vibration of axially moving yarn with time-dependent tension is investigated. Yarn material is modeled as Kelvin model. A partial differential equation governing the transversal vibration is derived from the Newton's second law.Galerkin method is used to truncate the governing nonlinear differential equation,and thus the first-order ordinary differential equation is obtained. The periodic vibration equation and the natural frequency of moving yarn are received by applying homotopy perturbation method. As a result,the condition which should be avoided during the tufting process for resonance is obtained.

Four-Beam Model for Vibration Analysis of a Cantilever Beam with an Embedded Horizontal Crack

As one of the main failure modes, embedded cracks occur in beam structures due to periodic loads. Hence it is useful to investigate the dynamic characteristics of a beam structure with an embedded crack for early crack detection and diagnosis. A new four-beam model with local flexibilities at crack tips is developed to investigate the transverse vibration of a cantilever beam with an embedded horizontal crack; two separate beam segments are used to model the crack region to allow opening of crack surfaces. Each beam segment is considered as an Euler-Bernoulli beam. The governing equations and the matching and boundary conditions of the four-beam model are derived using Hamilton＇s principle. The natural frequencies and mode shapes of the four-beam model are calculated using the transfer matrix method. The effects of the crack length, depth, and location on the first three natural frequencies and mode shapes of the cracked cantilever beam are investigated. A continuous wavelet transform method is used to analyze the mode shapes of the cracked cantilever beam. It is shown that sudden changes in spatial variations of the wavelet coefficients of the mode shapes can be used to identify the length and location of an embedded horizontal crack. The first three natural frequencies and mode shapes of a cantilever beam with an embedded crack from the finite element method and an experimental investigation are used to validate the proposed model. Local deformations in the vicinity of the crack tips can be described by the proposed four-beam model, which cannot be captured by previous methods.

Vibration of fluid-conveying pipe with nonlinear supports at both ends

The axial fluid-induced vibration of pipes is very widespread in engineering applications.The nonlinear forced vibration of a viscoelastic fluid-conveying pipe with nonlinear supports at both ends is investigated.The multi-scale method combined with the modal revision method is formulated for the fluid-conveying pipe system with nonlinear boundary conditions.The governing equations and the nonlinear boundary conditions are rescaled simultaneously as linear inhomogeneous equations and linear inhomogeneous boundary conditions on different time-scales.The modal revision method is used to transform the linear inhomogeneous boundary problem into a linear homogeneous boundary problem.The differential quadrature element method(DQEM)is used to verify the approximate analytical results.The results show good agreement between these two methods.A detailed analysis of the boundary nonlinearity is also presented.The obtained results demonstrate that the boundary nonlinearities have a significant effect on the dynamic characteristics of the fluid-conveying pipe,and can lead to significant differences in the dynamic responses of the pipe system.

Atomistic simulation of free transverse vibration of graphene,hexagonal SiC, and BN nanosheets

Free transverse vibration of monolayer graphene, boron nitride (BN), and silicon carbide (SiC) sheets is investigated by using molecular dynamics finite element method. Eigenfrequencies and eigenmodes of these three sheets in rectangular shape are studied with different aspect ratios with respect to various boundary conditions. It is found that aspect ratios and boundary conditions affect in a similar way on natural frequencies of graphene, BN, and SiC sheets. Natural frequencies in all modes decrease with an increase of the sheet’s size. Graphene exhibits the highest natural frequencies, and SiC sheet possesses the lowest ones. Missing atoms have minor effects on natural frequencies in this study.

STEADY-STATE RESPONSES AND THEIR STABILITY OF NONLINEAR VIBRATION OF AN AXIALLY ACCELERATING STRING

The steady-state transverse vibration of an axially moving string with geometric nonlinearity was investigated. The transport speed was assumed to be a constant mean speed with small harmonic variations. The nonlinear partial-differential equation that governs the transverse vibration of the string was derived by use of the Hamilton principle. The method of multiple scales was applied directly to the equation. The solvability condition of eliminating the secular terms was established. Closed form solutions for the amplitude and the existence conditions of nontrivial steady-state response of the two-to-one parametric resonance were obtained. Some numerical examples showing effects of the mean transport speed, the amplitude and the frequency of speed variation were presented. The Liapunov linearized stability theory was employed to derive the instability conditions of the trivial solution and the nontrivial solutions for the two-to-one parametric resonance. Some numerical examples highlighting influences of the related parameters on the instability conditions were presented.

A FINITE DIFFERENCE METHOD FOR SIMULATING TRANSVERSE VIBRATIONS OF AN AXIALLY MOVING VISCOELASTIC STRING

A finite difference method is presented to simulate transverse vibrations of an axially moving string. By discretizing the governing equation and the equation of stress strain relation at different frictional knots, two linear sparse finite difference equation systems are obtained. The two explicit difference schemes can be calculated alternatively, which make the computation much more efficient. The numerical method makes the nonlinear model easier to deal with and of truncation errors, O（△t^2 ＋ △x^2）. It also shows quite good stability for small initial values. Numerical examples are presented to demonstrate the efficiency and the stability of the algorithm, and dynamic analysis of a viscoelastic string is given by using the numerical results.

Transverse vibration characteristics of axially moving viscoelastic plate

The dynamic characteristics and stability of axially moving viscoelastic rectangular thin plate are investigated. Based on the two dimensional viscoelastic differential constitutive relation, the differential equations of motion of the axially moving viscoelastic plate are established. Dimensionless complex frequencies of an axially moving viscoelastic plate with four edges simply supported, two opposite edges simply supported and other two edges clamped are calculated by the differential quadrature method. The effects of the aspect ratio, moving speed and dimensionless delay time of the material on the trans- verse vibration and stability of the axially moving viscoelastic plate are analyzed.

Complex-mode Galerkin approach in transverse vibration of an axially accelerating viscoelastic string

Under the consideration of harmonic fluctuations of initial tension and axially velocity, a nonlinear governing equation for transverse vibration of an axially accelerating string is set up by using the equation of motion for a 3-dimensional deformable body with initial stresses. The Kelvin model is used to describe viscoelastic behaviors of the material. The basis function of the complex-mode Galerkin method for axially accelerating nonlinear strings is constructed by using the modal function of linear moving strings with constant axially transport velocity. By the constructed basis functions, the application of the complex-mode Galerkin method in nonlinear vibration analysis of an axially accelerating viscoelastic string is investigated. Numerical results show that the convergence velocity of the complex-mode Galerkin method is higher than that of the real-mode Galerkin method for a variable coefficient gyroscopic system.

Transverse Vibration and Stability Analysis of Circular Plate Subjected to Follower Force and Thermal Load

Transverse vibration and stability analysis of circular plate subjected to follower force and thermal load are analyzed.Based on the thin plate theory in involving the variable temperature,the differential equation of transverse vibration for the axisymmetric circular plate subjected to follower force and thermal load is established.Then,the differential equation of vibration and corresponding boundary conditions are discretized by the differential quadrature method.Meanwhile,the generalized eigenvalue under three different boundary conditions are calculated.In this case,the change curve of the first order dimensionless complex frequency of the circular plate subjected to the follower force in the different conditions with the variable temperature coefficient and temperature load is analyzed.The stability and corresponding critical loads of the circular plate subjected to follower force and thermal load with simply supported edge,clamped edge and free edge are discussed.The results provide theoretical basis for improving the dynamic stability of the circular plate.

TRANSVERSE VIBRATION CHARACTERISTICS OF VISCOELASTIC RECTANGULAR PLATE WITH CRACK

Based on the two-dimensional viscoelastic differential constitutive relation and the thin plate theory, the differential equations of motion of the viscoelastic plate with an all-over part-through crack are established and the expression of additional rotation induced by the crack is derived. The complex eigenvalue equations of the viscoelastic plate with crack are derived by the differential quadrature method, and the 8method is used at the crack continuity conditions. Dimensionless complex frequencies of a crack viscoelastic plate with four edges simply supported, two opposite edges simply supported and other two edges clamped are calculated. The effects of the crack parameter, the aspect ratio and dimensionless delay time of the material on the transverse vibration of the viscoelastic plate are analyzed.

Transverse vibrations of arbitrary non-uniform beams

Free and steady state forced transverse vibrations of non-uniform beams are investigated with a proposed method, leading to a series solution. The obtained series is verified to be convergent and linearly independent in a convergence test and by the non-zero value of the corresponding Wronski determinant, respectively. The obtained solution is rigorous, which can be reduced to a classical solution for uniform beams. The proposed method can deal with arbitrary non-uniform Euler-Bernoulli beams in principle, but the methods in terms of special functions or elementary functions can only work in some special cases.

TRANSVERSE VIBRATION OF RECTANGULAR PLATES ELASTICALLY SUPPORTED AT POINTS ON EDGES

This paper studies transverse vibration of rectangular plates with two opposite edges simply supperted other two edges arbitrarily supported and free edges elaslically supported at points,A highly accurate solution is presented for calculating inherent frequencies and mode shape of rectangular platen elaslically supported at points. The number and location of these points on free edges may be completely arbitrary. This paper uses impulse function to represent reaction and moment at points. Fourter series is used to expand the impulse function along the edges. Characteristic equations satisfying all boundary conditions are given.Inherent frequencies and mode shape with any accutacy can be gained.

AN APPROXIMATE SOLUTION OF EIGEN－FREQUENCIES OF TRANSVERSE VIBRATION OF RECTANGULAR PLATES WITH ELASTICAL RESTRAINTS

This poper presents an approximate solution for calculating eigen-frequencies of transverse vibration of rectangular plates elastically restrained against rotation along edges. The formulae are not only very simple and easily programmed but also have high accuracy. Finally, some numerical results are given and compared with other results obtained.

Energy transfer in double plate system dynamics

The study of energy transfer between coupled subsystems in a hybrid system is very important for applications. This paper presents an analytical analysis of energy transfer between plates of a visco-elastically connected double-plate system in free transversal vibrations. The analytical analysis shows that the visco-elastic connection between plates is responsible for the appearance of two-frequency regime in the time function, which corresponds to one eigen amplitude function of one mode, and also that time functions of different vibration modes are uncoupled, but energy transfer between plates in one eigen mode appears. It was shown for each shape of vibrations. Series of the two Lyapunov exponents corresponding to the one eigen amplitude mode are expressed by using the energy of the corresponding eigen amplitude time component.

Double plate system with a discontinuity in the elastic bonding layer

The double plate system with a discontinuity in the elastic bonding layer of Winker type is studied in this paper. When the discontinuity is small, it can be taken as an interface crack between the bi-materials or two bodies （plates or beams）. By comparison between the number of multifrequencies of analytical solutions of the double plate system free transversal vibrations for the case when the system is with and without disconti- nuity in elastic layer we obtain a theory for experimental vibration method for identification of the presence of an interface crack in the double plate system. The analytical analysis of free transversal vibrations of an elastically connected double plate systems with discontinuity in the elastic layer of Winkler type is presented. The analytical solutions of the coupled partial differential equations for dynamical free and forced vibration processes are obtained by using method of Bernoulli＇s particular integral and Lagrange＇s method of variation constants. It is shown that one mode vibration corresponds an infinite or finite multi-frequency regime for free and forced vibrations induced by initial conditions and one- frequency or corresponding number of multi-frequency regime depending on external excitations. It is shown for every shape of vibrations. The analytical solutions show that the discontinuity affects the appearance of multi-frequency regime of time function corresponding to one eigen amplitude function of one mode, and also that time functions of different vibration basic modes are coupled. From final expression we can separate the newgeneralized eigen amplitude functions with correspon- ding time eigen functions of one frequency and multifrequency regime of vibrations.

Coupled Dynamic Modeling of Rolls Model and Metal Model for Four High Mill Based on Strip Crown Control

The crown is a key quality index of strip and plate, the rolling mill system is a complex nonlinear system, the strip qualities are directly affected by the dynamic characteristics of the rolling mil. At present, the studies about the dynamic modeling of the rolling mill system mainly focus on the dynamic simulation for the strip thickness control system, the dynamic characteristics of the strip along the width direction and that of the rolls along axial direction are not considered. In order to study the dynamic changes of strip crown in the roiling process, the dynamic simulation model based on strip crown control is established. The work roll and backup roll are considered as elastic continuous bodies and the work roll and backup roll are joined by a Winkler elastic layer. The rolls are considered as double freely supported beams. The change rate of roll gap is taken into consideration in the metal deformation, based on the principle of dynamic conservation of material flow, the two dimensional dynamic model of metal is established. The model of metal deformation provides exciting force for the rolls dynamic model, and the roils dynamic model and metal deformation model couple together. Then, based on the two models, the dynamic model of rolling mill system based on strip crown control is established. The Newmark-13 method is used to solve the problem, and the dynamic changes of these parameters are obtained as follows： （1） The bending of work roll and backup roll changes with time; （2） The strip crown changes with time; （3） The distribution of rolling force changes with time. Take some cold tandem rolling mill as subject investigated, simulation results and the comparisons with experimental results show that the dynamic model built is rational and correct. The proposed research provides effective theory for optimization of device and technological parameters and development of new technology, plays an important role to improve the strip control precision and strip shape quality.

Dynamic response of axially moving Timoshenko beams: integral transform solution

The generalized integral transform technique （GITT） is used to find a semianalytical numerical solution for dynamic response of an axially moving Timoshenko beam with clamped-clamped and simply-supported boundary conditions, respectively. The implementation of GITT approach for analyzing the forced vibration equation eliminates the space variable and leads to systems of second-order ordinary differential equations （ODEs） in time. The MATHEMATICA built-in function, NDSolve, is used to numerically solve the resulting transformed ODE system. The good convergence behavior of the suggested eigenfunction expansions is demonstrated for calculating the transverse deflection and the angle of rotation of the beam cross-section. Moreover, parametric studies are performed to analyze the effects of the axially moving speed, the axial tension, and the amplitude of external distributed force on the vibration amplitude of axially moving Timoshenko beams.

ON GALERKIN DISCRETIZATION OF AXIALLY MOVING NONLINEAR STRINGS

A computational technique is proposed for the Galerkin discretization of axially moving strings with geometric nonlinearity. The Galerkin discretization is based on the eigenfunctions of stationary strings. The discretized equations are simplified by regrouping nonlinear terms to reduce the computation work. The scheme can be easily implemented in the practical programming. Numerical results show the effectiveness of the technique. The results also highlight the feature of Galerkin＇s discretization of gyroscopic continua that the term number in Galerkin＇s discretization should be even. The technique is generalized from elastic strings to viscoelastic strings.

Elastically restrained Bernoulli-Euler beams applied to rotary machinery modelling

Facing the lateral vibration problem of a machine rotor as a beam on elastic supports in bending,the authors deal with the free vibration of elastically restrained Bernoulli-Euler beams carrying a finite number of concentrated elements along their length.Based on Rayleigh’s quotient,an iterative strategy is developed to find the approximated torsional stiffness coefficients,which allows the reconciliation between the theoretical model results and the experimental ones,obtained through impact tests.The mentioned algorithm treats the vibration of continuous beams under a determined set of boundary and continuity conditions, including different torsional stiffness coefficients and the effect of attached concentrated masses and rotational inertias, not only in the energetic terms of the Rayleigh’s quotient but also on the mode shapes,considering the shape functions defined in branches.Several loading cases are examined and examples are given to illustrate the validity of the model and accuracy of the obtained natural frequencies.

Zero thermal expansion in metal-organic framework with imidazole dicarboxylate ligands

Exploring new abnormal thermal expansion materials is important to understand the nature of thermal expansion.Metal-organic framework(MOF)with unique structure flexibility is an ideal material to study the thermal expansion.This work adopts the high-resolution variable-temperature powder x-ray diffraction to investigate the structure and intrinsic thermal expansion in Sr-MOF([Sr(DMPhH_(2)IDC)_(2)]_n).It has the unique honeycomb structure with one-dimensional(1 D)channels along the c-axis direction,the a-b plane displays layer structure.The thermal expansion behavior has strong relationship with the structure,ZTE appears in the a-b plane and large PTE along the c-axis direction.The possible mechanism is that the a/b layers have enough space for the transverse thermal vibration of polydentate ligands,while along the c-axis direction is not.This work not only reports one interesting zero thermal expansion material,but also provides new understanding for thermal expansion mechanism from the perspective of the structural model.