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, t...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.展开更多
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 s...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.展开更多
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 equation...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.展开更多
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 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.展开更多
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 establishe...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.展开更多
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 ...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.展开更多
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...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.展开更多
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 mo...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.展开更多
Transmission error(TE)in geared rotors is a predominant source of inherent excitation at the pitch point of the gear meshing.In this paper,a transverse vibration analysis is presented to study the effect of TE on gear...Transmission error(TE)in geared rotors is a predominant source of inherent excitation at the pitch point of the gear meshing.In this paper,a transverse vibration analysis is presented to study the effect of TE on geared rotors.Due to asymmetry in the TE,it is expected to have both forward and backward whirls excited during rotor whirling,which could be used for its detection.This aspect has been envisioned first time in the present work.To capture this,an approach of orienting the line of action of a gear-pair along oblique plane is considered and the mathematical modeling has been performed of a simple spur gear-pair connecting two parallel shafts at its mid-span with an asymmetric TE.To capture the forward and backward whirls,equations of motion are converted into a complex form that facilitates obtaining response in full spectrum.The response of system model with assumed transmission error and gear-pair parameters has been obtained through a numerical simulation,which shows distinctly the forward and backward whirls due to the TE.Through a simple test rig experimentation,a similar behaviour was observed in transverse vibrations of geared rotors in the full spectrum,which validate the proposed model.展开更多
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 an...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.展开更多
In this paper,a fractional differential equation is introduced to describe the transverse vibrations of an axially moving viscoelastic string.An iterative algorithm is constructed to analyze the dynamical behavior.By ...In this paper,a fractional differential equation is introduced to describe the transverse vibrations of an axially moving viscoelastic string.An iterative algorithm is constructed to analyze the dynamical behavior.By conveying the memory effect of the fractional differential terms step by step,the computation cost can be greatly reduced.As a numerical example,the effects of the viscoelastic parameters on a moving string are investigated.展开更多
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 crac...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.展开更多
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 ...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.展开更多
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...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.展开更多
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 ...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.展开更多
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 ...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.展开更多
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 co...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.展开更多
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 discre...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.展开更多
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 con...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.展开更多
The initial boundary value problem of the transverse vibration of a taut string is a classic that can be found in many vibration and acoustics textbooks. It is often used as the basis for derivations of elementary num...The initial boundary value problem of the transverse vibration of a taut string is a classic that can be found in many vibration and acoustics textbooks. It is often used as the basis for derivations of elementary numerical models, for instance finite element or finite difference schemes. The model of axial vibration of a prismatic elastic bar also serves in this capacity, often times side-by-side with the first model. The stored (potential) energy for these two models is derived in the literature in two distinct ways. We find the potential energy in the taut string model to be derived from a second-order expression of the change of the length of the string. This is very different in nature from the corresponding expression for the elastic bar, which is predictably based on the work of the internal forces. The two models are mathematically equivalent in that the equations of one can be obtained from the equations of the other by substitution of symbols such as the primary variable, the resisting force and the coefficient of the stiffness. The solutions also have equivalent meanings, such as propagation of waves and standing waves of free vibration. Consequently, the analogy between the two models can and should be exploited, which the present paper successfully undertakes. The potential energy of deformation of the string was attributed to the seminal work of Morse and Feshbach of 1953. This book was also the source of a misunderstanding as to the correct expression for the density of the energy of deformation. The present paper strives to settle this question.展开更多
基金Project supported by the National Natural Science Foundation of China(No.10472060)the Natural Science Foundation of Shanghai Municipality(No.04ZR14058)the Shanghai Leading Academic Discipline Project(No.Y0103)
文摘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.
文摘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.
基金Project supported by the National Natural Science Foundation of China(No.50575180)
文摘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.
基金supported by the National Natural Science Foundation of China(11472211)the Natural Science Foundation of Education Department of Shaanxi Province of China(2013JK1042).
文摘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.
基金supported by National Natural Science Foundation of China(No.10872163).
文摘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.
基金Project supported by the National Natural Science Foundation of China(Nos.11072157,11272219,11227201,and 10932006)the National Basic Research Program of China(No.2012CB723301)the Training Program for Leading Talent in University Innovative Research Team in Hebei Province(No.LJRC006)
文摘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.
文摘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.
基金Project supported by the National Natural Science Foundation of China (No. 10472060)Shanghai Leading Academic Discipline Project (No.Y0103)the Natural Science Foundation of Shanghai (No.04ZR14058)the Outstanding Youth Program of Shanghai Municipal Commission of Educatio(No.04YQHB088)
文摘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.
文摘Transmission error(TE)in geared rotors is a predominant source of inherent excitation at the pitch point of the gear meshing.In this paper,a transverse vibration analysis is presented to study the effect of TE on geared rotors.Due to asymmetry in the TE,it is expected to have both forward and backward whirls excited during rotor whirling,which could be used for its detection.This aspect has been envisioned first time in the present work.To capture this,an approach of orienting the line of action of a gear-pair along oblique plane is considered and the mathematical modeling has been performed of a simple spur gear-pair connecting two parallel shafts at its mid-span with an asymmetric TE.To capture the forward and backward whirls,equations of motion are converted into a complex form that facilitates obtaining response in full spectrum.The response of system model with assumed transmission error and gear-pair parameters has been obtained through a numerical simulation,which shows distinctly the forward and backward whirls due to the TE.Through a simple test rig experimentation,a similar behaviour was observed in transverse vibrations of geared rotors in the full spectrum,which validate the proposed model.
文摘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.
基金supported by the National Natural Science Foundation of China(Nos.11072120 and 11002075)supported partially by Brazilian National Council for Scientific and Technological Development(CNPq).
文摘In this paper,a fractional differential equation is introduced to describe the transverse vibrations of an axially moving viscoelastic string.An iterative algorithm is constructed to analyze the dynamical behavior.By conveying the memory effect of the fractional differential terms step by step,the computation cost can be greatly reduced.As a numerical example,the effects of the viscoelastic parameters on a moving string are investigated.
基金Supported by National Natural Science Foundation of China(Grant Nos.51035008,51304019)National Science Foundation of USA(Grant Nos.CMMI-1000830,CMMI-1229532)+1 种基金the University of Maryland Baltimore County Directed Research Initiative Fund ProgramFundamental Research Funds for the Central Universities,China(Grant No.FRF-TP-14-123A2)
文摘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.
基金supported by the National Natural Science Foundation of China(Nos.12072181 and 12121002)the State Key Laboratory of Mechanical System and Vibration of China(No.MSV202105)。
文摘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.
文摘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.
基金Natural Science Foundation of Inner Mongolia,China(No.2012MS0811)
文摘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.
基金supported by Hebei Provincial Natural Science Foundation of China (Grant No. E2012203177)National Science and Technology Support Plan of China (Grant No. 2011BAF15B01)+1 种基金Hebei Provincial Funds for Distinguished Young Scientists of China (Grant No.E2006001038)Open Project Program of National Engineering Research Center for Equipment and Technology of Cold Strip Rolling(Grant No. NECSR-201202)
文摘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.
基金Project supported by the Science Foundation of China University of Petroleum in Beijing(No.2462013YJRC003)
文摘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.
基金supported by the National Outstanding Young Scientists Fund of China(No.10725209)the National Natural Science Foundation of China(No.10672092)+3 种基金Shanghai Subject Chief Scientist Project(No.09XD1401700)Shanghai Municipal Education Commission Scientific Research Project(No.07ZZ07)Shanghai Leading Academic Discipline Project (No.S30106)Changjiang Scholars and Innovative Research Team in University Program(No.IRT0844).
文摘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.
基金supported by the Portuguese Foundation for Science and Tech-nology(FCT),under the project POCI 2010 and the PhD grant SFRH/BD/44696/2008
文摘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.
文摘The initial boundary value problem of the transverse vibration of a taut string is a classic that can be found in many vibration and acoustics textbooks. It is often used as the basis for derivations of elementary numerical models, for instance finite element or finite difference schemes. The model of axial vibration of a prismatic elastic bar also serves in this capacity, often times side-by-side with the first model. The stored (potential) energy for these two models is derived in the literature in two distinct ways. We find the potential energy in the taut string model to be derived from a second-order expression of the change of the length of the string. This is very different in nature from the corresponding expression for the elastic bar, which is predictably based on the work of the internal forces. The two models are mathematically equivalent in that the equations of one can be obtained from the equations of the other by substitution of symbols such as the primary variable, the resisting force and the coefficient of the stiffness. The solutions also have equivalent meanings, such as propagation of waves and standing waves of free vibration. Consequently, the analogy between the two models can and should be exploited, which the present paper successfully undertakes. The potential energy of deformation of the string was attributed to the seminal work of Morse and Feshbach of 1953. This book was also the source of a misunderstanding as to the correct expression for the density of the energy of deformation. The present paper strives to settle this question.