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Steady-state responses of axially accelerating viscoelastic beams: Approximate analysis and numerical confirmation

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Science China(Physics,mechanics & Astronomy) 2008年第11期51卷 1707-1721页

作者： CHEN LiQun1,2.& DING Hu2.1 Department of mechanics, shanghai University, shanghai 2.0444, China 2.shanghai institute of applied mathematics and mechanics, shanghai 2.0070, China Department of mechanics Shanghai University Shanghai 200444 China shanghai institute of {2. mathematics and mechanics Shanghai 200070 China

Nonlinear parametric vibration of axially accelerating viscoelastic beams is inves-tigated via an approximate analytical method with numerical confirmations. Based on nonlinear models of a finite-small-stretching slender beam moving at a speed with a periodic fluctuation, a solvability condition is established via the method of multiple scales for subharmonic resonance. Therefore, the amplitudes of steady-state periodic responses and their existence conditions are derived. The amplitudes of stable steady-state responses increase with the amplitude of the axial speed fluctuation, and decrease with the viscosity coefficient and the nonlinear coefficient. The minimum of the detuning parameter which causes the existence of a stable steady-state periodic response decreases with the amplitude of the axial speed fluctuation, and increases with the viscosity coefficient. Nu-merical solutions are sought via the finite difference scheme for a nonlinear par-tial-differential equation and a nonlinear integro-partial-differential equation. The calculation results qualitatively confirm the effects of the related parameters pre-dicted by the approximate analysis on the amplitude and the existence condition of the stable steady-state periodic responses. Quantitative comparisons demonstrate that the approximate analysis results have rather high precision.

关键词： nonlinearity parametric vibration axially accelerating beam method of multiple scales numerical confir-mation

Variational Iteration Method for Delay Differential Equations

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Communications in Nonlinear Science and Numerical Simulation 1997年第4期2卷 235-236页

The variational iteration method is shown to be applicable to delay differential equations for analytical approximate solutions

关键词： Variational Iteration Method for Delay Differential Equations

Boundary element-free method for elastodynamics

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Science China(Physics,mechanics & Astronomy) 2005年第6期48卷 641-657页

作者： CHENG Yumin1 & PENG Miaojuan2.1. shanghai institute of {2. mathematics and mechanics, shanghai University, shanghai 2.0072. China 2. Department of Civil Engineering, shanghai University, shanghai 2.0072. China 1. shanghai institute of {2. mathematics and mechanics Shanghai University 200072 Shanghai China 2. Department of Civil Engineering Shanghai University 200072 Shanghai China

The moving least-square approximation is discussed first. Sometimes the method can form an ill-conditioned equation system, and thus the solution cannot be obtained correctly. A Hilbert space is presented on which an orthogonal function system mixed a weight function is defined. Next the improved moving least-square approximation is discussed in detail. The improved method has higher computational efficiency and precision than the old method, and cannot form an ill-conditioned equation system. A boundary element-free method (BEFM) for elastodynamics problems is presented by combining the boundary integral equation method for elastodynamics and the improved moving least-square approximation. The boundary element-free method is a meshless method of boundary integral equation and is a direct numerical method compared with others, in which the basic unknowns are the real solutions of the nodal variables and the boundary conditions can be applied easily. The boundary element-free method has a higher computational efficiency and precision. In addition, the numerical procedure of the boundary element-free method for elastodynamics problems is presented in this paper. Finally, some numerical examples are given.

关键词： moving least-square approximation, improved moving least-square approximation, elastodynamics, boundary integral equation, meshless method, boundary element-free method, Fourier eigen transform.

On two transverse nonlinear models of axially moving beams

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Science China(Technological Sciences) 2009年第3期52卷 743-751页

作者： DING Hu1 & CHEN LiQun1,2.1 shanghai institute of {2. mathematics and mechanics, shanghai University, shanghai 2.0072. China 2.Department of mechanics, shanghai University, shanghai 2.0444, China 1. shanghai institute of {2. mathematics and mechanics Shanghai University Shanghai 200072 China 2. Department of mechanics Shanghai University Shanghai 200444 China

Nonlinear models of transverse vibration of axially moving beams are computationally investigated. A partial-differential equation is derived from the governing equation of coupled planar motion by omit- ting its longitudinal terms. The model can be reduced to an integro-partial-differential equation by av- eraging the beam disturbed tension. Numerical schemes are respectively presented for the governing equations of coupled planar and the two governing equations of transverse motion via the finite dif- ference method and differential quadrature method under the fixed boundary and the simple support boundary. A steel beam and a copper beam are treated as examples to demonstrate the deviations of the solutions to the two transverse equations from the solution to the coupled equation. The numerical results indicate that the differences increase with the amplitude of vibration and the axial speed. Both models yield almost the same precision results for small amplitude vibration and the inte- gro-partial-differential equation gives better results for large amplitude vibration.

关键词： axially moving beam nonlinearity transverse vibration finite difference method differential quadrature method

A complex variable meshless method for fracture problems

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Science China(Physics,mechanics & Astronomy) 2006年第1期49卷 46-59页

作者： CHENG Yumin1 & LI Jiuhong2.1. shanghai institute of {2. mathematics and mechanics, shanghai University, shanghai 2.0072. China 2. Department of Building Engineering, Xi’an University of Technology, Xi’an 710048, China shanghai institute of {2. mathematics mechanics Shanghai University Shanghai 200072 China Department of Building Engineering Xi’an University of Technology Xi’an 710048 China

Based on the moving least-square (MLS) approximation, the complex variable moving least-square approximation (CVMLS) is discussed in this paper. The complex variable moving least-square approximation cannot form ill-conditioned equations, and has greater precision and computational efficiency. Using the analytical solution near the tip of a crack, the trial functions in the complex variable moving least-square approxi- mation are extended, and the corresponding approximation function is obtained. And from the minimum potential energy principle, a complex variable meshless method for fracture problems is presented, and the formulae of the complex variable meshless method are obtained. The complex variable meshless method in this paper has greater precision and computational efficiency than the conventional meshless method. Some examples are given.

关键词： moving least-square approximation, complex variable moving least-square approximation, meshless method, complex variable meshless method, fracture.

An Approximate Solution Technique Depending on an Artificial Parameter: A Special Example

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Communications in Nonlinear Science and Numerical Simulation 1998年第2期3卷 40-45页

作者： Jihuan HE (shanghai University, shanghai {2. of applied mathematics and mechanics, shanghai 2.0072. China) shanghai University Shanghai Institute of Applied Mathematics and Mechanics Shanghai 200072 China

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关键词： perturbation methods homotopy Duffing equation nonlinearity

The complex variable reproducing kernel particle method for elasto-plasticity problems

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Science China(Physics,mechanics & Astronomy) 2010年第5期53卷 954-965页

作者： CHEN Li1 & CHENG YuMin2.1 Department of Engineering mechanics,Chang’an University,Xi’an 710064,China 2.shanghai institute of applied mathematics and mechanics,{2. University,{2. 2.0072.China 1. Department of Engineering mechanics Chang’an University Xi’an 710064 China2. Shanghai Institute of Applied Mathematics and Mechanics Shanghai University Shanghai 200072 China

On the basis of reproducing kernel particle method(RKPM),using complex variable theory,the complex variable reproducing kernel particle method(CVRKPM) is discussed in this *** advantage of the CVRKPM is that the correction function of a two-dimensional problem is formed with one-dimensional basis function when the shape function is *** the CVRKPM is applied to solve two-dimensional elasto-plasticity *** Galerkin weak form is employed to obtain the discretized system equation,the penalty method is used to apply the essential boundary *** then,the CVRKPM for two-dimensional elasto-plasticity problems is formed,the corresponding formulae are obtained,and the Newton-Raphson method is used in the numerical *** numerical examples are given to show that this method in this paper is effective for elasto-plasticity analysis.

关键词： meshless method reproducing kernel particle method(RKPM) complex variable reproducing kernel particle method(CVRKPM) correction function elasto-plasticity

An interpolating boundary element-free method (IBEFM) for elasticity problems

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Science China(Physics,mechanics & Astronomy) 2010年第4期53卷 758-766页

作者： REN HongPing 1 , CHENG YuMin 2.& ZHANG Wu 1 1 School of Computer Engineering and Science, shanghai University, shanghai 2.0072. China {2., shanghai University, shanghai 2.0072. China 1. School of Computer Engineering and Science Shanghai University Shanghai 200072 China2. Shanghai Institute of Applied Mathematics and Mechanics Shanghai University Shanghai 200072 China

The paper begins by discussing the interpolating moving least-squares (IMLS) method. Then the formulae of the IMLS method obtained by Lancaster are revised. On the basis of the boundary element-free method (BEFM), combining the boundary integral equation method with the IMLS method improved in this paper, the interpolating boundary element-free method (IBEFM) for two-dimensional elasticity problems is presented, and the corresponding formulae of the IBEFM for two-dimensional elasticity problems are obtained. In the IMLS method in this paper, the shape function satisfies the property of Kronecker δ function, and then in the IBEFM the boundary conditions can be applied directly and easily. The IBEFM is a direct meshless boundary integral equation method in which the basic unknown quantity is the real solution to the nodal variables. Thus it gives a greater computational precision. Numerical examples are presented to demonstrate the method.

关键词： moving least-squares (MLS) approximation interpolating moving least-squares (IMLS) method boundary integral equation meshless method boundary element-free method (BEFM) interpolating boundary element-free method (IBEFM) elasticity problem

Weakly nonlinear stability of ultra-thin slipping films

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Progress in Natural Science:Materials International 2005年第9期15卷 827-831页

作者： HU Guohui(shanghai institute of {2. mathematics and mechanics, shanghai University, shanghai 2.0072. China) shanghai institute of {2. mathematics and mechanics Shanghai University Shanghai 200072 China

A weakly nonlinear theory is presented to study the effects of slippage on the stability of the ultra-thin polymer films. The nonlinear mathematical model is constructed for perturbations of small finite amplitude based on hydrodynamic equations with the long wave approximation. Results reveal that the nonlinearity always accelerates the rupture of the films. The influences of the slip length, film thickness, and initial amplitude of perturbations on the rupture of the films are investigated.

关键词： ultra-thin film weakly nonlinear theory hydrodynamic stability rupture of thin film.