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Nonlocal and strain gradient effects on nonlinear forced vibration of axially moving nanobeams under internal resonance conditions

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Applied Mathematics and Mechanics(English Edition) 2020年第2期41卷 261-278页

作者： Jing WANG Yilin ZHU Bo ZHANG Huoming SHEN Juan LIU School of Mechanical and Electrical Engineering Southwest Petroleum UniversityChengdu 610500China School of Civil Engineering and Architecture Southwest Petroleum UniversityChengdu 610500China School of Mechanics and Engineering Southwest Jiaotong UniversityChengdu 611756China Applied Mechanics and Structure Safety Key Laboratory of Sichuan Province Chengdu 611756China

Based on the nonlocal strain gradient theory(NSGT), a model is proposed for an axially moving nanobeam with two kinds of scale effects. The internal resonanceaccompanied fundamental harmonic response of the external excitation frequency in the vicinities of the first and second natural frequencies is studied by adopting the multivariate Lindstedt-Poincaré(L-P) method. Based on the root discriminant of the frequencyamplitude equation under internal resonance conditions, theoretical analyses are performed to investigate the scale effects of the resonance region and the critical external excitation amplitude. Numerical results show that the region of internal resonance is related to the amplitude of the external excitation. Particularly, the internal resonance disappears after a certain critical value of the external excitation amplitude is *** is also shown that the scale parameters, i.e., the nonlocal parameters and the material characteristic length parameters, respectively, reduce and increase the critical amplitude,leading to a promotion or suppression of the occurrence of internal resonance. In addition,the scale parameters affect the size of the enclosed loop of the bifurcated solution curves as well by changing their intersection, divergence, or tangency.

关键词： scale effect axially moving nanobeam internal resonance critical amplitude

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Dynamic stiffness method for free vibration of an axially moving beam with generalized boundary conditions

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Applied Mathematics and Mechanics(English Edition) 2019年第7期40卷 911-924页

作者： Hu DING Minhui ZHU Liqun CHEN Shanghai Institute of Applied Mathematics and Mechanics Shanghai University Shanghai 200072 China Shanghai Key Laboratory of Mechanics in Energy Engineering Shanghai University Shanghai 200072 China Department of Mechanics Shanghai University Shanghai 200444 China

axially moving beams are often discussed with several classic boundary conditions, such as simply-supported ends, fixed ends, and free ends. Here, axially moving beams with generalized boundary conditions are discussed for the first time. The beam is supported by torsional springs and vertical springs at both ends. By modifying the stiffness of the springs, generalized boundaries can replace those classical boundaries. Dynamic stiffness matrices are, respectively, established for axially moving Timoshenko beams and Euler-Bernoulli (EB) beams with generalized boundaries. In order to verify the applicability of the EB model, the natural frequencies of the axially moving Timoshenko beam and EB beam are compared. Furthermore, the effects of constrained spring stiffness on the vibration frequencies of the axially moving beam are studied. Interestingly, it can be found that the critical speed of the axially moving beam does not change with the vertical spring stiffness. In addition, both the moving speed and elastic boundaries make the Timoshenko beam theory more needed. The validity of the dynamic stiffness method is demonstrated by using numerical simulation.

关键词： axially moving beam natural frequency Timoshenko beam model dynamic stiffness matrix generalized boundary condition

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Free vibration of non-uniform axially functionally graded beams using the asymptotic development method

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Applied Mathematics and Mechanics(English Edition) 2019年第1期40卷 85-96页

作者： Dongxing CAO Yanhui GAO College of Mechanical Engineering Beijing University of Technology Beijing Key Laboratory of Nonlinear Vibrations and Strength of Mechanical Structures

The asymptotic development method is applied to analyze the free vibration of non-uniform axially functionally graded(AFG) beams, of which the governing equations are differential equations with variable coefficients. By decomposing the variable flexural stiffness and mass per unit length into reference invariant and variant parts, the perturbation theory is introduced to obtain an approximate analytical formula of the natural frequencies of the non-uniform AFG beams with different boundary ***, assuming polynomial distributions of Young's modulus and the mass density, the numerical results of the AFG beams with various taper ratios are obtained and compared with the published literature results. The discussion results illustrate that the proposed method yields an effective estimate of the first three order natural frequencies for the AFG tapered beams. However, the errors increase with the increase in the mode orders especially for the cases with variable heights. In brief, the asymptotic development method is verified to be simple and efficient to analytically study the free vibration of non-uniform AFG beams, and it could be used to analyze any tapered beams with an arbitrary varying cross width.

关键词： axially functionally graded(AFG)beam non-uniform natural frequency asymptotic development method

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Stress distribution and fatigue life of nonlinear vibration of an axially moving beam

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Science China(Technological Sciences) 2019年第7期62卷 1123-1133页

作者： DING Hu HUANG LingLu DOWELL Earl CHEN LiQun Shanghai Institute of Applied Mathematics and Mechanics Shanghai University Shanghai 200072 China Shanghai Key Laboratory of Mechanics in Energy Engineering Shanghai University Shanghai 200072 China Department of Mechanical Engineering and Materials Science Duke University Durham 27708 USA Department of Mechanics Shanghai University Shanghai 200444 China

In this paper, the effects of the nonlinear vibration on stress distribution and fatigue life of the axially moving beam are *** parametric excitation of the flexible material is created by the pulsating moving speed. Three-to-one internal resonance condition is satisfied. The three-parameter model is adopted in the viscoelastic constitutive relation. The nonlinear vibration of the axially moving beam with parametric and internal resonance are studied by using the direct multiple scales method(MSM)with numerical simulation confirmation. Based on the approximate analytical solution, the distribution of tensile stress and bending stress on the axially moving beam is presented by adopting a V-belt as the prototype. Based on the maximum stable cyclic stress, the limit cycle response of the V-belt is utilized to evaluate the effect of the resonance on the fatigue life. Also, the influences of the internal resonance on the steady-state responses and the fatigue life of the V-belt are revealed. Numerical examples illustrate that large unwanted resonances occur and the second-order mode receives vibration energy from to the firstorder mode. The numerical results demonstrate that the nonlinear vibration significantly reduces the fatigue life of the *** fatigue life analysis method in this paper can be applied to the excited vibration of other axially moving systems and even static continuum.

关键词： axially moving beam nonlinear vibration parametric excitation internal resonance stress distribution V-belt

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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 Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072, China 2 Department of Mechanics, Shanghai University, Shanghai 200444, China Shanghai Institute of Applied Mathematics and Mechanics Shanghai University Shanghai China Department of Mechanics Shanghai University Shanghai 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

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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 200444, China 2 Shanghai Institute of Applied Mathematics and Mechanics, Shanghai 200070, China Department of Mechanics Shanghai University Shanghai China Shanghai Institute of Applied Mathematics and Mechanics Shanghai 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

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Large deformation analysis of a cantilever beam made of axially functionally graded material by homotopy analysis method

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Applied Mathematics and Mechanics(English Edition) 2019年第10期40卷 1375-1386页

作者： Xin LIN Yixin HUANG Yang ZHAO Tianshu WANG School of Astronautics Harbin Institute of TechnologyHarbin 150001China School of Aerospace Engineering Tsinghua UniversityBeijing 100084China

Large deformation of a cantilever axially functionally graded (AFG) beam subject to a tip load is analytically studied using the homotopy analysis method (HAM). It is assumed that its Young’s modulus varies along the longitudinal direction according to a power law. Taking the solution of the corresponding homogeneous beam as the initial guess and obtaining a convergence region by adjusting an auxiliary parameter, the analytical expressions for large deformation of the AFG beam are provided. Results obtained by the HAM are compared with those obtained by the finite element method and those in the previous works to verify its validity. Good agreement is observed. A detailed parametric study is carried out. The results show that the axial material variation can greatly change the deformed configuration, which provides an approach to control and manage the deformation of beams. By tailoring the axial material distribution, a desired deformed configuration can be obtained for a specific load. The analytical solution presented herein can be a helpful tool for this procedure.

关键词： large deformation beam axially functionally graded (AFG) material Euler-Bernoulli beam homotopy analysis method (HAM)

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On axially Symmetric Vibrations of Fluid Filled Poroelastic Spherical Shells

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Open Journal of Acoustics 2011年第2期1卷 15-26页

作者： Syed Ahmed Shah Mohammed Tajuddin 不详

Employing Biot’s theory of wave propagation in liquid saturated porous media, waves propagating in a hollow poroelastic closed spherical shell filled with fluid are studied. The frequency equation of axially symmetric vibrations for a pervious and an impervious surface is obtained. Free vibrations of a closed spherical shell are studied as a particular case when the fluid is vanished. Frequency as a function of ratio of thickness to inner radius is computed in absence of dissipation for two types of poroelastic materials each for a pervious and an impervious surface. Results of previous works are obtained as a particular case of the present study.

关键词： Biot’s Theory axially Symmetric Vibrations Radial Vibrations Rotatory Vibrations Spherical Shell Elastic Fluid Pervious Surface Impervious Surface Frequency

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The Tangential Force Distribution on Inner Cylinder of Power Law Fluid Flowing in Eccentric Annuli with the Inner Cylinder Reciprocating axially

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Open Journal of Fluid Dynamics 2015年第2期5卷 183-187页

作者： Junzhe Ma Yu Wei Jianan Zhang Hongjian Liu Xianjun Meng Enhanced Oil and Gas Recovery Key Laboratory of Ministry of Education Northeast Petroleum University Daqing China

Based on the governing equations of the inner cyinder of the unsteady flow of the power law fluid in eccentric annuli with the inner cylinder reciprocating axially in bipolar coordinate system, the calculation formulae of tangential force were established, and the relevant numerical calculation method was given. Taking the aqueous solution of partially hydrolyzed polyacrylamides (HPAM) for examples, the tangential forces were calculated by using the formulae and numerical calculation method mentioned above;the curves of the tangential force on the wall of the inner cylinder of HPAM aqueous solution were plotted;and the effects on the tangential force of the flow behavior index of the power law fluid, the stroke and the stroke frequency of the inner cylinder were analyzed.

关键词： Tangential Force Power Law Fluid Eccentric Annuli Reciprocate axially

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Gyroscopic modes decoupling method in parametric instability analysis of gyroscopic systems

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Acta Mechanica Sinica 2018年第5期34卷 963-969页

作者： Y.J.Qian X.D.Yang H.Wu W.Zhang T.Z.Yang College of Mechanical Engineering Beijing University of TechnologyBeijing 100124China Department of Mechanics Tianjing UniversityTianjing 300072China

Traditional procedures to treat vibrations of gyroscopic continua involve direct application of perturbation methods to a system with both a strong gyroscopic term and other weakly coupled *** this study,a gyroscopic modes decoupling method is used to obtain an equivalent system with decoupled gyroscopic modes having only weak *** the axially moving string as an example,the instability boundaries in the vicinity of parametric resonances are detected using both the traditional coupled gyroscopic system and our system with decoupled gyroscopic modes,and the results are compared to show the advantages and disadvantages of each method.

关键词： axially moving material Decoupling of gyroscopic modes Parametric instability Perturbation method Gyroscopic system

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