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1.
Nonlinear parametric vibration of axially accelerating viscoelastic strings is investigated via an approximate analytical
approach. The standard linear solid model using the material time derivative is employed to describe the string viscoelastic
behaviors. A coordinate transformation is introduced to derive Mote’s model of transverse motion from the governing equation
of the stationary string. Mote’s model leads to Kirchhoff’s model by replacing the tension with the averaged tension over
the string. An asymptotic perturbation approach is proposed to study principal parametric resonance based on the two models.
The amplitude and the existence conditions of the steady-state responses are determined by locating the nonzero fixed points
in the modulation equations resulting from the solvability condition. Numerical results are presented to highlight the effects
of the material parameters, the axial-speed fluctuation amplitude, and the initial stress on steady-state responses. 相似文献
2.
This paper analytically and numerically investigates the nonlinear vibration in parametric and internal resonances of in-plane
accelerating viscoelastic plates subjected to plane stresses. An approximate nonlinear plate theory was developed under the
Kirchoff assumptions. The in-plane translating speed is characterized as a simple harmonic variation about the constant mean
axial speed. The governing equation with the associated boundary conditions is derived from the generalized Hamilton principle
and the Kelvin constitutive relation. The method of multiple scales is applied to establish the solvability conditions in
principal parametric and internal resonances. The steady-state responses are predicted in three possible patterns: trivial,
single-mode, and two-mode solutions. The stabilities of the steady-state responses are determined based on the Routh-Hurwitz
criterion. The effects of the mean in-plane translating speed, the in-plane translating speed fluctuation amplitude, the viscosity
coefficient, and the nonlinear coefficient on the steady-state responses are examined. The differential quadrature schemes
are developed for the two-dimensional full plate model and the one-dimensional reduced plate model to solve the nonlinear
governing equations numerically. The numerical calculations confirm the approximate analytical results regarding the trivial
and single-mode solutions of the steady-state responses. 相似文献
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A classical engineering approach to thin film flow problems with localised geometric step features is to use the Reynolds equation. For applications to new generation hydrodynamic bearings with very small gap clearances and lift-generating features, any abrupt changes in the thickness of a film will break the validity of the Reynolds equation, which is based on lubrication theory. In this work, formal asymptotic expansions are used to match a numerical solution of a local formulation of the full Navier–Stokes equations near a step feature to a Reynolds equation model which is valid sufficiently far from the step, i.e. with smooth film thickness variation. The approach is used to model a pressurised bearing with an axisymmetric Rayleigh step feature. An efficient and accurate mathematical model is presented using matched asymptotic expansions for both incompressible and compressible fluid flows. This work quantifies the effect of inertia at the step and considers the validation of the classical approach of patching lubrication solutions across the step with specified compatibility conditions. A parametric study is undertaken to evaluate cases where the classical engineering approach is justified. 相似文献
5.
Two-to-one parametric resonance in transverse vibration of an axially accelerating viscoelastic string with geometric nonlinearity is investigated. The transport speed is assumed to be a constant mean speed with small harmonic variations. The nonlinear partial differential equation that governs transverse vibration of the string is derived from Newton's second law. The method of multiple scales is applied directly to the equation, and the solvability condition of eliminating secular terms is established. Closed-form solutions for the amplitude of the vibration and the existence conditions of nontrivial steady-state response in two-to-one parametric resonance are obtained. Some numerical examples showing effects of the mean transport speed, the amplitude and the frequency of speed variation are presented. Lyapunov's linearized stability theory is employed to analyze the stability of the trivial and nontrivial solutions for two-to-one parametric resonance. Some numerical examples highlighting the effects of the related parameters on the stability conditions are presented. 相似文献
6.
研究非齐次边界条件和1∶3内共振下面内平动黏弹性板的横向非线性1∶2主参数振动的稳态响应。考虑黏弹性对边界条件的影响,建立了面内平动板的偏微分运动方程和相应的非齐次边界条件。采用直接多尺度法建立了次谐波参数共振时的可解性条件,并根据Routh-Hurvitz判据判别了系统幅频响应的稳定性。讨论了速度扰动幅值和黏弹性系数对幅频响应的影响,对比了齐次和非齐次边界条件下稳态响应的差异。最后,引入微分求积法验证直接多尺度法的近似解析结果。 相似文献
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强迫Duffing振动系统的主共振鞍结分岔控制 总被引:4,自引:0,他引:4
设计了非线性参数控制器来改变非线性系统的稳态响应,减小了系统的响应幅值并消除了共振时的鞍结分岔。首先由多尺度法得到系统的近似频响方程,再由奇异性理论来分析分岔特性,从而实现非线性控制的目标。最后对强迫Duffing系统的主共振形式进行了分析,由数值模拟来确定分岔控制是可行的和有效的。 相似文献
9.
四种群食物链方程的整体渐近稳定性 总被引:3,自引:1,他引:2
本文考虑具有时滞的四种群食物链反应扩散方程。给出了在一定条件下解的存在性和平衡态方程正解的整体渐近稳定性。由此可以得到解的持久性,平凡解和所有半平凡解的不稳定性。 相似文献
10.
In the present work, the nonlinear response of a vertically moving viscoelastic beam subjected to a periodically varying contact load is investigated. The generalized Galerkin??s method is used to discretize the nonlinear partial differential equation of motion into the temporal equation of motion. The temporal equation of motion contains many nonlinear terms such as cubic geometric and inertial nonlinear terms, nonlinear damping term, and nonlinear parametric excitation terms in addition to forced excitation and parametric excitation terms. The first-order approximate solutions are obtained by using the method of multiple scales, and the stability and bifurcations of the obtained steady-state responses are studied. Extensive numerical simulations are presented to illustrate the influences of various types of system parameters for different resonance conditions. A significant amount of vibration reduction is obtained with the increase in the material loss factor. The results obtained by numerically solving the temporal equation of motion are found to be in good agreement with the results determined by the method of multiple scales. The obtained results are useful for reduction in the vibration of the viscoelastic flexible beam with prismatic joint or single-link viscoelastic Cartesian manipulator with payload subjected to a sinusoidally varying contact load. 相似文献
11.
面内弹性绳系卫星系统的内共振 总被引:1,自引:0,他引:1
研究了绳系卫星系统因系绳弹性和俯仰运动相耦合而引起的非线性内共振问题。首先应用Kane方程建立了面内椭圆轨道两体弹性绳系卫星系统在状态保持阶段的非线性动力学模型,分析了可能发生的内共振条件。然后,基于多尺度方法获得内共振下的调制方程及其Jacobi椭圆函数表示的解析解。结果表明,轨道摄动因素不会影响内共振,系统参数引起的内共振耦合振动会在两个模态之间相互传递,其调制周期取决于初始模态幅值。在一定的调谐参数上,模态振幅出现饱和现象。 相似文献
12.
We demonstrate frequency modulation Kelvin probe force microscopy operated in lift-mode under ambient conditions. Frequency modulation detection is sensitive to force gradients rather than forces as in the commonly used amplitude modulation technique. As a result there is less influence from electric fields originating from the tip's cone and cantilever, and the recorded surface potential does not suffer from the large lateral averaging observed in amplitude modulated Kelvin probe force microscopy. The frequency modulation technique further shows a reduced dependence on the lift-height and the frequency shift can be used to map the second order derivative of the tip-sample capacitance which gives high resolution material contrast of dielectric sample properties. The sequential nature of the lift-mode technique overcomes various problems of single-scan techniques, where crosstalk between the Kelvin probe and topography feedbacks often impair the correct interpretation of the recorded data in terms of quantitative electric surface potentials. 相似文献
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The general class of models proposed by Peña and Hollander for recurrent event data is considered under a fully parametric specification of the baseline hazard rate function and under the two cases where the model does and does not incorporate frailty components. Estimators of model parameters are presented, and their finite and asymptotic properties are ascertained. For the asymptotic properties, the results of Borgan concerning maximum likelihood estimators in counting process models are used to obtain weak convergence to Gaussian distributions of estimators. However, the required regularity conditions are reformulated into conditions involving gap times, which make it more feasible to obtain explicit theoretical expressions of asymptotic covariances. The procedures are applied to fit the general class of models with a parametric baseline hazard rate function to a dataset on hydraulic subsystems of “load-haul-dump” machines in mining. 相似文献
15.
研究轴向运动导电梁在平行导线产生的磁场环境中的主-内联合共振问题。基于电磁场基本理论和哈密顿原理,导出轴向运动梁在外激励和磁场共同作用下的非线性振动方程。针对一端夹支一端铰支的导电梁,采用多尺度法求解方程,得到非线性方程的近似解析解和幅频响应方程,并对稳态解的稳定性进行了分析。通过算例,得到系统前两阶幅值随频率调谐参数、外激励力、轴向速度、电流强度等参数的变化规律。结果表明:系统发生主-内联合共振时一阶和二阶响应都被激发,且存在不同的多解区域;一阶和二阶幅值的稳态解个数在几个多解区域同步变化,其个数取决于外激励力、运动速度和电流强度值。 相似文献
16.
Abstract The photorefractive holographic recording is a dynamic grating formulation process, where the grating is induced by the interference modulation of two beams, and the energies of the beams are exchanged with each other owing to their interaction in the induced grating. The grating modulation envelope is a comprehensive analysis for holographic dynamic recording and non-destructive readout, but analytic solutions can only be obtained by simultaneously solving the materials equations and the coupled wave equations. In this paper, based on Kukhtarev's model and Moharam's field function exact steady-state analytic expressions for dynamic grating envelopes are deduced for any constant values of light modulation depth and light excitation efficiency. Thus their approximations for straight formulation instead of parametric formulation are then obtained. These steady-state analytic solutions to envelopes describe the dynamic holographic recording and readout more clearly and exactly. 相似文献
17.
Sheridan JT Lawrence JR 《Journal of the Optical Society of America. A, Optics, image science, and vision》2000,17(6):1108-1114
The standard one-dimensional diffusion equation is extended to include nonlocal temporal and spatial medium responses. How such nonlocal effects arise in a photopolymer is discussed. It is argued that assuming rapid polymer chain growth, any nonlocal temporal response can be dealt with so that the response can be completely understood in terms of a steady-state nonlocal spatial response. The resulting nonlocal diffusion equation is then solved numerically, in low-harmonic approximation, to describe grating formation. The effects of the diffusion rate, the rate of polymerization, and a new parameter, the nonlocal response length, are examined by using the predictions of the model. By applying the two-wave coupled-wave model, assuming a linear relationship between polymerized concentration and index modulation, the resulting variation of the grating diffraction efficiency is examined. 相似文献
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Abbas Rastgoo Farzad Ebrahimi Ali Feyz Dizaji 《International Journal of Mechanics and Materials in Design》2006,3(1):29-38
A cantilever beam having arbitrary cross section with a lumped mass attached to its free end while being excited harmonically
at the base is fully investigated. The derived equation of vibrating motion is found to be a non-linear parametric ordinary
differential equation, having no closed form solution for it. We have, therefore, established the sufficient conditions for
the existence of periodic oscillatory behavior of the beam using Green’s function and employing Schauder’s fixed point theorem.
The derived equation of vibration motion is found to be a non-linear parametric ordinary differential equation, having no
closed form solution for it. To formulate a simple, physically correct dynamic model for stability and periodicity analysis,
the general governing equations are truncated to only the first mode of vibration. Using Green’s function and Schauder’s fixed
point theorem, the necessary and sufficient conditions for periodic oscillatory behavior of the beam are established. Consequently,
the phase domain of periodicity and stability for various values of physical characteristics of the beam-mass system and harmonic
base excitation are presented. 相似文献
20.
Kaptay G 《Journal of nanoscience and nanotechnology》2012,12(3):2625-2633
The Kelvin equation, the Gibbs equation and the Gibbs-Thomson equation are compared. It is shown that the Kelvin equation (on equilibrium vapor pressure above nano-droplets) can be derived if the inner pressure due to the curvature (from the Laplace equation) is substituted incorrectly into the external pressure term of the Gibbs equation. Thus, the Kelvin equation is excluded in its present form. The Gibbs-Thomson equation (on so-called equilibrium melting point of a nano-crystal) is an analog of the Kelvin equation, and thus it is also excluded in its present form. The contradiction between the critical nucleus size (from the Gibbs equation) and the so-called equilibrium melting point of nano-crystals (from the Gibbs-Thomson equation) is explained. The contradiction is resolved if the Gibbs equation is applied to study both nucleation and equilibrium of nano-crystals. Thus, the difference in the behavior of nano-systems compared to macro-systems is due to their high specific surface area (Gibbs) and not to the high curvature of their interface (Kelvin). Modified versions of the Kelvin equation and the Gibbs-Thomson equation are derived from the Gibbs equation for phases with a general shape and for a spherical phase. 相似文献