Lagrange-Maxwell方程与发电机组磁饱和参数共振

推广了Lagrange-Maxwell方程,使之适用于分析发电机组磁饱和振动问题．应用机电分析动力学方法和电机学理论,找到了考虑磁饱和时发电机气隙磁场能,建立了发电机组电磁激发振动的非线性常微分方程组．发现在磁饱和状态下发电机电磁干扰力包括倍频参数激励成分．应用平均法求得系统主参数共振时的解,分析了发电机组电磁参数对共振特性的影响,揭示了一些新现象．     

Nonlinearly coupled in-plane and transverse vibrations of a spinning disk

Previous nonlinear spinning disk models neglected the in-plane inertia of the disk since this permits the use of a stress function. This paper aims to consider the effect of including the in-plane inertia of the disk on the resulting nonlinear dynamics and to construct approximate solutions that capture the new dynamics. The inclusion of the in-plane inertia results in a nonlinear coupling between the in-plane and transverse vibrations of the spinning disk. The full nonlinear partial differential equations are simplified to a simpler nonlinear two degrees of freedom model via the method of Galerkin. A canonical perturbation approach is used to derive an approximate solution to this simpler nonlinear problem. Numerical simulations are used to evaluate the effectiveness of the approximate solution. Through the use of these analytical and numerical tools, it becomes apparent that the inclusion of in-plane inertia gives rise to new phenomena such as internal resonance and the possibility of instability in the system that are not predicted if the in-plane inertia is ignored. It is also demonstrated that the canonical perturbation approach can be used to produce an effective approximate solution.     

求强非线性系统次谐共振解的MLP方法

定义了一个新的参数变换α=α(ε,nω₀/m,ω₁),扩展了改进的LP方法的应用范围,使该方法能够求强非线性系统的次谐共振解.研究了Duffing方程的1/3亚谐和3次超谐共振解以及Vander Pol-Mathieu方程1/2亚谐共振解,这些例子说明近似解和数值解相当吻合.     

Convergence of Fourier spectral method for resonant long-short nonlinear wave interaction

In this paper, the evolution equations with nonlinear term describing the resonance interaction between the long wave and the short wave are studied. The semi-discrete and fully discrete Crank-Nicholson Fourier spectral schemes are given. An energy estimation method is used to obtain error estimates for the approximate solutions. The numerical results obtained are compared with exact solution and found to be in good agreement.     

Analytical approximations to primary resonance response of harmonically forced oscillators with strongly general nonlinearity

This paper presents an innovative analytical approximate method for constructing the primary resonance response of harmonically forced oscillators with strongly general nonlinearity. A linearization process is introduced prior to harmonic balancing (HB) of the nonlinear system and a linear system is derived by which the accurate analytical approximation procedure is easily and innovatively implemented. The main cutting edge of the proposed method is that complicated and coupled nonlinear algebraic equations obtained by the classical HB method is avoided. With only one iteration, very accurate analytical approximate primary resonance response can be determined, even for significantly nonlinear systems. Another advantage is the direct determination of the maximum oscillation amplitude. This is due to the appropriate form chosen for the approximation with no extra processing required. It is concluded that the result of an initial approximate solution from the two-term (constant plus the first harmonic term) harmonic balance is not reliable especially for strongly nonlinear systems and a correction to the initial approximation is necessary. The proposed method can be applied to general oscillators with mixed nonlinearities, such as the Helmholtz-Duffing oscillator. Two examples are presented to illustrate the applicability and effectiveness of the proposed technique.     

Limiting phase trajectories and nonstationary resonance oscillations of the Duffing oscillator. Part 2. A dissipative oscillator

This paper extends the concept of limiting phase trajectories (LPT) to systems with dissipation. The Duffing oscillator with linear dissipation and periodic excitation is studied. Under condition of 1:1 resonance, we obtain an approximate analytic solution describing two stages of motion: at the first stage the trajectory is close to the non-dissipative LPT, while at the second stage the trajectory approaches stationary oscillations near the steady state.     

Triple scale analysis of periodic solutions and resonance of some asymmetric non linear vibrating systems

We consider small solutions of a vibrating mechanical system with smooth non-linearities for which we provide an approximate solution by using a triple scale analysis; a rigorous proof of convergence of the triple scale method is included; for the forced response, a stability result is needed in order to prove convergence in a neighbourhood of a primary resonance. The amplitude of the response with respect to the frequency forcing is described and it is related to the frequency of a free periodic vibration.     

A Note on a Nonlinear Model of a Piezoelectric Rod

If piezoceramics are excited by weak electric fields a nonlinear behavior can be observed, if the excitation frequency is close to a resonance frequency of the system. To derive a theoretical model nonlinear constitutive equations are used, to describe the longitudinal oscillations of a slender piezoceramic rod near the first resonance frequency. Hamilton's principle is used to receive a variational principle for the piezoelectric rod. Introducing a Rayleigh Ritz ansatz with the eigenfunctions of the linearized system to approximate the exact solution leads to nonlinear ordinary differential equations. These equations are approximated with the method of harmonic balance. Finally it is possible to calculate the amplitudes of the displacements numerically. As a result it is shown, that the Duffing type nonlinearities found in measurements can be described with this model.     

无解的模糊关系方程的最优近似解

对无解的模糊关系方程给出了最优近似解的定义,证明了最优近似解的存在性,给出了求最优近似解的算法     

On the averaging method for rod equations with quadratic non-linearity

The averaging method is used to approximate solutions of systems of linearly coupled, (quadratic) non-linear dispersive wave equations, which describe extensional–torsional dynamics of a rod. Existence and uniqueness results are established. Error estimates confirm the asymptotic validity of the approximation method on a long time-scale. The linear couplings between the equations imply that resonance can occur inside a single mode of the solution, but energy can also be transferred to other modes.     

幂律流体边界层方程的近似解析解和壁摩擦因数的近似值

对幂率流体层流平板边界层的解析解进行了研究．对该问题提供了Adomian分解方法并且推导出了问题的级数形式的近似解析解,该近似解析解具有快速收敛性和易于计算性．对不同的幂率给出了方程的近似解析解和相应的壁摩擦因数近似值,最后对近似解所推出结果和所得壁摩擦因数与文献中的数值解进行了比较验证,证实了该文提出的解析近似方法的准确性和可靠性,说明了该近似解能够应用于提供所研究问题的壁摩擦因数．     

A Spectral Method for Neutral Volterra Integro-Differential Equation with Weakly Singular Kernel

This paper is concerned with obtaining an approximate solution and an approximate derivative of the solution for neutral Volterra integro-differential equation with a weakly singular kernel. The solution of this equation, even for analytic data, is not smooth on the entire interval of integration. The Jacobi collocation discretization is proposed for the given equation. A rigorous analysis of error bound is also provided which theoretically justifies that both the error of approximate solution and the error of approximate derivative of the solution decay exponentially in $L^∞$ norm and weighted $L^2$ norm. Numerical results are presented to demonstrate the effectiveness of the spectral method.     

Intuitive approach to the approximate analytical solution for the Blasius problem

For the Blasius problem, we propose an approximate analytical solution in the form of a logarithm of the hyperbolic cosine function which satisfies the given boundary conditions and some known properties of the exact solution. Furthermore, adding some hyperbolic tangent functions to this solution, we obtain much more accurate approximate solution with the relative error less than 0.16% over the whole region. The superiority of the proposed solutions is shown by comparison with the existing approximate analytical solution.     

Near-field and far-field approximations by the Adomian and asymptotic decomposition methods

The Adomian decomposition method and the asymptotic decomposition method give the near-field approximate solution and far-field approximate solution, respectively, for linear and nonlinear differential equations. The Padé approximants give solution continuation of series solutions, but the continuation is usually effective only on some finite domain, and it can not always give the asymptotic behavior as the independent variables approach infinity. We investigate the global approximate solution by matching the near-field approximation derived from the Adomian decomposition method with the far-field approximation derived from the asymptotic decomposition method for linear and nonlinear differential equations. For several examples we find that there exists an overlap between the near-field approximation and the far-field approximation, so we can match them to obtain a global approximate solution. For other nonlinear examples where the series solution from the Adomian decomposition method has a finite convergent domain, we can match the Padé approximant of the near-field approximation with the far-field approximation to obtain a global approximate solution representing the true, entire solution over an infinite domain.     

基于近似惯性流形的后验Galerkin的方法

１．引言 为提高用数值方法解非线性发展方程及非线性椭圆边值问题的逼近阶,许多学者例如Ｊ．Ｎｏｖｏ和 Ｅ．Ｔｉｔｉ［４］, Ｍａｒｉｏｎ和 Ｔｅｍａｎ［６］,Ｊ．Ｘｕ［７］以及 Ｗ．Ｌａｙｔｏｎ［９］等人,提出了后验Ｇａｌｅｒｋｉｎ方法、近似惯性流形方法、非线性Ｇａｌｅｒｋｉｎ方法、各种区域分裂法、多重网格法等等．本文根据［１］提出了一种新的高精度的后验 Ｇａｌｅｒｋｉｎ方法．它的逼近阶是经典 Ｇａｌｅｒｋｉｎ方法逼近阶的两倍． 考虑非线性椭圆边值问题这里ｎ是按ｄ＝２,３）上具有分段光滑边界ｒ的有界区域,…     

On the approximate solution to an initial boundary valued problem for the Cahn–Hilliard equation

The particular approximate solution of the initial boundary valued problem to the Cahn–Hilliard equation is provided. The Fourier Method is combined with the Adomian’s decomposition method in order to provide an approximate solution that satisfy the initial and the boundary conditions. The approximate solution also satisfies the mass conservation principle.     

二维连续信号的近似采样定理

利用加细方程的面具,给出该方程的一个近似解,并根据这个近似解构造出二维连续信号的近似采样定理.其近似采样函数是所求加细方程的近似解,它是由加细方程的面具唯一确定的逐段线性函数,且有显示的计算公式.因此可以根据需要选择加细方程的面具,从而达到控制近似采样函数的衰减速度.     

An approximate solution for linear boundary-value problems with slowly varying coefficients

In this paper, an approximate closed-form solution for linear boundary-value problems with slowly varying coefficient matrices is obtained. The derivation of the approximate solution is based on the freezing technique, which is commonly used in analyzing the stability of slowly varying initial-value problems as well as solving them. The error between the approximate and the exact solutions is given, and an upper bound on the norm of the error is obtained. This upper bound is proportional to the rate of change of the coefficient matrix of the boundary-value problem. The proposed approximate solution is obtained for a two-point boundary-value problem and is compared to its solution obtained numerically. Good agreement is observed between the approximate and the numerical solutions, when the rate of change of the coefficient matrix is small.     

Semicontinuity of the Approximate Solution Sets of Multivalued Quasiequilibrium Problems

We consider two kinds of approximate solutions and approximate solution sets to multivalued quasiequilibrium problems. Sufficient conditions for the lower semicontinuity, Hausdorff lower semicontinuity, upper semicontinuity, Hausdorff upper semicontinuity, and closedness of these approximate solution sets are established. Applications in approximate quasivariational inequalities, approximate fixed points, and approximate quasioptimization problems are provided.     

On the complexity of solving feasible systems of linear inequalities specified with approximate data

An algorithm that gives an approximate solution of a desired accuracy to a system of linear inequalities specified with approximate data is presented. It uses knowledge that the actual instance is feasible to reduce the data precision necessary to give an approximate solution to the actual instance. In some cases, this additional information allows problem instances that are ill-posed without the knowledge of feasibility to be solved.The algorithm is computationally efficient and requires not much more data accuracy than the minimal amount necessary to give an approximate solution of the desired accuracy. This work aids in the development of a computational complexity theory that uses approximate data and knowledge.