Application of ultraspherical polynomials to non-linear autonomous systems

This paper deals with the approximate solutions of non-linear autonomous systems by the application of ultraspherical polynomials. From the differential equations for amplitude and phase, set up by the method of variation of parameters, the approximate solutions are obtained by a generalized averaging technique based on the ultraspherical polynomial expansions. The method is illustrated with examples and the results are compared with the digital and analog computer solutions. There is a close agreement between the analytical and exact results.     

Response of non-linear non-conservative systems of two degrees of freedom to transient excitations

This paper deals with the approximate analysis of non-linear non-conservative systems of two degrees of freedom subjected to transient excitations. By using a transformation of the co-ordinates, the governing differential equations of the system are brought into a form to which the method of averaging of Krylov and Bogoliubov can be applied. The response of a representative spring-mass-damper system to typical pulses like a blast pulse and a half sinusoidal pulse is determined. The validity of the approach is demonstrated by comparison of the approximate solutions with numerical results obtained on a digital computer.     

Dynamic stability analysis of torsional vibrations of a shaft system connected by a Hooke׳s joint through a continuous system model

Stability of parametrically excited torsional vibrations of a shaft system composed of two torsionally elastic shafts interconnected through a Hooke?s joint is studied. The shafts are considered to be continuous (distributed-parameter) systems and an approximate discrete model for the torsional vibrations of the shaft system is derived via a finite element scheme. The stability of the solutions of the linearized equations of motion, consisting of a set of Mathieu–Hill type equations, is examined by means of a monodromy matrix method and the results are presented in the form of a Strutt–Ince diagram visualizing the effects of the system parameters on the stability of the shaft system.     

Stability analysis of systems with periodic coefficients: An approximate approach

The paper deals with an approximate method of stability analysis for second order linear systems with periodic coefficients. The periodic functions are approximated during the first period of motion by a constant, a linear or a quadratic function of time such that the resulting approximate equations have known closed form solutions. The approximate equivalent equations are generated through an expansion of periodic coefficients into ultraspherical polynomials. The stability criteria is determined from the solution of approximate equivalent system and the generalized Floquet theory. The technique is quite general and does not require any restriction on the magnitudes of system parameters. In particular, the method has been applied to construct approximate stability chart for the Mathieu equation. A close agreement between the approximate and the exact result is found even for large values of system parameters.     

A nonlinear system under combined periodic and random excitation

The anharmonic oscillator under combined sinusoidal and white noise excitation is studied using the Gaussian closure approximation. The mean response and the steady-state variance of the system is obtained by the WKBJ approximation and also by the Fokker-Planck equation. The multiple steadystate solutions are obtained and their stability analysis is presented. Numerical results are obtained for a particular set of system parameters. The theoretical results are compared with a digital simulation study to bring out the usefulness of the present approximate theory.     

The premelt dynamics of an exploding conductor circuit

The system of nonlinear differential equations that model a simple exploding conductor circuit in its initial stages up to the melting point of the conductor is analyzed. Using the method of variation of parameters, approximate analytical solutions are derived that reflect the role of the circuit parameters and initial conditions in shaping the dynamics. A comparison with solutions obtained by numerical integration shows that the approximate solutions are quite accurate over the entire domain of validity of the mathematical model     

Semi-analytical model of Raman generation in silicon-on-insulator rib waveguide with DBR/F-P resonator

An approximate method of modelling of Raman generation in silicon-on-insulator (SOI) rib waveguide with DBR/F-P resonator including spatial field distribution and nonlinear effects such as Raman amplification and two photon absorption (TPA), is developed. In threshold analysis of steady-state Raman laser operation, an analytical formula relating threshold pump power to the system parameters is obtained. The analysis of the above threshold operation is based on an energy theorem. In exact energy conservation relation, we approximate the Stokes field distributions by that existing at the threshold, whereas the approximate pump field distributions are obtained by integrating the equations for the pump signal using the linear (threshold) pump field distributions and the threshold Stokes field distributions. An approximate, semi-analytical expression related the Raman output power to the pump power and system parameters is derived. Our calculations remain in a good agreement with the exact numerical solutions.     

微重力下圆管毛细流动解析近似解研究

应用同伦分析法研究微重力环境下圆管毛细流动解析近似解问题, 给出了级数解的表达公式. 不同于其他解析近似方法, 该方法从根本上克服了摄动理论对小参数的过分依赖, 其有效性与所研究的非线性问题是否含有小参数无关, 适用范围广. 同伦分析法提供了选取基函数的自由, 可以选取较好的基函数, 更有效地逼近问题的解, 通过引入辅助参数和辅助函数来调节和控制级数解的收敛区域和收敛速度, 同伦分析法为圆管毛细流动问题的解析近似求解开辟了一个全新的途径. 通过具体算例, 将同伦分析法与四阶龙格库塔方法数值解做了比较, 结果表明, 该方法具有很高的计算精度. 关键词： 圆管 微重力 毛细流动 同伦分析法     

A general solution procedure for vibrations of systems with cubic nonlinearities and nonlinear/time-dependent internal boundary conditions

The subject of this paper is the development of a general solution procedure for the vibrations (primary resonance and nonlinear natural frequency) of systems with cubic nonlinearities, subjected to nonlinear and time-dependent internal boundary conditions—this is a commonly occurring situation in the vibration analysis of continuous systems with intermediate elements. The equations of motion form a set of nonlinear partial differential equations with nonlinear, time-dependent, and coupled internal boundary conditions. The method of multiple timescales, an approximate analytical method, is applied directly to each partial differential equation of motion as well as coupled boundary conditions (i.e. on each sub-domain and the corresponding internal boundary conditions for a continuous system with intermediate elements) which ultimately leads to approximate analytical expressions for the frequency-response relation and nonlinear natural frequencies of the system. These closed-form solutions provide direct insight into the relationship between the system parameters and vibration characteristics of the system. Moreover, the suggested solution procedure is applied to a sample problem which is discussed in detail.     

Analytical solutions for the spin-1 Bose-Einstein condensate in a harmonic trap

The homotopy analysis method and Galerkin spectral method are applied to find the analytical solutions for the Gross-Pitaevskii equations, a set of nonlinear Schrödinger equation used in simulation of spin-1 Bose-Einstein condensates trapped in a harmonic potential. We investigate the one-dimensional case and get the approximate analytical solutions successfully. Comparisons between the analytical solutions and the numerical solutions have been made. The results indicate that they are in agreement well with each other when the atomic interaction is weakly. We also find a class of exact solutions for the stationary states of the spin-1 system with harmonic potential for a special case.     

自行车系统的计算机模拟

自行车系统是一个具有非完整约束的复杂系统,可以根据最小作用量原理通过欧拉-拉格朗日方程求解,利用差分代替微分及一系列修正截断误差的方法计算近似值.利用MATLAB,只要给定任意质点系的拉格朗日函数以及约束系数,即可求出系统的近似解,并通过这个方法为分析自行车的运动与稳定性提供一种新的思路.     

Large deflection of clamped circular plate and accuracy of its approximate analytical solutions

A different set of governing equations on the large deflection of plates are derived by the principle of virtual work(PVW), which also leads to a different set of boundary conditions. Boundary conditions play an important role in determining the computation accuracy of the large deflection of plates. Our boundary conditions are shown to be more appropriate by analyzing their difference with the previous ones. The accuracy of approximate analytical solutions is important to the bulge/blister tests and the application of various sensors with the plate structure. Different approximate analytical solutions are presented and their accuracies are evaluated by comparing them with the numerical results. The error sources are also analyzed. A new approximate analytical solution is proposed and shown to have a better approximation. The approximate analytical solution offers a much simpler and more direct framework to study the plate-membrane transition behavior of deflection as compared with the previous approaches of complex numerical integration.     

THE RESPONSE OF A DYNAMIC VIBRATION ABSORBER SYSTEM WITH A PARAMETRICALLY EXCITED PENDULUM

The vibration response of a spring-mass-damper system with a parametrically excited pendulum hinged to the mass is investigated using the harmonic balance method. The approximate results are found to be fairly consistent with those obtained by the numerical calculation. The vibrating regions of the pendulum system are obtained which are similar to those given by Mathieu's equation. Based on the analysis of three parameters in the response equation, the characteristics of response of the system are clarified. The stabilities of the harmonic solutions are analyzed, and finally our proposed approximation is verified compared with the numerical results.     

Modeling and study of dynamic performance of a valveless micropump

This work presents an approximate nonlinear analytical model for the problem of fluid-structural interaction in a valveless micropump. The model is constructed using the lumped-mass approach and takes into account the inertial force and time variation of mass density of the working fluid within the micropump chamber, pressure viscous losses of the flow through the diffuser/nozzle elements and the structural geometric nonlinearity due to the membrane mid-plane stretching. It consists of a set of coupled partial integro-differential equations which is reduced to a third order nonlinear coupled fluid-plate vibration equation by using the assumed mode method to approximate the plate dynamic deflection. An approximate analytical solution for the nonlinear vibration model is carried out using the harmonic balance method and is used to investigate the effect of various system parameters on the performance of the micropump. The obtained model and approximate analytical results are compared with those available in the open literature. The approximate analytical results show that, depending on the micropump physical parameters and membrane driving frequency, the working fluid stiffness, which arise in the present model as a result of taking into account the variation of the fluid density with time, and the membrane geometric nonlinearity can have significant effects on the predicted micropump performance and can lead to a complex nonlinear dynamic behavior. The accuracy of these results is subject to a future numerical validation of the presented approximate theoretical model.     

The response of a beam to a transient pressure wave load

The dynamic behaviour of beam structures under pressure waves is investigated. The propagation of the bending waves under a moving single load is first studied for three types of beam: a Bernoulli-Euler beam, a beam with shear deflection and a Timoshenko beam. Then the responses of the Bernoulli-Euler and the Timoshenko beam are studied under moving pressure wave excitation. The results are presented as dynamic amplification factors (DAF). The influence of the load parameters (load shape, propagation speed, pressure wave duration, etc.) and the beam parameters (slenderness, damping, etc.) is discussed. The load shape (symmetrical, asymmetrical) and the propagation speed strongly influence the response. The results are compared with available approximate solutions for the corresponding lumped element, single degree of freedom model of the structure.     

大平板瞬态热传导问题的一种新的近似解法

１引言大平板瞬态热传导问题有着广泛的工程应用背景。对于复杂的初边值条件或含内热源问题,以及工程上常见的多层复合平壁对象,分析求解难度很大甚至无法求解。在此类情况下往往采用数值方法。但是单纯的数值解不便于理解影响该问题的各种参数的物理意义。因此,各种近似分析方法得到了发展［１,２］。但在近似精度上,往往难以对整个时间坐标范围都达到较高的精度,这就使得近似解更多地局限于定性分析。此外,对于不同的初边值条件或含内热源问题,近似解的形式相异,降低了解的通用性。增加了求解的工作量。本文提出一种基于矩阵理论…     

Approximate Solutions of Perturbed Nonlinear Schr(o)dinger Equations

By applying Lou's direct perturbation method to perturbed nonlinear Schr(o)dinger equation and the critical nonlinear Schr(o)dinger equation with a small dispersion, their approximate analytical solutions including the zero-order and the first-order solutions are obtained. Based on these approximate solutions, the analytical forms of parameters of solitons are expressed and the effects of perturbations on solitons are briefly analyzed at the same time. In addition, the perturbed nonlinear Schr(o)dinger equations is directly simulated by split-step Fourier method to check the validity of the direct perturbation method. It turns out that the analytical results given by the direct perturbation method are well supported by numerical calculations.     

Practical use of variational principles for modeling water waves

This paper describes a method for deriving approximate equations for irrotational water waves. The method is based on a ‘relaxed’ variational principle, i.e., on a Lagrangian involving as many variables as possible. This formulation is particularly suitable for the construction of approximate water wave models, since it allows more freedom while preserving a variational structure. The advantages of this relaxed formulation are illustrated with various examples in shallow and deep waters, as well as arbitrary depths. Using subordinate constraints (e.g., irrotationality or free surface impermeability) in various combinations, several model equations are derived, some being well-known, other being new. The models obtained are studied analytically and exact traveling wave solutions are constructed when possible.     

Infinite series symmetry reduction solutions to the modified KdV--Burgers equation

From the point of view of approximate symmetry, the modified Korteweg--de Vries--Burgers (mKdV--Burgers) equation with weak dissipation is investigated. The symmetry of a system of the corresponding partial differential equations which approximate the perturbed mKdV--Burgers equation is constructed and the corresponding general approximate symmetry reduction is derived; thereby infinite series solutions and general formulae can be obtained. The obtained result shows that the zero-order similarity solution to the mKdV--Burgers equation satisfies the Painlevé II equation. Also, at the level of travelling wave reduction, the general solution formulae are given for any travelling wave solution of an unperturbed mKdV equation. As an illustrative example, when the zero-order tanh profile solution is chosen as an initial approximate solution, physically approximate similarity solutions are obtained recursively under the appropriate choice of parameters occurring during computation.     

Estimation of resonant peak amplitudes

A simple method is proposed for estimating the resonant peak response levels of damped structures, by using an analysis of undamped vibration. The method is based on a simple identity which is exact for a single degree of freedom system and approximate for more general systems. However, since the level of damping in practical structures can seldom be described with any precision, it is argued that a simple approximate method for predicting resonant vibration levels is the most appropriate. A number of examples and applications are described.