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文本是文[1]的继续.文[1]中,提出和建议使用非线性偏微分方程直接摄动与加权积分方程法,计算连续介质系统的共振与非共振周期解.本文中,应用该方法计算了定跨度弹性梁在各种常见边界条件下强迫振动的共振与非共振周期解,方板在集中周期荷载作用下的共振周期解.指出了,非主振型对非线性振动周期解的影响及静荷载对幅频特性曲线的影响. 相似文献
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两自由度非线性振动系统周期运动及其稳定性研究 总被引:1,自引:0,他引:1
运用Liapunov函数方法,对一类两自由度非线性振动系统周期运动及其稳定性进行了研究,得到了存在唯一渐近稳定的周期解的充分条件. 相似文献
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将不等距离分割方法与人工鱼群算法相结合,提出一种基于人工鱼群算法求任意函数数值积分的方法,该方法除能计算通常意义下任意函数的定积分外,还能计算奇异函数积分、振荡函数积分以及原函数不易求得的被积函数的积分.最后给出几个数值积分算例,并与传统数值积分方法作了比较,仿真结果分析表明,该算法十分有效,能够快速有效地获得任意函数的数值积分值. 相似文献
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§1.引言 1.问题的提出在工程技术上,很多问题归结为解常微分方程组的初值问题,而求得具有分析表达式的解,通常是不可能的,必须借助数值积分法求其近似解。在数值积分常微分方程,特别是小参数微分方程(最高阶导数项含有小参数的微分方程)时,步长的选择是一个复杂的问题:步长大了,就会引起计算的不稳定;而步长取得过小,又会花费大量的机器时间。通常所谓的“小参数问题”就是由于这个原因而著称的。而在自动控制系统 相似文献
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渐近法在一类强非线性系统中的应用 总被引:2,自引:0,他引:2
本文采用文[1、2]的渐近解形式,将渐近法推广到如下较为广泛一类的强非线性振动系统式中g和f为x,的非线性解析函数,ε>0为小参数,并假设对应于ε=0的派生系统有周期解.本文推得系统(0.1)的渐近解递推方法,并应用于实例. 相似文献
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研究了一个激光脉冲放大器增益通量系统解的问题.首先讨论了较一般的系统, 然后引入一个同伦映射.再利用映射的性质, 引进一个人工参数, 将求解非线性问题转化为求解一系列线性问题.再逐次地求出对应的线性问题的解, 最后得到了原模型解的近似展开式.可以看出, 同伦映射方法是一个解析的方法.它是通过函数的解析运算并用初等函数来表达近似解,其不同于用离散数值运算的数值计算方法.因此通过同伦映射解, 还可以对它继续进行解析运算, 从而可以进行微分和积分等运算来得到与激光脉冲放大器增益通量相关的其他物理量的性态. 相似文献
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长水波近似方程组的新精确解 总被引:3,自引:0,他引:3
依据齐次平衡法的思想 ,首先提出了求非线性发展方程精确解的新思路 ,这种方法通过改变待定函数的次序 ,优势是使求解的复杂计算得到简化 .应用本文的思路 ,可得到某些非线性偏微分方程的新解 .其次我们给出了长水波近似方程组的一些新精确解 ,其中包括椭圆周期解 ,我们推广了有关长波近似方程的已有结果 . 相似文献
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Presented herein is to establish the asymptotic analytical solutions for the fifth-order Duffing type temporal problem having strongly inertial and static nonlinearities. Such a problem corresponds to the strongly nonlinear vibrations of an elastically restrained beam with a lumped mass. Taking into consideration of the inextensibility condition and using an assumed single mode Lagrangian method, the single-degree-of-freedom ordinary differential equation can be derived from the governing equations of the beam model. Various parameters of the nonlinear unimodal temporal equation stand for different vibration modes of inextensible cantilever beam. By imposing the homotopy analysis method (HAM), we establish the asymptotic analytical approximations for solving the fifth-order nonlinear unimodal temporal problem. Within this research framework, both the frequencies and periodic solutions of the nonlinear unimodal temporal equation can be explicitly and analytically formulated. For verification, numerical comparisons are conducted between the results obtained by the homotopy analysis and numerical integration methods. Illustrative examples are selected to demonstrate the accuracy and correctness of this approach. Besides, the optimal HAM approach is introduced to accelerate the convergence of solutions. 相似文献
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《Chaos, solitons, and fractals》2005,23(1):249-258
This paper investigates bifurcation and chaos of an axially accelerating viscoelastic beam. The Kelvin–Voigt model is adopted to constitute the material of the beam. Lagrangian strain is used to account for the beam's geometric nonlinearity. The nonlinear partial–differential equation governing transverse motion of the beam is derived from the Newton second law. The Galerkin method is applied to truncate the governing equation into a set of ordinary differential equations. By use of the Poincaré map, the dynamical behavior is identified based on the numerical solutions of the ordinary differential equations. The bifurcation diagrams are presented in the case that the mean axial speed, the amplitude of speed fluctuation and the dynamic viscoelasticity is respectively varied while other parameters are fixed. The Lyapunov exponent is calculated to identify chaos. From numerical simulations, it is indicated that the periodic, quasi-periodic and chaotic motions occur in the transverse vibrations of the axially accelerating viscoelastic beam. 相似文献
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In this paper, we predict the accurate bifurcating periodic solution for a general class of first-order nonlinear delay differential equation with reflectional symmetry by constructing an approximate technique, named residue harmonic balance. This technique combines the features of the homotopy concept with harmonic balance which leads to easy computation and gives accurate prediction on the periodic solution to the desired accuracy. The zeroth-order solution using just one Fourier term is applied by solving a set of nonlinear algebraic equations containing the delay term. The unbalanced residues due to Fourier truncation are considered iteratively by solving linear equations to improve the accuracy and increase the number of Fourier terms of the solutions successively. It is shown that the solutions are valid for a wide range of variation of the parameters by two examples. The second-order approximations of the periodic solutions are found to be in excellent agreement with those obtained by direct numerical integration. Moreover, the residue harmonic balance method works not only in determining the amplitude but also the frequency of the bifurcating periodic solution. The method can be easily extended to other delay differential equations. 相似文献
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This paper applies He’s Energy balance method (EBM) to study periodic solutions of strongly nonlinear systems such as nonlinear
vibrations and oscillations. The method is applied to two nonlinear differential equations. Some examples are given to illustrate
the effectiveness and convenience of the method. The results are compared with the exact solution and the comparison showed
a proper accuracy of this method. The method can be easily extended to other nonlinear systems and can therefore be found
widely applicable in engineering and other science. 相似文献
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用自适应步长积分格式结合打靶技巧,可以有效地求解比较困难的常微分方程边界层型奇异摄动问题.本文给出了若干计算实例,说明了这种方法应用于线性问题时的一次收敛性,以及应用于单端、双端边界层、转向点和多个边界层时的效果,特别是能方便地求出多解.最后并与习用的差分方法作了比较. 相似文献
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A flow induced system, consisting of an elastically mounted body with a pendulum attached, is considered here. The stability of the semi-trivial solution, representing the vibration of the body with the non-oscillating pendulum, is investigated. The analytical investigation shows that at a certain flow velocity, higher than the critical one, the pendulum begins to oscillate due to autoparametric resonance. For a convenient tuning, the vibration of the system can be substantially reduced. The analysis of both semi-trivial and non-trivial solutions is complemented by numerical integration of the differential equations of motion. A mapping technique based on Poincaré section, suitable for investigating the non-periodic vibrations occuring at higher flow velocities, is proposed. 相似文献
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Walter V. Wedig 《PAMM》2015,15(1):561-564
When vehicles ride on uneven roads, they are excited to vertical random vibrations whose stationary rms-values (root-mean-square) strongly depend on the velocity of the vehicle. To investigate this vibration behavior, it is appropriate to introduce road models in way domain which are based on the theory of stochastic differential equations and transformed from way to time by means of velocity-dependent way and noise increments. The random base excitations by roads are applied to nonlinear quarter car models. They lead to stationary rms-values of the vertical vehicle vibrations which become resonant for critical velocities and show jump phenomena similar to those of the Duffing oscillator under harmonic excitations. In the stochastic case, jump phenomena are only observable for narrow-banded road excitations. They vanish for increasing car damping and excitation bandwidth. For efficient simulations of the road-vehicle model, the n state equations are utilized to derive n(n + 1)/2 stochastic covariance equations. For small step sizes, their numerical mean square solutions coincide with the nonlinear results of fix-point iterations obtained when the noise terms of the covariance equations are omitted. It can easily be shown, that this deterministic approach leads to the correct stationary covariances in the linear case. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
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Here, we propose a method to obtain local analytic approximate solutions of ordinary differential equations with variable coefficients, or even some nonlinear equations, inspired in the Lyapunov method, where instead of polynomial approximations, we use truncated Fourier series with variable coefficients as approximate solutions. In the case of equations admitting periodic solutions, an averaging over the coefficients gives global solutions. We show that, under some restrictive condition, the method is equivalent to the Picard-Lindelöf method. After some numerical experiments showing the efficiency of the method, we apply it to equations of interest in physics, in which we show that our method possesses an excellent precision even with low iterations. 相似文献
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《Chaos, solitons, and fractals》2000,11(9):1349-1363
Periodic motions of the nonlinear system representing the escape equation with cosine and sine parametric excitations and external harmonic excitations are obtained by the incremental harmonic balance (IHB) method. The system contains quadratic stiffness terms. The Jacobian matrix and the residue vector for the type of nonlinearity with parametric excitation are explicitly derived. An arc length path following procedure is used in combination with Floquet theory to trace the response diagram and to investigate the stability of the periodic solutions. The system undergoes chaotic motion for increase in the amplitude of the harmonic excitation which is investigated by numerical integration and represented in terms of phase planes, Poincaré sections and Lyapunov exponents. The interpolated cell mapping (ICM) method is used to obtain the initial condition map corresponding to two coexisting period 1 motions. The periodic motions and bifurcation points obtained by the IHB method compare very well with results of numerical integration. 相似文献