利用线化和校正法求非线性单摆运动的周期

应用线化和校正方法,研究了单摆的非线性振动,作出了周期比和相对误差随摆角的变化曲线.将所得近似解与精确解比较可知,该方法具有简单实用,精度高,相对误差低等优点,对于求解非线性振动问题具有一定的实用价值.     

对称双弹簧振子横向振动的椭圆函数解

通过分析对称双弹簧振子横向振动的特点,在立方非线性振动精确解的基础上提出振动方程的椭圆函数型近似解析解,并与数值解结果进行了比较.     

组合线性弹簧振子中的非线性振动

从拉格朗日方程出发,分析了几种常见的线性弹簧组合,对作非线性振动弹簧振子进行了数值求解.当作微小振动时,正好是几种典型的非线性振动.通过计算得出解析解并与数值解进行了对比.     

对称非线性弹簧振子的周期特性

通过计算机编程(Quick Basic)描绘对称非线性弹簧振子振动的特性曲线,使难懂的物理过程变得直观、形象。对称非线性弹簧振子的振动是一种周期性振动,但不是严格的简谐振动,其振动周期随非线性系数、振幅的变化而偏离简谐振动的周期。     

在水平面上受稳定约束的弹簧振子运动研究

在水平面上受稳定约束的弹簧振子运动模型是理论力学中不可积系统的典型模型,同时也是工程学中很重要的非线性运动之一。本文利用拉格朗日方程建立了该模型的动力学方程,采用Matlab软件对该模型的动力学方程进行了数值计算,作出了在不同初始条件下相应的坐标随时间变化的演化曲线、运动相图、运动轨迹。通过比较,具体地分析了其运动情况,研究发现:受稳定约束的弹簧振子在水平面上的运动是一种准周期运动。     

弹簧振子在光滑水平面内的运动规律

在非线性双自由度运动的弹簧振子中,光滑水平面内运动的模型具有运动微分方程简单、变量可分离的特点.本文通过极坐标描述法,建立了弹簧振子在光滑水平面内的运动微分方程.将与径向运动有关的变量分离出来,建立了径向运动微分方程,并基于其非线性的特点进行了定性分析和近似求解.使用MATLAB数值求解,模拟弹簧振子的运动轨迹,验证了本文的定性分析结论和两种特殊情形下的近似解的精度.     

同伦分析法在求解非线性演化方程中的应用

利用同伦分析法求解了(2+1)维改进的 Zakharov-Kuznetsov方程, 得到了它的近似周期解,该解与精确解符合很好. 结果表明，同伦分析法在求解高维非线性演化方程时, 仍然是一种行之有效的方法. 同时,还对该方法进行了一定的扩展， 经过扩展后的方法能够更方便地求解更多非线性演化方程的高精度近似解析解. 关键词： 同伦分析法 改进的 Zakharov-Kuznetsov方程 周期解     

与r的一次方成正比有心力作用下 质点的运动研究

在水平面上受稳定约束的弹簧振子运动模型, 实质上是一个与距离r的一次方成正比有心力作用下质 点的运动问题. 本文利用拉格朗日方程建立了该运动模型在极坐标系中的动力学方程, 分别采用泰勒级数展开的方 法和 Ma t l a b数值模拟的方法对该模型的动力学方程进行了计算, 作出了相应的坐标随时间的演化曲线、 运动相图、 运动轨迹, 并将两种方法得出的结果进行了比较, 研究发现, 当初速度较小时, 弹簧振子在径向的运动是周期性的简 谐运动, 在横向的运动是非线性增大的, 在平面上的运动是准周期的     

基于 MAT LA B的强迫振动达芬 方程的非线性幅频响应分析

利用MAT LA B研究了强迫振动达芬( D u f f i n g) 方程的非线性幅频响应特性, 分析了达芬方程非线性幅 频响应近似解析求解和数值求解的方法和步骤, 给出了相应的 MAT LA B求解程序, 并将解析解与数值解结果进行 了比较. 仿真程序和结果能够加深学生对非线性振动相关知识的理解, 提高大学物理及相关力学课程的课堂教学效 果     

对称非线性弹簧振子的振动周期的级数解

用积分法求出了对称非线性水平弹簧振子的振动周期的级数解,并对所得结果进行讨论,得到了线形弹簧振子的振动周期以及x~3振荡器的振荡周期.     

Exact and Approximate Values of the Period for a "Truly Nonlinear" Oscillator: $\ddot{x} + x + x^{1/3} = 0$

We investigate the mathematical properties of a "truly nonlinear" oscillator differential equation. In particular, using phase-space methods, it is shown that all solutions are periodic and the fixed-point is a nonlinear center. We calculate both exact and approximate analytical expressions for the period, where the exact solution is given in terms of elliptic functions and the method of harmonic balance is used to calculate the approximate formula.     

Solution of a mixed parity nonlinear oscillator: Harmonic balance

The method of harmonic balance is used to calculate first-order approximations to the periodic solutions of a mixed parity nonlinear oscillator. First, the amplitude in the negative direction is expressed in terms of the amplitude in the positive direction. Then the two auxiliary equations, where the restoring force functions are odd, are solved by using a first harmonic term (without a constant). Therefore, we obtain the two approximate solutions to the mixed parity nonlinear oscillator. One is expressed in terms of the exact amplitude in the negative direction, the other in terms of the approximate amplitude. These solutions are more accurate than the second approximate solution obtained by the Lindstedt–Poincaré method for large amplitudes.     

Pure odd-order oscillators with constant excitation

In this paper the excited vibrations of a truly nonlinear oscillator are analyzed. The excitation is assumed to be constant and the nonlinearity is pure (without a linear term). The mathematical model is a second-order nonhomogeneous differential equation with strong nonlinear term. Using the first integral, the exact value of period of vibration i.e., angular frequency of oscillator described with a pure nonlinear differential equation with constant excitation is analytically obtained. The closed form solution has the form of gamma function. The period of vibration depends on the value of excitation and of the order and coefficient of the nonlinear term. For the case of pure odd-order-oscillators the approximate solution of differential equation is obtained in the form of trigonometric function. The solution is based on the exact value of period of vibration. For the case when additional small perturbation of the pure oscillator acts, the so called ‘Cveticanin's averaging method’ for a truly nonlinear oscillator is applied. Two special cases are considered: one, when the additional term is a function of distance, and the second, when damping acts. To prove the correctness of the method the obtained results are compared with those for the linear oscillator. Example of pure cubic oscillator with constant excitation and linear damping is widely discussed. Comparing the analytically obtained results with exact numerical ones it is concluded that they are in a good agreement. The investigations reported in the paper are of special interest for those who are dealing with the problem of vibration reduction in the oscillator with constant excitation and pure nonlinear restoring force the examples of which can be found in various scientific and engineering systems. For example, such mechanical systems are seats in vehicles, supports for machines, cutting machines with periodical motion of the cutting tools, presses, etc. The examples can be find in electronics (electromechanical devices like micro-actuators and micro oscillators), in music instruments (hammers in piano), in human voice producing folds (voice cords), etc.     

广义Duffing扰动振子随机共振机理的渐近解

利用非线性方法研究了一类广义Duffing扰动方程.首先求得了典型的Duffing方程的解.然后利用泛函广义变分迭代原理得到了广义Duffing扰动振子随机共振机理的近似解,并论述了解的一致有效性.     

Harmonic balancing approach to nonlinear oscillations of a punctual charge in the electric field of charged ring

The harmonic balance method is used to construct approximate frequency-amplitude relations and periodic solutions to an oscillating charge in the electric field of a ring. By combining linearization of the governing equation with the harmonic balance method, we construct analytical approximations to the oscillation frequencies and periodic solutions for the oscillator. To solve the nonlinear differential equation, firstly we make a change of variable and secondly the differential equation is rewritten in a form that does not contain the square-root expression. The approximate frequencies obtained are valid for the complete range of oscillation amplitudes and excellent agreement of the approximate frequencies and periodic solutions with the exact ones are demonstrated and discussed.     

An explicit approximate solution to the Duffing-harmonic oscillator by a cubication method

The nonlinear oscillations of a Duffing-harmonic oscillator are investigated by an approximated method based on the ‘cubication’ of the initial nonlinear differential equation. In this cubication method the restoring force is expanded in Chebyshev polynomials and the original nonlinear differential equation is approximated by a Duffing equation in which the coefficients for the linear and cubic terms depend on the initial amplitude, A. The replacement of the original nonlinear equation by an approximate Duffing equation allows us to obtain explicit approximate formulas for the frequency and the solution as a function of the complete elliptic integral of the first kind and the Jacobi elliptic function, respectively. These explicit formulas are valid for all values of the initial amplitude and we conclude this cubication method works very well for the whole range of initial amplitudes. Excellent agreement of the approximate frequencies and periodic solutions with the exact ones is demonstrated and discussed and the relative error for the approximate frequency is as low as 0.071%. Unlike other approximate methods applied to this oscillator, which are not capable to reproduce exactly the behaviour of the approximate frequency when A tends to zero, the cubication method used in this Letter predicts exactly the behaviour of the approximate frequency not only when A tends to infinity, but also when A tends to zero. Finally, a closed-form expression for the approximate frequency is obtained in terms of elementary functions. To do this, the relationship between the complete elliptic integral of the first kind and the arithmetic-geometric mean as well as Legendre's formula to approximately obtain this mean are used.     

扰动厄尔尼诺/拉尼娜-南方涛动模型的解

厄尔尼诺/拉尼娜-南方涛动(ENSO)是一个包括在赤道太平洋中海-气交互作用的年际现象. 本文创建一个求解ENSO模型非线性方程的近似方法. 并在这基础上借助同伦映射方法研究了一类ENSO模型振子模型的近似解与精确解. 同时讨论了近似解的精度. 从得到的结果看出,同伦映射方法能够用于分析ENSO模型在赤道东太平洋中的SST异常和气-海振荡的温跃层深度异常.     

Shape Analysis and Solution to a Class of Nonlinear Wave Equation with Cubic Term

In this paper, we analyze the relation between the shape of the bounded traveling wave solutions and dissipation coefficient of nonlinear wave equation with cubic term by the theory and method of planar dynamical systems. Two critical values which can characterize the scale of dissipation effect are obtained. If dissipation effect is not less than a certain critical value, the traveling wave solutions appear as kink profile; while if it is less than this critical value, they appear as damped oscillatory. All expressions of bounded traveling wave solutions are presented, including exact expressions of bell and kink profile solitary wave solutions, as well as approximate expressions of damped oscillatory solutions. For approximate damped oscillatory solution, using homogenization principle, we give its error estimate by establishing the integral equation which reflects the relations between the exact and approximate solutions. It can be seen that the error is an infinitesimal decreasing in the exponential form.     

Accurate analytic presentation of solution of the Schrödinger equation with arbitrary physical potential: Excited states

The quasilinearization method (QLM) is used to approximate analytically, both the ground state and the excited state solutions of the Schrödinger equation for arbitrary potentials. The procedure of approximation was demonstrated on examples of a few often used physical potentials such as the quartic anharmonic oscillator, the Yukawa and the spiked harmonic oscillator potentials. The accurate analytic expressions for the ground and excited state energies and wave functions were presented. These high-precision approximate analytic representations are obtained by first casting the Schrödinger equation into a nonlinear Riccati form and then solving that nonlinear equation analytically in the first QLM iteration. In the QLM the nonlinear differential equation is treated by approximating the nonlinear terms by a sequence of linear expressions. The QLM is iterative but not perturbative and gives stable solutions to nonlinear problems without depending on the existence of a smallness parameter. The method provides final and reasonable results for both small and large values of the coupling constant and is able to handle even super singular potentials for which each term of the perturbation theory is infinite and the perturbation expansion does not exist. The choice of zero iteration is based on general features of solutions near the boundaries. In order to estimate the accuracy of the QLM solutions, the exact numerical solutions were found as well. The first QLM iterate given by analytic expression allows to estimate analytically the role of different parameters and the influence of their variation on different characteristics of the relevant quantum systems.     

强阻尼广义sine-Gordon方程特征问题的变分迭代法

研究了在数学、力学中广泛出现的一类非线性强阻尼广义sine-Gordon扰动微分方程问题. 首先, 引入行波变换, 求出退化方程的精确解. 再构造一个泛函, 创建了一个变分迭代算法, 最后, 求出原非线性强阻尼广义sine-Gordon扰动微分方程问题的近似行波解析解. 用变分迭代法可得到的各次近似解, 具有便于求解、精度高等特点. 求得的近似解析解弥补了单纯用数值方法的模拟解的不足.