夹层椭圆形板的1/3亚谐解

研究了夹层椭圆形板的非线性强迫振动问题。在以5个位移分量表示的夹层椭圆板的运动方程的基础上,导出了相应的非线性动力方程。提出一类强非线性动力系统的叠加-叠代谐波平衡法。将描述动力系统的二阶常微分方程,化为基本解为未知函数的基本微分方程和派生解为未知函数的增量微分方程。通过叠加-叠代谐波平衡法得出了椭圆板的1/3亚谐解。同时,对叠加-叠代谐波平衡法和数值积分法的精度进行了比较。并且讨论了1/3亚谐解的渐近稳定性。     

杜芬方程的1/3纯亚谐解及过渡过程的分形特征研究

通过谐波平衡法和数值积分法研究了杜芬方程的1／3纯亚谐解．提出假设解，找出了亚谐频域，并对参数变化的过渡过程的敏感性和初始值扰动的过渡过程进行了研究．考察了亚谐响应幅值系数对阻尼的敏感性及亚谐振动谐波成分的渐近稳态性．此外，运用广义分形理论对杜芬方程纯亚谐解过渡过程进行了分析．分析表明，广义维数的敏感维数能清楚地描述杜芬方程纯亚谐解过渡过程特征；并对改变初始扰动、阻尼系数、激励幅值情况下，其两个不同频域的杜芬方程纯亚谐解过渡过程的不同分形特性显现出敏感性．     

多自由度强非线性颤振分析的增量谐波平衡法

对多个自由度上含有强非线性项系统的颤振问题,推广应用增量谐波平衡法进行分析．考虑带有强非线性立方平移和俯仰刚度项的二元机翼颤振方程,首先将方程用矩阵形式表示,然后把振动过程分解成为振动瞬态的持续增量过程,再采用振幅作为控制参数应用谐波平衡法,以这种推广的增量谐波平衡法求得方程解的表达式,并由此分析系统的分岔现象、极限环颤振现象和谐波项数的取值问题,最后用龙格-库塔数值方法进行验算,结果表明:分析多个自由度的强非线性颤振,增量谐波平衡法是精确有效的．     

谐和与随机噪声联合作用下的粘弹系统

研究了粘弹系统在谐和与随机噪声联合作用下的响应和稳定性问题.用谐波平衡法和随机平均法分析了系统在确定性谐和激励和随机激励联合作用下的响应,讨论了粘弹项、随机扰动项对系统响应的影响.结果表明,在一定条件下,系统具有两个均方响应值和跳跃现象.数值模拟表明,谐波平衡法与随机平均法相结合的研究方法是有效的.     

二端面转轴相对转动非线性动力学系统的主共振与次共振解

本文研究在简谐激励力作用下二端面弹性转轴相对转动的主共振、超谐波共振和亚谐波共振.用平均法研究了系统的主共振,得到了系统的渐进稳态周期解,采用多尺度法求得了系统的3次超谐波共振解和1/3次亚谐波共振解.     

基于时域配点法的充液储箱系统多模态方程求解

采用时域配点法研究了充液储箱系统多模态方程的稳态周期解.在模型求解过程中,利用牛顿迭代法求解了配点法得到的非线性代数方程组,而牛顿迭代的初值来自谐波平衡法求解得到的低阶谐波近似.数值仿真结果验证了时域配点法的有效性,并验证以二倍激励频率为基频的第二模态的假设形式更为有效.最终通过对比谐波系数数量级提出一种更为简洁有效的模态表达形式.     

非线性颤振系统中既是超临界又是亚临界的Hopf分岔点研究

研究了二元机翼非线性颤振系统的Hopf分岔．应用中心流形定理将系统降维,并利用复数正规形方法得到了以气流速度为分岔参数的分岔方程．研究发现,分岔方程中一个系数不含分岔参数的一次幂,故使得分岔具有超临界和亚临界双重性质．用等效线性化法和增量谐波平衡法验证了所得结果．     

无限域上一维热传导方程的Gauss-Laguerre数值解

针对无限域上一维热传导方程的解析解为反常积分形式,直接计算往往比较困难.首先采用Fourier变换给出问题解析解,其次结合解析解的形式和无限域上Gauss型数值积分法精度高的优点,将半无限域上的一维热传导方程问题利用Gauss-Laguerre数值积分计算数值解,对无限域上的一维热传导方程的解析解转化为半无限域上的形式后用Gauss-Laguerre数值积分计算.实验结果表明,本文给出的数值解方法具有很高的精度.     

L~1空间中第二类Fredholm积分方程算法优越性分析

在L¹空间中对第二类Fredholm积分方程提出了一种好似使其近似解和精确解的误差达到最小的方法,对离散算法与数值积分法进行对比,并给出了误差估计,用实际的例子来进一步探究.     

数值积分法研究近况综述

在我们的文章[1]中,曾着重介绍了近代高维数值积分方法的研究概貌.本文将进一步综述各种数值积分法研究的最近情况.为论述方便起见,本文共分这样四个部份:(一)略论一般概况;(二)关于一维数值积分方法;(三)关于高维数值积分方法;(四)数值积分中的几个特殊问题.     

Bifurcation and chaos in escape equation model by incremental harmonic balancing

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.     

非线性弹性梁在谐波激励下的次谐和超次谐响应

本文研究受横向周期载荷作用的梁的动力响应,梁的本构关系具有三次非线性项·　轴向载荷作用下已屈曲的梁受到横向激励后,谐波是不稳定的,将分叉出次谐波、超次谐波,以Ｍｅｌｎｉｋｏｖ法确定了次谐轨道、超次谐轨道产生的条件·     

Bifurcating periodic solution for a class of first-order nonlinear delay differential equation after Hopf bifurcation

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.     

Scattering of surface gravity waves over a pair of trenches

A numerical model based on boundary element method is developed to study the scattering of surface gravity waves over a pair of trenches of varied configurations under the assumption of small amplitude water wave theory. Both the cases of symmetric and asymmetric trenches are considered in the present study. The accuracy of the numerical results is validated by comparing the reflection and transmission coefficients with energy identity, and the known results associated with single trench available in the literature. The study reveals that wave reflection decreases in an oscillatory manner with an increase in trench width. Moreover, Bragg resonance in wave reflection is observed for wave number corresponding to waves in shallow and intermediate depths in the case of a pair of trenches. Further, Bragg reflection increases with an increase in the number of trenches. In the case of multiple trenches, subharmonic peaks in Bragg reflection are depicted and the number of subharmonic peaks between two harmonic peaks is found to be two less than the number of trenches. However, for triangular trenches, the occurrence of the subharmonic peak is invariant of the number of trenches and the same vanishes for larger trench depth. Irrespective of trench configurations, wave reflection follows certain uniform oscillatory pattern with an increase in the gap between the trenches in case of deep water.     

Nonlinear transversely vibrating beams by the improved energy balance method and the global residue harmonic balance method

Mathematical modeling of many engineering systems such as beam structures often leads to nonlinear ordinary or partial differential equations. Nonlinear vibration analysis of the beam structures is very important in mechanical and industrial applications. This paper presents the high order frequency-amplitude relationship for nonlinear transversely vibrating beams with odd and even nonlinearities using the improved energy balance method and the global residue harmonic balance method. The accuracy of the energy balance method is improved based on combining features of collocation method and Galerkin–Petrov method, and an improved harmonic balance method is proposed which is called the global residue harmonic balance method. Unlike other harmonic balance methods, all the former global residual errors are introduced in the present approximation to improve the accuracy. Finally, preciseness of the present analytic procedures is evaluated in contrast with numerical calculations methods, giving excellent results.     

Solution of transient markovian state probabilities and rate of approach to equilibrium states

The differential equations for transient state probabilities for Markovian processes are examined to derive the rate of convergence of transient states to equilibrium states. There is an acute need to solve the balance equations for large states, particularly for handling computer per- formance modeling with a network of queues that do not satisfy product form solutions or can- not be cast into the forms convenient for mean value analysis. The rate of convergence to equilibrium states is derived for irreducible aperiodic homogeneous Markov chains on the basis of a geometrical interpretation. A numerical integration method with dynamic step-size adjustments is applied and compared against the power method of Wallace and Rosenberg.