本文用一维Logistic映射代替平均场,全局耦合构造了一类三维混沌映射,并利用坐标旋转变换对耦合系统的复杂动力学和隐藏动力学分析和识别.通过展示耦合系统的吸引子、分岔图及共存吸引域在坐标旋转前后的对比,我们发现原耦合系统经由坐标旋转后,其吸引子及其共存吸引域在旋转平面上的投影均具有很好的三角对称性.此外,我们在双参数稳态分布图上增加了余维1与余维2分岔线,提供了一种在参数平面域内发现可能存在的吸引子共存的方法.双参数稳态分布虽然与坐标旋转变换无关,但其正交变换的特性可大幅提高运算效率.

The nonstationary probability densities of system response of a single-degree-of-freedom system with lightly nonlinear damping and strongly nonlinear stiffness subject to modulated white noise excitation are studied. Using the stochastic averaging method based on the generalized harmonic functions, the averaged Fokker-Planck-Kolmogorov equation governing the nonstationary probability density of the amplitude is derived. The solution of the equation is approximated by the series expansion in terms of a set of properly selected basis functions with time-dependent coefficients. According to the Galerkin method, the time-dependent coefficients can be solved from a set of first-order linear differential equations. Then, the semi-analytical formulae of the nonstationary probability density of the amplitude response as well as the nonstationary probability density of the state response and the statistic moments of the amplitude response can be obtained. A van der Pol-Duffing oscillator subject to modulated white noise is given as an example to illustrate the proposed procedures. The effects of the system parameters, such as the linear damping coefficient and the nonlinear stiffness coefficient, on the system response are discussed.

To estimate the power flow transmission from the vibrating internal combustion engines to the complex elastic foundation accurately, a general analytical model is presented by extending the classical one. In the proposed model, the Mindlin plate, instead of the classical thin plate based on Kirchihnff's hypothesis, is used to model the elastic isolation foundations. Substructure technique and variational principle are employed to derive the governing equation and solve the power flow transmitted into the thick or orthotropic beam foundations with different configurations. Comparative study on the transmission characteristics of the power flow of the general isolation systems with the classical cases is conducted. Numerical simulation shows that the presented model is valid and it can obtain more accurate power flow solutions than the classical one.

引入谐波平衡近似解的符号运算与同伦延拓法,获得了强Van der Pol振子的解析近似极限环与稳定响应.为提高其近似精度,给出了谐波解的修正算法,并与数值模拟比较.结果表明,该算法非常精确可靠,是研究强非线性动力学系统的有效分析方法.

This research aims to test the housing price dynamics when considering heterogeneous boundedly rational expectations such as naive expectation, adaptive expectation and biased belief. The housing market is investigated as an evolutionary system with heterogeneous and competing expectations. The results show that the dynamics of the expected housing price varies substantially when heterogeneous expectations are considered together with some other endogenous factors. Simulation results explain some stylized phenomena such as equilibrium or oscillation, convergence or divergence, and over-shooting or under-shooting. Furthermore, the results suggest that variation of the proportion of groups of agents is basically dependent on the selected strategies. It also indicates that control policies should be chosen carefully in consistence with a unique real estate market during a unique period since certain parameter portfolio may increase or suppress oscillation.

China’s first interest rate hike during the last decade, aiming to cool down the seemingly overheated real estate market, had aroused more caution on housing market. This paper aims to analyze the housing price dynamics after an unanticipated economic shock, which was believed to have similar properties with the backward-looking expectation models. The analysis of the housing price dynamics is based on the cobweb model with a simple user cost affected demand and a stock-flow supply assumption. Several nth-order delay rational difference equations are set up to illustrate the properties of housing dynamics phenomena, such as the equilibrium or oscillations, overshoot or undershoot and convergent or divergent, for a kind of heterogeneous backward-looking expectation models. The results show that demand elasticity is less than supply elasticity is not a necessary condition for the occurrence of oscillation. The housing price dynamics will vary substantially with the heterogeneous backward-looking expectation assumption and some other endogenous factors.

A health monitoring and damage detection method using the time-frequency characteristic of the wavelet transform was presented based on the results of free vibration and random vibration tests of a system consisting of a cantilever plate and several springs.During vibration of the system,burning out the string connecting the spring to the plate resulted sudden damage to the structure.Discrete wavelet transform was used to recognize the damage at the moment it occurs.Subsequently,the shifts of frequencies and damping ratios were identified by continuous wavelet transform to study the amplitude of stiffness loss of the structure and make sure that the sudden change or impulse in the signal from the discrete wavelet transform was resulted from the damage but not the impulse noise.Test results showed that the proposed method is effective for health monitoring and damage detection of the considered structure.For the random forced vibration,the random decrement technique can be performed on the original signal to obtain the free decaying responses,and then the continuous wavelet transform can be applied to identify the structural parameters.The present method is free of modelling error and can be used for on-line health monitoring.

This paper applies a Hamiltonian method to study analytically the stress distributions of orthotropic two-dimensional elasticity in (x, z) plane for arbitrary boundary conditions without beam assumptions. It is a method of separable variables for partial differential equations using displacements and their conjugate stresses as unknowns. Since coordinates (x, z) can not be easily separated, an alternative symplectic expansion is used. Similar to the Hamiltonian formulation in classical dynamics, we treat the x coordinate as time variable so that z becomes the only independent coordinate in the Hamiltonian matrix differential operator. The exponential of the Hamiltonian matrix is symplectic. There are homogenous solutions with constants to be determined by the boundary conditions and particular integrals satisfying the loading conditions. The homogenous solutions consist of the eigen-solutions of the derogatory zero eigenvalues (zero eigen-solutions) and that of the well-behaved nonzero eigenvalues (nonzero eigen-solutions). The Jordan chains at zero eigenvalues give the classical Saint-Venant solutions associated with averaged global behaviors such as rigid-body translation, rigid-body rotation or bending. On the other hand, the nonzero eigen-solutions describe the exponentially decaying localized solutions usually ignored by Saint-Venant’s principle. Completed numerical examples are newly given to compare with established results.

The basic equations of antiplane crack problem for orthotropic media were transformed by introducing the parameter in terms of those of isotropic media.The analogy relationship between the displacements and stresses of an anti-plane crack in an isotropic and an orthotropic plate is very simple so that the inverse transform is convenient.For explaining this analogy method,the William′s general solutions to mode Ⅲ two-dimensional crack problem of orthotropic plates with internal and edge cracks are derived,respectively.This analogy method is efficient for solving anti-plane crack problems of orthotropic plates.

The exterior acoustic problems are calculated by fractal finite element method. The infinite domain is divided into two parts by an artificial boundary,the conventional finite element method is adopted within the artificial boundary, and the fractal finite element method is employed in the outer region beyond the artificial boundary. The computer time can be reduced greatly by the fractal finite element method, because of the facts that the relationships of element matrix between the adjacent fractal layers are very simple and the node degrees of freedom are transformed into a set of general coordinates by using the global interpolation functions,which satisfy automatically Sommerfeld's radiation conditions. The numerical results show that this method is effective in solving exterior acoustics problems.

In this paper,the Young's modulus of single-walled carbon nanotubes(SWCNT) are investigated by using energy approach.First,based on the principle of molecular mechanics,the total system potential energy of SWCNT,subjected to axial loading only,is obtained.The computing formulae of the Young's moduli of SWCNT are obtained by comparing the total molecular potential energy of a SWCNT with the strain energy of a corresponding cylindrical thin shell.Computations of the Young's modulus reveal that present results agree well with existing results.

The shear moduli of single-walled carbon nanotubes(SWCNT) are investigated by using energy theory.Based on the principle of molecular mechanics,the total system potential energy of SWCNT,subjected to torsion only,is obtained.The computing formulae of the shear moduli of SWCNT are obtained by comparing the total molecular potential energy of a SWCNT with the strain energy of a corresponding thin cylinder.Computations of the shear moduli reveal that present results agree well with existing results,so the validity of these computing formulae is verified.

分析了发生横、纵振动的大运动梁的非线性动力响应,引用梁横向和纵向振动的精确模态描述变形场,利用拉格朗日方程建立了运动梁刚柔耦合非线性动力学方程.该非线性方程自动计及了动力刚化的影响.基于Newmark直接积分法和Newton-Raphson迭代法,给出了求解该非线性方程的数值方法.仿真算例证明了本文方法的正确性和有效性.

The stress intensity factor(SIF) of plane crack on orthotropic plate equal-parameter was evaluated by using a fractal two-level finite element method(F2LFEM).The general solution of an orthotropic crack problem was obtained by assimilating the problem with isotropic crack problem,and employed as the global interpolation function in F2LFEM.The plane crack plate was discredited by F2LFEM,reducing degrees of freedom to be determined greatly.The numerical exampled showed that the method was efficient and accurate in solving the stress intensity factor by using finite coarse mesh and simple interpolating element.

The nonlinear oscillation of damped mechanical systems is applied to investigate the electrodynamics of long Josephson junction. The latter has been shown as equivalent to the weakly damped mechanical nonlinear oscillation system. This paper applies the combination of linearized governing equation and harmonic balancing method (LHBM) to yield the dispersion relation. The application of the method of averaging is then employed to achieve very accurate transient responses of the weekly dissipative system.The proposed approach herein not only accounts for energy dissipation but also presents the relationship of simple linear algebraic equations instead of lengthy and complicated analytical approximate solutions.

This paper analyzes the characteristics of utilizing shape memory effect (SME) of shape memory alloy (SMA) in improving the low velocity impact resistance performance of composite plate by using finite element method. The constitutive relation for SMA hybrid composite plates is presented. The analytic model of finite element for SMA composite plate subjected to low velocity impact is established. The modified Hertz's contact law is used to determine the impact contact force. The computing procedures for solving the finite element equation using Newmark direct integration method are given. The numerical modelling results show that the SMA can effectively improve the low velocity impact resistance performance of composite plate.

The mixed method of boundary element and fractal finite element was adopted to solve the cracked structure-acoustic coupling problems. The fractal two-level finite element method was employed to model the cracked structure, which can reduce the degree of freedoms greatly, and the boundary element method was used to evaluate the exterior acoustic field which could satisfy automatically infinity radiation condition. Numerical examples show that the resonance frequency is lower with the depth of the crack increase, and the effect on the acoustic field by the crack is particularly pronounced in vicinity of the crack tip.

通过采用状态变量把具有stiff性质的二阶柔性操纵器动力学方程降为一阶,基于向前和向后Euler积分法,提出了一种非线性数值积分方法,该方法具有A0稳定性及无限稳定性.引用Blajer提出的修正方法对数值积分过程中约束方程的破坏进行违约修正.对平面双连杆柔性操纵器进行的仿真计算表明了本文方法的有效性.

The acoustical scattering by a cracked elastic structure is studied. The mixed method of boundary element and fractal finite element is adopted to solve the cracked structure-acoustic coupling problem. The fractal two-level finite element method is employed for the cracked structure, which can reduce the degree of freedoms (DOFs) greatly, and the boundary element method is used for the exterior acoustic field which can automatically satisfy Sommerfeld's radiation condition. Numerical examples show that the resonance frequency is lower with the crack's depth increase, and that the effect on the acoustical field by the crack is particularly pronounced in the vicinity of the crack tip. This mixed method of boundary element and finite element is effective in solving the scattering problem by a cracked structure.

提出了两种分别适用于板的面内振动和圆环板弯曲振动模态分析的Fourier级数形式的插值形函数.数值计算结果表明:对于结构振动模态分析,Fourier p_单元法能够有效地提高计算精度及收敛速度,特别是对于较高阶的模态效果更为明显.