应用数学和力学

2003 Vol. 24, No. 6

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Safety Margin Criterion of Nonlinear Unbalance Elastic Axle System

CHEN Yu-shu, LI Yin-shan, XUE Yu-sheng

2003, 24(6): 551-558.

Abstract(1931) PDF(651)

Abstract:
The safety margin criterion of nonlinear dynamic question of an elastic rotor system are given. A series of observing spaces were separated from integral space by resolving and polymerizing method. The stable-state trajectory of high dimensional nonlinear dynamic systems was got within integral space. According to international standard of rotor system vibration, energy limits of safety criterion were determined. The safety margin was calculated within a series of observing spaces by comparative positive-area criterion(CPAC) method. A quantitative example calculating safety margin for unbalance elastic rotor system was given by CPAC. The safety margin criterion proposed includes the calculation of current stability margin in engineering. This criterion is an effective method to solve quantitative calculation question of safety margin and stability margin for nonlinear dynamic systems.

Research on the Flow Stability in a Cylindrical Particle Two-Phase Boundary Layer

LIN Jian-zhong, NIE De-ming

2003, 24(6): 559-564.

Abstract(2274) PDF(529)

Abstract:
Based on the momentum and constitutive equations, the modified Orr-Sommerfeld equation describing the flow stability in a cylindrical particle two-phase flow was derived. For a cylindrical particle two-phase boundary layer, the neutral stability curves and critical Reynolds number were given with numerical simulation. The results show that the cylindrical particles have a suppression effect on the flow instability, the larger the particle volume fraction and the particle aspect-ratio are, the more obvious the suppression effect is.

Bifurcation of a Shaft With Hysteretic-Type Internal Friction Force of Material

DING Qian, CHEN Yu-shu

2003, 24(6): 565-571.

Abstract(2242) PDF(718)

Abstract:
The bifurcation of a shaft with hysteretic internal friction of material was analysed. Firstly, the differential motion equation in complex form was deduced using Hamilton principle. Then averaged equations in primary resonances were obtained using the averaging method. The stability of steady-state responses was also determined. Lastly, the bifurcations of both normal motion (synchronous whirl) and self-excited motion(non-synchronous whirl)were investigated using the method of singularity. The study shows that by a rather large disturbance, the stability of the shaft can be lost through Hopf bifurcation in case the stability condition is not satisfied. The averaged self-excited response appears as a type of unsymmetrical bifurcation with high orders of co-dimension. The second Hopf bifurcation, which corresponds to double amplitude-modulated response, can occur as the speed of the shaft increases. Balancing the shaft carefully to decrease its unbalance level and increasing the external damping are two effective methods to avoid the appearance of the self-sustained whirl induced by the hysteretic internal friction of material.

On Reich’s Open Question

ZHANG Shi-sheng

2003, 24(6): 572-578.

Abstract(1955) PDF(732)

Abstract:
Under more general form and more general conditions an affirmative answer to Reich's open question is given. The results presented also extend and improve some recent results of Reich, Shioji, Takahashi and Wittmann.

Simplest Differential Equation of Stock Price,Its Solution and Relation to Assumption of Black-Scholes Model

YUN Tian-quan, LEI Guang-long

2003, 24(6): 579-582.

Abstract(2020) PDF(1071)

Abstract:
Two kinds of mathematical expressions of stock price, one of which based on certain description is the solution of the simplest differential equation(S. D. E.) obtained by method similar to that used in solid mechanics, the other based on uncertain description(i. e., the statistic theory)is the assumption of Black-Scholes s model(A. B-S. M.) in which the density function of stock price obeys logarithmic normal distribution, can be shown to be completely the same under certain equivalence relation of coefficients. The range of the solution of S. D. E. has been shown to be suited only for normal cases (no profit, or lost profit news, etc.) of stock market, so the same range is suited for A. B-S. M. as well.

Maximal Elements for G_B-Majorized Mappings in Product G-Convex Spaces and Applications(Ⅰ)

DING Xie-ping

2003, 24(6): 583-594.

Abstract(2053) PDF(712)

Abstract:
A new family of set-valued mappings from a topological space into generalized convex spaces was introduced and studied. By using the continuous partition of unity theorem and Brouwer fixed point theorem, several existence theorems of maximal elements for the family of set-valued mappings were proved under noncompact setting of product generalized convex spaces. These theorems improve, unify and generalize many important results in recent literature.

The Existence of Time Periodic Solutions for a Damped Generalized Coupled Nonlinear Wave Equations

FANG Shao-mei, GUO Bo-ling

2003, 24(6): 595-604.

Abstract(2234) PDF(862)

Abstract:
The time periodic solution problem of damped generalized coupled nonlinear wave equations with periodic boundary condition was studied. By using the Galerkin method to construct the approximating sequence of time periodic solutions, a priori estimate and Laray-Schauder fixed point theorem to prove the convergence of the approximate solutions, the existence of time periodic solutions for a damped generalized coupled nonlinear wave equations can be obtained.

Fracture Calculation of Bending Plates by Boundary Collocation Method

WANG Yuan-han, WU You-lun, YU Fei

2003, 24(6): 605-610.

Abstract(2221) PDF(782)

Abstract:
Fracture of Kirchhoff plates is analyzed by the theory of complex variables and boundary collocation method. The deflections, moments and shearing forces of the plates are assumed to be the functions of complex variables. The functions can satisfy a series of basic equations and governing conditions, such as the equilibrium equations in the domain, the boundary conditions on the crack surfaces and stress singularity at the crack tips. Thus, it is only necessary to consider the boundary conditions on the external boundaries of the plate, which can be approximately satisfied by the collocation method and least square technique. Different boundary conditions and loading cases of the cracked plates are analyzed and calculated. Compared to other methods, the numerical examples show that the present method has many advantages such as good accuracy and less computer time. This is an effective semi-analytical and semi-numerical method.

Propagation of Electromagnetic Wave in the Three Phases Soil Media

CHEN Yun-min, BIAN Xue-cheng, CHEN Ren-peng, LIANG Zhi-gang

2003, 24(6): 611-618.

Abstract(2130) PDF(736)

Abstract:
The fundamental parameters such as dielectric permittivity and magnetic permeability are required to solve the propagation of electromagnetic wave(EM Wave) in the soil. Based on Maxwell equations, the equivalent model is proposed to calculate the dielectric permittivity of mixed soil. The results of calculation fit the test data well and will provide solid foundation for the application of EM wave in the soil moisture testing, CT analyzing of soil and the inspecting of geoenvironment.

A Variation Method for Acoustic Wave Imaging of Two Dimensional Targets

FENG Wen-jie, ZOU Zhen-zhu

2003, 24(6): 619-623.

Abstract(2267) PDF(563)

Abstract:
A new way of acoustic wave imaging was investigated. By using the Green function theory a system of integral equations, which linked wave number perturbation function with wave field, was firstly deduced. By taking variation on these integral equations an inversion equation, which reflected the relation between the little variation of wave number perturbation function and that of scattering field, was further obtained. Finally, the perturbation functions of some identical targets were reconstructed, and some properties of the novel method including converging speed, inversion accuracy and the abilities to resist random noise and identify complex targets were discussed. Results of numerical simulation show that the method based on the variation principle has great theoretical and applicable value to quantitative nondestructive evaluation.

The Controllability of Delay Degenerate Control Systems With Independent Subsystems

JIANG Wei

2003, 24(6): 624-630.

Abstract(2160) PDF(783)

Abstract:
The controllability of delay degenerate differential control systems is discussed. Firstly, delay degenerate differential control system was transformed to be canonical form, and the connected terms were gotten rid of, had delay degenerate differential control systems with independent subsystems. For the general delay degenerate differnetial control systems, it was gotten that the necessary and sufficient condition of that they are controllable is that their reachable set is equal to the whole space. For the delay degenerate differential control systems with independent subsystems, it was gotten that the necessary and sufficient conditions of that they are controllable are that their reachable sets are equal to their corresponding subspaces. Then some algebra criteria were gotten. Finally, an example was given to illustrate the main results.

Soliton-Like Thermal Source Forcing and Singular Response of Atmosphere and Oceans to It

ZHANG Ren, SHA Wen-yu, JIANG Guo-rong, WANG Ji-guang

2003, 24(6): 631-636.

Abstract(2293) PDF(618)

Abstract:
Non-linear dynamic study is undertaken of the response of atmospheric and oceanic flow fields to local thermal source forcing in the context of a generalized geophysical fluid dynamic barotropic quasi-geostrophic model, discovering a good relation between thermal disturbance and flow field response to it, both having similar modes, and that the soliton-like responding field is a great deal larger in extent than the analogous-form forcing field, which implies that a "narrow" thermal disturbance can excite a "wide" response field, in some cases the particular structure of a thermal source may give rise to singular response of atmospheric and oceanic flow fields, thus displaying their abnormalities(for example the blocking situation in the atmosphere), the atmospheric and oceanic stream fields at mid-high latitudes respond to thermal forcing in a much more pronounced manner compared to those at low latitudes. The said research results that is in agreement with studies from mid-low latitude atmospheric experiments and observations and can be used to partially interpret the circulation singularity due to heat source anomaly on a local basis in the context of earch fluid flows.

Crack Problem Under Shear Loading in Cubic Quasicrystal

ZHOU Wang-min, FAN Tian-you, YIN Shu-yuan

2003, 24(6): 637-643.

Abstract(2161) PDF(535)

Abstract:
The axisymmetric elasticity problem of cubic quasicrystal is reduced to a single higher-order partial differential equation by introducing a displacement function. Based on the work, the analytic solutions of elastic field of cubic quasicrystal with a penny-shaped crack under the shear loading are found, and the stress intensity factor and strain energy release rate are determined.

The Dynamic Model of Crop Growth System and Numerical Simulation of Crop Growth Process Under the Multi-Environment External Force Action

LI Zi-zhen, WANG Wan-xiong, XU Cai-lin

2003, 24(6): 644-652.

Abstract(1998) PDF(845)

Abstract:
According to the biomechanic theory and method, the dynamic mechanism of crop growth under the external force action of multi-environment factors (light, temperature, soil and nutrients etc.) was comprehensively explored. Continuous-time Markov(CTM) approach was adopted to build the dynamic model of the crop growth system and the simulated numerical method. The growth rate responses to the variation of the external force and the change of biomass saturation value were studied. The crop grew in the semiarid area was taken as an example to carry out the numerical simulation analysis, therefore the results provide the quantity basis for the field management. Comparing the dynamic model with the other plant growth model, the superiority of the former is that it displays multi-dimension of resource utilization by means of combining macroscopic with microcosmic and reveals the process of resource transition. The simulation method of crop growth system is advanced and manipulated. A real simulation result is well identical with field observational results.

Global Linear and Quadratic One-Step Smoothing Newton Method for Vertical Linear Complementarity Problems

ZHANG Li-ping, GAO Zi-you

2003, 24(6): 653-660.

Abstract(1823) PDF(689)

Abstract:
A one-step smoothing Newton method is proposed for solving the vertical linear complementarity problem based on the so-called aggregation function. The proposed algorithm has the following good features: (i) it solves only one linear system of equations and does only one line search at each iteration; (ⅱ) it is well-defined for the vertical linear complementarity problem with vertical block P₀ matrix and any accumulation point of iteration sequence is its solution. Moreover, the iteration sequence is bounded for the vertical linear complementarity problem with vertical block P₀+R₀ matrix; (ⅲ) it has both global linear and local quadratic convergence without strict complementarity. Many existing smoothing Newton methods do not have the property(ⅲ).