应用数学和力学

2005 Vol. 26, No. 9

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Polynomial Solutions to Piezoelectric Beams(Ⅰ)——Several Exact Solutions

DING Hao-jiang, JIANG Ai-min

2005, 26(9): 1009-1015.

Abstract(3037) PDF(825)

Abstract:
For the orthotropic piezoelectric plane problem,a series of piezoelectric beams is solved and the corresponding exact solutions are obtained with the trial-and-error method on the basis of the general solution in the case of three distinct eigenvalues,in which all displacements,electrical potential,stresses and electrical displacements are expressed by three displacement functions in terms of harmonic polynomials.These problems are rectangular beams having rigid body displacements andidentical electrical potential,rectangular beams under uniform tension and electric displacement as well as pure shearing and pure bending,beams of two free ends subjected to uniform electrical potential on the upper and lower surfaces.

Polynomial Solutions to Piezoelectric Beams(Ⅱ)——Analytical Solutions to Typical Problems

DING Hao-jiang, JIANG Ai-min

2005, 26(9): 1016-1021.

Abstract(2377) PDF(765)

Abstract:
For the orthotropic piezoelectric plane problem,a series of piezoelectric beams is solved and the corresponding analytical solutions are obtained with the trial-and-error method on the basis of the general solution in the case of three distinct eigenvalues,in which all displacements,electrical potential,stresses and electrical displacements are expressed by three displacement functions in terms of harmonic polynomials.These problems are cantilever beam with cross force and point charge at free end,cantilever beam and simply-supported beam subjected to uniform loads on the upper and lower surfaces,and cantilever beam subjected to linear electrical potential.

Research on Reliability Growth for Synchronously Developed Multi-Systems

MA Xiao-ning, LÜ Zhen-zhou, YUE Zhu-feng

2005, 26(9): 1022-1026.

Abstract(2349) PDF(649)

Abstract:
An advanced reliability growth model,exponential model,was presented to estimate the model parameters for multi-systems,which was synchronously tested,synchronously censored,and synchronously improved.In the presented method,the data during the reliability growth process were taken into consideration sufficiently,including the failure numbers,safety numbers and failure times at each censored time.If the multi-systems were synchronously improved many times,and the reliability growth of each system fitted AMSAA model,the failure time of each system could be considered rationally as an exponential distribution between two adjoining censored times.The nonparametric method was employed to obtain the reliability at each censored time of the synchronous multi-systems.The point estimations of the model parameters,a and b,were given by the least square method.The confidence interval for the parameter b was given as well.An engineering illustration was used to compare the result of the presented method with those of the available models.The result shows that the presented exponential growth model fits AMSAA-BISE model rather well,and two models are suitable to estimate the reliability growth for the synchronously developed multi-systems.

Theoretical Study of Three-Dimensional Numerical Manifold Method

LUO Shao-ming, ZHANG Xiang-wei, LÜ Wen-ge, JIANG Dong-ru

2005, 26(9): 1027-1032.

Abstract(2666) PDF(787)

Abstract:
The three-dimensional Numerical Manifold Method(NMM)is studied on the basis of two-dimensional numerical manifold method.The three-dimensional cover displacement function is studied.The mechanical analysis and Hammer integral method of three-dimensional numerical manifold method are put forward,the stiffness matrix of three-dimensional manifold element are derived and the dissection rules are given.The theoretical system and numerical realizing method of three-dimensional numerical manifold method are systematically studied.As an example,the cantilever with load on the end is calculated,the results show that the precision and efficiency is agreeable.

Partial Synchronization Between Different Systems

CHEN Jun, LIU Zeng-rong

2005, 26(9): 1033-1037.

Abstract(2171) PDF(605)

Abstract:
A new method for partial synchronization between different systems was obtained.The definition of partial synchronization under which the problem works is given.The stability of the method is analyzed by the Liapunov function method and the condition of choosing the control term is derived.The reliability of this method is proved by some numerical examples,in which the dynamical behaviors of the synchronized systems are observed and it is found that whatever state the response system is partial synchronization can be always achieved by adding some proper control term.

Quantitative Methodology for the Stability Analysis of Nonlinear Rotor Systems

ZHENG Hui-ping, XUE Yu-sheng, CHEN Yu-shu

2005, 26(9): 1038-1044.

Abstract(2534) PDF(779)

Abstract:
Rotor-bearings systems applied widely in industry are nonlinear dynamic systems of multi-degree-of-freedom.Modern concepts on design and maintenance call for quantitative stability analysis. Using trajectory based stability-preserving,dimensional-reduction,a quantitative stability analysis method for rotor systems is presented.At first,a n-dimensional nonlinear non-autonomous rotor system is decoupled into n subsystems after numerical integration.Each of them has only one-degree-of-freedom and contains time-varying parameters to represent all other state variables.In this way,n dimensional trajectory is mapped into a set of one-dimensional trajectories.Dynamic central point (DCP)of a subsystem is then defined on the extended phase plane,namely force-position plane. Characteristics of curves on the extended phase plane and the DCP's kinetic energy difference sequence for general motion in rotor systems are studied.The corresponding stability margins of trajectory are evaluated quantitatively.By means of the margin and its sensitivity analysis,the critical parameters of the period doubling bifurcation and the Hopf bifurcation in a flexible rotor supported by two short journal bearings with nonlinear suspensionare determined.

Dynamic Interaction between Elastic Thick Circular Plate and Transversely Isotropic Saturated Soil Ground

HUANG Yi, WANG Xiao-gang

2005, 26(9): 1045-1054.

Abstract(2699) PDF(669)

Abstract:
A study of the dynamic interaction between foundation and the underlying soil has been presented in a recent paper based on the assumption of saturated ground and elastic circular plate excited by the axisymmetical harmonic source.However,the assumption may not always by valid.The work is extended to the case of a circular plate resting on transversely isotropic saturated soil and subjected to a non-axisymmetical harmonic force.The analysis is based on the theory of elastic wave in transversely isotropic saturated poroelastic media established.By the techuique of Fourier expansion and Hankel transform,the governing different equation for transversely isotropic saturated soil are easily solved and the cooresponding Hankel transformed stress and displacement solutions are obtained.Then,under the contact conditions,the problem leads to a pair of dual integral equations which describes the mixed boundary-value problem.Furthermore,the dual integral equations can be reduced to the Fredholm integral equations of the second kind and solved by numerical procedure.At the end,a numerical result is presented which indicates that on a certain frequency range,the displacement amplitude of the surface of the foundation is increased with the increase of the frequency of the exciting force,and decreased in vibration form with the increase of the distance.

Reliability Analysis for Implicit Limit State Equation

WANG Yan-ping, LÜ Zhen-zhou, YUE Zhu-feng

2005, 26(9): 1055-1060.

Abstract(2490) PDF(591)

Abstract:
In order to obtain the failure probability of the implicit limit state equation accurately,advanced mean value second order(AMVSO)method was presented,and advanced mean value(AMV) in conjunction with the response surface method(RSM)was also presented.The implementations were constructed on the basis of the advanced mean value first order(AMVFO)method and the RSM. The examples show that the accuracy of the AMVSO is higher than that of the AMVFO.The results of the AMV in conjunction with the RSM are not sensitive to the positions of the sampling points for determining the response surface equation,which illustrates the robustness of the presented method.

XIE Xin-yu, ZHANG Ji-fa, ZENG Guo-xi

2005, 26(9): 1061-1066.

Abstract(2590) PDF(801)

Abstract:
More general assumption than that in the classical one dimensional large strain consolidation theory is adopted and the exact analytical solution of nonlinear finite strain self-weight consolidation based on this assumption is obtained.By applying the same experimental data,the comparison of the solutions of linear and nonlinear finite strain theory,as well as the numerical calculating results from finite element method is presented.The results of the comparison show that the analytical solution obtained here takes on better agreement with practical cases than that of linear one,and they also show that,compared with the solutions from nonlinear theory,the settlement and the consolidation degree from linear theory are smaller.

Nonlinear Krylov Subspace Methods for Solving Nonsmooth Equations

MENG Ze-hong, ZHANG Jian-jun

2005, 26(9): 1067-1075.

Abstract(2540) PDF(871)

Abstract:
Newton-FOM algorithm and Newton-GMRES algorithm for solving nonsmooth equations are presented.It is proved that these Krylov subspace algorithms have locally quadratic convergence.Numerical experiments demonstrate the effectiveness of the algorithms.

Analysis of Dynamical Buckling and Post Buckling for Beams by Finite Segment Method

YIN Xue-gang, DU Si-yi, HU Ji-yun, DING Jian-ping

2005, 26(9): 1076-1082.

Abstract(2695) PDF(683)

Abstract:
Based on the multi-rigid body discretization model,namely finite segment model,a chain multi-rigid body-hinge-spring system model of a beam was presented,then a nonlinear parametrically exacted vibration equation of multi-degrees of freedom system was established using the coordination transformation method,and its resonance fields were derived by the restriction parameter method, that is the dynamical buckling analysis of the beam.Because the deformer of a beam isn't restricted by discrete model and dynamic equation,the post buckling analysis can be done in above math model.The numerical solutions of a few examples were obtained by direct integrated method,which shows that the mechanical and math model gotten is correct.

Mathematical Foundation of C₀ Complexity

SHEN En-hua, CAI Zhi-jie, GU Fan-ji

2005, 26(9): 1083-1090.

Abstract(2255) PDF(812)

Abstract:
For many continuous bio-medical signals with both strong nonlinearity and non-stationarity,two criterions were proposed for their complexity estimation:1)Only short data set is enough for robust estimation;2)No over-coarse graining preprocessing,such as transferring the original signal into a binary time series,is needed.C₀ complexity measure proposed by us previously is one of such measures.However,it lacked solid mathematical foundation and thus its use was limited.A modified version of this measure is proposed,and some important properties are proved rigorously.According to these properties,this measure can be considered as an index of randomness of time series in some senses,and thus also a quantitative index of complexity under the meaning of randomness finding complexity.Compared with other similar measures,this measure seems more suitable for estimating a large quantity of complexity measures for a given task,such as studying the dynamic variation of such measures in sliding windows of a long process,owing to its fast speed for estimation.

Mechanical Analysis of the Orbit Tracking Movement of Feed System in Large Spherical Radio Telescope

LIU Ming-zhi, SHEN Yu-ru, LIU Jun, NA Bai

2005, 26(9): 1091-1096.

Abstract(2302) PDF(576)

Abstract:
The curve equation and its mechanics analysis of suspended-cable under the condition of end load are given.Then on the basis of it,the mechanical analysis of suspended-cable system for large spherical radio telescope is studied,procedures of the control for the orbit tracking movement of the line feed in large spherical radio-telescope are given.The validity of the results mentioned above is confirmed by means of computer simulations.

Uniform Normal Structure and Solutions of Reich’s Open Question

ZENG Lu-chuan

2005, 26(9): 1097-1104.

Abstract(2269) PDF(618)

Abstract:
The open question raised by Reich is studied in a Banach space with uniform normal structure,whose norm is uniformly Gateaux differentiable.Under more suitable assumptions imposed on an asymptotically nonexpansive mapping,an affirmative answer to Reich's open question is given. The results presented extend and improve ZHANG Shi-sheng's recent ones in the following aspects: (i)ZHANG's stronger condition that the sequence of iterative parameters converges to zero is removed;(ii)ZHANG's stronger assumption that the asymptotically nonexpansive mapping has a fixed point is removed;(iii)ZHANG's stronger condition that the sequence generated by the Banach Contraction Principle is strongly convergent is also removed.Moreover,these also extend and improve the corresponding ones obtained previously by several authors including Reich,Shioji,Takahashi,Ueda and Wittmann.

Dynamic Propagation Problem on Dugdale Model of Mode Ⅲ Interface Crack

Lu Nian-chun, CHENG Yun-hong, TIAN Xiu-bo, CHENG Jin

2005, 26(9): 1105-1113.

Abstract(2596) PDF(611)

Abstract:
By the the ory of complex functions,dynamic propagation problem on Dugdale model of mode Ⅲ interface crack for nonlinear characters of materials was studied.The general expressions of analytical solutions are obtained by the methods of self-similar functions.The problems dealt with can be easily transformed into Riemann-Hilbert problems and their closed solutions are attained rather simple by this approach.After those solutions were utilized by superposition theorem,the solutions of arbitrarily complex problems could be obtained.

Aerodynamic Forces Acting on an Albatross Flying Above Sea-Waves

SHENG Qi-hu, WU De-ming, ZHANG Liang

2005, 26(9): 1114-1120.

Abstract(2408) PDF(844)

Abstract:
Numerical investigation on the dynamic mechanism has been made for an albatross to fly effectively in the region near sea surface.Emphasizing on the effect of the sea wave,the albatross is simplified as a two-dimensional airfoil and the panel method based on the potential flow theory is employed to calculate the wave effect on the aerodynamic forces.The numerical results have been presented for the states of:1)flying at different constant speeds with constant heights above sea level; 2)flying at different constant speeds with the combined oscillations of pitching and free heaving.The study on the cases shows that the albatross.flight efficiency depends on not only the speed and height of flight but also the wave amplitude and the wavelength.The albatross benefits by wave effect to get thrust,so as to reduce the resistance in the circumstances of rough sea.

Approximate Solution to a Class of Multivalued Nonlinear Mixed Variational Inclusions

ZHANG Cong-jun, SUN Min

2005, 26(9): 1121-1127.

Abstract(2475) PDF(618)

Abstract:
Concerned with the existence and convergence properties of approximate solution to multi-valued nonlinear mixed variational inclusion problem in a Hilbert space.We established the equivalence between the varitional inclusion and the general resolvent equations,obtained three iter ative algorithms,provided the convergence analysis of the algorithms.The results obtained improved and generalized a number of rsent results.

Optimal Error Bound in a Sobolev Space of Regularized Approximation Solutions for a Sideways Parabolic Equation

LI Hong-fang, FU Chu-li, XIONG Xiang-tuan

2005, 26(9): 1128-1134.

Abstract(2566) PDF(757)

Abstract:
The inverse heat conduction problem(IHCP)is severely ill-posed problem in the sense that the solution(if it exists)does not depend continuously on the data.But now the results on inverse heat conduction problem are mainly devoted to the standard inverse heat conduction problem.Some optimal error bounds in a sobolev spaceof regularized approximation solutions for a sideways parabolic equation,i.e.,a non-standard inverse heat conduction problem with convection term which appears in some applied subject are given.