EXTENSION OF POINCARE’S NONLINEAR OSCILLATION THEORY TO CONTINUUM MECHANICS (I)-BASIC THEORY AND METHOD

In this paper we extend Poincare’s nonlinear oscillation theory of discrete system to continuum mechanics. First we investigate the existence conditions of periodic solution for linear continuum system in the states of resonance and non-resonance. By applying the results of linear theory, we prove that the main conclusion of Poincare’s nonlinear oscillation theory can be extended to continuum mechanics. Besides, in this paper a new method is suggested to calculate the periodic solution in the states of both resonance and nonresonance by means of the direct perturbation of partial differential equation and weighted integration.     

EXTENSION AND APPLICATION OF NEWTON''''S METHOD IN NONLINEAR OSCILLATION THEORY

In this paper we suggest and prove that Newton's method may calculate the asymptotic analytic periodic solution of strong and weak nonlinear nonautonomous systems, so that a new analytic method is offered for studying strong and weak nonlinear oscillation systems. On the strength of the need of our method, we discuss the existence and calculation of the periodic solution of the second order nonhomogeneous linear periodic system. Besides, we investigate the application of Newton's method to quasi-linear systems. The periodic solution of Duffing equation is calculated by means of our method.     

EXTENSION OF POINCAR(?)’S NONLINEAR OSCILLATION THEORY TO CONTINUUM MECHANICS(Ⅱ)-APPLICATIONS

This is a continuation of[1].In [1] we suggested a method of direct perturbation ofpartial differential equation and weighted integration to calculate the periodic solution forcontinuum mechanics. In this paper, by using the above method we calculate the resonantand nonresonant periodic solutions of beam with fixed span and different boundaryconditions and the resonant periodic solution of square plate under the action ofconcentrated periodic load.Besides, the influences of non-principal mode upon periodicsolution and of static load upon amplitude-frequency curve are given.     

AN ANALYTICAL SOLUTION OF G. I. TAYLOR’S THEORY OF PLASTIC DEFORMATION IN IMPACT OF CYLINDRICAL PROJECTILES AND ITS IMPROVEMENT

The theory of plastic deformation in the impact of cylindrical projectiles on rigid targets was first introduced by G. I. Taylor(1948)^[1]. The importance of this theory lies in the fact that the dynamic yield strength of the materials can be determined from the measurement of the plastic deformation of flat-ended cylindrical projectiles. From the experimental results^[2] we find that the dynamic yield strength is independent of impact velocity, and that it is higher than the static yield strength in general, and several times higher than the static yield strength in certain cases. This gives an important foundation for the study of elastoplastic impact problems in general. However, it is well known that the complexity of differential equations in Taylor’s theory compelled us to use the troublesome numerical solution. In this paper, the analytical solution of all the equations in Taylor’s theory is given in parametrical form and the results are discussed in detail.In the latter part of this paper, the method of calculation of impulse of impact is improved by considering the processes of radial’ movement of materials. The analytical solution of the improved theory shows that it gives better agreement with the experimental results than that of original Taylor’s theory.     

AN ANALYTICAL SOLUTION OF G.I.TAYLOR’S THEORY OF PLASTIC DEFORMATION IN IMPACT OF CYLINDRICAL PROJECTILES AND ITS IMPROVEMENT

The theory of plastic deformation in the impact of cylindrical projec-tiles on rigid targets was first introduced by G.I.Taylor(1948).The importance of this theorg lies in the fact that the dgnamic yieldstrength of the materials can be determined from the measurement ofthe plastic deformation of flat-ended cylindrical projectiles.Fromthe experimental results.we find that the dynamic gield strengthis independent of impact velocity.and that it is higher than the sta-tic yield strength in general,and several times higher than the sta-tic yield strength in certain cases.This gives an important founda-tion for the studg of elastoplastic impact problems in qeneral.How-ever,it is well known that the complexity of differential equationsin Taylor’s theory compelled us to use the troublesome numerical so-lution.In this paper,the analgtical solution of all the equationsin Taylor’s theory is given in parametrical form and the results arediscussed in detail.In the latter part of this paper,the method of calculatio     

SLIP-LINE FIELD THEORY OF PLANE PLASTIC STRAIN DEALING WITH MOHR’S CRITERION EXPRESSED BY QUADRATIC LIMITING CURVES

In this paper, the slip-line field theory of plane plastic strain dealing with Mohr’s criterion expressed by quadratic limiting curves is preliminarily established. It takes the classical slip-line field theory as its special case, and it can be applied to the analysis of plane-strain problems in metal processing, rock and soil mechanics and tectonomechanics. As preliminary application, the slip-line field and limiting loads of flat punch indenting problem are determined by numerical solution, and the slip-line field of bedded medium gravity-sliding problem is determined and discussed.     

SUBHARMONIC SOLUTION OF A PIECEWISE LINEAR OSCILLATOR WITH TOW DEGREES OF FREEDOM

In the present paper subharmonic resonance solution of a piecewise linear oscillatorwith two degrees of freedom is studied.It is shown that in this system there exist a series ofsubharmonic resonance solutions,among them there are 1/2.1/3,1/4,1/5.1/6,... ...subharmonic resonance solutions.The calculated results by the analogy computer and thefield experiments in the factory partly verify this theory.Under certain circumstances,thegeneration of chaotic states of the oscillation is observed in analogy computer solutions.     

FREE VIBRATION OF A CLASS OF HILL’S EQUATION HAVING A SMALL PARAMETER

In this paper,we consider the initial value problem of a class of Hill’s equation having asmall parameter.Using the solvable condition of boundary value problem and the stretchedparameter method in the perturbation techniques,we present the method which can beapplied to obtain asymptotic periodic solution of the initial value problem.As an example,we consider Mathieu equation and present its computational result.     

Free vibration of a class of Hill's equation having a small parameter

In this paper, we consider the initial value problem of a class of Hill’s equation having a small parameter. Using the solvable condition of boundary value problem and the stretched parameter method in the perturbation techniques, we present the method which can be applied to obtain asymptotic periodic solution of the initial value problem. As an example, we consider Mathieu equation and present its computational result.     

THE RELATION BETWEEN THE MARKOV PROCESS THEORY AND KOLMOGOROFF’S THEORY OF TURBULENCE AND THE EXTINSION OF KOLMOGOROFF’S LAWS Ⅱ.The Extension of Kolmogoroff’s “2/3 LAW”and“-5/3 LAW”

By using the physical analysis described in paper I(part I ofthis paper),we shall establish.in a certain way,the quanti-tative relation between the markov process theory of two par-ticle dispersion in a turbulence of very large Reynolds num-ber and the Kolmogoroff’s theory.In terms of this relationand the results of two-particle dispersion.we shall obtain thestructure functions.the correlation functions and the energyspectrum.which are applicable not only to the inertial sub-range.but also to the whole range of the wave number less thanthat in the inertial subrange.The kolmogoroff’s“2/3 law’and“-5/3 Law”are the asymptotic cases of the present result forlarge k.Thus,the present resuit is an extension of kolmogo-roff’s laws.     

Subharmonic solution of a piecewise linear oscillator with tow degrees of freedom

In the present paper subharmonic resonance solution ofapiecewise linear oscillator with two degrees of freedom is studied. It is shown that in this system there exist a series of subharmonic resonance solutions, among them there are 1/2, 1/3, 1/4, 1/5, 1J6,……subharmonic resonance solutions. The calculated results by the analogy computer and the field experiments in the factory partly verify this theory. Under certain circumstances, the generation of chaotic states of the oscillation is observed in analogy computer solutions.     

Theory of nonlocal asymmetric elastic solids

In this paper, a nonlinear theory of nonlocal asymmetric, elastic solids is developed on the basis of basic theories of nonlocal continuum fieM theory and nonlinear continuum mechanics. It perfects and expands the nonlocal elastic fiteld theory developed by Eringen and others. The linear theory of nonlocal asymmetric elasticity developed in [1] expands to the finite deformation, We show that there is the nonlocal body moment in the nonlocal elastic solids. The noniocal body moment causes the stress asymmetric and itself is caused by the covalent bond formed by the reaction between atoms. The theory developed in this paper is applied to explain reasonably that curves of dispersion relation of one-dimensional plane longitudinal waves are not similar with those of transverse waves.     

ON DISCONTINUOUS PERIODIC SOLUTION AND DISCONTINUOUS SOLITARY WAVE OF TWO-DIMENSION SHALLOW WATER EQUATION

In this papepr we discues discomunucus peridic solution and discontinuous solitarywave of the shallow water model of geophysical fiuid dynamics.When we consider theproperties of vajeciory necr non一epuiulmium nom point.i.e.singular we fund that if weintraduce the concept of gendralized solution smoothing continuous solution),then the system will produce discontinueus periodle solution and the condition ofdiscontinuous periodic solution can be obtahed When the system is dagenerated.we findthat the discontinucus solitary wave is existent in the system. In this paper we consider aserdes of problems and obtain analyvic expression of discontinuous solution.This result iscompared with squall lime in the anmosphere, and both of them have many thing in common.     

THE RELATION BETWEEN THE MARKOV PROCESS THEORY AND KOLMOGOROFF’S THEORY OF TURBULENCE AND THE EXTENSION OF KOLMOGOROFF’S LAWS II.The Extension of Kolmogoroff’s “2/3 LAW” and “-5/3 LAW”

By using the physical analysis described in Paper I (Part I of this paper), we shall establish, in a certain way,the quantitative relation between the Markov process theory of two particle dispersion in a turbulence of very large Reynolds number and the Kolmogoroff’s theory. In terms of this relation and the results of two-particle dispersion,we shall obtain the structure functions, the correlation functions and the energy spectrum, which are applicable not only to the inertial subrange, but also to the whole range of the wave numberless than that in the inertial subrange. The Kolmogoroff ’s"2/3 law" and"-5/3 Law" are the asymptotic cases of the present result for large k. Thus, the present resuit is an extension of Kolmogoroff s laws.     

ASYMPTOTIC SOLUTION TO A NONLINEAR DIFFUSION PROCESS BY CHIEN-LATTA’S METHOD

In this paper,by using Chien Wei-zang—Latta’s compositeexpansion method[5],we have obtained the first-order asymptoticsolution to a system of equations for a nonlinear diffusion pro-cess,thus simplifying and improving the previous work[4]consi-derably.Moreover,a kind of complete analytical solution hasbeen given for a special case,and the periodic solution at thebifurcation point has been discussed,the related results beingin agreement with the experiments.     

ON VARIOUS VARIATIONAL PRINCIPLES IN NONLINEAR THEORY OF ELASTICITY

In the present paper functionals for the various possiblemain variational principles in the nonlinear theory of e-lasticity are derived from the“full energy principle” andseveral of them are not found yet in the literatures avail-able. Through the derivation of this paper we suggest aconjecture on the nonexistence of the eleventh and thesixth classes for the variational principles in Table 6.1of H. C. Hu’s monograph [1].     

Melnikov function and Poincaré map

In this paper we give the relationship between Melnikov function and Poincare map, and a new proof for Melnikov’s method. The advantage of our paper is to give a more explicit solution and to make Melnikov function for the subharmonics bifurcation and Melnikoy function which the stable manifolds and unstable manifolds intersect transversely into a formula.     

The solution of large deflection problem of thin circular plate by the method of composite expansion

In this paper, the method of composite expansion in perturbation theory is used for the solution of large deflection problem of thin circular plate. In this method. the outer field solution and the inner boundary layer solution are combined together to satisfy all the boundary conditions. In this paper, Hencky’s membrane solution is used for the first approximation in outer field solution, and then the second approximate solution is obtained. The inner boundary layer solution is found on the bases of boundary layer coondinate. In this paper, the reciprocal ratio of maximum deflection and thickness of the plate is used as the small parameter. The results of this paper improves quite a bit in comparison with the results obtained in 1948 by Chien Wei-zang.     

NOETHER’S THEORY OF VACCO DYNAMICS

In this paper,we first study the latent relation between the conservative quantityand the symmetry of nonholonomic dynamical systems without any additionalrestrictive conditions to its virtual displacement,and then establish Noether’s theoremand Noether’s inverse theorem of Vacco dynamics.Lastly,we give two examples toillustrate the application of result of this paper.     

Existence of traveling wave solutions for a nonlinear dissipative-dispersive equation

In this paper, we consider a dissipative-dispersive nonlinear equation appliable to many physical phenomena. Using the geometric singular perturbation method based on the theory of dynamical systems, we investigate the existence of its traveling wave solutions with the dissipative terms having sufficiently small coefficients. The results show that the traveling waves exist on a two-dimensional slow manifold in a three-dimensional system of ordinary differential equations （ODEs）. Then, we use the Melnikov method to establish the existence of a homoclinic orbit in this manifold corresponding to a solitary wave solution of the equation. Furthermore, we present some numerical computations to show the approximations of such wave orbits.