应用数学和力学

1992 Vol. 13, No. 9

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General Analytic Solution of Dynamic Response of Beams with Nonhomogeneity and Variable Cross Section

Yeh Kai-yuan, Tong Xiao-hua, Ji Zhen-yi

1992, 13(9): 753-764.

Abstract(1850) PDF(647)

Abstract:
In this paper, a new method, the step-reduction method, is proposed to investigate the dynamic response of the Bernoulli-Euler beams with arbitrary nonhomogeneity and arbitrary variable cross-section under arbitrary loads. Both free vibration and forced vibration of such beams are studied. The new method requires to discretize the space domain into a number of elements. Each element can be treated as a homogeneous one with uniform thickness. Therefore, the general analytical solution of homogeneous beams with uniform cross-section can be used in each element. Then, the general analytic solution of the whole beam in terms of initial parameters can be obtained by satisfying the physical and geometric continuity conditions at the adjacent elements. In the case of free vibration, the frequency equation in analytic form can be obtained, and in the case of forced vibration, a final solution in analytical form can also be obtained which is involved in solving a set of simultaneous algebraic equations with only two unknowns which are independent of the numbers of elements divided. The present analysis can also be extended to the study of the vibration of such beams with viscous and hysteretic damping and other kinds of beams and other structural elements with arbitrary nonhomogeneity and arbitrary variable thickness.

Theory of Nonlocal Asymmetric Elastic Solids

Gao Jian, Chen Zhi-da

1992, 13(9): 765-774.

Abstract(2131) PDF(526)

Abstract:
In this paper, a nonlinear theory of nonlocal asymmetric, elastic solids is developed on the basis of basic theories of nonlocal continuum field theory and nonlinear continuum mechanics. It perfects and expands the nonlocal elastic field 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 nonlocal 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.

Unsteady Transonic Aerodynamic Loadings on the Airfoil Caused by Heaving, Pitching Oscillations and Control Surface

Gao Zheng-hong

1992, 13(9): 775-783.

Abstract(2287) PDF(430)

Abstract:
An implicit upwind finite volume solver for the Euler equations using the improved flux-splitting method is established and used to calculate the transonic flow past the airfoils with heaving, pitching oscillations and the control surface. Results are given for the NACA64A-10 airfoil which is in harmonic heaving and pitching oscillation and with the control surface in the transonic flow field. Some computational results are compared with the experiment data and the good agreements are shown in the paper.

Perturbation Formulation of Continuation Method Including Limit and Bifurcation Points

Wu Yi, Ren Wen-min, Zhang Wei

1992, 13(9): 785-794.

Abstract(1834) PDF(466)

Abstract:
This paper gives the perturbation formulation of continuation method for nonlinear equations. Emphasis is laid on the discussion of searching for the singular points on the equilibrium path and of tracing the paths over the limit or bifurcation points. The method is applied to buckling analysis of thin shells. The pre-and post-buckling equilibrium paths and deflections can be obtained, which are illustrated in examples of buckling analysis of cylindrical and toroidal shells.

Some General Theorems and Generalized and Piecewise Generalized Variational Principles for Linear Elastodynamics

Xing Jing-tang, Zheng Zhao-chang

1992, 13(9): 795-810.

Abstract(2456) PDF(515)

Abstract:
From the concept of four-dimensional space and under the four kinds of time limit conditions, some general theorems for elastodynamics are developed, such as the principle of possible work action, the virtual displacement principle, the virtual stress-momentum principle, the reciprocal theorems and the related theorems of time terminal conditions derived from it. The variational principles of potential energy action and complementary energy action, the H-W principles, the H-R principles and the constitutive variational principles for elastodynamics are obtained. Hamilton's principle, Toupin's work and the formulations of Ref. [5],[17]-[24] may be regarded as some special cases of the general principles given in the paper. By considering three cases: piecewise space-time domain, piecewise space domain, piecewise time domain, the piecewise variational principles including the potential, the complementary and the mixed energy action fashions are given. Finally, the general formulation of piecewise variational principles is derived. If the time dimension is not considered, the formulations obtained in the paper will become the corresponding ones for elastostatics.

The Holder Continuity for the Gradient of Solutions to One-Sided Obstacle Problems

Wang Xiang-dong, Liang Xi-ting

1992, 13(9): 811-820.

Abstract(2418) PDF(476)

Abstract:
There is a gap in case 1<p<2 to the C^1+a regularity for solutions of variational inequalities with degenerate ellipticity.In this paper, basad on the fundamental of the C^0+a. reglarity of solutions,the Holder continuity for the gradient of solutions is proved in case 1<p<2 to a one-sided obstacle problem for variational inequalities ∫_G{∇v·A(x,u,∇u)+vB(x,u,∇u)}dx≥0,(∀v∈(G),v≥ψ-u).

The Principles of Least Action of Vairable Mass Nonholonomic Nonconservative Systems in Noninertial Reference Frames

Luo Shao-kai, Mei Feng-xiang

1992, 13(9): 821-828.

Abstract(2260) PDF(663)

Abstract:
This paper presents the generalized principles of least action of variable mass nonholonomic nonconservative system in noninertial reference frame, proves the equivalence between Holder form and Suslov form, and then obtains differential equations of motion of variable mass nonholonomic nonconservative system in noninertial reference frame.

Extension and Application of Newton’s Method in Nonlinear Oscillation Theory

Huo Lin-chun

1992, 13(9): 829-842.

Abstract(2386) PDF(578)

Abstract:
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.

A Kind of General Influence Measure on the Linear Weighted Regression

Yang Hu

1992, 13(9): 843-847.

Abstract(1688) PDF(463)

Abstract:
The linear weighted regression model is one of the models studied in many articles in recent years. Some further problems, such as disturbation, influence measure and estimate efficiency, have been discussed in this paper on the basis of the regression diagnosties. The partial conclusions of this paper are the extension of the familiar concepts in the regression diagnosties theory^[2,3,7] because they are representative of this kind of model.

Topological Structure of the Singular Points of the Third Order Phase Locked Loop Equations with the Character of Detected Phase Being g(φ)=(1+k)sinφ/(1+kcosφ)

Jin Jun

1992, 13(9): 849-854.

Abstract(1737) PDF(461)

Abstract:
In this paper, we study the topological structure of the singular points of the third order phase locked loop equations with the character of detected phase being g(φ)=(1+k)sinφ/1+kcosφ.