Nonlinear analysis of the forced response of a beam with three mode interaction

An analysis is presented for the primary resonance of a clamped-hinged beam, which occurs when the frequency of excitation is near one of the natural frequencies,_n . Three mode interaction (₂ 3₁ and _{3 1} + 2₂) is considered and its influence on the response is studied. The case of two mode interaction (₂ 3₁) is also considered to compare it with the case of three mode interaction. The straight beam experiencing mid-plane stretching is governed by a nonlinear partial differential equation. By using Galerkin's method the governing equation is reduced to a system of nonautonomous ordinary differential equations. The method of multiple scales is applied to solve the system. Steady-state responses and their stability are examined. Results of numerical investigations show that there exists no significant difference between both modal interactions' influences on the responses.     

The mutual variational principle of free wave propagation in periodic structures

By taking infinite periodic beams as examples, the mutual variational principle for analyzing the free wave propagation in periodic structures is established and demonstrated through the use of the propagation constant in the present paper, and the corresponding hierarchical finite element formulation is then derived. Thus, it provides the numerical analysis of that problem with a firm theoretical basis of variational principles, with which one may conveniently illustrate the mathematical and physical mechanisms of the wave propagation in periodic structures and the relationship with the natural vibration. The solution is discussed and examples are given. Supported by Doctorate Training Fund of National Education Commission of China     

Rotating bunsen flames as solutions of the Kuramoto-Sivashinsky equation

Steadily rotating solutions of the Kuramoto-Sivashinsky equationu _t + ² u++¦u¦ ² =c ² are studied. These solutions bifurcate from the steady radial solution of the above equation. For large values ofc and angular velocities such thatN¦¦<2c<(N+1)¦¦, we show that there exists a 2N-1 family of bifurcating solutions. The proof is based on a certain generic transversality assumption. A computer-assisted proof of this assumption is given for 1N10.     

Method of homogenization applied to dispersion in porous media

The theory of homogenization which is a rigorous method of averaging by multiple scale expansions, is applied here to the transport of a solute in a porous medium. The main assumption is that the matrix has a periodic pore structure on the local scale. Starting from the pores with the Navier-Stokes equations for the fluid motion and the usual convective-diffusion equation for the solute, we give an alternative derivation of the three-dimensional macroscale dispersion tensor for solute concentration. The original result was first found by Brenner by extending Brownian motion theory. The method of homogenization is an expedient approach based on conventional continuum equations and the technique of multiple-scale expansions, and can be extended to more complex media involving three or more contrasting scales with periodicity in every but the largest scale.     

Apollonius coordinates,theN-body problem,and continuation of periodic solutions

This paper treats theN-body problem and its relation to various restricted problems. For each solution of the Kepler problem a generalization of the pulsating coordinates used to express the Hamiltonian of the elliptic restricted three-body problem is given. These coordinates are called Apollonius coordinates. The method of symplectic scaling is used to give a precise derivation of the elliptic restricted problem showing the precise asymptotic relationship between the restricted problem and the full three-body problem. This derivation obviates the proof of the fact that a nondegenerate periodic solution of the elliptic restricted three-body problem can be continued into the full three-body problem under mild nonresonance assumptions. Also, the method of symplectic scaling is used to give a precise derivation of the elliptic Hill lunar equation showing the precise relationship between the elliptic Hill lunar equation and the full three-body problem. A similar continuation theorem is established.     

模差渐进投影法解析重叠色谱峰

提出了一种鉴别重叠色谱峰类型的化学计量学方法——模差渐进投影(MDEP)法。该方法利用二维色谱数据中相邻两个色谱流出点的光谱矢量模的和与该两个矢量和的模之差,建立模差-时间(dm-t)曲线判断非纯色谱峰的纯组分区,同时判定第二组分色谱流出点和倒数第二组分色谱结束点。分别以纯组分区的光谱矢量建立正交阵对该色谱峰进行投影运算,得到投影后转换色谱。该转换色谱消除了纯组分区所对应化合物的色谱和光谱信息。对转换色谱重复上述操作至整个色谱峰的模差-时间曲线均为0时终止,得到重叠色谱峰各组分的色谱流出点和结束点。利用上述结果,采用不包含待测组分光谱信息的该重叠峰二维色谱数据建立正交矩阵对该色谱峰进行投影运算,获得待测组分的相似色谱图。通过比对分析即可得到该组分在重叠色谱峰中的光谱和色谱信息。依次操作,即可获得重叠色谱峰中各组分的光谱和色谱信息。采用模差渐近投影法对4组分模拟重叠二维色谱峰及杜香挥发油GC-MS的重叠色谱峰进行了解析,方法简便、结果准确。     

The study of the crystalline phosphates of kosnarite type structure containing different alkali metals

The incorporation possibilities of different alkali elements into crystalline phosphates A_1−xA′_xHf₂(PO₄)₃ (A=Li, Na, K, Rb, Cs) were studied, the formation regions of kosnarite solid solutions were determined. Na_0.5K_0.5Hf₂(PO₄)₃ crystal structure was studied by powder X-ray diffraction, and the distribution of alkali metals in kosnarite structure was found out. The phosphate crystallizes in the space group R3?c, with a=8.7295(1) Å, c=23.2023(4) Å, V=1531.24(4) Å³, Z=6; R_wp=6.15, R_p=4.43. The concentration region knowledge of the kosnarite phase existence and peculiarities of their phase formation in the A_1−xA′_xM₂(PO₄)₃ (M=Ti, Zr, Hf) systems allow us to choose phosphate matrice compositions suitable for solidification of reprocessing wastes of spent U-Pu nuclear fuels.     

Synthesis,characterization, and crystallization of thermoplastic aromatic semi‐crystalline poly(ester‐imide)s based on bis(trimellitic acid anhydride) phenyl ester

A high molecular weight aromatic homopoly(ester‐imide) (homoPEI) was synthesized from homopoly(ester‐amic acids) (homoPEAA), which was obtained from the reaction of bis(trimellitic acid anhydride) phenyl ester (BTAH) with 4‐4'‐oxydianiline ether (ODA). This homoPEI was melt‐processable semi‐crystalline polymer and displayed dual endothermic transitions which were attributed to the different levels of crystal perfection and size in the crystal structures. Four high molecular weight aromatic copoly(ester‐imide)s (coPEIs) were synthesized via copoly(ester‐amic acids)s (coPEAA) from the reaction of 4, 4'‐oxydiphthalic anhydride (ODPA) and BTAH with ODA. The molar percentage of BTAH varied from 10 to 40%. When the molar percentages of BTAH were 30 and 40%, the resulting two coPEIs were crystallinable and their melting temperatures were 361°C and 356/371°C, respectively. Differential scanning calorimetry (DSC) and wide angle X‐ray diffraction (WAXD) results indicated that the crystal structures included short segments of BTAH/ODA and ODPA/ODA. The initial crystallization of the two coPEIs took place during imidization process. Thermogravimetric (TA) and mechanical analysis confirmed that both homoPEI and coPEIs showed almost no weight loss until 400°C in N₂ and good mechanical properties. Copyright © 2009 John Wiley & Sons, Ltd.