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Solving nonlinear differential equations of Vanderpol, Rayleigh and Duffing by AGM

null

《机械工程前沿（英文）》 2014年第9卷第2期页码 177-190 doi: 10.1007/s11465-014-0288-8

摘要：

In the present paper, three complicated nonlinear differential equations in the field of vibration, which are Vanderpol, Rayleigh and Duffing equations, have been analyzed and solved completely by Algebraic Method (AGM). Investigating this kind of equations is a very hard task to do and the obtained solution is not accurate and reliable. This issue will be emerged after comparing the achieved solutions by numerical method (Runge-Kutte 4th). Based on the comparisons which have been made between the gained solutions by AGM and numerical method, it is possible to indicate that AGM can be successfully applied for various differential equations particularly for difficult ones. The results reveal that this method is not only very effective and simple, but also reliable, and can be applied for other complicated nonlinear problems.

关键词： Algebraic Method (AGM) Angular Frequency Vanderpol Rayleigh Duffing

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Numerical analysis of strongly nonlinear oscillation systems using He’s max-min method

H. BABAZADEH, G. DOMAIRRY, A. BARARI, R. AZAMI, A. G. DAVODI

《机械工程前沿（英文）》 2011年第6卷第4期页码 435-441 doi: 10.1007/s11465-011-0243-x

摘要：

Nonlinear functions are crucial points and terms in engineering problems. Actual and physical problems can be solved by solving and processing such functions. Thus, most scientists and engineers focus on solving these equations. This paper presents a novel method called the max-min method for presenting an accurate approximate analytical solution to strong nonlinear oscillators. It can solve many linear or nonlinear differential equations without the tangible restriction of sensitivity to the degree of the nonlinear term. It is also quite convenient due to the reduction in the size of calculations. The algorithm suggests a promising approach and is systematically illustrated step by step.

关键词： min-max method nonlinear oscillation duffing equation mathematical pendulum He’s energy balance method

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Comparison of the performance of traditional advection-dispersion equation and mobile-immobile model

Haizhu HU,Xiaomin MAO

《农业科学与工程前沿（英文）》 2016年第3卷第3期页码 241-248 doi: 10.15302/J-FASE-2016108

摘要： The traditional advection-dispersion equation (ADE) and the mobile-immobile model (MIM) are widely used to describe solute transport in heterogeneous porous media. However, the fitness of the two models is case-dependent. In this paper, the transport of conservative, adsorbing and degradable solutes through a 1 m heterogeneous soil column under steady flow condition was simulated by ADE and MIM, and sensitivity analysis was conducted. Results show that MIM tends to prolong the breakthrough process and decrease peak concentration for all three solutes, and tailing and skewness are more pronounced with increasing dispersivity. Breakthrough curves of the adsorbing solute simulated by MIM are less sensitive to the retardation factor compared with the results simulated by ADE. The breakthrough curves of degradable solute obtained by MIM and ADE nearly overlap with a high degradation rate coefficient, indicating that MIM and ADE perform similarly for simulating degradable solute transport when biochemical degradation prevails over the mass exchange between mobile and immobile zones. The results suggest that the physical significance of dispersivity should be carefully considered when MIM is applied to simulate the degradable solute transport and/or ADE is applied to simulate the adsorbing solute transport in highly dispersive soils.

关键词： solute transport advection dispersion equation (ADE) mobile-immobile model (MIM) conservative solute adsorbing solute degradable solute PHREEQC-2

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Optimal design of extractive dividing-wall column using an efficient equation-oriented approach

Yingjie Ma, Nan Zhang, Jie Li, Cuiwen Cao

《化学科学与工程前沿（英文）》 2021年第15卷第1期页码 72-89 doi: 10.1007/s11705-020-1977-y

摘要： The extractive dividing-wall column (EDWC) is one of the most efficient technologies for separation of azeotropic or close boiling-point mixtures, but its design is fairly challenging. In this paper we extend the hybrid feasible path optimisation algorithm (Ma Y, McLaughlan M, Zhang N, Li J. Computers & Chemical Engineering, 2020, 143: 107058) for such optimal design. The tolerances-relaxation integration method is refined to allow for long enough integration time that can ensure the solution of the pseudo-transient continuation simulation close to the steady state before the required tolerance is used. To ensure the gradient and Jacobian information available for optimisation, we allow a relaxed tolerance for the simulation in the sensitivity analysis mode when the simulation diverges under small tolerance. In addition, valid lower bounds on purity of the recycled entrainer and the vapour flow rate in column sections are imposed to improve computational efficiency. The computational results demonstrate that the extended hybrid algorithm can achieve better design of the EDWC compared to those in literature. The energy consumption can be reduced by more than 20% compared with existing literature report. In addition, the optimal design of the heat pump assisted EDWC is achieved using the improved hybrid algorithm for the first time.

关键词： design extractive dividing-wall column equation-oriented optimisation pseudo-transient continuation model hybrid algorithm

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深水钻井隔水导管挠曲方程和固有频率的计算及其研究

姜伟

《中国工程科学》 2011年第13卷第5期页码 66-73

摘要：

从深水钻井应用的隔水导管的使用实际情况出发，应用弹塑性力学中的位移复分原理的Ritz法，考虑了隔水导管所受到的轴向张力T和隔水导管受力产生的倾角α以及由于自重q所产生的自重横向分力qsin α和轴向分力qcos α，建立了较为合理的挠曲方程，并且由此得到隔水导管的固有振动频率的简易计算求解方法。由于该挠曲方程结合了钻具的实际情况，其挠曲方程的约束条件采用的是一端铰支、一端自由的形式，因此其研究更接近实际情况，研究旨在为深水钻井隔水导管的合理使用设计和选择，提供一种更快捷实用的方法，对深水钻井作业有一定的指导意义。

关键词：深水钻井隔水导管挠曲方程固有频率形状函数

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Parametric analysis and design equation of ultimate capacity for unstiffened overlapped CHS K-joints

CHEN Yu, ZHAO Xianzhong, CHEN Yiyi

《结构与土木工程前沿（英文）》 2008年第2卷第2期页码 107-115 doi: 10.1007/s11709-008-0014-x

摘要： A finite element model simulating an experiment on unstiffened, overlapped circular hollow structure (CHS) K-joints was generated and validated by comparing the ultimate capacities, deformation processes and failure modes of the experimental results. Using this model, the stress distribution, propagation of plasticity and the failure modes of overlapped joints with through-brace-in-compression and welded hidden seams were analyzed. The effect of geometric parameters, with or without hidden welds, and the loading hierarchy reversal of braces on the ultimate capacity of the joints were also studied. The results of finite element parametric analysis indicate that the brace-to-chord thickness ratio has relatively large effects on the failure mechanism and ultimate capacity of overlapped joints. It was also found that the absence of hidden welds has less significance on the ultimate capacity of through-brace-in-compression joints than through-brace-in tension joints. Finally, based on the design equation of gap joints, a formula predicting the ultimate capacity of overlapped CHS K-joints was derived by applying multivariate regression analysis. Results from the proposed design equation are consistent with experimental results.

关键词： regression analysis consistent unstiffened plasticity overlapped

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一种损伤演化方程的建立和几种层裂准则的推导

曹结东,李永池

《中国工程科学》 2006年第8卷第1期页码 40-45

摘要：

通过唯象分析和细观统计相结合的方法建立了一种韧性材料的损伤演化方程，从物理上对方程中的参数进行了合理的解释，并以此为依据推导了材料的便于工程应用的几种层裂准则，根据准则总结了一些有意义的层裂规律；通过层裂实验和数值模拟相结合的方法来确定材料参数，计算结果与实验结果较好的一致性验证了所建的损伤演化方程和推导的层裂准则是合理的，决定材料参数的方法是有效的。

关键词：损伤演化方程层裂准则数值模拟

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抛物形缓坡方程的数值研究

沈永明,郑永红,吴修广

《中国工程科学》 2004年第6卷第3期页码 30-35

摘要：

对2种典型的抛物形缓坡方程进行比较系统的数值研究。通过对4种典型地形上的波浪变形的数值模拟，详细讨论了网格节点数对数值解精度的影响、模型对初始入射角的敏感程度、非线性项对数值结果的影响等。研究结果可为实际应用抛物形缓坡方程研究大区域复杂地形上的波浪传播提供一定的理论指导。

关键词：抛物形缓坡方程数值研究波浪

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聚酯材料状态方程的实验研究

林鹏,王长利,李焰,刘文祥,王等旺

《中国工程科学》 2009年第11卷第4期页码 63-66

摘要：

在单级轻气炮上对聚酯材料的激波状态方程进行了实验研究。最高撞击速度为526 m/s，相应在靶中得到的最高撞击压力为2.14 GPa。压力测试采用锰铜计，PVDF产生的压电信号用于判读激波速度。由实验结果拟合得到了聚酯材料的Hugoniot线性关系和多项式状态方程以及Grüneisen状态方程。

关键词：聚酯材料状态方程 Hugoniot 激波

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用改进的Boussinesq方程模拟潜堤上的波浪变形

郑永红,沈永明,吴修广,游亚戈

《中国工程科学》 2004年第6卷第4期页码 34-40

摘要：

研究了一种改进的Boussinesq方程，采用预报校正格式对该方程进行了数值离散，并对淹没潜堤上的波浪变形进行了数值模拟，从数值模拟结果和实测值的比较结果来看，该方程能较好地模拟波浪在潜堤上传播时波面的变形过程，可以用于实际中的波浪问题计算。这种改进的Boussinesq方程本身及其求解方法需做进一步的完善。研究结果为实际应用Boussinesq方程来研究复杂地形上的波浪传播提供一定的理论指导。

关键词：改进的Boussinesq方程数值模拟波浪变形潜堤

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计算管内湍动流体摩擦因数的显式新方程

王勇,阮奇

《中国工程科学》 2006年第8卷第6期页码 83-88

摘要：

管内湍动流体摩擦因数是雷诺数和相对粗糙度的二元非线性函数，由Colebrook隐式方程计算摩擦因数要用迭代的方法求解，很不方便。为了得到形式简单、精度高的计算摩擦因数的显式方程，提出了二元非线性多项式智能拟合法。该法将二元非线性多项式转化成多元线性多项式并建立线性最小二乘法标准矩阵，用遗传算法结合矩阵法对多项式的项数、项型式项指数及项系数进行搜索得到最优的拟合函数式。用该法拟合了Colebrook方程解的数据，得到一个计算管内湍动流体摩擦因数的显式新方程。在雷诺数3.000≤Re≤10⁸、相对粗糙度0≤e/d≤0.05的范围内，该方程计算结果与Colebrook方程的平均偏差为0.5%，最大偏差不超过1.8%，与实验数据偏差为2.3%。新方程具有形式简单、精度高、适用范围广的优点，且便于简化成光滑管或阻力平方区等情况下的计算摩擦因数的方程。

关键词：湍动流动摩擦因数 Colebrook隐式方程遗传算法智能拟合法显式方程

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零方程湍流模型在列车车厢内气流数值模拟中的应用

杨培志,顾小松

《中国工程科学》 2006年第8卷第1期页码 26-29

摘要：

采用k-ε湍流模型对列车车厢内气流进行数值模拟需要消耗大量的计算时间，为此，提出了采用零方程湍流模型对列车车厢内的气流组织进行数值模拟；分别采用零方程湍流模型与k-ε湍流模型对列车车厢内的空气流动及传热进行了数值计算，经分析比较可知，该两种湍流模型的数值计算结果吻合程度较好，采用零方程湍流模型可大大缩短计算时间，利用其简单、快捷的特点，可以为列车空调系统的工程设计提供简便的数值模拟方法。

关键词：空调零方程湍流模型数值模拟列车

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Iteration framework for solving mixed lubrication computation problems

《机械工程前沿（英文）》 2021年第16卷第3期页码 635-648 doi: 10.1007/s11465-021-0632-8

摘要： The general discrete scheme of time-varying Reynolds equation loses the information of the previous step, which makes it unreasonable. A discretization formula of the Reynolds equation, which is based on the Crank–Nicolson method, is proposed considering the physical message of the previous step. Gauss–Seidel relaxation and distribution relaxation are adopted for the linear operators of pressure during the numerical solution procedure. In addition to the convergent criteria of pressure distribution and load, an estimation framework is developed to investigate the relative error of the most important term in the Reynolds equation. Smooth surface with full contacts and mixed elastohydrodynamic lubrication is tested for validation. The asperity contact and sinusoidal wavy surface are examined by the proposed discrete scheme. Results show the precipitous decline in the boundary of the contact area. The relative error suggests that the pressure distribution is reliable and reflects the accuracy and effectiveness of the developed method.

关键词： mixed lubrication discretization formula relative error Reynolds equation asperity

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Analyzing the nonlinear vibrational wave differential equation for the simplified model of Tower Cranes

M.R. AKBARI,D.D. GANJI,A.R. AHMADI,Sayyid H. Hashemi KACHAPI

《机械工程前沿（英文）》 2014年第9卷第1期页码 58-70 doi: 10.1007/s11465-014-0289-7

摘要：

In the current paper, a simplified model of Tower Cranes has been presented in order to investigate and analyze the nonlinear differential equation governing on the presented system in three different cases by Algebraic Method (AGM). Comparisons have been made between AGM and Numerical Solution, and these results have been indicated that this approach is very efficient and easy so it can be applied for other nonlinear equations. It is citable that there are some valuable advantages in this way of solving differential equations and also the answer of various sets of complicated differential equations can be achieved in this manner which in the other methods, so far, they have not had acceptable solutions. The simplification of the solution procedure in Algebraic Method and its application for solving a wide variety of differential equations not only in Vibrations but also in different fields of study such as fluid mechanics, chemical engineering, etc. make AGM be a powerful and useful role model for researchers in order to solve complicated nonlinear differential equations.

关键词： Algebraic Method (AGM) tower crane oscillating system angular frequency

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Discrete ordinates method for three-dimensional neutron transport equation based on unstructured-meshes

JU Haitao, WU Hongchun, YAO Dong, XIAN Chunyu

《能源前沿（英文）》 2008年第2卷第2期页码 179-182 doi: 10.1007/s11708-008-0024-3

摘要： A discrete ordinates method for a three-dimensional first-order neutron transport equation based on unstructured-meshes that avoids the singularity of the second-order neutron transport equation in void regions was derived. The finite element variation equation was obtained using the least-squares method. A three-dimensional transport calculation code was developed. Both the triangular-z and the tetrahedron elements were included. The numerical results of some benchmark problems demonstrated that this method can solve neutron transport problems in unstructured-meshes very well. For most problems, the error of the eigenvalue and the angular flux is less than 0.3% and 3.0% respectively.

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标题作者时间类型操作

Solving nonlinear differential equations of Vanderpol, Rayleigh and Duffing by AGM

null

期刊论文

Numerical analysis of strongly nonlinear oscillation systems using He’s max-min method

H. BABAZADEH, G. DOMAIRRY, A. BARARI, R. AZAMI, A. G. DAVODI

期刊论文

Comparison of the performance of traditional advection-dispersion equation and mobile-immobile model

Haizhu HU,Xiaomin MAO

期刊论文

Optimal design of extractive dividing-wall column using an efficient equation-oriented approach