Optimum dynamic design for beam shapes subjected to a moving concentrated load

We are concerned with an optimum dynamic design of beams subjected to a moving concentrated load with constant speed. The influence of the dynamic behaviour of the beams is considered in a proposed optimum design problem. The optimum shapes of beams are determined by the minimization of two kinds of performance indices. The optimization procedure is performed by non-linear programming on the basis of the exterior penalty function and BFGS methods. Optimization is calculated by the modal coordinates transformation and the numerical integration method     

Integrating rotation and angular velocity from curvature

The problem of integrating the rotational vector from a given angular velocity vector is met in such diverse fields as the navigation, robotics, computer graphics, optical tracking and non-linear dynamics of flexible beams. For example, if the numerical formulation of non-linear dynamics of flexible beams is based on the interpolation of curvature, one needs to derive the rotation from the assumed curvature field. The relation between the angular velocity and the rotation is described by the first-order quasi-linear differential equation. If the rotation is given, the related angular velocity is obtained by the differentiation. By contrast, if the angular velocity is given, the related rotations are obtained by the integration. The exact closed-form solution for the rotation is only possible if the angular velocity is constant in time. In dynamics of non-linear flexible spatial beams, the problem of integrating rotations from a given angular velocity becomes even more complex because both the angular velocity and the curvature need simultaneously be integrated and are both functions of space and time. As the angular velocity and the curvature are assumed to be analytic functions, they must satisfy certain integrability conditions to assure the unique rotation is obtained from the two differential equations. The objective of the present paper is to derive approximate, yet closed-form solutions of the following problem: for a given curvature vector, determine both the rotation and the angular velocity. In order to avoid the singularity of kinematic relations, the quaternions are used for the parametrization of rotations, and the integrations are partly performed in the four-dimensional quaternion space. The resulting closed-form expressions for the rotational and angular velocity quaternions are ready to be used in the finite-element formulations of the dynamics of flexible spatial beams as interpolating functions. The present novel solution is assessed by comparisons of the numerical results with analytical solutions for variety of oscillating curvature functions, as well as with the solutions of the quaternion-based midpoint integrator and the Runge–Kutta-based Crouch–Grossman geometrical methods CG3 and CG4.     

Inelastic stability of laterally unsupported i-beams

A computer method to study the inelastic stability of laterally unsupported steel I-beams and based on a general non-linear theory is presented.Traditionally, the problem of flexural-torsional stability of beams is treated as a lateral buckling problem. Some of the draw-backs of these earlier studies are given below:The classical theory assumes that the deformations are small. In addition the deformation field is linearized. This theory is therefore valid only when the major axis flexural rigidity is much greater than its minor axis rigidity, so that deformations before the onset of lateral buckling are negligible.The lateral buckling theory is valid for straight beams, with loads applied rigorously in the plane of symmetry. Practical beams have initial imperfections and unavoidable load eccentricities. So the true behavior is better described by the stability phenomenon.For beams of intermediate length for which buckling occurs in the inelastic range, the tangent modulus theory is generally used. For ideally straight beams the tangent modulus theory provides an estimate for the collapse load which is slightly conservative. However, for practical beams with initial deformations, this need not be the case.In the majority of existing studies on inelastic lateral buckling, the differential equations for beams under uniform moment are used without modification for beams under moment gradient. In the later case the shear center line is inclined to the centroidal and geometrical axes. The differential equations for beams under uniform moment should therefore be modified by adding additional terms.The majority of the existing studies are limited to the behavior of isolated beams with simple end-conditions and so the beneficial effect of adjacent members on the beam collapse load cannot be studied accurately.A general non-linear theory to describe the spatial behavior of beams and that doesn't have the deficiencies mentioned above, is developed in the present paper.The paper also presents a computer method of solving these non-linear equations using the method of finite differences. Several numerical examples presented and comparison with the existing theoretical and experimental results show the applicability of the theory to a wide range of problems.     

换热网络的分析

通过对换热网络的研究发现，当物流有分流时，换热网络分析的数学模型为非线性规划问题。本文给出换热网络分析的非线性规划模型，同时优化各路分流的流量及换热单元进出口温度。经模型转化，用复合形法与线性规划法的双层组合求解，成功地解决了换热网络分析中非线性规划的求解问题。经实例考核计算证明，本文所建立的数学模型是正确可靠的，使用的双层组合法与其它方法相比，方法简单实用，结果准确，可在工程研究、设计中应用，分析后所得的最佳参数可大大提高换热网络的热回收量，节能效益十分显著。     

A unified wavelet-based modelling framework for non-linear system identification: the WANARX model structure

A new unified modelling framework based on the superposition of additive submodels, functional components, and wavelet decompositions is proposed for non-linear system identification. A non-linear model, which is often represented using a multivariate non-linear function, is initially decomposed into a number of functional components via the well-known analysis of variance (ANOVA) expression, which can be viewed as a special form of the NARX (non-linear autoregressive with exogenous inputs) model for representing dynamic input–output systems. By expanding each functional component using wavelet decompositions including the regular lattice frame decomposition, wavelet series and multiresolution wavelet decompositions, the multivariate non-linear model can then be converted into a linear-in-the-parameters problem, which can be solved using least-squares type methods. An efficient model structure determination approach based upon a forward orthogonal least squares (OLS) algorithm, which involves a stepwise orthogonalization of the regressors and a forward selection of the relevant model terms based on the error reduction ratio (ERR), is employed to solve the linear-in-the-parameters problem in the present study. The new modelling structure is referred to as a wavelet-based ANOVA decomposition of the NARX model or simply WANARX model, and can be applied to represent high-order and high dimensional non-linear systems.     

Optimum design of statically indeterminate beams under multiple loads

The paper presents minimum-weight designs of statically indeterminate beams under multiple loads. Under each load system consisting of external loads and self-weight the normal and shear stresses in the beams are restricted from above. The variation of the stiffness along the span of the beams is restricted to splines of order zero, one or two, thus reducing the optimization problem to a linear or non-linear programming problem depending on the cross-sectional shape. The latter is reduced to a sequence of linear programming problems which are solved by the highly-efficient active set method. Several examples are given to illustrate the potential of this method. A computer code in standard FORTRAN 77 is also provided.     

Controlling the Wrapping Effect in the Solution of ODEs for Asteroids

During the last decade, substantial progress has been made in fighting the wrapping effect in self-validated integrations of linear systems. However, it is still the main problem limiting the applicability of such methods to the long-term integration of non-linear systems. Here we show how high-order self-validated methods can successfully overcome this obstacle.We study and compare the validated integration of a Kepler problem with conventional and high-order methods represented by AWA and Taylor models, respectively. We show that this simple model problem exhibits significant wrapping that is particularly difficult to control for conventional first-order methods. It will become clear that utilizing high-order methods with shrink wrapping allows the system to be analyzed in a fully validated context over large integration times. By comparing high-order Taylor model integrations with Taylor model methods subjected to an artificial wrapping effect, we show that utilizing high-order methods to propagate initial conditions is indeed the foremost reason for the successful suppression of the wrapping effect.To further demonstrate that high-order Taylor model methods can be used for the integration of complicated non-linear systems, we summarize results obtained from a fully verified and self-validated orbit integration of the near earth asteroid 1997 XF11. Since this asteroid will have several close encounters with Earth, its analysis is an important application of reliable computations.     

Deformable strut and tie model for the calculation of the plastic rotation capacity

When a plastic analysis, a non-linear analysis or a linear analysis followed by redistribution of bending moments is used to predict the structural behaviour of beams, the critical sections should have the necessary plastic rotation capacity to allow the predicted behaviour at failure. When some doubts may arise, then an explicit calculation of this capacity must be carried out. This paper presents a theoretical model for the calculation of plastic rotation, considering the influence of the main factors. Some results are presented on the basis of the model, and conclusions are drawn.     

Review and comparison of finite element flexural–torsional models for non-linear behaviour of thin-walled beams

The authors have developed a beam finite element model in large torsion context for thin-walled beams with arbitrary cross sections [1]. In the model, the trigonometric functions of the twist angle θ_x (c = cos θ_x − 1 and s = sin θ_x) were included as additional variables in the whole model without any assumption. In the present paper, three other 3D finite element beams are derived according to three approximations based on truncated Taylor expansions of the functions c and s (cubic, quadratic and linear). A finite element approach of these approximations is carried out. Finally, it is worth mentioning that the promising results obtained in [1], [2] encourage the authors to extend the formulation of the model in order to include load eccentricity effects. Solution of the non-linear equations is made possible by Asymptotic Numerical Method (ANM) [3]. This method is used as an alternative to the classical incremental iterative methods. Many comparison examples are considered. They concern the non-linear behaviour of beams under twist moment and the post buckling behaviour of struts under axial loads or the beam lateral buckling under eccentric bending loads. The obtained results highlight the discrepancies between the various approximations often employed in thin-walled beams literature for the geometrically non-linear analysis of beams in flexural–torsional behaviour.     

Robust control design of non-linear processes using empirical state affine models

A methodology for the design of robust stable controllers for non-linear processes is proposed. The non-linear process is initially identified in the form of a non-linear ARMA polynomial. This model is then transformed into a non-linear state affine model which is especially convenient for robustness analysis and design. The model uncertainty is calculated from the non-linear terms of the non-linear state affine model. The uncertainty elements are shown to be a function of the input only and this facilitates the calculation of the uncertainty bounds. The technique is illustrated through a CSTR (continuously stirred tank reactor) control problem.     

Solving facilities location problem in the presence of alternative processing routes using a genetic algorithm

Facilities location problem deals with the optimization of location of manufacturing facilities like machines, departments, etc. in the shop floor. This problem greatly affects performance of a manufacturing system. It is assumed in this paper that there are multiple products to be produced on several machines. Alternative processing routes are considered for each product and the problem is to determine the processing route of each product and the location of each machine to minimize the total distance traveled by the materials within the shop floor. This paper presents a mixed-integer non-linear mathematical programming formulation to find optimal solution of this problem. A technique is used to linearize the formulated non-linear model. However, due to the NP-hardness of this problem, even the linearized model cannot be optimally solved by the conventional mathematical programming methods in a reasonable time. Therefore, a genetic algorithm is proposed to solve the linearized model. The effectiveness of the GA approach is evaluated with numerical examples. The results show that the proposed GA is both effective and efficient in solving the attempted problem.     

Improving neural network models of physical systems through dimensional analysis

The paper presents a technique for generating concise neural network models of physical systems. The neural network models are generated through a two-stage process. The first stage uses information embedded in the dimensions or units in which the data is represented. Dimensional analysis techniques are used initially to make this information explicit, and a limited search in the neural network architecture space is then conducted to determine dimensionless representations of variables/parameters that perform well for a given model complexity. The second stage uses information available in the numerical values of the data to search for high-level dimensionless variables/parameters, generated from simple combinations of dimensionless quantities generated in the first stage and which result in concise neural network models with improved performance characteristics. The search for these high-level dimensionless variables/parameters is conducted in an enhanced representation space using functional link networks with flat or near flat architectures. The use and effectiveness of the technique is demonstrated for three applications. The first is the design and analysis of reinforced concrete beams, which is representative of the class of problems associated with the design and analysis of composites. The second is the classical elastica problem, for predicting non-linear post-buckled behaviour of columns and the third, the analysis of a bent bar under a specified combination of loads.     

基于非线性力反馈模型的软组织变形仿真

虚拟手术仿真研究中反馈力一般采用线性模型,但软组织的反馈力往往呈现出非线性的特点.通过对软组织生物力学特性的分析,将非线性性质应用到软组织力反馈模型的研究.在保持原线性反馈力方向不变的前提下,加入非线性扰动,建立了反馈力的非线性模型,从而使反馈力更加真实.首先构建了人体肝脏物理模型,然后给出了非线性力反馈模型方程,并对人体肝脏物理模型变形和反馈力进行了仿真.为了减小计算量采用了整体刚度矩阵缩减算法.仿真结果表明该模型有较好的非线性和可调节性.     

Non-linear generalised minimum variance control state space design for a second-order Volterra series model

This paper presents a non-linear generalised minimum variance (NGMV) controller for a second-order Volterra series model with a general linear additive disturbance. The Volterra series models provide a natural extension of a linear convolution model with the nonlinearity considered in an additive term. The design procedure is entirely carried out in the state space framework, which facilitates the application of other analysis and design methods in this framework. First, the non-linear minimum variance (NMV) controller is introduced and then by changing the cost function, NGMV controller is defined as an extended version of the linear cases. The cost function is used in the simplest form and can be easily extended to the general case. Simulation results show the effectiveness of the proposed non-linear method.     

Guaranteed set-point computation with application to the control of a sailboat

The problem of characterizing in a guaranteed way the set of all feasible set-points of a control problem is known to be difficult. In the present work, the problem to be solved involves non-linear equality constraints with variables affected by logical quantifiers. This problem is not solvable by current symbolic methods like quantifier elimination, which is commonly used for solving this class of problems. We propose the utilization of guaranteed set-computation techniques based on interval analysis, in particular a solver referred to as Quantified Set Inversion (QSI). As an application example, the problem of simultaneously controlling the speed and the orientation of a sailboat is presented. For this purpose, the combination of QSI solver and feedback linearization techniques is employed.     

基于最小二乘支持向量机的汽轮机故障诊断

提出一种小波包分析与最小二乘支持向量机相结合的汽轮机故障诊断模型．对故障信号功率谱进行小波分解,简化了故障特征向量的提取．用二次损失函数取代支持向量机中的不敏感损失函数,将不等式约束条件变为等式约束．从而将二次规划问题转变为线性方程组的求解．选用RBF函数作为核函数。并提出对核函数的参数进行动态选取。提高了诊断的准确率．仿真结果表明该模型具有较强的非线性处理和抗干扰能力．     

PID和Fuzzy控制相结合的分段复合控制

采用常规PID控制很难实现对大滞后、非线性等难以建立精确数学模型的被控对象的良好控制.为了解决这一难题,该文针对一个非线性二阶系统设计出PID和模糊控制相结合的分段复合控制器,该控制器不需要被控对象的精确数学模型.由于在整个控制过程中的不同阶段采用不同的控制方式,既继承了常规PID控制无静差、静态稳定性好的特点,同时又兼有模糊控制适应能力强、动态性能好的优势.通过MATLAB仿真实验表明,该分段复合控制器能够满足二阶非线性被控对象的控制要求,并取得了良好的控制效果.     

Non-linear finite element analysis of composite beams with interlayer slips

This article presents a new non-linear finite element formulation for the analysis of two-layer composite plane beams with interlayer slips. The element is based on the corotational method. The main interest of this approach is that different linear elements can be automatically transformed to non-linear ones. To avoid curvature locking that may occur for low order element(s), a local linear formulation based on the exact stiffness matrix is used. Five numerical applications are presented in order to assess the performance of the formulation.     

Approximate feedback linearization of discrete-time non-linear systems using virtual input direct design

The design of feedback-linearization and poles-placement controllers for discrete-time non-linear plants, using Input/Output/State measurements only, is typically addressed via indirect design. In this paper we propose the use of a new technique, based on a Virtual Input Direct Design (VID²) approach. The main feature of such a technique is to reduce the control design problem into a standard non-linear mapping approximation problem, without calling for the preliminary construction of an appropriate model of the plant. As compared with the existing methods, the new one requires less computational effort, while taking full advantage of the non-linear approximation software tools already available. In this paper, the new method is described, a simple theoretical analysis is given, and some numerical examples are presented.     

Optimal design of rigid-plastic beams subjected to dynamical loading

In the present paper the optimization problem of dynamically loaded simply supported beams is handled. The concept of a rigid-plastic body is used. The shape of the beam is sought, for which the integral residual deflection for a given time-instant and load is minimal. Two numerical methods for solving the problem are proposed. The numerical results are compared with those obtained by Lepik (1982).