An integral-equation solution for the finite deflection of clamped skew sandwich plates

The large-deflection behaviour of skew sandwich plates is governed by a system of five coupled nonlinear partial differential equations which are highly complex in nature. In the reported study, this problem is analysed using an integral-equation approach. The integral equations of beams along the skew directions is used with appropriate boundary conditions to transform the governing nonlinear partial differential equations into a set of nonlinear algebraic equations. These equations are then solved using an iterative scheme suggested by Brown. The results obtained by this method are compared with available results of other investigators and the agreement is found to be good. Load-deflection characteristics have been presented for clamped skew sandwich plates.     

Large deflection analysis of skew plates by lumped triangular element formulation

A finite difference scheme with triangular mesh is presented for the analysis of skew plate problems with large deflections. The suggested formulation is independent of the boundary condition and uses energy principles to derive a set of nonlinear algebraic equations which are solved by using Newton-Raphson iterative method with incremental loading. The investigation is concerned with the behaviour of constant thickness clamped and simply supported isotropic skew plates with immovable edges and subjected to uniformly distributed transverse load. The effects of skew on plates with large deflections are investigated and comparisons are made with existing results; good agreement is shown.     

Reduction of stable differential–algebraic equation systems via projections and system identification

Most large-scale process models derived from first principles are represented by nonlinear differential–algebraic equation (DAE) systems. Since such models are often computationally too expensive for real-time control, techniques for model reduction of these systems need to be investigated. However, models of DAE type have received little attention in the literature on nonlinear model reduction. In order to address this, a new technique for reducing nonlinear DAE systems is presented in this work. This method reduces the order of the differential equations as well as the number and complexity of the algebraic equations. Additionally, the algebraic equations of the resulting system can be replaced by an explicit expression for the algebraic variables such as a feedforward neural network. This last property is important insofar as the reduced model does not require a DAE solver for its solution but system trajectories can instead be computed with regular ODE solvers. This technique is illustrated with a case study where responses of several different reduced-order models of a distillation column with 32 differential equations and 32 algebraic equations are compared.     

Nonlinear bending of beams using the finite element method

In this paper a finite element formulation for determining the finite deflection of thin bars is presented. The nonlinear stiffness equations are generated after simple approximate expressions involving the nodal parameters are used to replace the nonlinear terms in the energy functional. The procedure used results in a simplified set of nonlinear algebraic equations which are more amenable to solution than the equations usually presented. The applicability and accuracy of the method together with an evaluation of three incremental solution techniques, a step by step method, a one step Newton-Raphson procedure, and a variable interpolation technique is demonstrated by solving a cantilever beam with a point load acting on the end. Curves showing the sensitivity to increment size and to the number of elements are also presented. The results indicate that the formulation is accurate and inexpensive in terms of computational effort.     

Design of nonlinear regulators for linear plants

A procedure for designing stable nonlinear control laws for linear plants is presented. The design is based on an inverse optimal control problem with the closed form controller being directly given by the solution of a set of linear algebraic equations. The nonlinear controllers can be used to produce saturation effects on particular states, and an example is included to illustrate the results.     

基于简化ASM1模型的污水处理厂曝气系统优化控制

本文建立了基于ASM1模型的曝气系统简化数学模型,在此基础上提出了以曝气能量消耗最小为目标函数的曝气系统优化控制问题。采用联立配置法进行优化问题的求解,把非线性微分代数方程组的DAE系统转化为非线性代数方程组,将动态优化问题转化为非线性规划问题,最后调用IPOPT解法器求解。在动态入水的条件下进行曝气池的优化控制仿真,其结果显示比传统定值PID控制可节约近40%的能耗。     

A finite element large displacement analysis of stiffened plates

A finite element analysis of the large deflection behaviour of stiffened plates using the isoparametric quadratic stiffened plate bending element is presented. The evaluation of fundamental equations of the stiffened plates is based on Mindlin's hypothesis. The large deflection equations are based on von Kármán's theory. The solution algmrithm for the assembled nonlinear equilibrium equations is based on the Newton-Raphson iteration technique. Numerical solutions are presented for rectangular plates and skew stiffened plates.     

Thermal post-buckling characteristics of clamped skew plates

Thermal post-buckling characteristics of clamped skew plates with restrained edges subjected to planar temperature distributions are studied. The problem is formulated in terms of Von Kármán equations expressed in terms of displacements generalised to include thermal effects. A perturbation method is employed to obtain a linear set of partial differential equations. The solution to this linear set is then obtained by using the Galerkin method. Numerical results are presented for different skew plate configurations for both uniform and two-dimensional temperature distributions. Plots of central deflection, membrane and bending stress distributions are presented and the post-buckling characteristics are discussed in detail.     

Tracing post-limit-point paths with reduced basis technique

A reduced basis technique and a problem-adaptive computational algorithm are presented for predicting the post-limit-point paths of structures. In the proposed approach the structure is discretized by using displacement finite element models. The nodal displacement vector is expressed as a linear combination of a small number of vectors and a Rayleigh-Ritz technique is used to approximate the finite element equations by a small system of nonlinear algebraic equations.To circumvent the difficulties associated with the singularity of the stiffness matrix at limit points, a constraint equation, defining a generalized arc-length in the solution space, is added to the system of nonlinear algebraic equations and the Rayleigh-Ritz approximation functions (or basis vectors) are chosen to consist of a nonlinear solution of the discretized structure and its various order derivatives with respect to the generalized arc-length. The potential of the proposed approach and its advantages over the reduced basis-load control technique are outlined. The effectiveness of the proposed approach is demonstrated by means of numerical examples of structural problems with snap-through and snap-back phenomena.     

Large elastic deformations of clamped skewed plates by dynamic relaxation

This investigation is concerned with the nonlinear behaviour of clamped Isotropic skew plates of constant thickness subjected to a uniformly distributed transverse load. The recently developed numerical technique of dynamic relaxation has been adopted for the analysis. A detailed study of the large deflection behaviour of skew plates has been made by varying three parameters, viz. skew angle, load, and aspect ratio. Numerical results have been compared with the available solutions. Representative nondimensional solutions are presented in the form of graphs to elucidate the nonlinear effect due to large deflection at higher loads.     

不确定线性系统混合H2/H∞鲁棒输出反馈控制

解决了在系统状态空间模型的状态与输出矩阵中含有范数有界 参数不确定线性系统的混合Ｈ２／Ｈ∞鲁棒输出反馈控制问题,所推导的满阶控制器对于所有呆容许的参数不确定都能满足给定的Ｈ∞干扰衰减水平,且为最坏情形Ｈ２代价函数提供了一个最优的上界,所得的结果需要求解一个含有尺度参数的修正代数Ｒｉｃｃａｔｉ方程以及三个含有尺度参数的交叉耦合非线性方程,而且也给出了一个求解这些含有尺度参数非线性方程的数值算法。     

Accelerating generators of iterative methods for finding multiple roots of nonlinear equations

Two accelerating generators that produce iterative root-finding methods of arbitrary order of convergence are presented. Primary attention is paid to algorithms for finding multiple roots of nonlinear functions and, in particular, of algebraic polynomials. First, two classes of algorithms for solving nonlinear equations are studied: those with a known order of multiplicity and others with no information on multiplicity. We also demonstrate the acceleration of iterative methods for the simultaneous approximations of multiple roots of algebraic polynomials. A discussion about the computational efficiency of the root-solvers considered and three numerical examples are given.     

A triangular finite difference scheme for large deflection problems

A lumped triangular element formulation is developed based on a finite difference approach for the large deflection analysis of plates and shallow shells. The presented formulation is independent of the boundary condition (unlike the finite difference formulation) and uses energy principles to derive a set of nonlinear algebraic equations which are solved by using an incremental Newton-Raphson iterative procedure. A study of the large deflection behaviour of thin plates is made for various edge conditions and aspect ratios, and the results obtained are compared with those using a finite element scheme. Representative nondimensional solutions for deflections and stresses are presented in the form of graphs.     

Hybrid extended Fourier series for optimal control of nonlinear algebraic dynamical systems

The paper introduces a new method for finding optimal control of algebraic dynamic systems. The structure of algebraic dynamical systems is nonlinear with quadratic and bilinear terms. A new hybrid extended Fourier series is introduced, and state and control variables of the system are expanded by this series. Moreover, properties of new series are presented, and integration and product operational matrices are obtained. Using operational matrices, optimal control of the systems is converted to a set of simultaneous nonlinear algebraic relations. An illustrative example is included to compare our results with those in the literature.     

Large deflection analysis of clamped skew sandwich plates by parametric differentiation

The differential equations governing the large deflection behavior of skew sandwich plates are highly complex in nature and do not lend themselves for easy solution. In the study reported herein, this problem is solved by a numerical technique known as “parametric differentiation”. The non-linear differential equations in terms of displacements are transformed into a set of linear differential equations with variable coefficients. These are solved to give the gradients of the displacements in the load direction. The subsequent solution of a set of initial value problems yield the displacements proper. The results obtained by this method are compared with available results of other investigators and the agreement is found to be good. Load deflection characteristics have been presented for clamped skew sandwich plates.     

Robust nonlinear control of spacecraft formation flying using constraint forces

A robust nonlinear control method is presented for spacecraft precise formation flying.With the constraint forces and consid-ering nonlinearity and perturbations,the problem of the formation keeping is changed to the Lagrange systems with the holonomic constraints and the differential algebraic equations (DAE).The nonlinear control laws are developed by solving the DAE.Because the traditional numerical solving methods of DAE are very sensitive to the various errors and the resulting con-trol laws are not ro...     

Finite‐time sliding mode observer for uncertain nonlinear systems based on a tunable algebraic solver

In this paper, a finite‐time sliding mode observer for nonlinear systems with unknown inputs is proposed. The observer is based on a method for the solution of time‐varying algebraic equations. This algebraic solver is shown to converge in finite time by means of Lyapunov analysis; furthermore, a way to tune it so that it converges after a user‐defined amount of time is presented. Through the use of this technique and sliding mode differentiators, the state variables and unknown inputs of a class of nonlinear systems, which do not need to be affine in the inputs, can be estimated without the explicit use of state transformations. Both the algebraic solver and the proposed observer are illustrated through simulation examples. Copyright © 2017 John Wiley & Sons, Ltd.     

Scaling of general quadratic matrix equations

A scaling framework for general quadratic algebraic matrix equations is presented. All algebraic quadratic equations can be considered as special cases of a single generalized algebraic quadratic matrix equation (GQME). Hence, the paper is focused on the analysis and solution of the scaling problem of that GQME. The presented scaling method is based on the assignment of predetermined values of the coefficients and the unknown matrices of the GQME. The proposed framework is independent of any numerical method and therefore its use is general. Implementations are presented for the special case of matrix algebraic Riccati equations (AREs). Some new results of matrix algebraic identities considering Kronecker and Hadamard products are also reported.     

Smooth function approximation using neural networks

An algebraic approach for representing multidimensional nonlinear functions by feedforward neural networks is presented. In this paper, the approach is implemented for the approximation of smooth batch data containing the function's input, output, and possibly, gradient information. The training set is associated to the network adjustable parameters by nonlinear weight equations. The cascade structure of these equations reveals that they can be treated as sets of linear systems. Hence, the training process and the network approximation properties can be investigated via linear algebra. Four algorithms are developed to achieve exact or approximate matching of input-output and/or gradient-based training sets. Their application to the design of forward and feedback neurocontrollers shows that algebraic training is characterized by faster execution speeds and better generalization properties than contemporary optimization techniques.