Numerical solution of systems of mixed partial differential equations associated with transport phenomena with exchange

The so-called “percolation operations” which involve simultaneous heat and mass transport are defined in terms of a system of mixed partial differential equations which are nonlinear and include variable coefficients. Second-order interpolation followed by integration over an increment of the time or space variable is proposed as a method for arriving at finite difference equation equivalents of the original partial differential equations. A model change is also used to give physical significance to finite difference equivalents associated with derivatives in the space dimension. It is shown that second-order interpolation has the advantage over linear interpolation that it introduces a factor β which in a stability analysis based on cencentrations offers a range of choices that will insure stable calculation on a digital machine.     

Non-linear vibration analysis of laminated composite plates resting on non-linear elastic foundations

In this study, the homotopy analysis method (HAM) is used to obtain an approximate analytical solution for geometrically non-linear vibrations of thin laminated composite plates resting on non-linear elastic foundations. Geometric non-linearity is considered using von Karman’s strain-displacement relations. Then, the effects of the initial deflection, ply properties, aspect ratio of the plate and foundation parameters on the non-linear free vibration is studied. Comparison between the obtained results and those available in the literature demonstrates the potential of HAM for the analysis of such vibration problems, whose governing differential equations include the quadratic and cubic non-linear terms. This study shows that only a first-order approximation of the HAM leads to highly accurate solutions for this type of non-linear problems.     

Dynamics of impulsive neutral-type BAM neural networks

In this letter, the existence and the global exponential stability of piecewise pseudo almost periodic solutions (PAP_T) for bidirectional associative memory neural networks (BAMNNs) with time-varying delay in leakage (or forgetting) terms and impulsive are investigated by applying contraction mapping fixed point theorem, the exponential dichotomy of linear differential equations and differential inequality techniques. Furthermore, we give an explanatory example to illustrate the efficiency of the theoretical predictions.     

Nonlinear dynamic modeling and responses of a cable dragged flexible spacecraft

The space debris removal system (SDRS) of tethered space tug is modelled as a cable dragged flexible spacecraft. The main goal of this paper is to develop a dynamic modeling approach for mode characteristics analysis and forced vibration analysis of the planar motion of a cable dragged flexible spacecraft. Solar arrays of the spacecraft are modelled as multi-beams connected by joints with additional rotating spring where the nonlinear stiffness, damping and friction are considered. Using the Global mode method (GMM), a novel analytical and low-dimensional nonlinear dynamic model is developed for vibration analysis of SDRS to enhance the design capacity for better fulfillment of space tasks. The linear and nonlinear partial differential equations that governing transverse vibration of solar arrays, transverse and longitudinal vibrations of cable are derived, along with the matching and boundary conditions. The natural frequencies and analytical global mode shapes of SDRS are determined, and orthogonality relations of the global mode shapes are established. Dynamical equations of the system are truncated to a set of ordinary differential equations with multiple-DOF. The validity of the method is verified by comparing the natural frequencies obtained from the characteristic equation with those obtained from FEM. Interesting mode localization and mode shift phenomena are observed in mode analysis. Dynamic responses of the system excitated by fluctuation of attitude control torque and short-time attitude control torque are worked out, respectively. Nonlinear behaviors are observed such as hardening, jump and super-harmonic resonances. Residual vibration of the overall system with considering the varous values of nonlinear stiffness, damping coefficient and friction coefficient has shown that the nonlinearity of joints has a great influence on the vibration of the overall system.     

Representation and realization of operational differential equations with time-varying coefficients

An algebraic treatment of operational differential equations with time-varying coefficients is presented in terms of skew rings of differential polynomials defined over a Noetherian ring. Included in this framework are delay differential equations with time- varying coefficients. The operator equations are characterized by transfer matrices which are utilized to construct realizations given by first-order vector differential equations with operator coefficients. It is shown that the realization of matrix equations can be reduced to the realization of scalar equations. Finally, a simple procedure is derived for realizing scalar equations.     

Analysis of positive fractional-order neutral time-delay systems

In this paper, we consider an initial value problem for linear matrix coefficient systems of the fractional-order neutral differential equations with two incommensurate constant delays in Caputo’s sense. Firstly, we introduce the exact analytical representation of solutions to linear homogeneous and non-homogeneous neutral fractional-order differential-difference equations system by means of newly defined delayed Mittag–Leffler type matrix functions. Secondly, a criterion on the positivity of a class of fractional-order linear homogeneous time-delay systems has been proposed. Furthermore, we prove the global existence and uniqueness of solutions to non-linear fractional neutral delay differential equations system using the contraction mapping principle in a weighted space of continuous functions with regard to classical Mittag–Leffler functions. In addition, Ulam–Hyers stability results of solutions are attained based on fixed-point approach. Finally, we present an example to demonstrate the applicability of our theoretical results.     

Uncertain impulsive functional differential systems of fractional order and almost periodicity

The problem of existence of almost periodic solutions of uncertain impulsive functional differential systems of fractional order is investigated. Using the Lyapunov method combined with the concept of uniformly positive definite matrix functions and Hamilton–Jacobi–Riccati inequalities new criteria are presented. The robust stability of the almost periodic solution is also discussed. We apply our results to an impulsive Lasota–Wazewska type model of fractional order. Our results extend the theory of almost periodic solutions for impulsive delay differential equations to the fractional-order case under uncertainty.     

Modal vibration decomposition method and its application on multi-mode vibration control of high-speed railway car bodies

The vibration of a railway car body is a superposition of the vibrations of its various modes. It is typically easy to obtain the physical vibration of the car body using sensors in an in situ or a simulated test vehicle. However, it is difficult to determine the modal vibration of the body and its contribution. There are no effective multi-mode vibration control methods for the car bodies. This study proposes a modal vibration decomposition method (MVDM) based on singular value decomposition (SVD) and least squares fitting (LSF). Accordingly, the physical vibration of a railway car body is decomposed into modal vibrations. A method for calculating the modal contribution factor (MCF) is presented, and the dominant flexible modes of the car body are determined and considered the target for the vibration control method. Several pieces of equipment are considered as dynamic vibration absorbers (DVAs) to control the multi-mode vibration of the car body using the dynamic vibration absorption theory and determine the installation parameters of the individual equipment. Finally, the effectiveness of vibration control is verified through dynamic simulations. The results demonstrate the effective decomposition of the physical vibration of the car body into various modal vibrations using the MVDM. This provides accurate data for the MCF calculation and determination of the flexible modes of the car body. The proposed method reduces the vibration of the target modes and improves the ride quality of the railway vehicle. At the optimal damping ratio, the vibration of the DVA-based equipment itself is acceptable. This allows for multi-mode vibration control without requiring extensive modification to the car body structure or suspension system parameters of the vehicle.     

Approximations to the solution of linear Fredholm integrodifferential-difference equation of high order

There are few techniques available to numerically solve linear Fredholm integrodifferential-difference equation of high-order. In this paper we show that the Taylor matrix method is a very effective tool in numerically solving such problems. This method transforms the equation and the given conditions into the matrix equations. By merging these results, a new matrix equation which corresponds to a system of linear algebraic equation is obtained. The solution of this system yields the Taylor coefficients of the solution function. Some numerical results are also given to illustrate the efficiency of the method. Moreover, this method is valid for the differential, difference, differential-difference and Fredholm integral equations. In some numerical examples, MAPLE modules are designed for the purpose of testing and using the method.     

Response and stability of SDOF viscoelastic system under wideband noise excitations

The response and stability of a single degree-of-freedom (SDOF) viscoelastic system with strongly nonlinear stiffness under the excitations of wideband noise are studied in this paper. Firstly, terms associated with the viscoelasticity are approximately equivalent to damping and stiffness forces; the viscoelastic system is approximately transformed to SDOF system without viscoelasticity. Then, with application of the method of stochastic averaging, the averaged Itô differential equation is obtained. The stationary response and the largest Lyapunov exponent can be analytically expressed. The effects of different system parameters on the response and stability of the system are discussed as well.     

An integral-equation approach to problems of vibrating beams

In this paper integral equations are applied for the calculation of the normal modes of vibrating beams. Both exact and approximate methods of solving the integral equation are considered. The Green's function, or kernel, of the integral equation is constructed for both uniform and nonuniform beams. Solutions for the normal modes of a uniform cantilever are given. A nonuniform, naturally-twisted turbine blade is studied in detail and the first and second normal modes are calculated by the integral-equation method.     

Approximate solution (with error bounds) to a nonlinear,nonautonomous second-order differential equation

Two approximations are developed to the solution of an important nonlinear, nonautonomous second-order differential equation that arises in various fields of science and technology such as operations research, mathematical ecology and epidemiology. The origin of the second-order differential equation from a system of two nonlinear first-order differential equations modelling, for example, Lanchester-type combat between two homogeneous military forces is discussed. Extension of our results to a more general system of nonlinear first-order differential equations is indicated. Error bounds that do not require that the exact solution be known are developed. Some connections between our results and those for the Liouville-Green (or WKB) approximation to the solution of the linear second-order equation are indicated.     

pth Moment exponential stability of impulsive stochastic functional differential equations and application to control problems of NNs

This paper investigates the pth moment exponential stability of impulsive stochastic functional differential equations. Some sufficient conditions are obtained to ensure the pth moment exponential stability of the equilibrium solution by the Razumikhin method and Lyapunov functions. Based on these results, we further discuss the pth moment exponential stability of generalized impulsive delay stochastic differential equations and stochastic Hopfield neural networks with multiple time-varying delays from the impulsive control point of view. The results derived in this paper improve and generalize some recent works reported in the literature. Moreover, we see that impulses do contribute to the stability of stochastic functional differential equations. Finally, two numerical examples are provided to demonstrate the efficiency of the results obtained.     

Switching fault-tolerant control of a moving vehicle-mounted flexible manipulator system with state constraints

In this brief, a switching fault-tolerant control (FTC) scheme is presented for a moving vehicle-mounted flexible manipulator subject to state constraints. The dynamic characteristics of the system are represented by coupled ordinary differential equations and partial differential equations (ODEs–PDEs). When actuators are healthy, vibration control and position regulations can be realized without violation of the given constraints based on a Barrier Lyapunov Function (BLF). Moreover, a switching strategy is introduced to prevent the transgression of constraints even under actuator failure by detecting actuator faults as-assisted by the proposed monitoring functions. The closed-loop states are kept within the given bounds under FTC laws. By extending LaSalle's Invariance Principle to an infinite dimension, the asymptotic stability of the fault-free closed-loop system is strictly verified. Simulation results demonstrate the effectiveness of the proposed approach.     

Non-linear dynamic response of circular plates subjected to transient loads

The Chebyshev polynomials have been applied to the large amplitude motions of circular plates under transient loads, with and without damping. The non- linear differential equations are linearized by using Taylor series expansions for one of the terms. It is shown that there is good agreement between the results obtained by the present technique and the available results. The advantage of this technique is essentially due to the fact that the Chebyshev polynomials are rapidly converging polynomials. It is shown that very accurate results can be obtained with only four terms of the Chebyshev series which may not be possible with conventional methods.     

Practical stability in terms of two measures for fractional order dynamic systems in Caputo's sense with initial time difference

The initial time difference practical stability in terms of two measures has been investigated for nonlinear fractional differential equations in Caputo's sense and these properties have been unified with Lyapunov-like functions to establish a comparison result.     

Optimal location and controller design of STATCOM for power system stability improvement using PSO

The optimal location of a static synchronous compensator (STATCOM) and its coordinated design with power system stabilizers (PSSs) for power system stability improvement are presented in this paper. First, the location of STATCOM to improve transient stability is formulated as an optimization problem and particle swarm optimization (PSO) is employed to search for its optimal location. Then, coordinated design problem of STATCOM-based controller with multiple PSS is formulated as an optimization problem and optimal controller parameters are obtained using PSO. A two-area test system is used to show the effectiveness of the proposed approach for determining the optimal location and controller parameters for power system stability improvement. The nonlinear simulation results show that optimally located STATCOM improves the transient stability and coordinated design of STATCOM-based controller and PSSs improve greatly the system damping. Finally, the coordinated design problem is extended to a four-machine two-area system and the results show that the inter-area and local modes of oscillations are well damped with the proposed PSO-optimized controllers.     

Finite-time boundedness and control of positive coupled differential-difference equations with bounded time-varying delay

This paper is concerned with the problems of finite-time boundedness and finite-time control for positive coupled differential-difference equations (CDDEs) with bounded time-varying delay. The finite-time stability of such systems is analyzed by constructing an estimate of the solutions over a finite time interval. And, sufficient conditions based on linear programming (LP) are provided for finite-time stability of positive CDDEs with bounded time-varying delay. Then, by coordinate transformation, the obtained results are extended to the finite-time bounedness of positive CDDEs with bounded time-varying delay. By the obtained result of finite-time boundedness, static output-feedback controllers and static state-feedback controllers are designed and a sufficient condition is derived to ensure the positivity and finite-time boundedness of closed-loop system. Three illustrative examples are given to show the validity of our results.     

Parameter plane control design for a stirred-tank chemical system with slow-fast multirate sampling

The control of a multirate sampled-data, stirred-tank chemical reactor system using a parameter plane method is considered. Due to wide acceptance of proportional-plus- integral-plus-derivative (PID) control in the chemical process industries, a PID controller with a “slow-fast” multirate scheme is used for the chemical reactor system. Based on two related stability equations and using the PID gains as the adjustable parameters, the set of all possible PID gains to maintain the chemical reactor system's stability, and at the same time, to ensure the system has a specified gain margin, phase margin, damping ratio and damping factor is determined. The effects of changing the integer N (which is the ratio of the sampling rates between a slow-and a fast-sampler) and the basic sampling period T on the set of PID gains are examined and the results for single-rate and multirate cases are also studied.     

启示性方法用于分析有限差分方程的计算稳定性（英文）

这篇文章通过一些典型例子讨论了在用启示性方法时从原偏微分方程推导来的稳定性条件与从差分方程展开式推导来的稳定性条件间的不同点。结果表明，对于部分有限差分方程，在用启示性方法分析其计算稳定性的过程中最好采用从差分方程推导来的展开式以期得到较合理的结果。在文章的另一部分，反证法的运用表明了从启示性方法推导来的稳定性条件并非全都是必要条件，在应用中应引起注意。