Improved analytical solutions to a stagnation-point flow past a porous stretching sheet with heat generation

This paper investigates the steady laminar flow in a porous medium of an incompressible viscous fluid impinging on a permeable stretching surface with heat generation. The resulting system of coupled non-linear ordinary differential equations is solved analytically via homotopy analysis method (HAM). Analytical results are presented for the wall shear stress and the wall heat transfer coefficient as well as the velocity and temperature profiles for some values of governing parameters such as Prandtl number, stretching parameter, porosity parameter and the heat generation/absorption parameter. Appropriate auxiliary parameter, ?, is determined by minimizing Euclidean norm of residual. The convergence of the obtained series solutions is explicitly studied.     

A bifurcation method to calculate flutter characteristics of a helicopter blade having complex damping

A new method is presented for determining the stability and vibration modes of a class of parametric oscillations where the damping terms in the Mathieu-Hill type of differential equations are complex as well as periodic. The imaginary terms are converted to phase lagged terms in a set of simultaneous first-order differential equations used to express these equations with first-order derivatives treated as additional variables, thereby doubling the number of equations but allowing matrix methods to be conveniently used in a bifurcation procedure to obtain solutions and stability boundaries. The method has advantages over methods using series expansions in determining stability boundaries and vibration modes where convergence and summability become important considerations in the case of differential-difference equations with periodic damping terms. A particular example of a fluttering helicopter blade in slow forward flight is studied and stability boundaries shown to be displaced by up to 10% from those when the nature and extent of the rows of wakes below the rotor are neglected. The range of the parameters governing the stability are given for a specific numerical example. Application to an actual blade motion study shows reconciliation with previous experimental results and theory when the new method is applied. The importance of the effect of lagged arguments in simultaneous differential-difference equations is discussed with reference to two other examples in structural vibrations.     

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.     

功能梯度夹层板弯曲问题的微分求积法

本文基于一阶剪切变性板理论,运用能量交分得到问题的控制方程以及自然边界条件,并运用二维问题的微分求积法对其进行了求解.可以看出对于线性弯曲问题,微分求积法的收敛性很好.     

The graph-theoretic field model—I. Modelling and formulations

The Graph-Theoretical Field Model provides a unifying approach for developing numerical models of field and continuum problems. The methodology examines the field problem from the first stages of conceptualization without recourse to the governing differential equations of the field problem; this is accomplished by deriving discrete statements of the physical laws which govern the field behaviour. There are generally three laws, and these are modelled by the “cutset equations”, the “circuit equations”, and the “terminal equations”. In order to establish these three sets of equations it is expedient first to spatially discretize the field in a manner similar to the finite difference method and then to associate a linear graph (denoted as the field graph) with the spatial discretization. The concept of “through” and “across” variables, which underlies the cutset and circuit equations respectively, enables one to define the graph in an unambiguous manner such that each “edge” of the graph identifies a pair of complementary variables. From a knowledge of the constitutive properties and the boundary conditions of the field it is possible to associate terminal equations with sets of edges. Since the resulting sets of equations represent the field equations, these equations provide the basis for a complete (but approximate) solution to the field or continuum problem. In fact, this system approach uses a two part model: one for the components and another for the interconnection pattern of the components which renders the formulation procedures totally independent of the solution procedure.This paper presents the theoretical basis of the model and several graph-theoretic formulations for steady-state problems. Examples from heat conduction and small- deformation elasticity are included.     

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.     

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.     

The anti-Lagrangian equations: A missing network description

The explicit topological formulation of dynamic equations in terms of scalar functions for RM (M-memristor) networks, called the anti-Lagrangian equations, is introduced. In general, two scalar functions are needed to set the anti-Lagrangian equations. The differential operators acting on these functions bear a certain anti-symmetry relationship with respect to the operators occuring in the standard Lagrangian equations for LC networks. The well-known stationary principles for pure R networks, as well as new quasi-stationary principles for pure M networks are shown to emerge naturally from anti-Lagrangian equations. The form of transformation of variables leaving the form of equations invariant is also studied.     

Stability analysis of a class of stochastic differential delay equations with nonlinear impulsive effects

In this paper, we study stability of a class of stochastic differential delay equations with nonlinear impulsive effects. First, we establish the equivalent relation between the stability of this class of stochastic differential delay equations with impulsive effects and that of a corresponding stochastic differential delay equations without impulses. Then, some sufficient conditions ensuring various stabilities of the stochastic differential delay equations with impulsive effects are obtained. Finally, two examples are also discussed to illustrate the efficiency of the obtained results.     

An adaptive mesh method with variable relaxation time

Moving mesh partial differential equations have been widely used in the last decade for solving differential equations exhibiting large solution variations such as shock waves and boundary layers.In this paper, we have applied a dynamic adaptive method for solving time-dependent differential equations. The mesh velocities are governed by an equation in which a relaxation time is employed to move nodes in such a way that they remain concentrated in regions of rapid variation of the solution. A numerical example involving a blow-up problem shows the advantage of using a variable relaxation time over a fixed one.     

Nonlinear Vibration of Elastic Skew Plates Exhibiting Rectilinear Orthotropy

The governing equations for large amplitude flexural vibrations of orthotropic skew plates are obtained from the corresponding static equations derived in this paper. Making use of an approximation originally due to Berger, corresponding simplified equations are also derived. Considering the large amplitude free flexural vibration of orthotropic skew plates clamped along all the edges, it is shown that the Berger approximation leads to results good enough for engineering purposes. Amplitude vs period curves are presented for different aspect ratios and skew angles of the plate under two in-plane edge conditions. It is observed that the amplitude vs period behaviour is of the hardening type, i.e. period decreases with increasing amplitude.     

上覆单相弹性层的饱和地基上刚性圆板的摇摆振动分析

采用解析的方法研究了上覆单相弹性层的饱和地基上刚性圆板的摇摆振动.首先运用积分边变换技术分别求解了单相弹性介质和饱和介质情况时的控制方程,然后按混合边值条件建立了部分饱和地基上刚性圆板振动的对偶积分方程,并把对偶积分方程化为易于数值求解的第二类Fredholm积分方程,数值算例给出动力柔度系数和摇摆振动转角幅值随无量纲频率变化的曲线,并与完全饱和介质情况进行了对比.数值结果表明,在共振频率附近,弹性层的存在可减弱其振动.     

A more rational approach to the problem of an elastoplastic thick-walled cylinder

A new incremental theory has been developed for solving the problem of partially yielded thick-walled cylinders. Incremental stresses and strains are directly used as variables, hence numerical differentiation in the evaluation of stresses and strains is not required. The stresses and strains in all principal directions can be computed at the same time from governing equations for each increment of loading. Since the consideration of loading history is involved, the present theory is particularly suitable for predicting stress and strain distribution of a thick-walled cylinder subjected to nonproportionate loading.     

Optimal stabilization for differential systems with delays - Malkin’s approach

The paper considers a process controlled by a system of delayed differential equations. Under certain assumptions, a control function is determined such that the zero solution of the system is asymptotically stable and, for an arbitrary solution, the integral quality criterion with infinite upper limit exists and attains its minimum value in a given sense. To solve this problem, Malkin’s approach to ordinary differential systems is extended to delayed functional differential equations, and Lyapunov’s second method is applied. The results are illustrated by examples, and applied to some classes of delayed linear differential equations.     

Analysis of trajectory errors in integrating ordinary differential equations

A method of analyzing and interpreting trajectory errors in the numerical solution of ordinary differential equations by digital computers is discussed. Truncation in integrating a set of differential equations leads to errors in the trajectory of the solution. An explanation is given for the use of diagrams in the complex plane to evaluate errors in the trajectory, with a discussion of the properties of a number of frequently used integration formulas via the diagrams. The diagrams portray the characteristics of an integration method in more detail than do the absolutely stable regions presented by Dahlquist. Based on the diagrams, guidelines are listed as to how to choose a proper integration formula for the given set of differential equations. A method is presented to check whether or not the numerical solution is satisfactory.     

Formulation of stable difference schemes for systems of initial-value partial differential equations

A general system of initial-value partial differential equations is classified into four categories based on the partial differential operators which define the equations. Specific combinations of the operators are termed “invariants” since they are common to all finite difference approximations to the system of equations. The “invariants” are used to a priori determine if one may formulate a stable difference approximation to a system of partial differential equations. This is in essence a numerical existence theory.     

Numerical investigation on a new local preconditioning method for solving the incompressible inviscid, non-cavitating and cavitating flows

A locally power-law preconditioning algorithm is developed. This is applied to compute incompressible inviscid, steady-state, non-cavitating and cavitating flows. The preconditioning parameters are adapted automatically from the pressure of computational domain. This method suggests better convergence rates rather than the standard artificial compressibility and the standard preconditioning method. Single-fluid Euler equations, cast in their conservative form, along with the barotropic cavitation model are employed. The cell-centred Jameson's finite volume discretization technique is used to solve the preconditioned governing equations. The stabilization is achieved via the second and fourth, order artificial dissipation scheme. Explicit four-stage Runge-Kutta time integration is applied to find the steady-state condition. In this paper, the method is assessed through simulations of incompressible inviscid, steady-state, non-cavitating and cavitating flows over a 2D NACA0012 and a 2D NACA66(MOD)+a=0.8 hydrofoil section. The results show satisfactory agreement with others numerical and experimental works in pressure distribution and hydrodynamic forces. Using the power-law preconditioner decreases the convergence rate significantly. In addition, information such as the effects of the new locally power-law preconditioner, the effects of the artificial dissipation terms, and the effects of the artificial compressibility parameter, on convergence speed and solution accuracy is highlighted.     

Orthogonal functions approach to optimal control of delay systems with reverse time terms

Using block-pulse functions (BPFs)/shifted Legendre polynomials (SLPs) a unified approach for computing optimal control law of linear time-varying time-delay systems with reverse time terms and quadratic performance index is discussed in this paper. The governing delay-differential equations of dynamical systems are converted into linear algebraic equations by using operational matrices of orthogonal functions (BPFs and SLPs). The problem of finding optimal control law is thus reduced to the problem of solving algebraic equations. One example is included to demonstrate the applicability of the proposed approach.     

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.