Green's functions for forced vibration analysis of bending-torsion coupled Timoshenko beam

A method based on Green's functions is proposed for the analysis of the steady-state dynamic response of bending-torsion coupled Timoshenko beam subjected to distributed and/or concentrated loadings. Damping effects on the bending and torsional directions are taken into account in the vibration equations. The elastic boundary conditions with bending-torsion coupling and damping effects are derived and the classical boundary conditions can be obtained by setting the values of specific stiffness parameters of the artificial springs. The Laplace transform technology is employed to work out the Green's functions for the beam with arbitrary boundary conditions. The Green's functions are obtained for the beam subject to external lateral force and external torque, respectively. Coupling effects between bending and torsional vibrations of the beam can be studied conveniently through these analytical Green's functions. The direct expressions of the steady-state responses with various loadings are obtained by using the superposition principle. The present Green's functions for the Timoshenko beam can be reduced to those for Euler–Bernoulli beam by setting the values of shear rigidity and rotational inertia. In order to demonstrate the validity of the Green's functions proposed, results obtained for special cases are given for a comparison with those given in the literature and they agree with each other exactly. The influences of external loading frequency and eccentricity on Green's functions of bending-torsion coupled Timoshenko beam are investigated in terms of the numerical results for both simply supported and cantilever beams. Moreover, the symmetric property of the Green's functions and the damping effects on the amplitude of Green's functions of the beam are discussed particularly.     

Classification of stability-like concepts and their study using vector Lyapunov functions

In the first section, stability-like definitions for ordinary differential equations are derived from a general qualitative concept. It is shown that the classical definitions of stability in the sense of Lyapunov, and their extensions can easily be deduced from this general formulation. A classification of all the definitions which may be derived is proposed.The second section contains the main results of this paper. It deals with the “comparison method” based upon one of T. Wazewski's theorems on differential inequalities. Several authors have used this method in order to investigate stability-like properties. We display the structure of this method, in order to state and prove some general comparison principles. These apply to the class of concepts considered earlier.In the last section some new results about stability and attractivity of sets are obtained as examples for the comparison principles. A theorem on stability in tube-like domains is proved in order to emphasize the generality and the flexibility of the comparison method.     

Reduced order modeling of nonlinear time periodic systems subjected to external periodic excitations

In this work, new methodologies for order reduction of nonlinear systems with periodic coefficients subjected to external periodic excitations are presented. The periodicity of the linear terms is assumed to be non-commensurate with the periodicity of forcing vector. The dynamical equations of motion are transformed using the Lyapunov–Floquet (L–F) transformation such that the linear parts of the resulting equations become time-invariant while the forcing and nonlinearity takes the form of quasiperiodic functions. The techniques proposed here construct a reduced order equivalent system by expressing the non-dominant states as time-varying functions of the dominant (master) states. This reduced order model preserves stability properties and is easier to analyze, simulate and control since it consists of relatively small number of states in comparison with the large scale system.Specifically, two methods are discussed to obtain the reduced order model. First approach is a straightforward application of linear method similar to the ‘Guyan reduction’. The second novel technique proposed here extends the concept of ‘invariant manifolds’ for the forced problem to construct the fundamental solution. Order reduction approach based on this extended invariant manifold technique yields unique ‘reducibility conditions’. If these ‘reducibility conditions’ are satisfied only then an accurate order reduction via extended invariant manifold approach is possible. This approach not only yields accurate reduced order models using the fundamental solution but also explains the consequences of various ‘primary’ and ‘secondary resonances’ present in the system. One can also recover ‘resonance conditions’ associated with the fundamental solution which could be obtained via perturbation techniques by assuming weak parametric excitation. This technique is capable of handling systems with strong parametric excitations subjected to periodic and quasi-periodic forcing. It is anticipated that these order reduction techniques will provide a useful tool in the analysis and control system design of large-scale parametrically excited nonlinear systems subjected to external periodic excitations.     

A general population growth model with density dependence

A model for the growth of a population with p+q+r age groups in which there is competition for limited resources is considered. The steady-state solution is obtained and its stability is discussed. The existence of a time-invariant structure in which the ratios of the populations of the various age groups do not change with time is established under very general conditions, and its relation with the steady-state solution is discussed. The conditions under which we can treat the population as homogeneous with a common birth rate, a common death rate and a common inhibiting constant are also discussed.     

Probability,Defectives and Mail Ordering

Probabilistic methods are applied to the problems facing the customer attempting to order items from a distributor where the probability of any item being faulty is a constant, q. The model involves only faulty items being returned by the customer in each round, and it is suggested that the situation being analysed is fairly general, conferming, for example, to that of the mail order customer exemplified by the record collector. Calculations are provided showing the extent of benefit from delaying purchase until a large order has accumulated.     

The relaxation-time limit in the quantum hydrodynamic equations for semiconductors

The relaxation-time limit from the quantum hydrodynamic model to the quantum drift-diffusion equations in R³ is shown for solutions which are small perturbations of the steady state. The quantum hydrodynamic equations consist of the isentropic Euler equations for the particle density and current density including the quantum Bohm potential and a momentum relaxation term. The momentum equation is highly nonlinear and contains a dispersive term with third-order derivatives. The equations are self-consistently coupled to the Poisson equation for the electrostatic potential. The relaxation-time limit is performed both in the stationary and the transient model. The main assumptions are that the steady-state velocity is irrotational, that the variations of the doping profile and the velocity at infinity are sufficiently small and, in the transient case, that the initial data are sufficiently close to the steady state. As a by-product, the existence of global-in-time solutions to the quantum drift-diffusion model in R³ close to the steady-state is obtained.     

Structural elastic and creep models of a UD composite in longitudinal shear

The transient creep of a UD composite with a quadratic arrangement of elastic fibers of quadratic cross section is investigated. The deformational properties of the composite are determined from the known properties of its constituents. A structural model of the UD composite is developed, whose minimal elementary cell contains four elements. The stress-strain state of the elements is assumed homogeneous. Two types of basic and resolving governing equations of transient creep are deduced, which are based on static or kinematic assumptions. In each of the cases, a formula for the longitudinal elastic shear modulus of the composite is found. The stationary solutions of creep equations allow one to obtain formulas of the steady-state creep of the composite in a form similar to Norton’s law. Numerical calculations are also performed, and a comparison of the results with data given in the literature bears witness to the efficiency of the models developed and the solutions obtained. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 43, No. 4, pp. 437–448, July–August, 2007.     

Modelling of closed-chain manipulators on an excavator vehicle

The main focus of this paper is to develop a physics-based model for a closed-chain manipulator in an excavator vehicle. The derivation of closed-chain manipulator dynamic equations with a structure similar to open-chain manipulator equations is an important research problem, particularly with reference to controller design. In this paper, an approach for deriving closed-chain manipulator equations with an open-chain structure, based on trigonometric t-formulae, is presented. Holonomic loop closure constraints are employed in order to derive the closed-chain mechanism dynamics from the reduced system dynamics. The closed-chain equations, with a structure similar to serial link equations, are presented. The model incorporates the dynamic properties of the manipulator and bucket. The dynamic model for the excavation system is validated against measured data obtained from a full-scale closed-chain excavator vehicle. A dynamic model is important for the design of control strategies for trajectory tracking, a key requirement for automating the excavation task. It is noted that even though the results presented in this paper are focused on a particular excavator vehicle, the research is generic and can be adapted to any closed-chain manipulator.     

A two-regime traffic flow model and the consistency of its parameters

An attempt is made to demonstrate the traffic behaviour and phenomena under normal morning peak period conditions, and to examine the suitability of a two-regime traffic flow model for these conditions. This paper has three main parts. First, the consistency of flow and concentration patterns of a 9-mile freeway section is examined and provides a basis for distinguishing between the free-flow and the congested-flow regimes. This distinction clearly indicates the data points obtained from traffic flow situations, which, in time-sequence, approach maximum flow conditions, congested conditions, and through a recovery process backwards to free-flow conditions.

In the second part, a car-following model for the two-regime approach is introduced. By using the analysis of driver performance as a sensitivity measurement, model parameters are defined and evaluated. An overall comparison between the proposed and known generalized car-following models emphasizes the advantages of the proposed model, particularly its simplicity and clarity at both the micro- and macroscopic levels, for the two-regime phenomenon.

In the third part, the steady-state formulation, derived from the proposed car-following model, is evaluated by using the time-sequence data points. The consistency of the two-regime model parameters is apparently well preserved regarding data sets of 3-year period (1972–1975) with respect to three independent variables: years, workdays and locations.     

Analysis on existence of bifurcation solutions for a predator-prey model with herd behavior

This paper is concerned with a diffusive predator-prey model with herd behavior. The local and global stability of the unique homogeneous positive steady state U* is obtained. Treating the conversion or consumption rate γ as the bifurcation parameter, the steady-state bifurcations both from simple and double eigenvalues are studied near U*. The techniques include the Lyapunov function, the spectrum analysis of operators, the bifurcation theory, space decompositions and the implicit function theorem.     

Linear system approximation via covariance equivalent realizations

A truncation technique for model reduction to simultaneously match a specified number of output covariance derivatives is described. These COVariance Equivalent Realizations which match q + 1 covariance derivatives are called “q-COVERs.” In general q-COVERs are not unique. The additional freedom is used herein so that the q-COVER obtained also matches q Markov parameters. The truncation technique uses a form of the observability matrices of the full-order system to determine a priori the order required of the reduced order model to match a specified number of output covariance derivatives and Markov parameters. The resulting realization is shown to be independent of the basis of the complete model to within a unitary transformation. Stability conditions for the reduced order model are also described, and the relationships are established between stochastically equivalent realizations and the q-COVERs.     

Mathematical modelling and dynamic simulation of multi-effect falling-film evaporator for milk powder production

Multiple-effect evaporators are widely used in dairies and food industries because they are appropriately suited for concentrating food solutions. Some mathematical models for multi-effect evaporators are reported in previous studies. But most of them are steady-state models, and there are no extensive studies on the dynamic behaviour of these evaporators. In this paper, two types of dynamic model, lumped and distributed, are developed for an industrial four-effect falling-film evaporator which is used to concentrate whole milk. These models are validated with data from an industrial unit. The results show that the distributed model has slightly better predictions than the lumped model, but the lumped model has comparable performance because its structure is simple and the needed simulation time is short in comparison with the distributed model.     

Multiphase Jet Flow in Abrasive Water Jet Cutting

A numerical model for ultra-high velocity abrasive water jets (AWJ) is developed and jet dynamic characteristics are calculated under steady-state, turbulent, compressible, multi-phase flow conditions. The model is tested by comparison with analytical solutions for related theoretical problems, generally with very good agreement. Simulations of more realistic flows produce the expected values and behavior. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dynamic stability of a porous cylindrical shell

This paper is devoted to a closed cylindrical shell made of a porous-cellular material. The mechanical properties vary continuously on the thickness of a shell. The mechanical model of porosity is as described as presented by Magnucki, Stasiewicz. A shell is simply supported on edges. On the ground of assumed displacement functions the deformation of shell is defined. The displacement field of any cross section and linear geometrical and physical relationships are assumed in cylindrical coordinate system. The components of deformation and stress state were found. Using the Hamilton's principle the system of differential equations of dynamic stability is obtained. The forms of unknown functions are assumed and the system of a differential equations is reduced to a simple ordinary equation of dynamic stability of shell (Mathieu's equation). The derived equation are used for solving a problem of dynamic stability of porous-cellular shell with intensity of load directed in generators of shell. The critical loads are derived for a family of porous shells. The unstable space of family porous shells is found. The influence a coefficient of porosity on the stability regions in Figures is presented. The results obtained for porous shell are compared to a homogeneous isotropic cylindrical shell. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Analysis of a finite buffer model with two servers and two nonpreemptive priority classes

In this paper, we analyze a finite buffer queueing model with two servers and two nonpreemptive priority service classes. The arrival streams are independent Poisson processes, and the service times of the two classes are exponentially distributed with different means. One of the two servers is reserved exclusively for one class with high priority and the other server serves the two classes according to a nonpreemptive priority service schedule. For the model, we describe its dynamic behavior by a four-dimensional continuous-time Markov process. Applying recursive approaches we present the explicit representation for the steady-state distribution of this Markov process. Then, we calculate the Laplace–Stieltjes Transform and the steady-state distribution of the actual waiting times of two classes of customers. We also give some numerical comparison results with other queueing models.     

Well-posedness and stability of the repairable system with N failure modes and one standby unit

The well-posedness and stability of the repairable system with N failure modes and one standby unit were discussed by applying the c₀ semigroups theory of function analysis. Firstly, the integro-differential equations described the system were transformed into some abstract Cauchy problem of Banach space. Secondly, the system operator generates positive contractive c₀ semigroups T(t) and so the well-posedness of the system was obtained. Finally, the spectral distribution of the system operator was analyzed. It was proven that 0 is strictly dominant eigenvalue of the system operator and the dynamic solution of the system converges to the steady-state solution. The steady-state solution was shown to be the eigenvector of the system operator corresponding to the eigenvalue 0. At the same time the dynamic solution exponentially converges to the steady-state solution.     

Modeling,simulation and optimization of general solar updraft towers

A model to describe a solar chimney power plant with a generally sloped collector field and for the general situation of humid air is presented. This is a significant development of existing simple models for solar updraft towers with planar collector fields for the situation of purely dry air. The model describing the gas dynamics in the collector and in the chimney includes a turbine model, friction and heat transfer losses, evaporation and condensation models etc. However, the relevant physics can be modeled in one space dimension. It is the result of a fully compressible gas dynamic model in the small Mach number limit. A numerical algorithm is defined which admits very fast simulations. Therefore optimization procedures can easily be applied. Numerical results on optimization with respect to geometric and physical parameters which may be considered both in the planning and the operational phase are presented. The results are compared qualitatively and – if available – quantitatively to prototype data and to simulations from the literature.     

模糊集对分析用于脱硫除尘器的综合评价

提出了锅炉脱硫除尘器综合评价的模糊集对分析方法.评述了方法的基本原理和计算步骤,通过实例介绍了方法的应用.评价实例结果表明模糊集对分析用于脱硫除尘器综合评价是可行的.与密切值法相比,所得评价结果相同.该法具有计算简便和实用的特点.     

Modelling of dynamic interaction between structure and trolley for mega container cranes

This article deals with the analysis of trolley impact on the dynamic behaviour of the flexible structure of the mega quayside container crane (QCC) boom, identified as the most relevant structural part. It develops a modelling method for the dynamic response of the large flexible structure of the QCC boom under a moving trolley. By using FEM the original structure of the whole crane structure is reduced to an equivalent model of the boom. The boom is in this way modelled as a system with distributed parameters, comprising reduced stiffnesses and lumped masses from other parts of the upper structure. The article looks at the moving mass approach to achieve the desired performance of the QCC. Differential equations of the mathematical model are obtained by using Lagrange's equations and the assumed mode method. The continuum is discretized by a finite number of admissible functions. Deterministic simulation gives the dynamic response of the boom for quay-to-ship container transfer. Results are obtained for the boom deflection and bending moment values, as well as for the dynamic amplification factor of deflection.