Set Of Linear Differential Equations Research Articles

Nonlinear optimal vibration suppression of Timoshenko beams traversed by a moving mass

An instantaneous optimal closed loop control approach incorporating a general numerical procedure is developed in this work for suppressing the excessive vibration of Timoshenko beams due to the severe dynamic impact of a moving mass load. The geometric nonlinearity due to large deformations of the support beam is taken into account by applying the approach of fictitious loads for kinematic corrections. The complex nonlinear strain‐displacement relationship is alleviated by solving a set of linear differential equations and transforming the coordinates successively until the solution converges to a specified small quantity. The effect of actuator location on the performance of the control system designed is examined. The saturation effect of the control input is also addressed. This study shows that excessive vibration of the beam structure traversed by a moving mass load can be effectively suppressed by using the approach developed.

Analysis and modeling of the steady state behavior of the static Kramer induction generator

A steady state analysis of the static Kramer variable speed double output induction generator is conducted using a hybrid dq model wherein stator variables only are transformed onto a reference frame fixed to the rotor. Rotor variables are retained in their natural abc form, allowing diode bridge overlap together with switching effects in the rectifier and inverter to be catered for. For constant speed operation, the system equations are reduced to a set of linear differential equations with constant coefficients which are solved analytically to predict generator performance. Simulated performance is verified experimentally showing very good agreement with measurement.

FIT-formulation for non-linear dispersive media

A new approach using FIT-formulation (Finite Integration Technique) (T. Weiland, Electron. Commun., 31, 116–120 (1977); Int. J. Numer. Model., 9, 295–319 (1996)) for simulating waveguide propagation of optical pulses is presented. FIT-methods are widespread in use for broadband linear simulations. In recent years, several attempts have been made to describe different dispersive material-characteristics such as Drude, Debye or Lorentz dispersion. Today advanced FDTD-formulations (Finite Difference Time Domain) also consider non-linear effects (P. M. Goorjian and A. Taflove, IEEE Opt. Lett., 17(3), 180–182 (1992); D. M. Sullivan, IEEE Trans. Microwave Theory Techniques, 43(3), 676–682 (1995)). In the following presented method third-order non-linear effects were described, which can be observed in isotropic media in frequency ranges of optical pulses, by updating material polarization terms using classical descriptions of Lorentz dispersion, Raman scattering and the Kerr effect. The basic idea is transforming these description formulas into sets of linear differential equations and solving them with the help of the general exponential solution. Copyright © 1999 John Wiley & Sons, Ltd.

Transient analysis of reliability with and without repair for K-out-of- N : G systems with M failure modes

This paper represents Markov models for transient analysis of reliability with and without repair for K-out-of- N : G systems subject to M failure modes. The reliability and the mean time between failures of repairable systems can be calculated as a result of numerical solution of simultaneous set of linear differential equations. Closed form solutions of the transient probabilities are used to obtain the reliability and the mean time to failure for nonrepairable systems.

TORSIONAL VIBRATION OF CRANKSHAFTS: EFFECTS OF NON-CONSTANT MOMENTS OF INERTIA

The dynamic analysis of the running hardware of reciprocating machines is complex and is usually dealt with by using a number of simplifying assumptions. Usually an “equivalent dynamic system” is built for performing the torsional analysis: such equivalent system has inertial properties which are assumed to be constant and the variation of the actual configuration is taken into account only by adding suitable “inertia torques” to the driving torques. The aim of the present paper is that of studying the torsional vibration of crankshafts with account taken of the variation of the geometry of the system with the crank angle; both the free behaviour and the response to external excitation are dealt with. The analysis is linearized and a mathematical model having the form of a set of linear differential equations with periodic coefficients is obtained. The solution of the free behaviour is obtained through a formulation similar to Hill's infinite determinant, whose truncated forms yield an approximated solution of accuracy increasing with the number of harmonics which are retained. A similar method allows the forced response to be computed. Two examples, one related to a simplified single-cylinder machine and one to an actual aircraft engine, conclude the work.

R-Deformed Heisenberg Algebra, Anyons and D=2+1 Supersymmetry

A universal minimal spinor set of linear differential equations describing anyons and ordinary integer and half-integer spin fields is constructed with the help of deformed Heisenberg algebra with reflection. The construction is generalized to some d=2+1 supersymmetric field systems. Quadratic and linear forms of action functionals are found for the universal minimal as well as for supersymmetric spinor sets of equations. A possibility of constructing a universal classical mechanical model for d=2+1 spin systems is discussed.

R-DEFORMED HEISENBERG ALGEBRA

It is shown that the deformed Heisenberg algebra involving the reflection operator R (R-deformed Heisenberg algebra) has finite-dimensional representations which are equivalent to representations of para-Grassmann algebra with a special differentiation operator. Guon-like form of the algebra, related to the generalized statistics, is found. Some applications of revealed representations of the R-deformed Heisenberg algebra are discussed in the context of OSp(2|2) supersymmetry. It is shown that these representations can be employed for realizing (2+1)-dimensional supersymmetry. They also give a possibility to construct a universal spinor set of linear differential equations describing either fractional spin fields (anyons) or ordinary integer and half-integer spin fields in 2+1 dimensions.

Transient analysis of reliability with and without repair for K-out-of- N: G systems with two failure modes

This paper presents Markov models for transient analysis of reliability with and without repair for K-out-of- N: G systems subject to two failure modes. The reliability of repairable systems can be calculated as a result of the numerical solution of a simultaneous set of linear differential equations. Closed form solutions of the transient probabilities are used to obtain the reliability for nonrepairable systems.

Deformed Heisenberg Algebra, Fractional Spin Fields, and Supersymmetry without Fermions

Within a group-theoretical approach to the description of (2+1)-dimensional anyons, the minimal covariant set of linear differential equations is constructed for the fractional spin fields with the help of the deformed Heisenberg algebra (DHA), [a−, a+]=1+νK, involving the Klein operatorK, {, a±}=0,K2=1. The connection of the minimal set of equations with the earlier proposed “universal” vector set of anyon equations is established. On the basis of this algebra, a bosonization of supersymmetric quantum mechanics is carried out. The construction comprises the cases of exact and spontaneously brokenN=2 supersymmetry allowing us to realize a Bose-Fermi transformation and spin- 12 representation of SU(2) group in terms of one bosonic oscillator. The construction admits an extension to the case of OSp(2∣2) supersymmetry, and, as a consequence, both applications of the DHA turn out to be related. The possibility of “superimposing” the two applications of the DHA for constructing a supersymmetric (2+1)-dimensional anyon system is discussed. As a consequential result we point out that theosp(2∣2) superalgebra is realizable as an operator algebra for a quantum mechanical 2-body (nonsupersymmetric) Calogero model.

Structural design sensitivity analysis for composites undergoing elastoplastic deformation

Nonlinear structural design sensitivity analysis for structures undergoing elastoplastic deformation is developed in this paper. The reference volume concept is used to unify the shape and nonshape design problems. The rate (time-independent) constitutive model is employed to account for the plastic material behavior. In the response analysis, a higher order approximation procedure of the integration of the rate constitutive equations is proposed. The direct differentiation approach (DDA) is adopted to obtain the design sensitivity equation for the response variables. A method of partial differentiation of the rate constitutive equations, which yields a set of linear differential equations in the partial derivatives of stresses and internal variables with respect to the design variable, is included in the DDA procedure. In Part I of this paper, the general theory is described. In Part II, the theory is applied to a composite laminated beam problem.

Asymptotic analysis of nonlinear mode localization in a class of coupled continuous structures

An asymptotic methodology is developed for studying the localized nonlinear normal modes of a class of conservative periodic continuous structures. Localized modes are synchronous free periodic oscillations which are spatially localized to only a limited number of components of the structure. Nonlinear mode localization is analytically studied by defining modal functions to relate the motion of an arbitrary particle of the structure to the motion of a reference point. Conservation of energy is imposed to construct a set of singular nonlinear partial differential equations governing the modal functions. This set is asymptotically solved using a perturbation technique. Criteria for existence of strongly or weakly localized modes are formulated, and are subsequently used for detecting nonlinear mode localization in the system. The orbital stability of the detected localized modes is then investigated by expanding the corresponding variational equations in bases of orthogonal polynomials and using Floquet theory to analyze the resulting set of linear differential equations with periodic coefficients. Application of the general theory is given by studying the localized modes of a system of two weakly coupled, simply supported beams resting on nonlinear elastic foundations. The use of mode localization for the active or passive spatial confinement of impulsive responses of flexible structures is discussed.

Dextrous manipulation with multifingered robot hands including rolling and slipping of the fingertips

This paper treats two fundamental problems which occur when manipulating objects with an articulated multifingered robot hand: determination of the joint motions to perform the manipulation according to a given object trajectory and optimization of the joint torques needed to ensure a secure gripping situation. The consideration of rolling and slipping of the fingertip on the object leads to a set of linear differential equations when computing the joint angles and to a partly non-linear optimization problem for the torques. The presented removal of redundant information will decrease computation time significantly. A possible implementation of the developped equations will be given as well as an example demonstrating the need to consider rolling and slipping: rotation of an ordinary egg with the Karlsruhe Dextrous Hand.

Linear systems excited by polynomial forms of non-Gaussian filtered processes

A method for the evaluation of the statistical moments of linear one-dimensional or multi-dimensional systems subjected to non-Gaussian input in polynomial forms of filtered normal or non-normal delta-correlated processes is presented. The proposed procedure allows one to avoid the solution of a set of non-linear differential equations, requiring the solution of three sets of linear differential equations. The latter sets are: the equations governing the evolution of the moments of the response forced by input-output cross moments; the equations useful for the evaluation of the input-output cross-moments and the equations governing the evolution of the statistical moments of the filtered input.

Analysis of the Diametral Response of a Czochralski Crystal (III) Solution of the Equation Set. Transfer Functions

AbstractThe paper is the third in a series which presents results on the small‐signal diametral response of a Czochralski crystal grown in a resistively heated furnace to variations in heater temperature and pull rate. For typical growth conditions and a wide range of materials it has been shown that this response is obtained by solving a set of linear differential equations. Conditions for normal response, i.e. the diameter increasing on heater temperature and/or pull rate decrease, have been established and shown to hold for most practical cases. The transfer function obtained is consistent with previous results for an inductively heated growth apparatus.

Stabilisation Characteristics of Ventilation in Buildings

Abstract Indoor radon concentrations depend strongly on ventilation between the rooms and the atmosphere. Here a set of linear differential equations is used to describe this transport. The solution for a multi-room building is calculated numerically. A house with three storeys and three by three rooms on each storey was chosen to achieve some generality. The air in each room is open to exchange with the air in adjacent rooms. All rooms except the middle rooms ventilate with the atmosphere. The middle rooms represent a staircase and ventilate with all adjacent rooms. The calculations were made for radon sources in one, some and all rooms, and different winds. Special emphasis was laid on the description of stabilisation times. Each room in between the source and the room under consideration in the multi-room model reduces the maximum concentration by a factor of about 2.6 and delays the stabilisation by factor of about 1.7.

Parallel‐in‐time method based on shifted‐picard iterations for power system ty‐ansient stability analysis

AbstractThe transient stability analysis is the most time‐consuming computer simulation in power system studies. For this reason, in recent years, parallel processing has been applied for implementing on‐line time‐domain simulations of power system transient behaviour. As is well known, the problem of power system transient simulation consists, basically, in the solution of an initial value problem for a mixed set of Ordinary Differential Equations (ODE) and algebraic equations. The objective of this paper is to present an algorithm based on Shifted‐Picard dynamic iterations, which solves the above mentioned set of non‐linear Differential Algebraic Equations (DAE) by the iterative solution of a linear set of DAE. This feature is important because the time behaviour of the linear set of differential equations can be obtained by the solution of an integral. In this case, a multiprocessor parallel‐in‐time implementation of such algorithms is possible because each processor is devoted to the evaluation of the required variables relative to each time step. Moreover, parallelism‐in‐time allows multigrid techniques to be implemented in a natural way. Test results on realistic‐sized power systems are presented.

An Energy-Based Formulation for Computing Nonlinear Normal Modes in Undamped Continuous Systems

The nonlinear normal modes of a class of one-dimensional, conservative, continuous systems are examined. These are free, periodic motions during which all particles of the system reach their extremum amplitudes at the same instant of time. During a nonlinear normal mode, the motion of an arbitrary particle of the system is expressed in terms of the motion of a certain reference point by means of a modal function. Conservation of energy is imposed to construct a partial differential equation satisfied by the modal function, which is asymptotically solved using a perturbation methodology. The stability of the detected nonlinear modes is then investigated by expanding the corresponding variational equations in bases of orthogonal polynomials and analyzing the resulting set of linear differential equations with periodic coefficients by Floquet analysis. Applications of the general theory are given by computing the nonlinear normal modes of a simply-supported beam lying on a nonlinear elastic foundation, and of a cantilever beam possessing geometric nonlinearities.

A new method for determination of normal‐stress differences in highly viscoelastic substances using a modified Weissenberg rheometer

This paper presents a new experimental approach for determining both normal‐stress differences using a modification of the Weissenberg rheometer. The method eliminates both wall slippage and rim fracture that so far have been major obstacles in attaining steady‐state flow conditions with highly consistent fluids in rotational instruments. This is achieved by extending the upper plate with a separately moving exterior ring that is closed on the outside. For determination of both normal‐stress differences with this modification, the wall pressure at the radius of separation is to be measured in addition to the integral force on the inner plate, yielding a set of linear differential equations in N2 with linear terms in N1. Combining two geometrical configurations from the cone‐and‐plate, cone‐plate‐distance, and parallel‐plates systems, a differential equation in N2 only is obtained whose solution can be determined from a quadrature of the measured mechanical quantities. N1 results from the equations after reintroduction of the solution for the second normal‐stress difference.

Parallel-in-time implementation of transient stability simulations on a transputer network

The most time consuming computer simulation in power system studies is the transient stability analysis. Parallel processing has been applied for time domain simulations of power system transient behavior. In this paper, a parallel implementation of an algorithm based on Shifted-Picard dynamic iterations is presented. The main idea is that a set of nonlinear Differential Algebraic Equations (DAEs), which describes the system, can be solved by the iterative solution of a linear set of DAEs. The time behavior of the linear set of differential equations can be obtained by the evaluation of the convolution integral. In the parallel-in-time implementation of the proposed algorithm, each processor is devoted to the evaluation of the complete set of variables relative to each time step. The quadrature formula, adopted for the integral evaluation, can be easily parallelized by using a number of processors equal to the number of time steps. The algorithm, implemented on a transputer network with 32 Inmos T800/20 adopting a uni-directional ring topology, has been tested on standard power systems.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Overtone State‐Selective Isomerization by a Series of Picosecond Infrared Laser Pulses: Model Simulations for Be2H3D‐(C2v) → C3v)

AbstractThe vibrational state‐selective isomerization of Be2H3D− in the electronic ground state by a series of picosecond infrared laser pulses is simulated, using a one‐dimensional model in the framework of the Born‐Oppenheimer and semiclassical dipole approximations. Three pulses pumping the sequential overtone transitions serve to excite the anion from its vibrational ground state, representing the stable C2v configuration, to a delocalized vibrational state with energy close to the potential energy barrier. These three pump‐pulses are followed by a dump‐pulse which induces the overtone transition from the delocalized state to a vibrationally‐excited state of the slightly less stable isomer of Be2H3D− with C2v symmetry. The overall reaction probability for optimal laser pulses with sin2‐shapes is about 95%.The model is based on ab initio calculations of the potential energy surface and the dipole function for the electronic ground state of Be2H3D− at the MP4/6–31 ++G* level, with corresponding vibrational energies and dipole transition matrix elements. The laser‐stimulated dynamics of the ultrafast state‐selective isomerization is described by a representative, time‐dependent wave packet which is driven by the laser pulses, according to the time‐dependent Schrödinger equation, equivalent to a set of linear differential equations for the time‐dependent amplitudes of vibrational eigenstates which constitute the wave packet. This set ofdifferential equations is solved by using both standard numerical techniques and an efficient quasiresonant smoothing algorithm.