A General Time Dependent Constitutive Model: Part I— Theoretical Developments1

2000 ◽  
Vol 123 (1) ◽  
pp. 51-64 ◽  
Author(s):  
A. F. Saleeb ◽  
S. M. Arnold
Keyword(s):  
Response Characteristic ◽  
Structure Type ◽  
Internal Variable ◽  
Integral Representations ◽  
State Variables ◽  
Qualitative Response ◽  
Response Range ◽  
Starting Point ◽  
Potential Structure ◽  
Hereditary Integral

Using an internal-variable formalism as a starting point, we describe the viscoelastic complement of a previously-developed viscoplasticity formulation of the complete potential structure type. It is mainly motivated by experimental evidence for the presence of rate/time effects in the so-called quasilinear, reversible, material response range. Several possible generalizations are described, in the general format of hereditary-integral representations for nonequilibrium, stress-type, state variables, both for isotropic as well as anisotropic materials. In particular, thorough discussions are given on the important issues of thermodynamic admissibility requirements for such general descriptions, resulting in a set of explicit mathematical constraints on the associated kernel (relaxation and creep compliance) functions. In addition, a number of explicit, integrated forms are derived, under stress and strain control to facilitate the parametric and qualitative response characteristic studies reported here, as well as to help identify critical factors in the actual experimental characterizations from test data that will be reported in Part II.

scholarly journals Determination of synchronous generator nonlinear model parameters based on power rejection tests using a gradient optimization algorithm

2017 ◽  
Vol 65 (4) ◽  
pp. 479-488 ◽  
Author(s):  
A. Boboń ◽  
A. Nocoń ◽  
S. Paszek ◽  
P. Pruski
Keyword(s):  
Parameter Estimation ◽  
Objective Function ◽  
Optimization Algorithm ◽  
Least Squares Method ◽  
Synchronous Generator ◽  
Operating Conditions ◽  
Model Parameters ◽  
State Variables ◽  
Starting Point ◽  
Gradient Optimization

AbstractThe paper presents a method for determining electromagnetic parameters of different synchronous generator models based on dynamic waveforms measured at power rejection. Such a test can be performed safely under normal operating conditions of a generator working in a power plant. A generator model was investigated, expressed by reactances and time constants of steady, transient, and subtransient state in the d and q axes, as well as the circuit models (type (3,3) and (2,2)) expressed by resistances and inductances of stator, excitation, and equivalent rotor damping circuits windings. All these models approximately take into account the influence of magnetic core saturation. The least squares method was used for parameter estimation. There was minimized the objective function defined as the mean square error between the measured waveforms and the waveforms calculated based on the mathematical models. A method of determining the initial values of those state variables which also depend on the searched parameters is presented. To minimize the objective function, a gradient optimization algorithm finding local minima for a selected starting point was used. To get closer to the global minimum, calculations were repeated many times, taking into account the inequality constraints for the searched parameters. The paper presents the parameter estimation results and a comparison of the waveforms measured and calculated based on the final parameters for 200 MW and 50 MW turbogenerators.

Internal Variable Theory Formulated by One Extended Potential Function

2020 ◽  
Vol 45 (3) ◽  
pp. 311-318
Author(s):  
Qiang Yang ◽  
Zhuofu Tao ◽  
Yaoru Liu
Keyword(s):  
Potential Function ◽  
Internal Variable ◽  
The State ◽  
Internal Variables ◽  
State Variables ◽  
Kinetic Rate ◽  
Variable Theory ◽  
Rate Laws ◽  
Flow Potential ◽  
Internal Variable Theory

AbstractIn the kinetic rate laws of internal variables, it is usually assumed that the rates of internal variables depend on the conjugate forces of the internal variables and the state variables. The dependence on the conjugate force has been fully addressed around flow potential functions. The kinetic rate laws can be formulated with two potential functions, the free energy function and the flow potential function. The dependence on the state variables has not been well addressed. Motivated by the previous study on the asymptotic stability of the internal variable theory by J. R. Rice, the thermodynamic significance of the dependence on the state variables is addressed in this paper. It is shown in this paper that the kinetic rate laws can be formulated by one extended potential function defined in an extended state space if the rates of internal variables do not depend explicitly on the internal variables. The extended state space is spanned by the state variables and the rate of internal variables. Furthermore, if the rates of internal variables do not depend explicitly on state variables, an extended Gibbs equation can be established based on the extended potential function, from which all constitutive equations can be recovered. This work may be considered as a certain Lagrangian formulation of the internal variable theory.

scholarly journals Comparative Simulation Study on Synchronous Generators Sudden Short Circuits

2009 ◽  
Vol 2009 ◽  
pp. 1-11 ◽  
Author(s):  
Lucian Lupşa-Tătaru
Keyword(s):  
Mathematical Model ◽  
Time Domain ◽  
Short Circuit ◽  
Structural Equations ◽  
Process Models ◽  
State Variables ◽  
Synchronous Generators ◽  
State Equations ◽  
Starting Point ◽  
Short Circuits

Although of a great extent in time, the research works directed at studying transients in synchronous generators have not yet provided fully sufficient comparative studies in respect to sudden short circuits of the machine. The present paper puts forward novel and comprehensive process models for dynamic simulation of short circuit faults of initially unloaded synchronous generators, using the generalizedd-q-0 mathematical model as starting point in derivation. Distinct from the time-domain analysis, the technique proposed here allows an effective comparative overview by employing a specialized procedure to perform repeated time-domain simulations accompanied by peak values recording for the various circumstances. The time consuming matrix numerical inversion at each step of integration, usually performed when selecting currents as state variables, is eliminated by advancing the process models in a convenient split matrix form that allows the symbolic processing. Also, the computational efficiency is being increased by introducing a set of auxiliary variables common to different state equations. The models derivation is carried out without altering the structural equations of the generalizedd-q-0 mathematical model of synchronous generators whilst the simulation results are both compared and discussed in detail.

scholarly journals Entropy Production Rates of the Multi-Dimensional Fractional Diffusion Processes

Entropy ◽  
2019 ◽  
Vol 21 (10) ◽  
pp. 973 ◽  
Author(s):  
Yuri Luchko
Keyword(s):  
Fundamental Solution ◽  
Diffusion Process ◽  
Entropy Production ◽  
Production Rate ◽  
Fractional Diffusion ◽  
Integral Representations ◽  
Entropy Production Rate ◽  
Time Space ◽  
Starting Point ◽  
Fractional Pde

Our starting point is the n-dimensional time-space-fractional partial differential equation (PDE) with the Caputo time-fractional derivative of order β , 0 < β < 2 and the fractional spatial derivative (fractional Laplacian) of order α , 0 < α ≤ 2 . For this equation, we first derive some integral representations of the fundamental solution and then discuss its important properties including scaling invariants and non-negativity. The time-space-fractional PDE governs a fractional diffusion process if and only if its fundamental solution is non-negative and can be interpreted as a spatial probability density function evolving in time. These conditions are satisfied for an arbitrary dimension n ∈ N if 0 < β ≤ 1 , 0 < α ≤ 2 and additionally for 1 < β ≤ α ≤ 2 in the one-dimensional case. In all these cases, we derive the explicit formulas for the Shannon entropy and for the entropy production rate of a fractional diffusion process governed by the corresponding time-space-fractional PDE. The entropy production rate depends on the orders β and α of the time and spatial derivatives and on the space dimension n and is given by the expression β n α t , t being the time variable. Even if it is an increasing function in β , one cannot speak about any entropy production paradoxes related to these processes (as stated in some publications) because the time-space-fractional PDE governs a fractional diffusion process in all dimensions only under the condition 0 < β ≤ 1 , i.e., only the slow and the conventional diffusion can be described by this equation.

scholarly journals Dynamic energy budget theory and population ecology: lessons from Daphnia

2010 ◽  
Vol 365 (1557) ◽  
pp. 3541-3552 ◽  
Author(s):  
Roger M. Nisbet ◽  
Edward McCauley ◽  
Leah R. Johnson
Keyword(s):  
Population Dynamics ◽  
Population Ecology ◽  
Energy Budget ◽  
Large Body ◽  
Energy Utilization ◽  
State Variables ◽  
Bioenergetic Model ◽  
Dynamic Energy Budget ◽  
Individual Organism ◽  
Starting Point

Dynamic energy budget (DEB) theory offers a perspective on population ecology whose starting point is energy utilization by, and homeostasis within, individual organisms. It is natural to ask what it adds to the existing large body of individual-based ecological theory. We approach this question pragmatically—through detailed study of the individual physiology and population dynamics of the zooplankter Daphnia and its algal food. Standard DEB theory uses several state variables to characterize the state of an individual organism, thereby making the transition to population dynamics technically challenging, while ecologists demand maximally simple models that can be used in multi-scale modelling. We demonstrate that simpler representations of individual bioenergetics with a single state variable (size), and two life stages (juveniles and adults), contain sufficient detail on mass and energy budgets to yield good fits to data on growth, maturation and reproduction of individual Daphnia in response to food availability. The same simple representations of bioenergetics describe some features of Daphnia mortality, including enhanced mortality at low food that is not explicitly incorporated in the standard DEB model. Size-structured, population models incorporating this additional mortality component resolve some long-standing questions on stability and population cycles in Daphnia . We conclude that a bioenergetic model serving solely as a ‘regression’ connecting organismal performance to the history of its environment can rest on simpler representations than those of standard DEB. But there are associated costs with such pragmatism, notably loss of connection to theory describing interspecific variation in physiological rates. The latter is an important issue, as the type of detailed study reported here can only be performed for a handful of species.

Singularity-Reduced Integral Representations for Discontinuity in Smart Materials

2008 ◽  
Vol 47-50 ◽  
pp. 789-792
Author(s):  
Jaroon Rungamornrat ◽  
Teerapong Senjuntichai
Keyword(s):  
Integral Equation ◽  
Smart Materials ◽  
Numerical Procedure ◽  
Integral Representations ◽  
The Body ◽  
Material Behavior ◽  
Constitutive Law ◽  
State Variables ◽  
Weakly Singular ◽  
Integral Relations

This paper presents a complete set of singularity-reduced integral relations for isolated discontinuity embedded in a three-dimensional infinite medium. The development is carried out in a broad context such that the constitutive law governing the material behavior assumes a general form and the discontinuity surface possesses general configuration and jump distribution. The former feature allows the treatment of a well-known class of smart materials (e.g. piezoelectric and piezomagnetic materials) as a special case while the latter renders the treatment of particular types of discontinuity such as cracks and dislocations possible. The key elements utilized in the regularization procedure are special decompositions of two involved kernels in a form well-suited for integration by parts to be performed via Stokes’ theorem. The weakly singular kernels appearing in these representations are obtained in a concise form appropriate for numerical evaluation. A set of integral relations is subsequently specialized to cracks and dislocations. For dislocations, the field quantities such as state variables, the body flux, and the generalized interaction energy are given in terms of line integral representations. The obtained expressions are fundamental and useful in the context of dislocation mechanics and modeling. For cracks, a weakly singular, weak-form integral equation for the surface flux is established. Such integral equation constitutes a basis for a wellknown numerical procedure, a symmetric Galerkin boundary element method (SGBEM). The crucial feature of using the derived integral equation as the key governing equation is its weakly singular nature that allows low order interpolations to be used in the numerical approximation.

PAIRING AND SPONTANEOUS POLARIZATION IN DENSE HYDROGEN

1999 ◽  
Vol 13 (05n06) ◽  
pp. 595-605 ◽  
Author(s):  
N. W. ASHCROFT
Keyword(s):  
Charge Density ◽  
Electron Gas ◽  
Density Wave ◽  
Functional Integral ◽  
Integral Representations ◽  
Many Body ◽  
Starting Point ◽  
The Many ◽  
A Charge ◽  
The Continuum

The many-body problem associated with dense hydrogen can represented by the standard interacting electron gas problem (complete with compensating charge continuum) augmented by the proton equivalent and a coupling between the two. This viewpoint is useful as a starting point in the pursuit of ground-state instabilities (particularly the onset of a charge density wave) as a function of the continuum charge density, and even more generally in the identification of forms for the fields required to establish a thermodynamic treatment via coherent state functional integral representations of the partition function for dense hydrogen.

Two methods to find analytical redundancy relations for fault diagnosis

2017 ◽  
Vol 65 (4) ◽  
Author(s):  
Jan Lunze
Keyword(s):  
Structural Analysis ◽  
Linear System ◽  
State Variables ◽  
Vector Representation ◽  
New Methods ◽  
Starting Point ◽  
Analytical Redundancy ◽  
Matrix Vector ◽  
Analytical Expressions ◽  
Derivatives Of

AbstractThe paper deals with methods to derive analytical redundancy relations for diagnosable systems. The starting point is an over-determined set of constraints, which is obtained by a structural analysis of the linear system under consideration. The problem is to find analytical expressions for residuals in terms of the measured signals and the derivatives of these signals. Two new methods are described. The first method provides a representation of the residual in terms of the observability matrix of the diagnosable subsystem. The second method replaces the derivative constraints by a series of equations that include the derivatives of the state variables and the outputs. The result is a matrix-vector representation of the over-determined set of equations, which leads to the analytical redundancy relations. The methods are illustrated by their application to an automotive example.

Effect of Modal Interaction on Transient Behavior in Double-Input SEPIC DC–DC Converters

2020 ◽  
Vol 30 (10) ◽  
pp. 2050145
Author(s):  
Hao Zhang ◽  
Min Jing ◽  
Shuai Dong ◽  
Wei Liu ◽  
Zhaohua Cui
Keyword(s):  
Transient Behavior ◽  
Approximate Solutions ◽  
Underlying Mechanism ◽  
Response Characteristic ◽  
State Variables ◽  
Modal Interaction ◽  
Oscillation Modes ◽  
Dynamic Response Characteristic ◽  
The One ◽  
Averaged Model

In this paper, we exploit an idea of nonlinear modal series method to investigate the effects of modal interaction in the one-cycle controlled (OCC) double-input SEPIC DC–DC converter. Based on the proposed nonlinear averaged model, the analytical approximate solutions are obtained to characterize the dynamic response characteristic of the transient behaviors in the double-input SEPIC converters. The fundamental modal analysis is utilized to identify the dominant oscillation modes and discover the relationship between parameters, fundamental modes and state variables. Furthermore, the second-order interaction indices are proposed to uncover the underlying mechanism of nonlinear interaction behaviors. In particular, the correlation between parameters and modal interaction are derived to optimize the transient process of the double-input SEPIC converters. Finally, numerical simulations are performed to verify the theoretical analysis.

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