Method Of Integrodifferential Relations Research Articles

Optimization of Longitudinal Motions of an Elastic Rod Using Periodically Distributed Piezoelectric Forces

The longitudinal vibrations of an elastic rod controlled by a distributed force, which is applied to individual sections of the rod, are studied. It is assumed that the force varies in space in a piecewise constant manner. Such a mechanical system can be implemented using piezoactuators attached along the rod. The dynamics of the system is determined from the solution of the variational problem following the method of integrodifferential relations. The variational problem is solved analytically. To do this, traveling waves of the d’Alembert type are introduced on the space-time mesh, which determine continuous displacements and a dynamic potential. The latter relates the momentum density and stresses. A control problem is posed under the condition of the weighted minimization of the vibrational energy stored by the rod at the terminal time instant, and the mean potential energy generated by the control actions. The extremal motion and the corresponding control law are found explicitly by solving the Euler–Lagrange equations. As an example, the control capabilities for certain configurations of piezoelectric elements are studied.

On Free Bending Vibrations Of Concrete Beams With Variable Cross Section

The paper consists of an introduction and three sections. The introduction discusses the relevance of issues related to the study of inhomogeneous beams vibrations. The analysis of publications and the results obtained in this area is performed. The first section is devoted to the formulation of the boundary-value problem of finding natural-vibration frequencies for an inhomogeneous beam under the Euler-Bernoulli hypotheses. By introducing new variables, the problem originally formulated in displacements reduces to an identical one, but formulated in terms of bending moment. The next section describes the method of integro-differential relations, which is an alternative to the classical variational approaches. Further, the possibilities of constructing various bilateral energy estimates of the quality of approximate solutions arising from the method of integro-differential relations are investigated. In the third section, using the example of free vibrations of a supported concrete beam, numerical aspects of constructing an approximate solution for boundary problems described by an ordinary differential equation with variable coefficients are studied. The proposed bilateral quality criteria of the approximate solution make it possible to obtain high-precision solutions for mathematical models of small dimension.

Analysis of Dynamic Behavior of Beams with Variable Cross-section

A formulation of a boundary value problem to find natural frequencies of an inhomogeneous beam in the framework of the Euler–Bernoulli hypotheses are represented. Questions related to various classical variational formulations for a spectral problem arising in the beam theory are discussed. Particularities of the application of the Hamiltonian principles to boundary-value problems are considered. The method of integro-differential relations, which is an alternative to the classical variational approaches is discussed. Various bilateral energy quality estimates for approximate solutions that follow from the method of integro-differential relations are proposed. In the final part of the paper advantages of the variational technique in problems of free vibrations of inhomogeneous beams are discussed based on a numerical example.

Model reduction and optimal control for an electromechanical structure with viscoelastic links

Motions of a flexible electromechanical structure consisting of two joined beams clumped on a mobile carriage controlled with an electric driver are studied. An optimal control problem is considered to transfer the structure from its initial state to terminal one in a fixed time and to minimize the mean mechanical energy of the beams. The numerical solution of the inverse dynamic problem, which is based on the method of integrodifferential relations and a model reduction technique, is presented and discussed.

An integrodifferential approach to modeling, control, state estimation and optimization for heat transfer systems

Abstract In this paper, control-oriented modeling approaches are presented for distributed parameter systems. These systems, which are in the focus of this contribution, are assumed to be described by suitable partial differential equations. They arise naturally during the modeling of dynamic heat transfer processes. The presented approaches aim at developing finite-dimensional system descriptions for the design of various open-loop, closed-loop, and optimal control strategies as well as state, disturbance, and parameter estimation techniques. Here, the modeling is based on the method of integrodifferential relations, which can be employed to determine accurate, finite-dimensional sets of state equations by using projection techniques. These lead to a finite element representation of the distributed parameter system. Where applicable, these finite element models are combined with finite volume representations to describe storage variables that are—with good accuracy—homogeneous over sufficiently large space domains. The advantage of this combination is keeping the computational complexity as low as possible. Under these prerequisites, real-time applicable control algorithms are derived and validated via simulation and experiment for a laboratory-scale heat transfer system at the Chair of Mechatronics at the University of Rostock. This benchmark system consists of a metallic rod that is equipped with a finite number of Peltier elements which are used either as distributed control inputs, allowing active cooling and heating, or as spatially distributed disturbance inputs.

Optimal control of a viscoelastic rack feeder based on the method of integrodifferential relations

In this paper, control-oriented models are derived for an experimental setup representing the structure of a typical high bay rack feeder. To develop a real-time applicable control algorithm, a frequency analysis is performed for the original viscoelastic double-beam structure. This leads to a simplified Bernoulli beam model with specific boundary conditions. On the basis of the proposed model, a feedforward control strategy is designed. The control objective under consideration is to move the flexible structure to a desired position in a given time interval and to minimize the relative mean energy stored in the beams during the process. A modification of the method of integrodifferential relations, which is based on a projection approach and a suitable finite element technique, is employed to optimize the controlled motions. Results of numerical simulations are presented and compared with experimentally measured data from the experimental setup.

Finite Element Approaches for Real-Time Control and Observer Design of Flexible Rack Feeder Systems

The basis for the design of real-time capable control and state estimation strategies for mechanical systems with distributed parameters is the derivation of suitable system models. These models lead to state-space representations that are, on the one hand, sufficiently accurate to capture the dominant features of the system dynamics. On the other hand, the dimensionality of these models has to be limited to a reasonable size to make sure that the resulting control and state estimation approaches can be implemented in real time. Therefore, a huge amount of work has been performed in recent years which is related to early lumping techniques. By using these techniques, the governing partial differential equations are replaced by sets of ordinary differential equations before the control and estimator design is performed. In this paper, both local and global approximations of the system dynamics are compared in simulations. These approximations are determined either by using a finite element representation that is based on the Method of Integrodifferential Relations (MIDR) or, alternatively, by a global Ritz ansatz. Especially the MIDR approach allows for the derivation of system models which exactly fulfill energy conservation laws as well as boundary and inter-element conditions.

Optimal Design of Elastic Rod Motions Based on a Projection Approach

Dynamics modelling and control design for elastic rods with distributed parameters are under study. The constitutive laws are specified by an integral equality according with the method of integro-differential relations. A numerical algorithm is developed to solve direct and inverse dynamic problems in linear elasticity based on the Ritz method and finite element technique. The control problem is to obtain the rod motion from some initial state to terminal one at a fixed time with the minimal mean energy. The exact solution is found with a specific triangular space-time mesh and a piecewise polynomial displacement input.

Optimal control of a viscoelastic rack feeder based on the method of integrodifferential relations

Optimal control of a viscoelastic rack feeder based on the method of integrodifferential relations

Integrodifferential approaches to frequency analysis and control design for compressible fluid flow in a pipeline element

In this study, modelling, frequency analysis, and optimization of control processes are considered for the fluid flow in pipeline systems. A mathematical model of controlled pipeline elements with distributed parameters is proposed to describe the dynamical behaviour of compressible fluid which is transported in a long rigid tube. By exploiting specific functions representing cross-sectional forces and effective displacements as well as linear approximations of fluidic resistances, the original problem with non-uniform parameters is reduced to a partial differential equation (PDE) system with constant coefficients and homogeneous initial and boundary conditions. Three numerical approaches are applied to an efficient analysis of natural vibrations and reliable control-oriented modelling of pipeline elements. The conventional Galerkin method is compared with the method of integrodifferential relations based on a weak formulation of the constitutive laws. In the latter approach, the original initial-boundary value problem is reduced to the minimization of an error functional which provides explicit energy estimates of the solution quality. A novel projection approach is implemented on the basis of the Petrov–Galerkin method combined with the method of integrodifferential relations. This technique benefits from the advantages of the above-mentioned projection and variational approaches, namely sufficient numerical stability, a lower differential order, and an explicit quality estimation. Numerical optimization procedures, making use of a modified finite element technique, are proposed to obtain a feedforward control strategy for changing the pressure and mass flow inside the pipeline system to a desired operating state. At this given finite point of time, residual elastic oscillations inside the pipeline are minimized. Numerical results, obtained for ideal as well as viscous fluid models, are analysed and discussed.

Modelling of the forced motions of an elastic beam using the method of integrodifferential relations

A variational approach to the numerical modelling of forced lateral motions of an Euler–Bernoulli elastic beam is developed for a number of linear boundary conditions using the method of integrodifferential relations. A class of linear boundary actions is considered. A family of quadratic functionals, connecting the displacement field of points of the beam with the bending-moment functions in the cross section and the momentum density is proposed. Variational formulations of the original initial-boundary value problem on the motion of the beam are given and the necessary conditions for the functionals introduced to be stationary are analysed. The integral and local quality characteristics of the admissible approximate solutions are determined. The relation between the variational problems, formulated for the beam model, with the classical Hamilton–Ostrogradskii variational principles is demonstrated. An algorithm for constructing approximate systems of ordinary differential equations is developed, the solution of which yields stationary (minimum) values of the functionals introduced on a specified set of displacement fields, moments and momenta. Examples of calculations of the displacements for an elastic beam and an analysis of the quality of the numerical solutions obtained are presented.

An Integrodifferential Approach to Adaptive Control Design for Heat Transfer Systems with Uncertainties

AbstractOpen-loop and closed-loop control problems for distributed parameter systems, described by parabolic partial differential equations, are considered in this contribution. The goal of the study is the development of strategies for control and estimation of states, disturbances, and parameters. These strategies are based on the method of integrodifferential relations, a projection approach, and a suitable finite element technique. A real-time applicable control algorithm is proposed and its specific features are discussed. A verification of the control laws derived in this contribution is performed taking into account explicit error estimates resulting directly from the integrodifferential approach. The parameters, geometry, and actuation principles of a heat transfer system available at the Chair of Mechatronics, University of Rostock, are used for the numerical simulation and experimental validation. The test setup consists of a metallic rod equipped with a finite number of Peltier elements which are used as distributed control inputs allowing for active cooling and heating.

Modelling and analysis of the natural oscillations of a prismatic elastic beam based on a projection approach

A regular approach to the construction of mathematical models describing the natural motions of beam-type elastic bodies within the limits of the linear theory of elasticity is developed using the method of integrodifferential relations. By employing the integral form of the equations of state, relating the stresses and strains and also the velocities and momenta, the system of partial differential equations is reduced to a denumerable system of ordinary differential-algebraic equations. A polynomial representation of the unknown functions of the displacements, stresses and momentum density along two spatial coordinates is used for this purpose. The effect of the geometric and mechanical parameters of the system on the frequencies and modes of the natural oscillations of a rectilinear elastic beam is investigated.

Approaches to control design and optimization in heat transfer problems

In this paper, boundary control problems are considered for a distributed heating system. The dynamical model of the heating system under consideration is given by a parabolic partial differential equation. In the first stage, the implementation of the Fourier method is discussed for the problem of heat convection and conduction. In the second stage, two alternative solutions to the design of tracking controllers are discussed. On the one hand, an optimal control problem is solved based on the method of integrodifferential relations. On the other hand, this procedure is used to verify the quality of a flatness-based control strategy. The results obtained by the integrodifferential approach are compared with finite-node Fourier approximations. After derivation of suitable, general-purpose solution procedures for the design of open-loop as well as closed-loop boundary control strategies, experimental results are presented. These results highlight the applicability of these procedures in a real-world experiment. For the experimental validation, a test setup at the University of Rostock has been used.

Asymptotic Approach to Free Beam Vibration Analysis

The equations described free longitudinal and lateral vibrations of the rectilinear elastic beams with the rectangular cross section are derived based on the theory of linear elasticity and the method of integrodifferential relations. The original system of partial differential equations is reduced to the system of ordinary differential equations with constant coefficients. The influence of geometrical and elastic properties of the beam on eigenvalues and eigenfunctions are investigated. The existence of the different displacement and internal stress types of the longitudinal motions are shown. The presence of two frequency zones corresponded to different kinds of the characteristic solutions for the lateral vibrations is detected.

Variational approaches to solving initial-boundary-value problems in the dynamics of linear elastic systems

Problems of the controlled motion of an elastic body are considered in the linear theory. Using the method of integrodifferential relations, a family of quadratic functionals is introduced, which define the state of the elastic body, and variational formulations of the initial-boundary-value problem of dynamics are given. Euler's equations and boundary and terminal relations corresponding to them are obtained from the condition for the functionals to be stationary. It is shown that there is a relation between the proposed formulations and the Hamilton variational principle in the case of boundary-value and time-periodic problems of dynamics. A numerical algorithm is developed for finding the motions of an elastic body, based on piecewise-polynomial approximations and a criterion is proposed for estimating the quality of the approximate solutions. An example of the calculation and analysis of the forced transverse motions of a rectilinear beam with a square cross section is given for the three-dimensional model.

The method of integrodifferential relations for analysing the natural oscillations of membranes

An approach to solving of problems of the free oscillations of membranes, based on the method of integrodifferential relations, is proposed. The variational properties of the integrodifferential problems are investigated and their relation to classical variational principles is demonstrated. Integral and local criteria for the quality of the approximate solutions are proposed. A numerical-analytical algorithm is developed for finding the characteristic frequencies and modes of oscillations. Problems involving the free transverse motions of circular and elliptic membranes are considered as an example.

Variational analysis in dynamical problems of linear elasticity

AbstractThe initial–boundary value problems in the linear theory of elasticity is considered. Based on the method of integrodifferential relations (MIDR) two dynamical variational principles is proposed and discussed. It is shown that the Hamilton principle as well as the corresponding complementary principle stated for dynamic boundary value problems follow out the variational formulations proposed. To minimize the nonnegative functional under algebraic and differential constraints a regular finite element algorithm is worked out. The algorithm allows us to estimate explicitly the local and integral quality of numerical solutions obtained. A 3D problem of lateral motions of a rectilinear elastic prism with a rectangular cross section are considered. The numerical results are presented and discussed. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

An asymptotic approach to analysis of 3D beam stress‐strain elastic states

AbstractA regular asymptotic approach to analysis of 3D beam stress–strain states is proposed based on the linear theory of elasticity and the method of integrodifferential relations. Using the integral formulation of Hooke's law and polynomial expansions of unknown stress and displacement functions with respect to transversal Cartesian coordinates the initial system of partial differential equations is reduced to a countable system of ordinary differential equations with constant coefficients. For rectilinear beams with rectangular cross–sections the consistent boundary value problems describing independently the compression and stretch, bends, and torsion states are derived. To find equilibrium stress and admissible displacement fields satisfying boundary conditions an effective numerical algorithm is worked out. Integral and local criteria for explicit bilateral estimates of resulted solution quality are proposed. The numerical results are presented and discussed. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A variational formulation in fracture mechanics

A variational approach to linear elasticity problems is considered. The family of variational principles is proposed based on the linear theory of elasticity and the method of integrodifferential relations. The idea of this approach is that the constitutive relation is specified by an integral equality instead of the local Hooke’s law and the modified boundary value problem is reduced to the minimization of a nonnegative functional over all admissible displacements and equilibrium stresses. The conditions of decomposition on two separated problems with respect to displacements and stresses are found for the variational problems formulated and the relation between the approach under consideration and the minimum principles for potential and complementary energies is shown. The effective local and integral criteria of solution quality are proposed. A numerical algorithm based on the piecewise polynomial approximations of displacement and stress fields over an arbitrary domain triangulation are worked out to obtained numerical solutions and estimate their convergence rates. Numerical results for 2D linear elasticity problems with cracks are presented and discussed.