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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.

The paper discusses the problem of optimal design (stationary optimal control) for multi body dynamic system. The system includes a train and track superstructure interacting with a beam bridge. It is regarded that the speed of the train can reach 400 km/h. The problem of optimal stiffness distribution for continual subsystem – rail bed – is considered. The aim of the optimization is decreasing of irregularity of rail bed loading under constraint on minimal vertical wheel-rail contact force that prevents derailment.

The paper studies the modeling and optimization of the dynamic behavior of heterogeneous beam structures. Usually, a boundary-value problem is formulated as a general differential equation with variable coefficients. One of the common characteristic features inherent in boundary-value problems of mathematical physics is a certain ambiguity in their formulation. By introducing new variables that characterize the behavior of the system, the boundary-value problem reduces to three ordinary differential equations with variable coefficients. New variables have a clear physical meaning. One function is the linear momentum density, and the other a bending moment in the beam cross section. Such a formulation of the problem of free vibrations of a beam of variable cross section allows us to reduce the system of differential equations to one fourth-order equation, but written in terms of the momentum or moments functions. This equations are equivalent to the original one formulated in displacements, but have different forms. The state equations are taken into account integrally in accordance with the ideas of the method of integrodifferential relations. A numerical algorithm is developed for solving direct and inverse problems of the beam dynamics with variable cross section based on the Ritz method and the technique of semidiscrete polynomial approximations of the desired functions. The effectiveness of the approach is demonstrated by the example of controlled movements of a thin rectilinear elastic inhomogeneous rod. The control problem is to optimally transfer the rod from the initial to the given final state. The results of numerical analysis are presented.

The topicality of the issues related to studying vibrations of elastic bodies and structures is discussed. The publications and results obtained in this field have been analyzed. It is noted that one of the common characteristics of all the approximate methods of analyzing boundary-value problems is certain ambiguity in formulating finite-dimensional approximations of the solution. A boundary-value problem of determining eigen frequencies of a homogeneous membrane has been formulated. The basic idea of the approaches in question is that variables used in equations of mathematical physics can always be divided into two groups, one of which consists of the so-called measureable variables, such as displacement, velocity, temperature etc., and the other one includes non-measureable ones, such as stress, pulse, heat flow etc. Issues related to various classical formulations of spectral problems arising in the theory of elasticity have been investigated. The method of integral-differential relations is described, which is an alternative to classical numerical approaches. The possibility of constructing various bilateral energy-based evaluations of the accuracy of approximate solutions, following from the method of integral-differential relations has been studied. A one-parameter family of quadratic non-negative functionals has been introduced, the stationarity condition of which together with integral-differential constraints form a complete equation set describing dynamic behavior of elastic bodies. A projection approach to analyzing spectral problems of linear plasticity theory has been considered. Using the example of the problem of free vibrations of a circular membrane, the effectiveness of the method of integral-differential relations is demonstrated. Various energy-based evaluations of an approximate solution constructed using polynomial approximations of the sought functions are proposed. Application of the standard technique of Bubnov-Galerkin's method to the problem of free vibrations is shown to lead to the appearance of complex eigen frequencies, the real part of the eigenvalue being its approximate value, whereas its imaginary part being able to serve as an evaluation of the accuracy of the solution. The proposed numerical algorithm makes it possible to uniquely evaluate the integral quality of the obtained numerical solutions.

The main part of the paper is devoted to the formulations of the boundary value problem to finding natural frequencies of an inhomogeneous beam in the framework of the Euler-Bernoulli hypotheses. Three different formulations of the beam equation in displacements, momentums, and bending moments are discussed. Next, the possibilities to constructing various bilateral energy quality estimates for approximate solutions that follow from the method of integro-differential relations are investigated. In the final part, based on a numerical model example, the advantages of the variational technique in problems of free vibrations of inhomogeneous beams are discussed.

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.

For the forced motion of elastic bodies, we provide the variational and projection statements of initial-boundary problems. In the framework of the spatial linearmodel, we investigate the optimal control problem for an elastic rectilinear beam with a rectangular cross-section. Using the proposed generalized formulations, we develop a design algorithm for optimal displacements of elastic beams. Results of the numerical simulation and the analysis of the dynamics are provided.

This paper describes the issue of validity of running safety criterion on high-speed railways (HSR). In this research, the 2D model of ‘car–track–bridge’ system is used. A feature of HSR is the practical reaching of the critical speeds that cause resonance of bridge superstructures. The Eurocode establishes the limit of bridge superstructure acceleration below 5[Formula: see text]m/s2 in case of nonballasted track. This limit does not suit in certain conditions such as resonance of the superstructure that is permitted in Eurocode. In this case, there is no correlation between acceleration limit and firm wheel–rail contact. The numerical analysis shows repeated wheel lift-off during train passing a bridge zone. It is resonance of the bridge deck that causes this effect because without resonance, the running train is safe. Lower damping in nonballasted track increases resonant vibration influence.

The possibility of reducing an initial-boundary value problem formulated initially in partial derivatives to a system of ordinary differential equations of small dimension is considered. The properties of various quadratic energy relations arising in different problems of mathematical physics are studied and discussed. As an example, integro-differential formulations of a control problem of elastic body motion are presented in the framework of the two-dimensional theory of elasticity. It is shown that in this case the control problem using semi-discrete approximations with polynomials as basis functions can be reduced to a system of ordinary differential equations. In conclusion, the results of numerical simulation are presented and discussed.

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.

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.

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.

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.

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...

Controlled motions of a flexible rack feeder, which is a viscoelastic structure consisting of two joined beams clumped on a mobile carriage, are studied. The optimal control problem is to transfer the rack feeder from its initial state to terminal one in a fixed time and to minimize the mean mechanical energy of the system. The numerical solution of the control problem is based on the method of integro-differential relations.

Spatial motions of an elongated elastic body of cuboid shape (beam) are studied. The optimal control problem is to transfer the beam by moving its boundary cross section from the initial state to terminal one in a fixed time interval with minimization of a quadratic cost function. Based on the integral statement for the constitutive relations of linear elasticity, the generalized formulation of an initial-boundary value problem are given and a numerical approach to the optimization problem is developed.

Rack feeders are of high practical importance as automated conveying systems. In this paper, control-oriented models are derived for an experimental setup representing the structure of a typical high bay rack feeder. On the basis of these models, feedforward control strategies are designed. The rack feeder is a viscoelastic structure consisting of two identical flexible beams which are attached to a horizontally movable carriage. The beams are rigidly connected at their tip by a pulley block which is necessary for the vertical positioning of a payload. To develop a real-time applicable control algorithm, a frequency analysis is performed for the original double-beam structure. As a consequence, a simplified Bernoulli beam model is derived with specific boundary conditions. The control objective under consideration is the positioning of the flexible beam structure at a desired position in such a way that the terminal mechanical energy stored in the beam is minimized. A modification of the Galerkin method which is based on an integrodifferential approach and a suitable finite element technique are employed to describe the viscoelastic structural vibrations and to design optimal control strategies. Results of numerical simulations are presented and compared with measured data.

Modeling and optimization of control processes for compressible fluid flow in pipeline systems are considered. A mathematical model of controlled elements with distributed parameters is proposed to describe the dynamical behavior of a long rigid tube with transported liquid supplied by a control pump. Functions describing effective displacements and linear approximations of fluidic resistances are used to reduce the original initial-boundary value problem to a variational formulation on the basis of the method of integrodifferential relations. Moreover, this approach makes use of a novel finite element technique. Optimal control procedures are derived for changing either the outlet pressure or the mass flow to desired operating points and to damp undesirable elastic oscillations at the end of the control process. A projection approach is implemented on the basis of a modification of the Galerkin method in combination with the semi-discretization of unknown functions over the spatial coordinate. This procedure is applied to reliable numerical modeling of a simple pipeline system and to the design of an efficient control procedure. Numerical results obtained for different system parameters and control objectives are analyzed and discussed.

Open-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.

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.

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.

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.

Boundary-value problems in linear elasticity can be solved by a method based on introducing integral relations between the components of the stress and strain tensors. The original problem is reduced to the minimization problem for a nonnegative functional of the unknown displacement and stress functions under some differential constraints. We state and justify a variational principle that implies the minimum principles for the potential and additional energy under certain boundary conditions and obtain two-sided energy estimates for the exact solutions. We use the proposed approach to develop a numerical analytic algorithm for determining piecewise polynomial approximations to the functions under study. For the problems on the extension of a free plate made of two different materials and bending of a clamped rectangular plate on an elastic support, we carry out numerical simulation and analyze the results obtained by the method of integro-differential relations.

The paper studies dynamics modelling and control design for elastic systems with distributed parameters. The constitutive laws are specified by an integral equality according with the method of integro-differential relations. The original initial-boundary value problem is reduced to a constrained minimization problem for a nonnegative quadratic functional. 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 minimized functional is used to define an energy type criteria of solution quality. The efficiency of the approach is demonstrated on the example of a thin rectilinear elastic rod. The control problem is to find motion of a rod from an initial state to a terminal one at a fixed time with the minimal mean energy. The control input is presented by piecewise polynomial displacements on one end of the rod. It is possible to find the exact solution of the problem for a specific relation of the space-time mesh steps. The results of numerical analysis are presented and discussed.