Integrodifferential Relations in Linear Elasticity

Kostin, Georgy; Saurin, Vasily V.

doi:10.1515/9783110271003

Cited by 27 publications

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“…The presented paper is devoted to an approach in which the original problem in partial differential equations (PDEs) is approximated by a system of ordinary differential equation (ODEs) based on the method of integro-differential relations (MIDR) [4]. The projection technique developed in [2] was applied to 3D linear elasticity problems with semi-discrete approximations for the displacement vector and the stress tensor.…”

Section: Introductionmentioning

confidence: 99%

“…As compared to the classical Galerkin method, the variational approach described in other studies of the authors [4] possesses some advantages: it guarantees optimality properties of the numerical solution for some approximations, provides explicit error estimates, and supposes the exact implementation of the initial, boundary, and momentum balance conditions. Some drawbacks of the variational method are doubling the dimension of the resulting ODE system and worsening the numerical stability of the results.…”

Section: Introductionmentioning

confidence: 99%

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A Projection Approach to Analysis of Natural Vibrations for Beams with Non-symmetric Cross Sections

Saurin

Kostin

2015

Computational Methods in Applied Sciences

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A projection approach based on the method of integro-differential relations and semi-discretization technique is applied to analyze natural variations of rectilinear elastic beams with non-symmetric cross sections. A numerical algorithm is proposed to compose compatible approximating systems of ordinary differential equations. It is shown that the beam vibrations cannot be separated into four independent types of longitudinal, bending, and torsional motions if a non-symmetric cross section is considered. In this case, all these motions can interact with one another. Nevertheless, only one type of displacement and stress fields makes the largest contribution in the amplitudes of the corresponding vibrations. Several eigenfrequencies and eigenforms of a beam with the isosceles cross section are presented and analyzed.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A Projection Approach to Analysis of Natural Vibrations for Beams with Non-symmetric Cross Sections

Saurin

Kostin

2015

Computational Methods in Applied Sciences

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“…Both of these publications are the basis for the experimental case study for a spatially one-dimensional heat transfer process in this paper. For further information concerning the theoretical background of the MIDR, see the work of Kostin and Saurin (2012). Possible extensions of this approach to a problem-oriented modeling of higher-dimensional applications can be found in the works of Rauh et al (2013b) and Kersten et al (2014).…”

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confidence: 99%

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

Rauh

Senkel

Aschemann

et al. 2016

International Journal of Applied Mathematics and Computer Science

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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 finitedimensional 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.

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“…In addition, the beams are coupled hingedly by means of two rigid rods. The driver force applied to the carriage is the control input of the structure.The projection approach is based on an integro-differential formulation [3] for the initial-boundary value problem of structure motions with the constitutive relations generalized in the dimensionless form according tow(0, y) = w 0 (y) and p(0, y) = p 0 (y) ; A a w(t, a) + B a s(t, a) = u(t)c a + f a (t) with a ∈ {0, 1} .Here t denotes the time and y is the common coordinate for each structure element. The function w(t, y) ∈ R n is generalizes displacements, p(t, y) ∈ R n is the generalized momentum density, and s(t, y) ∈ R n is the generalized forces.…”

mentioning

confidence: 99%

“…The projection approach is based on an integro-differential formulation [3] for the initial-boundary value problem of structure motions with the constitutive relations generalized in the dimensionless form according to…”

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confidence: 99%

A variational approach to modelling and optimization of viscoelastic structure motions

Kostin

2016

Proc Appl Math and Mech

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Locomotion of a flexible rack feeder consisting of two viscoelastic beams clumped on a mobile carriage are studied. The optimal control problem is to move 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 solution of the problem is based on the method of integro-differential relations. 1 The statement of an optimal control problem for viscoelastic structure dynamicsThe method of integro-differential relations (MIDR) was proposed in [1] to design the optimal control of flexible structure motions. This technique is extended to modelling and optimization of a rack feeder system with distributed parameters, which has already been considered by using an alternative representation in [2], where the system parameters are given. The experimental setup, built up at the Chair of Mechatronics of the University of Rostock, represents the structure of typical rack feeders as shown in Fig. 1. The structure consists of two identical beams clamped to a movable carriage. Both beams are rigidly connected at their tips by a pulley block, which is necessary for the positioning of a cage. In addition, the beams are coupled hingedly by means of two rigid rods. The driver force applied to the carriage is the control input of the structure.The projection approach is based on an integro-differential formulation [3] for the initial-boundary value problem of structure motions with the constitutive relations generalized in the dimensionless form according tow(0, y) = w 0 (y) and p(0, y) = p 0 (y) ; A a w(t, a) + B a s(t, a) = u(t)c a + f a (t) with a ∈ {0, 1} .Here t denotes the time and y is the common coordinate for each structure element. The function w(t, y) ∈ R n is generalizes displacements, p(t, y) ∈ R n is the generalized momentum density, and s(t, y) ∈ R n is the generalized forces. The functions v(t, y) ∈ R n and r(t, y) ∈ R n are arbitrarily chosen virtual velocities and displacements. The constitutive vector-functions q := p − M · w t and g := s − K · Dw − R · Dw t of t and y are introduced. The components of the operators D and D are [D] i,j = δ i,j ∂ α /∂y α and [D] i,j = (−1) α+1 δ i,j ∂ α /∂y α with α = 2 for i ≤ m (lateral motions) and α = 1 for i > m (longitudinal and torsional motions). The mass, stiffness, and damping matrices M(y), K(y), R(y) include structural parameters. The functions w 0 (y) and p 0 (y) define the initial state. The form of the linear operators A a and B a depends on the boundary and interelement conditions. The function u(t) ∈ R is the control input, c a ∈ R m+n is the input vector with the number m ≤ n of the lateral degrees of freedom, f (t, y) ∈ R n and f a (t) ∈ R m+n are the functions of external disturbances.The control objective is to move the structure to a desired state in a given time interval T and to minimize the cost functionsubject to the constraints w(T, y) = w 1 (y) and p(T, y) = p 1 (y). The equation of the carriage driver is τv(t) = u(t) − v(t), where v(t) is the carriage velocity, γ ≥ 0 is the weig...

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Integrodifferential Relations in Linear Elasticity

Cited by 27 publications

References 0 publications

A Projection Approach to Analysis of Natural Vibrations for Beams with Non-symmetric Cross Sections

A Projection Approach to Analysis of Natural Vibrations for Beams with Non-symmetric Cross Sections

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

A variational approach to modelling and optimization of viscoelastic structure motions

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