Integro-differential approach to solving problems of linear elasticity theory

2006

Arch Appl Mech

Some possible modifications of the governing equations of the linear theory of elasticity are considered. The stress-strain relation is specified by an integral equality instead of the local Hooke's law. The modified integrodifferential boundary value problem is reduced to the minimization of a nonnegative functional under differential constraints. A numerical algorithm based on polynomial approximations of unknown functions (stresses and displacements) is developed and applied to linear elasticity problems. The bilateral estimation criteria of solution errors are proposed in order to analyze the algorithm convergence rate. The numerical results obtained by applying the integrodifferential relation method and the conventional variational method are compared and discussed.

Section: Fig 1 Elastic Bodymentioning

confidence: 99%

“…The basic ideas of this approach are discussed in [13]. In the next section, the modified boundary value problem is reduced to a constrained variational problem.…”

Section: Introductionmentioning

confidence: 99%

The Method of Integrodifferential Relations for Linear Elasticity Problems

2006

Arch Appl Mech

“…The general method of integro-differential relations (IDR) for solving a wide class of boundary value problems is developed and criteria of solution quality are proposed [1]. A numerical algorithm for discrete approximation of controlled motions is worked out [2] and applied to design the optimal control low steering an elastic system to the terminal position and minimizing the given objective function [3].…”

mentioning

confidence: 99%

“…To solve the boundary value problem (1)- (3), we apply the method of integro-differential relations (IDR), described in [1], in which some strict local relations are replaced by an integral relation. In this case, it possible to reduce problem (1)-(3) to a variational problem.…”

mentioning

confidence: 99%

Method of Integro‐Differential Relations for Optimal Beam Control

Proc Appl Math and Mech

2006

An approach to modelling and optimization of controlled dynamical systems with distributed elastic and inertial parameters are considered. The general method of integro-differential relations (IDR) for solving a wide class of boundary value problems is developed and criteria of solution quality are proposed [1]. A numerical algorithm for discrete approximation of controlled motions is worked out [2] and applied to design the optimal control low steering an elastic system to the terminal position and minimizing the given objective function [3]. The polynomial control of plane motions of a homogeneous cantilever beam is investigated. The optimal control problem of beam transportation from the initial rest position to given terminal states, in which the full mechanical energy of the system reaches its minimal value, is considered.Consider the plane controlled motions of a homogeneous rectilinear elastic beam. One end of the beam is free, and the other is clamped on a truck that can move in a horizontal plane. In the strainless state, the beam is in a vertical position. The control action for the beam is the horizontal acceleration u of the truck. Initially, we are given the shape of the beam transverse deflection (displacement) w and its relative linear momentum density p in a coordinate system tied to the truck moving at a velocity v. The location of the truck in a stationary coordinate system is specified by x; here,ẋ = v andv = u. Without loss of generality, it can be assumed that the coordinate and velocity of the truck are initially zero. The equations of motion of the beam with initial and boundary conditions have the forṁHere, m is the bending moment in the beam cross section; and ρ are the length and linear density of the beam, respectively; EI is its flexural rigidity; and T is the terminal time instant of the controlled motion. The dotted symbols denote the partial derivatives with respect to t, and the primed symbols stand for the partial derivatives with respect to y. The problem is to find an optimal control u(t) that drives the truck from its initial to terminal states in the given time T and minimizes a performance index J [u] in the class U of admissible controls:To solve the boundary value problem (1)- (3), we apply the method of integro-differential relations (IDR), described in [1], in which some strict local relations are replaced by an integral relation. In this case, it possible to reduce problem (1)-(3) to a variational problem. If there exists a solution then the following functional Φ reaches on this solution its absolute minimumNote that the integrand in (3.5) has the dimension of the energy density and is nonnegative. Hence, the corresponding integral is nonnegative for any arbitrary functions p, m, and w (Φ ≥ 0). To find an approximate solution to the optimization problem defined by (1), (3)- (5), we use a polynomial representation of the desired functions. The functions p, m, and w are approximated by polynomials in two variables and the control u is defined by a polynomial in time of ...

“…In the IDR method [2,3], the equations of Hooke's law (1) are replaced by a nonnegative quadratic integral and the unknown functions are the components of the stress tensor σ and the displacement vector u. To solve elasticity problems, we introduce the equivalent variational formulation…”

mentioning

confidence: 99%

Variational Approaches in the Beam Theory

Proc Appl Math and Mech

2006

Equations governing the stress-strain state of an elastic beam are derived on the basis of the method of the integro-differential relations (IDR). The influence of Poisson's ratio and shear modulus on the stiffness characteristics of an elastic beam with a rectangular cross section is investigated. Explicit expressions are obtained for the stress fields in an isotropic or anisotropic beam for an arbitrary type of loading. A system of differential equations for displacement fields in these beams is presented.Consider an elastic rectangular prism (beam) with boundary γ. Introduce a Cartesian coordinate system Oxy with origin O located at the middle point of one of the smallest beam sides and the x-axis perpendicular to this side and pointing toward the opposite side. Let h and be the height and the length of the beam. The thickness of the body is equal to unity. The plane stress-strain state of an isotropic beam is governed by the system of differential equations of 2D elasticity [1]