Nikolay Banichuk

Optimization and Analysis of Processes with Moving Materials Subjected to Fatigue Fracture and Instability

We study systems of traveling continuum modeling the web as a thin elastic plate of brittle material, traveling between a system of supports at a constant velocity, and subjected to bending, in-plane tension and small initial cracks. We study crack growth under cyclic in-plane tension and transverse buckling of the web analytically. We seek optimal in-plane tension that maximizes a performance vector function consisting of the number of cycles before fracture, the critical velocity and process effectiveness. The present way of applying optimization in the studies of fracture and stability is new and affords an analytical tool for process analysis.

On Structural Optimization with Incomplete Information

Abstract In this paper we investigate the problems of optimal design of structures under condition of incomplete information regarding the external action, material properties and arisen damages. In formulating and solving problems, we shall apply the worst case scenario (minimax guaranteed approach). Special attention is devoted to transformation of optimal design problems with incomplete information to the conventional problems of structural optimization.

Mechanics of moving materials

1 Introduction 2 Travelling strings, beams, panels, membranes and plates 3 Stability analysis 4 Non-homogeneous tension profile 5 Travelling panels made of viscoelastic material 6 Travelling panels interacting with external flow 7 Fracture and fatigue of travelling plates 8 Some optimization problems References Index

Vibrations of a continuous web on elastic supports

ABSTRACT We consider an infinite, homogenous linearly elastic beam resting on a system of linearly elastic supports, as an idealized model for a paper web in the middle of a cylinder-based dryer section. We obtain closed-form analytical expressions for the eigenfrequencies and the eigenmodes. The frequencies increase as the support rigidity is increased. Each frequency is bounded from above by the solution with absolutely rigid supports, and from below by the solution in the limit of vanishing support rigidity. Thus in a real system, the natural frequencies will be lower than predicted by commonly used models with rigid supports.

Some Problems of Multipurpose Optimization for Deformed Bodies and Structures

Some problems of multipurpose analysis and optimization of deformed structures and thin-walled structural elements are studied in this paper under some constraints including incomplete data. The first problem is the multipurpose optimization of layered plate made from given set of materials in context of optimization of ballistic limit velocity. Incomplete data concerning the thickness of layers of optimized multilayered shield structure are taken into account. The Pareto-approach and numerical evolutionary method (genetic algorithm) are used for solving of the considered multipurpose problem. The second problem studied in the paper is the shape optimization problem for rigid punch moving on the surface of elastic half-space, which is solved analytically in multipurpose formulation taking into account friction of contacted surfaces, wear of materials and arising pressure distributions. The relative movement is considered in frame of quasi-static formulation. Formulated optimization problem is studied analytically using the developed decomposition approach and exact solutions are obtained for the punch which has a rectangular contact region and moves translationally with a constant velocity.

On Bifurcation Analysis of Implicitly Given Functionals in the Theory of Elastic Stability

In this paper, we analyze the stability and bifurcation of elastic systems using a general scheme developed for problems with implicitly given functionals. An asymptotic property for the behaviour of the natural frequency curves in the small vicinity of each bifurcation point is obtained for the considered class of systems. Two examples are given. First is the stability analysis of an axially moving elastic panel, with no external applied tension, performing transverse vibrations. The second is the free vibration problem of a stationary compressed panel. The approach is applicable to a class of problems in mechanics, for example in elasticity, aeroelasticity and axially moving materials (such as paper making or band saw blades).

Safety Analysis and Optimization of Travelling Webs Subjected to Fracture and Instability

The problems of safety analysis and optimization of a moving elastic web travelling between two rollers at a constant axial velocity are considered in this study. A model of a thin elastic plate subjected to bending and in-plane tension (distributed membrane forces) is used. Transverse buckling of the web and its brittle and fatigue fracture caused by fatigue crack growth under cyclic in-plane tension (loading) are studied. Safe ranges of velocities of an axially moving web are investigated analytically under the constraints of longevity and instability. The expressions for critical buckling velocity and the number of cycles before the fracture (longevity of the web) as a function of in-plane tension and other problem parameters are used for formulation and investigation of the following optimization problem. Finding the optimal in-plane tension to maximize the performance function of paper production is required. This problem is solved analytically and the obtained results are presented as formulae and numerical tables.

Analysis and optimization against buckling of beams interacting with elastic foundation

ABSTRACT We consider an infinite continuous elastic beam that interacts with linearly elastic foundation and is under compression. The problem of the beam buckling is formulated and analyzed. Then the optimization of beam against buckling is investigated. As a design variable (control function) we take the parameters of cross-section distribution of the beam from the set of periodic functions and transform the original problem of optimization of infinite beam to the corresponding problem defined at the finite interval. All investigations are on the whole founded on the analytical variational approaches and the optimal solutions are studied as a function of problems parameters.

Optimization of Axially Moving Layered Web

The stability analysis and optimization of elastic web travelling between two rollers with a constant velocity are presented. The mathematical model for a layered travelling web (continuous isotropic composite plate) is developed restricting the consideration to one open draw. The layered plate with various mechanical properties of layers is considered and analytical expressions for the effective characteristics are derived. As a result the composed structure can be considered as an isotropic homogeneous plate and the obtained formulas for computation of critical velocity can be applied. Then the isoperimetric optimization problem is formulated and studied. The total mass of the layered plate is considered as an isoperimetric condition. The critical divergence velocity is taken as an optimized quality criterion. To this end consisted in maximization of the web stability and for maximization of the divergence velocity with respect to material distribution, the evolutionary optimization method (genetic algorithm) is applied. The number of materials is supposed to be given. Applying the genetic algorithm these materials are distributed on the plate thickness (provide the optimal plate consisted of some layers of different thickness) and the critical velocity is maximized under the constraint on the total mass of the structure. Numerical results are presented for different sets of problem parameters.

Travelling Panels Interacting with External Flow

This chapter is devoted to the analysis of the travelling panel, submerged in axially flowing fluid. In order to accurately model the dynamics and stability of a lightweight moving material, the interaction between the material and the surrounding air must be taken into account somehow. The light weight of the material leads to the inertial contribution of the surrounding air to the acceleration of the material becoming significant. In the small displacement regime, the geometry of the vibrating panel is approximately flat, and hence flow separation is unlikely. We will use the model of potential flow for the fluid. The approach described in this chapter allows for an efficient semi-analytical solution, where the fluid flow is solved analytically in terms of the panel displacement function, and then strongly coupled into the partial differential equation describing the panel displacement. The panel displacement, accounting also for the fluid–structure interaction, can then be solved numerically from a single integrodifferential equation. In the first section of this chapter, we will set up and solve the problem of axial potential flow obstructed by the travelling panel. In the second section, we will use the results to solve the fluid–structure interaction problem, and give so me numerical examples.