0000000000922249
AUTHOR
Alexander Barsuk
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 (su…
Stability of Axially Moving Plates
This chapter focuses on the stability analysis of axially moving materials, in the context of two-dimensional models. There are many similarities with the classical stability analysis of structures, such as the buckling analysis of plates. However, the presence of axial motion introduces inertial effects to the model. We consider the stability of an axially moving elastic isotropic plate travelling at a constant velocity between two supports and experiencing small transverse vibrations. We investigate the stability of the plate using an analytical approach. We also look at elastic orthotropic plates, and an elastic isotropic plate subjected to an axial tension distribution that varies in th…
Analysis and optimization against buckling of beams interacting with elastic foundation
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. peerReviewed
Bifurcation method of stability analysis and some applications
In this paper a new approach to the analysis of implicitly given function- als is developed in the frame of elastic stability theory. The approach gives an effective procedure to analyse stability behaviour, and to determine the bifur- cation points. Examples of application of the proposed approach for analysis of stability are presented, more precisely we consider the stability problem of an axially moving elastic panel, with no external applied tension, performing transverse vibrations. The analysis is applicable for many practical cases, for example, paper making and band saw blades.
Variational approach for analysis of harmonic vibration and stabiligy of moving panels
In this paper, the stability of a simply supported axially moving elastic panel (plate undergoing cylindrical deformation) is considered. A complex variable technique and bifurcation theory are applied. As a result, variational equations and a variational principle are derived. Analysis of the variational principle allows the study of qualitative properties of the bifurcation points. Asymptotic behaviour in a small neighbourhood around an arbitrary bifurcation point is analyzed and presented. It is shown analytically that the eigenvalue curves in the (ω, V0) plane cross both the ω and V0 axes perpendicularly. It is also shown that near each bifurcation point, the dependence ω(V0) for each m…
An analytical-numerical study of dynamic stability of an axially moving elastic web
This paper is devoted to a dynamic stability analysis of an axially moving elastic web, modelled as a panel (a plate undergoing cylindrical deformation). The results are directly applicable also to the travelling beam. In accordance with the dynamic approach of stability analysis, the problem of harmonic vi- brations is investigated via the study of the dependences of the system’s nat- ural frequencies on the problem parameters. Analytical implicit expressions for the solution curves, with respect to problem parameters, are derived for ranges of the parameter space where the natural frequencies are real-valued, corresponding to stable vibrations. Both axially tensioned and non-tensioned tra…
On some bifurcation analysis techniques for continuous systems
This paper is devoted to techniques in bifurcation analysis for continuous mechanical systems, concentrating on polynomial equations and implicitly given functions. These are often encountered in problems of mechanics and especially in stability analysis. Taking a classical approach, we summarize the relevant features of the cubic polynomial equation, and present some new aspects for asymptotics and parametric representation of the solutions. This is followed by a brief look into the implicit function theorem as a tool for analyzing bifurcations. As an example from mechanics, we consider bifurcations in the transverse free vibration problem of an axially compressed beam. peerReviewed
Variational principle and bifurcations in stability analysis of panels
In this paper, the stability of a simply supported axially moving elastic panel is considered. A complex variable technique and bifurcation theory are applied. As a result, variational equations and a variational principle are derived. Anal- ysis of the variational principle allows the study of qualitative properties of the bifurcation points. Asymptotic behaviour in a small neighbourhood around an arbitrary bifurcation point is analyzed and presented. It is shown analytically that the eigenvalue curves in the (ω, V0) plane cross both the ω and V0 axes perpendicularly. It is also shown that near each bifur- cation point, the dependence ω(V0) for each mode approximately follows the shape of …