Malyshev S. Bifurcations and stability of nonlinear vibrations of beam structures with fatigue cracks

The dissertation is devoted to solution of an actual scientific and practical problem related to the analysis of bifurcations and stability of nonlinear vibrations of beams with breathing fatigue cracks. The goal is to develop mathematical models and numerically analyze nonlinear vibrations of beams with breathing cracks under conditions of interaction of nonlinearities of various nature. Research Object is the nonlinear dynamics of beam structures with fatigue cracks. The research Subject is the construction of mathematical models for describing nonlinear vibrations of beams with fatigue cracks and analyzing their bifurcation behavior. The crack breathing causes changes in the stiffness of the structure, leading to significant nonlinear effects. The possibility of transition to chaotic dynamics in the vibrations of beams with two fatigue cracks at small amplitudes is shown. The problem of geometrically nonlinear vibrations of a beam with a fatigue crack is considered. Using the Hamilton and Hu–Washizu variational principles, differential equations for the vibrations of a beam with a fatigue crack are derived. Nonlinear normal modes of free vibrations, their stability, and bifurcations are investigated. An analysis of forced vibrations of a beam with a fatigue crack under geometrically nonlinear deformation is conducted. The possibility of closed loops (isolas) creation on frequency responses is shown. The established vibration regimes in the region of Neimark–Sacker bifurcation is analyzed. Parametric vibrations of a beam with a fatigue crack are studied, taking into account the effects of nonlinear curvature and nonlinear inertia. The dynamics in the region of Neimark–Sacker bifurcation is analyzed, and the possibility of transitioning to chaotic vibrations of the beam with a fatigue crack is established. The effect of linear dissipation on the bifurcation behavior of the beam with a fatigue crack is investigated. The introduction provides a justification for the relevance of the chosen research topic, indicates the research methods, highlights the scientific novelty and practical significance of the results obtained. It notes the personal contribution of the author to the dissertation work, and provides brief information on the testing of the dissertation materials. Also the differences from previously known results are shown. In the first chapter, theoretical and practical investigation aspects of the nonlinear vibrations of beams with fatigue cracks are considered. The current state of the problem is analyzed, and various models used to describe nonlinear vibrations of beams with fatigue cracks are compared. Chapter 2 presents numerical methods that were developed and improved during the research. General information about automatic differentiation is provided, along with examples of applying dual numbers for calculating derivatives. A method for calculatingthe Jacobian matrix for two-point boundary value problems is proposed. Proposed method allows the variational equations not to be derived by hand. A method for calculating the Lyapunov spectrum for systems of ordinary differential equations via automatic differentiation is also suggested. Chapter 3 is devoted to the construction of mathematical models of vibrations of beams with opened fatigue cracks using variational principles. Using the Hamilton principle, a vibration model of a beam with a fatigue crack under geometrically nonlinear deformation is developed. Using the Hu–Washizu principle, differential equations of geometrically nonlinear vibrations for a beam with a fatigue crack are obtained. A model of parametric vibrations of a beam with a fatigue crack is developed using the variational principle of Hu–Washizu. The effects of nonlinear curvature and inextensionality of the beam’s midline are taken into account. The Lagrange multipliers method is used to take into account the inextensionality phenoma. Chapter 4 conducts a numerical analysis of vibrations of beams with fatigue cracks. A contact parameter is used to account for the breathing phenoma. The forced vibrations of a cantilever beam with one and two fatigue cracks at small amplitudes are studied. The emergence of multivaluedness and chaos in the vibrations of a cantilever beam with breathing fatigue cracks are shown. The free vibrations nonlinear normal mode analysis of a beam with a fatigue crack under geometrically nonlinear deformation is conducted. In-phase and antiphase modes are studied. The possibility of isolas creation in frequency responses of a beam with a fatigue crack is shown. The established vibration regimes in the region of Neimark–Sacker bifurcation are ananalyzed. Parametric vibrations of a beam with a fatigue crack are considered. Large curvature and nonlinear inertia phenoma are taken into account. The effect of dissipation on the parametric vibrations of a beam with a fatigue crack is analyzed. The possibility of transitioning to chaotic vibrations is shown.