Study On The Dynamics Of A Class Of Geometric Nonlinear Dry Friction Oscillators

The study of dynamics of non-smooth systems is an important subject in the field of modern science and technology.Dry friction is a typical non-smooth factor,which will lead to complex dynamics problems.At the same time,geometric nonlinearity is a kind of nonlinear problem in practical engineering applications.Practical engineering problems that co-exist with dry friction and geometric nonlinearities,such as bowstring instruments and aircraft carrier fighter interceptors,are also widespread.Stick-slip vibrations may occur in these systems due to the existence of dry friction.However,the current research on friction system is more focused on the case of trajectory crossing the switching surface,and more inclined to numerical calculation.Due to the particularity of dry friction,the numerical calculation of the system is also need special numerical processing.In this paper,the dynamical system theory is combined with the event-driven method for calculating the dry friction system,and a series of compli-cated stick-slip nonlinear dynamics phenomena of a class of geometric nonlinear dry friction oscillators are studied,which provides important theoretical basis and numerical calculation method for engineering application research.The main contents and results are as follows:In Chapter 1,the engineering background,research purpose and significance of geomet-ric nonlinear dry friction oscillators are introduced.The latest research achievements of this kind of system are summarized.The basic theory of Filippov system and various numerical calculation methods of Filippov system are introduced.In Chapter 2,a class of geometric nonlinear dry friction oscillators with dry friction is es-tablished.The event-driven method and sliding bifurcations are briefly introduced.An event-driven method is used to calculate the oscillators under two dry friction characteristics,and four sliding bifurcations and period-added bifurcations are studied.In Chapter 3,the Fundamental matrix solutions theory of smooth system is presented and the dry friction system is smoothened by jump matrix,the stability of stick-slip periodic solutions are studied.At the same time,the detachment phase map generated by the dimension reduction was introduced to study the stability of the stick-slip periodic orbits.Compared with the Fundamental matrix solutions theory,the detachment phase map achieved the same result.At the end of this chapter,the Floquet multiplier calculated by the fundamental matrix solution theory is used to replace the detachment phase map.The stick-slip periodic orbits are calculated by shooting method and continuation algorithm.The delayed bifurcation phenomenon of dry friction system is studied.In Chapter 4,the Lyapunov exponent of the dry friction system is calculated by the com-bination of the cocycle theory and the event-driven method,and the reasons for the existence of different types of periodic motions are verified by the theory of circle homeomorphism and the detachment phase map.Meanwhile,the coexisting attractors' basins of attraction and evolution are studied based on the simple cell mapping method.