Self-excited vibrations in gyroscopic systems

Self-excited vibrations are an unwanted phenomenon in many technical applications. Frequently they occur in moving media that are in frictional contact with other structures. Important examples for these kind of structures are moving belts, band saws, disk brakes, clutches and many more. The aim of this thesis is to make a contribution towards a convenient mathematical treatment of these problems. In a first part of the thesis linear discrete gyroscopic systems under the influence of nonconservative forces are studied. Following ideas of Seyranian and his coworkers perturbation theory is applied in order to obtain analytical approximations to the stability boundaries of these kinds of systems. The subsequent chapters of the thesis deal with the introduction of frictional contact into continuous models originating from the theory of elasticity. As a preliminary example an elastic rod is considered to prepare the derivation of the equations of motion using variational principles in the context of non material transition conditions originating from the frictional contact. The following chapter deals with a moving beam sliding through frictional pins. The equations of motion are derived from the basics of the theory of elasticity yielding a boundary value problem and a discretization approach for numerical calculations. Inspection of the equations of motion yields a mathematical and a mechanical explanation for the occurrence of self-excited vibrations. Using a perturbation approach stability boundaries for the system are considered and results are discussed in the context of Arnold’s singularity theory. The next system under consideration is rotating Kirchhoff plate in frictional contact with idealized friction pads. The methods developed for the moving beam are extended to the rotating plate and in particular the coupling of in- and out-of-plane vibrations is discussed. Furthermore the influence of nonlinearities of the pad characteristics is studied. The model can be seen as a minimal model for disk brake squeal but changes of the parameters and slight modifications would allow for conclusions about other mechanical systems. With regard to brake squeal problems in the treatment with commercial FEM codes are addressed and a method for investigating asymmetric brake rotors is given. In the last part of the thesis the discretization approaches used are studied mathematically. Using ideas from the theory of hyperbolic partial differential equations and introducing the concept of weak solutions, existence and uniqueness of the solutions for the problems studied are proven, at the same time establishing convergence of the discretization methods used.