Frictional Vibration Transmission From a Laterally Moving Surface to a Traveling Beam

As the density of information stored in automated magnetic tape libraries continues to increase, greater requirements are placed on the precision of mechanical positioning in order to successfully read and write data bits. The location of the read/write head in the direction across the tape’s width (termed the lateral direction) is actively controlled in order to maintain alignment between the head and data tracks, even in the presence of the tape’s lateral vibration. However, during repositioning, vibration is undesirably transmitted from the laterally moving head structure to the axially-moving tape because of frictional contact between the two adjacent surfaces. As an analog of that interaction, a model is developed here to describe frictional vibration transmission from a surface having prescribed lateral motion to a tensioned beam that travels and slides over it. The beam is divided into contiguous regions corresponding to free spans and the beam’s portion that contacts the surface. A critical engagement length between the beam and the surface exists for which vibration transmission at a particular natural frequency can be substantially reduced, and for a given mode, that length depends weakly on the surface’s position along the beam’s span. By contouring the surface to have portions of differing radii of curvature, the extent of vibration transmission can be reduced over a broad range of frequency.

Friction Induced Vibrations in Moving Continua and Their Application to Brake Squeal

Considerable effort is spent in the design and testing of disk brake systems installed in modern passenger cars. This effort can be reduced if appropriate mathematical–mechanical models are used for studying the dynamics of these brakes. In this context, the mechanism generating brake squeal in particular deserves closer attention. The present paper is devoted to the modeling of self-excited vibrations of moving continua generated by frictional forces. Special regard is given to an accurate formulation of the kinematics of the frictional contact in two and three dimensions. On the basis of a travelling Euler–Bernoulli beam and a rotating annular Kirchhoff plate with frictional point contact the essential properties of the contact kinematics leading to self-excited vibrations are worked out. A Ritz discretization is applied and the obtained approximate solution is compared to the exact one of the traveling beam. A minimal disk brake model consisting of the discretized rotating Kirchhoff plate and idealized brake pads is analyzed with respect to its stability behavior resulting in traceable design proposals for a disk brake.

Post-Critical Response of an Axially Moving Beam

In this paper the dynamic response of a simply supported traveling beam, subjected to a pointwise transversal load, is investigated. The motion is described by means of a high dimensional system of ordinary differential equations with linear gyroscopic part and cubic nonlinearities obtained through the Galerkin method. The system is studied in the super-critical speed range with emphasis on the stability and the global dynamics that exhibits special features after the first bifurcation. A sample case of a physical beam is developed and numerical results are presented concerning bifurcation analysis and stability, and direct simulations of global postcritical dynamics. In the supercritical speed range a regular motion around a bifurcated equilibrium position becomes chaotic for particular values of frequency and force. The bifurcation diagram for varying force intensity is shown, it can be noticed that a chaotic motion occurs in a wide range of the forcing parameter, co-existing with a 3T periodic solution in a limited window.

Classical Vibration Analysis of Axially Moving Continua

Axially moving continua, such as high-speed magnetic tapes and band saw blades, experience a Coriolis acceleration component which renders such systems gyroscopic. The equations of motion for the traveling string and the traveling beam, the most common models of axially moving materials, are each cast in a canonical state space form defined by one symmetric and one skew-symmetric differential operator. When an equation of motion is represented in this form, the eigenfunctions are orthogonal with respect to each operator. Following this formulation, a classical vibration theory, comprised of a modal analysis and a Green’s function method, is derived for the class of axially moving continua. The analysis is applied to the representative traveling string and beam models, and exact closed-form expressions for their responses to arbitrary excitation and initial conditions result. In addition, the critical transport speed at which divergence instability occurs is determined explicitly from a sufficient condition for positive definiteness of the symmetric operator.