Transverse Response of an Axially Moving Beam with Intermediate Viscoelastic Support

This study presented the transverse vibration of an axially moving beam with an intermediate nonlinear viscoelastic foundation. Hamilton’s principle was used to derive the nonlinear equations of motion. The finite difference and state-space methods transform the partial differential equations into a system of coupled first-order regular differential equations. The numerical modeling procedures are utilized for evaluating the effects of parameters, such as axial translation velocity, flexure rigidities of the beam, damping, and stiffness of the support on the transverse response amplitude and frequencies. It is observed that the dimensionless fundamental frequency and magnitude of axial speed had an inverse correlation. Furthermore, increasing the flexure rigidity of the beam reduced the transverse displacement, but at the same instant, fundamental frequency rises. Vibration amplitude is found to be significantly reduced with higher damping of support. It is also observed that an increase in the foundation damping leads to lower fundamental frequencies, whereas increasing the foundation stiffness results in higher frequencies.

Nonlinear Phenomena in Axially Moving Beams with Speed-Dependent Tension and Tension-Dependent Speed

This paper investigates some nonlinear dynamical behaviors about domains of attraction, bifurcations, and chaos in an axially accelerating viscoelastic beam under a time-dependent tension and a time-dependent speed. The axial speed and the axial tension are coupled to each other on the basis of a harmonic variation over constant initial values. The transverse motion of the moving beam is governed by nonlinear integro-partial-differential equations with the rheological model of the Kelvin–Voigt energy dissipation mechanism, in which the material derivative is applied to the viscoelastic constitutive relation. The fourth-order Galerkin truncation is employed to transform the governing equation to a set of nonlinear ordinary differential equations. The nonlinear phenomena of the system are numerically determined by applying the fourth-order Runge–Kutta algorithm. The tristable and bistable domains of attraction on the stable steady state solution with a three-to-one internal resonance are analyzed emphatically by means of the fourth-order Galerkin truncation and the differential quadrature method, respectively. The system parameters on the bifurcation diagrams and the maximum Lyapunov exponent diagram are demonstrated by some numerical results of the displacement and speed of the moving beam. Furthermore, chaotic motion is identified in the forms of time histories, phase-plane portraits, fast Fourier transforms, and Poincaré sections.

A Strategy for Achieving Smooth Filamentation Cutting of Transparent Materials with Ultrafast Lasers

A strategy is proposed for achieving a practically important mode of laser processing—a so-called “smooth” laser filamentation cutting (LFC) of transparent materials by a moving beam of a pulse-periodic pico- or subpicosecond laser. With such cutting, the surface of the sidewalls of the cuts have a significantly improved smoothness, and, as a result, the laser-cut plates have increased resistance to damage and cracking due to mechanical impacts during their subsequent use. According to the proposed analytical model, for the case when each filament is formed only by a single laser pulse, the strategy of such cutting is defined by a set of necessary conditions, whose implementation requires, in turn, a certain selection and matching with each other of irradiation parameters (pulse duration and energy, repetition rate, pitch of filaments following) and of material parameters—thermal, optical and mechanical strength constants. The model shows good agreement with experiments on sapphire and tempered glass. Besides, LFC modes are also predicted that provide very high cutting speeds of the order of 1–25 m/s or more, or allow cutting with an extremely low average laser power, but still at a speed acceptable for practical applications.

ON TRANSVERSAL VIBRATIONS OF AN AXIALLY MOVING BEAM UNDER INFLUENCE OF VISCOUS DAMPING

In this paper, a transversal vibration of an axially moving beam under the influence of viscous damping has been studied. The axial velocity of the beam is assumed to be positive, constant and small compared to wave-velocity. The beam is moving in a positive horizontal direction between the pair of pulleys and the length between the two pulleys is fixed. From a physical viewpoint, this model describes externally damped transversal motion for a conveyor belt system. The beam is assumed to be externally damped, where there is no restriction on the damping parameter which can be sufficiently large in contrast to much research material. The straightforward expansion method is applied to obtain approximated analytic solutions. It has been shown that the obtained solutions have not been broken out for any parametric values of the small parameter 𝜀. The constructed solutions are uniform and have been damped out. Even though there are several secular terms in the solutions, but they are small compared to damping.