Endless elastic beam travelling on a moving rough surface with zones of stick and sliding

We study the planar deformation of a beam that travels across a given control domain supported by a moving rough plane, which is a prototype for various technological processes. A sufficiently small misalignment between the guideways at the ends of the domain results in a stationary regime of motion, which features a zone of sticking contact near the entry to the control domain, followed by infinitely many segments of transverse sliding with alternating directions. Self-similarity of this solution of an essentially nonlinear boundary value problem is the primary novel result of the present contribution. Closed-form analytic results are validated against a finite element simulation of the transient evolution process, which demonstrates stability of the obtained solution and provides insights regarding the characteristic time scales of establishing of subsequent zones of sliding.

Endless Elastic Beam Travelling on a Moving Rough Surface With Zones of Stick and Sliding

We study the planar deformation of a beam that travels across a given control domain supported by a moving rough plane, which is a prototype for various technological processes. A sufficiently small misalignment between the guideways at the ends of the domain results into a stationary regime of motion, which features a zone of sticking contact near the entry to the control domain followed by infinitely many segments of transverse sliding with alternating directions. Self-similarity of this solution of an essentially nonlinear boundary value problem is the primary novel result of the present contribution. Closed-form analytic results are validated against a finite element simulation of the transient evolution process, which demonstrates stability of the obtained solution and provides insights regarding the characteristic time scales of establishing of subsequent zones of sliding.

THE MOTION OF A PARTICLE ON A WAVY SURFACE DURING ITS TRANSLATIONAL CIRCULAR OSCILLATIONS IN HORIZONTAL PLANES

The rough plane is a universal structural element of many machines and devices for sifting and separation of parts of technological material. The motion of particles on the horizontal plane, which performs oscillating rectilinear or circular motion, is the most studied. A wavy surface with a sinusoidal cross-sectional line as a working surface will significantly change the trajectories of their motion. The mathematical description of such a motion will change accordingly. The sliding of a particle in a plane will be a partial case of sliding on a wavy surface when the amplitude of the sinusoid is equal to zero. When the wavy surface oscillates and all its points describe circles, the motion of the technological material changes significantly. The regularities of the motion of material particles on such a surface during its circular translational oscillations in the horizontal planes are investigated in the article. Differential equations of relative particle displacement are compiled and solved by numerical methods. The trajectories of the particle sliding on the surface and the graphs of its reaction are constructed. A partial case of a surface is a plane, and the sliding trajectory of a particle is a circle. An analytical expression to determine its radius is found. During circular oscillations of a wavy linear surface with a cross section in the form of a sinusoid relative trajectory of a particle after stabilization of the motion can be closed or periodic spatial curves. To avoid the breakaway of the particle from the surface, the oscillation mode should be set, which takes into account the shape of the surface and the kinematic parameters of oscillations. With the diameter of the circle, which is described by all points of the surface during its oscillation, is equal to the period of the sinusoid, the trajectory of the relative motion of the particle can be a periodic curve. In this case, the particle moves in a direction close to the transverse, overcoming depressions and ridges. In other cases, the trajectory is a closed spatial curve, the horizontal projection of which is close to a circle. The found analytical dependencies allow determining the influence of structural and technological parameters of the surface on the trajectory of the particle.