Abstract
In various machines of the manufacturing industry, and in particular in paper converting machinery, there are shafts operating under conditions similar to that of a slender beam subjected to a transverse load moving in the axial direction. This condition can lead to vibrations and consequent deterioration of the machine performance and of the product quality. The problem has been theoretically studied in the literature since the 1990s. While shaft mass and stiffness are universally considered among the most influential parameters on its vibratory behavior, less obvious and not investigated in the literature is the influence of the spatial interval between two successive loads, an aspect that should be considered in the shaft design phase. In fact, if that is less than the length of the shaft, i.e., if there is more than one transverse load on the shaft at a given time, the vibration level may decrease with respect to the single-load configuration. This work describes the development of a mathematical model of a slender shaft hinged at its ends, representing the rotor of a paper roll perforating unit, with the SW Mathematica. The effect of a load moving axially at a given speed followed by similar loads after given spatial intervals was simulated investigating the influence of speed and load interval on shaft vibrations and resonance. The results showed how reducing the load interval can lead to a reduction of the shaft vibration which is a useful indication on possible design corrective actions.
Highlights
Rayleigh or Timoshenko beam in [8,9,10], assuming the clamped-hinged supports and including as in [11] the e ect of support elasticity. Starting from this background, this article presents a mathematical model for slender shafts subject to moving transverse loads to ideally simulate the industrial perforation process of paper rolls. e rotor of a paper roll perforating unit is a slender shaft with a given number of helicoidal blades that come into contact with a countershaft to perforate the paper sheet
Due to the helical shape of the blades, the contact is ideally a point that moves axially as the shaft rotates. ere can be more than one blade simultaneously in contact with the counteracting rotor depending on the number of blades and on the helix angle. e modelling and simulation of the dynamic behavior of such a system is of interest for the designer in order to avoid vibrations and consequent deterioration of the machine performance and of the product quality. e in uence of various parameters including speed and load interval on shaft vibrations and resonance was investigated
An analytical model of a shaft subject to a moving load was developed starting from the literature
Summary
Rayleigh or Timoshenko beam in [8,9,10], assuming the clamped-hinged supports and including as in [11] the e ect of support elasticity. Starting from this background, this article presents a mathematical model for slender shafts subject to moving transverse loads to ideally simulate the industrial perforation process of paper rolls. E shaft, schematically represented, in this rst analysis, is assumed of constant cross section, of length L, and directly hinged to the frame, neglecting the compliance and inertia of the end parts of reduced cross section, which are schematically represented using the dashed lines. Where EI is the exural sti ness of the shaft section, with E Young’s modulus of the shaft material, I the moment of Perforating blades
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