Discrete differential geometry modeling on the fully coupled vibration of the viscoelastic rod interacting with rigid cylinders

Abstract

The coupled vibration of flexible rods in contact with rigid cylinders has been observed to generate disruptive noise and significantly decrease the service life. Although the coupling effect of cylindrical rotation with transverse vibration or axial motion of slender rods has been well studied, the full coupling of the three motions remains inadequately understood. To address this issue, we propose an improved discrete differential geometry model with a tri-linear friction method to deliver accurate reproduction of resonant frequencies and vibration responses. Our model features several advances: (i) Viscous forces associated with stretching and bending strain rates are incorporated into the motion equation of the viscoelastic rod, replacing the assumption of quasi-static axial motion; (ii) A tri-linear friction method is introduced to obtain the relative motion between the slender rod and cylinders on the contact zone, thus eliminating errors stemming from pre-prescribed contact regions; (iii) The model fully couples the axial and transverse motion over the entire rod as well as the rotation of cylinders, surpassing the limitation of the coupled rotational-transverse model which only establishes equations of transverse motion on the non-contact zone of the rod. The numerical results indicate that the axial and transverse vibrations of the upper and lower span of the slender rod and the cylindrical rotational vibration are mutually excited, with their resonant frequencies having overlapping sites, which validates the proposed coupling model. Moreover, we are able to capture the phenomenon of contact surface sliding during the coupled vibration, confirm that the sliding limits the vibration amplitude, and determine the nonlinear relationship between the excitation and vibration response. This work provides a theoretical basis for further optimization of composite transmission structures, particularly in terms of suppressing vibration and improving efficiency of power transmission.