On the effects of damping on the dynamics of axially moving spatially periodic strings

Abstract

The paper concerns the analysis of the effects of damping on the linear dynamics of an axially moving non-uniform structure, string with periodically varying mass and damping properties. The string, in particular models transmission chains for drive engines. An expression for the threshold velocity of the string axial motion is determined; if the string moves with a velocity higher than the threshold one, then waves travelling in it are unstable. For a uniform string, the threshold velocity equals to the transverse wave speed. It is shown that pure modulations of the string mass can increase the threshold velocity for waves with certain wavenumbers. By contrast, combined modulations of mass and damping properties lead to reduction of the threshold velocity as compared to the case of the uniform string. And the larger the amplitudes of the modulations, the more pronounced the reduction. This indicates that due to periodic spatial modulations of the string properties, the string dynamics can become unstable even if its axial velocity is considerably lower than the transverse wave speed. Thus, periodic distribution of damping and mass properties can considerably affect design of axially moving structures used in applications.