This study is motivated by the vibrations that plague coating processes used in the manufacturing of coated sheet metal. These vibrations arise from time-dependent tension fluctuations within the sheet metal plate as well as from the eccentricity of the rollers used to transport the plate. The time-dependent tension is observed to be rather broad-band and creates multi-frequency parametric excitation. By contrast, the roller eccentricity is largely single-frequency (synchronized with the roller speed) and creates single-frequency external excitation. The plate and excitation sources are studied herein using a single-degree-of-freedom model with a cubic nonlinearity, subject to combined parametric and external excitation. In our study, we investigate the resonances that arise from the synergistic effects of multi-frequency parametric excitation and single-frequency external excitation. For the simpler case of single-frequency parametric excitation, we observe both sum and difference combination resonances in addition to principal parametric resonance. For the case of multi-frequency parametric excitation, we observe a frequency shift for the parametric resonance that derives from the cubic nonlinearity and external excitation. Moreover, the phase relationships of the external and each parametric excitation source have a significant effect on the resulting response amplitude. We use these analyses to explain the resonance mechanisms observed in experiments conducted on an example sheet metal coating process.
Detection of different dynamics of two coupled oscillators including a time-dependent cubic nonlinearity
