Nonlinear Vibration of Sheet Metal Plates Under Interacting Parametric and External Excitation During Manufacturing

Abstract

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.

Optimizing the dynamical behavior of a dual-frequency parametric amplifier with quadratic and cubic nonlinearities

“…Single degree of freedom (SDOF) systems consisting nonlinear stiffness terms can exhibit a wide range of dynamical phenomena, as has already been reported by several researchers [7][8][9][10][14][15][16][17][18][19][20][21][22][23][24][25][26][27]. Models describing continuous systems with multiple modes of vibration can still be adequately described as SDOF system with quadratic and cubic nonlinear terms, under specific operating conditions (e.g., Nayfeh and Lacabonara [14,15]).…”

A symbolic algorithm for the automatic computation of multitone-input harmonic balance equations for nonlinear systems

“…For the computation of first main harmonic vectors become scalars and works like (15). First sum selects the specified combination m vector from the set to generate possible new rows according to the contribution of the row's relevant main frequency.…”

“…Also, chaotic behaviors of nonlinear systems under multitone excitations are of interest [14]. In [15], multifrequency parametric excitation was used for the analysis of a sheet metal coating process model. There are numerous papers to cite about multitone analysis for nonlinear systems on various application areas.…”

The Effects of Nonlinearity on Parametric Amplifiers

“…Also note that though equations of motion similar to that presented in Eqn. (1) have been explored in other research contexts, including sheet metal rolling and rotating discs, this work is believed to be the first to consider the system's dynamics within the context of parametric amplification [6][7][8][9][10][11].…”