Nonlinear time-delay feedback controllability for vertical parametrically excited vibration of roll system in corrugated rolling mill

This paper investigates vibration characteristics of the corrugated roll system and designs a time-delay feedback controller to control the parametrically excited vibration of system. The model of parametrically excited nonlinear vertical vibration of roller system is established by considering the nonlinear damping and nonlinear stiffness within corrugated interface of corrugated rolling mill. The approximate analytical solution and amplitude-frequency characteristic equations of principal resonance and sub-resonance of roller system are obtained by using the multiple-scale method. The influences of nonlinear stiffness coefficient, nonlinear damping coefficient, system damping coefficient and rolling force amplitude on vibration are further analyzed. The time-delay feedback controller is designed to eliminate the jump and hysteresis phenomenon of the roll system and numerical simulation results demonstrate the effectiveness of the controller. The analysis results provide some theoretical guidance for vibration suppression of roller system of corrugated rolling mill.

Parametrically Excited Stability of Periodically Supported Beams Under Longitudinal Harmonic Excitations

A direct eigenvalue analysis approach for solving the stability problem of periodically supported beams with multi-mode coupling vibration under general harmonic excitations is developed based on the Floquet theorem, Fourier series and matrix eigenvalue analysis. The transverse periodic supports are considered for improving the parametrically excited stability of beams under longitudinal periodic excitations. The dynamic stability of parametrically excited vibration of the beam with transverse spaced supports under longitudinal harmonic excitations is studied. The partial differential equation of motion of the beam with spaced supports under harmonic excitations is given and converted into ordinary differential equations with time-varying periodic parameters using the Galerkin method, which describe the parametrically excited vibration of the beam with coupled multiple modes. The fundamental solution to the equations is expressed as the product of periodic and exponential components based on the Floquet theorem. The periodic component and periodic parameters are expanded into Fourier series, and the matrix eigenvalue equation is obtained which is used for directly determining the parametrically excited stability. The dynamic stability of parametrically excited vibration of the beam with spaced supports under harmonic excitations is illustrated by numerical results on unstable regions. The influence of the periodic supports and excitation parameters on the parametrically excited stability is explored. The parametrically excited stability of the beam with multi-mode coupling vibration can be improved by the periodic supports. The developed analysis method is applicable to more general period-parametric beams with multi-mode coupling vibration under various harmonic excitations.