Stability Analysis of parameter-excited linear Vibration Systems with Time Delay, using the Example of a Sheetfed Offset Printing Press

Abstract

This article describes stability studies on parameter-excited linear vibration systems with time delay. A method for stability analysis is presented. Therefore, the transcendental transmission element of the time delay e^-st is approximated as an all-pass element with the rational transfer function by means of the so-called Padé approximation. The system can be represented in the state space and the methods of the Floquet theory can also be applied to the system with approximated time delay. The process can be implemented without great effort in a standardized simulation environment such as MATLAB/SIMULINK, whereby existing models and methods can be reused. The suitability of the method is shown in the well-known example of the Mathieu differential equation with time delay. Variations between different solvers and approximation orders are described. An extended view and the transfer to an industrial application take place with the example of the drive of a sheetfed offset printing machine. The relevant vibration system is represented by an oscillator with several degrees of freedom. The belt, which couples the degrees of freedom of the drive motor and the machine, leads to a periodic (harmonic) parameter excitation of the system due to its inhomogeneous nature. The speed and position control of the drive motor (PI controller) is associated with a time delay, resulting in a system of the type described above.