Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 216197, 9 pages

http://dx.doi.org/10.1155/2015/216197

## Nonlinear Vibration Analysis of Moving Strip with Inertial Boundary Condition

^{1}Key Laboratory of Mechanics Reliability for Heavy Equipment and Large Structure of Hebei Province, Yanshan University, Qinhuangdao, Hebei 066004, China^{2}Department of Automation, Institute of Electrical Engineering, Yanshan University, Qinhuangdao, Hebei 066004, China

Received 21 April 2015; Accepted 21 June 2015

Academic Editor: Michael Vynnycky

Copyright © 2015 Chong-yi Gao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

According to the movement mechanism of strip and rollers in tandem mill, the strip between two stands was simplified to axially moving Euler beam and the rollers were simplified to the inertial component on the fixed axis rotation, namely, inertial boundary. Nonlinear vibration mechanical model of Euler beam with inertial boundary conditions was established. The transverse and longitudinal motion equations were derived based on Hamilton’s principle. Kantorovich averaging method was employed to discretize the motion equations and the inertial boundary equations, and the solutions were obtained using the modified iteration method. Depending on numerical calculation, the amplitude-frequency responses of Euler beam were determined. The axial velocity, tension, and rotational inertia have strong influences on the vibration characteristics. The results would provide an important theoretical reference to control and analyze the vertical vibration of moving strip in continuous rolling process.

#### 1. Introduction

Vibration phenomenon of moving strip is extremely complicated in continuous rolling process. It may lead to strip quality decline, equipment trouble, and production efficiency reduction [1, 2]. For rolling mill vibrations, there are more studies on vertical, axial, torsional, horizontal, cross, and swing vibration modes of roller [3]. However, the strip vibration exists as well in practical production and scholars provided a little focus for this aspect. Therefore, it is significant to study the vibration of moving strip between two stands during rolling.

The moving strip during continuous rolling process can be similar to an axially moving beam if the vertical vibration of mill can be neglected [4]. Using the theory of moving beam, the vertical vibration of strip can be equivalent to transverse vibration of axially moving beam. Some studies have been already conducted. Sun et al. [5, 6] established two-dimensional dynamic model and three-dimensional dynamic model of moving strip between two mills with time-dependent tension, and the stability of principle parametric resonances was analyzed. Huang and Chen [7, 8] investigated nonlinear vibration of the axially moving beam with coupled transverse and longitudinal motions, and the incremental harmonic balance method was used for analysis. Świa̧toniowski and Bar [9, 10] analyzed chatter vibration phenomena in tandem rolling mill with use of mathematical models of parametrical self-excited vibration and the nonlinear mathematical model of oscillated system was given: the continuous group of the rolling stands coupled by transferring strip. Miranker [11] put forward differential equations of transverse vibration on moving strings firstly, and the corresponding moving string was analyzed. Hu and Zhang [12] investigated an axially accelerating rectangular thin plate subjected to parametric excitations resulting from the axial time-varying tension and axial time-varying speed in the magnetic field. The parametric resonance of plates with axis tension during zinc coasting process was studied by Kim et al. [13]. Stability of transverse vibration in axial variable motion system was studied by Mote Jr. [14]. The mathematical model of a strip-roller-flexible-supporting hybrid system in coating section of a continuous hot-dip galvanizing line was established by Wang et al. [15]. The above studies considered some aspects of moving strip but did not give the influences of rotational inertia and other factors on the strip. It will not be able to analyze the vibration characteristics of strip accurately.

Based on the movement mechanism of rolling and theory of axially moving beam, the vertical vibration of moving strip can be similar to a transverse vibration of axially moving Euler beam with tension [16–18]. The author established a dynamic model of interaction between inertial component and Euler beam. When the rotational inertia is considered, the nonlinear vibration mechanical model of the Euler beam with inertial boundary conditions is established. The modified iteration method is employed to solve the motion equations and used Matlab for simulation and analysis [19, 20]. Hereby, the relationships of axial velocity, tension, and rotational inertia are investigated, and the results may provide an important theoretical reference to control and analyze the vertical vibration of strip during continuous rolling process.

#### 2. Mechanical and Mathematical Models

The strip moves between two stands during rolling process; the schematic diagram is shown in Figure 1. The moving strip can be simplified to moving Euler beam which is supported by both rollers. The rolling direction can be considered as axial motion direction, and it is assumed that the motion is uniform movement, is axial velocity, and is tension. Rollers can be equivalent to symmetric and rigid inertial components and regarded as the disc of fixed axis rotation along the width of beam; is rotational inertia. Therefore, Figure 2 shows an axially moving Euler beam at velocity and acted tension with inertial boundary conditions. The origin of coordinate locates as a crossing point of the center line of the first stand and the passing line. The transverse displacement and the longitudinal displacement of Euler beam are and , respectively. is the distance of Euler beam between two stands.