Bifurcation and chaos for piecewise nonlinear roll system of rolling mill

Abstract

A non-smooth cold roll system of rolling mill is studied to reveal the bifurcation of the piecewise-smooth and discontinuous system. To examine the influence of the parameters on the dynamics, the bifurcation diagram is constructed when it is unperturbed. Hamilton phase diagrams of the non-smooth system are detected, which differ significantly from the ones obtained in the smooth system. Non-smooth homoclinic, heteroclinic, and periodic orbits are determined, which depend on the classical heteroclinic periodic orbits, periodic orbits, and a necessary condition. A extended Melnikov function is employed to obtain the criteria for the non-smooth homoclinic bifurcation in this class of piecewise-smooth and discontinuous system, which implies that the existence of homoclinic bifurcation arises from the breaking of homoclinic orbits under the perturbation of damping. The results reveal that this kind of non-smooth factor has little influence on the chaotic threshold apart from calculating piecewise integrals. The efficiency of the criteria for non-smooth homoclinic bifurcation mentioned above is verified by the phase portraits, Poincaré section, and bifurcation diagrams, which laid a theoretical foundation for parameter design, stable operation, and fault diagnosis of rolling mills.

An Application of the Melnikov Method to a Piecewise Oscillator

“…In particular, we find that the results in [22,23] fits well with our situation, where the manifold separating the regions of smoothness is just a vertical line which is transversal to the homoclinic orbit (see Figure 4). A similar geometric configuration (for a different nonlinearity) has been also discussed in [25]. the vector field is not smooth.…”

“…Concerning the Melnikov method, in the recent years several interesting extensions have been achieved by different authors. The articles [13][14][15][16][17][18][19][20][21][22][23][24][25][26] and the references therein (just to mention a few main recent contributions in this area of research), show the great deal of interest in this topic both from the theoretical and the applied point of view. Most of the achieved results are general enough to cover the case of piecewise smooth differential systems which present a manifold of discontinuity.…”

“…For this system we will assume the existence of a hyperbolic saddle point at the origin with a homoclinic trajectory. In Section 3 we apply the Melnikov theory, in the variant for non-smooth systems described in [22,23,25], to equation (1). Finally, Section 4 provides some numerical examples illustrating the chaotic nature of the solutions whose existence has been proved analytically in Section 3.…”

A generalized approach for implicit time integration of piecewise linear/nonlinear systems

“…As shown in Figure 5, the rolling mill system, 2 a typical piecewise nonlinear problem, is considered. The governing equation can be written as follows: The equivalent form of Equation () can be expressed as follows: …”

“…Piecewise linear and nonlinear systems, such as gap‐activated springs in vibrating machines, 1,2 structures with damage or clearance, 3 gear backlashes, 4 and drag torques, 5 are commonly used in civil engineering, aerospace, mechanical engineering, and infrastructures. Because of the piecewise linear and nonlinear characteristics, these systems exhibit very complex and diverse dynamic behaviors.…”