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.

Introduction

With the development of science and technology, the investigations on vibration and control of a non-smooth system have been of great significance inengineering.^1–3 Melnikov’s method, a tool to obtain analytical results, has been extended to piecewise-smooth dynamical systems by geometrical or analytical technique.^4–14 It is important to note that the particular piecewise-smooth dynamical system was investigated in Tian et al.^11–13 and Niziol and Swiatoniowski. ¹⁴ Although the known works have given some good ideas, these results were limited in terms of the theoretical results, which encourage continued research on the practical application.

Rolling mill is the core equipment of rolling industry. Recently, modern industry is rapidly developed and the demand for the specifications of steel products becomes more stringent. The vibration of the rolling mill affects the accuracy of products, which becomes one of the key factors. It can be divided into vertical vibration,^3,14,15 free transverse vibration, ¹⁶ vertical–horizontal coupling vibration, ¹⁷ torsional vibration, ¹⁸ and so on.^19–26 Although the characteristics of the vibration of the rolling mill have been investigated, the complicated bifurcations still remain unknown. It is worthwhile noting that a piecewise nonlinear dynamic model of rolling mill was constructed in Hou et al. ³ and the characteristics of bifurcation were considered by means of singular point stability theory and singularity theory. However, due to the non-smooth characteristic, the research was not focused on the global bifurcation of the piecewise nonlinear dynamic model of rolling mill.

The motivation of this article is to explore the global bifurcation based on the non-smooth roll system of rolling mill constructed in Hou et al. ³ Hamilton phase diagrams of the non-smooth system are obtained in the next section. The results show that non-smooth homoclinic, heteroclinic, and periodic orbits depend on the classical heteroclinic orbits, periodic orbits, and a necessary condition. In order to obtain the criteria for non-smooth homoclinic bifurcation, Melnikov function is extended to this kind of non-smooth system by the use of Hamiltonian function and the characteristics of the non-smooth homoclinic orbits in section “Non-smooth homoclinic orbits bifurcation.” The efficiency of the criteria for homoclinic bifurcation mentioned above is verified by the phase portraits, Poincaré section, and bifurcation diagrams in section “Numerical simulation.” These results might be a step toward promoting the interest in the global bifurcations directly for non-smooth system in the practical application.

Preliminary dynamic analysis

Consider a cold rolling mill shown in Figure 1 and suppose the system subjected viscous damping $c\stackrel{·}{x}$ and the external excitation $F\cos \omega t$ , the governing equation can be written as ³

$$\stackrel{··}{x}+{\omega }_0^{2}x+k(x)+\xi \stackrel{·}{x}=f\cos \omega t$$

(1)

where ${\omega }_0=\sqrt{(k_1/m)}>0$ , $k(x)=\{\begin{array}{c}\hfill a_{1}x+a_2{x}^{3}x\le 0\hfill \\ \multicolumn{1}{c}{a_{3}x+a_4{x}^{3}x>0}\end{array}$ , $a_1=(k_2/m)$ , $a_2=(k_3/m)$ , $a_3=(k_4/m)$ , $a_4=(k_5/m)$ , $\xi =(c/m)$ , $f=(F/m)$ , and $x$ is the displacement. $m$ is the equivalent mass of the upper working roller, supporting roller and bearing. $k_1$ is the equivalent linear elastic coefficient between roller and rolled products

$$K_1(x)=\{\begin{array}{c}\hfill k_{2}x+k_3{x}^3,x\le 0\hfill \\ \multicolumn{1}{c}{0,x>0}\end{array}$$

(2)

is the nonlinear elastic force between pressure cylinder and roller

$$K_2(x)=\{\begin{array}{c}\hfill 0,x\le 0\hfill \\ \multicolumn{1}{c}{k_{4}x+k_5{x}^3,x>0}\end{array}$$

(3)

is the nonlinear elastic force between balance cylinder and roller.

Figure 1.

The piecewise nonlinear model of rolling mill system.

Although the characteristics of bifurcation of the rolling mill have been investigated preliminarily in Hou et al., ³ the complicated bifurcations of system (1) still remain unknown. Therefore, in the following sections, we investigate the complicated dynamics such as equilibrium bifurcation and non-smooth Homoclinic bifurcation.

Let

$$\frac{dx}{dt}=y$$

(4)

system (1) can be written as two first-order differential equations

$$\{\begin{array}{c}\stackrel{·}{x}=y\hfill \\ \multicolumn{1}{c}{\stackrel{·}{y}=-{\omega }_0^{2}x-\xi \stackrel{·}{x}+f\cos \omega t+\{\begin{array}{c}\hfill -a_{1}x-a_2{x}^{3}x\le 0\hfill \\ \multicolumn{1}{c}{-a_{3}x-a_4{x}^{3}x>0}\end{array}}\end{array}$$

(5)

The restoring force function

$$-{\omega }_0^{2}x+\{\begin{array}{c}\hfill -a_{1}x-a_2{x}^3(x\le 0)\hfill \\ \multicolumn{1}{c}{-a_{3}x-a_4{x}^3(x>0)}\end{array}$$

(6)

is the piecewise nonlinear function, which leads Hamiltonian function $H(x,y)$ to the piecewise function. The unperturbed system (5) can be rewritten in the following form

$$\{\begin{array}{c}\stackrel{·}{x}=\frac{\partial H(x,y)}{\partial y}=y\hfill \\ \stackrel{·}{y}=-\frac{\partial H(x,y)}{\partial x}=-{\omega }_0^{2}x-k(x)\hfill \end{array}$$

(7)

where

$$H(x,y)=\{\begin{array}{c}\hfill H_{-}(x,y),x\le 0\hfill \\ \multicolumn{1}{c}{H_{+}(x,y),x>0}\end{array}$$

(8)

$$H_{-}(x,y)=\frac{1}{2}{y}^2+\frac{1}{2}{\omega }_0^2{x}^2+\frac{1}{2}{a}_1{x}^2+\frac{1}{4}{a}_2{x}^4$$

(9)

$$H_{+}(x,y)=\frac{1}{2}{y}^2+\frac{1}{2}{\omega }_0^2{x}^2+\frac{1}{2}{a}_3{x}^2+\frac{1}{4}{a}_4{x}^4$$

(10)

Obviously, due to $a_1>0$ , $a_3>0$ , ${\omega }_0^2+a_1>0$ , and ${\omega }_0^2+a_3>0$ , the equilibrium of system (7) certainly depends on the values of $a_2$ and $a_4$ . In order to show the influences of parameters of $a_2$ and $a_4$ on the dynamics of equation (7), we construct the bifurcation diagram, depicted in Figure 2. The system undergoes a bifurcation at both $a_2=0$ and $a_4=0$ (Figure 2), where three regions exist as follows

• I: $a_2\ge 0,a_4\ge 0$ ;

• II: $a_2>0,a_4<0;(a_2<0,a_4>0)$ ;

• III: $a_2<0,a_4<0$ .

• Region I: the origin is the only center (0,0).

• Region II: there exists two fixed points. The origin $(0,0)$ is center equilibrium and $(\sqrt{-(({\omega }_0^2+a_3)/a_4)},0)$ (or $(-\sqrt{-(({\omega }_0^2+a_1)/a_2)},0)$ is a saddle.

• Region III: system (5) has three fixed points. The origin $(0,0)$ is center equilibrium and $(-\sqrt{-(({\omega }_0^2+a_1)/a_2)},0)$ and $(\sqrt{-(({\omega }_0^2+a_3)/a_4)},0)$ are saddle points.

Figure 2.

Equilibrium bifurcation diagram of system (7): (a) $x-a_2-a_4$ and (b) $a_2-a_4$ .

Conventionally, according to equilibrium bifurcation diagram and Hamiltonian function, we can obtain the phase portraits for the Hamiltonian system. But the method is invalid in region III for the piecewise Hamiltonian system (7). Of course, the existence of non-smooth orbits in this system depends on both the classical heteroclinic orbits and the periodic orbits. Meanwhile, the more conditions of bifurcation would be detected.

Assume that heteroclinic orbits exist in region III, which connect a pair of saddle points. Hence, the heteroclinic orbits should satisfy

$$H_{+}(\sqrt{-\frac{{\omega }_0^2+a_3}{a_4}},0)=H_{-}(-\sqrt{-\frac{{\omega }_0^2+a_1}{a_2}},0))$$

(11)

Obviously

$$H_{-}(-\sqrt{-\frac{{\omega }_0^2+a_1}{a_2}},0)=-\frac{{({\omega }_0^2+a_1)}^2}{4a_2}$$

(12)

and

$$H_{+}(\sqrt{-\frac{{\omega }_0^2+a_3}{a_4}},0)=-\frac{{({\omega }_0^2+a_3)}^2}{4a_4}$$

(13)

So, we have

$$a_4=\frac{{({\omega }_0^2+a_3)}^2}{{({\omega }_0^2+a_1)}^2}·a_2$$

(14)

which is the necessary and sufficient condition of the existence of the heteroclinic orbits for system (7).

Furthermore

$$H_{-}(0,y_1^{*})=-\frac{{({\omega }_0^2+a_1)}^2}{4a_2}$$

(15)

and

$$H_{+}(0,y_2^{*})=-\frac{{({\omega }_0^2+a_3)}^2}{4a_4}$$

(16)

where $(0,y_1^{*})$ and $(0,y_2^{*})$ are the intersections of the heteroclinic orbits and y-axis in the $x-y$ plane for the unperturbed system (7). From equation (14), we have $y_1^{*}=y_2^{*}$ . Hence, the non-smooth heteroclinic orbits are the continuous orbits and the equilibrium bifurcation should be depicted in Figure 3, E. In particular, from the characteristics of Hamiltonian system, the saddle points in region $D_1$ or $D_2$ are connected to itself by a homoclinic orbits (see Figure 3; $D_1$ and $D_2)$ . The results show that it is not conductive to control the movement of the unperturbed rolling mill in regions II and III. Hence, the parameters would be probably best to be chosen in region (I) in practice.

Figure 3.

Phase portraits for the system (7).

Non-smooth homoclinic orbits bifurcation

The damping and the external excitation are unavoidable in practice, which lead to some more complicated phenomena in regions II and III. In this section, we will detect non-smooth Homoclinic orbits bifurcation for system (5).

Let $\xi \to \delta \xi$ and $f\to \delta f$ . System (5) can be rewritten as follows with the perturbation

$$\{\begin{array}{c}\stackrel{·}{X}=F_{-}(X)+\delta G_{-}(X,t),x\le x_0\hfill \\ \multicolumn{1}{c}{\stackrel{·}{X}=F_{+}(X)+\delta G_{+}(X,t),x>x_0}\\ \multicolumn{1}{c}{{\mathrm{\Delta }x|}_{x=x_0}=0}\\ \multicolumn{1}{c}{{\mathrm{\Delta }\stackrel{·}{x}|}_{x=x_0}=0}\\ \multicolumn{1}{c}{{\mathrm{\Delta }\stackrel{·}{y}|}_{x=x_0}\ne 0}\end{array}$$

(17)

where $f_{-}(x)=-{\omega }_0^{2}x-a_{1}x-a_2{x}^3\in C^r(r>1)$ , $f_{+}(x)=-{\omega }_0^{2}x-a_{3}x-a_4{x}^3\in C^r(r>1)$ , $X=\left(\begin{array}{c}\hfill x\hfill \\ \multicolumn{1}{c}{y}\end{array}\right)$ , $g_{1,\pm }(x)=0\in C^r(r>1)$ , $g_{2,\pm }(x)=-\xi y+f\cos \omega t\in C^r(r>1)$ , $0<\delta <<1$ , ${\mathrm{\Delta }x|}_{x=x_0}=x_{\delta ,x_0}^{+}-x_{\delta ,x_0}^{-}$ , ${\mathrm{\Delta }\stackrel{·}{x}|}_{x=x_0}={\stackrel{·}{x}}_{\delta ,x_0}^{+}-{\stackrel{·}{x}}_{\delta ,x_0}^{-}$ , $x_0=0$ , $F\pm (X)=\left(\begin{array}{c}\hfill y\hfill \\ \multicolumn{1}{c}{f\pm (x)}\end{array}\right)$ , $G\pm (X,t)=\left(\begin{array}{c}\hfill g_{1,\pm }(X,t)\hfill \\ \multicolumn{1}{c}{g_{2,\pm }(X,t)}\end{array}\right)$ , $F(X)=\{\begin{array}{c}\hfill F_{-}(X),x\le x_0\hfill \\ \multicolumn{1}{c}{F_{+}(X),x>x_0}\end{array}$ , $G(X)=\{\begin{array}{c}\hfill G_{-}(X),x\le x_0\hfill \\ \multicolumn{1}{c}{G_{+}(X),x>x_0}\end{array}$

$$f_{\pm }(x_0)=0$$

(18)

and

$${\mathrm{\Delta }\stackrel{·}{y}|}_{x=x_0}={\stackrel{·}{y}}_{\delta ,x_0}^{+}-{\stackrel{·}{y}}_{\delta ,x_0}^{-}$$

(19)

Furthermore, $x_{\delta ,x_0}^{+}$ and $x_{\delta ,x_0}^{-}$ represent the right-hand limit and left-hand limit of $x$ at $x=x_0$ , respectively. ${\stackrel{·}{x}}_{\delta ,x_0}^{+}$ and ${\stackrel{·}{x}}_{\delta ,x_0}^{-}$ denote the right-hand limit and left-hand limit of ${\stackrel{·}{x}|}_{x=x_0}$ , respectively. ${\stackrel{·}{y}}_{\delta ,x_0}^{+}$ and ${\stackrel{·}{y}}_{\delta ,x_0}^{-}$ are the right-hand limit and left-hand limit of ${\stackrel{·}{y}|}_{x=x_0}$ , respectively.

Clearly, for $\delta =0$ , system (19) undergoes the heteroclinic orbit bifurcation at $a_4=(({\omega }_0^2+a_3{)}^2/({\omega }_0^2+a_1{)}^2)·a_2$ . Meanwhile, homoclinic orbit bifurcation occurs at $a_2=0$ or $a_4=0$ or $a_4=(({\omega }_0^2+a_3{)}^2/({\omega }_0^2+a_1{)}^2)·a_2$ .

For $\delta \ne 0$ and $\delta <<1$ , the classical Melnikov method could not be effective in the investigation of heteroclinic and homoclinic bifurcation for this non-smooth system. Previously, the criteria for chaotic motion have been constructed in some non-smooth systems.^4–14 A lot of effort will be made to extend the Melnikov method to this kind of non-smooth system. Homoclinic bifurcation is detected here and heteroclinic bifurcation will be carried out in a separate paper.

For $\delta =0$ , we know

• (A1) The functions $F_{\pm }(X)$ , $G_{\pm }(X,t)$ are sufficiently smooth and bounded.

• (A2) When $\delta =0$ , system (17) is a piecewise Hamilton system. In region II, it has one center equilibrium $(0,y_0={\stackrel{·}{x}|}_{x=x_0})$ and a hyperbolic equilibrium $p_{01}$ that are connected by a non-smooth homoclinic orbit (Figure 4). We know system (17) possesses a saddle point $p_{01}^{\delta }=p_{01}+o(\delta )$ .

Figure 4.

Non-smooth homoclinic orbits.

For convenience, $X_h(t)=\left(\begin{array}{c}\hfill x_h(t)\hfill \\ \multicolumn{1}{c}{y_h(t)}\end{array}\right)$ denotes the non-smooth homoclinic orbit. $t_{a,\pm }$ and $t_{b,\pm }$ are the times at which the $X_h(t)$ reaches the constraint surface $\sum =\{X\in {\mathrm{R}}^2|x=x_0\}$ . Clearly, $t_{a,+}=t_{a,-}$ , $t_{b,+}=t_{b,-}$ . Let

$$X_{\delta }^{u(s),\pm }(t+t_0,t_0,\delta )=\left(\begin{array}{c}\hfill x_{\delta }^{u(s),\pm }(t+t_0,t_0,\delta )\hfill \\ \multicolumn{1}{c}{y_{\delta }^{u(s),\pm }(t+t_0,t_0,\delta )}\end{array}\right)$$

(20)

be the trajectories of perturbed system (17), which lies in the unstable manifolds $W_{\delta }^u$ and the stable manifolds $W_{\delta }^s$ of $p_{01}^{\delta }$ , respectively. Denote $L$ a line segment in the $\delta ={\delta }_0$ plane that is perpendicular to the non-smooth homoclinic orbit $X_h(t)$ . Let $X_{\delta }^u(t+t_0,t_0,\delta )$ and $X_{\delta }^s(t+t_0,t_0,\delta )$ cross $L$ at short distance. $t_{\delta ,a,\pm }$ and $t_{\delta ,b,\pm }$ are the times at which the $X_{\delta }^{u,s}(t+t_0,t_0,\delta )$ reaches the constraint surface $\sum =\{X\in {\mathrm{R}}^2|x=x_0\}$ , which are shown in Figure 5. Clearly, $t_{\delta ,a,+}=t_{\delta ,a,-}$ , $t_{\delta ,b,+}=t_{\delta ,b,-}$ and

$$t_{\delta ,a(b),\pm }=t_{a(b),\pm }+\delta t_{a(b),1,\pm }+O({\delta }^2)$$

(21)

Figure 5.

Stable and unstable for the perturbed system.

Let

$${\mathrm{\Delta }}_1^{u(s),\pm }(t+t_0,t_0)=\delta F_{\pm }(X_h(t))\wedge X_1^{u(s),\pm }(t+t_0,t_0)$$

(22)

where the operator “∧” is defined as

$$a\wedge b=a_1{b}_2-a_2{b}_1$$

(23)

for any $a=(a_1,a_2{)}^T$ , $b=(b_1,b_2{)}^T$ .

As in the smooth case, we obtain

$${\stackrel{·}{\mathrm{\Delta }}}_1^{u(s),\pm }(t+t_0,t_0)=\delta F_{\pm }(X_h(t))\wedge G_{\pm }(X_h(t),t+t_0)$$

(24)

and

$${\mathrm{\Delta }}_1^{u,+}(-\infty ,t_0)={\mathrm{\Delta }}_1^{s,+}(+\infty ,t_0)=0$$

(25)

due to $F_{+}(p_{01})=(0,0{)}^T$ .

Using the Hamiltonian function to measure the distance between the perturbed stable and unstable manifolds yields^11–14

$$\begin{array}{c}\hfill H_{\delta }(t_0)=\delta {\int }_{-\infty }^{+\infty }F(X_h(t))\wedge G(X_h(t),t+t_0)dt\hfill \\ \multicolumn{1}{c}{+[{\mathrm{\Delta }}_1^{u,+}(t_0+t_{b,+},t_0)-{\mathrm{\Delta }}_1^{u,-}(t_0+t_{b,-},t_0)]}\\ \multicolumn{1}{c}{+[{\mathrm{\Delta }}_1^{u,-}(t_0+t_{a,+},t_0)-{\mathrm{\Delta }}_1^{u,+}(t_0+t_{a,-},t_0)]}\end{array}$$

(26)

Moreover, ${\mathrm{\Delta }}_1^{u,+}(t_0+t_{b,+},t_0)-{\mathrm{\Delta }}_1^{u,-}(t_0+t_{b,-},t_0)$ and ${\mathrm{\Delta }}_1^{u,-}(t_0+t_{a,+},t_0)-{\mathrm{\Delta }}_1^{u,+}(t_0+t_{a,-},t_0)$ are the contributions to the non-smooth factors, which lead to a barrier for investigating the homoclinic bifurcation applying conventional nonlinear techniques. In the following, we will give a reasonable compromise based on the characteristics of these terms and we can obtain a concise result.

Let

$${\mathrm{\Delta }}_{\delta }^{u(s),\pm }(t+t_0,t_0)=F_{\pm }(X_h(t))\wedge X_{\delta }^{u(s),\pm }(t+t_0,t_0,\delta )$$

(27)

and

$${\mathrm{\Delta }}_h(t+t_0,t_0)=F_{\pm }(X_h(t))\wedge X_h(t)$$

(28)

Clearly

$$\begin{array}{c}\hfill {\mathrm{\Delta }}_1^{u,-}(t_0+t_{a,+},t_0)-{\mathrm{\Delta }}_1^{u,+}(t_0+t_{a,-},t_0)\hfill \\ \multicolumn{1}{c}{={\mathrm{\Delta }}_{\delta }^{u,-}(t_0+t_{a,+},t_0)-{\mathrm{\Delta }}_{\delta }^{u,+}(t_0+t_{a,-},t_0)}\end{array}$$

(29)

Clearly, the perturbed solution $X_{\delta }^{u,-}(t_{\delta ,a,+}+t_0,t_0,\delta )$ and $F_{-}(X_{\delta }^{u,-}(t_0+t_{a,+},t_0,\delta ))$ can be expressed

$$\begin{array}{c}{X}_{\delta }^{u,-}(t_{\delta ,a,+}+t_0,t_0,\delta )\hfill \\ =X_{\delta }^{u,-}(t_{a,+}+t_0,t_0,\delta )+\hfill \\ \delta t_{a,1,+}{F}_{-}(X_{\delta }^{u,-}(t_0+t_{a,+},t_0,\delta ))+O({\delta }^2)\hfill \end{array}$$

(30)

and

$$\begin{array}{c}{F}_{-}(X_{\delta }^{u,-}(t_0+t_{a,+},t_0,\delta ))\hfill \\ =F_{-}(X_h(t_{a,+}))+O(\delta )\hfill \end{array}$$

(31)

Substituting equations (18), (30), and (31) into equation (27), we have

$$\begin{array}{c}{\mathrm{\Delta }}_{\delta }^{u,-}(t_0+t_{a,+},t_0)=F_{-}(X_h(t_{a,+}))\hfill \\ \wedge X_{\delta }^{u,-}(t_{a,+}+t_0,t_0,\delta )\hfill \\ =y_h(t_{a,+})y_{\delta }^{u,-}(t_0+t_{\delta ,a,+},t_0,\delta )\hfill \\ -f_{-}(x_h(t_{a,+}))x_{\delta }^{u,-}(t_0+t_{\delta ,a,+},t_0,\delta )+O({\delta }^2)\hfill \\ =y_h(t_{a,+})y_{\delta }^{u,-}(t_0+t_{\delta ,a,+},t_0,\delta )+O({\delta }^2)\hfill \end{array}$$

(32)

Similarly, ${\mathrm{\Delta }}_{\delta }^{u,+}(t_0+t_{a,-},t_0)$ yields

$$\begin{array}{c}{\mathrm{\Delta }}_{\delta }^{u,+}(t_0+t_{a,-},t_0)=y_h(t_{a,-})y_{\delta }^{u,+}\hfill \\ (t_0+t_{\delta ,a,-},t_0,\delta )+O({\delta }^2)\hfill \end{array}$$

(33)

Then, equations (29), (32), and (33) lead to

$${\mathrm{\Delta }}_1^{u,-}(t_0+t_{a,-},t_0)-{\mathrm{\Delta }}_1^{u,+}(t_0+t_{a,-},t_0)=0$$

(34)

By a similar procedure, we can obtain

$${\mathrm{\Delta }}_1^{u,+}(t_0+t_{b,+},t_0)-{\mathrm{\Delta }}_1^{u,-}(t_0+t_{b,-},t_0)=0$$

(35)

From equations (26), (34), and (36), we obtain

$$H_{\delta }(t_0)=\delta \underset{-\infty }{\overset{+\infty }{\int }}F(X_h(t))\wedge G(X_h(t),t+t_0)dt+O({\delta }^2)$$

(36)

That is

$$M(t_0)=\underset{-\infty }{\overset{+\infty }{\int }}F(X_h(t))\wedge G(X_h(t),t+t_0)dt$$

(37)

which can be used to detect the homoclinic tangency in the non-smooth system mentioned above.

Remark 1

The results reveal that this kind of non-smooth factor has little influence on the chaotic threshold apart from calculating piecewise integrals for this class of nonlinear dynamical system (17).

Numerical simulation

In this section, using equation (37) and the numerical method, the homoclinic bifurcation and the chaos for system (17) are detected and verified.

Let $a_2>0,a_4<0$ , ${\omega }_0^2+a_1=c_1$ , $a_2=c_2$ , ${\omega }_0^2+a_3=c_3$ , and $a_4=c_4$ . One homoclinic-type orbit for the unperturbed system of (17) connecting the saddle

$$\{\begin{array}{c}(\sqrt{-\frac{c_3}{c_4}}\frac{c{e}^{\sqrt{2c_3}t}-1}{c{e}^{\sqrt{2c_3}t}+1},-\sqrt{\frac{c_3^3}{2c_4^2}}{[\left(\frac{c{e}^{\sqrt{2c_3}t}-1}{c{e}^{\sqrt{2c_3}t}+1}\right)}^2-1],\hfill \\ \multicolumn{1}{c}{t\in (-\infty ,-T]}\\ \multicolumn{1}{c}{(-BCn(-\mathrm{\Omega }t,k),-B\mathrm{\Omega }Sn(-\mathrm{\Omega }t,k)dn(-\mathrm{\Omega }t,k)),}\\ \multicolumn{1}{c}{t\in (-T,0]}\\ \multicolumn{1}{c}{(-BCn(\mathrm{\Omega }t,k),B\mathrm{\Omega }Sn(\mathrm{\Omega }t,k)dn(\mathrm{\Omega }t,k)),}\\ \multicolumn{1}{c}{t\in (0,T]}\\ \multicolumn{1}{c}{(\sqrt{-\frac{c_3}{c_4}}\frac{c{e}^{\sqrt{2c_3}t}-1}{c{e}^{\sqrt{2c_3}t}+1},\sqrt{\frac{c_3^3}{2c_4^2}}{[\left(\frac{c{e}^{\sqrt{2c_3}t}-1}{c{e}^{\sqrt{2c_3}t}+1}\right)}^2-1],}\\ \multicolumn{1}{c}{t\in (T,+\infty ]}\end{array}$$

(38)

where

$$h=-\frac{c_3^2}{4c_4},B=-{\left(\frac{\sqrt{c_1^2+4h{c}_2}-c_1}{c_2}\right)}^{1/2}$$

$$\mathrm{\Omega }=-(\sqrt{c_1^2+4h{c}_2}{)}^{1/2},k=-{\left(\frac{\sqrt{c_1^2+4h{c}_2}-c_1}{2\sqrt{c_1^2+4h{c}_2}}\right)}^{1/2}$$

$$c=e^{-\sqrt{2c_3}\frac{F(-k,\pi /2)}{\mathrm{\Omega }}},T=\frac{F(k,\pi /2)}{\mathrm{\Omega }}$$

and $F(k)$ is the complete elliptic integrals of the first kind. $Sn$ , $Cn$ , and $dn$ are Jacobian elliptic function.

Thus, the corresponding Melnikov function for system (17) is given by equation (37) as the following form

$$M(t_0)=\xi N-2f\sin \omega t_{0}M$$

(39)

where

$$N=\frac{2\sqrt{2c_3^2}}{3c_4}-B^2{\mathrm{\Omega }}^2\underset{-T}{\overset{T}{\int }}S{n}^2(\mathrm{\Omega }t,k)d{n}^2(\mathrm{\Omega }t,k)dZ$$

(40)

$$\begin{array}{c}\hfill M=\underset{T}{\overset{+\infty }{\int }}{\sqrt{\frac{c_3^3}{2c_4^2}\left[\left(\frac{c{e}^{\sqrt{2c_3}t}-1}{c{e}^{\sqrt{2c_3}t}+1}\right)\right]}}^2-1]\sin \omega tdZ\hfill \\ \multicolumn{1}{c}{+B\omega \underset{0}{\overset{T}{\int }}Cn(\mathrm{\Omega }t,k)\cos \omega tdZ}\end{array}$$

(41)

Hence, if and only if the following inequality holds

$$f>\frac{\xi \left|N\right|}{2·\left|M\right|}$$

(42)

$M(t_0)$ might be yield

$$M(t_0)=0$$

(43)

Taking $\omega =3{\omega }_0$ , $a_1=6.092$ , $a_2=0.0609$ , $a_3=0.1916$ , $a_4=-0.0192$ , $\xi =0.5$ , that is, $m=2.16\times {10}^4\mathrm{kg}$ , ${\omega }_0=\sqrt{(k_1/m)}$ , $k_1=1.17\times {10}^4\mathrm{kN}/\mathrm{m}$ , $k_2=1.59\times {10}^5\mathrm{kN}/\mathrm{m}$ , $k_3=1.59\times {10}^4\mathrm{kN}/\mathrm{m}$ , $k_4=5\times {10}^3\mathrm{kN}/\mathrm{m}$ , $k_5=-5\times {10}^2\mathrm{kN}/\mathrm{m}$ , we know that $f=(F/m)=0.7636$ is the chaos threshold. The bifurcation diagrams under the controlled variation of the amplitude $f$ of the external excitation and the Melnikovian detected chaotic boundary from equation (42) are obtained, as shown in Figure 6. Figure 7 represents the phase portraits and Poincaré section and the details are described as follows:

1. We can show the existence of period-2 motion $(f=3.5)$ and period-4 motion $(f=4.6)$ , as shown in Figure 7(a) and (b), respectively. In this case, although inequality (42) holds, no chaotic motions can exist, which confirms that the Melnikovian detected chaotic boundary is the necessary and insufficient condition for chaos.

2. The chaotic attractor is presented in Figure 7(c) $(f=4.75)$ and (d) $(f=5)$ , respectively. Obviously, inequality (42) is found for the chosen parameters.

Figure 6.

Bifurcation diagrams for $x$ versus the external forcing amplitude $f$ .

Figure 7.

The phase trajectory and Poincar’e section: (a) period-2 motion for $f=3.5$ , (b) period-4 motion for $f=4.6$ , (c) chaotic motion for $f=4.75$ , and (d) chaotic motion for $f=5$ .

The results presented above show that the vibrations of the non-smooth roll system of rolling mill have rich dynamic behaviors depending on $f$ , which may lead to the potential hazard. Hence, we can adjust the values of parameters to influence the motion state of the system, which laid a theoretical foundation for parameter design, stable operation, and fault diagnosis of the non-smooth roll system of rolling mill.

Conclusion

The bifurcations of the non-smooth roll system of rolling mill were detected to reveal the characteristics of this class of the piecewise-smooth and discontinuous system. The bifurcation diagram constructed for the unperturbed roll system of rolling mill was used to examine the influence of the parameters on the dynamics. Hamilton phase diagrams showed the non-smooth homoclinic, heteroclinic, and periodic orbits depended on the classical heteroclinic periodic orbits, periodic orbits, and a necessary condition. The necessary condition $a_4=(({\omega }_0^2+a_3{)}^2/({\omega }_0^2+a_1{)}^2)·a_2$ was deduced, which was distinctive from the smooth system. Although the piecewise-smooth and discontinuous restoring force led to a barrier for the well-known Melnikov function, the comprehensive analysis based on the characteristics of these non-smooth factors was applied to obtain the concise results. The expanded Melnikov function revealed that this kind of non-smooth factors has little influence on the chaotic threshold apart from calculating piecewise integrals. The obtained results were employed to study the chaotic boundary for the class of nonlinear dynamical systems (17), which was in excellent agreement with the numerical investigations. The investigation might be a step toward promoting the interest in the piecewise-smooth and discontinuous system.