**2. Vertical Vibration Model of Load**

The external load of the HAGC system consists of several rollers, which present the symmetrical structure [43]. The basic load structure of the commonly used, four-high mill's HAGC system is shown in Figure 1.

**Figure 1.** Structural schematic diagram of load.

In addition, the structure and vibration of load possess symmetry [44]. Moreover, some nonlinear factors such as nonlinear stiffness, nonlinear damping and nonlinear excitation are also considered. Then, a nonlinear load vertical vibration model of the HAGC system is built, as displayed in Figure 2.

**Figure 2.** Vertical vibration model of load.

*Processes* **2019**, *7*, 718

According to Newton's second law, the nonlinear load vertical vibration equation of a HAGC system can be represented as:

$$m\_1\ddot{y} + c\_1\dot{y} + k\_1y + aF\_k(y) + \beta F\_c(\dot{y}) = \Delta F \tag{1}$$

$$
\Delta F = p\_L A\_p - p\_b A\_b - F\_L \tag{2}
$$

where, *m*<sup>1</sup> is equivalent mass; *c*<sup>1</sup> and *k*<sup>1</sup> are linear damping coefficient and linear stiffness coefficient, respectively; *y* is vibration displacement. *Fk*(*y*) and *Fc*( . *y*) are nonlinear elastic force and nonlinear friction of hydraulic cylinder, respectively; α and β are the action coefficient of nonlinear stiffness and nonlinear damping, respectively; *Ap* and *Ab* are the effective working area of rodless cavity and rod cavity, respectively; *pL* and *pb* are the working pressure of rodless cavity and rod cavity, respectively; *FL* is external load force. Δ*F* is external disturbance excitation. Δ*F* is mainly caused by the factors such as the pulsation of oil source, the fluctuations of rolled metal thickness or tension etc., which can be expressed by *F* cos ω*t* [45]. *F* is the amplitude of external excitation.

Among them, the expression of nonlinear elastic force *Fk*(*y*) can be expressed by [46]:

$$F\_k(y) = \left(\frac{A\_p\beta\_\mathcal{E}}{L\_1} + \frac{A\_b\beta\_\mathcal{E}}{L - L\_1}\right)y + \left[\frac{A\_p\beta\_\mathcal{E}}{L\_1^3} + \frac{A\_b\beta\_\mathcal{E}}{\left(L - L\_1\right)^3}\right]y^3\tag{3}$$

where *L*<sup>1</sup> is the liquid column height of control cavity of hydraulic cylinder.

The expression of nonlinear friction *Fc*( . *y*) can be displayed as [47,48]:

$$F\_c(\dot{y}) = \begin{cases} F\_N(\mu\_s - \mu\_1 \dot{y} + \mu\_2 \dot{y}^3) & \dot{y} > 0 \\ F & \dot{y} = 0 \\ F\_N(-\mu\_s - \mu\_1 \dot{y} + \mu\_2 \dot{y}^3) & \dot{y} < 0 \end{cases} \tag{4}$$

where

$$
\mu\_1 = \frac{\Im(\mu\_s - \mu\_m)}{2v\_m} \tag{5}
$$

$$
\mu\_2 = \frac{\mu\_\kappa - \mu\_m}{2v\_m^3} \tag{6}
$$

where μ*<sup>m</sup>* is the maximum dynamic friction factor; μ*<sup>s</sup>* is static friction factor; *vm* is vibration velocity; *FN* is the positive pressure of piston acts on cylinder wall, which mainly depends on the factors such as the tightness degree of assembly, the hardness of sealing material, and the radial component of load.
