Coupling vibration model for hot rolling mills and its application

3.1. Mathematic model of rolling process

The slab method is one of the most widely used methods to describe the plastic deformation of rolled materials. Sims derived the specific roll force equation, which is often applied in hot rolling mills. In this section, based on Sims’ rolling force model, a dynamic rolling process model with vibration is proposed.

3.1.1. Strip entry and exit position

The roll bite geometry with vibration is illustrated in Fig. 4. It is assumed that the exit plane does not coincide with the centerline of work rolls once the rolls begin to vibrate. In addition, work rolls vibrate out of phase.

Fig. 4Roll bite geometry with vibration

${\stackrel{-}{o}}_1$ and ${\stackrel{-}{o}}_2$ are the original work roll centers; $o_1$ and $o_2$ the work roll centers with vibration; $h_1$, $h_2$ and $h_n$ are the strip entry thickness, the strip exit thickness and the strip thickness of the neutral plane, respectively; $x_1$, $x_2$ and $x_n$ are the strip entry position, the strip exit position and the neutral point position, respectively; $R\mathrm{\text{'}}$ is the deformed radius of the work roll; $u_1$ and $u_2$ are the strip speed at entry and exit, respectively; ${\varphi }_1$, ${\varphi }_2$ and ${\varphi }_n$ are the strip entry angle, the strip exit angle and the neutral angle, respectively; $v_r$ is the roll speed; $h_c$ is the roll gap spacing along the centerline of the work rolls; and ${\dot{h}}_c$ is the rate of change of the roll gap spacing.

The high rolling force causes the work roll deformation. According to the Hitchcock equation, the deformed radius is calculated as follows:

2

$R^{\mathrm{\text{'}}}=R\left(1+\frac{8\left(1-{\nu }^2\right)F_R}{\pi E\left(h_1-h_2\right)}\right),$

where $R$ is the undeformed radius of the work roll; $\nu$ is Poisson’s ratio; $E$ is the Young’s modulus; and $F_R$ is the total rolling force.

In the roll bite, the roll surface is usually assumed to be a parabolic curve. From Fig. 4, the strip thickness at an arbitrary cross-section can be calculated by:

3

$h=h_c+\frac{x^2}{R\mathrm{\text{'}}}.$

By combining Eq. (3), $x_1=R\mathrm{\text{'}}\mathrm{s}\mathrm{i}\mathrm{n}\left({\varphi }_1\right)$ and $\mathrm{s}\mathrm{i}\mathrm{n}\left({\varphi }_1\right)\approx {\varphi }_1$, the strip entry angle can be derived from:

4

${\varphi }_1=\sqrt{\frac{h_1-h_c}{R\mathrm{\text{'}}}}.$

By applying the principle of mass conservation to the roll bite, the flow of metal through an arbitrary cross-section can be written as:

5

$uh=u_1{h}_1-\left(x_1-x\right){\dot{h}}_c.$

By applying Eq. (5) to the neutral point position and the exit position, the corresponding mass conservation equations are listed as:

6

$\left\{\begin{array}{l}{u}_2{h}_2=u_1{h}_1-\left(x_1-x_2\right){\dot{h}}_c,\\ u_n{h}_n=u_1{h}_1-\left(x_1-x_n\right){\dot{h}}_c.\end{array}\right.$

From Fig. 4, the following equation can be found:

7

${\varphi }_2\approx \mathrm{t}\mathrm{a}\mathrm{n}\left({\varphi }_2\right)=\frac{{\dot{h}}_c}{2u_2}.$

By substituting Eq. (3) and Eq. (6) into Eq. (7), the strip exit angle can be obtained:

8

${\varphi }_2=\frac{h_c{\dot{h}}_c}{2\left(v_r{h}_c+v_r{x}_n^2/R\mathrm{\text{'}}-x_n{\dot{h}}_c\right)}.$

3.1.2. Neutral point position

Hot rolling mills roll with low tension which can be neglected. The stress distribution in a volume element in the roll gap is illustrated in Fig. 5, where the circumferential thickness of the slice is $d\varphi$; the horizontal composite force acting on a cross-section is $Q$; the normal roll pressure is $p_{\varphi }$; the specific shear stress is ${\tau }_{\varphi }$; and the surface angularity is $\varphi$.

Fig. 5Stresses distribution acting on a volume element in the roll gap

According to Sims’ method, the horizontal force equilibrium equation can be derived as:

9

$\frac{d}{d\varphi }\left(\frac{p_{\varphi }}{k}-\frac{\pi }{4}\right)=\frac{R\mathrm{\text{'}}\pi \varphi }{2\left(h_c+R\mathrm{\text{'}}{\varphi }^2\right)}\pm \frac{R\mathrm{\text{'}}}{h_c+R\mathrm{\text{'}}{\varphi }^2},$

where the constant $k$ is the deformation resistance of the arc of contact.

Eq. (9) is solved by starting at the plane of exit and extending towards the plane of entry. Thus, the normal roll pressure between the exit position and the neutral point position is obtained:

10

$p_{\varphi }=k\left[\frac{\pi }{4}\mathrm{l}\mathrm{n}\left(\frac{h_{\varphi }}{h_2}\right)+\frac{\pi }{4}+\sqrt{\frac{R\mathrm{\text{'}}}{h_c}}\mathrm{t}\mathrm{a}{\mathrm{n}}^{-1}\left(\sqrt{\frac{R\mathrm{\text{'}}}{h_c}}\varphi \right)-\sqrt{\frac{R\mathrm{\text{'}}}{h_c}}\mathrm{t}\mathrm{a}{\mathrm{n}}^{-1}\left(\sqrt{\frac{R\mathrm{\text{'}}}{h_c}}{\varphi }_2\right)\right].$

Eq. (9) is solved by starting at the plane of entry and extending towards the plane of exit. Thus, the normal roll pressure between the entry position and the neutral point position is obtained:

11

$p_{\varphi }=k\left[\frac{\pi }{4}\mathrm{l}\mathrm{n}\left(\frac{h_{\varphi }}{h_1}\right)+\frac{\pi }{4}+\sqrt{\frac{R\mathrm{\text{'}}}{h_c}}\mathrm{t}\mathrm{a}{\mathrm{n}}^{-1}\left(\sqrt{\frac{R\mathrm{\text{'}}}{h_c}}{\varphi }_1\right)-\sqrt{\frac{R\mathrm{\text{'}}}{h_c}}\mathrm{t}\mathrm{a}{\mathrm{n}}^{-1}\left(\sqrt{\frac{R\mathrm{\text{'}}}{h_c}}\varphi \right)\right].$

The neutral angle can be calculated by the equalization of the normal roll pressure obtained from Eq. (10) and Eq. (11) at the neutral point:

12

${\varphi }_n=\sqrt{\frac{h_c}{R\mathrm{\text{'}}}}\mathrm{t}\mathrm{a}\mathrm{n}\left[\frac{\pi }{8}\sqrt{\frac{h_c}{R\mathrm{\text{'}}}}\mathrm{l}\mathrm{n}\left(\frac{h_2}{h_1}\right)+\frac{1}{2}\mathrm{t}\mathrm{a}{\mathrm{n}}^{-1}\left(\sqrt{\frac{R\mathrm{\text{'}}}{h_c}}{\varphi }_1\right)+\frac{1}{2}\mathrm{t}\mathrm{a}{\mathrm{n}}^{-1}\left(\sqrt{\frac{R\mathrm{\text{'}}}{h_c}}{\varphi }_2\right)\right].$

3.2. Dynamic model of mill structure

The lumped-mass model is suitable for our research purpose. To balance the trade-off between full consideration and facilitative study, the mill structure, which includes the rack, rollers and hydraulic cylinder, is described as a 5-dof (five degrees of freedom) lumped-mass model. Notably, the up back roll and the cylinder body are assumed as a whole, and the piston rod and the mill rack are assumed as a whole. As the servo valve only adjusts the pressure of piston side, and does not change the pressure of rod side, then which can be assumed to be constant. In addition, the lower part of the rack is assumed to be the ground. The schematic diagram of the mill structure is depicted in Fig. 6.

To study the vibration only dynamic parts of parameters are considered in our model. According to the vibration and the fluid theory, the kinetic equations of the mill structure are written as follows:

where $m_1$ and $x_1$ are the equivalent mass and the displacement of the rack, respectively; $m_2$ and $x_2$ are the equivalent mass and the displacement of the up back roll, respectively; $m_3$ and $x_3$ are the equivalent mass and the displacement of the up work roll, respectively; $m_4$ and $x_4$ are the equivalent mass and the displacement of the bottom work roll, respectively; $m_5$ and $x_5$ are the equivalent mass and the displacement of the bottom back roll, respectively; $k_1$ and $c_1$ are the equivalent stiffness and damping, respectively, between the rack and the ground; $k_2$ and $c_2$ are the equivalent stiffness and damping, respectively, between the up work roll and the up back roll; $k_3$ and $c_3$ are the equivalent stiffness and damping, respectively, between the bottom work roll and the bottom back roll; $k_4$ and $c_4$ are the equivalent stiffness and damping, respectively, between the bottom back roll and the ground; $A_p$ is the area of the piston side; $p_p$ is the pressure of the piston side; $c_h$ is the internal leakage coefficient of the cylinder; and $d{F}_R$ is the dynamic part of the vertical rolling force.

The dynamic roll force from the rolling process acts on the work rolls and causes them to vibrate. The vibration of the work roll reacts to the rolling process. Therefore, there is coupling effect between the rolling process and the structure of rolling mill. The coupling relationship between the work roll vibration and the dynamic roll gap spacing is shown as follows:

Fig. 6Schematic diagram of mill structure

3.3. Hydraulic servo system model

The hydraulic servo system is an important component to adjust the delivery thickness. The dynamic response of a hydraulic servo system is a determinant of gauge control and plays an important role in rolling mill vibration due to the closed-loop control system. Therefore, in the rolling mill vibration, it is necessary to take into account the hydraulic servo system. In general, the hydraulic servo system consists of a double-acting single-rod hydraulic cylinder and a three-position four-way spool valve as shown in Fig. 7.

During rolling, the servo valve always works near zero. Thus, the variation in variable is equal to the absolute variation. Furthermore, the bandwidth of the hydraulic AGC system is approximately 15-20 Hz. As a consequence, the dynamic response of the servo valve can be described as a second-order system:

where $Q$ is the flow of the servo valve; $I$ is the input current of the servo valve; $K_v$ is the amplification gain of the servo valve; ${\xi }_v$ is the damping ratio of the servo valve; $W_v$ is the natural frequency of the servo valve; $s$ is the complex frequency.

Fig. 7Diagram of hydraulic servo system

As the servo valve only adjusts the pressure of piston side, and does not change the pressure of rod side, then which can be assumed to be constant. The flow continuity equation is applied to the piston side and can be written as follows:

where $V_p$ is the volume of the piston cavity; ${\beta }_e$ is the elastic modulus of the liquid; and $c_h$ is the internal leakage coefficient of cylinder; $A_p$ is the area of the piston side; $p_p$ is the pressure of the piston side; $Q$ is the flow of the servo valve; $x_1$ is the displacement of the rack; $x_2$ is the displacement of the up back roll.

3.4. Overview of coupling dynamic model

The roll system, the hydraulic servo system, the rolling process and the control system constitute the coupling dynamic model, which is depicted in Fig. 8.

In Fig. 8, there are two closed loops. The inner control loop is HAPC, and the outer control loop is mill modulus control AGC. The mill modulus control modifies the gap position to maintain a constant delivery strip thickness from each stand and uses the gauge meter equation to estimate change in the stand exit strip thickness [22]. Thus, the mill modulus control equation can be expressed as:

where $K$ is the natural mill modulus and $\alpha$ is the mill modulus control gain.

The mill modulus control modifies the modulus of the mill according to the dynamic rolling force. The modified modulus of the mill can be expressed as:

when $\alpha <$ 0, the modified mill modulus is less than the natural mill modulus. When $\alpha >$ 0, the mill modified modulus is greater than the natural mill modulus. To ensure that the system remains stable, the mill modulus control gain $\alpha$ must be limited in $\left(-\frac{K+Q}{Q},\frac{K+Q}{Q}\right)$, where $Q$ is the plasticity coefficient.

Fig. 8Diagram of the coupling dynamic model

4.1. Model validity

MATLAB/SIMULINK software is used to simulate the mill vibration based on the coupling model mentioned above. The standard parameter conditions of F2 (the second finish rolling stand), which correspond to SPA-H with a 1.52 mm exit thickness of F7 (SPA-H (1.52 mm)), are listed in Table 1.

On the one hand, because the system response is very important for the control system, the step response and system bandwidth of the coupling model can be a proof of the model validity. The Pos_ref is assumed as the system input, and the up back roll displacement is assumed as the system output. It is examined with a 20 μm step input, and the calculated step response is shown in Fig. 9. To obtain the system response bandwidth, the Bode diagram from Pos_ref to the up back roll displacement is depicted in Fig. 10.

Fig. 9Step response of back roll displacement

Fig. 10Bode diagram from Pos_ref to up back roll displacement

From Fig. 9, it is obvious that the settling time of the step response is approximately 60 ms. In general, the bandwidth is 15-20 Hz. From Fig. 10, the response bandwidth from Pos_ref to the up back roll displacement is approximately 16 Hz, which is identical to the field value.

On the other hand, the threading response is also treated as proof of the model validity. Under standard parameters, the threading response of the vibration velocity of the up work roll is shown in Fig. 11. From Fig. 11, the dominant frequency is 33.7 Hz. The threading response of F2 when rolling SPA-H (1.52 mm) during field manufacturing is shown in Fig. 12. From Fig. 12, the dominant frequency is 32 Hz, which is roughly consistent with the numerical result.

Fig. 11Threading response of the coupling dynamic model with the standard parameters

a)

b)

Fig. 12Threading response of F2 when rolling SPA-H (1.52 mm) during field manufacturing

a)

b)

4.2. Application of coupling vibration model

To facilitate the research, a harmonic disturbance must be introduced into the coupling dynamic model. Let us assume that the incoming thickness harmonic deviation is the harmonic disturbance. The influence of the mill modulus control gain $\alpha$ in AGC on mill vibration is studied. The vibration velocity of the work roll is used as a criterion to evaluate mill vibrations.

Let us assume that the amplitude of the incoming thickness harmonic deviation is 40 μm and the frequency is 35 Hz. The simulation time is set as 1 s, and the mill modulus control gain is equal to 0.8. Furthermore, the mill modulus control AGC is OFF at the beginning and switches to ON at 0.5 s. The comparison between AGC ON and OFF is shown in Fig. 13.

From Fig. 13, we can see that when the mill modulus control AGC is OFF, the vibration velocity amplitude of the up work roll is 1.82 mm/s. Moreover, when the mill modulus control AGC is ON, the vibration velocity amplitude of the up work roll is 2 mm/s. The increase in the vibration velocity amplitude of the up work roll is 10 %. This phenomenon that the vibration of up work roll is increased with mill modulus control gain agrees qualitatively with the experiment result in section 2.

Fig. 13Comparison between AGC ON and OFF when α= 0.8

Fig. 14Comparison between AGC ON and OFF when α= –0.8

The comparison between AGC ON and OFF when $\alpha =$ –0.8 is shown in Fig. 14. Fig. 14 shows that, the vibration velocity amplitude of the work roll is 1.82 mm/s when the mill modulus control AGC is OFF and that the vibration velocity amplitude of the work roll is 1.67 mm/s when the mill modulus control AGC is ON. The reduction in the vibration velocity amplitude of the work roll is 8.2 %.

To research the dynamic response of the vibration velocity of the work roll to various excitation frequencies, the transfer function from the strip thickness at the entry to the vibration velocity of the work roll is derived, and its magnitude spectrum is shown in Fig. 15.

Fig. 15Magnitude spectrum from the strip thickness at entry to the vibration velocity of the work roll

Fig. 15 shows that in the low-frequency range, magnitude is reduced with an increase in mill modulus control gain; in the mid-frequency range, magnitude is increased with an increase in mill modulus control gain; and in the high-frequency range, mill modulus control gain has little effect on magnitude. The phenomenon can be explained as follows. For $\alpha =$ 0.8, the compensation of mill modulus control AGC has low-frequency attenuation characteristics; that is, mill modulus control AGC can remove the low-frequency incoming thickness deviation. The dynamic response of mill modulus control AGC presents a phase lag as the frequency of the incoming thickness deviation increases, which reverses compensation of mill modulus control AGC and increases vibration. As the frequency of the incoming thickness deviation increases further, it will exceed the mill modulus control AGC dynamic response capabilities, which results in noncompensated mill modulus control AGC performance. For $\alpha =$ –0.8, the situation is contrary to that for $\alpha =$ 0.8. Notably, the frequency of the cross-point is 5-7 Hz, which is less than half of the bandwidth. In addition, the mid-frequency range which is more than the system bandwidth can still be affected by the mill modulus control gain, which indicates that the vibration frequency which the control system can affect is more than the system bandwidth.

Comparing Fig. 2 and Figs. 13 and 14, we can see that there is a great difference, that is, the experiment result is characteristic by multiple frequency and the numerical result is characteristic by single frequency. This is because that the real system is nonlinear and the simulation model is linear. But this difference does not affect the research on key issue, i.e., the influence of mill modulus control gain. Comparing Fig. 3(a) and Fig. 15, we can see that there are some quantitative differences, but the influence of mill modulus control gain on mill vibration is qualitatively the same. The effect of mill modulus control gain on hot rolling mill vibration is positive compensation, negative compensation and no compensation, respectively.

Based on the above analysis, a practical measure to relieve mill vibration can be proposed. The vibration frequency for the hot rolling mill in general is 10-70 Hz. As a result, the vibration increases with mill modulus control AGC gain for the large sensitivity frequency range of hot rolling mills. For these frequencies, the mill modulus control gain can be decreased to relieve mill vibration.