Effect of Friction on Tandem Cold Rolling Mills Chattering

Abstract

Chatter phenomenon in the rolling mills is the undesired mechanical vibrations which reduces the productivity and quality of the products. Many studies have led to the point that friction is one of the most effective factors incurring chatter. In this paper, a chatter simulation model for the tandem cold rolling mills based on the well-known Coulomb and the Tresca friction models is proposed. By considering the effects of the material strain-hardening and the elastic deformation of the work roll, the model relaxes certain assumptions. The simulation results are verified by comparing to experimental data obtained from Mobarakeh Steel Company. In addition, a numerical method is developed to introduce a relationship between the Tresca friction factor and the Coulomb friction coefficient. A parametric study on the effects of friction coefficients of the two stand rolling mill on the chatter critical speed is conducted. The results show that an optimal setting of friction coefficients improves the process efficiency. Under the operating conditions stated in this paper, the friction coefficient of the second stand should be increased as much as possible in order to get closer to its optimum point. The optimum friction coefficient of the first stand depends on the friction coefficient of the second stand.

1. Introduction

Cold rolling is the most prominent procedure in manufacturing metallic plates and sheets. Reduction in productivity due to chatter vibration has significant effects on the price of the rolled strips. Chatter phenomenon is a particular case of self-excited vibrations, which has three basic types: third octave, fifth octave, and torsional chatter.^1,2) Among them, the first type is the most detrimental since it can lead to surface defects, large gauge fluctuations, and serious damages to mills. Four major mechanisms for third octave chatter can be classified as: model matching, negative damping, regenerative, and mode coupling.³⁾ Many researchers have recognized the friction as the key factor causing mill vibration.^4,5,6,7) Although many investigations have been conducted to study the influence of friction settings on chatter, this effect is still not clearly understood.

Johnson and Qi^4,8) and Chen et al.^6,9) considered friction coefficient in the roll bite as a function of relative slip between the strip and the work rolls. Their research showed one factor that plays an important role in incurring rolling chatter is friction. Kimura et al.⁵⁾ pointed out that neither too high nor too low friction at the roll-strip interface cannot stabilize chatter vibration in tandem rolling mills. They found that there exists an optimal range of friction conditions to achieve maximum rolling velocity and prevent the onset of chatter.

Yun et al.^2,10,11,12) conducted a comprehensive study on the self-exited vibrations. Based on the Tresca model, they offered a dynamic model to study the chatter in the cold rolling of flat products. Hu et al.^{3,13,14,15,16)} improved the work of Yan et al. and declared that a suitable assessment of friction is crucial to make an accurate chatter model. Based on the traditional Coulomb model, Zhao et al.^17,18,19) built a chatter model and investigated on the stability of tandem rolling mills. Particularly, they studied the effects of friction on chatter and gave an explanation for the existence of an optimal friction conditions. Heidari et al.²⁰⁾ developed a new chatter model of the cold strip rolling with consideration of unsteady lubrication. As a result, they showed that the lubricant’s limiting shear stress is directly proportional to the chatter critical speed. Also Kim et al.^21,22) built a mathematical model for the rolling chatter containing the driving system.

In this research, a numerical model based on the Coulomb and Tresca friction models is offered to study the chatter vibration in the cold rolling process. In order to verify the results of the model, experimental data obtained from the cold strip tandem mill unit of Mobarakeh Steel Company (MSC) are used. Also, a relationship between the friction factor in the Tresca model and the friction coefficient in the Coulomb model is presented. Then, the effects of friction conditions of the two stand rolling mill on chatter critical speed are examined through numerical simulations.

2. Rolling Process Fundamentals

2.1. Rolling Process Geometry

The basic geometry of the roll bite is shown in Fig. 1. A strip of width w enters the roll bite with thickness y₁ and speed u₁, and exits with thickness y₂ and speed u₂. Because of high ratio of strip width-to-strip thickness, it can be assumed that the rolling process operates under the plane strain conditions. The work roll radius is a and the peripheral speed of the work roll is v_r. Since the focus of current study is on the third octave chatter which is symmetric about the x-axis,²³⁾ the work rolls are assumed to vibrate only along the y-axis at a rate of ${\dot{y}}_c/2$ .

Fig. 1.

Rolling process geometry.

2.2. Friction Model

The most traditional friction models utilized in industry are the friction coefficient model or the Coulomb model and the friction factor or the Tresca model. The first model may be described by the coefficient of friction, μ, defined as the ratio of the friction stress τ_f to the normal pressure p where   

$${\tau }_f=\mu \hspace{0.17em}p$$(1)

The second model assumes that the friction stress τ_f is proportional to the multiplication of the shear strength k of the strip and the constant friction factor m.   

$${\tau }_f=m\hspace{0.17em}k\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}0\le m\le 1$$(2)

For m = 0, the frictionless interface occurs; for m = 1, the contacting surfaces will stick together.

2.3. Strain Hardening Effect

The Ludwik equation is widely used in mathematical models for the cold rolling process.^16,24)   

$${\mathrm{\sigma }}_{yp}={\mathrm{\sigma }}_0\left(1+A{\overline{\epsilon }}^n\right)$$(3)

Where σ_yp is the material yield stress, σ₀, A and n are the material constants which have to be specified via experiments, and $\overline{\epsilon }$ is the effective strain under the plain strain conditions which can be expressed as:   

$$\overline{\epsilon }=\frac{2}{\sqrt{3}}ln\left(\frac{y_0}{y}\right)$$(4)

In this equation, y is the local strip thickness and y₀ is the initial strip thickness entering the first stand in a tandem rolling mill.

2.4. Work Roll Flattening Effect

An equation for the effective radius of the work roll (a′) was presented by Hitchcock in 1935. Later, Roberts^25,26) showed that the quantity of a′ calculated from Hitchcock’s formula is somewhat conservative and has modified this equation by:   

$$a^{\prime }=a\left(1+2\sqrt{\frac{f}{E_{w}r{y}_1}}+2\frac{f}{E_{w}r{y}_1}\right)$$(5)

Where f is the rolling force per unit width, r is the reduction in the thickness and E_w is the elastic modulus of the work roll. Note that the effective work roll radius is obtained iteratively during computations under steady-state conditions.

3. Methodology

Rolling chatter is the result of interactions between the rolling process and the mill structure, as well as between consecutive stands on a tandem mill. Based on two friction models which are widely employed in industry, the rolling process model is conducted in this research by considering the material strain-hardening effect and the work roll flattening effect.

3.1. Rolling Process Model

The roll surface is usually approximated by a parabolic curve.^3,17)   

$$y(x)=y_c+\frac{x^2}{a^{\prime }}$$(6)

Where y_c is the roll gap spacing along the centerline of the work rolls. From the above equation, the strip entry plane position, x₁, can be determined as:   

$$x_1=\sqrt{a^{\prime }(y_1-y_c)}$$(7)

With applying the mass conservation law to the control volume of the strip within the roll bite, the continuity equation can be deduced as:   

$$uy=u_1{y}_1-(x_1-x){\dot{y}}_c$$(8)

The strip exit plane position, x₂, can be derived by using the Eqs. (6) and (8) and the condition of zero strain rates:   

$$x_2=\frac{a^{\prime }{y}_c{\dot{y}}_c}{2[u_1{y}_1-x_1{\dot{y}}_c]}$$(9)

The force balance equation in the x-axis can be expressed by the slab element.²⁰⁾   

$$\frac{dy}{dx}(p+{\mathrm{\sigma }}_x)+y\frac{d{\mathrm{\sigma }}_x}{dx}\underset{\_}{+}2{\tau }_f=0$$(10)

In the above equation, if position x is larger than the neutral point position, x_n, the minus sign is used, otherwise the plus sign should be used. Assuming that the rolling pressure, p, and the strip tensile stress, σ_x, are the principal stresses, it is concluded that σ_y = –p and with the yield criterion under the plane strain conditions (σ_x – σ_y = σ_k), Eq. (10) can be written as:   

$${\mathrm{\sigma }}_k\frac{dy}{dx}+y\frac{d({\mathrm{\sigma }}_k-p)}{dx}\underset{\_}{+}2{\tau }_f=0$$(11)

Where σ_k is the yield strength in the plane strain conditions. The pressure distribution within the roll bite and the neutral point position are computed numerically based on the above equation, Eqs. (1), (2), (3), (4), Eqs. (5), (6), (7), (8), and the following boundary conditions.   

$$\begin{array}{l}x=x_2\hspace{0.17em},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}p={\mathrm{\sigma }}_{k,2}-{\mathrm{\sigma }}_{x,2}\\ x=x_1\hspace{0.17em},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}p={\mathrm{\sigma }}_{k,1}-{\mathrm{\sigma }}_{x,1}\end{array}$$(12)

Integrating the pressure distribution from the entry to the exit plane position gives an estimation of the rolling force. The rolling torque can be obtained with the use of integration of the friction force along the contact length between the work roll and the strip. Also, using the mass conservation principle and given the fact that the strip speed at the neutral point position is equal to the roll peripheral velocity, the strip speed at entry and exit plane can be calculated by:^3,17)   

$$u_1=\frac{1}{y_1}\left[v_r\left(y_c+\frac{x_n{\hspace{0.17em}}^2}{a^{\prime }}\right)+(x_1-x_n){\dot{y}}_c\right]$$(13)

  

$$u_2=\frac{u_1{y}_1+(x_2-x_1){\dot{y}}_c}{y_c+\frac{x_2{\hspace{0.17em}}^2}{a^{\prime }}}$$(14)

3.2. Mill Structural Model

Due to the symmetry of the third-octave-mode chatter about the stand pass line,^8,19,23) the work roll and the backup roll are allowed to only vibrate independently in the y direction as shown in Fig. 2.

Fig. 2.

Mill structural model.

The displacements of the work roll center (y_w) and the backup roll center (y_b) caused by the rolling force variation (df_y) can be written as:²⁰⁾   

$$\{\begin{array}{l}{M}_w{\ddot{y}}_w+C_w({\dot{y}}_w-{\dot{y}}_b)+K_w(y_w-y_b)=wd{f}_y\\ M_b\ddot{y}{}_b+C_b{\dot{y}}_b+K_b{y}_b=C_w({\dot{y}}_w-{\dot{y}}_b)+K_w(y_w-y_b)\end{array}$$(15)

The subscripts ‘w’ and ‘b’ signify the work and backup roll, respectively. M is the roll mass, C is the damping coefficient, K is the spring constant, and w is the strip width. Note that the displacement of work roll center and the variation of the roll gap spacing at the centerline, dy_c, can be related as y_w = dy_c/2.

3.3. Chatter Model

The rolling vibration model in the single stand is achieved by combining the rolling process model with the mill structural model. The inputs of the rolling process model are the strip thickness at entry, the strip tensile stress at entry, and the strip tensile stress at exit. The outputs of this model are the strip thickness at exit, the strip speed at entry and exit, and the rolling force. The dynamic rolling force enters to the mill structural model. It leads to variation of the roll gap spacing and this variation enters to the process model. So the relationship between the rolling process and the mill structural model continues. The negative damping mechanism often occurs in the single stand rolling while the regenerative effect is also activated in the tandem mills. So the chatter is more likely to happen in the multi stands.^17,18) Figure 3 shows the block diagram of the chatter model for the tandem configuration based on the single stand chatter model.

Fig. 3.

Block diagram of chatter model.

In the tandem rolling mills, the inter stand tension effects and the transport of the strip thickness variation between the adjacent stands play important role in the regenerative chatter. The strip tension stress variation is proportional to the integral of the difference between the variations of the entry speed of the downstream stand and the exit speed of the upstream stand.²⁰⁾   

$${\left(d{\mathrm{\sigma }}_{x,1}\right)}_i=\frac{E_s}{L_i}{\int }_t(d{u}_{1,i}-d{u}_{2,i-1})dt$$(16)

  

$${\left(d{\mathrm{\sigma }}_{x,\hspace{0.17em}2}\right)}_i=\frac{E_s}{L_{i+1}}\underset{t}{\int }(d{u}_{1,\hspace{0.17em}i+1}-d{u}_{2,i})dt$$(17)

Where E_s is the elastic modulus of the strip and L_i is the distance between stands i–1 and i. It is noted that the time delay term, Δ, in Fig. 3 can be expressed as:   

$${\mathrm{\Delta }}_i=\frac{L_i}{u_{2,i-1}}$$(18)

4. Experimental Equipment

The experimental data of this research are from a two-stand tandem mill unit of Mobarakeh Steel Company (MSC). Figure 4 depicts a view of this unit.

Fig. 4.

A view of two-stand tandem mill unit of Mobarakeh Steel Company. (Online version in color.)

Incoming coils to this unit have a thickness of 2 mm. Coils open up and pass through two 4-high stands two or three times depending on the final product thickness. Outgoing strips from the second stand in any pass are coiled by a coiler machine (Pickup reel). The pickup reel of the previous pass plays role of the payoff reel for the current pass which is in the reverse direction of the previous pass. In this mill, chatter usually occurs in the second stand during the third pass. Specifications of the tandem rolling mill are reported in Table 1.²⁰⁾

Table 1. Specifications of two-stand tandem mill unit.

Parameter	Stand 1	Stand 2
Work roll mass, Mw, (Kg)	14000	14000
Backup roll mass, Mb, (Kg)	38000	38000
Work roll damping coefficient, Cw, (N.s/m)	0	0
Backup roll damping coefficient, Cb, (N.s/m)	1.254e6	1.254e6
Work roll spring constant, Kw, (N/m)	4.97e10	4.97e10
Backup roll spring constant, Kb, (N/m)	1.22e10	1.22e10
Work roll radius, aw, (m)	0.245	0.245
Backup roll radius, ab, (m)	0.675	0.675

All rolling conditions in this unit are recorded in a data collection system (IBA) through several analog and digital signals. Characteristics of five coils under consideration in the third pass are presented in Table 2. In this unit, the coefficient of friction is obtained using an empirical equation taking into account the bite angle and the initial work roll surface roughness. Also the change of roughness, grinding by grinding, is considered, as well the variation of the friction coefficient due to wear during rolling.

Table 2. Characteristics of the coils in the third pass.

Parameter	Coil #1		Coil #2		Coil #3		Coil #4		Coil #5
	Stand 1	Stand 2	Stand 1	Stand 2	Stand 1	Stand 2	Stand 1	Stand 2	Stand 1	Stand 2
Strip width, w, (mm)	783	783	829	829	818	818	843	843	762	762
Entry thickness, y1, (mm)	0.431	0.340	0.498	0.415	0.541	0.476	0.432	0.354	0.409	0.317
Exit thickness, y2, (mm)	0.340	0.240	0.415	0.270	0.476	0.300	0.354	0.230	0.317	0.210
Coefficient of friction, μ	0.0087	0.0096	0.0083	0.0124	0.0071	0.0141	0.0079	0.0111	0.0088	0.0099

Inter-stand parameters are also listed in Table 3. The strip constants in the Ludwik equation are σ₀ = 298 MPa, A = 1.8742, n = 0.5844. These values are determined by fitting the Eq. (3) with data of several tensile tests. Since the Ludwik equation describes only the strain hardening effect, the data of plastic zone in stress-strain curve are used to fit them.

Table 3. Inter-stand parameters of the two-stand tandem mill unit.

Parameter		Payoff real-Stand 1	Stand 1-Stand 2	Stand 2-Pickup real
Distance, L, (m)		5.675	4.725	6.6
Tensile Stress, σx, (MPa)	Coil #1	95	147	79
	Coil #2	96	142	77
	Coil #3	98	139	76
	Coil #4	93	147	79
	Coil #5	95	151	81

5. Simulation Algorithm

Figure 5 shows the chatter simulation algorithm for the two-stand tandem rolling mill unit of MSC during the third pass. Each of the “Stand” blocks represents the single-stand chatter model including the rolling process model, the mill structural model and the inter-stand model. The last model deals with the time delay of the strip transportation and the strip tensile stress effects. The “Payoff” and “Pickup” blocks are employed to consider the tension variations before the entry to the “Stand 1” and after the exit from the “Stand 2”. Also, the “Payoff” block provides an initial excitation to the “Stand 1”. Before running the chatter simulation program, another program under steady-state condition is applied to initialize all parameters used in the chatter simulation model.

Fig. 5.

Chatter simulation program.

6. Results and Discussions

The forces and torques obtained from the experiment and the program under the static conditions have been compared for five strips rolled in the two-stand tandem mill unit of MSC (Fig. 6). The results of this program have been found to be in good agreement with those of the experimental data.

Fig. 6.

Comparison of the rolling forces and torques obtained from simulations and experiments (a) Rolling force of the first stand (b) Rolling torque of the first stand (c) Rolling force of the second stand (d) Rolling torque of the second stand.

A procedure through numerical simulation programs is proposed in this paper in order to determine the constant friction factor, m, using the coefficient of friction, μ. First the program under the steady-state conditions is run based on the Coulomb model and friction stress distribution along the contact length between the strip and the work roll is derived. Then, the area under this curve is calculated and compared to that of the Tresca model with an unknown constant friction factor and thus m is computed. By running this procedure for enough samples, an approximately linear relationship between m and μ has been obtained. The equation is $m\cong \sqrt{3}\hspace{0.17em}\mu$ . For the physical justification of this relationship, it can be noted that in the cold rolling, the pressure within the roll bite is about the material yield stress. So ${\tau }_f=\mu \hspace{0.17em}p\cong \mu \hspace{0.17em}{\mathrm{\sigma }}_{yp}$ and on the other hand ${\tau }_f=m\hspace{0.17em}k=m\hspace{0.17em}\left({\mathrm{\sigma }}_{yp}/\sqrt{3}\right)$ . As a result, ${\tau }_f=m\left({\mathrm{\sigma }}_{yp}/\sqrt{3}\right)\cong \mu \hspace{0.17em}{\mathrm{\sigma }}_{yp}\Rightarrow m\cong \sqrt{3}\mu$ . This equation is consistent with results of tests of Vicente et al.²⁷⁾ One of the advantages of the proposed numerical method is that this efficient approach can be used in both cold and hot rolling.

The most important characteristic in numerical simulations of the chatter phenomena is their dependency on the rolling speed.^5,13,24) To investigate this issue, vibration in the work roll of the Stand 2 during the third pass has been inquired. The results for the Coil #1 in the two different speeds (u_1,stand1 = 12 m/s and 13 m/s) are presented in Figs. 7 and 8. Figure 7 indicates that the system is stable at low speed (12 m/s), but with increasing the velocity to 13 m/s, oscillations increase versus time (Fig. 8).

Fig. 7.

Work roll fluctuations in the stable case, u_1,stand1 = 12 m/s, for the Coil #1. (Online version in color.)

Fig. 8.

Work roll fluctuations in the unstable case, u_1,stand1 = 13 m/s, for the Coil #1. (Online version in color.)

With trial and error, a case can be obtained where fluctuations amplitude with time is unchanged. This case represents the critical status of the system. Accordingly, the critical velocity for the Coil #1 has been found to be 12.4 m/s.

Figure 9 illustrates the FFT spectrum of the fluctuations signals for the speed of 13 m/s in the unstable case. Dominant frequency is computed 191 Hz, which is within the frequency range of third octave chatter.

Fig. 9.

FFT spectrum of the vibration signals in the unstable case for the Coil #1. (Online version in color.)

Based on the Coulomb model and the Tresca model, the above procedure has been repeated for the five coils of the Table 2. Then, the critical speeds obtained from these simulations and experiments have been compared as shown Fig. 10. As observed under the operating conditions studied in this paper, the results of the Coulomb model are closer to experiments than those of the Tresca model. This can be expressed by the fact that during vibrations, the pressure distribution within the roll bite is changed and the Coulomb model can better reflect this issue. Also the friction stress in practice usually exceeds the lubricant limiting shear stress²⁰⁾ and in the other hand, the amount of lubricant limiting shear stress is a function of the lubricant pressure.²⁸⁾

Fig. 10.

Comparison of critical speeds obtained from simulations and experiments.

In the following, as one of the most effective and controllable process parameters, the effect of friction on chatter in tandem mills is examined. To pave the way for discussion on this issue, it should be noted that in the regenerative mechanism, initially one stand of the tandem mills is closer to its stability limit and dominates in determining the stability of the whole multi-stand mill.^17,18) In order to find out which of two stands is closer to its stability limit, the negative damping mechanism is only enabled in each stand and then the chatter simulation program is executed. Vibration in the work rolls for u_1,stand1 = 15 m/s is shown in Fig. 11 for the Coil #1. The values of the friction coefficient of Stand 1, μ₁, and the friction coefficient of Stand 2, μ₂, are 0.0087 and 0.0096, respectively. As shown, under this condition the Stand 2 becomes unstable and the Stand 1 remains stable. This means that the stability of the second stand determine the stability of the overall system.

Fig. 11.

Work rolls fluctuations under negative damping mechanism, u_1,stand1 = 15 m/s, μ₁ = 0.0087, μ₂ = 0.0096. (Online version in color.)

Again the regenerative mechanism in the program is activated. Figure 12 shows the effect of μ₂ on the chatter critical rolling speed related to the Coil #1 for three different values of μ₁. Each point in this figure is the result of several runs of the chatter simulation program to converge to the critical speed where each run takes a couple of hours. As shown in Fig. 12, the trend of the curves for three different friction coefficients of Stand 1 is similar.

Fig. 12.

Effect of μ₂ on chatter critical speed for three values of μ₁. (Online version in color.)

The general behavior of the critical speed versus μ₂ is shown in Fig. 13. It is divided into two regions. In the Region I, the stability of the Stand 2 determines the stability of the tandem mills. By increasing μ₂ in this region, the stability boundary of the second stand is enhanced and results in increasing the critical speed. In the Region II, the Stand 1 instead of the Stand 2 plays dominant role in determining the mill stability. So the sensitivity of the critical rolling speed with respect to μ₂ must be eliminated. But still the critical rolling speed increases slightly with an increasing μ₂. This is due to the shift of the neutral point of Stand 2 toward its entry point, so the distance between the entry point and the neutral point becomes shorter. Based on the continuity equation (Eq. (8)), the variation of the entry speed of Stand 2 decreases, leading to a smaller inter stand tension variation according to the Eqs. (16) and (17). As a result the variation of the rolling force of Stand 1 decreases and the stability boundary of Stand 1 is partially increased.^17,18) The values of μ₂ at the beginning of the Region II for three curves shown in Fig. 12 are 0.019, 0.021 and 0.029, respectively.

Fig. 13.

General behavior of chatter critical speed versus μ₂. (Online version in color.)

As can be seen in Fig. 12, an increase in μ₁ has two outcomes. First, the rolling force of Stand 1 increases and therefore more energy is absorbed by the strip with the same amount of deformation. In other words, the material damping coefficient in the Stand 1 increases with increasing μ₁.^17,18) Consequently, the stability limit of Stand 1 is enhanced and in the Region II where the stability of Stand 1 is predominant, increasing μ₁ results in a higher chatter speed. Second, the neutral point of Stand 1 shifts toward its entry point, hence the distance between the neutral point and the exit point becomes longer. Based on the continuity equation, the variation of the exit speed of Stand 1 increases, leading to an increase in the inter stand tension variation. Thus the variation of the rolling force of Stand 2 increases and the stability boundary of Stand 2 is decreased somewhat. As a result the critical speed decreases slightly in the Region I where the stability of Stand 2 is overcoming.

Also, it is interesting to observe what happens when the value of μ₂ is fixed while μ₁ is altering as shown in Fig. 14. The general behavior of the critical speed versus μ₁ is shown in Fig. 15. It is divided into three regions. In the Region I, the Stand 1 turns out to be unstable and determines the stability of the whole system. In this region, an increase in μ₁ results in an increase in the stability limit of Stand 1 and thus critical speed is increased. In the Region II of the Fig. 15, the Stand 2 instead of the first Stand is dominant and excessive increasing of μ₁ do not have considerable effect on the critical rolling speed. In the Region III, as a result of shifting the neutral point of the Stand 1 toward its entry point, the inter stand tension variation increases; and so the stability limit of the second stand is decreased.

Fig. 14.

Effect of μ₁ on chatter critical speed for three values of μ₂. (Online version in color.)

Fig. 15.

General behavior of chatter critical speed versus μ₁. (Online version in color.)

As can be seen in Fig. 14, When μ₂ has a constant low value, only the Region III is observed and increasing of μ₁ might even reduce the critical rolling speed. On the other hand when μ₂ is high, all three regions can occur.

Based on Figs. 12 and 15, it can be found that to achieve the maximum critical speed, the value of μ₂ should be increased as much as possible. Also, the optimal value of μ₁ changes based on the value of μ₂.

In Fig. 16, the values of μ₁ and μ₂ related to the characteristics of Coil #1 and Coil #5 have been changed by using a new lubricant. Under these conditions, the chatter critical speeds obtained from simulation have been compared to the experimental data. As can be seen, the trend of changes in the critical speed for both the simulation and experiment is similar to the Region I in the Figs. 12 and 13.

Fig. 16.

Comparison of critical speeds obtained from simulation and experiment versus friction coefficients for (a) the Coil #1 (b) the Coil #5.

7. Conclusion

In this work, a chatter simulation program for the tandem cold rolling mills based on the two widely-accepted friction models of Coulomb and Tresca is presented. The program has been run for five coils rolled in the two-stand tandem mill unit of Mobarakeh Steel Company. The simulation results have been verified by the forces, the torques, and the critical speeds obtained from the experimental data. The results show that the Coulomb friction model is more accurate than the Tresca model based on reported critical rolling speed in real operating conditions. Also, a relationship between the Tresca friction factor and the Coulomb friction coefficient is determined by means of numerical method. The effect of the friction coefficient on the chatter critical speed has been investigated. Based on the results, it can be concluded that an optimal setting of friction coefficients tends to result in higher critical speeds. For this purpose, the friction coefficient of Stand 2 under the current data should be increased to the extent possible and selection of the optimum friction coefficient of Stand 1 depends on the friction coefficient of Stand 2.

Acknowledgment

The authors are grateful to the Mobarakeh Steel Company for their assistance.

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