Development and evaluation of friction models for chatter simulation in

The International Journal of Advanced Manufacturing Technology https://doi.org/10.1007/s00170-018-1658-x

ORIGINAL ARTICLE

Development and evaluation of friction models for chatter simulation in cold strip rolling Ali Heidari 1 & Mohammad Reza Forouzan 2 & Mohammad Reza Niroomand 3 Received: 6 November 2016 / Accepted: 17 January 2018 # Springer-Verlag London Ltd., part of Springer Nature 2018

Abstract High-speed rolling and high reduction usually are desirable as increase mill productivity in the cold strip rolling. However, most often, these conditions lead to the creation of chatter vibration which has a significant effect on the price of the rolling products. Experimental results show that friction conditions play an important role in the chatter. In this research, in order to stimulate chatter in tandem cold strip rolling mill, a new friction model has been provided based on unsteady mixed lubrication. In addition, chatter modeling has been done based on the simple friction models of Coulomb and Tresca. The friction model of Tresca has been simulated both linearly and nonlinearly, and work roll flattening and strain hardening effects have been undertaken as well. From the viewpoint of four output parameters, i.e., chatter critical speed, dominant frequency, rolling force, and rolling torque, the simulation results have been compared with experimental data taken from an industrial two-stand tandem rolling mill. Also, a parametric study on the effect of some of the major characteristics of rolling lubricant on the four output parameters is conducted. The key result of the research is that unsteady lubrication model, much better than simple friction models, can simulate friction conditions governing the chatter phenomenon. Keywords Tandem cold rolling . Dynamic model . Chatter critical speed . Frequency analysis . Unsteady mixed lubrication

Nomenclature A Fractional contact area Cw Work roll damping coefficient Backup roll damping coefficient Cb c Adhesion coefficient Ew Elastic modulus of the work roll f Rolling force per unit width h0 Inlet film thickness Nominal surface separation hn Average thickness of lubricant film ht Kw Work roll spring constant Backup roll spring constant Kb k Shear strength

* Ali Heidari [email protected] 1

Young Researchers and Elite Club, Khomeinishahr Branch, Islamic Azad University, Khomeinishahr, Isfahan, Iran

2

Department of Mechanical Engineering, Isfahan University of Technology, Isfahan 8415683111, Iran

3

Department of Mechanical Engineering, Payame Noor University, Tehran 193953697, Iran

Li Mw Mb m p pr R R′ Rw Rb Rq r t u u1 u2 vr w y y0 y1 y2

Distance between stands (i) and (i + 1) Work roll mass Backup roll mass Constant friction factor Normal pressure Constant coefficient Roll radius Deformed roll radius Work roll radius Backup roll radius Composite surface roughness Reduction in the thickness Time Strip speed within the roll bite Strip speed at entry Strip speed at exit Peripheral speed of the work roll Strip width Strip thickness Initial strip thickness entering first stand of tandem rolling mill Strip thickness at entry Strip thickness at exit

Int J Adv Manuf Technol

yc yc Z z

˙

α μ η σx, 1 σx, 2 σyp ε σ0, c1, c2 τf τa τb τL Δ f(δ)

Roll gap spacing along the centerline of the rolls Rate of change of roll gap spacing Dimensionless parameter Viscosity pressure coefficient in the Roelands equation Viscosity pressure coefficient in the Barus equation Coefficient of friction Lubricant viscosity Back tension at entry Front tension at exit Material yield stress Effective strain under the plain strain conditions Material constants Friction stress Shear stress in contact area Shear stress in valleys Limiting shear stress Time delay of the strip transportation Probability density function

1 Introduction Rolling industry is considered as one of the most prevalent industries in the production of metal products. Chatter, which is a special type of self-excited vibration in rolling is one of the most destructive types of vibration, usually occurs in the cold rolling of thin strip at high speeds [1]. Given the high cost of rolling machines, reducing production rates in order to avoid chatter has made this phenomenon not only a technical problem but also an economic problem in rolling mills. Researches conducted in the field of rolling mill chatter can be classified into several different categories. Since chatter is a complex phenomenon, a large number of previous studies have examined the nature of chatter vibration [2–4]. Chatter has three basic types: third octave, fifth octave, and torsional chatter. Among them, the first type is the most detrimental. Four major mechanisms for the third octave chatter are divided: model matching, negative damping, regenerative, and mode coupling [5]. Other researchers have conducted practical tests [6–9]. Having laboratory or industrial rolling machines, they have examined behavior of the system at the time of chatter incidence. Many researchers have tried to find ways to prevent or control chatter [10–12]. Some of these studies have led to the installation of chatter detection systems in rolling stands. Furthermore, many researches have been done to model selfexcited vibration in rolling [13–16]. The friction condition plays a key role in the chatter modeling [17]. It has a great contribution in all of the mentioned four mechanisms. Friction in cold rolling has two parts of Coulomb and viscose. In the second part, the lubricant flow between the

work roll and the strip surface becomes extremely unsteady as a result of chatter. Owing to the complexity of the problem, most researchers have used simple friction models in their analysis, and only very few researches have described how chatter occurs through unsteady lubrication phenomenon [17–19]. The studies accompanied in the field of lubrication and friction in chatter can be divided into three main categories: the first category focuses on the importance of friction and lubrication in chatter vibration, the second category focuses on the chatter simulation using the simple friction models, and the third category focuses on the researches in improving chatter analysis by means of the unsteady lubrication model. Johnson [20] investigated the effect of friction on the chatter in strip rolling. In his equations, friction has been considered as a function of the relative speed between the strip and work roll. He concluded that to predict chatter correctly, exact friction model is necessary. Lin et al. [21] investigated the relationship between the chatter phenomenon and emulation. They showed that under certain conditions, by controlling the concentration of the emulsion, system vibrations may be removed at high speeds. Chen et al. [22] considered friction coefficient as the nonlinear function of the relative speed of the strip and work roll and showed that vibration stability is strongly affected by the friction coefficient. Yun [23], Hu [3], and Zhao [24] improved some effective chatter models based on simple friction models. In these researches, after obtaining governing equations, they linearized the equations and, then, used them in the form of transfer function. Yang et al. [25] proposed a model which was a combination of the rolling process model, the mill structural model, and the hydraulic system model. They evaluated the influence of various rolling factors on the stability of the system. Kim et al. [26] proposed another mathematical model of chatter in rolling which was a combination of the rolling process model, the mill structural model, and the driving system. They did this by adding a number of masses, springs, and damping to the preceding models and compared the results of the model with experimental results in terms of chatter frequency. Liu et al. [27] studied the effects of inter-stand tension variation and strip variation transportation between adjacent stands with time delay on tandem cold rolling mills chattering. Results of this research demonstrated that time delay effect is essential for the regenerative chatter model. For preventing chatter, Fujita et al. [17] proposed an intelligent lubrication control system. Chatter stability improved when the friction coefficient was suitably controlled within a certain range in the high-speed rolling region. Lingqiang et al. [28] offered a vertical–torsional–horizontal coupled vibration model by considering nonlinear friction. Their results showed that when rolling thin strips, the system stability boundary may be only enclosed by the critical domains of vertical and torsional vibration modal. Tung [29] suggested an unsteady

Int J Adv Manuf Technol Fig. 1 Rolling process geometry

lubrication model for the strip rolling process in two regimes of full film and mixed. He evaluated the effects of timedependent variables such as the strip tensile stress and the work roll and strip speeds on lubrication. Heidari et al. [18] proposed a chatter model in the single stand rolling based on unsteady full film lubrication. In the present research, an improved friction model is offered based on the unsteady mixed lubrication to simulate tandem cold rolling mill chattering. Effect of some usual assumptions in rolling chatter simulations such as friction models, i.e., Coulomb and Tresca, linearization of the output parameters, work hardening, and strain hardening has been studied on critical chattering speed, dominant frequency, steady-state rolling force, and steady-state rolling torque. Concluding on effect of any assumption becomes possible

Fig. 2 Illustration of pressure sharing in mixed lubrication regime [29]

after comparison of the results with experimental data of an industrial plant.

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 thickness, it can be assumed that the rolling process operates under the plane strain conditions. The work roll radius is R and the peripheral speed of the work roll is vr. Since the focus of current study is on the third octave chatter

Int J Adv Manuf Technol Fig. 3 Block diagram of chatter model

Stand i-1

Stand i

Mill Structural Model

Mill Structural Model

Rolling Process Model

Rolling Process Model

which is symmetric about the x-axis [3], the work rolls are assumed to vibrate only along the y-axis at a rate of ˙yc =2. As shown in Fig. 1, the roll bite is divided into three zones namely the inlet, the work, and the outlet zones. The first zone is where the lubricant pressure increases from the atmospheric pressure to the pressure required incurring strip plastic deformation. The second zone is where the rolling force and torque are generated. The variation of pressure in the last zone is reverse of that in the first zone. The lubricant pressure in the first and last zone only causes strip elastic deformation.

2.2 Friction models The friction coefficient model or the Coulomb model and the friction factor or the Tresca model are the most traditional friction models utilized in industry. The first model may be described by the coefficient of friction, μ, defined as the ratio of the friction stress τf to the normal pressure p where τ f ¼ μp

Fig. 4 A view of two-stand tandem mill unit of Mobarakeh Steel Company

ð1Þ

The second model assumes that the friction stress τf is proportional to the multiplication of the shear strength k of the strip and the constant friction factor m. τ f ¼ mk

0 ≤m ≤ 1

ð2Þ

For m = 0, the frictionless interface occurs; for m = 1, the contacting surfaces will stick together. The dominant mechanism of lubrication in the rolling process is mixed lubrication regime [30]. In this regime, the nominal surface separation (hn) is decreased to less than triple of the composite surface roughness (Rq). Then, a part of applied pressure is tolerated by the asperity contacts and the remaining pressure is borne by the lubricant in asperity valleys. Figure 2 illustrates how the pressure shares in mixed lubrication regime schematically. Fractional contact area (A) is equal to the ratio of actual contact area to nominal contact area. The ratio of lubricant volume to the nominal contact area is defined as the average thickness of the lubricant film and is denoted by ht, which is not shown in Fig. 2. The distance between the mean lines of the undeformed surfaces, i.e., hn, is the nominal surface

Int J Adv Manuf Technol

Fig. 5 A view of the data collection system

separation. If the surfaces are not contact with each other, then ht and hn are the same; but, if there is contact, ht is bigger than hn. Having determined the variation of the inlet film thickness (h0(t)) from the inlet zone analysis using the Reynolds equation and applying the boundary conditions, the work zone can be analyzed. The output of this analysis is the work zone film thickness distribution (hn(x, t)) which can be obtained from the continuity equation in the fluid mechanics [18]. Assuming the Gaussian height distribution surface, the fractional contact area and the average thickness of the lubricant film can be expressed by ∞

ht ¼ ∫ ðhn þ δÞf ðδÞ dδ −hn

ð3Þ

∞

A ¼ ∫ f ðδÞ dδ

ð4Þ

hn

where f(δ) is the probability density function and is defined as follows: "
 #3 35 1 δ 2 1− for jδj≤ 3Rq f ðδ Þ ¼ ð5Þ 96Rq 9 Rq ¼0 for jδj > 3Rq ht and A can also be represented in terms of the dimensionless parameter Z = hn/3Rq as follows:

ht ¼

 3Rq  35 þ 128Z þ 140Z 2 −70Z 4 þ 28Z 6 −5Z 8 256

ð6Þ

Rolling Direction

Fig. 6 A schematic view of the placement of magnetic accelerometer in the two-stand tandem

Magnetic accelerometer sensor

Stand 1

Stand 2

Int J Adv Manuf Technol Fig. 7 Block diagram of chatter detector

A ¼ 1 for Z < −1  A ¼ 16−35Z þ 35Z 3 −21Z 5 þ 5Z 7 =32 for −1 ≤ Z ≤ 1 A¼0 for 1 < Z

where pr has a constant coefficient equal to 1.962 × 108 and z is the coefficient of pressure–viscosity. ð7Þ

Frictional shear stress in mixed regime is considered as follows [31]: τ f ¼ A τ a þ ð1−AÞ τ b

ð8Þ

where τa and τb are the shear stress in the contact area and the valleys respectively and are calculated through the following equations: τa ¼ ck u−vr τb ¼ η ht

ð9Þ ð10Þ

2.3 Strain hardening effect To express the strain hardening effect of metals, the Ludwik equation is widely used in mathematical models for cold rolling process [5].

c2  σyp ¼ σ0 1 þ c1 ε ð12Þ where σyp is the material yield stress; σ0, c1, and c2 are the material constants which have to be specified via experiments; and ε is the effective strain under the plain strain conditions which can be expressed as follows:
 2 y ð13Þ ε ¼ pffiffiffi ln 0 y 3

where c is the adhesion coefficient, η is lubricant viscosity, and k is the shear strength of the strip. In cold rolling of steel strips by steel rolls, c is considered as 0.2 by most researchers [32, 33]. Since the pressure is high in the work zone, viscosity, as a function of pressure, is calculated using the Roelands equation [34]:

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.

   
p z ηðpÞ ¼ η0 exp flnðη0 Þ þ 9:67g 1 þ −1 pr

An equation for the effective radius of the work roll (R′) was presented by Hitchcock in 1935. Later, Robert showed

ð11Þ

2.4 Work roll flattening effect

Fig. 8 Strip speed at the entrance to the second stand during the three rolling passes

Int J Adv Manuf Technol Table 1

Specifications of the MSC two-stand tandem mill unit

Parameter

Stand 1

Stand 2

Work roll mass, Mw (kg) Backup roll mass, Mb (kg) Work roll damping coefficient, Cw (N s/m) Backup roll damping coefficient, Cb (N s/m) Work roll spring constant, Kw (N/m) Backup roll spring constant, Kb (N/m) Work roll radius, Rw (m) Backup roll radius, Rb (m)

14,000 38,000 0 1.254e6 4.97e10 1.22e10 0.245 0.675

14,000 38,000 0 1.254e6 4.97e10 1.22e10 0.245 0.675

that the quantity of R′, calculated from Hitchcock’s formula, is somewhat conservative and modified the equation as follows [35]: sffiffiffiffiffiffiffiffiffiffiffiffi ! f f 0 R ¼R 1þ2 ð14Þ þ2 Ew ry1 E w ry1 where f is the rolling force per unit width, r is the reduction in the thickness, and Ew is the elastic modulus of the work roll. Note that the effective work roll radius is obtained iteratively during computations under steady-state conditions.

relationship between the rolling process and the mill structural model continues in a loop. The negative damping mechanism often occurs in the single stand rolling while the regenerative effect is also activated in the tandem mills [5, 24]. Therefore, the chatter is more likely to happen in the multi-stands. Figure 3 shows the block diagram of the chatter model for the tandem configuration based on the single stand 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 an 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 [5]. 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. 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. Under this condition, the stand, which becomes first unstable, determines the stability limit of the overall system [36]. The authors previously have offered full details of the chatter model based on simple friction models in references [4, 36, 37] and also based on the unsteady full film lubrication regime [18].

3 Chatter model 4 Experimental equipment 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. Whole of these outputs fluctuate over time dynamically. The dynamic rolling force is the main input of the mill structural model. It leads to variation of the roll gap spacing, and this variation changes the inputs of the process model. Thus, the

Table 2

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. Incoming coils to this unit have a thickness of 2 mm. Coils open up and pass through two four-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

Characteristics of the coils in the third pass

Parameter

Strip width, w (mm) Entry thickness, y1 (mm) Exit thickness, y2 (mm) Coefficient of friction, μ

Coil no. 1

Coil no. 2

Coil no. 3

Coil no. 4

Coil no. 5

Stand 1

Stand 2

Stand 1

Stand 2

Stand 1

Stand 2

Stand 1

Stand 2

Stand 1

Stand 2

783

783

829

829

818

818

843

843

762

762

0.431 0.340 0.0087

0.340 0.240 0.0096

0.498 0.415 0.0083

0.415 0.270 0.0124

0.541 0.476 0.0071

0.476 0.300 0.0141

0.432 0.354 0.0079

0.354 0.230 0.0111

0.409 0.317 0.0088

0.317 0.210 0.0099

Int J Adv Manuf Technol 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

Coil no. 1

5.675 95

4.725 147

6.6 79

Coil no. 2 Coil no. 3

96 98

142 139

77 76

Coil no. 4

93

147

79

Coil no. 5

95

151

81

Distance, L (m) Tensile stress, σx (MPa)

previous pass. In this mill, chatter usually occurs in the second stand during the third pass [7]. Strain hardening effect begins to play its role reducing strip plastic deformation and increasing elastic component of deformation. This effect along with reducing the thickness and increasing speed makes mill more susceptible to vibration. All rolling data in this unit are recorded in a data collection system through several analog and digital signals; a view of this system is shown in Fig. 5. The rolling force and the rolling torque are obtained by averaging the force and torque values in the steady-state conditions from the data collection system. As soon as the chatter occurs, a special sound similar to mobile phone vibration is heard even in the control room. This sound is the main sign of chatter. Hearing it, the operator should immediately reduce the rolling speed. In this method, the operator’s carelessness or lack of timely action may lead to heavy damages of the unit. This problem pushes the operators to do the rolling much slower than the limit speed of chatter. In order to solve this problem, a chatter detector is used. In the chatter detector, a magnetic accelerometer sensor is used which is permanently located at the top housing of one of the stands. The location of the magnetic accelerometer in the two-stand tandem is schematically shown in Fig. 6. In this research, the critical rolling speed is gained using the chatter detector. Figure 7 shows the block diagram of the chatter detector. Signals derived from accelerometer are first filtered by a high-pass filter; then, they are amplified and filtered by a low-pass filter. Then, analog signals are converted into

Table 4 Characteristics of the used lubricants

digital ones and stored in the machine’s computer. According to the standards defined for it, chatter detector notifies the operator in two levels: chatter warning and chatter alarm. Figure 8 shows the strip speed at the entrance to the second stand during the three rolling passes. As can be seen, the rolling process is done in three passes. In each of the passes, the rolling speed has increased gradually from zero. In the first and second passes, after reaching a maximum speed, the rolling has continued with this speed for a while. At the end of these passes, speed is steadily reduced to zero. As the thickness of strip is high enough in these passes, there is no potentiality for chatter occurrence. Also, in the third pass, the speed is increased gradually from zero till its instability speed. At this moment, the chatter detector alarms the occurrence of the chatter; this speed is recorded as the critical speed and the system automatically reduces the rolling speed. In Table 1, specifications of the MSC tandem rolling mill are given. Characteristics of five coils under consideration in the third pass are presented in Table 2. Inter-stand parameters are also listed in Table 3. And finally, the lubricant characteristics are reported in Table 4.

5 Simulation algorithm Figure 9 shows the chatter simulation block diagram for the two-stand tandem rolling mill unit of MSC. Each of the “stand” blocks represents the single-stand chatter model

Parameter

Dynamic viscosity at 50 °C, η50 (Pa s) Viscosity pressure coefficient in the Barus equation, α (MPa−1) Viscosity pressure coefficient in the Roelands equation, z Limiting shear stress, τL (MPa)

Value Oil A

Oil B

0.0178 0.015 0.5216 24.3

0.0344 0.0164 0.5118 33.2

Int J Adv Manuf Technol Fig. 9 Chatter simulation program for two-stand tandem cold rolling mills

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. It should be noted that depending on the type of friction model used in the program, the equations of the rolling process model inside the stand block are selected.

Fig. 10 Work roll fluctuation of the second stand in the speeds of a 6 m/s and b 8 m/s

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 is applied under steady-state condition to initialize all parameters used in the chatter simulation model.

Int J Adv Manuf Technol Fig. 11 Work roll fluctuation of the second stand in the critical speed case

The clock used in this program follows all dynamic time-dependent responses of the system by recording them. In the present Simulink model, the selected solver is ode45 with variable time steps. Given the fact that the third octave chatter frequency can vary from 120 to 240 Hz in various rolling mills, for finding appropriate results, time steps should be at least less than 0.0021 s. In order to increase the accuracy, the largest time step that the solver can take in the present program has been set on 0.0001 s.

6 Results and discussions In this section, the results of simulation program will be evaluated from the viewpoint of four output parameters: the rolling

Fig. 12 Critical speed values for five coils in seven different cases. First column: experimental results; second column: Tresca friction model– linear model; third column: Tresca friction model–nonlinear model; fourth column: Tresca friction model–nonlinear model–work roll flattening; fifth column: Tresca friction model–nonlinear model–work

force, the rolling torque, the critical speed of chatter, and the dominant frequency. Then, the simulation results are compared with the experimental data of Mobarakeh Steel Company.

6.1 Chatter critical speed The most important characteristic of a simulation program of rolling dynamic model which has been prepared for analyzing the system in terms of chatter is its dependency to the rolling speed. When the rolling speed is low, vibrations become damped and the system is stable. With the increase of the speed, the system is expected to move towards instability [7, 24]. The speed, on which vibrations are not removed as time passes and the vibration amplitude remains constant, is named chattering critical speed. If the rolling speed

roll flattening–strain hardening; sixth column: Coulomb friction model– nonlinear model–work roll flattening–strain hardening; seventh column: unsteady mixed lubrication–nonlinear model–work roll flattening–strain hardening

Int J Adv Manuf Technol

(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

Fig. 13 Comparison of experimental values of rolling force and torque of the first and second stands with the results of the simulated program in steady-state a rolling force of the first stand, b rolling torque of the first stand, c rolling force of the second stand, and d rolling torque of the second stand. First column: experimental results; second column:

Tresca friction model; third column: Tresca friction model–work roll flattening; fourth column: Tresca friction model–work roll flattening– strain hardening; fifth column: Coulomb friction model–work roll flattening–strain hardening; sixth column: mixed lubrication model– work roll flattening–strain hardening

exceeds this value, the system becomes unstable and vibration amplitude increases. Rolling critical speed in simulations presented here is determined by trial and error. The dependency of the present simulation program on the rolling speed is investigated in continue. In the first step to display the results, the program is used in which the simulation is based on Tresca friction model, the output parameters of the dynamic model are linearized, and the strain hardening and work roll flattening effects are not considered. In order to investigate the system’s vibrational state or its stability, the work roll fluctuations of the second stand have been examined. The results in two different speeds (6 and 8 m/s) are presented in Fig. 10. These results are related to the first coil, and the used lubricant is oil A. As can be observed in the Fig. 10, at the speed of 6 m/s, the vibrations are damped and the system is stable. With the increase of speed to 8 m/s, vibration amplitude increases with time and the system becomes unstable. Obviously, the vibration divergence is indicative of the system instability. Therefore, the dependency of the program on rolling speed

is seen in Fig. 7 clearly. Through trial and error, a situation is achieved where vibrations are not damped and vibration amplitude remains constant over time. This shows the situation of the system in the critical mode. Critical speed for the above condition is equal to 7.3 m/s. Figure 11 shows work roll fluctuation of the second stand in the critical speed mode. It is worth noting that as the model is a two-stand model and there is modulation in the vibrations of Figs. 10 and 11, the dominant mechanism for the occurrence of chatter is the regenerative mechanism. The above procedure has been repeated for all five investigated coils whose characteristics are given in Tables 1, 2, 3, and 4. Figure 12 shows the values of critical speed for all coils in seven different states. In this figure, the first column for all coils is related to experimental results. To find the critical speed in practice, the chatter detector has been used. The second column is related to the program simulated based on the Tresca friction model, and the output parameters of rolling process dynamic model have been linearized based on input parameters; the third column is also related to the program simulated

Int J Adv Manuf Technol

Average error of rolling force (%)

25 Tresca 20 Tresca-Work roll flaening 15 Tresca-Work roll flaening-Strain hardening

10

Coulomb-Work roll flaening-Strain hardening

5

Mixed lubricaon-Work roll flaening-Strain hardening 0

(a) Average error of rolling torque (%)

30 25

Tresca

20

Tresca-Work roll flaening

15

Tresca-Work roll flaeningStrain hardening

10 Coulomb-Work roll flaeningStrain hardening 5 Mixed lubricaon-Work roll flaening-Strain hardening 0

(b) Fig. 14 Average errors in the calculation of a rolling force and b rolling torque, for five different states. First column: Tresca friction model; second column: Tresca friction model–work roll flattening; third column: Tresca friction model–work roll flattening–strain hardening;

forth column: Coulomb friction model–work roll flattening–strain hardening; fifth column: mixed lubrication model–work roll flattening– strain hardening

based on the Tresca friction model but with no linearization for the rolling process parameters. The fourth column is related to the program which is based on the Tresca friction model and nonlinear rolling model in which work roll flattening effect is considered. In the fifth column, compared with the fourth one, strain hardening effect is also added. In the sixth column, compared with the fifth one, the friction model has changed from the Tresca model to the Coulomb model; however, rolling process is still nonlinear and work roll flattering and strain hardening effects are active. In the seventh column, the values of critical speed have been obtained based on unsteady mixed lubrication. The results of different columns are compared in continuation.

of the results of the second and third columns in Fig. 12 suggests that there is a significant difference between the linear and nonlinear results; thus, henceforth, the results are presented based on nonlinear model.

6.1.1 Comparing the results of the second and third columns The results of these two columns are related to the program which is based on the Tresca friction model; however, the difference is that rolling process model is linear in the second model and nonlinear in the third. The advantage of the linear model over the nonlinear model is the running of program in a much shorter time. However, comparison

6.1.2 Comparing the results of the third and fourth columns The results of these two columns are related to the program which is based on the Tresca friction model and the nonlinear model of the rolling process, but the difference is in considering the nonlinearities due to the work roll flattening effect in the fourth column. Comparison of the results of these two columns in Fig. 12 shows that the work roll flattening effect increases the critical speed calculated by the program significantly. This is due to the direct relationship between the equivalent damping coefficient of the system and the flatted radius of the work roll [1]. 6.1.3 Comparing the results of the fourth and fifth columns The results of these two columns are related to the program which is again based on the Tresca friction model with considering the nonlinearities of roll flattering, while in

Int J Adv Manuf Technol

(a) 146 Hz

(b) Fig. 15 Filtered signal of the vertical vibrations of the upper backup roll when chatter occurs (cutoff frequency = 2 kHz). a Acceleration signal. b FFT spectrum

the fourth column, only work roll flattening effect is considered based on the Roberts formula; in the fifth one, both the work roll flattening and strain hardening effects are considered. It is expected that by adding strain hardening effect to the model and, hence, reducing the equivalent damping coefficient of the system, the critical speed calculated by the simulation program would be reduced; however, comparing the results of the fourth and fifth columns in Fig. 9, shows that considering strain hardening increases

Fig. 16 FFT spectrum of the vibration signals in the unsteady state by unsteady lubrication model

the critical speed to some extent. To justify this, it must be said that, despite by adding strain hardening effect to single stand model, the critical speed calculated by the simulation program is reduced which is due to the reduction of the equivalent system damping coefficient; the neutral point of the first stand transfers towards its exit point and leads to reduction the inter-stand tension variation [1]. Therefore, stability boundary of the second stand is increased; on the other hand, the second stand is dominant

Int J Adv Manuf Technol Fig. 17 Dominant frequency values in seven different states. First column: experimental results; second column: Tresca friction model–linear model; third column: Tresca friction model– nonlinear model; fourth column: Tresca–nonlinear–work roll flattening; fifth column: Tresca– nonlinear–work roll flattening– strain hardening; sixth column: Coulomb friction model– nonlinear–work roll flattening– strain hardening; seventh column: unsteady mixed lubrication– nonlinear–work roll flattening– strain hardening

in the conditions of Fig. 12. As a result, considering strain hardening, stability boundary of the second stand improves and, consequently, through the dominance of the second stand, stability boundary of the system increases; this effect, therefore, increases the critical speed in the examined two-stand tandem. 6.1.4 Comparing the results of the fifth and sixth columns The results of these two columns are related to a program based on nonlinear model of rolling process in which the work roll flattening and the strain hardening effects are active; these two columns are different as the simulation program of the fifth column uses the Tresca friction model while the simulation program of the sixth column uses the Coulomb friction model. From the comparison of these two columns in Fig. 12, it is observed that under the conditions of this study, the results of Coulomb friction model, compared with the Tresca friction

Table 5

model, are closer to the experimental results. This may be explained by the fact that during the vibrations of the rolling structure, pressure distribution changes in the work zone and the Coulomb friction model reflects this better than the Tresca friction model. The critical speed achieved by the simulation program based on the Tresca and the Coulomb friction models is different from experimental values (first column); however, it should be noted that such a difference in the results of critical speed seems reasonable. This is due to unsteady lubrication regime during the vibrations which makes the friction between the strip and the work roll function of time and position. In such a condition, other simple friction models may no longer be a good estimation for the friction. 6.1.5 Comparing the results of the sixth and seventh columns The important point of this comparison is a significant reduction of the critical speed in the unsteady mixed lubrication

Average errors of simulation programs from the viewpoint of the four output parameters

Conditions of simulation chatter program

Tresca–linear Tresca–nonlinear Tresca–nonlinear–work roll flattening Tresca–nonlinear–work roll flattening–strain hardening Coulomb–nonlinear–work roll flattening–strain hardening Unsteady mixed lubrication–nonlinear–work roll flattening–strain hardening

Average error (%)

RMS of average error (%)

Critical speed

Rolling force

Rolling torque

Dominant frequency

24.3 19.2 38.3 42

22 22 12 10.4

22.2 22.2 24.9 26.7

15 21.2 24.7 28.1

22.2 22.2 24.9 26.7

30.5

10

21.8

26.7

21.8

7.5

8

9.6

10.3

9.6

Int J Adv Manuf Technol Fig. 18 Root mean square of average error in six different states. First column: Tresca friction model–linear model; second column: Tresca friction model–nonlinear model; third column: Tresca–nonlinear–work roll flattening; fourth column: Tresca–nonlinear–work roll flattening–strain hardening; fifth column: Coulomb friction model–nonlinear–work roll flattening–strain hardening; sixth column: unsteady mixed lubrication–nonlinear–work roll flattening–strain hardening

model compared with the latest program of the previous section; hence, the results of the unsteady mixed lubrication model are closer to the experimental results. This shows that the unsteady mixed lubrication model can simulate the friction conditions better than simple friction models in chattering conditions.

6.2 Rolling force and rolling torque In order to verify the simulation program from another viewpoint, the steady-state rolling force and torque of both stands for whole the five coils of Table 4 have been obtained from the simulations and compared with the experimental results of two-stand tandem cold rolling of the Mobarakeh Steel Company. This evaluation is presented in Fig. 13 for all cases of the program. Simulation conditions of all columns in Fig. 13% are the same as Fig. 12 and can be found in the figure caption. From Fig. 13, it can be seen that considering strain hardening effect and, particularly, work roll flattening effect will

6.3 Dominant frequency Figure 15 shows the filtered acceleration signal of the vertical vibrations of the upper backup roll and its FFT spectrum when

45 40 35 Average error (%)

Fig. 19 Average errors of the four output parameters in six different states. Tresca–linear (TL); Tresca–nonlinear (TN); Tresca– nonlinear–work roll flattening (TNW); Tresca–nonlinear–work roll flattening–strain hardening (TNWS); Coulomb–nonlinear– work roll flattening–strain hardening (CNWS); unsteady mixed lubrication–nonlinear– work roll flattening–strain hardening (UNWS)

reduce error in calculation of rolling force and rolling torque; on the other hand, in the study’s condition, there is no significant difference between the Tresca and the Coulomb friction models in calculating rolling force and torque. In other words, this figure emphasizes the necessity of considering strain hardening and work roll flattening effects in the calculations of cold rolling process. The average errors in calculating rolling force and torque for different programs of this research are presented in Fig. 14. The figure shows that error values in the program simulated based on lubrication model are less than the errors in the program simulated based on the Coulomb friction model. Therefore, even from the viewpoint of the rolling force and torque, lubrication model is much more accurate than simple friction models.

TL

30

TN 25

TNW

20

TNWS

15

CNWS

10

UNWS

5 0 Crical speed

Rolling force

Rolling torque

Dominant frequency

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11

Output parameters

Fig. 20 Effect of rolling lubricant viscosity on output parameters

10 Crical speed (m/s) 9 Rolling force (MN) 8 Rolling torque (kN.m) 7 Dominant frequency (20 Hz) 6 5 0.016

0.018

0.02

0.022

0.024

0.026

Viscosity (Pa.s) 11

Output parameters

Fig. 21 Effect of viscosity pressure coefficient on output parameters

10 Crical speed (m/s)

9

Rolling force (MN)

8

Rolling torque (kN.m) 7 Dominant frequency (20 Hz) 6 5 0.5

0.6

0.7

Viscosity pressure coefficient

chatter occurs in practice. The increase in the amplitude of the vibrations due to chatter is seen in Fig. 15a. Figure 15b shows that the prominent frequency is 146 Hz which is in the range of the third octave chatter. Figure 16 shows the FFT spectrum of the vibration signals in the unsteady state by unsteady lubrication model for the conditions of the Table 4 first coil at the rolling speed of 13 m/s. Its dominant frequency is equal to 161 Hz which is in the range of third octave chatter. For all five coils, the

dominant frequency obtained by simulation is almost similar and equal to 161 Hz. Using the different friction models in chatter simulation alters the equivalent damping coefficient and the equivalent spring stiffness of the system. As a result, dominant frequency obtained from these models will differ. In Fig. 17, the values of the dominant frequency for different programs of this research have been compared with each other and with the experimental value that is 146 Hz. It is observed that even from the viewpoint of the

13

Fig. 22 Effect of limiting shear stress on output parameters Output parameters

12

Crical speed (m/s)

11 Rolling force (MN)

10 9

Rolling torque (kN.m)

8 Dominant frequency (20 Hz)

7 6 5 23

25

27

29

31

33

Liming shear stress (MPa)

35

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(a) Oil A

(b) Oil B Fig. 23 Output parameters obtained from the data collection system versus lubricant type. a Oil A. b Oil B

dominant frequency, unsteady lubrication model is more accurate than the simple friction models.

6.4 Comparing the unsteady lubrication friction with the simple friction models In Table 5, the average errors of the chatter simulation programs are compared from the viewpoint of the four parameters of critical speed, dominant frequency, rolling force,

and the rolling torque. As can be seen, the average error of unsteady lubrication model program is much less than simple friction models for whole the parameters. This is more evident for the parameter of dominant frequency and, particularly, critical speed. However, it is worth noting that for the computation of the steady-state rolling force and torque, simple friction models have the necessary accuracy as long as work roll flattening and strain hardening effects are active.

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(a) Oil A

(b) Oil B

Fig. 24 Dominant frequency versus lubricant type. a Oil A. b Oil B

Root mean square (RMS) of the average error for the six different states has been offered in the last column of the above table; these values also are shown in Fig. 18. It can be seen in this figure that unsteady lubrication model has been able to reflect chatter phenomenon better than simple friction models. It can be explained that pressure distribution in the roll-bite, the relative speed between the strip and the work roll, lubricant’s film thickness, and fractional contact area have been changed during the vibrations that, in turn, alter the distribution of shear stress between the strip and the work roll. On the other hand, considering these effects is almost impossible in simple friction models; in contrast, the unsteady lubrication model can consider these changes in the simulation of chatter. In Fig. 19, the average errors of the four output parameters, i.e., critical speed, rolling force, rolling torque, and dominant frequency, are compared from the viewpoint of the six different conditions of simulation chatter program. As can be observed in this figure, the error of all four output parameters is lowest in the state of the unsteady mixed lubrication model.

Fig. 25 Distribution of fractional contact area over the work roll– coil interface

6.5 Effect of lubricant characteristics Using the unsteady mixed lubrication model, effects of some main characteristics of the rolling lubricant on the output parameters have been studied. Figures 20, 21, and 22 indicate the effects of the rolling lubricant viscosity (η50), the viscosity pressure coefficient in the Roelands equation (z), and the limiting shear stress (τL) on the output parameters, respectively. The limiting shear stress concept means that a viscoplastic behavior is assumed for the lubricant. The results demonstrate that excessive increase of the rolling lubricant viscosity and the viscosity pressure coefficient does not have considerable effect on the chatter critical speed and the rolling force, but the limiting shear stress variation does. This can be expressed by the fact that in the investigated conditions, the friction stress exceeds the limiting shear stress; hence, the friction stress is set equal to the limiting shear stress. As a result, the effect of the rolling lubricant viscosity and viscosity pressure coefficient variations in the

Int J Adv Manuf Technol Fig. 26 SEM Images of the coil no. 1 before rolling (a, b) and after rolling (c, d) at two different magnifications

(a) before rolling (50x magnification)

(b) before rolling (1600x magnification)

(c) after rolling (50x magnification)

(d) after rolling (1600x magnification)

studied conditions on the chatter critical speed and the rolling speed is negligible. In Fig. 23 and 24, the friction condition related to the coil no. 1 has been changed by using an alternative lubricant (oil B). Accordingly, the output parameters obtained from the experimental data have been compared versus lubricant type. Properties of oil A and oil B are presented in Table 4. The limiting shear stress of oil A and oil B is equal to 24.3 and 33.2 MPa, respectively. The values of critical speed, rolling force, and rolling torque of one of the motors before the gearbox obtained from the data collection system are presented at the time of chatter alarm for oil A and oil B in Fig. 23. Since the limiting shear stress of oil B is higher than that of oil A, and the friction stress in the investigated conditions is set equal to the limiting shear stress, by changing the lubricant type from oil A to oil B, the critical speed and the rolling force have increased. As a result of the lubricant characteristics variations, the neutral point position and possibly the friction stress distribution change, and so the rolling torque value varies. In Fig. 24, the value of dominant frequency obtained from the chatter detector is provided for oil A and oil B. The dominant frequency variation is not significant due to lubricant characteristics variations because it mainly depends on the structural specifications of the rolling mill and the process damping due to plastic deformation of the strip. Based on Figs. 20, 21, 22, 23, and 24, it can be found that to achieve the maximum critical speed in the studied

conditions, the value of friction stress should be increased as much as possible. So if the friction stress exceeds the limiting shear stress, a lubricant which has a higher limiting shear stress is more suitable for chatter conditions. But if the friction stress is less than the limiting shear stress, the rolling lubricant viscosity and viscosity pressure coefficient will be effective. Using the proposed friction model, the ratio of actual contact area to nominal contact area, fractional contact area (A), has been calculated in the work roll–coil interface. For the coil no. 1 rolled using the oil B, the distribution of fractional contact area is presented in Fig. 25. As can be seen in the figure, the value of A over the work zone is larger than zero. It can be concluded that optimum lubrication regime of the cold rolling process, i.e., the mixed film lubrication regime, occurred. This suggests that the lubrication conditions with the used oil are suitable. To further explore the issue, the surface of coil no. 1 rolled using the oil B was evaluated by the scanning electron microscope (SEM). Figure 26 shows the SEM images of the coil no. 1 before and after rolling at two different magnifications (50× and 1600×). Reducing the surface roughness of the coil indicates the existence of the asperities contact at the work zone. In other words, the lubrication regime in practice is the mixed film lubrication. This also confirms the results of the proposed friction model. In addition, the SEM images demonstrate that the grains were stretched in the rolling direction; so, the creation of a

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preferred direction improves the mechanical properties of the coil in the deformation direction.

References 1.

7 Conclusions 2.

Focusing on friction models, the chatter phenomenon in two-stand tandem of cold strip rolling is examined in this research. To this end, a developed friction model has been provided based on unsteady mixed lubrication regime and the results have been compared with the results of simple friction models and experimental results from the viewpoint of chatter speed, dominant frequency, rolling force, and rolling torque. Tresca simple friction model has been simulated both linearly and nonlinearly. Furthermore, work roll flattening and strain hardening effects have been investigated. According to the results: – –

– –

–

–

–

The critical speed obtained from the linear model is less than the critical speed of the nonlinear model. Taking into account the work roll flattening effect, the critical speed calculated by the simulation program is increased. Also, adding the strain hardening effect to the two-stand model increases critical speed. However, the work roll flattening, more than the strain hardening, influences the critical speed. Under the conditions of this research, the results of the Coulomb friction model, compared with the Tresca model, are closer to the experimental results. Considering the work roll flattening effect, the strain hardening effect, and the lubrication model in the friction model reduces the error in the calculation of the rolling force and rolling torque. From the all four viewpoints of the rolling force, the rolling torque, the dominant frequency, and, particularly, the critical speed, the unsteady lubrication model is much more precise than the simple friction models. In the calculation of the rolling force and the rolling torque, simple friction models have sufficient precision as long as the strain hardening effect and, particularly, the work roll flattening effect are considered. However, the key and fundamental point of the research is that unsteady lubrication model can simulate the friction conditions of the chatter better than simple friction models. Under the reported operating conditions, the rolling lubricant viscosity and the viscosity pressure coefficient do not have important effect on the chatter critical speed and the rolling force, but the limiting shear stress is directly proportional to them.

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

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