A study of nonlinear coupling dynamic characteristics ... - SAGE Journals

Research Article

A study of nonlinear coupling dynamic characteristics of the cold rolling mill system under different rolling parameters

Advances in Mechanical Engineering 2017, Vol. 9(7) 1–15 Ó The Author(s) 2017 DOI: 10.1177/1687814017713706 journals.sagepub.com/home/ade

Hai Xu, Ling-Li Cui and De-Guang Shang

Abstract The dynamic characteristics of the mill and the drive system are mutually coupled and affected closed-loop system. However, most research has considered only the vibration of the drive system or the vibration of the mill to determine the cause of the accident in the equipment condition monitoring and fault diagnosis process. Condition monitoring and fault diagnosis based on this type of approach can lead to misdiagnosis or missed diagnosis in determining faults in actual systems. So, in this study, a dynamic model of the coupling between a mill and its drive system was developed to study the interaction of the mill and the drive system with the goal of increasing the accuracy of diagnostic methods and to improve the quality of the rolled material. A nonlinear coupling dynamic model was formulated to represent the relation between the gearbox vibration amplitude and various time-varying parameters to study the effects of various parameters on the drive system vibration characteristic under unsteady lubrication. Simulations results showed that increasing the strip speed, the input strip thickness, or the output strip thickness or decreasing the lubricating oil temperature or the roller radius caused the vibration amplitude of the drive system to increase. The vibration frequency caused by variations in the strip inlet or outlet thickness can be transmitted to the drive system, and gear meshing frequency of the gearbox can be transmitted to the mill. Test data from an actual cold rolling mill verified the accuracy of the model. The model was shown to be capable of simulating the mutually coupled and affected mechanism between a mill and its drive system. Keywords Nonlinear coupling dynamic characteristics, cold rolling mill system, different rolling parameters, fault diagnosis

Date received: 24 January 2017; accepted: 11 May 2017 Academic Editor: Kai Bao

Introduction With advances in the industry, the requirements for cold rolling mills in terms of strip yield and strip surface quality have become more stringent. In cold rolling mills, increasing the speed increases the yield, but the vibrations must be reduced to improve the quality of the product. High speeds increase vibrations and thus the probability of failure, which decreases the profitability of a mill. For example, in a cold rolling mill operated by Wuhan Iron and Steel Corporation (WISCO), a roller locked fault in the fourth stand

caused a bolt that connects the upper and lower sections of the gearbox housing to break. The mill had to be shut down for 48 h, resulting in a loss of tens of millions of RMB and may have caused additional,

College of Mechanical Engineering and Applied Electronics Technology, Beijing University of Technology, Beijing, P.R. China Corresponding author: Hai Xu, College of Mechanical Engineering and Applied Electronics Technology, Beijing University of Technology, Beijing 100124, P.R. China. Email: [email protected]

Creative Commons CC-BY: This article is distributed under the terms of the Creative Commons Attribution 4.0 License (http://www.creativecommons.org/licenses/by/4.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/ open-access-at-sage).

2 undetected problems. Therefore, methods for reducing the failure rate and improving product quality and production efficiency are important to the industry. One effective method to reduce equipment failure rates is through fault detection and diagnostics, namely, detecting anomalies and accurately determining the source of the problem at the early stages of a failure, taking effective interventions to prevent severe damage. However, the mill and its drive system are mutually coupled and affected closed-loop system. Thus, if the mill or the drive system develops a problem, it is very difficult to determine the source of the problem. Because the coupling mechanism between the two is not clear, making fault detection and diagnosis is difficult. Therefore, it is of great practical significance to study the nature of coupled vibration characteristic and failure mechanism in a cold rolling mill. The mechanism of rolling mill vibrations has attracted significant attention from the research community. Various studies have investigated the mechanism of drive system vibrations. CY Gao et al.1 studied torsional vibration in the main drive system of a mill with multiple clearances and simulated the dynamic response of a system with various clearances. The study also compared the effects of various types of simplified clearances on the dynamic response of the system and the torque amplification coefficient of each shaft section. In addition, the effects of the amount of clearance and the position on the system torque amplification coefficient were studied. SP Ming and LJ Zhao2 studied the nonlinear dynamics characteristic for torsional vibration of the main drive system with various parametric excitations. He points out that the joint angle, damping coefficients, and nonlinear stiffness have a strong effect on vibration characteristics of the main drive system. Some scholars have studied the mill’s selfexcited vibration characteristics, which consider the effects of different lubrication conditions and rolling parameters on the self-vibration and strip quality.3–5 XK Wang and H Lin6 proposed an on-line measurement method for various guide modes of vertical ring rolling mill. YP Ding et al.7 studied the effect of rolling speed on the mechanical properties and quality of strip. Y Kimura and N Fujita8 studied the effect of lubrication on the tandem cold rolling mill. And a method is proposed to reduce the vibration of rolling mill by controlling the lubrication condition in high-speed rolling. Partial film lubrication characteristics of inlet zone in cold strip rolling have been studied by K Fu et al.9 He simulated the influence of surface topography on the inlet film thickness and inlet zone length. The effect of the film thickness and surface roughness on the loads has been studied by K Dick and JG Lenard.10 He pointed out that the roll roughness plays an integral role affecting the dependent rolling parameters during cold rolling of steel strips. As expected, an increased

Advances in Mechanical Engineering roughness leads to an increase in the rolling force on the rolling mill. An accurate dynamic model for the rolling process had been developed by PH Hu and KF Ehmann11 based on the understanding of the unsteady lubrication mechanism. He studied the effect of the strip exit velocity on the rolling mill. N Fujita et al.12 pointed out that the stability of the rolling mill decreases with the increase in the rolling speed, and through control, the friction coefficient can reduce the self-excited vibration. Compared with the model proposed by Hu, QY Wang and ZY Jiang13 proposed a multi-factor coupling dynamic model of rolling mill. He studied the influence of main process parameters (the input or output strip thickness and the roller radius) on the critical speed and the amplitude of the vertical self-excited vibration of rolling mill. Previous studies have focused on the vibration characteristic of only the drive system or the mill. Actually, the mill’s self-excited vibrations can affect not only the mill but also the drive system, and the vibration characteristic of the drive system can also affect the vibration characteristic of the mill. The vibration frequency from gear meshing in the drive system can affect the strip thickness at the entry or the exit or the oil temperature. The vibration frequency changes in the inlet or outlet strip thickness or the oil temperature can also affect the drive system and generate corresponding vibration frequency or sub-resonant frequency or frequency doubling in the gearbox. Therefore, developing a model of the coupling between a rolling mill and its drive system is very important. Such a model would not only allow the source of a fault in a cold rolling mill to be determined more accurately but also reduce misdiagnosis or missed diagnosis in fault detection that result from ignoring certain factors. Furthermore, the vibration of the mill can be determined indirectly, which avoids the problem of unreliable and discontinuous signals in the measurement of vibrations. This problem occurs because vibration sensors must be moved whenever the rollers are replaced, which occurs frequently. Therefore, the results of this study can be used to improve the effectiveness of fault diagnoses.

Coupling mathematical model of the mill To study coupled vibrations in a cold rolling mill, a dynamic model of a 2800 cold rolling mill operated by WISCO was referenced. A simplified model of the mill, which includes a motor, a gearbox, and rollers, is shown in Figure 1.

Model of the drive system The drive system is composed of several components with inertia including the motor, the gearbox, and a coupler. The drive system transmits torque from the

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Figure 1. 14-DOF model of a cold rolling mill.

motor to the rollers and thus vibration characteristics in the drive system directly affect the surface quality of the rolled product. Therefore, this article presents a model of 12-degreeof-freedom (DOF) lumped parameters drive system model. The following mathematical model of the drive system was used: assumed that the intermediate shaft is rigid shaft ðu2 ¼ u3 Þ m1€x1 + kx1 x1 + cx1 x_ 1 ¼ ½kt12 ðRb1 u1 Rb2 u2 y1 + y2 Þ + ct12 ðRb1 u_ 1 Rb2 u_ 2 y_ 1 + y_ 2 Þtanb1 m2€x2 + kx2 x2 + cx2 x_ 2 ¼ m3€x3 + kx3 x3 + cx3 x_ 3

I1 €u1 kp ðum u1 Þ cp ðu_ m u_ 1 Þ ¼ ðkt12 ðRb1 u1 Rb2 u2 y1 + y2 Þ + ct12 ðRb1 u_ 1 Rb2 u_ 2 y_ 1 + y_ 2 ÞÞ I2 €u2 kg ðu2 u3 Þ cg ðu_ 2 u_ 3 Þ ¼ ðkt12 ðRb1 u1 Rb2 u2 y1 + y2 Þ + ct12 ðRb1 u_ 1 Rb2 u_ 2 y_ 1 + y_ 2 ÞÞ

ð8Þ

ð9Þ

I3 €u3 kg ðu3 ub Þ cg ðu_ 3 u_ b Þ ð10Þ

ð1Þ

¼ ðkt34 ðRb3 u3 Rb4 u4 y3 + y4 Þ + ct34 ðRb3 u_ 3 Rb4 u_ 4 y_ 3 + y_ 4 ÞÞ

ð2Þ

Im €um + kp ðum u1 Þ + cp ðu_ m u_ 1 Þ ¼ M1

ð11Þ

m4€x4 + kx4 x4 + cx4 x_ 4 ¼ ½kt34 ðRb3 u3 Rb4 u4 y3 + y4 Þ + ct34 ðRb3 u_ 3 Rb4 u_ 4 y_ 3 + y_ 4 Þtanb3 ð3Þ m1€y1 + ky1 y1 + cy1 y1 ¼ ½kt12 ðRb1 u1 Rb2 u2 y1 + y2 Þ + ct12 ðRb1 u_ 1 Rb2 u_ 2 y_ 1 + y_ 2 Þðtan an1 =cos b1 Þ ð4Þ m2€y2 + ky2 y2 + cy2 y2 ¼ ½kt12 ðRb1 u1 Rb2 u2 y1 + y2 Þ + ct12 ðRb1 u_ 1 Rb2 u_ 2 y_ 1 + y_ 2 Þðtan an1 =cos b1 Þ ð5Þ m3€y3 + ky3 y3 + cy3 y3 ¼ ½kt34 ðRb3 u3 Rb4 u4 y3 + y4 Þ + ct34 ðRb3 u_ 3 Rb4 u_ 4 y_ 3 + y_ 4 Þðtan an3 =cos b3 Þ ð6Þ m4€y4 + ky4 y4 + cy4 y4 ¼ ½kt34 ðRb3 u3 Rb4 u4 y3 + y4 Þ + ct34 ðRb3 u_ 3 Rb4 u_ 4 y_ 3 + y_ 4 Þðtan an3 =cos b3 Þ ð7Þ

I3 €u3 kg ðu3 ub Þ cg ðu_ 3 u_ b Þ ¼ M2

ð12Þ

where Kt and Ct are the gear meshing stiffness and meshing damping coefficient, respectively. Kx1 and Cx1 are the bearing support stiffness and damping coefficient along the X-axis of input shaft, while Ky1 and Cy1 are the bearing support stiffness and damping coefficient along the Y-axis of input shaft, respectively. Kx3 and Cx3 are the bearing support stiffness and damping coefficient along the X-axis of output shaft, while Ky3 and Cy3 are the bearing support stiffness and damping coefficient along the Y-axis of output shaft, respectively. Kg and Cg represent the torsional rigidity and damping coefficient of output shaft, respectively. m1 is the mass of big gear of the first-stage gear system. m2 is the mass of pinion of the first-stage gear system. m3 is the mass of big gear of the second-stage gear system. m4 is the mass of pinion of the second-stage gear system. I1 is mass moment inertia of the gear of the first-stage gear system. I2 is mass moment inertia of the pinion of the first-stage gear system. I3 is mass moment inertia of the

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Table 1. Parameter values in the transmission system.

~xð2

Items

Amount

Mass of big gear of the first-stage gear system (kg) Mass of pinion of the first-stage gear system (kg) Mass of big gear of the second-stage gear system (kg) Mass of pinion of the second-stage gear system (kg) Mass moment inertia of the gear of the first-stage gear system (kg m2) Mass moment inertia of the pinion of the first-stage gear system (kg m2) Mass moment inertia of the motor (kg m2) Pressure angle of the first-stage gear (°) pressure angle of the second-stage gear (°) Helix angle of the first-stage gear (°) Helix angle of the second-stage gear (°) Mass moment inertia of the output load of the gearbox (kg m2) Mass moment inertia of the roller (kg m2)

8400 6000 7200

1103.16 421.88 279 20 20 14 10 298.6 913

The dynamic rolling force in a mill is affected by multiple factors such as the thickness of the strip at the entry and the exit, the entry speed, the roller speed, the lubricating oil viscosity, the temperature of the emulsion (lubricant), and the roller radius. A mathematical model was developed to study the relation between various mill parameters and the rolling force, that is, to determine the changes in the rolling force caused by changes in the values of the various mill parameters. To simplify the calculations, the parameters were dimensionless in the equations relating each parameter to a rolling force variation coefficient. The rolling force variation coefficient is multiplied by the average rolling force to obtain the time-varying rolling force. The relations between each of the rolling parameters and the rolling force were studied. The derivation was as follows f ¼ a0 b0 v + c0 v2

Pd;

~xP ¼

1 ln f gs

ð14Þ

~x1 gs

f ¼ ~y 8 9 ð~x < m0 v0 gx10 gs1 ½ðu2 =vÞ~y1 ð1 RÞ ~yV = gs gsð1SÞ ~y ~y e 2 d~x : 1 ; ~yH0 hm y10 ~x1

Model of rolling torque and rolling force in the mill with different rolling parameters

Ftv ¼ Ff ;

j¼

7200

gear of the second-stage gear system. Im is mass moment inertia of the motor. an1 is the pressure angle of the first-stage gear. an3 is the pressure angle of the secondstage gear. b1 is the helix angle of the first-stage gear. b3 is the helix angle of the second-stage gear. The values of these parameters are shown in Table 1.

Mtv ¼ Ftv R;

F ¼ jFmean ;

ð13Þ

where Mtv is the time-varying torque, Ftv is the timevarying rolling force, f is the friction coefficient of the roller, and v is the rolling speed. The rolling force can be expressed as

ð15Þ where Fmean is the mean rolling force; j is a varying coefficient of the rolling force; S ¼ s=s is the dimensionless tension; ~y ¼ y=y10 is the dimensionless strip thickness in the rolling zone; y10 is the average thickness of the strip; hm is the average thickness of the oil film; V ¼ v=v0 is the dimensionless rotational speed of the roller; v0 is the average rotation speed of the roller; R is the dimensionless percent reduction; u2 is the strip outlet speed; u1 is the average surface speed; g is the pressure–viscosity coefficient; s is the yield stress of the material; y1 is the inlet strip thickness; y is the strip thickness at an arbitrary point; y2 is the outlet strip thickness; x1 and x2 are the dimensionless starting and ending positions of the rolling zone, respectively; x10 is the dimensionless mean length of the rolling zone; m0 is the viscosity at atmospheric pressure; and H is the dimensionless oil film thickness. Variations in the oil film thickness in the deformation zone play an important role in the change of rolling force. Thus, variations in the oil film thickness were modeled, and the effect of various rolling process parameters such as the rolling speed and the lubricant viscosity and concentration on the oil film thickness were investigated. The oil film thickness can be described with the relation shown in equation (16) u2 h0 ð1 egðssÞ Þ _ R + u2 h0 uC h_ 0 ¼ uu1 6gm0 CR ¼

ð16Þ

a2 uð2au2 h0 Þ a2 ðau2 + h0 Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi + 2 2 h0 ðau 2h0 Þ ðau2 2h0 Þ aðau2 2h0 Þ 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 au aðau2 2h0 Þ 7 6 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5 q ð17Þ ln4 au + aðau2 2h0 Þ

where u is the inlet angle; u_ is the rate of change of the inlet angle; a is roller radius; m0 is the viscosity at atmospheric pressure; and u1 is the average surface speed given by u1 ¼ ðu1 + vÞ=2, where u1 is the strip speed at the inlet and v is the roller rotational speed. The parameters were dimensionless as follows H0 ¼ h0 =hm ;

U1 ¼ u1 =v0 ;

Y ¼ u=u0

where hm is the inlet oil film thickness in the steady state, v0 is the roller average rotational speed, and u1 is

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the speed of the work piece (strip) at the inlet. The following equations13 for variations in the nondimensional film thickness were obtained gðssÞ _ R1 ð18Þ 1 U 10 Y2 H0 1 e + Y2 H0 YC H_ 0 ¼ YU gðss 0Þ 1e

CR1 ¼

uð2u2 ðhm =au20 ÞH0 Þ ðhm =au20 ÞH0 ðu2 2ðhm =au20 ÞH0 Þ u + ðhm =au20 ÞH0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ð2ðhm =au20 ÞH0 u2 Þ ð1 2Hðhm =au20 Þ0 Þ 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 u 1 2ðhm =au20 ÞH0 Þ 7 6 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5 ð19Þ ln4 u + ð1 2ðhm =au20 ÞH0 Þ +

d~x=dT ¼ U

ð20Þ

3 ð∂U =∂~xÞ dH=d~x ¼ H=U 1 1 U ¼ U1 + U1 =2 1 + Rð~x2 1Þ

ð21Þ ð22Þ

where the initial conditions for equation (20)13 are ~ ¼ H0 ¼ 1. ~x ¼ ~x1 ¼ 1 and H The preceding equations can be combined to derive an expression for the dynamic rolling force as a function of the time-varying parameters.

Model of mill and drive system coupling The mill is connected to the drive system via a universal coupler, which the 2-DOF lumped parameters model between the mill and the drive system is shown in Figure 2. The rolling force and the torque are generated by the mill while in the process of rolling strip and transmitted to the gearbox by the coupler, thereby imparting a counter torque at the output of the gearbox. The torque at the output of the gearbox drives the mill. Therefore, the mill and the drive system are coupled, and equations (23) and (24) express the coupling relation ub + Cðu_ b u_ q Þ + K0 ðub uq Þ + M12 ¼ M2 Ib €

ð23Þ

uq + Cðu_ b u_ q Þ + K0 ðub uq Þ + Mtv ¼ M12 Iq €

ð24Þ

where Ib is mass moment inertia of the output load of the gearbox, and Iq is mass moment inertia of the roller. The coupler has been considered to have nonlinear stiffness K0 , which is assumed to have a value of 9.5408 3 106 (N m)/rad, and dissipative damping C, which is assumed to have a value of 339 N/(m s). Equations (23) and (24) represent the coupling between the mill and the drive system and were used to study the coupled vibration characteristic. The rolling force and torque, which vary with time, are transmitted to the output of the drive system, and the output torque

Figure 2. 2-DOF dynamic model of the coupler between the mill and the drive system.

of the gearbox drives the mill. The dynamic response of the drive system can be obtained by solving the coupling equations.

Simulation of coupled vibrations of the mill and drive system Effect of rolling speed on drive system vibration characteristic To study the effect of the rolling speed on the drive system vibration characteristic using the coupling model, equations (13)–(15) were used to solve for the torque at the output of the gearbox. Equations (1)–(12), (23), and (24) were used to derive the amplitude of the gearbox vibration for different dimensionless rolling speeds. To simplify the calculations, the rolling speed was dimensionless as shown in equation (25) V ¼ v=v0

ð25Þ

where v0 is the average rolling speed, v is the timevarying rolling speed, and V is the dimensionless timevarying rolling speed. The simulated gearbox vibration amplitudes for various rolling speeds are shown in Figure 3, where the abscissa is the dimensionless rolling speed and the ordinate is the vibration amplitudes of the gearbox. Figure 3 shows that the dimensionless rolling speed had a significant effect on the vibrations of the gearbox. At low speeds (i.e. less than 0.8), the amplitude of the gearbox vibration remained nearly unchanged, but for dimensionless rolling speeds greater than 1, the amplitude increased dramatically with the rolling speed. Therefore, the greater the rolling speed, the more intense the gearbox vibration and the greater the amplitude. The noise also increased with the speed. Therefore, an appropriate rolling speed can be selected based on the amplitude of the gearbox vibration.

Effect of inlet strip thickness on drive system vibration characteristic Changes in the inlet strip thickness cause changes in the rolling force of the mill and thus affect the vibration

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Advances in Mechanical Engineering factors. The variation in the thickness of the strip at the entry was modeled using a cosine function with a frequency of 60.99 Hz, as shown in equation (26) y1 ¼ y0 + yi1 cosð2p 3 60:99 3 t + f1 Þ

Figures 3. Gearbox vibration amplitude under different rolling speeds.

ð26Þ

where y0 is the average inlet strip thickness, yi1 is the magnitude of the variation in the inlet strip thickness, and f1 is the initial phase angle. At some moment, the vibration spectrum diagram of the gearbox under the time-varying inlet strip thickness is shown in Figure 5. Figure 4 shows the effect of inlet strip thickness on vibration characteristic of the gearbox. It can be seen that the gearbox vibration amplitude increased with increasing inlet strip thickness. Equations (12) and (13) indicate that the magnitude of the rolling force variation increased with increasing inlet strip thickness. The increase in rolling force fluctuation amplitude causes an increase in the reverse excitation at the gearbox output, resulting in an increase in the gearbox vibration amplitude. Figure 5 shows that the vibration frequency (60.99 Hz) caused by variations in the strip inlet thickness was transmitted to the drive system and caused a corresponding vibration frequency in the gearbox.

Effect of outlet strip thickness on drive system vibration characteristic

Figure 4. Gearbox vibration amplitude under different inlet strip thicknesses.

characteristic of the gearbox. The vibration characteristic of the gearbox with various inlet strip thicknesses was studied with all other parameter values held constant. An expression for the rolling force as a function of the inlet strip thickness can be derived using equations (14) and (15), and an expression for the gearbox output torque can be derived using equation (13). At some moment, the gearbox vibration amplitudes for various inlet strip thicknesses and the vibration spectrum can be derived using equations (1)–(12), (23), and (24). The relation between the inlet strip thickness and the gearbox vibration amplitude obtained from simulations is shown in Figure 4. The inlet strip can vary in thickness because of surface roughness and other

Changes in the strip thickness at the outlet lead to changes in the mill rolling force, and changes in the rolling force are transmitted to the gearbox by the coupler and thus have a major influence on the vibration and the fatigue of the gearbox. Equations (14) and (15) were used to derive an expression for the rolling force as a function of the outlet strip thickness. The gearbox output torque was calculated using equation (13). The gearbox vibration amplitudes for various outlet strip thicknesses and the gearbox vibration spectrum were derived using equations (1)–(12), (23), and (24). The relation between the outlet strip thickness and the gearbox vibration amplitude obtained from simulations is shown in Figure 6. There are variations in the outlet strip such as changes in the thickness and surface roughness. The variations in the strip thickness at the outlet were modeled using a cosine function with a frequency of 62.45 Hz, as shown in equation (27) y2 ¼ ym + yi2 cosð2p 3 62:45 3 t + f2 Þ

ð27Þ

where ym is the average outlet strip thickness, yi2 is the magnitude of the variations in the strip outlet thickness, and f2 is the initial phase angle. At some moment, the vibration spectrum diagram of the gearbox under the time-varying outlet strip thickness is shown in Figure 7. Figure 6 shows the effect of outlet strip thickness on vibration characteristic of the gearbox. It can be seen that the gearbox vibration amplitude decreased with

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Figure 5. Response diagram of gearbox with a time-varying inlet strip thickness: (a) time-domain diagram and (b) frequency-domain diagram.

consequently cause changes in the rolling force. The relation between the temperature and the viscosity of the emulsion is given by equation (28).14 Where we usually take the value as h40 ¼ 120 cSt and h100 ¼ 15:9 cSt14 logðlogm0 + 1:2Þ log h100 + 1:2 tm ¼ 0:128 log log 1 + 135 log h40 + 1:2 S0 ! 175 ð28Þ + log ðlogh40 + 1:2Þ 135

Figure 6. Gearbox vibration amplitude under different outlet strip thicknesses.

increasing outlet strip thickness. Equations (10) and (11) indicate that the magnitude of the changes in the rolling force decreased as the strip outlet thickness decreased. Smaller variations in the rolling force result in less excitation at the gearbox output, resulting in smaller gearbox vibration amplitudes. Figure 7 indicates that the vibration frequency (62.45 Hz) caused by variations in the strip outlet thickness can affect the fluctuation characteristic of gearbox and caused a corresponding vibration frequency in the gearbox.

Effect of emulsion temperature on drive system vibration characteristic Changes in the emulsion temperature during operation can cause changes in the viscosity of the emulsion and

The relation between the emulsion temperature and emulsion viscosity can be obtained using equation (28). Through the joint solution of equations (18)–(22), the relation between the emulsion viscosity and the dimensionless film thickness can be obtained. The dimensionless oil film thickness obtained is used in equations (14) and (15) to derive a relation between the dimensionless film thickness and the rolling force. So, the relation between the emulsion temperature and the rolling force can be obtained. The rolling force obtained is used in equations (13) to derive a relation between the rolling force and the gearbox output torque. At some moment, equations (1)–(13), (23), and (24) were combined to solve the gearbox vibration amplitude under different emulsion temperatures and to obtain the vibration spectrum of the gearbox. The results of simulations are shown in Figure 8. Variations in the strip surface roughness, the rolling force, and other factors affect the emulsion temperature. The emulsion temperature was modeled using a cosine function with a frequency of 65 Hz. The maximum amplitude of the temperature


Vibration amplitude (m/s2)

Vibration amplitude (m/s2)

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Figure 7. Response diagram of gearbox with a time-varying outlet strip thickness: (a) time-domain diagram and (b) frequencydomain diagram.

increasing oil temperature. The reason is that an increase in the oil temperature causes the lubricant viscosity to decrease. And the decrease in lubricant viscosity causes the film thickness to decrease, resulting in smaller amplitude in the rolling force and thus a reduced excitation force in the tangential direction. These effects reduce the torque excitation acting on the gearbox and thus the gearbox vibration amplitude. Figure 9 indicates that the vibration frequency (65 Hz) caused by variations in oil temperature was transmitted by the coupler to the drive system and caused a corresponding vibration frequency in the gearbox.

Effect of roller radius on drive system vibration characteristic

Figure 8. Gearbox vibration amplitude versus emulsion temperature.

changes was 10°C. The temperature model is given by equation (29) tm ¼ t0 + tA cosð2p 3 65 3 t + fm Þ

ð29Þ

where t0 is the average emulsion temperature, which was assumed to have a value of 55°C; tA is the amplitude of the emulsion temperature variation (10°C); and fm is the initial phase angle. At some moment, the vibration spectrum diagram of the gearbox under the time-varying emulsion temperature is shown in Figure 9. Figure 8 shows the effect of oil temperature on vibration characteristic of the gearbox. It can be seen that the gearbox vibration amplitude decreased with

Using equations (18)–(21), a relation between the dimensionless roller radius and the dimensionless oil film thickness was derived. The resulting expression for the oil film thickness was used in equations (14) and (15) to derive a relation between the roller radius and the rolling force. An expression for the gearbox output torque as a function of the roller radius was derived by substituting equation (14) into equation (13). Equations (1)–(13), (23), and (24) were combined to solve for the gearbox vibration amplitude as a function of the roller radius and to obtain the vibration spectrum of the gearbox. The relation between the roller radius and the gearbox vibration amplitude obtained from simulations is shown in Figure 10. The rollers are not perfectly round, so variations in the radius were modeled. The variations in the roller radius were modeled using a cosine function with a frequency of 60 Hz, as shown in equation (30)

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Figure 9. Response diagram of gearbox with a time-varying emulsion temperature: (a) time-domain diagram and (b) frequencydomain diagram.

gearbox and thus the gearbox vibration amplitude. Figure 11 indicates that the vibration frequency (60 Hz) caused by variations in roller radius was transmitted by the coupler to the drive system and caused a corresponding vibration frequency in the gearbox.

Field verification of coupled vibration characteristic under various operating conditions

Figure 10. Gearbox vibration amplitude under different nondimensional roller radii.

a ¼ a0 + aA cosð2p 3 60 3 t + fa Þ

ð30Þ

where a0 is the average roller radius, which was assumed to have a value of 0.29; aA is the amplitude of the variations in the roller radius, which was assumed to have a value of 0.058; and fa is the initial phase angle. At some moment, the vibration spectrum diagram of the gearbox under the time-varying roller radius is shown in Figure 11. Figure 10 shows that the gearbox vibration amplitude decreased with increasing dimensionless roller radius. The reason is that an increase in the roller radius causes the inlet oil film thickness to decrease, resulting in smaller amplitude in the rolling force and thus a reduced excitation force in the tangential direction. These effects reduce the torque excitation acting on the

Vibration signals were taken from one of the cold rolling mills in the #2 plant of WISCO. Acceleration sensors were placed on the bearing at the output of the gearbox, as shown in Figures 12 and 13. The mill stand was monitored using Siemens online monitoring system, which measured the strip inlet thickness, the emulsion temperature, the rolling force, and the rolling speed, as shown in Figures 12 and 14. The drive system was monitored using a two-channel Emerson offline monitoring system with a sampling frequency of 2000 Hz and a sampling time of 1 s.

Field verification of the effect of roller speed on the vibration characteristic of drive system The on-line monitoring data of rolling mill and out-line monitoring data of the gearbox bearing were analyzed. With the other parameters held constant, the vibration amplitudes at the gearbox bearing for various rolling speeds were obtained. The actual values are shown in Figure 15. From comparison between Figures 3 and 15, it can be seen that the simulation results of the coupled model were consistent with the field measurement. For a

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Figure 11. Response diagram of gearbox with time-varying roller radii: (a) time-domain diagram and (b) frequency-domain diagram.

Figure 12. Diagram of the high-speed cold rolling mill.

dimensionless rolling speed greater than 0.5, the measured vibration amplitudes were slightly greater than the simulated values. The difference may be caused by the various field conditions: for rolling speeds greater than a certain level, various vibration sources are superimposed, resulting in slightly higher measured vibration amplitudes. However, the measurements verified the correctness and the validity of the coupling model.

Field verification of the effect of inlet strip thickness on the vibration characteristic of drive system Vibration sensors were placed on the bearing at the output of the gearbox. The parameters were held constant using the mill control interface, and the vibration characteristics of the gearbox were measured for various inlet strip thicknesses. The measured values are shown in Figure 16. Figures 17 and 18 show the vibration

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Figure 15. Gearbox bearing vibration amplitude under different non-dimensional rolling speeds. Figure 13. Location of sensors at the gearbox output.

Figure 14. Parameter monitoring and control interface for the mill.

spectrum diagrams of gearbox bearing in the vertical and axial directions, respectively, and Figure 19 shows the variation in the inlet strip thickness. From comparison between Figures 4 and 16, it can be seen that the simulation results were consistent with the field test data, demonstrating the accuracy of the model of gearbox vibrations as a function of the inlet strip thickness. In the diagrams of the vibration signals shown in Figures 17 and 18, the vibration at 60.99 Hz can be observed in the vertical direction of the gearbox but not in the axial direction. The excitation frequency

Figure 16. Gearbox vibration amplitude versus inlet strip thickness.

from the variation in the inlet strip thickness was mainly at 60.99 Hz and its ½ sub-resonant harmonic. Therefore, we can conclude that these two frequencies caused by variations in the inlet strip thickness transmit to the gearbox. Because of the structure of the universal coupler, the excited vibration in the mill mainly affected the vibrations of the gearbox in the direction of vertical axis, whereas the axial vibrations were largely attenuated by the universal coupler. It can be seen from Figures 17 and 19 that the gear meshing frequency (424.8 Hz) of the gearbox was transmitted to the mill by the coupler and generated a corresponding fluctuation frequency in the spectrum diagram of the

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Figure 17. Gearbox vibration response diagram in the vertical axis under the time-varying strip inlet thickness: (a) time-domain diagram and (b) frequency-domain diagram.

Figure 18. Gearbox vibration response diagram in the axial direction for a time-varying strip inlet thickness: (a) time-domain diagram and (b) frequency-domain diagram.

Figure 19. Variation diagram in the strip inlet thickness: (a) time-domain diagram and (b) frequency-domain diagram.

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Field verification of the effect of outlet strip thickness on the vibration characteristic of drive system

Figure 20. Gearbox vibration amplitude under different strip outlet thicknesses.

inlet strip thickness. The effect of the strip inlet thickness variation frequency (60.99 Hz) on the mill was also transmitted through the coupler to the gearbox and generated a corresponding frequency in the gearbox vibration spectrum diagram. A comparison of the spectra in Figures 4 and 5 with those in Figures 16, 17, and 19, which were obtained from the field test data, confirm the accuracy of the simulation model. These analysis results demonstrate that the mill and the drive system are mutually coupled and affected closed-loop system.

Vibration sensors were placed on the bearing at the output end of the gearbox in field test. With the other parameter values held constant using the mill control interface, the vibrations of the gearbox were measured for various strip outlet thicknesses. The measured results are shown in Figure 20. The vibration spectrum diagram of the gearbox bearing in the vertical directions is shown in Figure 21. Figure 22 shows the variation spectrum diagram in the outlet strip thickness. As shown in Figure 20, the gearbox vibration amplitude decreased with increasing strip outlet thickness. Compared with Figure 6, it can be seen that the simulation results were consistent with the measured data. The measurements verified the accuracy of the derived relation between the strip outlet thickness and the gearbox vibration amplitude. From Figures 21 and 22, it can be seen that the gear meshing frequency in the gearbox (425.3 Hz) was transmitted to the mill by the coupler and generated the corresponding frequency in the frequency spectrum of the strip outlet thickness. The strip outlet thickness variation frequency (62.45 Hz) of the mill was also transmitted through the coupler to the gearbox and generated a corresponding variation frequency in the frequency spectrum of the gearbox. A comparison of the spectra in Figures 20–22, which were obtained from the measurements, with those in Figures 6 and 7 verified the correctness of the simulation model, indicating the important practical significance of the coupling model.

Figure 21. Gearbox vibration response diagram in the vertical direction under the time-varying strip outlet thickness: (a) timedomain diagram, (b) frequency-domain diagram.

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Figure 22. Variation diagram in the strip outlet thickness: (a) time-domain diagram and (b) frequency-domain diagram.

Conclusion

Declaration of conflicting interests

A quantitative analysis was conducted on the effects of changes in various parameters of the rolling process in a cold rolling mill on the vibration of the drive system. Both the simulation model and measurements taken from an actual mill showed that increasing the rolling speed or the inlet strip thickness or decreasing the outlet strip thickness, the lubrication oil temperature, or the roller radius caused the vibration amplitude of the gearbox in the drive system to gradually increase. The vibration frequency caused by variations in various parameters of the mill was transmitted to the drive system through the universal coupler and generated a corresponding vibration frequency at the drive system. The vibration frequency from gear meshing in the drive system was transmitted to the mill and caused variations frequency in various parameters. The measured data confirmed that the mill and the drive system are mutually coupled and affected closed-loop system. The analysis results verified the correctness and the usefulness of the coupling model. The coupling model provides a theoretical basis for further study. This coupling model has very important practical significance. The model not only allows more accurate determinations of the source of a fault in a mill system, thus avoiding a misdiagnosis or a missed fault, but also indirectly detects vibration characteristic of the mill. Therefore, unreliable and discontinuous vibration signals in actual vibration measurements, which are caused by the necessity to move the vibration sensors each time the rollers are replaced, an event that occurs frequently, are avoided. Hence, the fault diagnosis is more effective.

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Natural Science Foundation of China (grant nos 51575007 and 51375019).

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Appendix 1 Notation a cx1 cx2 cy1 cy1 Cp f F Fmean h0 hm H0 I1 I2 Ib Im Iq kx1 kx2 ky1 ky1

roller radius support damping coefficient along the 1Xaxis support damping coefficient along the 2Xaxis support damping coefficient along the 1Yaxis support damping coefficient along the 2Yaxis damping coefficient of the shaft friction coefficient of the roller dynamic rolling force average rolling force inlet film thickness average inlet film thickness dimensionless inlet film thickness mass moment inertia of the gear of the first-stage gear system mass moment inertia of the pinion of the first-stage gear system mass moment of inertia of output load of the gearbox mass moment of inertia of the motor mass moment of inertia of the roller of the rolling mill support stiffness along the 1X-axis support stiffness along the 2X-axis support stiffness along the 1Y-axis support stiffness along the 2Y-axis

K0 Kg Kp m1 m2 m3 m4 M1 M2 M12 Mtv P R Rbi S u1 u2 v0 V ~x1 ~x2 y1 y10 ~y ~y1 u u_ u0 u1 u2 u3 u4 ub um uq g m0 j s

torsional stiffness of the connecting shaft torsional stiffness of input shaft of the gearbox torsional stiffness of output shaft of the gearbox mass of big gear of the first-stage gear system mass of pinion of the first-stage gear system mass of bid gear of the second-stage gear system mass of pinion gear of the second-stage gear system torque force acting on the driving gears torque force acting on the driven gears torque of joint shaft acting on the roller time-varying torque of the roller acting on the joint shaft dimensionless compressive stress dimensionless reduction rate base circle radius of the gear i dimensionless back tension inlet speed of the strip outlet speed of the strip mean speed of roller dimensionless roller speed starting position of the dimensionless working area ending position of the dimensionless working area inlet thickness of the strip average thickness of the strip dimensionless strip thickness in work zone dimensionless inlet thickness of the strip inlet angle of the roller variation rate of the inlet angle of the roller mean inlet angle of the roller angular displacement of pinion of the first-stage gear system angular displacement of gear of the firststage gear system angular displacement of gear of the second-stage gear system angular displacement of gear of the second-stage gear system angular displacement of the connecting shaft angular displacement of the of the motor angular displacement of the roller pressure coefficient of viscosity viscosity at atmospheric pressure rolling force fluctuation coefficient material yield stress