Issue 
Metall. Res. Technol.
Volume 117, Number 2, 2020



Article Number  210  
Number of page(s)  12  
DOI  https://doi.org/10.1051/metal/2020020  
Published online  24 April 2020 
Regular Article
Nonlinear timedelay feedback controllability for vertical parametrically excited vibration of roll system in corrugated rolling mill
College of Mechanical and Vehicle Engineering, Taiyuan University of Technology,
Taiyuan
030024, PR China
^{*} email: tyutwt@163.com
Received:
22
September
2019
Accepted:
16
March
2020
This paper investigates vibration characteristics of the corrugated roll system and designs a timedelay feedback controller to control the parametrically excited vibration of system. The model of parametrically excited nonlinear vertical vibration of roller system is established by considering the nonlinear damping and nonlinear stiffness within corrugated interface of corrugated rolling mill. The approximate analytical solution and amplitudefrequency characteristic equations of principal resonance and subresonance of roller system are obtained by using the multiplescale method. The influences of nonlinear stiffness coefficient, nonlinear damping coefficient, system damping coefficient and rolling force amplitude on vibration are further analyzed. The timedelay feedback controller is designed to eliminate the jump and hysteresis phenomenon of the roll system and numerical simulation results demonstrate the effectiveness of the controller. The analysis results provide some theoretical guidance for vibration suppression of roller system of corrugated rolling mill.
Key words: corrugated rolling mill / parametrically excited vibration / principal resonance / subresonance / timedelay feedback control
© EDP Sciences, 2020
1 Introduction
The rolling mill vibration phenomenon is common in actual production, which not only adversely affects the surface roughness and rolling precision, but also causes serious damage to equipment [1–3]. Traditional composite plate rolling technology is that flat rolls are usually chosen as work roll to realize the horizontal and vertical extension of two kinds of metal. However, flat rolls easily causes large curvature warp and can’t realize continuous production in the actual production as the two kinds of metal have different rates of elongation, as is shown in Figure 1. The composite sheet corrugated roll forming technology (as is shown in Fig. 2) can promote the better metallurgical bonding of dissimilar materials, improve the bonding strength and achieve continuous production. The experimental results of Cu/Al composite plate prepared by corrugated roller show that the grains refinement at the interface is obvious. Grains refinement degree on the aluminum side is obviously higher than the copper side. The interface bonding result is good, meanwhile, there are dimples at the interface during the tensile process at the condition of equivalent reduction rate [4].
During the rolling process of composite plate, the nonlinear damping and the nonlinear stiffness between corrugated roll and flat roll may cause parametrically excited vibration, which affects the production quality of the composite plates and reduces the online service time of rolls. Over the past several decades, some researchers have theoretically investigated the vertical vibration phenomenon of rolling mill. Yarita proposed a fourdegreeoffreedom symmetric model with linear elastic deformation and linear damping between the components [5]. Tamiya simplified the analysis based on the Yarita model, assuming that the mass of the working roller was much less than the mass of the supporting roller, and established a twodegree of freedom vibration system [6]. Tlusty developed a basic theory of rolling mill chatter based on the Tamiya vibration model, believing that the third octave chatter of rolling mill is a selfexcited vibration phenomenon caused by the tension phase delay of rolling mill [7]. Yun proposed third octave chatter model caused by negative damping effect, modal effect and regeneration effect on the basis of Tlusty model, and found that modal coupling may cause third octave chatter of rolling mill [8]. Hu further developed Yun model and found that the fifth octave chatter of rolling mill lagged behind the third octave chatter [9–10]. Johnson analyzed the influence of the nonlinearity of the contact interface between the work roll and the supporting roll on the rolling mill dynamics, and found that the nonlinearity could cause high frequency harmonic vibration [11]. Swiatoniowski analyzed the dynamic behavior of vertical vibration of rolling system by considering the elastoplastic deformation as nonlinear elastic force [12]. Pawelski established a predictive control vibration model of rolling system, which can predict the influence of external disturbance on adjacent rolling systems of strip mill [13]. Kimura has installed a dynamic vibration absorber on the cold rolling mill, which can counteract the vibration energy by detecting the vibration signal of the rolling mill system and producing a matching response [14]. Wehr installed a hydraulic lining between the rolling mill stand and the backup roll to reduce the vibration frequency of the rolling mill and enhance the vibration damping capacity of the rolling mill system [15]. Younes assumed that the workpiece is an elastic part with linear stiffness, and the linear vertical vibration model of rolling mill stand is established based on the linear vibration theory [16]. Panjkovi thought that the specification of steel and the surface oxidation of the roll surface are the causes of the variation of the friction coefficient of roll gap. Furthermore, it was pointed out that the change of friction environment was one of the causes of rolling mill system chatter [17]. Hou took the dynamic changes of vibration displacement of cold rolling mill roller in vertical direction into consideration, based on the BlandFordHill rolling force model, and established a dynamic rolling force model [18].
Timedelay feedback control is a control method proposed by Pyragas for chaos phenomenon [19], which does not require phase space reconstruction. The application of control does not need the system state to approach the target state, which deduces the tracking calculation during the control process. Timedelay feedback control is widely used in the control of engineering vibration such as large rotor vibration, vibration of flexible beam, nonlinear vibration of rolling mill system and so on. Yu was concerned with the effect of time delayed feedbacks in a nonlinear oscillator with external forcing and found out that the double Hopf bifurcations with two pairs of purely imaginary eigenvalues at a critical point occur in the system [20]. Maccari investigated the primary resonance of an externally excited van der Pol oscillator under state feedback control with a time delay and discussed the vibration control and high amplitude response suppression can be performed with appropriate time delay and feedback gains [21]. Li studied on time delay feed back control and vibration performance of rolling mill’s main drive coupling system and discussed the dynamical behaviors of the system under multitime delay feedback with two state variables [22].
Although so far, much work about rolling mill abnormal vibration has been done, however, there are many factors affecting mill vibration, which involve the cross fusion of vibration theory, rolling theory and fault diagnosis theory. The corrugated rolling mill is a multivariable, strongly coupled, nonlinear, multiconstrained and timevarying system. The nonlinear vibration problem of corrugated rolling mill affects the quality of the product and the online service life of components. Therefore, it is important to research vibration problem of corrugated rolling mill. To the authors’ best knowledge, no study in the literature has been focused on vibration characteristics and control of roll system of corrugated rolling mill.
The present paper is organized into five sections as follows. The model of parametrically excited nonlinear vertical vibration of roller system is established in Section 2. The approximate analytical solution and amplitudefrequency characteristic equations of principal resonance are derived by using the multiplescale method in Section 3.1. The timedelay feedback controller is designed to eliminate the jump and hysteresis phenomenon of the roll system in Section 3.2. The influences of nonlinear stiffness coefficient, nonlinear damping coefficient, system damping coefficient and rolling force amplitude on amplitudefrequency curves of subresonance and superresonance are analyzed in Sections 4.1 and 4.3. In Sections 4.2 and 4.4, the timedelay feedback controller is designed to control the parametrically excited vibration of subresonance and superresonance in the corrugated roll system. Conclusion is drawn in Section 5.
Fig. 1 Warping phenomenon of flat roll. 
Fig. 2 Rolling diagram of corrugated roll. 
2 Nonlinear vibration mathematical model of corrugated roll system
Considering the nonlinear damping and nonlinear stiffness within corrugated interface of corrugated rolling mill, the roller systems of corrugated rolling mill can be simplified into a twofreedomdegree vertical nonlinear parametrically excited vibration model, as is shown in Figure 3.
m_{1} represents equivalent mass of the corrugated roll, m_{2} represents equivalent mass of the flat roll, k_{1} is stiffness between the corrugated roll and rack, k_{2} is stiffness between the flat roll and rack, k_{0} is average value of stiffness in steady state of corrugated rolling mill, F_{0} is steady rolling force, F_{n}cosωt is dynamic rolling force, c_{1} is average damping value between the corrugated roll and rack and c_{2} is average damping value between the flat roll and rack, c_{0} is average value of damping in steady state of corrugated rolling mill. Considering the nonlinear damping and stiffness characteristics of the corrugated rolling mill, the Van der Pol vibrator c_{0 }+ c_{0}(y_{1 }− y_{2})^{2} is defined as the nonlinear damping between the corrugated roll and flat roll, which is caused by the roll shape curve, the Duffing vibrator k + k^{,}y^{2} is defined as nonlinear stiffness between roller systems and rack.
The nonlinear parametrically excited vibration equations of the roller systems are expressed as follows: (1)
Roller system has approximate symmetry, set m_{1} = m_{2}, y_{1} = −y_{2}, then, equation (1) can be simplified as follows: (2)
Let’s simplify equation (2), set:
, , , , , and .
Then, the equation (2) can be written as follows: (3)
The roll shape curve of corrugated roller is a cosine function curve. The influence of dynamic rolling force F_{n}cosωt on rolling mill vibration is much more than steady rolling force F_{0}, so the steady rolling force Fm in equation (3) can be ignored. The equation (3) can be written as: (4)
Fig. 3 Parametrically excited vibration mathematical model of corrugated roller systems. 
3 Principal resonance and timedelay feedback control of corrugated roll system
The nonlinear vibration characteristics of roll system in corrugated rolling mill may cause various resonance phenomena during rolling processes, such as internal resonance, principal resonance and double resonance. The principal resonance is first analyzed when the exciting frequency of rolling force of corrugated roller is close to the natural frequency [23,24].
In order to use the multiplescale method, assume that ϵ is a small parameter and the nonlinear term of equation (4) is given a small parameter, the equation (4) has the following form: (5)
The parametrically excited vibration characteristics of the corrugated rolling mill are analyzed by multiscale method below. Introducing time variables representing different scales T_{n} = ϵ^{n}t, n = 0, 1, 2,… the vibration response of the system is a function of time variables with different scales and has the form: (6) where time variables of different scales are treated as independent variables. Thus Y(t, ϵ) is a function of m independent time variables. Through using the derivative method of the compound function, the derivative of the function Y(t, ϵ) with respect to time can be expanded in variable ϵ as: (7) (8)where, is a partial differential operator.
3.1 Principal resonance solution of corrugated roll system
Assume that ζ is the frequency modulation parameter. When the exciting frequency of the rolling force of the corrugated roll is close to the natural frequency of the corrugated rolling mill [21], set: (9)
Substituting equations (6)–(8) into equation (5), we can obtain: (10) (11)
Assume that the solution of the zeroth approximation equation (10) is: (12) where Δ represents the complex conjugate of the former.
Substituting equation (12) into equation (11), we can obtain: (13)
In order to eliminate the secular terms, assuming that the coefficient of e^{jωT0}equals to zero, the following equation can be obtained as: (14)
Suppose that B(T_{1}) is expressed in complex form as: (15)
Substituting equation (15) into equation (14) and separating the real part and the imaginary part, we can obtain: (16) (17)
Through solving the equations (16) and (17), the firstorder approximate solution of the principal resonance is obtained as: (18)
Based on the condition that the system has a steady solution means that , , then, the amplitudefrequency characteristic curve equation of principal resonance can be obtained as: (19)
By changing the nonlinear stiffness coefficient, nonlinear damping coefficient, system damping coefficient and rolling force amplitude of the corrugated roll mill, the different amplitudefrequency characteristic curves of principal resonance can be obtained as is shown in Figure 4 (the relationship between disturbance frequency ζ and amplitude μ).
As is shown in Figure 4a, when the nonlinear stiffness coefficient γ increases, the curve shifts to the right and the bending degree increases. When the disturbance frequency ζ changes from negative to positive, the jumping phenomenon of amplitude takes place in the system, which will cause the oscillation of system. As is shown in Figures 4b and 4c, with the nonlinear damping coefficient β and system damping coefficient α increase, the amplitude and the resonance domain decreases, but the bending degree of frequency response curve is not affected. As is shown in Figure 4d, with the amplitude of rolling force F increases, the amplitude and resonance region of the system increase obviously. Based on results of the analysis, a controller should be designed to reduce the influence of rolling force amplitude on primary resonance.
Fig. 4 The amplitudefrequency characteristic curve of principal resonance with different parameters. 
3.2 Principal resonance timedelay control of corrugated roll system
In order to eliminate the jump and hysteresis phenomena of the roll system, a nonlinear timedelay feedback controller is designed to control the principal resonance response of the roll system in the corrugated rolling mill. Set the feedback controller as follows: (20)
Application of the timedelay feedback controller (20) to the system (5) leads to the controlled system: (21) where,τ_{1} and τ_{2} are timedelay parameters, g_{1} is linear feedback gain, g_{2} is nonlinear feedback gain. If g_{1} and g_{2} are zero, the system is uncontrolled.
Substituting equations (6)–(8) into equation (21), we can obtain: (22) (23)
Assume that the solution of the zeroth approximation equation (22) is: (24) whereΔ represents the complex conjugate of the former.
Substituting equation (24) into equation (23), we can obtain: (25)
Through setting the coefficient of e^{jωT0} equal to zero to eliminate the secular terms, the following equation can be obtained as: (26)
Suppose that A is expressed in complex form: (27)
Substituting equation (27) into equation (26) and separating the real part and the imaginary part, we can obtain: (28) (29)
Based on the condition of steady solution, the amplitudefrequency characteristic curve equation of principal resonance with timedelay control can be obtained as: (30) where, , , , .As is shown in Figure 5, the control of the principal resonance amplitude value and the resonance region can be achieved by appropriately adjusting timedelay parameters (τ_{1}, τ_{2}) and feedback gains (g_{1}, g_{2}), and the principal resonance bifurcation can be reduced. By comparison, it follows from that Figure 5 that the control effect of simultaneously adjusting the timedelay parameters (τ_{1}, τ_{2}) and feedback gain (g_{1}, g_{2}) is better than the control effect of separately adjusting the linear feedback gain g_{1} or the nonlinear feedback gain g_{2}.
As is shown in Figure 6a, with the linear gain g_{1} and the nonlinear gain g_{2} increase gradually, the amplitude of the system and the resonance domain gradually decrease, and the bifurcation of curve is eliminated. Thus it can be seen that the principal resonance phenomenon of the system can be reasonably controlled if the linear and nonlinear feedback control gains simultaneously increase when the feedback control gains are only taken as the control parameters. As is shown in Figure 6b, with the gradual increase of the timedelay parameter (τ_{1}andτ_{2}), the amplitude of the system gradually decreases and the resonance domain gradually decreases, but the adjustment of the timedelay parameter does not affect the bending degree of the curve. By comparison, the bifurcation phenomenon of the principal resonance amplitudefrequency curve is easily eliminated by adjusting the feedback control gains g_{1} and g_{2}, which shows that the control effect with the feedback control gain as the control object is better than the control effect with the timedelay parameter.
Fig. 5 The timedelay control amplitudefrequency curve of principal resonance. 
Fig. 6 The timedelay control amplitudefrequency curve of principal resonance with different parameters. 
4 Subresonance and timedelay feedback control of corrugated roll system
When the excitation frequency ω of rolling force is three times of the natural frequency ω_{0} of the corrugated roll, the system will undergo subharmonic vibration. When the excitation frequency ω of rolling force is one third of the natural frequency ω_{0} of the corrugated roll, the system will undergo superharmonic vibration. Superharmonic vibration and subharmonic vibration are called subresonance [25–28].
Replacing ϵFcosωt with Fcosωt in equation (5) and substituting equation (6) into equation (5) to obtain: (31) (32)
Assume that the solution of the zeroth approximation equation (31) is: (33) where and Δ represent the complex conjugate of the previous two terms.
4.1 Subharmonic vibration solution of corrugated roll system
When ω = 3ω0, the system will generate subharmonic vibration response, suppose that: (34)
Substituting equations (33) and (34) into equation (32), we can obtain: (35) where, NST represents other items that do not produce a secular term, Δ represents the complex conjugate of the previous terms.
In order to eliminate the secular terms, assuming that the coefficient of e^{jω0T0} equals to zero, the following equation can be obtained as (36)
Suppose that B is expressed in complex form: (37)
Substituting equation (37) into equation (36) and separating the real part and the imaginary part, we can obtain (38) (39)
Through solving the equations (38) and (39), the firstorder approximate solution of the principal resonance is obtained as
Based on the condition of steady solution, the amplitudefrequency characteristic curve equation of subharmonic vibration can be obtained (40)
By changing the nonlinear stiffness coefficientγ, nonlinear damping coefficient β, system damping coefficient α and rolling force amplitude F of the corrugated roll mill, the different amplitudefrequency characteristic curves of subharmonic vibration can be obtained as is shown in Figure 7 (the relationship between disturbance frequency and amplitude).
As is shown in Figure 7a, with the nonlinear stiffness coefficient γ increases, the curve shifts to the right and the bending degree increases. The disturbance frequency ζ change from negative to positive not only causes the amplitude jumps but also causes the oscillation. As is shown in Figures 7b and 7c, with the nonlinear damping coefficient β and system damping coefficient α increase, the amplitude and the resonance domain decrease. However, the bending degree of frequency response curve is not affected by the variation of coefficients β and α. As is shown in Figure 7d, with the amplitude of rolling force F increases, the amplitude and resonance region of the system increase obviously.
Fig. 7 The amplitudefrequency characteristic curve of subharmonic vibration with different parameters. 
4.2 Subharmonic vibration timedelay control of corrugated roll system
Substituting equation (34) into equation (21), we can obtain: (41) (42)
Assume that the solution of the zeroth approximation equation (41) is: (43) where and Δ represent the complex conjugate of the previous terms.
Substituting equation (43) into equation (42), we can obtain: (44)
where, NST represents other items that do not produce a secular terms.
Through eliminating the secular terms, we have: (45)
Assume that B is expressed in complex form: (46)
Substituting equation (46) into equation (45), separating the real part and the imaginary part, we can get: (47) (48)
Based on the condition of steady solution, the amplitudefrequency characteristic curve equation of subharmonic vibration with timedelay control can be obtained: (49) where, , , , γ_{s} = γ − g_{2}cosω_{0}τ_{2}, .
As is shown in Figure 8, the control of the subharmonic vibration amplitude value and the resonance region can be achieved by appropriately adjusting timedelay parameters (τ_{1}, τ_{2}) and feedback gain (g_{1}, g_{2}), and the subharmonic vibration bifurcation can be reduced. By comparison, it is found that the control effect of simultaneously adjusting the timedelay parameters (τ_{1}, τ_{2}) and feedback gain (g_{1}, g_{2}) is better than the control effect of separately adjusting the linear feedback gain g_{1} or the nonlinear feedback gain g_{2}.
As is shown in Figure 9a, with the linear gain g_{1} and the nonlinear gain g_{2} increase gradually, the amplitude of the system gradually decreases and the resonance domain gradually decreases, eliminating the curve bifurcation. When the feedback control gain is taken as the control parameter, the subharmonic vibration phenomenon of the system can be reasonably controlled if the linear and nonlinear feedback control gains were simultaneously increased. As is shown in Figure 9b, with the gradual increase of the timedelay parameter (τ_{1}andτ_{2}), the amplitude of the system gradually decreases and the resonance domain gradually decreases, but the adjustment of the timedelay parameter does not affect the bending degree of the curve. By comparison, it can be found that the control effect with the feedback control gain as the control object is better than the control effect with the timedelay parameter as the control object.
Fig. 8 The timedelay control amplitudefrequency curve of subharmonic vibration. 
Fig. 9 The timedelay control amplitudefrequency curve of subharmonic vibration with different parameters. 
4.3 Superharmonic vibration solution of corrugated roll system
When 3ω = ω0, system will generate superharmonic vibration response, set: (50)
Through using the same method to eliminate secular terms, the amplitudefrequency characteristic curve equation of superharmonic vibration can be obtained as: (51)
By changing the nonlinear stiffness coefficient γ, nonlinear damping coefficient β, system damping coefficient α and rolling force amplitude F of the corrugated roll mill, the different amplitudefrequency characteristic curves of superharmonic vibration can be obtained as is shown in Figure 10 (the relationship between disturbance frequency and amplitude).
As shown in Figure 10a, with the nonlinear stiffness coefficient γ increases, the curve shifts to the right and the bending degree increases. When the disturbance frequency ζ changes from negative to positive, the amplitude jumps with the increase of coefficient γ. As shown in Figure 10b, with the nonlinear damping coefficient β increases, the amplitude decreases and the resonance domain decreases. As shown in Figure 10c, with the system damping coefficient α increases, the amplitude decreases and the resonance domain decreases. The bending degree of frequency response curve is not affected by the variation of nonlinear damping coefficient β and system damping coefficient α. As shown in Figure 10d, with the amplitude of rolling force increases, the amplitude and resonance region of the system increase obviously.
Fig. 10 The amplitudefrequency characteristic curve of superharmonic vibration with different parameters. 
4.4 Superharmonic vibration timedelay control of corrugated roll system
Substituting equations (50) and (43) into equation (42), we can obtain: (52) where, NST represents other items that do not produce a duration item, Δ represents the conjugate complex of the previous term.
Through eliminating secular terms, the amplitudefrequency characteristic curve equation of superharmonic vibration with timedelay control can be obtained as: (53) where, , , , γ_{s} = γ − g_{2}cosω_{0}τ_{2},
As is shown in Figure 11a, with the linear gain g_{1} and the nonlinear gain g_{2} increase gradually, the amplitude of the system gradually decreases and the resonance domain gradually decreases, eliminating the curve bifurcation. When the feedback control gain is taken as the control parameter, the superharmonic vibration phenomenon of the system can be reasonably controlled if the linear and nonlinear feedback control gains were simultaneously increased. As is shown in Figure 11b, with the gradual increase of the timedelay parameter (τ_{1} and τ_{2}), the amplitude of the system gradually decreases and the resonance domain gradually decreases, but the adjustment of the timedelay parameter does not affect the bending degree of the curve. By comparison, it can be found that the control effect is obvious in superharmonic vibration control no matter with the feedback control gain or the timedelay parameter as the control object.
Fig. 11 The timedelay control curve of superharmonic vibration with different parameters. 
5 Conclusion
The model of parametrically excited nonlinear vertical vibration of roll system has been established. The approximate analytical solution and amplitudefrequency characteristic equations of principal resonance and subresonance of roller system excited by corrugated interface have been obtained by using the multiplescale method.
The jump and hysteresis phenomena of the roll system have been revealed through analyzing the influences of nonlinear stiffness coefficient, nonlinear damping coefficient, system damping coefficient and rolling force amplitude on vibration. The controllability of the amplitude of the system and the resonance region have been realized by designing the timedelay feedback controller. Numerical experiments have been carried out to validate the feasibility of the control method in the roll system.
In the process of corrugated roll design, the roll profile curve should be optimized reasonably to avoid and reduce the influence of rolling force amplitude on primary resonance and subresonance. The further research is needed to combine the intelligent vibration suppression to analyze the vibration characteristics of the corrugated rolling mill.
Acknowledgement
This work was supported by the Major Program of National Natural Science Foundation of China (U1710254), National Natural Science Foundation of China (51974196 51804215, 51905372), Shanxi Province Graduate Education Innovation Project (2019BY052), Key Research and Development Program of Shanxi Province (201703D111003), and Applied Basic Research Program of Shanxi Province of China (201801D121021).
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Cite this article as: Dongping He, Huidong Xu, Tao Wang, Zhongkai Ren, Nonlinear timedelay feedback controllability for vertical parametrically excited vibration of roll system in corrugated rolling mill, Metall. Res. Technol. 117, 210 (2020)
All Figures
Fig. 1 Warping phenomenon of flat roll. 

In the text 
Fig. 2 Rolling diagram of corrugated roll. 

In the text 
Fig. 3 Parametrically excited vibration mathematical model of corrugated roller systems. 

In the text 
Fig. 4 The amplitudefrequency characteristic curve of principal resonance with different parameters. 

In the text 
Fig. 5 The timedelay control amplitudefrequency curve of principal resonance. 

In the text 
Fig. 6 The timedelay control amplitudefrequency curve of principal resonance with different parameters. 

In the text 
Fig. 7 The amplitudefrequency characteristic curve of subharmonic vibration with different parameters. 

In the text 
Fig. 8 The timedelay control amplitudefrequency curve of subharmonic vibration. 

In the text 
Fig. 9 The timedelay control amplitudefrequency curve of subharmonic vibration with different parameters. 

In the text 
Fig. 10 The amplitudefrequency characteristic curve of superharmonic vibration with different parameters. 

In the text 
Fig. 11 The timedelay control curve of superharmonic vibration with different parameters. 

In the text 
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