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Scientific Reports volume 14, Article number: 29503 (2024 ) Cite this article pre gi hollow section
In order to address the vertical vibration phenomenon of the working roll during the rolling process, taking into consideration the nonlinear factors of the rolling interface and the roll system, this study establishes a nonlinear excitation vertical vibration model based on dynamic rolling force. The impact of nonlinear factors on system stability and vibration characteristics is analyzed. The study reveals that nonlinear damping and linear stiffness exhibit significant effects through time delay characteristics. Nonlinear stiffness has a drastic impact on system stability, while increasing system damping effectively reduces the amplitude and narrows the resonance region. By employing averaging method and singular value theory, the stability of the cold rolling mill roll system is analyzed under non-autonomous and autonomous states. A combined time-delay feedback controller is proposed, which effectively suppresses the system’s large-scale vibration by adjusting the control gain and time delay parameters. MATLAB simulations are conducted to validate the accuracy of the control strategy, providing theoretical support for vibration prediction and system design in rolling mills.
Vibration in rolling mills has been widely studied in the field of rolling. In actual rolling processes, the vertical vibration of the mill housing, torsional vibration of the main drive system, and horizontal vibration of the mill foundation pose the greatest risks to the quality of the strip and the rolling mill equipment. These phenomena not only affect the thickness, accuracy, and surface roughness of the processed sheets but also, if not properly controlled, can lead to equipment damage and even harm to personnel1,2,3. Many scholars have explored the vibration issues in the vertical direction of the mill rolls to varying degrees and achieved corresponding results. The establishment of a vibration mechanism model is the basis for studying equipment vibration and vibration control. In terms of modeling, Reference4 demonstrated through roll models and simulations that the offset of the roll system and the thickness of the rolled strip have a significant impact on vertical vibration. Reference5 analyzed the amplitude-frequency characteristics of vertical vibration in a mill under primary resonance conditions using a mill model under external excitation. Reference6 considered the dynamic changes in vertical displacement and torsional vibration angle of the mill rolls and established relevant models to determine the conditions for the occurrence of periodic, subharmonic, and chaotic vibrations in the vertical and torsional directions of the mill. Based on a nonlinear seven-degree-of-freedom vibration model considering dynamic rolling forces, Reference7 analyzed the influence of various parameters on the amplitude-frequency characteristics, as well as the system’s bifurcation and chaotic behavior, through on-site data analysis. However, the aforementioned literature mainly focuses on the modeling and analysis of multi-degree-of-freedom nonlinear vibrations, which involve complex calculations. Therefore, it is necessary to simplify the model by reducing the degrees of freedom appropriately and establish a two-degree-of-freedom nonlinear parametric vertical vibration model for the cold rolling mill roll system based on the consideration of dynamic rolling forces.
In researching the nonlinear factors affecting equipment vibrations, scholars predominantly start by examining the system’s amplitude-frequency characteristics. Reference8 investigates the dynamic characteristics of rolling force affected by roll system vibrations, revealing that amplitude increases with external excitation amplitude and rolling speed. Additionally, the range of unstable frequencies decreases with increasing rolling speed but increases with larger external excitation amplitudes. Reference9 studies the resonance characteristics of the roller system under high and low-frequency excitation signals, demonstrating that the critical resonant amplitude of high-frequency signals in fractional-order systems depends on damping strength and is influenced by fractional-order damping. Reference10 considers various nonlinear factors and finds through simulations of subharmonic resonance amplitude-frequency responses that increasing system damping effectively reduces amplitude and narrows the resonance region, while nonlinear stiffness has a more severe impact on subharmonic vibrations, potentially leading to system instability. Reference11 discusses system stability under different parameter conditions using singular value theory, analyzing the effects of nonlinear parameters such as stiffness and damping on system stability and vibration characteristics. However, the research on system stability is somewhat limited, making it difficult to ensure effective vibration control strategies. Therefore, analyzing system stability under various states in the established model is necessary.
Time delay phenomena are common and even unavoidable in control systems. During the rolling process, fluctuations in rolling force lead to continuous loading and unloading of the rolled material, causing variations in the yield limit during plastic deformation. This results in non-overlapping loading and unloading curves, and various nonlinear factors at the contact interface exhibit time delay characteristics that require appropriate control strategies for intervention. Reference12 discusses “delay state feedback control” and proposes a posterior analysis of the primary closed-loop eigenvalues to ensure accurate position achievement through state feedback. Reference13 introduces a new method for solving the optimal control problem of linear time-delay systems with a quadratic cost function. Reference14 investigates feedback delay approaching the system’s inherent period and provides the parameter range for delayed feedback that allows control of these states. Reference15 finds that adjusting control gains and delays can avoid resonance zones, effectively suppress large vibrations in the system, and achieve good control performance. References16,17,18 study the vibration characteristics and damping effects of rolling mill roll systems, applying time-delay feedback control to the main resonance response, and optimizing the amplitude, irregular vibrations, and energy consumption of the vibration system by obtaining optimal feedback control gains and delays. Reference19 explores the capability of delayed position feedback control in suppressing vortex-induced vibrations of elastically mounted cylindrical structures, demonstrating that selecting appropriate delay control parameters can significantly increase or decrease system damping. The Duffing oscillator with time-delay position feedback20,21 and with time-delay position-velocity feedback22,23,24 has been studied. Discussions on magnetic suspension systems with time-delay position feedback control25 and time-delay displacement-velocity feedback control26 are also available. Comparative analysis of different models using time-delay feedback control reveals that resonance responses exhibit multiple solutions and jumping phenomena, with periodic variations as the time delay value increases. Rational selection of control gains and time delays can effectively suppress large vibrations in the system, achieving good control results27,28,29,30,31. Based on these studies, it is evident that scholars have focused more on nonlinear time-delay feedback control while often neglecting linear time-delay feedback control. This aspect is also crucial, and combining linear and nonlinear time-delay feedback control strategies remains a challenging issue.
Taking inspiration from the aforementioned motivation, this study focuses on a certain cold rolling mill roll system and establishes a nonlinear parametric excitation vertical vibration model based on dynamic rolling forces. By nondimensionalizing the equations, a nonlinear parametric excitation vertical vibration equation with perturbation parameters is obtained. The stability of the cold rolling mill roll system is analyzed using averaging method and singular value theory under non-autonomous and autonomous states. The effects of different nonlinear parameters on system vibration are examined when the excitation frequency of the dynamic rolling forces is similar to the primary resonance frequency. A time delay feedback controller is designed under a combination of linear and nonlinear control to adjust the values of various parameters, aiming to enhance system stability and reduce vibration amplitude. MATLAB simulations are conducted to validate the accuracy of the results, providing theoretical support and basis for the effective and precise prediction of rolling mill vibrations and the optimization of rolling mill system design, thereby reducing the likelihood of vertical vibrations.
Frictional forces develop between high-speed rotating rolls and the rolled material, guiding the material into the roll gap. Simultaneously, the upper and lower work rolls apply a pressing force to the material, causing it to undergo plastic deformation. This process is known as rolling. During rolling, the material exerts two forces on the rolls: a radial pressure (denoted as P) when entering the roll gap, and a frictional force (denoted as f) perpendicular to the radial pressure. The direction of the frictional force f is opposite to the direction of roll rotation. By decomposing the radial pressure P and the frictional force f along the rolling direction (x-axis) and perpendicular to the rolling direction (y-axis), the resultant force in the direction perpendicular to the rolling direction is the rolling force, denoted as \(F={P_y}+{f_y}.\) Figure 1 illustrates the force decomposition on the rolls during the rolling process.
Decomposition diagram of rolling force on roll during rolling.
To study the nonlinear vibration characteristics in the vertical direction of the cold rolling mill’s work roll mechanical structure, and considering the nonlinear damping properties of the cold rolling mill, c1 and c2 are defined as the equivalent damping between the upper roll system and the upper crossbeam, and between the lower roll system and the lower crossbeam, respectively. The Vanderpol oscillator \(C+C^{\prime}{\left( {{x_1} - {x_2}} \right)^2}\) , represented by Cv, defines the nonlinear damping between the upper and lower roll systems caused by the roll profile curve during the rolling process. C denotes the mean value of the damping in the steady state of the cold rolling mill. Considering the nonlinear stiffness characteristics of the cold rolling mill, the roll profile curve and rolling force introduce a nonlinear stiffness term \(\bar {K}\) between the upper and lower roll systems, with F0 representing the steady-state rolling force, and \({F_1}\cos \left( {\omega t} \right)\) approximating the dynamic rolling force during the process. K denotes the mean stiffness in the steady state, while \(\bar {K}\) is denoted as \(K\left( {{x_1} - {x_2}} \right) - {F_0} - {F_1}\cos \left( {\omega t} \right)\) . The Duffing oscillators \({k_1}+{k^{\prime}_1}x_{1}^{2}\) and \({k_2}+{k^{\prime}_2}x_{2}^{2}\) , represented by \({k_{D1}}\) and \({k_{D2}}\) , define the nonlinear stiffness terms between the upper roll system and the frame, and between the lower roll system and the frame, respectively. Here, k1 and k2 represent the equivalent stiffness between the upper roll system and the upper crossbeam, and between the lower roll system and the lower crossbeam. The equivalent masses m1 and m2 represent the upper and lower roll systems. The mathematical model for the vertical vibration of the cold rolling mill’s roll system is shown in Fig. 2.
Vertical vibration model of cold mill roller system.
The nonlinear factors of the roll system of the cold rolling mill are mainly related to the vibration displacement X and the vibration velocity \(X^{\prime}\) , and the main dynamic characteristics are also reflected in these two parameters. Using the mass concentration method and the generalized dissipative Langrange principle, the basic equation of the vertical vibration system of the cold rolling mill roll system is a general form considering nonlinear constraints32, where M, C, K and P represent the mass block, damping block, stiffness block and external disturbance force block of the rolling mill roll system, respectively. The specific expression is as follows:
By substituting the nonlinear modules involved in the model in Fig. 2 into Eq. (1), the nonlinear vertical vibration equation of cold rolling mill based on dynamic rolling force can be obtained as follows:
During the rolling process of the mill, due to the symmetric mechanical structure of the upper and lower roll systems, we can define \({m_1}={m_2}\) and \({x_1}= - {x_2}\) , simplifying Eq. (2) as:
By nondimensionalizing the above equation and defining:
Equation (3) can be simplified as:
The cold rolling mill roll system inevitably induces various resonance phenomena during the rolling process. Therefore, it is necessary to analyze the vibration characteristics of the primary resonance of the cold rolling mill roll system. Among the factors influencing the system’s nonlinear vibration, the steady-state rolling force value F0 is much smaller than the dynamic rolling force value F1, so Fm can be neglected33. Assuming ε is the disturbance parameter of the nonlinear parametric excitation vertical vibration system of the cold rolling mill, the nonlinear term in Eq. (4) can be multiplied by the disturbance parameter, resulting in:
This section employs the multiple scales method to solve the amplitude-frequency response equation of the primary resonance parametric excitation in the cold rolling mill. Let \({T_n}={\varepsilon ^n}t\) represent time variables at different scales, where \(n=0,1,2, \cdots\) . Taking Y as a function of the independent variables ε and t, with the highest order of the disturbance parameter being m, we have:
Where \({T_0}=t\) represents the slow time scale and \({T_1}=\varepsilon t\) represents the fast time scale, making \(Y\left( {t,\varepsilon } \right)\) a function of m independent time variables.
Expanding in powers of ε, differentiating with respect to time, and using the chain rule of differentiation, we obtain:
Where \({Q_n}\left( {n=0,1,2, \cdots ,m} \right)\) is the partial differential operator, specifically defined as:
Let Eq. (7), Eq. (8), and Eq. (9) be the set of time differential functions, substitute them into Eq. (5), expand, and remove the high-order terms of the disturbance parameter ε on both sides of the equation. The resulting approximate equations for each order are:
When the excitation frequency ω of the dynamic rolling force in the cold rolling mill is similar to the natural frequency ω0, the system exhibits primary resonance response. Assuming λ as the frequency modulation parameter, we have \(\omega ={\omega _0} - \varepsilon \lambda\) , where X represents the complex conjugate of the previous term. Let the zeroth-order approximate solution of Eq. (11) be:
By substituting Eq. (13) into Eq. (12), we obtain:
To eliminate the secular term, we set the coefficient of \({e^{i{\omega _0}{T_0}}}\) to zero, resulting in:
Let \(A=k{T_1}{e^{i\varphi {T_1}}}\) , substitute it into Eq. (15), and separate the real and imaginary parts. We can derive:
When \(\dot {k}=k\dot {\phi }=0\) , the system has a steady-state solution, and we obtain the amplitude-frequency characteristic equation of the cold rolling mill’s roll system based on the dynamic rolling force:
According to the obtained principal common amplitude frequency characteristic equation of the cold rolling mill roll system, when \({\omega _0}=1\) 、\(\delta =0.2\) 、\(\beta =0.155\) 、\(\gamma =0.012\) 、\(F=0.1\) , the principal common amplitude frequency characteristic curves under different conditions can be obtained by adjusting the coefficients of the system’s principal resonance nonlinear influencing factors:
Principal common amplitude-frequency characteristic curves under the influence of different nonlinear terms.
In Fig. 3(a), comparing the amplitude-frequency response curves for α = 1.0 and α = 2.0, it is observed that as the system’s linear stiffness α increases, the vibration amplitude decreases. Increasing the linear stiffness coefficient changes the system’s natural frequency, thereby widening the gap between the excitation frequency and the resonance frequency, which effectively reduces vertical vibration in the work rolls. In Fig. 3(b), as the nonlinear stiffness coefficient δ increases from 0.2 to 1.5, the dimensionless amplitude of the system remains nearly unchanged, but the peak of the vibration curve gradually shifts to the right. With increasing coordination parameter λ, the dimensionless amplitude experiences jumps, leading to system instability. Figure 3(c) and (d) show that both cases exhibit certain similarities: in the primary resonance system, as the damping coefficient increases, the system’s amplitude and resonance region gradually decrease, while the backbone curve of the vibration response does not shift or bend, and the dimensionless amplitude k does not experience jumps. Therefore, changes in the linear damping coefficient β and nonlinear damping coefficient γ do not affect the curvature of the curve, but only alter the amplitude and resonance region.
Amplitude-frequency characteristic curve of external disturbance force of main resonance.
From the amplitude-frequency response curves of the primary resonance under varying external disturbance forces shown in Fig. 4, it is evident that as the external disturbance force parameter F increases from 0.1 to 1.4, the system’s amplitude becomes more pronounced and the resonance region enlarges. This indicates that the involvement of dynamic rolling force significantly affects the system’s amplitude. Therefore, effective suppression of the vertical vibration phenomenon in the cold rolling mill can be achieved by properly controlling and reducing the influence of dynamic rolling force on the system.
The stability of the roll system in the cold rolling mill is crucial for the quality of sheet metal processing. During the sheet rolling process, the variations in dynamic rolling force, non-linear stiffness of the contact interface, and non-linear damping can all affect the stability of the system, leading to complex nonlinear dynamic behaviors such as bifurcation and chaos. Therefore, maintaining the stability of the system is of great significance for improving the production quality and efficiency of sheet metal.
In a non-autonomous state, let \({\omega ^2}=\alpha +\varepsilon \alpha {\sigma _1}\) , substitute it into Eq. (5), and we obtain:
When the perturbation parameter ε is sufficiently small or even close to zero, the approximate solution of the stable equation in the non-autonomous state is:
where a and φ are slow-varying functions of time t. For convenience in calculations, let \(\theta =\omega t - \varphi\) , and the second derivative of the approximate solution is:
Substituting the approximate solution and its second derivative into Eq. (18), we get:
The averaged equation of the system can be defined as:
From Eq. (23), we can obtain the Jacobian matrix of the non-autonomous system:
When \(\dot {a}=\dot {\varphi }=0\) , we obtain the equilibrium points \(\bar {a}\) and \(\bar {\varphi }\) for a and φ:
Based on the stability of the singular points, the stability conditions for the periodic solution of the nonlinear vertical vibration system in the cold rolling mill can be expressed as follows in Eq. (27):
In the autonomous state34, let Fn = 0, and from Eq. (4), we have:
The Jacobian matrix of the autonomous system is then:
When δ > 0, the singular point of the system is (0,0). When δ < 0, the singular points of the system are (0,0), \(\left( { - \sqrt { - \frac{\alpha }{\delta }} , 0} \right)\) , and \(\left( {\sqrt { - \frac{\alpha }{\delta }} , 0} \right)\) . Substituting the singular points \(\left( { - \sqrt { - \frac{\alpha }{\delta }} , 0} \right)\) and \(\left( {\sqrt { - \frac{\alpha }{\delta }} , 0} \right)\) into Eq. (28), we obtain:
The characteristic equation and its solutions are:
Substituting the singular point (0,0) into Eq. (28), we have:
The characteristic equation and its solutions are:
Discuss the motion states of the system with singular point (0,0) and system stiffness α = 1 under different conditions of damping β:
When β>2α, the characteristic roots are two real numbers with opposite signs, and the singular point is a saddle point. The system is stable. Figure 5 shows the motion state of the system when β = 4:
Motion state of the system when β = 4.
When β = 2α, the characteristic roots are both negative real numbers, and the singular point is a stable node. The system is stable. Figure 6 shows the motion state of the system when β = 2:
Motion state of the system when β = 2.
When 0<β<2α, the characteristic roots are conjugate complex numbers with negative real parts, and the singular point is a stable focus. The system is stable. Figure 7 shows the motion state of the system when β = 1:
Motion state of the system when β = 1.
When β = 0, the system is in an undamped state. The real parts of the characteristic roots are zero, and they are conjugate complex numbers. The singular point is a center point. Figure 8shows the motion state of the system when β = 0:
Motion state of the system when β = 0.
When -2α<β<0, the characteristic roots are conjugate complex numbers with positive real parts, and the singular point is an unstable focus. The system is unstable. Figure 9 shows the motion state of the system when β=-1:
Motion state of the system when β=-1.
When β =-2α, the characteristic roots are both positive real numbers, and the singular point is an unstable node. The system is unstable. Figure 10 shows the motion state of the system when β=-2:
Motion state of the system when β=-2.
When β<-2α, the characteristic roots are two real numbers with opposite signs, and the singular point is a saddle point. The system is unstable. Figure 11 shows the motion state of the system when β=-4:
Motion state of the system when β=-4.
When studying the nonlinear parametric resonance response system of cold rolling mill rolls, a combined time-delay feedback control is proposed to reduce or eliminate vibrations caused by nonlinear effects and external disturbances. This method incorporates both linear and nonlinear time delays and feedback gains to effectively adjust the system’s dynamic behavior, thereby controlling and stabilizing the system.
Linear feedback impacts the current state by introducing a delay in state variables such as displacement or velocity. By adjusting the linear time-delay parameter τ1 and feedback gain coefficient g1, high-frequency vibrations and resonance phenomena in the system can be suppressed. This control primarily addresses the system’s linear dynamic characteristics. For nonlinear characteristics, introducing corresponding time-delay parameters τ2 and feedback gain coefficients g2 can more effectively manage complex vibration behaviors resulting from nonlinear effects. Nonlinear time-delay feedback control offers unique advantages in handling nonlinear dynamic behaviors, such as chaos or large amplitude vibrations.In this section, a combined time-delay feedback control is proposed on the nonlinear parametric resonance response of the cold rolling mill roll system. It can simultaneously consider both linear and nonlinear time delays and establish the controlled equation for the roll system parametric resonance. In the combined time-delay feedback control, τ1 and τ2 are time delay parameters, and g1 and g2 are the feedback gains considering linear and nonlinear time delays, respectively. The combined time-delay feedback control is given as follows:
By adding the combined time-delay feedback controller to the nonlinear parametric vibration equation based on the dynamic rolling force of the cold rolling mill, we obtain:
Substituting the time differential function group into Eq. (36) and expanding it in terms of the perturbation parameter ε, while removing the higher-order terms, we obtain the approximate equations of each order:
Let X represent the conjugate complex of the previous term. Assuming the zeroth-order approximate solution of Eq. (39) is:
Substituting the zeroth order approximate solution into Eq. (38), we have:
To eliminate the secular term, we set the coefficient of \({e^{i{\omega _0}{T_0}}}\) to zero, which gives:
\(- 2i{\omega _0}{Q_1}A - \beta i{\omega _0}A - i\gamma {\omega _0}{A^2}\bar {A} - 3\delta {A^2}\bar {A}+\frac{1}{2}{F_n}{e^{i\lambda {T_1}}}\)
Let \(A=k{T_1}{e^{i\varphi {T_1}}}\) and substitute it into Eq. (41), separating the real and imaginary parts, we obtain:
In the time-delay feedback control system, some coefficients undergo changes:
The system has a steady-state solution condition \(\dot {k}=k\dot {\phi }=0\) , thus obtaining the amplitude-frequency characteristic curve equation for the time-delay feedback control of the cold rolling mill roll system’s parametric resonance based on dynamic rolling force:
\(\frac{1}{4}{\left[ {\left( {\beta +\frac{{{g_1}}}{{{\omega _0}}}\sin {\omega _0}{\tau _1}} \right)k - \left( { - \gamma - \frac{{{g_2}}}{{{\omega _0}}}\sin {\omega _0}{\tau _2}} \right){k^3}} \right]^2}\)
If Fn = 0, it can be obtained by averaging method35 and Eq. (43):
According to the stability condition of the periodic solution of the system in Eq. (26), when M < 0 or N > 0 is satisfied, the system remains stable36. Since Eq. (45) meets this stability condition, it indicates that the system is stable under the combined time-delay feedback control.
By incorporating the combined time-delay feedback control signal into the parametric vertical vibration equation of the system, the dynamic behavior of the system was analyzed through numerical simulations. Appropriate time-delay parameters and feedback gain coefficients were selected and adjusted based on the system’s specific characteristics, such as natural frequency and damping ratio, to achieve optimal control. The controlled response equations of the system’s primary resonance amplitude-frequency characteristics were obtained, and the time-delay feedback control curves for the primary resonance under different conditions are shown in Fig. 12.
Main resonance combined delay feedback control curve.
As shown in Fig. 12(a), without any control, the amplitude-frequency curve of the primary resonance exhibits bifurcation. When adjusting the linear time-delay τ1 = 0.2, feedback gain coefficient g1 = 0.2, or the nonlinear time-delay τ2 = 0.2 and feedback gain coefficient g2 = 0.2 separately, the control effects are nearly identical, effectively managing the primary resonance amplitude and resonance region, and mitigating the system’s bifurcation behavior. However, simultaneous adjustment of the combined control variables (τ1 = τ2 = g1= g2 = 0.2) yields stronger control effects compared to adjusting any single control variable alone. In Fig. 112b), keeping the combined time-delay parameters (τ1 = τ2 = 0.2) constant, and varying the linear feedback gain g1 and nonlinear feedback gain g2 equally (g1 = g2 = 0.2, 0.4, 0.6, 0.8), it is observed that as the gain coefficients increase, the system’s amplitude gradually decreases, and the resonance domain also diminishes, effectively controlling the vibration phenomena. Figure 12(c) demonstrates that, while keeping the combined gain coefficients (g1 = g2 = 0.2) constant, varying the two time-delay parameters equally (τ1 = τ2 = 0.2, 0.4, 0.6, 0.8) shows that as the time-delay parameters increase, the system’s amplitude decreases and the resonance domain also reduces.
Figure 12(d) explores the stability range of the feedback gain coefficients. By maintaining the combined time-delay parameters (τ1 = τ2 = 0.2) constant, it is observed that effective control of the bifurcation behavior and vibration phenomena occurs when the combined gain feedback coefficients τ1>0 and τ2>0 are applied. As these coefficients increase, the control effect improves. Comparatively, adjusting the feedback control gain yields better results than adjusting the time-delay parameters alone.
The new amplitude-frequency vibration form obtained by changing the original nonlinear vertical vibration amplitude-frequency characteristic equation of the roll system of cold rolling mill is studied deeply. The core of our method is to establish the dynamic expression of nonlinear constraint based on mass concentration method and generalized dissipative Langrange principle, and to solve the nonlinear dynamic equation strictly by multi-scale method and singular value theory.
It is worth noting that the proposed control strategy takes into account the time delay of both linear and nonlinear systems, thus avoiding the need for complex algorithms. This design improves computing efficiency and reduces time and cost. The results of this study are compared with the existing literature, which is helpful to fully understand the effectiveness of the proposed method. According to the research results, the following conclusions can be drawn:
Considering the effect of vertical vibration displacement of roll on rolling force of cold rolling mill and the nonlinear factors between rolling interfaces, the stability conditions of the periodic solution of the system under non-autonomous state and the motion trajectory of the system damping under different circumstances under autonomous state are obtained on the basis of the established model, which makes up for the shortcomings of the stability analysis of the nonlinear roll system in the existing literature.
In the cold mill roll system, nonlinear damping and linear stiffness significantly affect the vertical vibration characteristics, and the nonlinear stiffness term will obviously lead to the instability of the system. The increased damping of the system can effectively reduce the vibration amplitude and limit the expansion of the resonance region, thus improving the overall stability and performance.
According to the comparative analysis of different situations of the designed joint time-delay feedback controller, the resonance response shows multiple solutions and jumps. With the increase of delay parameters, these responses will undergo periodic changes. By adjusting the control gain coefficient and time delay parameters, large amplitude vibration can be suppressed and the control results can be significantly improved. This finding can verify the feasibility of the designed control strategy.
Time lag, as a common phenomenon in control systems, has proved to be crucial in rolling mills. Due to the alternating loading and unloading of materials in the plastic deformation stage, the nonlinear damping and linear stiffness show delay characteristics due to the fluctuating rolling force in the rolling process. This insight has far-reaching implications for the design of mill control strategies.
Te datasets used and/or analyzed during the current study are available from the corresponding author on reasonable request.
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This work was supported by Basic Research Funds for Universities directly under the Inner Mongolia Autonomous Region (Grant numbers 2023QNJS071). All authors has received research support from the fund.
School of Mechanical Engineering, Inner Mongolia University of Science and Technology, Baotou, 014010, China
Li Li & Chenhao Zhong
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All authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by Li Li and Chenhao Zhong. The first draft of the manuscript was written by Chenhao Zhong and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.
The authors declare no competing interests.
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Li, L., Zhong, C. Research on nonlinear dynamic vertical vibration characteristics and control of roll system in cold rolling mill. Sci Rep 14, 29503 (2024). https://doi.org/10.1038/s41598-024-79117-4
DOI: https://doi.org/10.1038/s41598-024-79117-4
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