Li Jiang, Tao Wang, Qingxue Huang

Abstract: In this paper, the main researches are focused on the horizontal nonlinear vibration characteristics of roll systems for rolling mill, mainly including the study of forced vibration and free vibration of the roller.Firstly, the nonlinear damping parameters and nonlinear stiffness parameters within interface of the rolling mill are both considered, and a fractional-order differential term is also introduced to model the horizontal nonlinear vibration.Secondly, the averaging method is introduced to solve the forced vibration system of the mill roll system, and the amplitude-frequency characteristic curves of the system are obtained for different orders, external excitation amplitudes, stiffness and fractional order coefficients.Thirdly, the amplitude-frequency and phasefrequency characteristics of the free vibration of the mill roll system are investigated at different fractional orders.Then, the accuracy of the averaging method for solving the dynamic characteristics of the system is verified by numerical analysis, and the effect of the fractional differential term coefficients and order on the dynamic characteristics of the roll system are investigated.Finally, the time-frequency characteristics and phase-frequency characteristics of free vibration systems at different fractional orders are studied.The validity of the theoretical study is also verified through experiments.

Keywords: roller system; fractional-order; average method; forced vibration; free vibration

1 Introduction

Rolling mill is a typically complex dynamic system coupled with structure and process.The vibration of the mill roll system not only affects the stable operation of the equipment, but also may cause vibration lines on the strip surface which restricts the product quality[1].The early research on roller vibration mostly focused on treating the roller system as steady-state system,and the main drive system, seat frame system and roller system are linear treatment.The vibration model is then built by considering the equivalent mass, the symmetry of the mechanical structure and the vibration symmetry.At last, the inherent characteristics of each component of the rolling mill system are obtained, and the optimization and checking of the mill system parameters are realized [2–11].

But in fact, there are many nonlinear factors in the rolling mill work process, such as friction, gap and collision factors.Linear treatment not only make the results differ greatly from the actual results but also make it difficult to explain the complex dynamic behavior of rolling mill systems due to nonlinear factors.Fortunately, with the development of the nonlinear theory, the study of nonlinear dynamic characteristics of rollers has achieved great results.In [12–15],Dongxiao Hou studied the nonlinear vibration characteristics of vertical-horizontal coupling of plate strip rolling mill roller system under variable friction force, and studied the nonlinear parameter vibration characteristics of cold rolling mill under dynamic rolling force.In [16, 17], Jinlei Huang analyzed the influence of the asymmetric structure parameters on the stability of the rolling mill and the asymmetry of the friction coefficient on the vibration and stability of the rolling mill.In [18], Yunyun Sun studied the influence of the rough morphology of the rolling interface on the nonlinear vibration characteristics of the rolling mill.In [19–21], Dongping He studied the nonlinear vibration characteristics of the corrugated roller system, defined the principal resonance characteristics and the sub-resonance characteristics of the roller system under the nonlinear parameter excitation, and proposed the vibration suppression measures.In [22, 23],Bin Liu studied the nonlinear vibration characteristics of strip mill affected by the vibration of rolling parts.In addition, many scholars have studied the nonlinear dynamic characteristics of rolling rollers, which covered the vertical nonlinear vibration, horizontal nonlinear vibration, coupling nonlinear vibration and so on[24, 25].

The previous studies on nonlinear vibration system always regarded the roller material as an ideal solid.But in fact, the steel material tends to be between an ideal solid and the ideal fluid.In recent years, many scholars began to study the behavior of fractional dynamical systems[26–33].The study shows that the fractional differential equations are more accurate than the traditional integer differential equations while kinetic solving, and the application of fractional differential terms in the dynamic characteristics of roller systems is still blank [34–36].In order to depict the true constitutive relationship of the material more accurately, fractional-order will be introduced in the model establishing, and which can be used to simulate materials containing memory properties of the engineering material.

In summary, there is a significant horizontal non-linear vibration phenomenon in the rolling mill roll system and the horizontal vibration will be coupled with the vertical vibration, intensifying the vertical vibration of the roll system,affecting the stability of the system and the quality of the rolled material.The main reason for this is that the theoretical research is not accurate enough to establish the vibration model of the roll system, and does not fully consider the non-linear factors in the vibration process of the roll system and the intrinsic relationship of the structural material is not reasonable.In order to further explore the horizontal vibration mechanism of the roll system, it is necessary to establish a more accurate horizontal vibration model of the mill roll system.

In this paper, the fractional differential term is introduced into the study of the dynamic characteristics of roller system.And the structure of this present paper is arranged as follows: In the second section, the friction nonlinear and stiffness nonlinear are both considered, and the nonlinear vibration model is established.The third section mainly uses the average method to obtain the approximate analytical solution of the forced vibration, and analyzes the stability of the solution.The fourth section mainly solves the amplitude-frequency characteristics of free vibration system.The fifth section mainly analyze the results of the numerical simulation and the validity of the theoretical study was also verified through experiments.The last section gives the conclusion of this paper.

2 Nonlinear Vibration Model of Roller System

Strip rolling is a complex working process.In order to study the nonlinear vibration characteristics of the work roll mechanical structure in the horizontal direction of the cold rolling mill, introduce the Duffing and the Van der Pol oscillators,and consider the nonlinear damping and nonlinear stiffness within interface of the rolling mill, a nonlinear model of the work roll mechanical structure of the mill is established as shown in Fig.1, wheremrepresents equivalent mass of the roll, (k1+k2x2(t)) represents nonlinear stiffness coefficient term between roller system and frame,c(x2(t)–1)represents nonlinear damping coefficient term between roller system and rolling piece,x(t)represents horizontal displacement of the roller, andKDp[x(t)]represents fractional differential term.Due to friction, clearance and the presence of additional bending moments, the mill roll system is subjected to a horizontal combined force, which is defined as a horizontal excitation forceFcos(ωt).

Fig.1 Physical model of horizontal vibration with fractionalorder

The nonlinear parametrically excited vibration equations of the roller systems are expressed as

wherek1is the linear stiffness coefficient,k2is the nonlinear stiffness coefficient,cis the nonlinear damping coefficient,Fis the amplitude of the excitation value,ωis the excitation frequency,Kis the fractional differential term coefficient andK>0, andpis the fractional differential term order andp(0≤p≤1).

The horizontal vibration of the roller system is a weak vibration.The weak nonlinear term of Eq.(1) is crowned with small parameters.Perform the following coordinate transformation on the system:

Then, the Eq.(1) can be written as

There are three definitions often used in describing the fractional-order derivative, i.e.the Riemann-Liouville(R-L) definition, the Grünwald-Letnikov(G-L) definition and the Caputo definition.In the roller vibration system,x(t) can be continuously first order differential in [0, T], andx¨(t)can be continuously first order integral in [0,T].So all three definitions for eachp(0≤p≤1)value are equivalent.This paper uses the G-L definition to handle the fractional differential terms.

Ifp=0 or 1, there is no fractional differential terms present.Lett0=0, for the value of 0

The fractional derivative term functions not only as a classical damping, but also as a restoring force [34–36].According to the generalized harmonic function method, the fractional derivative term can be decomposed into the sum of the linear recovery forces and the linear damping forces.

Whenε=0, suppose the steady-state response analytical solution of Eq.(3) is

whereφ=ωt+θ, the amplitudeaand the phaseθare constants determined by the initial conditions.

Introduce the fractional derivative formula for simple harmonic functions[37]

Then the fractional differential terms are

3 Forced Vibration of the Roller System

3.1 Response

There are various forms of vibration during the rolling mill works.Including the main resonance,sub-harmonic resonance, super-harmonic resonance, etc.And the main resonance ofω≈ω0is the most concerned form of vibration.It is also the main type of vibration studied in this section.

According to the perturbation theory, using the average method to solves the main resonance of roller system nonlinear vibration system.To meet the application form of the average method,the Eq.(3) is rewritten into the following form

To quantitatively describe the closing degree of the frequency of external excitation force and the natural frequency of the roller system.The principal resonanceω2≈ω02+εσis studied, whereεis small parameters.Substitute Eq.(7) into Eq.(8) to obtain the horizontal nonlinear vibration approximate system without the fractional differential term

According to the average method, one can get an approximate analytical solution of the roller system vibration.Whenε≠0, there are nonlinear interference forces present, nowαandθare determined by the following equations.

From Eq.(10), the first derivative of the roller system amplitude and phase is proportional toε, and the solution of the roller system

Displacementx(t)and velocityx˙(t) is converted into the solution of the roller system amplitudeaand phaseθthrough variable substitution.From that, the second-order equation is transformed into the first-order equation.

Solve Eq.(10), sinceaandθare slow change function about time, which changes very little in a period ofT=2π/ω0.It is feasible to indicate the change of the first part ofa˙ andθ˙ in period 2π by average value.Solvea˙ andθ˙

By substituting the initial parameters of the horizontal nonlinear vibration system into Eqs.(11a) and (11b), it can be obtained

From Eq.(12), it is obviously that both the coefficientsKand orderpof the fractional differential terms affect the equivalent damping and the equivalent stiffness of the roller vibration system.

Focus on the steady-state response of roller system forced vibration.Makinga˙=0 ,θ˙ =0, we can get

Then

where

Eq.(14) is the amplitude-frequency characteristic equation for the forced vibration of the roll system.According to Eq.(14), different main resonance amplitude-frequency characteristic curves can be obtained by differentk1,k2,c,F,K,p.

3.2 Stability Analysis of the Solutions

The stability of the stationary solution determines the stability of the roller system.For Eq.(13), the Jacobi matrix of the system is obtained.

Let Jacobi Matrix as

Characteristic determinant of the matrixJ

Secular equation

Characteristic root

WhileP1+P2<0, (P1+P2)2≥4[P1P2+(Q/2mω)2(1+3Qk2/2)]>0,λ1,2are negative values.This system is stable while rolling.

4 Free Vibration of the Roller System

The free vibration characteristics of the roller system are mainly studied in this section.From Eq.(8), letf=0, and the equation can be rewritten as

Similarly, according to the average method

Solve Eq.(21a) and Eq.(21b)

By substituting the initial parameters of the roller system into Eq.(22a) and Eq.(22b) , we get

Integrating both sides of Eq.(23a) and Eq.(23a), one can get

5 Numerical Simulation

The Fourier series method have been used to obtain the numerical solution of the amplitude of the system response at the excitation signal frequencyω, namely

whereasandacare the cosine and sine Fourier components of the system response at the excitation signal frequencyω, and the definition formula is

wheremis a sufficiently large positive integer,which generally means that the system will run for m cycles after the transient response disappears.For discrete time series, the calculation formula ofasandacis

where Δtis the calculated time step,Tis the cycle of external excitation,T= 2π/ω.

According to the superposition principle of the fractional differential term derivatives, reduce the order of Eq.(1) as

5.1 Verify the Analytical Solution

Fig.2 Numerical solution and analytical solution of amplitude-frequency characteristic curves under different fractional orders, m=5,k1=1, k2=3, c=0.2, K=2, F=2: (a)p=0.4; (b)p=0.6; (c)p=0.8; (d)p=1.0

To verify the correctness of the approximate analytical solution of the amplitude-frequency characteristics.Discrete Eq.(28) by an algorithm based on the G-L definition.Take a set of basic parameters, and take the fractional differential term order isp=0.4, 0.6, 0.8, 1.0.The analytical solution of the amplitude-frequency curve of Eq.(14) is shown by the solid line in Fig.2.The numerical solution of the roller system defined by G-L of Eq.(28) is shown by the discrete points in Fig.2.And the abscissa is the excitation frequency, and the ordinate is the vibration amplitude.Fig.2 shows that the numerical solution and analytical solution of amplitude-frequency characteristic curves under different fractional orders have a good coincidence, and as the order increases, the fit is better.It is proved feasible to establish the horizontal nonlinear vibration model and solve it through the averaging method.

5.2 Discussion

Fractional-order is introduced in the roller vibration system.There are two important parameters, namely coefficientKand orderp, which are studied below.

Fig.3 shows the amplitude-frequency characteristic curves of the forced vibration system under different fractional orders.Fig.3 shows that with the fraction order increases, the maximum amplitude gradually decreases, which is further shown that the fractional order of the fractional differential term acts as a damping.

Fig.3 Amplitude-frequency characteristic curves of the forced vibration under different fractional orders, m=5, k1=2,k2=3, c=0.2, K=2, F=2

Fig.4 shows the amplitude-frequency characteristic curves of the forced vibration system under different coefficients.Fig.4 shows that as the coefficient increase, the amplitude-frequency characteristic curve shifts downward and right,that is, the maximum amplitude of the roller system gradually decreases, and the frequency of the system resonance gradually increases gradually,which indicates that the fractional differential term coefficientKalso plays a damping role in the vibration system.

Fig.4 Amplitude-frequency characteristic curves of the forced vibration system under different coefficients, m=5, k1=2,k2=3, c=0.2, p=0.4, F=2

In conclusion, the fractional differential term plays a damping role in the nonlinear vibration system, and the vibration amplitude and resonance frequency of the system can be adjusted by changing the order and coefficient, and the resonancecan be effectively avoided in the design process.

Fig.5 shows the amplitude-frequency characteristic curves of the roll system for different external excitation amplitudes.It is clear from the figure that the higher the external excitation amplitude, the higher the resonance amplitude of the system.The amount of horizontal vibration of the roll system can therefore be controlled by varying the rolling force.

Fig.5 Amplitude-frequency characteristic curves of the forced vibration system under different damping, m=5, k1=2,k2=3, K=2, p=0.4, F=2

Fig.6 shows the amplitude-frequency characteristic curve of the roll system under different damping coefficients.It is obvious from the figure that with the increase of dampingc, the resonance amplitude of the system gradually decreases and the resonance frequency also gradually decreases.In engineering practice, the damping can be controlled by changing the friction factor at the rolling interface and the fluctuation of the strip.

Fig.6 Amplitude-frequency characteristic curves of the forced vibration system under different excitation amplitudes,m=5, k1=2, k2=3, c=0.2, K=2, p=0.4

Fig.7 and Fig.8 show the amplitude-frequency characteristic curves of the system when the linear stiffness factork1and the non-linear stiffness factork2are gradually increased, respectively.From the figures, it can be seen that with the increase ofk1andk2, the resonance amplitude of the system does not change significantly,but the resonance frequency gradually increases,which indicates that the greater the stiffness the better the stability of the system, and increasing the stiffness can improve the stability of the system.Therefore, it is possible to reduce the amplitude of the resonance by reasonably increasing the horizontal stiffness and weakening the nonlinearity of the system.

Fig.7 Amplitude-frequency characteristic curves of the forced vibration system under different stiffness coefficient k1,m=5, k2=3, c=0.2, K=2, p=0.4, F=2

Fig.8 Amplitude-frequency characteristic curves of the forced vibration system under different stiffness coefficient k2,m=5, k1=2, c=0.2, K=2, p=0.4, F=2

Fig.9 Numerical response of the roll system to forced vibration when ω = 0.5, and m=5, k1=2, k2=3, c=0.2, K=2,F=2, a=0.4

Fig.10 Phase trajectory of the roll forced vibration system when ω = 0.5, and m=5, k1=2, k2=3, c=0.2, K=2,F=2, a=0.4

Fig.12 Phase trajectory of the roll forced vibration system when ω = 1.2, and m=5, k1=2, k2=3, c=0.2, K=2,F=2, a=1.1

Fig.9 and Fig.10 show the time response and phase trajectory curves of the forced vibration system of the roll system when the external excitation frequency is a low frequency excitation, respectively.Fig.11 and Fig.12 show the time response and phase trajectory curves of the forced vibration system of the roll system when the external excitation frequency is a high frequency excitation, respectively.From the two sets of graphs, it can be seen that the low frequency signal amplitude is positively correlated with the value of the order, while the high frequency signal amplitude is negatively correlated with the value of the order and the overall amplitude range of the low frequency signal is smaller than that of the high frequency signal.The response of the high frequency signal is chaotic,while the response of the low frequency signal is smooth.The main reason for this is that the low frequency signal is weakly excited and the high frequency signal is strongly excited.This study has shown that a reasonably fractional differential term has a direct effect on the horizontal non-linear dynamic characteristics of the roll system.Therefore, in the design of the roll system,the system intrinsic structure relationship should be reasonably designed to avoid the appearance of chaos, which can effectively ensure the stability of the system.

Fig.13 shows the response curve of the free vibration system of the roll system.It is clear from Fig.13 that the response of the system is an under damped decaying waveform and the amplitude becomes smoother as the fractional order increases.Fig.14 shows the phase trajectory curve of the free vibration system of the roll system at orderp=0.3.It can be seen that the phase trajectory of the system is a spiral line close to the origin.Fig.15 shows the curve of the amplitude of the free vibration system of the roll system as a function of time.As we can see from the graph, an increase in the fractional order will result in a sharp decrease in the amplitude of the system.Fig.16 shows the phase versus time for a free vibration system on a roller system.Therefore, the fractional order differential term plays the role of equivalent stiffness and equivalent damping in the system, and is a mathematical description of the intrinsic relationship of the system, which directly affects the dynamic characteristics of the roll system, so the intrinsic parameters can be adjusted to ensure the stable operation of the roll system in the design of the roll system.

Fig.13 Numerical response of the roll system to free vibration,and m=5, k1=2, k2=3, c=0.2, K=2, a0=10

Fig.14 Phase trajectory of the free vibration system, and m=5, k1=2, k2=3, c=0.2, K=2, a0=10

Fig.15 Time vs.amplitude curve of free vibration system, and m=5, k1=2, k2=3, c=0.2, K=2, a0=0, ω =2

Fig.16 Time vs.phase curve of free vibration system, and m=5, k1=2, k2=3, c=0.2, K=2, a0=0, ω =2

In order to verify the accuracy of the theoretical study, a laboratory 150mm mill was used as the research object to collect the vibration data of the roll system during the rolling process,and the field sensors were installed as shown in Fig.17.The simulated signal of the roll system vibration equation with and without the fractional order differential term is compared with the measured signal, as shown in Fig.18.As can be seen from Fig.18, the time domain and spectral characteristics of the vibration equation with fractional order are closer to the measured signal,which proves the validity of the theoretical study and its superiority to the integer order theory.

Fig.17 Vibration testing

Fig.18 Vibration signal comparison: (a) time domain signal;(b) frequency domain signal

6 Conclusion

In this paper, the damping nonlinear and stiffness nonlinear of the mill system are both considered simultaneously when building the horizontal nonlinear vibration model of the mill, and the fractional-order derivative term is introduced.The amplitude-frequency characteristics and phase-frequency characteristics of the forced and free vibration systems of the roll system are investigated using the averaging method.Through numerical simulations, the analytical and numerical solutions of the amplitude-frequency characteristics of the forced vibration system are compared, and it is demonstrated that the results of both are in good agreement.The effects of fractional coefficients and fractional orders on the response of the forced vibration system are then investigated separately.The response curves of the forced vibration system

and the free vibration system are plotted for low and high external excitation frequencies, respectively.Finally, the time-frequency and phase-frequency characteristics of the free vibration are also investigated.The validity of the theoretical study is verified by the analysis of the measured data.It is shown that the introduction of the fractional order differential term can reveal the non-linear vibration characteristics of the roll system at a deeper level and provide new ideas for the study of the vibration theory of the roll system.