# Resonance Analysis of Horizontal Nonlinear Vibrations of Roll Systems for Cold Rolling Mills under Double-Frequency Excitations

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Horizontal Nonlinear Equation of the Roller System

_{1}is the linear stiffness coefficient, k

_{2}is the nonlinear stiffness coefficient, m is equivalent mass of the roll, F

_{1}is the low-frequency excitation amplitude, F

_{2}is the high-frequency excitation amplitudes, ω is the low excitation frequency, Ω is the high excitation frequency, e is the clearance between the bearing housing and the frame, and x(t) is the horizontal displacement of the roller system. Due to the presence of e, the roll system vibrates back and forth under horizontal excitation forces, which is defined as a horizontal excitation force of ${F}_{1}\mathrm{cos}(\omega t)+{F}_{2}\mathrm{cos}(\Omega t)$.

_{0}, which do not have a known physical meaning, thereby making the results problematic in terms of practical applications in the village. Therefore, this paper adopts a more engineeringly meaningful definition of Caputo in order to deal with fractional order differential terms.

_{0}= 0 and t = t

_{i}for the different values of p, then the derivative under the Caputo definition is as per [45].

## 3. Bifurcation and Vibration Resonance

#### 3.1. Pitchfork Bifurcation

_{1}> 0, then the ${V}_{e}$ is a single-well function and that the system has a unique stable equilibrium point ${X}_{1}^{*}=0$. When the S

_{1}< 0, ${V}_{e}$ is a double-well function and the system has two stable equilibrium points of ${\mathrm{X}}_{2,3}^{\ast}=\pm \sqrt{-{S}_{1}/{S}_{2}}$ and one unstable equilibrium point ${X}_{1}^{*}=0$. The influence of the system parameters on the equilibrium points are shown in Figure 2. The figures show that the system has a forked bifurcation and that different parameters have different effects on the degree of bifurcation. This phenomenon occurs because the high frequency signal softens the stiffness of the system, as well as alters the equilibrium point of the system, which thus causes the nonlinear system to become a pitchfork bifurcation.

#### 3.2. Vibration Resonance

## 4. Single-well and Double-well Systems

#### 4.1. Single-well System

_{VR}is present when ${\omega}^{2}>{\omega}_{0}^{2}+\delta {\omega}^{p}\mathrm{cos}(p\pi /2)$, and when $p\ne 1$; as such, F

_{VR}is not only related to $\delta $, but is also the value of p.

_{VR}and p. It is clear from the figure that the system has a resonance phenomenon when $p\ge 0.84$ and that there is no F

_{VR}when $p<0.84$. As such, it is necessary to investigate the resonance phenomenon in the range of $0.84\le p\le 2$. Figure 6b is a plot of Q versus F and p. It is clear that a change in the value of F or p may trigger a vibration resonance. Figure 6c shows the relationship curve between Q and F. As the value of p changes, the relationship curve changes from a monotonic curve to a peak curve. Resonance only occurs at p = 1.8. This implies that the resonance of the single-well function may be caused by fractional damping.

#### 4.2. Double-well System

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Yarita, I.; Furukawa, K.; Seino, Y.; Takimoto, T.; Nakazato, Y.; Nakagawa, K. Analysis of chattering in cold rolling for ultrathin gauge steel strip. Trans. Iron Steel Inst. Jpn.
**1978**, 18, 1–10. [Google Scholar] [CrossRef] - Tamiya, T.; Furui, K.; Lida, H. Analysis of chattering Phenomenon in cold Rolling. In Proceedings of the International Conference on Steel Rolling, Tokyo, Japan, 29 September–4 October 1980; pp. 1192–1202. [Google Scholar]
- Hou, D.X.; Peng, R.R.; Liu, H.R. Vertical-Horizontal coupling vibration characteristics of strip mill rolls under the variable friction. J. Northeast. Univ. (Nat. Sci.)
**2013**, 34, 1615–1619. [Google Scholar] - Hou, D.X.; Chen, H.; Liu, B. Analysis on parametrically excited nonlinear vertical vibration of roller system in rolling mills. J. Vib. Shock
**2009**, 28, 1–5. [Google Scholar] - Hou, D.X.; Zhu, Y.; Liu, H.R.; Liu, F.; Peng, R. Research on nonlinear vibration characteristics of cold rolling mill based on dynamic rolling force. J. Mech. Eng.
**2013**, 49, 45–50. [Google Scholar] [CrossRef] - Hou, D.X.; Wang, X.G.; Zhang, H.W.; Zhao, H.X. Parametrically excited vibration characteristics of cold rolling mill under nonlinear dynamic rolling process. J. Northeast. Univ. (Nat. Sci.)
**2017**, 38, 1754–1759. [Google Scholar] - Huang, J.L.; Zhang, Y.; Gao, Z.Y.; Zeng, L. Influence of asymmetric structure parameters on rolling mill stability. J. Vibroeng.
**2017**, 19, 4840–4853. [Google Scholar] [CrossRef][Green Version] - Huang, J.L.; Zang, Y.; Gao, Z.Y. Influence of friction coefficient asymmetry on vibration and stability of rolling mills during hot rolling. Chin. J. Eng.
**2019**, 41, 1465–14722. [Google Scholar] - Sun, Y.Y.; Xiao, H.F.; Xu, J.W. Nonlinear vibration characteristics of a rolling mill system considering the roughness of rolling interface. J. Vib. Shock
**2017**, 36, 113–120. [Google Scholar] - He, D.P.; Wang, T.; Xie, J.Q.; Ren, Z.; Liu, Y. An analysis on parametrically excited nonlinear vertical vibration of a roller system in corrugated rolling mills. J. Vib. Shock
**2019**, 38, 164–171. [Google Scholar] - He, D.P.; Wang, T.; Xie, J.Q.; Ren, Z.; Liu, Y.; Ma, X. Research on principal resonance bifurcation control of roller system in corrugated rolling mills. J. Mech. Eng.
**2020**, 56, 109–118. [Google Scholar] - He, D.P.; Xu, H.D.; Wang, T. Nonlinear time-delay feedback controllability for vertical parametrically excited vibration of roll system in corrugated rolling mill. Metall. Res. Technol.
**2020**, 117, 3–12. [Google Scholar] [CrossRef][Green Version] - Liu, B.; Jiang, J.H.; Liu, F.; Liu, H.; Li, P. Nonlinear vibration characteristic of strip mill under the coupling effect of roll-rolled piece. J. Vibroeng.
**2016**, 18, 5492–5505. [Google Scholar] [CrossRef][Green Version] - Liu, B.; Jiang, J.H.; Liu, F.; Liu, H.; Li, P. Nonlinear vibration characteristics of strip mill influenced by horizontal vibration of rolled piece. China Mech. Eng.
**2016**, 27, 2513–2520. [Google Scholar] - Zhang, R.C.; Chen, Z.K.; Wang, F.B. Study on parametrically excited horizontal nonlinear vibration in single-roll driving mill system. J. Vib. Shock.
**2010**, 29, 105–108. [Google Scholar] - Yang, X.; Li, J.Y.; Tong, C.N. Nonlinear vibration modeling and stability analysis of vertical roller system in cold rolling mill. J. Vib. Meas. Diagn.
**2013**, 33, 302–306. [Google Scholar] - Seilsepour, H.; Zarastv, M.; Talebitooti, R. Acoustic insulation characteristics of sandwich composite shell systems with double curvature: The effect of nature of viscoelastic core. J. Vib. Control
**2022**, 29, 5–6. [Google Scholar] [CrossRef] - Ghafouri, M.; Ghassabi, M.; Zarastvand, M.R. Sound Propagation of Three-Dimensional Sandwich Panels: Influence of Three-Dimensional Re-Entrant Auxetic Core. AIAA J.
**2022**, 60, 6374–6384. [Google Scholar] [CrossRef] - Ghayesh, M.H. Nonlinear transversal vibration and stability of an axially moving viscoelastic string supported by a partial viscoelastic guide. J. Sound Vib.
**2008**, 314, 757–774. [Google Scholar] [CrossRef] - Ghayesh, M.H.; Moradian, N. Nonlinear dynamic response of axially moving, stretched viscoelastic strings. Arch. Appl. Mech.
**2011**, 81, 781–799. [Google Scholar] [CrossRef] - Ghayesh, M.H.; Alijani, F.; Darabi, M.A. An analytical solution for nonlinear dynamics of a viscoelastic beam-heavy mass system. J. Mech. Sci. Technol.
**2011**, 25, 1915–1923. [Google Scholar] [CrossRef] - Ghayesh, M.H.; Amabili, M.; Farokhi, H. Coupled global dynamics of an axially moving viscoelastic beam. Int. J. Non-Linear Mech.
**2013**, 51, 54–74. [Google Scholar] [CrossRef] - Ghayesh, M.H.; Amabili, M. Nonlinear dynamics of axially moving viscoelastic beams over the buckled state. Comput. Struct.
**2012**, 112–113, 406–421. [Google Scholar] [CrossRef] - Ghayesh, M.H.; Amabili, M.; Farokhi, H. Two-dimensional nonlinear dynamics of an axially moving viscoelastic beam with time-dependent axial speed. Chaos Solitons Fractals
**2013**, 52, 8–29. [Google Scholar] [CrossRef] - Ghayesh, M.H. Parametrically excited viscoelastic beam-spring systems: Nonlinear dynamics and stability. Struct. Eng. Mech.
**2011**, 40, 705–718. [Google Scholar] [CrossRef] - Farokhi, H.; Ghayesh, M.H.; Hussain, S. Three-dimensional nonlinear global dynamics of axially moving viscoelastic beams. J. Vib. Acoust.
**2016**, 138, 011007. [Google Scholar] [CrossRef][Green Version] - Ghayesh, M.H.; Farokhi, H.; Hussain, S. Viscoelastically coupled size-dependent dynamics of microbeams. Int. J. Eng. Sci.
**2016**, 109, 243–255. [Google Scholar] [CrossRef] - Liu, L.; Wang, J.; Zhang, L.C.; Zhang, S. Multi-AUV Dynamic Maneuver Countermeasure Algorithm Based on Interval Information Game and Fractional-Order DE. Fractal Fract.
**2022**, 6, 235. [Google Scholar] [CrossRef] - Lu, Z.Q.; Gu, D.H.; Ding, H.; Lacarbonara, W.; Chen, L.Q. Nonlinear vibration isolation via a circular ring. Mech. Syst. Signal Process.
**2020**, 136, 106490. [Google Scholar] [CrossRef] - Li, X.; Dong, Z.Q.; Wang, L.P.; Niu, X.D.; Yamaguchi, H.; Li, D.C.; Yu, P. A magnetic field coupling fractional step lattice Boltzmann model for the complex interfacial behavior in magnetic multiphase flows. Appl. Math. Model.
**2023**, 117, 219–250. [Google Scholar] [CrossRef] - Mainardi, F. Fractional Calculus and Waves in Linear Viscoelasticity; Imperial College Press: London, UK, 2010. [Google Scholar]
- Meral, F.C.; Royston, T.J.; Magin, R. Fractional calculus in viscoelasticity: An experimental study. Commun. Nonlinear Sci. Numer. Simul
**2010**, 15, 939–945. [Google Scholar] [CrossRef] - Xu, H.Y.; Yang, X.Y. Creep constitutive models for viscoelastic materials based on fractional derivatives. Comput. Math. Appl.
**2017**, 73, 1377–1387. [Google Scholar] [CrossRef] - Shen, L.J. Fractional derivative models for viscoelastic materials at finite deformations. Int. J. Solids Struct.
**2020**, 190, 226–237. [Google Scholar] [CrossRef] - Dang, R.Q.; Chen, Y.M. Fractional modelling and numerical simulations of variable-section viscoelastic arches. Appl. Math. Comput.
**2021**, 409, 126376. [Google Scholar] [CrossRef] - Hashemizadeh, E.; Ebrahimzadeh, A. An efficient numerical scheme to solve fractional diffusion-wave and fractional Klein–Gordon equations in fluid mechanics. Physica A
**2018**, 503, 1189–1208. [Google Scholar] [CrossRef] - Odibat, Z.; Momani, S. The variational iteration method: An efficient scheme for handling fractional partial differential equations in fluid mechanics. Comput. Math. Appl.
**2009**, 58, 2199–2208. [Google Scholar] [CrossRef][Green Version] - Magin, R.L.; Ovadia, M. Modeling the cardiac tissue electrode interface using fractional calculus. IFAC Proc.
**2008**, 39, 1431–1442. [Google Scholar] [CrossRef] - Mainardi, F. Fractional Calculus. In Fractals and Fractional Calculus in Continuum Mechanics; Springer: Berlin, Germany, 1997; pp. 291–348. [Google Scholar]
- Rossikhin, Y.A.; Shitikova, M.V. Application of fractional calculus for dynamic problems of solid mechanics: Novel trends and recent results. Appl. Mech. Rev.
**2010**, 63, 10801. [Google Scholar] [CrossRef] - Yan, Z.; Wang, W.; Liu, X.B. Analysis of a quintic system with fractional damping in the presence of vibrational resonance. Appl. Math. Comput.
**2018**, 321, 780–793. [Google Scholar] [CrossRef] - Xie, J.Q.; Wang, H.J.; Chen, L. Dynamical analysis of fractional oscillator system with cosine excitation utilizing the average method. Math. Methods Appl. Sci.
**2022**, 45, 10099–10115. [Google Scholar] [CrossRef] - Wen, B.C.; Li, Y.N.; Han, Q.K. Analytical Methods in Nonlinear Vibration Theory and Engineering Applications; Northeastern University Press: Shijiazhuang, China, 2000. [Google Scholar]
- Podlubny, I. Fractional Differential Equations; IBT-M in S and E; Academic Press: New York, NY, USA, 1999. [Google Scholar]
- Li, C. Numerical Methods for Fractional Calculus; CRC Press: Boca Raton, FL, USA, 2015. [Google Scholar]
- Xu, M.; Tan, W. Intermediate processes and critical phenomena: Theory, method and progress of fractional operators and their applications to modern mechanics. Sci. Chin.
**2006**, 49, 257–272. [Google Scholar] [CrossRef] - Li, C.; Peng, G. Chaos in Chen’s system with a fractional order. Chaos Solitons Fract.
**2004**, 22, 443–450. [Google Scholar] [CrossRef] - French, M.; Rogers, J. A survey of fractional calculus for structural dynamics applications. IMAC
**2001**, 1, 305–309. [Google Scholar] - Yang, Y.; Xu, W.; Gu, X.; Sun, Y. Stochastic response of a class of self-excited systems with Caputo-type fractional derivative driven by Gaussian white noise. Chaos Solitons Fractals
**2015**, 77, 190–204. [Google Scholar] [CrossRef] - Yang, J.H. Vibrational resonance in fractional-order anharmonic oscillators. Chin. Phys. Lett.
**2012**, 29, 104501–104504. [Google Scholar] [CrossRef] - Gammaitoni, L.; Hänggi, P.; Jung, P. Stochastic resonance. Rev. Mod. Phys.
**1998**, 70, 223. [Google Scholar] [CrossRef]

**Figure 1.**A physical model of the horizontal nonlinear vibrations with the fractional order of roll systems.

**Figure 2.**The parameter changes cause a forked bifurcation of the roll system (

**a**) F versus X*.(

**b**) $\beta $ versus X*. (

**c**) ${\omega}_{0}^{2}$ versus X*.

**Figure 3.**The potential function $V(x)={\omega}_{0}^{2}{x}^{2}/2+\beta {x}^{4}/4$ are shown as (

**a**) the single-well system and (

**b**) the double-well system.

**Figure 4.**The potential function ${V}_{e}(x)={\omega}_{0}^{2}{x}^{2}/2+\beta {x}^{4}/4$ is shown as a (

**a**) single-well system and (

**b**) a double-well system.

**Figure 5.**(

**a**) The surface diagram of the relationship between ω

_{VR}versus F and p. (

**b**) The top view of (

**a**). (

**c**) The plot of the critical frequency ω

_{VR}versus F at three different fractional orders. (

**d**) ω

_{VR}versus p for F = 0.2. (

**e**) The analytical results for the response amplitude Q versus ω and p for F = 0.2. (

**f**) The response amplitude Q versus ω for the three different values of p, while ω

_{0}

^{2}= 1; u

^{2}= 1; β = 1; and δ = 1.5.

**Figure 6.**(

**a**) The cover of F

_{VR}versus p. (

**b**) The analytical results for the response amplitude Q versus F and p. (

**c**) The response amplitude Q versus F for the three different values of p, while ω

_{0}

^{2}= 1; u

^{2}= 1; β = 1; and δ = 1.5.

**Figure 7.**(

**a**) The cover of Q versus p. (

**b**) The phase portrait of system (3) for the three values of p, while ω

_{0}

^{2}= 1; u

^{2}= 1; β = 1; and δ = 1.5.

**Figure 8.**(

**a**) Surface diagram of the relationship between ω

_{VR}versus F and p. (

**b**) The upward view of (

**a**). (

**c**) The plot of critical frequency ω

_{VR}versus F at three different fractional orders. (

**d**) ω

_{VR}versus p for F = 0.5. (

**e**) The analytical results for response amplitude Q versus ω and p for F = 0.5. (

**f**) The response amplitude Q versus ω for the three different values of p, while ω

_{0}

^{2}= −1; u

^{2}= 1; β = 1; and δ = 1.5.

**Figure 9.**(

**a**) The F

_{VR}versus ω for p = 0.3. (

**b**) The F

_{VR}versus ω for p = 1.0. (

**c**) The F

_{VR}versus ω for p = 1.8. (

**d**) The analytical result of F

_{VR}versus p. (

**e**) The analytical results of the response amplitude Q versus F and p for ω = 1. (

**f**) The response amplitude Q versus F for the different values of p, while ω

_{0}

^{2}= −1; u

^{2}= 1; β = 1; and δ = 1.5.

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Jiang, L.; Wang, T.; Huang, Q.-X. Resonance Analysis of Horizontal Nonlinear Vibrations of Roll Systems for Cold Rolling Mills under Double-Frequency Excitations. *Mathematics* **2023**, *11*, 1626.
https://doi.org/10.3390/math11071626

**AMA Style**

Jiang L, Wang T, Huang Q-X. Resonance Analysis of Horizontal Nonlinear Vibrations of Roll Systems for Cold Rolling Mills under Double-Frequency Excitations. *Mathematics*. 2023; 11(7):1626.
https://doi.org/10.3390/math11071626

**Chicago/Turabian Style**

Jiang, Li, Tao Wang, and Qing-Xue Huang. 2023. "Resonance Analysis of Horizontal Nonlinear Vibrations of Roll Systems for Cold Rolling Mills under Double-Frequency Excitations" *Mathematics* 11, no. 7: 1626.
https://doi.org/10.3390/math11071626