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1. IntroductionThe condition monitoring of the sheet production pro-cess calls for a better understanding of dynamical processesduring rolling. This understanding can be gained from themathematical analysis of physical models of the system:rolling stand–strip. Whereas the structure and parameters ofthe models of the mechanical elements of rolling mills arewell known, the models of the strip treated as a continuoussystem are still not well developed. As a result, the strip be-haviour and in particular its influence on the dynamics ofthe full system, is not well known.The present paper discusses an inertial model of stripbehaviour. In this model, for the first time in the litera-ture, the strip longitudinal feed motion has been taken intoaccount. The available observations indicate that the stripfeed velocity is the parameter, which when it takes on toohigh a value, can cause self-excited vibration of the sys-tem rolling stand–strip. Excessive vibration levels can resultin the quality deterioration of the strip by introducing di-mension and shape errors as well as lowering the surfacequality.The non-linear partial differential equation of transversevibration used in the present study, which accounts for thestrip longitudinal feed motion, has a more complex form∗ Corresponding author.34146
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E-mail addresses: bar@ap.krakow.pl (A. Bar), swiatoni@imir.agh.edu.pl(A. ´ Swi˛ atoniowski).than in the classical approach, since the position co-ordinate(which is usually taken as an independent co-ordinate) isnow a function of time.As a result of taking into account the longitudinal motion,the following phenomena can be described:• The dependence of the frequency of rolling stand vibrationon the vibration of the working rolls, which in turn de-pends on the feed velocity. This distinguishes the presentsystem from classical self-excited systems, in which thisfrequency is in principle close to one of the natural fre-quencies.• Narrower than in the classical approach regions of pa-rameters for which steady-state vibrations can take place.Therefore, the possibility of the occurrence of vibrationswith increasing amplitudes is much higher than followsfrom the analysis which does not account for a constantstrip feed velocity.• The increase of the strip feed velocity results in increasedeffective damping. One of the consequences of thisproperty is that higher magnitudes of the disturbancesof the strip movement necessary to start vibrations arepredicted.The above conclusions have been verified by numericalsimulations. Additionally, from numerical analyses one candetermine the influence of the rolling stand damping on theregions in which vibrations get excited as well as on thevibration amplitudes. 2. Equation of the strip parametric vibrationIn the paper the model of the system rolling stand–stripdescribed in paper [1] is used. The derivation of differen-tial equations of the strip moving at a constant velocityv1, was done by means of the second Newton law, for anelement of (dx) length. The aerodynamic drag, associatedwith the transversal motion, was taken into account, since alarge space is affected by this drag.
To describe this aero-dynamic load, a linear approximate formula was adopted:dR = α(dw/dt) dx. In this case these equations have thefollowing form:ρFd2udt2= ∂S∂x ,ρFd2wdt2= ∂∂x ∂w∂xS − αdwdt(2.1)where u is the horizontal co-ordinate, w the verticalco-ordinate.The internal force S(x, t), present in this equation, can beexpressed by the formula:S(x, t) = EFε0 + EFε + E Fdεdt,ε(x, t) = u x + 12(w x)2(2.2)To calculate the horizontal and vertical accelerations of el-ement ρF dx, which constitute the time complex derivativesof the second order, it is necessary to regard the fact, thatthe x co-ordinate is linearly time-dependent. The followingformulas yield:d2udt2= v21u xx + 2v1u xt+ u tt,dwdt= v1w x + w t,d2wdt2= v21w xx + 2v1w xt+ w tt,˙ ε = v1u xx + u xt+ v1w xxw xx + w xw xt(2.3)Introducing (2.3) into (2.1) and using the Kirchoff hy-pothesis [2] and non-dimensional quantities one obtains theequation of the strip transverse vibration in the form:ε0[w 0ττ + 2v0w 0τξ − (1 − v20)w 0ξξ]= [u0(1,τ) − u0(0,τ)]w 0ξξ − 2ζ01ε0(w 0τ + v0w 0ξ)+ 10(w 0ξ)2dξ w 0ξξ (2.4)whereτ =˜ ω0t, ξ = xl,w = w0L, u = u0L,v1 = v0L˜ ω0, ˜ ω20 = Eε0ρL2, 2 ˜ ω0ζ01 = αρF(2.5)The determination of the function u0(0,τ) is possiblewhen the equation of stream continuity, represents the strippassing through the roll-gap, and the variability of the dis-tance between the working rolls, results fromthe stand longi-tudinal vibration, approximately of harmonic character, aretaken into account [1]. The simple calculations yield to theformulas:u0(0,τ) =−2y0v0ν0+ 2y0v0ν0cos(ν0τ),y0 = amhav,ν0 = ωm˜ ω0(2.6)where ak is the amplitude of rolls vibration; y0 thenon-dimensional amplitude of rolls vibration; hav the aver-age thickness of the strip on V-roll gap output.The transformation from non-linear partial differentialequation (2.4) to an ordinary differential equation was doneusing Ritz method [3]. The selection of approximating func-tions in that method is limited only by the postulate of con-formance with the boundary conditions. Considering the firstapproximation as satisfactory enough, i.e. taking only onefunction from the whole sequence, we have to impose morerigorous physical requirements on that function.Since in the problem being analysed we assume the stateof parametric resonance, occurring, as we know, at a fre-quency close to the double of the natural frequency selected,therefore the function ϕn must be similar to the eigenfunc-tion. In the case when a strip is moving with a feed velocityof v0, the equation of free vibration of the strip is describedby the left-hand side of relationship (2.4):w 0ττ + 2v0w 0τξ + (1 − v20)w 0ξξ = 0Substituting there w0(ξ, τ) = g(ξ) eiΩτwe obtain an or-dinary equation:−Ω2g(ξ) + 2iv0Ωg (ξ) − (1 − v20)g (ξ) = 0The real part of the solution is the desired approximatingfunction. Simple calculation shows thatϕn(ξ) = sin(nπξ) sin(nπv0ξ) (2.7)After replacing in (2.4) with w0 − ϕn(ξ)fn(τ), and aftertransposing all terms to the left-hand side which will befurther referred to as Φ(ϕn,fn), it is required (according tothe method) that L0Φ(ϕn,fn)ϕn dξ = 0This leads to an ordinary differential equation having theform of¨ fn + 2ζ0n ˙ fn + β2n(1 − 2δn) cos(ν0τ)fn + cnf 3n = 0 (2.8)whereζ0n = ζ01 + a3na1nv0, 2δn = 2y0v0ε0ν0β20,β20 = a2na1n 1 + 2y0ε0ν0v0 − v20 ,cn = a2na4na1nε0(2.9) anda1n = 10ϕ2n(ξ) dξ, a2n = 10ϕn(ξ)ϕ n (ξ) dξ,a3n = 10ϕn(ξ)ϕ n(ξ) dξ, a4n = 10[ϕ n(ξ)]2dξ (2.10)As the transformation from the partial equation to the or-dinary equation was made using an approximate method, itwas also decided to solve (2.8) in an approximate manner.Such an approach is also preferred because it gives an ad-ditional opportunity of drawing some general physical con-clusions, in contrast to numerical methods.To find the solution,