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the asymptoticmethod of Bogolubov–Krylov [3] based on the small parameter method proposedby Poincare was used in the paper. This method was se-lected because it determines exactly the first harmonic ofthe solution what is very important in case of parametricequations.With this assumption, the solution has the form:fn(τ) = an(τ) cos ν02τ + υn(τ)(2.11)In the steady statea0n = lim τ→∞ an(τ) = 4ν03cn ν02− βn ± δ2nβ4nν20− ζ20n,υ0n = lim τ→∞ υn(τ) =−arctg
ζ0nβn − (ν0/2) + (3c/4ν0)a20
(2.12)As it can be seen, the dimensionless amplitude of vibra-tion is inversely proportional to the coefficient cn. By per-forming integration as required in (2.10), it can be shownapproximately thatcn = 0.25(nπ)2ε0where the values of v0 correspond to the strip feed velocityin the range of (25, 30)m/s. The higher the mode number(n) is, the higher is also cn, and as a consequence, the ampli-tudes of strip vibration are smaller. For the same reason inthe actual systems only the fourth mode of transverse vibra-tion may appear. This conclusion is confirmed by numericalcomputations.Secondly, in order to ensure a positive value of the ex-pression under the root in (2.12), it is required to maintainthe appropriate relationship between the parameters of therolling stand, the strip, and the rolling process itself. By sub-stituting relationship (2.10) into: δnβ2nν0
2≥ ζ0nwe obtain an inequality in which dimensional expressionsappear. This inequality indicates that a steady-state paramet-ric vibration of the strip may be generated only if the initialamplitude of vertical vibration of the rolls ak in the rollingstand has a certain, adequate value:ak ≥ hav ˜ ω2kL2√ρ(ζ0n + (0.12πnv1/L˜ ω0))Eν√σ0In order to initiate a self-excited and parametric process,such vibration amplitude must be caused by a phenomenonwhich is different from the ones described above,
for exam-ple by strip material non-homogeneity, instantaneous distur-bances of constant feed velocity, slipping of rolls, etc.It can be noticed that the possibility of exciting vibrationof the type described is higher when:• the rolling stand has more compliant structure, ˜ ωk is itssecond natural frequency;• static tensile stress σ0 is higher;• lower modes of strip vibration arise, this takes place athigher values of σ0 and confirms the previous observation;strip thickness hav is relatively small, and the feed veloc-ity v1 is high. Both these conditions are fulfilled for thestrip segments situated between the last stands of the rollingline.3. Equation of self-excited vibration of the millThe phenomenon of vertical vibration observed in rollingstands features frequencies in the range of (120, 190Hz).An analysis of this vibration leads to the conclusion thatthe vibration represents the second mode, which means thatthe working roll and the supporting roll (forming a pair ofrolls) move in opposite directions. Assuming that the upperpart and the lower part of the stand are symmetrical, eachof them can be treated as a harmonic oscillator with onedegree of freedom. The mass (m) of it is equivalent to thecombined mass of the working roll and the supporting roll,and the stiffness of the spring element is replaced by theequivalent stiffness of the stand housing, the bearings and thepressure bolt (km). The reaction P of the strip being rolledcan be described using the relationship given by Hill andSims:P = (K − σx)b√R,hwhere K is the creep limit of the strip material; σx the ax-ial stress at the boundary of the area subjected to plasticdeformation; b the average width of the strip; R the re-duced radius of the working roll; ,h the change in the stripthickness between the inlet and the outlet of the rollingstand.The value of the axial stress is determined by three com-ponents: σ0 resulting from strip tension required by the pro-cess, σ1 caused by the change in the vertical gap between theworking rolls due to their vertical vibration, and σ2 being aconsequence of strip vibration between subsequent rolling stands. The stress σ0, according to (2.2), is described by thefollowing formula:σ0 = Eε0 (3.1)The value of σ1 determines horizontal displacement ofstrip cross sections in the proximity of the gap—u(0,t).Itcan be assumed approximately thatσ1 = Eu(0,t)LRelationship (2.6) leads to an expression containing di-mensionless quantities used beforeσ1 = EhavLu0(0,τ) (3.2)The dynamic component σ2 is defined by time-dependentrelative elongation of the axis of transversally vibrating strip−ε2. It can be calculated using the formula for curve lengthcomputation:ε2 = ln(t) − LL= 1L L0 1 + (w )2 dx − 1 ∼ = 12 10(w 0ξ)2dξ (3.2)Taking into account (2.7) and (2.11), after simple trans-formations we obtain an expression for the stress discussedabove:σ2 = Eε2 = 18a20Ebn 1 − cos ν02τ + υ0,bn = 14 1 − 1 − 2v202πnv0(1 − v20)sin(2nπv0)(3.3)According to the assumptions made and the discussiondone, the differential equation of motion of the workingroll axis described by variable (y) will have the followingform:m¨ y + αm ˙ y + kmy = P(t) − P0= (K − σ0 − σ1 − σ2)b R,h(t)−(K − σ0)b R,h0 (3.4)In this formula, the coefficient α stands for equivalentdamping (energy dissipation), ,h0 the rolling reduction, and,h(t) = ,h0 − 2y. Using (2.6) and (3.1)–(3.3), after sometransformations we obtain:¨ z0 + αm˜ ω0˙ z0 + ks + kmm˜ ω20z0=− ksm˜ ω20Ny0ν0σ(t) ,h0hav− z0 − 12hav,h0z20 (3.5)The formula contains the following new symbols definedas follows:z = yhav,ks = (K − σ0)b R,h0(strip equivalent stiffness),N = EK − σ0,σ(t) = 2v0Ey0πν0(1 − cos ν0τ) + nπ22Ea20nbn[1 − cos(ν0τ + 2υ0)] (3.6)The parameter y0 defined by relationship (2.6) is the am-plitudal value of the variable z0. Eq. (3.5) was solved analyt-ically using an approximate method, similarly as in the caseof the equation of strip vibration. Formulas (2.8) and (3.5)determine unambiguously the vibration of a system consist-ing of the rolling stand and the strip.4. Results of numerical analysisThe calculations were provided for a continuous systemof the rolling mil (tandem) consisting of six stands. TheFig. 1. The effect of the strip feed velocity on the amplitude of verticalvibrations of the working rolls, a and their frequency, fw as a function ofthe static tension σ0. Unstable solutions are shown by dashed line. Fig. 2. The amplitude and phase (relative to the vibration of the workingrolls) of points on the strip at distance equal to 1.75m from the vibratingstand, as a function of static tension.numerical simulation has shown that the analysed vibrationstake place only at the sixth stand and in the strip sectionconnecting the fifth and sixth stand. The second natural fre-quency of this stand was found to be about 140Hz. Thetransverse vibrations of the strip are with the third vibrationmode (n = 3). The results of the calculations are shown inthe form of plots in Figs. 1–3.Inertia and damping parameters of the rolling stand wereadjusted so as to obtain the natural frequency in the range of(120, 200Hz), and the dimensionless damping coefficient’svalues ranged between 0.05 and 0.1. The distance betweensubsequent rolling stands L and the strip width b were as-sumed as 3.5 and 0.9m, respectively.