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Other quantities per-taining to the strip are described in the succeeding drawings.Fig. 1 shows the relationship between numerically com-puted amplitudes and frequencies of the working roll axisvertical vibration and the strip tension required by the pro-cess, at various feed velocities. As it can be seen, the in-fluence of the feed velocity is significant and it cannot beneglected in the evaluation of amplitude levels. As regardsvibration frequency, this influence is not considerable.The computations show also that with lower values of striptension, the rolling stand vibration frequency is significantlydifferent from its natural frequency assumed as 125Hz. Inself-excited systems such a feature is relatively rare. A regionof stable vibration was obtained for the static tension valuesranging from 106 to 132MPa. At lower values of tension onecan expect even significantly increased amplitudes (unstableregion).Fig. 2 shows similar relationships but pertaining to thestrip itself. It can be noticed that, when the tension of thestrip is being reduced, the phase shift between the vibrationof the rolling stand and the strip approaches (−π/2). Atthat value the system becomes unstable. The influence ofthe strip feed velocity is significant as far as the phase isconcerned, but it is generally negligible in the evaluation ofstrip amplitudes (measured in the middle of the strip lengthfor its third vibration mode).Fig. 3 shows the influence of the rolling stand stiffness onits vibration intensity. Two rolling stands were compared forwhich the second natural frequencies are 140 and 199Hz,respectively. In the first case, third mode of strip vibrationoccurs, and in the second case, fourth mode is observed.It can be noticed that the areas where vibration occur aregetting narrower along with increasing rolling stand stiff-ness. The results obtained were compared with the results obtained fromthe “classical” approach that does not take intoaccount strip velocity. Dashed line indicates the minimummagnitude of disturbance that causes self-excited vibration.This disturbance may result, for example, from a micro-slipof the strip sliding over the rolls, local non-homogeneity ofthe strip material or the strip shape, etc.5. ConclusionsThe following conclusions can be formulated from theresults obtained in the paper:• An increase of the strip feed velocity results in highervalues of self-excited/parametric oscillations. This effectis also described by the classical approach, but with amuch smaller degree (Fig. 1).• The frequency of the vibration of the stand differs fromits natural frequency, the fact that is not typical of mostself-excited vibrations. This effect cannot be described bythe classical model.• The regions of the steady-state vibration regimes becomevery narrow (Fig. 1) and the possibility of the occurrenceof unstable vibrations with increasing amplitudes grows.• Our computation shows a linear dependence of the damp-ing coefficient from the strip transfer velocity.
Sincethis coefficient is related to the vibration mode number(Eqs. (2.4) and (2.5)), it is justifiable to assume thattaking into account the strip transfer velocity has thesame effect as adopting the material vibration dampingaccording to the Kelvin–Voigt model.The attempt to make the model reflect more closely thereal character of the rolling phenomenon is of much impor-tance in the analysis of data obtained from the vibration con-ditioning system. The observations of the past use of suchsystems indicate clearly that the conditioning system shouldnot be treated as a “black box”. On the contrary, the higheris the knowledge about the monitored system, the use of thecondition monitoring system can be made more efficient.AcknowledgementsThis work was subsidized by the State Committee for Sci-entific Research, KBN nr 5 T07C 029 22, Warsaw, Poland.References[1] Z. Drzymała, A. ´ Swi˛ atoniowski, A. Bar, Non-linear vibrations in coldrolling mills XIIIe, Colloque Vibrations, Ecole Centrale de Lyon,12–14 June 2002.[2] S. Timoshenko, D.H. Young, Vibration Problems in Engineering, VanNostrand, Princeton, NY, 1955, pp. 89–93.[3] W.J. Cunningham, Introduction to Non-linear Analysis, London, NY,1959, pp. 148–150 and 174–175.