Dimensional stability of thin extruded polymer sheets is analyzed as a function of draw ratio. It is assumed that the sheet keeps its rectangular cross section between the flat die and the chill roll and the flow is Newtonian. Through the analysis of linearized equations, it is shown that the sheet becomes unstable beyond a critical draw ratio as observed in experimental studies. The nonlinear model of sheet dynamics is then analyzed using the singular perturbation analysis to confirm the existence of a stable limit cycle. A feedback controller is designed to stabilize the sheet dynamics for draw ratios greater than the critical draw ratio.
Agasant, J.F., 1991, Polymer ProcessingPrinciples and Modelling, Hanser, Munich, Germany
2.
Barq, P., Haudin, J.M., Agassant, J.F., Roth, H., and Bourgin, P., 1990, "Instability phenomena in film casting process ," International Polymer Processing5, 264-271.
3.
Denn, M. and Fisher, R.J., 1975, "Finite amplitude stability and draw resonance in isothermal melt spinning," Chemical Engineering Science30, 1129-1134.
4.
Denn, M. and Fisher, R.J., 1976, "A theory of isothermal melt spinning and draw resonance ," AICh22, 236-246.
Ishihara, H. and Kase, S., 1975, "Studies on melt spinning. V. Draw resonance as a limit cycle," Journal of Applied Polymer Science an Introduction, Springer-Verlag, New York/ Berlin .
7.
Kase, S., 1974, "Studies on melt spinning. IV. On the stability of melt spinning," Journal of Applied Polymer Science18, 3279-3304.
8.
Kase, S., 1984, "Mathematical simulation of melt spinning dynamics ," in HighSpeed Fiber Spinning, A. Ziabicki and H. Kawai, eds., John Wiley, New York.
9.
Kase, S. and Araki, M., 1982, "Studies on melt spinning. VIII. transfer function approach ," Journal of Applied Polymer Science27, 4439-4465.
10.
Kase, S. and Matsuo, T., 1965, "Studies on melt spinning. I. Fundamental equations on the dynamics of melt spinning," Journal of Applied Polymer Science3A, 2541-2554.
11.
Kase, S. and Matsuo, T., 1967, "Studies on melt spinning. II. Steady state and transient solutions of fundamental equations compared with experimental results ," Journal of Applied Polymer Science11, 251-287.
12.
Khalil, K.K., 1992, Nonlinear Systems, Macmillan , New York.
13.
Mulpur, A. and Thompson, C., 1996, "Nonlinear control of optical fiber diameter variations ," IEEE Transactions on Control Systems Technology4(2), 152-162.
14.
Nayfeh, A.H. and Balachandran, B. , 1995, Applied Nonlinear Dynamics , John Wiley, New York.
15.
Nizami, J., 1997, Modeling, Stability Analysis and Control of DrawResonance in Polymer Sheet Extrusion, Ph.D. dissertation, University of Akron, OH.
16.
Nizami, J. and Batur, C., 1997, "Stabilizing controller for polymer sheet extrusion ," in Proceedings of the 36th IEEE Conference on Decision and Control Vol. 3(5), pp. 2543-2545.
17.
Pearson, J.R.A. and Shah, Y.T., 1974, "On the stability of isothermal and nonisothermal fiber spinning of power-law fluids," Ind. Eng. Chem. Fund.13(2), 234-238.
18.
Petrie, C.J.S. and Denn, M.M., 1976, "Instabilities in polymer processing," AIChE Journal22(2), 209-236.
19.
Saberi, A. and Khalil, H., 1985, "Stabilization and regulation of nonlinear singularly perturbed systems-Composite control," IEEE Transactions on Automatic ControlAC-30(8), 417-425.
20.
Schultz, W.W., Zebib, A., Davis, S.H., and Lee, Y., 1984, "Nonlinear stability of Newtonian fibres," Journal of Fluid Mechanics149, 455-475.
21.
Slotine, J.E. and Li, W., 1991, Applied Nonlinear Control, Prentice Hall, Englewood Cliffs, NJ.
22.
Straut, A.M. and Humphries, A.R., 1996, Dynamic Systems and Numerical Analysis, Cambridge University Press, Cambridge, UK.
23.
Vidyasagar, M., 1993, Nonlinear Systems Analysis, Prentice Hall, Englewood Cliffs, NJ.
