Acta Mechanica

, Volume 230, Issue 6, pp 2087–2104 | Cite as

Modeling and control of resonance effects in steel casting mold oscillators

  • O. Angatkina
  • V. Natarajan
  • Z. Chen
  • M. Ding
  • J. BentsmanEmail author
Original Paper


A methodology for capturing resonance in a continuous-casting mold oscillator software testbed incorporating two coupled hydraulically actuated Timoshenko beams is proposed. The mechanism of the mold motion distortion generation in the mold oscillator is clearly delineated. Nontrivial damped natural frequency and resonance frequency calculations are carried out. Then, for a finite-difference analytical beam model approximation, a discovery of monotonic dependence on spatial approximation accuracy of the beam resonance frequency under the fixed mass and of the mass under the fixed resonance frequency is demonstrated numerically. Based on these findings, a novel beam parameters selection procedure for the precise attainment of the desired resonance and damped natural frequencies by analytical and numerical models under the relevant boundary conditions and runtime constraints is developed. Using this procedure, fitting of the beam model parameters to match the actual resonance frequencies exhibited by thin and thick slab casting mold oscillators at, respectively, the Nucor Decatur and the AK Steel Dearborn steel mills, is demonstrated. The resulting resonance frequency value for the latter is then used to guide the internal-model-principle-based controller design for resonance suppression in the software testbed for the AK Steel caster mold oscillator.


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This work was supported by NSF Award 1300907, UIUC Strategic Instructional Initiatives Program, UIUC Grant for Advancement of Teaching in Engineering, and UIUC Continuous Casting Consortium.


  1. 1.
    Szekeres, E.: Overview of mold oscillation in continuous casting. Iron Steel Eng. 73(7), 29–37 (1996)Google Scholar
  2. 2.
    Suzuki, M., et al.: Computer simulation for mold oscillation. Hitachi Zosen Tech. Rev. (Jpn.) 54(2), 13–19 (1993)MathSciNetGoogle Scholar
  3. 3.
    Lopez, P.E.R., Mills, K.C., Lee, P.D., Santillana, B.: A unified mechanism for the formation of oscillation marks. Metall. Mater. Trans. B Process Metall. Mater. Process. Sci. 43(1), 109–122 (2012)CrossRefGoogle Scholar
  4. 4.
    Natarajan, V., Bentsman, J.: Robust rejection of sinusoids in stable nonlinearly perturbed unmodelled linear systems: theory and application to servo. In: Proceeding of the American Control Conference, San Francisco, pp. 3289–3294 (2011)Google Scholar
  5. 5.
    Han, S.M., Benaroya, H., Wei, T.: Dynamics of transversely vibrating beam using four engineering theories. J. Sound Vib. 225(5), 935–988 (1999)CrossRefzbMATHGoogle Scholar
  6. 6.
    Huang, T.C.: The effect of rotary inertia and of shear deformation on the frequency and normal mode equations of uniform beams with simple end conditions. ASME J. Appl. Mech. 28, 579–584 (1961)CrossRefzbMATHGoogle Scholar
  7. 7.
    Timoshenko, S.: On the correction for shear of the differential equation for transverse vibration of prismatic bars. Philos. Mag. 41, 744–746 (1921)CrossRefGoogle Scholar
  8. 8.
    Rensburg, N.V., Merwe, A.V.D.: Natural frequencies and modes of a Timoshenko beam. Wave Motion 44, 58–69 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Zietsman, L., Rensburg, N.F., Merwe, A.J.: A Timoshenko beam with tip body and boundary damping. Wave Motion 34, 199–211 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Li, F., Huang, K.: Numerical approximation and error analysis for the Timoshenko beam equations with boundary feedback. Numer. Math. J. Chin. Univ. (Engl. Ser.) 16(3), 233–252 (2007)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Jafarali, P., Ameen, M., Prathap, G., Mukherjee, S.: Variational correctness and Timoshenko beam finite element elastodynamics. J. Sound Vib. 299(1–2), 196–211 (2007)CrossRefGoogle Scholar
  12. 12.
    Więckowski, Z., Golubiewski, M.: Improvement in accuracy of the finite element method in analysis of stability of Euler–Bernoulli and Timoshenko beams. Thin-Walled Struct. 45(10–11), 950–954 (2007)CrossRefGoogle Scholar
  13. 13.
    Mukherjee, S., Jafarali, P., Prathap, G.: A variational basis for error analysis in finite element elastodynamic problems. J. Sound Vib. 285, 615–635 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Więckowski, Z., Golubiewski, M.: Improvement in accuracy of the finite element method in analysis of stability of Euler–Bernoulli and Timoshenko beams. Thin-Walled Struct. 45(10–11), 950–954 (2007)CrossRefGoogle Scholar
  15. 15.
    Almeida Jr., D.S.: Conservative semidiscrete difference schemes for Timoshenko systems. J. Appl. Math. 2014, 686421 (2014). MathSciNetGoogle Scholar
  16. 16.
    Dorf, R.C., Bishop, R.H.: Modern Control Systems. Prentice Hall, Upper Saddle River (2001)zbMATHGoogle Scholar
  17. 17.
    Natarajan, V., Bentsman, J.: Approximate local output regulation for nonlinear distributed parameter systems. Math. Control Signals Syst. 28(24), 1–44 (2016)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Angatkina, O., Natarajan, V., Chen, Z., Bentsman, J.: Capturing and suppressing resonance in steel casting mold oscillation systems using Timoshenko beam model. In: Proceedings of the 2015 American Control Conference (ACC), July 1–3, Chicago, Illinois, pp. 559–564 (2015)Google Scholar

Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mechanical Science and EngineeringUniversity of Illinois at Urbana-ChampaignUrbanaUSA
  2. 2.Systems and Control Engineering GroupIndian Institute of Technology, BombayPowai, BombayIndia

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