Advertisement

Acta Mechanica Sinica

, 27:578 | Cite as

Stochastic optimal control of cable vibration in plane by using axial support motion

  • Ming Zhao
  • Wei-Qiu ZhuEmail author
Research Paper

Abstract

A stochastic optimal control strategy for a slightly sagged cable using support motion in the cable axial direction is proposed. The nonlinear equation of cable motion in plane is derived and reduced to the equations for the first two modes of cable vibration by using the Galerkin method. The partially averaged Itô equation for controlled system energy is further derived by applying the stochastic averaging method for quasi-non-integrable Hamiltonian systems. The dynamical programming equation for the controlled system energy with a performance index is established by applying the stochastic dynamical programming principle and a stochastic optimal control law is obtained through solving the dynamical programming equation. A bilinear controller by using the direct method of Lyapunov is introduced. The comparison between the two controllers shows that the proposed stochastic optimal control strategy is superior to the bilinear control strategy in terms of higher control effectiveness and efficiency.

Keywords

Stay cable Active control Stochastic optimal control Dynamical programming principle 

References

  1. 1.
    Mastsumoto, M., Saitoh, T., Kitazawa, M., et al.: Response characteristics of rain-wind induced vibration of stay-cables of cable-stayed bridges. Journal of Wind Engineering and Industrial Aerodynamics 57(2–3), 323–333 (1995)CrossRefGoogle Scholar
  2. 2.
    Mastsumoto, M., Shirato, H., Yagi, T., et al.: Field observation of the full-scale wind-induced cable vibration. Journal of Wind Engineering and Industrial Aerodynamics 91(1–2), 13–26 (2003)CrossRefGoogle Scholar
  3. 3.
    Wu, Q., Takahashi, K., Okabayashi, T., et al.: Response characteristics of local vibrations in stay cables on an existing cablestayed bridge. Journal of Sound and Vibration 261(3), 403–420 (2003)CrossRefGoogle Scholar
  4. 4.
    Pacheco, B.M., Fujino, Y., Sulekh, A.: Estimation curve for modal damping in stay cables with viscous damper. ASCE Journal of Structural Engineering 119(6), 1961–1979 (1993)CrossRefGoogle Scholar
  5. 5.
    Yu, Z., Xu, Y.L.: Non-linear vibration of cable-damper systems Part I: formulation. Journal of Sound and Vibration 225(3), 447–463 (1999)CrossRefGoogle Scholar
  6. 6.
    Xu, Y.L., Yu, Z.: Non-linear vibration of cable-damper systems Part II: application and verification. Journal of Sound and Vibration 225(3), 465–481 (1999)CrossRefGoogle Scholar
  7. 7.
    Abedel-rohman, M., Spencer, B.F.: Control of wind-induced nonlinear oscillations in suspended cables. Nonlinear Dynamics 37(4), 341–355 (2004)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Lee, S.Y., Mote, C.D. Jr.: Vibration control of an axially moving string by boundary control. Journal of Dynamic System, Measurement and Control 118(1), 66–74 (1996)zbMATHCrossRefGoogle Scholar
  9. 9.
    Canbolat, H., Dawson, D., Rahn, C., et al.: Adaptive boundary control of out-of-plane cable vibration. Journal of Applied Mechanics 65(4), 963–969 (1998)MathSciNetCrossRefGoogle Scholar
  10. 10.
    de Queriroz, M.S., Dawson, D.M., Rahn, C.D., et al.: Adaptive vibration control of an axially moving string. Journal of Vibration and Acoustics 121(1), 41–49 (1999)CrossRefGoogle Scholar
  11. 11.
    Fujino, Y., Susumpow, T.: Active control of cables by axial support motion. Smart Material and Structures 4(1A), 41–51 (1995)CrossRefGoogle Scholar
  12. 12.
    Fujino, Y., Warnitchai, P., Pacheco, B.M.: Active stiffness control of cable vibration. Journal of Applied Mechanics 60(4), 948–953 (1993)CrossRefGoogle Scholar
  13. 13.
    Gattulli, V., Pasca, M., Verstroni, F.: Nonlinear oscillations of a nonresonant cable under in plane excitation with a longitudinal control. Nonlinear Dynamics 14(2), 139–156 (1997)zbMATHCrossRefGoogle Scholar
  14. 14.
    Gehle, R.W., Masri, S.F.: Active control of shallow, slack cable using the parametric control of end tension. Nonlinear Dynamics 17(1), 77–94 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Pasca, M., Vestroni, F., Gattulli, V.: Active longitudinal control of wind-induced oscillations of a suspended cable. Meccanica 33(3), 255–266 (1998)zbMATHCrossRefGoogle Scholar
  16. 16.
    Susumpow, T., Fujino, Y.: Active control of multimodal cable vibration by axial support motion. Journal of Engineering Mechanics 121(9), 964–972 (1995)CrossRefGoogle Scholar
  17. 17.
    Fujino, Y., Susumpow, T.: An experimental study on active control of in plane cable vibration by axial support motion. Earthquake Engineering and Structural Dynamics 23(12), 1283–1297 (1994)CrossRefGoogle Scholar
  18. 18.
    Zhu, W.Q., Ying, Z.G.: Optimal nonlinear feedback control of quasi-Hamiltonian systems. Science in China, Series A 42(11), 1213–1219 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Zhu, W.Q., Ying, Z.G., Soong, T.T.: An optimal nonlinear feedback control strategy for randomly excited structural systems. Nonlinear Dynamics 24(1), 31–51 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Ying, Z.G., Zhu, W.Q., Soong, T.T.: A stochastic optimal semiactive control strategy for ER/MR dampers. Journal of Sound and Vibration 259(1), 45–62 (2003)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Ying, Z.G., Zhu, W.Q.: A stochastically averaged optimal control strategy for quasi-Hamiltonian systems with actuator saturation. Automatica 42(9), 1577–1582 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Cai, Y., Chen, S.S.: Dynamics of elastic cable under parametric and external resonances. Journal of Engineering Mechanics 120(8), 1786–1802 (1994)CrossRefGoogle Scholar
  23. 23.
    Zhu, W.Q., Yang, Y.Q.: Stochastic averaging of quasi-nonintegrable-Hamiltonian systems. Journal of Applied Mechanics 64(1), 157–164 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Arnold, L.: Stochastic Differential Equations: Theory and Applications. Wiley, New York (1974)zbMATHGoogle Scholar
  25. 25.
    Caughey, T.K.: Nonlinear theory of random vibrations. Advances in Applied Mechanics 11, 209–253 (1971)CrossRefGoogle Scholar

Copyright information

© The Chinese Society of Theoretical and Applied Mechanics; Institute of Mechanics, Chinese Academy of Sciences and Springer-Verlag GmbH 2011

Authors and Affiliations

  1. 1.Department of Mechanics, State Key Laboratory of Fluid Power Transmission and ControlZhejiang UniversityHangzhouChina

Personalised recommendations