Vibration control of a viscoelastic flexible marine riser with vessel dynamics
Abstract
In this work, we investigate the asymptotic behavior of solutions of a viscoelastic flexible marine riser with vessel dynamics. Under a suitable control applied at the top end of the riser, we establish explicit decay rates for a large class of relaxation functions. In particular, exponentially and polynomially (or power type) decaying functions are included in this class. Our method is based on the multiplier technique. Numerical simulations justifying the effectiveness of the proposed boundary control to suppress the vibrations of the flexible marine riser are provided.
Keyword : stability, vibration control, flexible marine riser, boundary control, Euler-Bernoulli beam structure, viscoelasticity
How to Cite
Seghour, L., Berkani, A., Tatar, N.- eddine, & Saedpanah, F. (2018). Vibration control of a viscoelastic flexible marine riser with vessel dynamics. Mathematical Modelling and Analysis, 23(3), 433-452. https://doi.org/10.3846/mma.2018.026
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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[25] Y. Liu, Z. Zhao and W. He. Stabilization of an axially moving accelerated/decelerated system via an adaptive boundary control. ISA Transactions, 64:394–404, 2016. https://doi.org/10.1016/j.isatra.2016.04.006
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[27] J.Y. Park, Y.H. Kang and J.A. Kim. Existence and exponential stability for a Euler-Bernoulli beam equation with memory and boundary output feedback control term. Acta. Appl. Math., 104(3):287–301, 2008. https://doi.org/10.1007/s10440-008-9257-8
[28] J.Y. Park and J.A. Kim. Existence and uniform decay for EulerBernoulli beam equation with memory term. Math. Meth. Appl. Sci., 27(14):1629–1640, 2004. https://doi.org/10.1002/mma.512
[29] L. Seghour, A. Khemmoudj and N.-E. Tatar. Control of a riser through the dynamic of a vessel. Appl. Anal., 95(9):1957–1973, 2016. https://doi.org/10.1080/00036811.2015.1080249
[30] N.-E. Tatar. Uniform decay in viscoelasticity for kernels with small non-decreasingness zones. Appl. Math. Comput., 218(15):7939–7946, 2012. https://doi.org/10.1016/j.amc.2012.02.012
[2] A. Berkani, N.-E. Tatar and A. Kelleche. Vibration control of a viscoelastic translational Euler-Bernoulli beam. J. Dyn. Control Syst., 24(1):167–199, 2017. https://doi.org/10.1007/s10883-017-9364-9
[3] A. Berkani, N.-E. Tatar and A. Khemmoudj. Control of a viscoelastic translational Euler-Bernoulli beam. Math. Methods Appl. Sci., 40(1):237–254, 2017. https://doi.org/10.1002/mma.3985
[4] F.C.L. Borges, N. Roitman, C. Magluta, D. A . Castello and R. Franciss. A concept to reduce vibrations in steel catenary risers by the use of viscoelastic materials. Ocean Engineering., 77:1–11, 2014. https://doi.org/10.1016/j.oceaneng.2013.12.004
[5] M.S. de Querioz, D.M. Dawson, S.P. Nagarkatti and F. Zhang. Lyapunov-based control of mechanical system. Birkhauser, Boston, 2000. https://doi.org/10.1007/978-1-4612-1352-9
[6] K.D. Do. Global stabilization of three-dimensional flexible marine risers by boundary control. Ocean Systems Eng., 1(2):171–194, 2011. https://doi.org/10.12989/ose.2011.1.2.171
[7] M. Fabrizio and A. Morro. Mathematical Problems in Linear Viscoelasticity. SIAM Stud. Appl. Math. Philadelphia, 1992. https://doi.org/10.1137/1.9781611970807
[8] S.S. Ge, W. He, B.V.E. How and Y.S. Choo. Boundary control of a coupled nonlinear flexible marine riser. Trans. Control Syst. Tech., 18(5):1080–1091, 2010. https://doi.org/10.1109/TCST.2009.2033574
[9] B.Z. Guo and W. Guo. Adaptive stabilization for a Kirchhoff-type nonlinear beam under boundary output feedback control. Nonlinear Anal., 66(2):427–441, 2007. https://doi.org/10.1016/j.na.2005.11.037
[10] F. Guo and F. Huang. Boundary feedback stabilization of the undamped EulerBernoulli beam with both ends free. SIAM J. Control Optim., 43(1):341–356, 2004. https://doi.org/10.1137/S0363012901380961
[11] W. He and S.S. Ge. Cooperative control of a nonuniform gantry crane with constrained tension. Automatica, 66:146–154, 2016. https://doi.org/10.1016/j.automatica.2015.12.026
[12] W. He, S.S. Ge, B.V.E. How, Y.S. Choo and K.S. Hong. Robust adaptive boundary control of a flexible marine riser with vessel dynamics. Automatica, 47(4):722–732, 2011. https://doi.org/10.1016/j.automatica.2011.01.064
[13] W. He, S.S. Ge and S. Zhang. Adaptive boundary control of a flexible marine installation system. Automatica, 47(12):2728–2734, 2011. https://doi.org/10.1016/j.automatica.2011.09.025
[14] W. He, Y. Ouyang and J. Hong. Vibration control of a flexible robotic manipulator in the presence of input deadzone. IEEE Trans. Appl. Ind. Informatics, 13(1):48–59, 2017. https://doi.org/10.1109/TII.2016.2608739
[15] W. He and X. He ; C. Sun. Vibration control of an industrial moving strip in the presence of input deadzone. IEEE Trans. Ind. Electron, 64(6):4680–4689, 2017. https://doi.org/10.1109/TIE.2017.2674592
[16] W. He and S. Zhang. Control design for nonlinear flexible wings of a robotic aircraft. IEEE Trans. Control Syst. Technol., 25(1):351–357, 2017. https://doi.org/10.1109/TCST.2016.2536708
[17] B.V.E. How, S.S. Ge and Y.S. Choo. Active control of flexible marine risers. J. Sound Vib., 320(4–5):758–776, 2009. https://doi.org/10.1016/j.jsv.2008.09.011
[18] Y.H. Kang, J.Y. Park and J.A. Kim. A memory type boundary stabilization for an Euler-Bernoulli beam under boundary output feedback control. J. Korean Math. Soc., 49(5):947–964, 2012. https://doi.org/10.1016/j.jsv.2008.09.011
[19] A. Kelleche, A. Berkani and N.-E. Tatar. Uniform stabilization of a nonlinear axially moving string by a boundary control of memory type. J. Dyn. Control Syst., pp. 1–11, 2017. https://doi.org/10.1007/s10883-017-9370-y
[20] A.M. Krall. Asymptotic stability of the Euler-Bernoulli beam with boundary control. J. Math Anal. Appl., 137(1):288–295, 1989. https://doi.org/10.1016/0022- 247X(89)90289-8
[21] S. Li, Y. Wang and Z. Liang. Stabilization of vibrating beam with a tip mass controlled by combined feedback forces. J. Math. Anal. Appl., 256(1):13–38, 2001. https://doi.org/10.1006/jmaa.2000.7217
[22] Y. Liu and F. Guo. Output feedback boundary control of a flexible marine riser system. J. Vib. Control, 2017. https://doi.org/10.1177/1077546317708516
[23] Y. Liu, H. Huang, H. Gao and X. Wu. Modeling and boundary control of a flexible marine riser coupled with internal fluid dynamics. J. Control Theory Appl., 11(2):316–323, 2013. https://doi.org/10.1007/s11768-013-1245-5
[24] Y. Liu, Z. Zhao and W. He. Boundary control of an axially moving accelerated/decelerated belt system. Int. J. Robust Nonlinear Control, 26(17):3849– 3866, 2016. https://doi.org/10.1002/rnc.3538
[25] Y. Liu, Z. Zhao and W. He. Stabilization of an axially moving accelerated/decelerated system via an adaptive boundary control. ISA Transactions, 64:394–404, 2016. https://doi.org/10.1016/j.isatra.2016.04.006
[26] Y. Liu, Z. Zhao and W. He. Boundary control of an axially moving system with high acceleration/deceleration and disturbance observer. J. Franklin Inst., 354(7):2905–2923, 2017. https://doi.org/10.1016/j.jfranklin.2017.01.026
[27] J.Y. Park, Y.H. Kang and J.A. Kim. Existence and exponential stability for a Euler-Bernoulli beam equation with memory and boundary output feedback control term. Acta. Appl. Math., 104(3):287–301, 2008. https://doi.org/10.1007/s10440-008-9257-8
[28] J.Y. Park and J.A. Kim. Existence and uniform decay for EulerBernoulli beam equation with memory term. Math. Meth. Appl. Sci., 27(14):1629–1640, 2004. https://doi.org/10.1002/mma.512
[29] L. Seghour, A. Khemmoudj and N.-E. Tatar. Control of a riser through the dynamic of a vessel. Appl. Anal., 95(9):1957–1973, 2016. https://doi.org/10.1080/00036811.2015.1080249
[30] N.-E. Tatar. Uniform decay in viscoelasticity for kernels with small non-decreasingness zones. Appl. Math. Comput., 218(15):7939–7946, 2012. https://doi.org/10.1016/j.amc.2012.02.012