Distributed Control of a Vibrating String in Response to Pointwise Force Application

Yıl 2024, Cilt: 28 Sayı: 1, 167 - 173, 29.02.2024

Şule Kapkın Mehmet Şirin Demir Erol Uzal

https://doi.org/10.16984/saufenbilder.1356453

Öz

The problem of controlling the vibrations of a string by a discrete applied force is considered. The vibrations of the string are modeled by the linear wave equation and the control is provided by an added force term. The wave equation is solved for controlled and uncontrolled cases with and without control force term. The applied force is chosen to be proportional to string displacement at some specified point. In the controlled case; the wave equation involves a control parameter (gain) and related terms involving the value of the displcement at a single point and a delta function. This makes the equation quite different from the usual wave equation. The problem is solved analytically using a modified (compared to usual wave equation) solution procedure and an equation relating the string eigenfrequencies to the proportionality constant (gain) is derived. This allows the observation of the change in eigenfrequencies with the gain. Finally, examples of uncontrolled and controlled responses are presented, graphically. The results show that the resonances can be avoided by the applied control procedure.

Anahtar Kelimeler

Pointwise Control, Vibrations, Control of Partial Differential Equations, Analytical Solution

Kaynakça

• [1] Y. You, “Controllability and stabilizability of vibrating simply supported plate with pointwise control,”. Advances in Applied Mathematics, vol. 10, no. 3, pp. 324-343, 1989.
• [2] S. Sadek, “Optimal pointwise control of flexible structures,” Mathematical and Computer Modelling, vol. 17, no. 9, pp. 89-99, 1993.
• [3] C. M. Wang, “Linear quadratic optimal control of a wave equation with boundary damping and pointwise control input, ”Journal of Mathematical Analysis and Applications, vol. 192, no. 2, pp. 562-578, 1995.
• [4] A. Cherid, Y. A. Fiagbedzi, I. S. Sadek, “Stabilization of structurally damped systems by pointwise time-delayed feedback control, ”Journal of the Franklin Institute, vol. 336, no. 7, pp. 1175-1185, 1999.
• [5] J. Droniou, J. P. Raymond, "Optimal pointwise control of semilinear parabolic equations," Nonlinear Analysis: Theory, Methods Applications, vol. 39, no. 2, pp. 135-156, 2000.
• [6] I. Sadek, M. Abukhaled, T. Abualrub, "Optimal pointwise control for a parallel system of Euler–Bernoulli beams," Journal of Computational and Applied Mathematics, vol. 137, no. 1, pp. 83-95, 2001.
• [7] B. Z. Guo, Y. Xie, "Basis property and stabilization of a translating tensioned beam through a pointwise control force," Computers and Mathematics with Applications, vol. 47, no. 8-9, pp. 1397-1409, 2004.
• [8] K. Beauchard, "Local controllability and non-controllability for a 1D wave equation with bilinear control," Journal of Differential Equations, vol. 250, no.4, pp. 2064-98, 2011.
• [9] P. A. Nguyen, J. P. Raymond, "Pointwise control of the Boussinesq system," Systems and Control Letters, vol. 60, no. 4, pp. 249-255, 2011.
• [10] M. Ouzahra, "Controllability of the wave equation with bilinear controls," European Journal of Control, vol. 20, no. 2, pp. 57-63, 2014.
• [11] L. Sirota, Y. Halevi, "Fractional order control of the two-dimensional wave equation," Automatica, vol. 59, pp. 152-163, 2015.
• [12] W. Latas, "Active vibration suppression of axially moving string via distributed force," Vibrations in Physical Systems, vol. 31, no. 2020215, pp. 1-8, 2020.
• [13] M. Tucsnak, “On the pointwise stabilization of a string,” in Control and Estimation of Distributed Parameter Systems: International Conference in Vorau, Vorau, Austria, 1996, pp. 287-295.
• [14] G. Chen, M. Coleman, H. H. West, "Pointwise stabilization in the middle of the span for second order systems, nonuniform and uniform exponential decay of solutions," SIAM Journal on Applied Mathematics, vol. 47, no. 4, pp. 751-780, 1987.
• [15] L. F. Ho, "Controllability and stabilizability of coupled strings with control applied at the coupled points," SIAM Journal on Control and Optimization, vol. 31, no. 6, pp. 1416-1437, 1993.