Damping of longitudinal vibrations of an elastic rod by a piezoelectric element

Cover Page

Cite item

Full Text

Open Access Open Access
Restricted Access Access granted
Restricted Access Subscription Access

Abstract

Possible damping of longitudinal vibrations of a thin homogeneous elastic rod under the influence of a normal force in the cross section is studied. This time-varying force, which can be excited, for example, by using piezoelectric elements, is uniformly distributed along the length on a given segment of the cantilevered rod and is equal to zero outside it. Those placements of the ends of the segment are presented in which the excited force does not affect the amplitude of certain modes. The minimum time in which the oscillations of all other modes can be damped is found, and based on the Fourier method, the corresponding law of the damping force is obtained in the form of a series. A generalized formulation of the boundary value problem on moving the rod during this time to the zero terminal state is given, for which an algorithm for exact solution is proposed in the case of rational relations on the geometric parameters. Unknown functions of the rod state are sought in the form of a linear combination of the traveling wave and normal force functions, which are determined from a linear system of algebraic equations following from boundary relations and continuity conditions. The solutions obtained in series by the Fourier method and in the form of d’Alembert traveling waves are compared.

Full Text

Restricted Access

About the authors

G. V. Kostin

Ishlinsky Institute for Problems in Mechanics RAS

Author for correspondence.
Email: kostin@ipmnet.ru
Russian Federation, Moscow

References

  1. Butkovskii A.G. Distributed Control Systems. N.Y.: Elsevier, 1970. 446 p.
  2. Lions J.L. Optimal Control of Systems Governed by Partial Differential Equations. N.Y.: Springer, 1971. 400 p.
  3. Chernousko F.L., Ananievski I.M., Reshmin S.A. Control of Nonlinear Dynamical Systems: Methods and Applications. Berlin: Springer, 2008. 408 p.
  4. Chen G. Control and stabilization for the wave equation in a bounded domain. II // SIAM J. Control Optimization, 1981, vol. 19, no. 1, pp. 114–122.
  5. Romanov I.V., Shamaev A.S. On a boundary controllability problem for a system governed by the two-dimensional wave equation // J. Comput.&Syst. Sci. Int., 2019, vol. 58, no. 1, pp. 105–112.
  6. Gavrikov A.A., Kostin G.V. Bending vibrations of an elastic rod controlled by piezoelectric forces // Mech. of Solids, 2023, vol. 58, no. 8, pp. 2803–2817.
  7. IEEE Standard on Piezoelectricity // ANSI/IEEE Std 176-1987. 1988 https://doi.org/10.1109/IEEESTD.1988.79638
  8. Kucuk I., Sadek I., Yilmaz Y. Optimal control of a distributed parameter system with applications to beam vibrations using piezoelectric actuators // J. Franklin Inst., 2014, vol. 351, no. 2, pp. 656–666.
  9. Kostin G.V., Saurin V.V. Dynamics of Solid Structures. Methods Using Integrodifferential Relations. Berlin: De Gruyter, 2018.
  10. Kostin G., Gavrikov A. Controllability and optimal control design for an elastic rod actuated by piezoelements // IFAC-PapersOnLine, 2022, vol. 55, no. 16, pp. 350–355. https://doi.org/10.1016/j.ifacol.2022.09.049
  11. Gavrikov A.A., Kostin G.V. Optimization of longitudinal motions of an elastic rod using periodically distributed piezoelectric forces // J. Comput.&Syst. Sci. Int., 2023, vol. 62, no. 5, pp. 800–816.
  12. Kostin G., Gavrikov A. Modeling and optimal control of longitudinal motions for an elastic rod with distributed forces // ArXiv, 2022, arXiv: 2206.06139v2, pp. 1–11. https://doi.org/10.48550/arXiv.2206.06139
  13. Gavrikov A., Kostin G. Optimal LQR control for longitudinal vibrations of an elastic rod actuated by distributed and boundary forces // Mech.&Machine Sci., 2023, vol. 125, pp. 285–295. https://doi.org/10.1007/978-3-031-15758-5_28
  14. Ho L.F. Exact controllability of the one-dimensional wave equation with locally distributed control // SIAM J. Control&Optim., 1990, vol. 28, no. 3, pp. 733–748.
  15. Bruant I., Coffignal G., Lene F., Verge M. A methodology for determination of piezoelectric actuator and sensor location on beam structures // J. Sound&Vibr., 2001, vol. 243, no. 5, pp. 861–882. https://doi.org/10.1006/ jsvi.2000.3448
  16. Gupta V., Sharma M., Thakur N. Optimization criteria for optimal placement of piezoelectric sensors and actuators on a smart structure: A technical review // J. Intell. Mater. Syst.&Struct., 2010, vol. 21, no. 12, pp. 1227–1243. https://doi.org/10.1177/1045389X10381659
  17. Botta F., Rossi A., Belfiore N.P. A novel method to fully suppress single and bi-modal excitations due to the support vibration by means of piezoelectric actuators // J. Sound&Vibr., 2021, vol. 510, no. 13, pp. 116260. https://doi.org/10.1016/j.jsv.2021.116260
  18. Tikhonov A.N., Samarskii A.A. Equations of Mathematical Physics. Oxford: Pergamon, 1963. 800 p.
  19. Mikhlin S.G. Mathematical physics; an Advanced Course. Amsterdam: North-Holland, 1971. 576 p.
  20. Yosida K. Functional Analysis. Berlin: Springer, 1965. 504 p.

Supplementary files

Supplementary Files
Action
1. JATS XML
Download
2. Fig. 1. Diagram of a rod with a control element

Download (28KB)
Indexing metadata
3. Fig. 2. Grid in the space-time domain D for nx = 4

Download (213KB)
Indexing metadata
4. Fig. 3. Control u*(t) for λ = 1/4: x- = 0 (solid curve), x- =1/4 (dashed), x- =1/2 (dash-dotted), x- = 3/4 (dotted).

Download (64KB)
Indexing metadata
5. Fig. 4. Force f(t,188) for x- = 0 and λ =1/4: exact solution (solid curve) and 8-mode approximation (dashed curve).

Download (64KB)
Indexing metadata
6. Fig. 5. Distribution of the dynamic potential r(t, x) under critical control for x- = 0 and λ =1/4

Download (77KB)
Indexing metadata
7. Fig. 6. Distribution of elastic displacements v(t, x) under critical control for x- = 0 and λ =1/4

Download (82KB)
Indexing metadata

Copyright (c) 2024 Russian Academy of Sciences