Document Type : Article

**Authors**

Centre of Excellence in Design, Robotics & Automation, Department of Mechanical Engineering, Sharif University of Technology, Tehran, Iran.

**Abstract**

Due to flexibility of thin plates, high amplitude vibrations are observed when they are subjected to severe dynamic loads. Due to the extensive application of circular plates in industry, attenuating the undesired vibrations is of foremost importance. In this paper, adaptable vibration suppressors (AVSs) as a semi-active control approach, are utilized to suppress the vibrations in a free circular plate; under the concentrative harmonic excitation. Using mode summation method, mathematical model of the hybrid system, including the plate and an arbitrary number of vibration suppressors is analyzed. By developing a complex multiple-loops algorithm, optimum values for the AVSs’ parameters (stiffness and position) are achieved such that the plate deflection is comprehensively minimized. According to the results, AVSs act efficiently in suppressing the vibrations in resonance/non-resonance conditions. It is also observed that optimum AVSs reduce the plate deflection over a broad spectrum of excitation frequencies. Finally, since the algorithm is developed in a general user friendly style, AVSs’ design can be extended to other shapes of plates with various boundary conditions and excitations.

**Keywords**

- Circular plates
- Adaptable vibration suppressor
- Semi-active control
- Advanced algorithm
- Optimal design

**Main Subjects**

References:

1. Wang, Q., Shi, D., Liang, Q., and Shi, X. "A unified solution for vibration analysis of functionally graded circular, annular and sector plates with general boundary conditions", Composites Part B: Engineering, 88,pp. 264-294 (2016).

2. Minkarah, I.A. and Hoppmann, W.H. "Flexural vibrations of cylindrically aeolotropic circular plates", The Journal of the Acoustical Society of America, 36(3), pp. 470-475 (1964).

3. Narita, Y. "Natural frequencies of completely free annular and circular plates having polar orthotropy", Journal of Sound and Vibration, 92(1), pp. 33-38 (1984).

4. Kang, W., Lee, N.H., Pang, S., and Chung, W.Y. "Approximate closed form solutions for free vibration of polar orthotropic circular plates", Applied Acoustics, 66(10), pp. 1162-1179 (2005).

5. Leissa, A.W., Vibration of Plates, Vol. SP-160. NASA, Washington, DC: US Government Printing Office (1969).

6. Leissa, A.W. "Literature review: Survey and analysis of the shock and vibration literature: Recent studies in plate vibrations: 1981-85 Part I. Classical Theory", The Shock and Vibration Digest, 19(2), pp. 11-18 (1987).

7. Yamaki, N. "Influence of large amplitudes on flexural vibrations of elastic plates", ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift fur Angewandte Mathematik und Mechanik, 41(12), pp. 501-510 (1961).

8. Kung, G.C. and Pao, Y.H. "Nonlinear flexural vibrations of a clamped circular plate", Journal of Applied Mechanics, 39(4), pp. 1050-1054 (1972).

9. Tobias, S.A. and Arnold, R.N. "The influence of dynamical imperfection on the vibration of rotating disks", Proceedings of the Institution of Mechanical Engineers, 171(1), pp. 669-690 (1957).

10. Tobias, S.A. "Free undamped non-linear vibrations of imperfect circular disks", Proceedings of the Institution of Mechanical Engineers, 171(1), pp. 691-715 (1957).

11. Williams, C.J.H. and Tobias, S.A. "Forced undamped non-linear vibrations of imperfect circular discs", Journal of Mechanical Engineering Science, 5(4), pp. 325- 335 (1963).

12. Shi, X., Shi, D., Li, W.L., and Wang, Q. "A unified method for free vibration analysis of circular, annular and sector plates with arbitrary boundary conditions", Journal of Vibration and Control, 22(2), pp. 442-456 (2016).

13. Sridhar, S., Mook, D.T., and Nayfeh, A.H. "Non-linear resonances in the forced responses of plates, Part II: asymmetric responses of circular plates", Journal of Sound and Vibration, 59(2), pp. 159-170 (1978).

14. Ribeiro, P. and Petyt, M. "Non-linear free vibration of isotropic plates with internal resonance", International Journal of Non-Linear Mechanics, 35(2), pp. 263-278 (2000).

15. Ribeiro, P. and Petyt, M. "Geometrical non-linear, steady state, forced, periodic vibration of plates, part II: Stability study and analysis of multi-modal response", Journal of Sound and Vibration, 226(5), pp. 985-1010 (1999).

16. Wu, F., Liu, G.R., Li, G.Y., Cheng, A.G., and He, Z.C. "A new hybrid smoothed FEM for static and free vibration analyses of Reissner-Mindlin plates", Computational Mechanics, 54(3), pp. 865-890 (2014).

17. Haterbouch, M. "Effects of the geometrically nonlinearity on the free and forced response of clamped and simply supported circular plates", PhD Thesis, Universite Mohammed V-Agdal, Rabat (2003).

18. Haterbouch, M. and Benamar, R. "Geometrically nonlinear free vibrations of simply supported isotropic thin circular plates", Journal of Sound and Vibration, 280(3-5), pp. 903-924 (2005).

19. Liou, G.S. "Vibrations induced by harmonic loadings applied at circular rigid plate on half-space medium", Journal of Sound and Vibration, 323(1-2), pp. 257-269 (2009).

20. Kerlin, R.L. "Predicted attenuation of the platelike dynamic vibration absorber when attached to a clamped circular plate at a non-central point of excitation", Applied Acoustics, 23(1), pp. 17-27 (1988).

21. Snowdon, J.C. "Platelike dynamic vibration absorbers", Journal of Engineering for Industry, 97(1), pp. 88-93 (1975).

22. Kirk, C.L. and Leissa, A.W. "Vibration characteristics of a circular plate with a concentric reinforcing ring", Journal of Sound and Vibration, 5(2), pp. 278-284 (1967).

23. Azimi, S. "Axisymmetric vibration of point-supported circular plates", Journal of Sound and Vibration, 135(2), pp. 177-195 (1989).

24. Kunukkasseril, V.X. and Swamidas, A.S.J. "Vibration of continuous circular plates", International Journal of Solids and Structures, 10(6), pp. 603-619 (1974).

25. Avalos, D.R., Larrondo, H.A., and Laura, P.A.A. "Transverse vibrations of a circular plate carrying an elastically mounted mass", Journal of Sound and Vibration, 177(2), pp. 251-258 (1994).

26. Ray, M.C. and Shivakumar, J. "Active constrained layer damping of geometrically nonlinear transient vibrations of composite plates using piezoelectric fiberreinforced composite", Thin-Walled Structures, 47(2),pp. 178-189 (2009).

27. Vidoli, S. and Dell'Isola, F. "Vibration control in plates by uniformly distributed PZT actuators interconnected via electric networks", European Journal of Mechanics-A/Solids, 20(3), pp. 435-456 (2001).

28. Caruso, G., Galeani, S., and Menini, L. "Active vibration control of an elastic plate using multiple piezoelectric sensors and actuators", Simulation Modelling Practice and Theory, 11(5-6), pp. 403-419 (2003).

29. Wu, S.T., Chen, J.Y., Yeh, Y.C., and Chiu, Y.Y. "An active vibration absorber for a flexible plate boundarycontrolled by a linear motor", Journal of Sound and Vibration, 300(1-2), pp. 250-264 (2007).

30. Qiu, Z.C., Zhang, X.M., Wu, H.X., and Zhang, H.H. "Optimal placement and active vibration control for piezoelectric smart flexible cantilever plate", Journal of Sound and Vibration, 301(3-5), pp. 521-543 (2007).

31. Hu, Y.R. and Ng, A. "Active robust vibration control of flexible structures", Journal of Sound and Vibration, 288(1-2), pp. 43-56 (2005).

32. Wiciak, J. "Modelling of vibration and noise control of a submerged circular plate", Archives of Acoustics, 32(4(S)), pp. 265-270 (2014).

33. Khorshidi, K., Rezaei, E., Ghadimi, A.A. and Pagoli, M. "Active vibration control of circular plates coupled with piezoelectric layers excited by plane sound wave", Applied Mathematical Modelling, 39(3-4), pp. 1217- 1228 (2015).

34. Ji, H., Qiu, J., Badel, A., and Zhu, K. "Semiactive vibration control of a composite beam using an adaptive SSDV approach", Journal of Intelligent Material Systems and Structures, 20(4), pp. 401-412 (2009).

35. Badel, A., Sebald, G., Guyomar, D., Lallart, M., Lefeuvre, E., Richard, C., and Qiu, J. "Piezoelectric vibration control by synchronized switching on adaptive voltage sources: Towards wideband semi-active damping", The Journal of the Acoustical Society of America, 119(5), pp. 2815-2825 (2006).

36. Saadabad, N.A., Moradi, H., and Vossoughi, G.R. "Semi-active control of forced oscillations in power transmission lines via optimum tuneable vibration absorbers: with review on linear dynamic aspects", International Journal of Mechanical Sciences, 87, pp. 163-178 (2014).

37. Meirovitch, L., Principles and Techniques of Vibrations, (1), New Jersey: Prentice Hall (1997).

38. Xie, L., Qiu, Z.C., and Zhang, X.M. "Vibration control of a flexible clamped-clamped plate based on an improved FULMS algorithm and laser displacement measurement", Mechanical Systems and Signal Processing, 75, pp. 209-227 (2016).

2. Minkarah, I.A. and Hoppmann, W.H. "Flexural vibrations of cylindrically aeolotropic circular plates", The Journal of the Acoustical Society of America, 36(3), pp. 470-475 (1964).

3. Narita, Y. "Natural frequencies of completely free annular and circular plates having polar orthotropy", Journal of Sound and Vibration, 92(1), pp. 33-38 (1984).

4. Kang, W., Lee, N.H., Pang, S., and Chung, W.Y. "Approximate closed form solutions for free vibration of polar orthotropic circular plates", Applied Acoustics, 66(10), pp. 1162-1179 (2005).

5. Leissa, A.W., Vibration of Plates, Vol. SP-160. NASA, Washington, DC: US Government Printing Office (1969).

6. Leissa, A.W. "Literature review: Survey and analysis of the shock and vibration literature: Recent studies in plate vibrations: 1981-85 Part I. Classical Theory", The Shock and Vibration Digest, 19(2), pp. 11-18 (1987).

7. Yamaki, N. "Influence of large amplitudes on flexural vibrations of elastic plates", ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift fur Angewandte Mathematik und Mechanik, 41(12), pp. 501-510 (1961).

8. Kung, G.C. and Pao, Y.H. "Nonlinear flexural vibrations of a clamped circular plate", Journal of Applied Mechanics, 39(4), pp. 1050-1054 (1972).

9. Tobias, S.A. and Arnold, R.N. "The influence of dynamical imperfection on the vibration of rotating disks", Proceedings of the Institution of Mechanical Engineers, 171(1), pp. 669-690 (1957).

10. Tobias, S.A. "Free undamped non-linear vibrations of imperfect circular disks", Proceedings of the Institution of Mechanical Engineers, 171(1), pp. 691-715 (1957).

11. Williams, C.J.H. and Tobias, S.A. "Forced undamped non-linear vibrations of imperfect circular discs", Journal of Mechanical Engineering Science, 5(4), pp. 325- 335 (1963).

12. Shi, X., Shi, D., Li, W.L., and Wang, Q. "A unified method for free vibration analysis of circular, annular and sector plates with arbitrary boundary conditions", Journal of Vibration and Control, 22(2), pp. 442-456 (2016).

13. Sridhar, S., Mook, D.T., and Nayfeh, A.H. "Non-linear resonances in the forced responses of plates, Part II: asymmetric responses of circular plates", Journal of Sound and Vibration, 59(2), pp. 159-170 (1978).

14. Ribeiro, P. and Petyt, M. "Non-linear free vibration of isotropic plates with internal resonance", International Journal of Non-Linear Mechanics, 35(2), pp. 263-278 (2000).

15. Ribeiro, P. and Petyt, M. "Geometrical non-linear, steady state, forced, periodic vibration of plates, part II: Stability study and analysis of multi-modal response", Journal of Sound and Vibration, 226(5), pp. 985-1010 (1999).

16. Wu, F., Liu, G.R., Li, G.Y., Cheng, A.G., and He, Z.C. "A new hybrid smoothed FEM for static and free vibration analyses of Reissner-Mindlin plates", Computational Mechanics, 54(3), pp. 865-890 (2014).

17. Haterbouch, M. "Effects of the geometrically nonlinearity on the free and forced response of clamped and simply supported circular plates", PhD Thesis, Universite Mohammed V-Agdal, Rabat (2003).

18. Haterbouch, M. and Benamar, R. "Geometrically nonlinear free vibrations of simply supported isotropic thin circular plates", Journal of Sound and Vibration, 280(3-5), pp. 903-924 (2005).

19. Liou, G.S. "Vibrations induced by harmonic loadings applied at circular rigid plate on half-space medium", Journal of Sound and Vibration, 323(1-2), pp. 257-269 (2009).

20. Kerlin, R.L. "Predicted attenuation of the platelike dynamic vibration absorber when attached to a clamped circular plate at a non-central point of excitation", Applied Acoustics, 23(1), pp. 17-27 (1988).

21. Snowdon, J.C. "Platelike dynamic vibration absorbers", Journal of Engineering for Industry, 97(1), pp. 88-93 (1975).

22. Kirk, C.L. and Leissa, A.W. "Vibration characteristics of a circular plate with a concentric reinforcing ring", Journal of Sound and Vibration, 5(2), pp. 278-284 (1967).

23. Azimi, S. "Axisymmetric vibration of point-supported circular plates", Journal of Sound and Vibration, 135(2), pp. 177-195 (1989).

24. Kunukkasseril, V.X. and Swamidas, A.S.J. "Vibration of continuous circular plates", International Journal of Solids and Structures, 10(6), pp. 603-619 (1974).

25. Avalos, D.R., Larrondo, H.A., and Laura, P.A.A. "Transverse vibrations of a circular plate carrying an elastically mounted mass", Journal of Sound and Vibration, 177(2), pp. 251-258 (1994).

26. Ray, M.C. and Shivakumar, J. "Active constrained layer damping of geometrically nonlinear transient vibrations of composite plates using piezoelectric fiberreinforced composite", Thin-Walled Structures, 47(2),pp. 178-189 (2009).

27. Vidoli, S. and Dell'Isola, F. "Vibration control in plates by uniformly distributed PZT actuators interconnected via electric networks", European Journal of Mechanics-A/Solids, 20(3), pp. 435-456 (2001).

28. Caruso, G., Galeani, S., and Menini, L. "Active vibration control of an elastic plate using multiple piezoelectric sensors and actuators", Simulation Modelling Practice and Theory, 11(5-6), pp. 403-419 (2003).

29. Wu, S.T., Chen, J.Y., Yeh, Y.C., and Chiu, Y.Y. "An active vibration absorber for a flexible plate boundarycontrolled by a linear motor", Journal of Sound and Vibration, 300(1-2), pp. 250-264 (2007).

30. Qiu, Z.C., Zhang, X.M., Wu, H.X., and Zhang, H.H. "Optimal placement and active vibration control for piezoelectric smart flexible cantilever plate", Journal of Sound and Vibration, 301(3-5), pp. 521-543 (2007).

31. Hu, Y.R. and Ng, A. "Active robust vibration control of flexible structures", Journal of Sound and Vibration, 288(1-2), pp. 43-56 (2005).

32. Wiciak, J. "Modelling of vibration and noise control of a submerged circular plate", Archives of Acoustics, 32(4(S)), pp. 265-270 (2014).

33. Khorshidi, K., Rezaei, E., Ghadimi, A.A. and Pagoli, M. "Active vibration control of circular plates coupled with piezoelectric layers excited by plane sound wave", Applied Mathematical Modelling, 39(3-4), pp. 1217- 1228 (2015).

34. Ji, H., Qiu, J., Badel, A., and Zhu, K. "Semiactive vibration control of a composite beam using an adaptive SSDV approach", Journal of Intelligent Material Systems and Structures, 20(4), pp. 401-412 (2009).

35. Badel, A., Sebald, G., Guyomar, D., Lallart, M., Lefeuvre, E., Richard, C., and Qiu, J. "Piezoelectric vibration control by synchronized switching on adaptive voltage sources: Towards wideband semi-active damping", The Journal of the Acoustical Society of America, 119(5), pp. 2815-2825 (2006).

36. Saadabad, N.A., Moradi, H., and Vossoughi, G.R. "Semi-active control of forced oscillations in power transmission lines via optimum tuneable vibration absorbers: with review on linear dynamic aspects", International Journal of Mechanical Sciences, 87, pp. 163-178 (2014).

37. Meirovitch, L., Principles and Techniques of Vibrations, (1), New Jersey: Prentice Hall (1997).

38. Xie, L., Qiu, Z.C., and Zhang, X.M. "Vibration control of a flexible clamped-clamped plate based on an improved FULMS algorithm and laser displacement measurement", Mechanical Systems and Signal Processing, 75, pp. 209-227 (2016).

Transactions on Mechanical Engineering (B)

May and June 2019Pages 1358-1377