Skip to main content
SpringerLink
Log in

Optimal location of piezoelectric actuators for active vibration control of thin axially functionally graded beams

  • Published:
International Journal of Mechanics and Materials in Design Aims and scope Submit manuscript

Abstract

Up to now, optimal location for active control studies concern principally multilayers or homogeneous structures. In the case of functionally graded materials, very few papers exist and they only concern cross section variations. In this way, this paper deals with the optimization of piezoelectric actuators locations on axially functionally graded beams for active vibration control. For this kind of structures, the free vibration problem is more complicated as the governing equations have variable coefficients. Here, the eigenproblem is solved using Fredholm integral equations. The optimal locations of actuators are determined using an optimization criterion, ensuring good controllability of each eigenmode of the structure. The linear quadratic regulator, including a state observer, is used for active control simulations. Two numerical examples are presented for two kinds of boundary conditions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22

Similar content being viewed by others

References

  • Alshorbagy, A.E., Eltaher, M.A., Mahmoud, F.F.: Free vibration characteristics of a functionally graded beam by finite element method. Appl. Math. Model. 35, 412–425 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  • Aminbgahai, M., Murin, J., Hutis, V.: Modal analysis of the FGM-beams with continuous transversal symmetric and longitudinal variation of material properties with effect of large axial force. Eng. Struct. 34, 314–329 (2012)

    Article  Google Scholar 

  • Arbel, A.: Controllability measures and actuator placement in oscillatory systems. Int. J. Control 33(3), 565–574 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  • Balamurugan, V., Narayanan, S.: Shell finite element for smart piezoelectric composite plate/shell structures and its application to the study of active vibration control. Finite Elem. Anal. Des. 37, 713–738 (2001)

    Article  MATH  Google Scholar 

  • Biglar, M., Mirdamadi, H.R., Danesh, M.: Optimal locations and orientations of piezoelectric transducers on cylindrical shell based on gramians of contributed and undesired Rayleigh–Ritz modes using genetic algorithm. J. Sound Vib. 333, 1224–1244 (2014)

    Article  Google Scholar 

  • Bruant, I., Coffignal, G., Léné, F., Vergé, M.: Optimal location of piezoelectric actuators on a beam. In: Proceedings of Active 97, Budapest, pp. 635–649 (1997)

  • Bruant, I., Coffignal, G., Léné, F., Vergé, M.: Active control of beam structures with piezoelectric actuators and sensors: modeling and simulation. Smart Mater. Struct. 10, 404–408 (2001)

    Article  Google Scholar 

  • Bruant, I., Proslier, L.: Optimal location of actuators and sensors in active vibration control. J. Intell. Mater. Syst. Struct. 16, 197–206 (2005)

    Article  Google Scholar 

  • Bruant, I., Proslier, L.: Improved active control of a functionally graded material beam with piezoelectric patches. J. Vib. Control (2014). doi:10.1177/1077546313506926

  • Caddemi, S., Calio, I.: Exact closed-form solution for the vibration modes of the Euler–Bernoulli beam with multiple open cracks. J. Sound Vib. 327, 473–489 (2009)

    Article  Google Scholar 

  • Caddemi, S., Calio, I.: The exact explicit dynamic stiffness matrix of multi-cracked Euler–Bernoulli beam and applications to damaged frame structures. J. Sound Vib. 332, 3049–3063 (2013)

    Article  Google Scholar 

  • Devasia, S., Meressi, T., Paden, B., Bayo, E.: Piezoelectric actuator design for vibration suppression: placement and sizing. J. Guid. Control Dyn. 16(5), 859–864 (1993)

    Article  Google Scholar 

  • Dhingra, A., Lee, B.: Multiobjective design of actively controlled structures using a hybrid optimizatoon method. Int. J. Numer. Methods Eng. 38, 3383–3401 (1995)

    Article  MATH  Google Scholar 

  • Dhuri, K.D., Seshu, P.: Piezo actuator placement and sizing for good control effectiveness and minimal change in original system dynamics. Smart Mater. Struct. 15, 1661–1672 (2006)

    Article  Google Scholar 

  • Fakhari, V., Ohadi, A.: Nonlinear vibration control of functionally graded plate with piezoelectric layers in thermal environment. J. Vib. Control 17, 448–469 (2010)

    MathSciNet  MATH  Google Scholar 

  • Frecker, M.: Recent advances in optimization of smart structures and actuators. J. Intell. Mater. Syst. Struct. 14, 207–215 (2003)

    Article  Google Scholar 

  • Fu, Y., Wang, J., Mao, Y.: Nonlinear vibration and active control of functionally graded beams with piezoelectric sensors and actuators. J. Intell. Mater. Syst. Struct. 22, 2093–2102 (2013)

    Article  Google Scholar 

  • Gawronski, W.: Simultaneous placement of actuators and sensors. J. Sound Vib. 228(4), 915–922 (1999)

    Article  Google Scholar 

  • Gharib, A., Salehi, M., Fazeli, S.: Deflection control of functionally graded material beams with bonded piezoelectric sensors and actuators. Mater. Sci. Eng. A 498, 110–114 (2008)

    Article  Google Scholar 

  • Giunta, G., Crisafulli, D., Belouettar, S., Carrera, E.: Hierarchical theories for free vibration analysis of functionally graded beams. Compos. Struct. 94, 68–74 (2011)

    Article  Google Scholar 

  • Gney, M., Eskinat, E.: Optimal actuator and sensor placement in flexible structures using closed-loop criteria. J. Sound Vib. 312, 210–233 (2007)

    Article  Google Scholar 

  • Hac, A., Liu, L.: Sensor and actuator location in motion control of flexible structures. J. Sound Vib. 167, 239–261 (1993)

    Article  MATH  Google Scholar 

  • Halim, D., Reza Moheimani, S.O.: An optimization approach to optimal placement of collocated piezoelectric actuators and sensors on a thin plate. Mechatronics 13, 27–47 (2003)

    Article  Google Scholar 

  • He, X.Q., Ng, T.Y., Sivashanker, S., Liew, K.M.: Active control of FGM plates with integrated piezoelectric sensors and actuators. Int. J. Solids Struct. 38, 1641–1655 (2001)

    Article  MATH  Google Scholar 

  • Huang, Y., Li, X.F.: A new approach for free vibration of axially functionally graded beams with non-uniform cross-section. J. Sound Vib. 329, 2291–2303 (2010)

    Article  Google Scholar 

  • Huang, Y., Yang, L.E., Luo, Q.Z.: Free vibration of axially functionally graded Timoshenko beams with non-uniform cross-section. Compos. Part B 45, 1493–1498 (2013)

    Article  Google Scholar 

  • Hiramoto, K., Doki, H., Obinata, G.: Optimal sensor/actuator placement for active vibration control using explicit solution of algebraic Riccati equation. J. Sound Vib. 229(5), 1057–1075 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  • Jha, A.K., Inman, D.J.: Optimal sizes and placements of piezoelectric actuators and sensors for an inflated torus. J. Intell. Mater. Syst. Struct. 14, 563–576 (2003)

    Article  Google Scholar 

  • Kailath, T.: Linear Systems. Prentice Hall, Englewwod Cliffs, NJ (1980)

    MATH  Google Scholar 

  • Kargarnovin, M.H., Najafizadeh, M.M., Viliani, N.S.: Vibration control of functionally graded material plate patched with piezoelectric actuators and sensors under a constant electric charge. Smart Mater. Struct. 16, 1252–1259 (2007)

    Article  Google Scholar 

  • Kiani, Y., Sadighi, M., Eslami, M.R.M.: Dynamic analysis and active control of smart doubly curved FGM panels. Compos. Struct. 102, 205–216 (2013)

    Article  Google Scholar 

  • Kondoh, S., Yatomi, C., Inoue, K.: The positioning of sensors and actuators in the vibration control of flexible systems. JSME Int. J. 33, 145–152 (1990)

    Google Scholar 

  • Li, X.F., Kang, Y.A., Wu, J.X.: Exact frequency equations of free vibration of exponentially functionally graded beams. Appl. Acoust. 74, 413–420 (2013)

    Article  Google Scholar 

  • Liew, K.M., Sivashanker, S., He, X.Q., Ng, T.Y.: The modeling and design of smart structures using functionally graded materials and piezoelectrical sensor/actuator patches. Smart Mater. Struct. 12, 647–655 (2003)

    Article  Google Scholar 

  • Liu, W., Hou, Z., Demetriou, M.A.: A computational scheme for the optimal sensor/actuator placement of flexible structures using spatial H2 measures. Mech. Syst. Signal Process. 20, 881–895 (2006)

    Article  Google Scholar 

  • Liu, D.Y., Wang, C.Y., Chen, W.Q.: Free vibration of FGM plates with in-plane material inhomogeneity. Compos. Struct. 92, 1047–1051 (2010)

    Article  Google Scholar 

  • Mahamood, R.M., Akinlabi, E.T.: Functionally graded material: an overview. In: Proceedings of the World Congress on Engineering 2012, p. 3 (2012)

  • Markworth, A.J., Ramesh, K.S., Parks, W.P.: Review: modeling studies applied to functionally graded materials. J. Mater. Sci. 30, 2183–2193 (2012)

    Article  Google Scholar 

  • Mirzaeifar, R., Bahai, H., Shahab, S.: Active control of natural frequencies of FGM plates by piezoelectric sensor/actuator pairs. Smart Mater. Struct. 17, 8p (2008)

  • Murin, J., Aminbaghai, M., Kutis, V.: Exact solution of the bending vibration problem of FGM beams with variation of material properties. Eng. Struct. 32, 1631–1640 (2010)

    Article  Google Scholar 

  • Murin, J., Aminbaghai, M., Hrabovsky, J., Kutis, V., Kugler, S.: Modal analysis of the FGM beams with the effect of the shear correction function. Compos. Part B 45, 1575–1582 (2013)

    Article  Google Scholar 

  • Nam, C., Kim, Y., Weisshaar, T.: Optimal sizing and placement of piezo-actuators for active flutter suppression. Smart Mater. Struct. 5, 2216–2224 (1996)

    Article  Google Scholar 

  • Narayanan, S., Balamurugan, V.: Functionally graded shells with distributed piezoelectric sensors and actuators for active vibration control. In: IUTAM Symposium on Multi Functional Material Structures and Systems, Springer, Dordrecht (2010)

  • Peng, F., Ng, A., Hu, Y.R.: Actuator placement optimization and adaptive vibration control of plate smart structures. J. Intell. Mater. Syst. Struct. 16, 263–271 (2005)

    Article  Google Scholar 

  • Preumont, A.: Vibration Control of Active Structures. Kluwer, Dordrecht (1999)

    MATH  Google Scholar 

  • Qiu, Z.C., Zhang, X.M., Wu, H.X., Zhang, H.H.: Optimal placement and active vibration control for piezoelctric smart flexible cantilever plate. J. Sound Vib. 301, 521–543 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  • Ramesh Kumar, K., Narayanan, S.: Active vibration control of beams with optimal placement of piezoelectric sensors/actuator pairs. Smart Mater. Struct (2008). doi:10.1088/0964-1726/17/5/055008

  • Sarkar, K., Ganguli, R.: Closed-form solutions for axially functionally graded Timoshenko beams having uniform cross-section and fixed–fixed boundary condition. Compos. Part B 58, 361–370 (2014)

    Article  Google Scholar 

  • Schulz, S.L., Gomes, H.M., Awruch, A.M.: Optimal discrete piezoelectric patch allocation on composite structures for vibration control based on GA and modal LQR. Comput. Struct. 128, 101–115 (2013)

    Article  Google Scholar 

  • Shahba, A., Attarnejad, R., Tavanaie Marvi, M., Hajilar, S.: Free vibration and stability of axially functionally graded trapered Timoshenko beams with classical and non-classical boundary conditions. Compos. Part B 42, 801–808 (2011)

    Article  Google Scholar 

  • Shahba, A., Rajasekaran, S.: Free vibration and stability of tapered Euler–Bernoulli beams made of axially functionally graded materials. Appl. Math. Model. 36, 3094–3111 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  • Sheng, G.G., Wang, X.: Active control of functionally graded laminated cylindrical shells. Compos. Struct. 90, 448–457 (2009)

    Article  Google Scholar 

  • Simsek, M., Kocaturk, T., Akbas, S.D.: Dynamic behavior of an axially functionally graded beam under action of a moving harmonic load. Compos. Struct. 94, 2358–2364 (2012)

    Article  Google Scholar 

  • Uymaz, B., Aydogdu, M., Filiz, S.: Vibration analyses of FGM plates with in-plane material inhomogeneity by Ritz method. Compos. Struct. 94, 1398–1405 (2012)

    Article  Google Scholar 

  • Wang, Q., Wang, C.: A controllability index for optimal design of piezoelectric actuators in vibration control of beam structures. J. Sound Vib. 242(3), 507–518 (2001)

    Article  Google Scholar 

  • Wu, L., Wang, Q.S., Elishakoff, I.: Semi-inverse method for axially functionally graded beams with an anti-symmetric vibration mode. J. Sound Vib. 284, 1190–1202 (2005)

    Article  Google Scholar 

  • Xiang, H.J., Yang, J.: Free and forced vibration of a laminated FGM Timoshenko beam of variable thickness under heat conduction. Compos. Part B 39, 292–303 (2008)

    Article  Google Scholar 

  • Yang, S., Lee, Y.: Optimization of noncollocated sensor/actuator location and feedback gain in control systems. Smart Mater. Struct. 2, 96–102 (1993)

    Article  Google Scholar 

  • Yang, Y., Jin, Z., Kiong, C.: So Integrated optimal design of vibration control system for smart beams using genetic algorithms. J. Sound Vib. 282, 1293–1307 (2005)

    Article  Google Scholar 

  • Yiqi, M., Yiming, F.: Nonlinear dynamic response and active vibration control for piezoelectric functionally graded plate. J. Sound Vib. 329, 2015–2028 (2010)

    Article  Google Scholar 

  • Zheng, S.J., Dai, F., Song, Z.: Active control of piezothermoelastic FGM shells using integrated piezoelectric sensor/actuator layers. Int. J. Appl. Electromagn. Mech. 30, 107–124 (2009)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. LEME EA 4416, Université Paris Ouest Nanterre La Défense, 50 rue de Sèvres, 92410, Ville d’Avray, France

    Isabelle Bruant & Laurent Proslier

Authors
  1. Isabelle Bruant
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Laurent Proslier
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Isabelle Bruant.

Appendix: Expressions of \(K_{1}(\xi,s,p)\) and \(K_{2}(\xi,s,p)\)

Appendix: Expressions of \(K_{1}(\xi,s,p)\) and \(K_{2}(\xi,s,p)\)

The following expressions are given in Huang and Li (2010).

1.1 For simply supported beam

$$ \begin{aligned} K_{1}(\xi,s,p)&= (\xi -1)I\left[ E''(s,p)s+2E'(s,p)\right] , \qquad 0 \le s \le \xi \\&= \xi I\left[ (s-1)E''(s,p)+2E'(s,p)\right] , \qquad \xi \le s \le 1 \end{aligned} $$
(47)
$$ \begin{aligned} K_{2}(\xi , s,p)&= {\frac{S}{6} \rho (s,p)s(1-\xi )\left( \xi ^{2}+s^{2}-2\xi \right) , \qquad 0 \le s \le \xi } \\&= {\frac{S}{6} \rho (s,p)\xi (1-s)\left( \xi ^{2}+s^{2}-2s\right) , \qquad \xi \le s \le 1} \end{aligned} $$
(48)

1.2 For clamped–pinned beam

$$ \begin{aligned} K_{1}(\xi , s,p)&= {\frac{1}{ 2} \xi ^{2}(\xi -3)I\left[ E''(s,p)(1-s)-2E'(s,p)\right] +\,IE''(s,p) \left( \xi -s\right) -2IE'(s,p)}, \\ &\qquad 0\le s \le \xi \\&= {\frac{1}{2}\xi ^{2}\left( \xi -3\right) I\left[ E''(s,p)(1-s)-\,2E'(s,p) \right] }, \qquad \xi \le s \le 1 \end{aligned} $$
(49)
$$ \begin{aligned} K_{2}(\xi , s,p)&= {\frac{S}{ 12} \rho (s,p)s^{2}(\xi -1)\left[s(\xi ^{2}-2\xi -2)-3(\xi ^{2}-2\xi )\right], \qquad 0 \le s \le \xi } \\&= {\frac{S}{12} \rho (s,p)\xi ^{2}(s-1)\left[ \xi (s^{2}-2s-2)-3(s^{2}-2s)\right] , \qquad \xi \le s \le 1} \end{aligned} $$
(50)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bruant, I., Proslier, L. Optimal location of piezoelectric actuators for active vibration control of thin axially functionally graded beams. Int J Mech Mater Des 12, 173–192 (2016). https://doi.org/10.1007/s10999-015-9297-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10999-015-9297-y

Keywords

Search

Navigation