**4. Conclusion**

Deriving the shape of κ(x) , the influence both masses and stiffness of the actuators and glue on the shape of X(x) , and consistently on the shape of κ(x) , were omitted; if not, an adaptation method must be applied. But after determining shape of κ(x) , all these parameters were considered.

As can be note, the actuators optimal distribution is attained assuming that the added energy to excite actuators is constant. It is translated into constant sf . Having the optimal distribution, the reduction coefficient may be improved by adding more energy or in order words, by increasing sf . This way, presented optimal method corresponds to that one presented in (Q. Wang & C. Wang, 2001), namely "maximization of the control forces transmitted by the actuators to the structure".

Based on theoretical considerations, and numerical ones presented in own papers, the following conclusion may be formulated.


An Optimal Distribution of Actuators in

No.3, pp.519−533.

Vol.13, pp.27−47.

Vol.49, pp.55−63.

pp.563–576.

097X, Krakow.

pp.381−389.

*Sensors and Actuators*, Vol.84, pp*.*81−94. Kaliski, S. (1986). *Vibrations and Waves*, PWN, Warsaw.

London.

London.

*Structure*, Vol.15, pp.1661−1672.

*Method*, pp.69−79, Krakow,

Active Beam Vibration – Some Aspects, Theoretical Considerations 417

Dhuri, K.D. & Seshu, P. (2006). Piezo actuator placement and sizing for good control

Ercoli, L. & Laura, P.A.A. (1987). Analytical and experimental investigation on continuous

Frecker, M. (2003). Recent advances in optimization of smart structures and actuators.

Fuller, C.R.; Elliot, S.J. & Nielsen, P.A. (1997). *Active control of vibration*, Academic Press,

Gosiewski, Z. & Koszewnik, A. (2007). The influence of the piezoelements placement on the

Guney, M. & Eskinat, E. (2007). Optimal actuator and sensor placement in flexible structures using closed-loop criteria, *Journal of Sound and Vibrations*, Vol.312, pp.210−233. Halim, D. & Reza Moheimani, S.O. (2003). An optimization approach to optimal placement

Han, S.M., Benaroya, H. & Wei, T. (1999). Dynamics of transversely vibrating beams using

Hansen, C.H. & Snyder, S.D. (1997). *Active control of noise and vibration*, E&FN SPON,

Hernandes, J.A.; Almeida, S.F.M. & Nabarrete, A. (2000). Stiffening effects on the free

Hong, C.; Gardonio, P. & Elliott, S.J. (2007). Active control of resiliently mounted beams using triangular actuators, *Journal of Sound and Vibrations*, Vol.301, pp.297−318. Ip, K.H. & Tse, P.C. (2001). Optimal configuration of a piezoelectric path for vibration control of isotropic rectangular plate*, Smart Material Structure*, Vol.10, pp.395−403. Jha, A.K. & Inman, D.J. (2003). Optimal sizes and placement of piezoelectric actuators and

Jiang, T.Y.; Ng, T.Y. & Lam, K.Y. (2000). Optimization of a piezoelectric ceramic actuator.

Kasprzyk, S. & Wiciak, M. (2007). Differential equation of transverse vibrations of a beam

Kozień, M. (2006). *Acoustic radiation of plates and shallow shells,* PK, Monograph 331, ISS 0860-

Liu, W.; Hou, Z. & Demetriou, M.A. (2006). A computational scheme for the optimal

Low, K.H. & Naguleswaran, S. (1998). On the eigenfrequencies for mass loaded beams

*Mechanical Systems and Signal Processing*, Vol.20, pp.881–895.

Fichtenholtz, G.M. (1999). Differential and integral calculus, PWN, Warsaw.

*Journal Intelligent Material System Structures*, Vol.14, pp.207–215.

effectiveness and minimal change in original system dynamics. *Smart Material* 

beam carying elastically mounted masses*. Journal of Sound and Vibrations*, Vol.114,

active vibration damping system, *Proceedings Active Noise and Vibration Control* 

of collocated piezoelectric actuators and sensors on a thin plate. *Mechatronics*,

four engineering theories, *Journal of Sound and Vibrations*, Vol.225, No.5, pp.935−988.

vibration behavior of composite plates with PZT actuators, *Composite Structures*,

sensors for an inflated torus*. Juornal Inteligent Material System Structures*, Vol.14,

with a local stroke change of stiffness, *Opuscula Mathematica*, Vol.27, No.2, 245−252.

sensor/actuator placement of flexible structures using spatial H2 measures*,* 

under classical boundary conditions, *Journal of Sound and Vibrations*, Vol.215, No.2,


It seems that proposed optimization method is very simple and may be useful in many technical problems of active vibration reduction. This work is a starting point for many computer simulations and experiments.

### **5. References**


• to increase the value sf , through the energy increase which excites actuators, until the

It seems that proposed optimization method is very simple and may be useful in many technical problems of active vibration reduction. This work is a starting point for many

Bapat, C.N. & Bapat, C. (1987). Natural frequencies of a beam with non-classical boundary

Brański, A. & Szela, S. (2007). On the quasi optimal distribution of PZTs in active reduction

Brański, A. & Szela, S. (2008). Improvement of effectiveness in active triangular plate

Brański, A.; Borkowski, M. & Szela, S. (2010). The idea of the selection of PZT-beam

Brański, A. & Szela S. (2010). Quasi-optimal PZT distribution in active vibration reduction of

Brański, A. & Lipiński, G. (2011). Analytical determination of the PZT's distribution in active beam vibration protection problem. (in press in *Acta Physica Polonica*). Brański, A. & Szela, S. (2011). Evaluation of the active plate vibration reduction via the

Bruant, I.; Coffignal, G.; Lene, F. & Verge, M. (2001). A methodology for determination of

Bruant, I. & Proslier, L. (2005). Optimal location of actuators and sensors in active vibration control, *Journal Inteligent Material System Structures*, Vol.16, pp.197–206. Bruant, I.; Gallimard, L. & Nikoukar, S. (2010). Optimal piezoelectric actuator and sensor

Burke, S.E. & Hubbard, J.E. (1991). Distributed transducer vibration control of thin plate,

Charette, F.; Berry, A. & Guigou, C. (1998). Dynamic effect of piezoelectric actuators on the

Croker, M.J. (2007). *Handbook of noise and vibration control*, John Wiley & Sons.

vibration reduction*. Archives of Acoustics*, Vol.33, No.4, pp.521-530.

parameter of the acoustic field. (in press in *Acta Physica Polonica*).

conditions and concentrated masses, *Journal of Sound and Vibrations*, Vol.112, No.1,

of the triangular plate vibration. *Archives of Control Sciences*, Vol.17, No.4, pp. 427–

interaction forces in active vibration protection problem. *Acta Physica Polonica*,

the triangular plate with P-F-F boundary conditions. *Archives of Control Sciences*,

piezoelectric actuator and sensor location on beam structure*. Journal of Sound and* 

location for active vibration control, using genetic algorithm*. Journal of Sound and* 

vibration response of a plate. *Journal Inteligent Material System Structures*, Vol.8,

κ

(x) takes in turn its

• to search of stationary points {x of the beam curvature, s} • to search of {x ,x points of the beam curvature, max min}

• to bond the actuators at the {xQ} points,

computer simulations and experiments.

pp. 177−182.

Vol.118, pp.17-22.

Vol.20, No.2, pp.209−226.

*Vibrations*, Vol.243, No.5, pp.861−882.

*Vibrations*, Vol.329, pp.1615-1635.

*J.A.S.A*., Vol.90, No.2, pp.937−944.

pp.513–524.

437.

**5. References** 

• to determine the value of the reduction coefficient,

reduction coefficient will attain its maximum.

• n points among stationary Q {x and s} {x ,x max min} ones, at which

maximum absolute values, are selected, they are denoted by {xQ} ,


**Part 4** 

**Acoustic Wave Based Microdevices**

