Vibration Attenuation Timoshenko Beam Based on Optimal Placement Sensors/Actuators PZT Patches with LQR-MOPSO

Document Type : Research Paper


1 University of Guilan

2 university of guilan


The main objective of this study is to reduce optimal vibration suppression of Timoshenko beam under non-periodic step and impulse inputs. Cantilever beam was modeled by Timoshenko theory and finite element numerical method. Stiffness (K), mass (M), and damping (C) matrices are extracted. Then, in order to control structure vibration, piezoelectric patches were used due to simultaneous dual behavior, i.e. switching mechanical behavior to electrical behavior (sensor) and electrical behavior to mechanical behavior (actuator). Piezoelectric patches are used in two different arrays with equal dimensions and different elements for establishing feedback control. In the following, by using quadratic optimal controller (LQR), structure vibrations became attenuated. Weighting coefficients of R and Q matrices and piezoelectric patch location have been searched by multi-objective particle swarm optimization algorithm (MOPSO). Finally, the structure underwent standard inputs of impulse and step and the results are analyzed and compared.


Main Subjects

[1] Quek, S.T., Wang, S.Y., and Ang, K.K., “Vibration Control of Composite Plates via Optimal Placement of Piezoelectric Patches”, Journal of Intelligent Material Systems and Structures. Vol. 14, Issue. 4-5, pp. 229-245, (2003).
[2] Liu, W., Hou, Z., and Demtriou, M.A., “A Computational Scheme for the Optimal Sensor/Actuator Placement of Flexible Structures using Spatial  Measures”, Mechanical System and Signal Processing, Vol. 20, pp. 881-895, (2006).
[3] Gua, H.Y., Zhang, L., Zhang, L.L., and Zhou, J.X., “Optimal Placement of Sensors for Structural Health Monitoring using Improved Genetic Algorithms, Smart Material and Structures”, Vol. 13, No. 528, pp. 528-534, (2004).
[4] Da Rocha, T.L., Da Silva, S., and Lopes Jr, V., “Optimal Location of Piezoelectric Sensor and Actuator for Flexible Structures”, 11th International Congress on Sound and Vibration, 5-8, St. Petersburg, Russia, pp. 5-8, (2004).
[5] Dos.Santoes, E., Lucato, S.L., Mc Meeking, R.M., and Evans, A.G., “Actuator Placement Optimization in A Kagome Based High Authority Shape Morphing Structure”, Smart Materials and Structures, Vol. 14, No. 86, pp. 875, (2005).
[6] Brasseur, M., Boe, P.D., Gdinval, J.C., Tamaz, P., Caule, P., Embrechts, J.J., and Nemerlin, J., “Placement of Piezoelectric Laminate Actuator for Active Structural Acoustic Control”, International ISMA, KU Leuven, Belgium, pp. 1-14, (2004).
[7] Ning, H.H., “Optimal Number and Placements of Piezoelectric Patch Actuators in Structural Active Vibration Control”, Engineering Computations, Vol. 21, No. 6, pp. 601-665, (2004).
[8] De Oliveira, A.S., and Junior, J.J.L., “Placement Optimization of Piezoelectric Actuators in a Simply Supported Beam through SVD Analysis and Shape Function Critic Point”, 6th World Congress of Structural and Multidisciplinary Optimization, Rio de Juneiro, Brazil, (2005).
 [9] Wang, S.Y., Tai, K., and Quek, S.T., “Topology Optimization of Piezoelectric Sensors/Actuators for Torsional Vibration Control of Composite Plates, Smart Materials and Structures”, Vol. 15, pp. 253-269, (2006).
[10] Lottin, J., Formosa, F., Virtosu, M., and Brunetti, L., “About Optimal Location of Sensors and Actuators for the Control of Flexible Structures”, 7th International Workshop on Research and Education in Mechatronics, KTH University, Stockholm, Sweden, pp. 1-5, (2006).
[11] Lottin, J., Formosa, F., Virtosu, M., and Brunetti, L., “Optimization of Piezoelectric Sensor Location for Delamination Detection in Composite Laminates”, Engineering Optimization, Vol. 38, No. 5, pp. 511-528, (2006).
[12] Belloli, A., and Ermanni, P., “Optimum Placement of Piezoelectric Ceramic Modules for Vibration Suppression of Highly Constrained Structures”, Smart Materials and Structures, Vol. 16, pp. 1662-1671, (2007).
[13] 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, Vol. 301, pp. 521-543, (2007).
[14] Roy, T., and Chakraborty, D., “GA-LQR Based Optimal Vibration Control of Smart FRP Composite Structures with Bonded PZT Patches”, Journal of Reinforced Plastics and Composites, Vol. 28, pp. 1383, (2009).
[15] Safizadeh, M.R., Mat Darus, I.Z., and Mailah, M., “Optimal Placement of Piezoelectric Actuator for Active Vibration Control of Flexible Plate”, Faculty of Mechanical Engineering Universiti Technologi Malaysia (UTM) 81310 Skudai, Johor, Malaysia.
[16] Yang, J.Y., and Chen, G.P., “Actuator Placement and Configuration Direct Optimization in Plate Structure Vibration Control System”, International Conference on Measuring Technology and Mechatronics Automation, Beijing, China, Vol. 1, pp. 407-411, (2011).
[17] Yang, J., and Chen, G., “Optimal Placement and Configuration Direction of Actuators in Plate Structure Vibration Control System”, 2nd International Asia Conference on Informatics in Control, Automation and Robotics,Wuhan, China, DOI: 10.1109/CAR.2010.5456890, (2010).
[18] Gupta, V., Sharma, M., and Thakur, N., “Optimization Criteria for Optimal Placement of Piezoelectric Sensors and Actuators on a Smart Structure: A Technical Review”, Journal of Intelligent Material Systems and Structures, Vol. 21, Issue. 12, pp. 1-17, (2010).
[19] Trajkov, M., and Nestorvic, T., “Optimal Placement of Piezoelectric Actuators and Sensors for Smart Structures”, 15th International Conference on Experimental Mechanics, Porto/Portugal, pp. 1-13, (2012).
[20] Bachmann, F., Bergamini, A.E., and Ermanni, P., “Optimum Piezoelectric Patch Positioning: A Strain Energy-Based Finite Element Approach”, Journal of Intelligent Material Systems and Structures, Vol. 23, pp. 1575-1591, (2012).
[21] Zhang, J., He, L., and Wang, E., “Active Vibration Control of Piezoelectric Intelligent Structures”, Journal of Computers, Vol. 5, No. 3, pp. 401-409, (2010).
[22] Molter, A., Fonseca, J.S.O., and Bottega, V., “Simultaneous Piezoelectric Actuator and Sensors Placement Optimization and Optimal Control Design for Flexible Non-prismatic Beams, 20th International Congress of Mechanical Engineering”, November 15-20, Gramado, RS.Brazil, (2009).
[23] Nowak, L., and Zielinski, T.G., “Determining the Optimal Locations of Piezoelectric Transducers for Vibroacoustic Control of Structures with General Boundary Conditions”, Institute of Fundamental Technological Research, Polish Academy of Sciences ul.Pawinskiego 5B, 02-106 Warsaw, Poland.
[24] Rosi, G., Paccapeli, R., Ollivier, F., and Pouget, J., “Optimization of Piezoelectric Patches Positioning for Passive Sound Vibration Control of Plates”, Journal of Vibration and Control, Vol. 19, Issue. 5, pp. 1-16, (2012).
[25] Daraji, A.H., and Hale, J.M., “The Effect of Symmetry on Optimal Transducer Location for Active Vibration Control”, Proceedings of the ASME 2012 International Design Engineering Technical Conference & Computers and Information in Engineering Conference IDETC/CIE, Chicago. IL, USA, (2012).
[26] Hale, J.M., and Daraji, A.H., “Optimal Placement of Sensors and Actuators for Active Vibration Reduction of a Flexible Structure using a Genetic Algorithm Based on Modified ”, Modern Practice in Stress and Vibration Analysis, Journal of Physics, Conference Series. 382, pp. 1-7, (2012).
 [27] Zoric, N.D., Simonovic, A.M., Mitrovic, Z.S., and Stuper, S.N., “Multi-objective Fuzzy Optimization of Sizing and Location of Piezoelectric Actuators and Sensors”, FME Transactions Vol. 40, pp. 1-9, (2012).
[28] Schulz, S.L., Gomes, H.M., and Awruch, A.M., “Optimal Discrete Piezoelectric Patch Allocation on Composite Structures for Vibration Control Based on GA and Modal LQR”, Computers and Structures, Vol. 128, pp. 101-115, (2013).
[29] Chhabra, D., Bhushan, G., and Chandra, P., “Optimal Placement of Piezoelectric Actuators on Plate Structures for Active Vibration Control using Modified Control Matrix and Singular Value Decomposition Approach”, International Journal of Mechanical, Industrial Science and Engineering, Vol. 7, No. 3, pp. 1-6, (2013).
[30] Botta, F., Dini, D., Schwingshackl, C., Mare, L.D., and Cerri, G., “Optimal Placement of Piezoelectric Plates to Control Multimode Vibrations of a Beam”, Advances in Acoustics and Vibration, Vol. 2013, Article ID905160, pp. 1-8, (2013).
[31] Araujo, A.L., Madeira, J.F.A., Soares, C.M.M., and Soares, C.A.M., ”Optimal Design for Active Damping in Sandwich Structures using the Direct Multi Search Method”, Composite Structures Vol. 105, pp. 29-34, (2013).
[32] Wrona, S., and Pawelczyk, M., “Controllability Oriented Placement of Actuators for Active Noise Vibration Control of Rectangular Plate using A Memetic Algorithm”, The Journal of Institute of Fundamental Technological of Polish Academy of Science, Vol. 38, No. 4, pp. 529-526, (2013).
[33] Worna, S., and Pawelczyk, M., “Application of a Memetic Algorithm to Placement of Sensors for Active Noise-vibration Control”, Mechanics and Control, Vol. 32, No.3, (2013).
[34] Takezam, A., Makihara, K., Kogiso, N., and Kitramura, M., “Layout Optimization Methodology of Piezoelectric Transducers in Energy-recycling Semi-active Vibration Control Systems”, Journal of Sound and Vibration, Vol. 333, pp. 327-344, (2014).
[35] Dafang, W., Liang, H., Bing, P., Yuewu, W., and Shuang, W., “Experimental Study and Numerical Simulation of Active Vibration Control of a Highly Flexible Beam using Piezoelectric Intelligent Material”, Aerospace Science and Technology, Vol. 37, pp. 10-19, (2014).
[36] Li, F.M., and Lv, X.X., “Active Vibration Control of Lattice Sandwich Beams using the Piezoelectric Actuator/Sensor Pairs”, Composite Part B: Engineering, Vol. 67, pp. 571-578, S1359-8368-00327-8, DOI: , (2014).
 [37] Kwak, M.K., and Yang, D.H., “Dynamic Modelling and Active Vibration Control of a Submerged Rectangular Plate Equipped with Piezoelectric Sensor and Actuator”, Journal of Fluids and Structures, Vol. 54, pp. 848-867, (2015).
 [38] Khorshidi, K., Rezaei, E., Ghadimi, A.A., and Pagoli, M., “Active Vibration Control of Circular Plate Coupled with Piezoelectric Layers Excited by Plane Sound Wave”, Applied Mathematical Modelling, Vol. 39, Issues. 3-4, pp. 1217-1228, (2015).
 [39] Trindade, M.A., Pagani, Jr, C.C., and Oliveira, L.P.R., “Semi-model Active Vibration Control of Plates using Discrete Piezoelectric Model Filter, Journal Sound and Vibration”, Vol. 351, pp. 17-28, (2015).
[40] Zhang, S.Q., Li, Y.X., and Schmidt, R., “Active Shape and Vibration Control for Piezoelectric Bonded Composite Structures using Various Geometric Nonlinearities”, Composite Structures, Vol. 122, pp. 239-249, (2015).
[41] Yildirim, K., and Kucuk, I., “Active Piezoelectric Vibration Control for a Timoshenko Beam”, Journal of the Franklin Institude, Vol. 353, pp. 95-107, (2016).
[42] Selim, B.A., Zhang, L.W., and Liew, K.M., “Active Vibration Control of FGM Plates with Piezoelectric Layers Based on Reddy’s Higher Order Shear Deformation Theory”, Composite Structures, Vol. 155, pp. 118-134, (2016).
[43] Abedlijaber, O., Avci, O., and Inman, D.J., “Active Vibration Control of Flexible Cantilever Plate using Piezoelectric Materials and Artificial Neural Network”, Journal of Sound and Vibration, Vol. 363, pp. 33-53, (2015).
[44] Plattenburg, J., Dreyer, J.T., and Singh, R., “Vibration Control of a Cylindrical Shell with Concurrent Active Piezoelectric Patches and Passive Cardboard Liner”, Mechanical Systems and Signal Processing, Vol. 91, pp. 422-437, (2016).
[45] Selim, B.A., Zhang, L.W., and Liew, K.M., “Active Vibration Control of CNT-reinforced Composite Plates with Piezoelectric Layers Based on Reddy’s Higher-order Shear Deformation”, Composite Structures, Vol. 163, pp. 350-364, (2017).
[46] Song, Z.G., Zhang, L.W., and Liew, K.M., “Active Vibration Control of CNT-reinforced Composite Cylindrical Shells via Piezoelectric Patches”, Composite Structures, Vol. 158, pp. 92-100, (2016).
[47] Manjunath, T.C., and Bandyopadhyay, B., “Vibration Control of Timoshenko Smart Structure using Multirate Output Feedback Based Discrete Sliding Mode Control for SISO Systems”, Journal Sound and Vibration, Vol. 326, pp. 50-74, (2009).
[48] Rao, S.S., “Mechanical Vibrations”, 5th Edition, Prentice Hall Press, New Jersey, ISBN 978-0-13-212819-3, pp. 737, (2011).
[49] Tiersten, H.F., “Linear Piezoelectric Plate Vibrations-elements of the Linear Theory of Piezoelectricity and the Vibration of Piezoelectric Plates”, Plenum Press, New York, ISBN : 9781489964533, pp. 1-211, (1969).
[50] Clerc, M., “Particle Swarm Optimization”, Wiley Press, Brazil, ISBN-13:978-1-905209-04-0, (2005).
 [51] Clerc, M., and Kennedy, J., “The Particle Swarm-Explosion Stability and Convergence in a Multidimensional Complex Space”, IEEE Transaction on Evolutionary Computation, Vol. 6, Issue. 1, pp. 58-73, (2002).