Active Vibration Suppression of an elastic piezoelectric sensor and actuator fitted cantilevered beam configurations as a generic smart composite structure Harijono Djojodihardjo ⇑,1 , Mohammad Jafari 2 , Surjatin Wiriadidjaja 3 , Kamarul Arifin Ahmad 3 Universiti Putra Malaysia, 34400 UPM Serdang, Selangor Darul Ehsan, Malaysia article info Article history: Available online 8 July 2015 Keywords: Euler–Bernoulli theory Finite element method Hamiltonian mechanics Piezoelectric material Active vibration control Structural dynamics abstract An efficient analytical method for vibration analysis of a Euler–Bernoulli beam with Spring Loading at the Tip has been developed as a baseline for treating flexible beam attached to central-body space structure, followed by the development of MATLABÓ finite element method computational routine. Extension of this work is carried out for the generic problem of Active Vibration Suppression of a cantilevered Euler–Bernoulli beam with piezoelectric sensor and actuator attached as appropriate along the beam. Such generic example can be further extended for tackling light-weight structures in space applications, such as antennas, robot’s arms and solar panels. For comparative study, three generic configurations of the combined beam and piezoelectric elements are solved. The equation of motion of the beam is expressed using Hamilton’s principle, and the baseline problem is solved using Galerkin based finite ele- ment method. The robustness of the approach is assessed. Ó 2015 Elsevier Ltd. All rights reserved. 1. Introduction Vibration control of light-weight structures is of great interest of many studies and investigations [1–3]. The high cost of sending heavy masses and large volumes into space has prompted the wide utilization of light-weight structures in space applications, such as antennas, robot’s arms, solar panels. A model of such set-up is exemplified in Fig. 1 [2]. These kinds of structures are largely flex- ible, which results in lightly damped vibration, instability and fati- gue. To suppress the adverse effect of vibration, sophisticated controller is needed. Active control approaches are widely reported in the literatures for the vibration control of structures [4–10]. The active control approach makes use of actuators and sensors to find out some essential variables of the structure and suppress its vibration through minimizing the settling time and the maximum amplitude of the undesirable oscillation. This method requires a specific level of understanding about the dynamic behavior of continuous struc- tures via mathematical modeling [4,5]. Selecting adequate sensor and actuator is essential in active vibration control [11,12]. The conventional form of sensor and actuator, such as electro-hydraulic or electro-magnetic actuator, are not applicable to implement on the light-weight space structures. Thus, in recent years, a new form of sensor and actuator has been studied using smart materials, such as shape memory alloys and piezoelectric materials. The definition of smart material may be expressed as a material which adapts itself in response to environmental changes. Among smart materials, piezoelectric materials are widely studied in literatures, since they have many advantageous such as ade- quate accuracy in sensing and actuating, applicable in the wide fre- quency range of operations, applicable in distributed or discrete manner and available in different size, shape and arrangement. Space structures can be simplified using beam and plate. The present investigation is based on the vibration analysis of simple beam as a generic structure. Without loss of generalities, the theo- retical development utilizes Euler–Bernoulli beam approximation, which can readily be extended to other refined models. Euler– Bernoulli beam theory is applicable to thin and long span, for which plane sections can be assumed to remain plane and perpendicular to the beam axis, and shear stress and rotational inertia of the cross section can be neglected. Solar panel and antenna are very flexible and slender, so that Euler–Bernoulli beam theory can be considered. The equation of motion of the beam will be developed using Hamilton’s method and Lagrange method [13,14]. Hamiltonian http://dx.doi.org/10.1016/j.compstruct.2015.06.054 0263-8223/Ó 2015 Elsevier Ltd. All rights reserved. ⇑ Corresponding author. E-mail addresses: harijono@djojodihardjo.com (H. Djojodihardjo), al.m.jafari@ gmail.com (M. Jafari), surjatin@hotmail.com (S. Wiriadidjaja), aekamarul@eng.upm. edu.my (K.A. Ahmad). 1 Professor, Department of Aerospace Engineering. 2 Graduate Student, Department of Mechanical and Manufacturing Engineering. 3 Associate Professor, Department of Aerospace Engineering. Composite Structures 132 (2015) 848–863 Contents lists available at ScienceDirect Composite Structures journal homepage: www.elsevier.com/locate/compstruct