Efficiency of Active Control of Beam Vibration Using PZT Patches

This study is aimed to explore efficiency problems of using PZT patches in active vibration control of beams. The PZT patches are surface-bonded as actuators and strain gages are used as sensors. Distributed moment forcing mode is adopted to derive the control authority of actuators. Uniform beam model is applied when the mass and stiffness of PZT patches are ignored. Stepped beam model is then derived to describe the beam-actuator structure when the effect of patches is taken into consideration. Modal expansion method is used to determine the efficiency index of PZT patches in each mode. It is found that the structural modification caused by attaching PZT patches may degrade the control efficiency. It is also found that the optimal positions from stepped beam model are similar to those from the uniform beam model. The ratio of control efficiency from the stepped beam model to that from the uniform beam model is associated with stiffness of host structure and actuators. Base on this finding, the optimal thickness of PZT patches can be obtained. Because the factor is a constant when the structure is determined, we propose a modified model to estimate the control efficiency. This modified model, while taking advantage of simplicity of the uniform beam model, yet accounts for the effect of the inertia and stiffness the PZT patch. Experiments on vibration control of beams were conducted to verify our efficiency estimate. INTRODUCTION The use of PZT (lead zirconate titanate piezoceramic materials) in active vibration control has attracted a great deal attentions in the last twenty years. Extensive studies have been conducted in this area. These include, but not limited to, modeling control mechanism of PZT, implementation and verification of using PZT as actuating elements, and optimization problems in control systems involving PZT as actuators and/or sensors. Two models have been proposed to represent the static interaction between PZT and structures. Crawley and de Louis [1] presented uniform strain model. It assumed uniform strain within the PZT patch and pure shear state in adhesive layer while linear strain in host structure. Crawley and Anderson [2] proposed a Bernoulli-Euler model assuming the strain distributions within the actuator, the adhesive layer and the substrate were linear variation. It was found that uniform strain model was not correct when the beam was getting thinner while the Bernoulli-Euler model predicted the interaction relatively well. When extending these two models to dynamic cases, they are called the pin-force model and the distributed moment forcing model, respectively. Park et al. [3] extended uniform stain model and Bernoulli-Euler model to a bonded piezoelectric patch case. They also presented coupled bending-extension mode and coupled bending-extension-torsion model. Strambi et al. [4] presented pin-force and Euler-Bernoulli models in a general fashion; they discussed the case of one-side only actuation layer. Although Chaudhry and Rogers [5] proposed an enhanced pin-forced model, which improves the accuracy when PZT patches get thicker, most majority of researchers adopted the distributed moment forcing model because the latter provides better prediction and the assumptions are generally reasonable. There are many researchers investigated the optimization of positions and dimensions of PZT patches in vibration reduction. They implemented the available models to introduce the control authority into structures. It is well known that misplaced sensors and actuators may cause some problems such as lack of observability and controllability and spillover [6-8]. Tzou and Fu [9] showed that fully covered piezoelectric sensors and actuators might not control some modes of vibration because of poor observability and controllability. They concluded that segmentation of the sensors and actuators is necessary to control most vibrational modes. Crawley and Louis [1] used pin-force model to investigate the optimal position of PZT pairs. They suggest the optimal position of a certain mode be the anti-node of that mode. Barboni et al. [10] extended this work using the method of modal analysis. They obtained similar results as Crawley Louis [1]. Kondoh et al. [7] presented a strategy of positioning sensors and actuators by minimizing the quadratic cost function in the standard optimal control. Hac and Liu [11] studied the problem of sensor and actuator locations in motion control of flexible structures. They obtained optimization criteria by considering the energy input and output. Steffen et al. [12] used distributed moment forcing model to study the optimal problem of PZT surface bonded beams. Sadri et al. [13] obtained two criteria for the optimal placement of piezoelectric actuators using controllability and observability of the system. Han and Lee [11] used a generic algorithm to seek the optimal locations of piezoelectric sensors and actuators of a smart composite plate from the perspective of controllability and observability. Aldraihem et al. [14] proposed an optimization criterion based on modal cost and controllability index to investigate the size and placement of actuators. Ball and Jones [15] investigated the characteristics and effectiveness of the shape of piezo-actuators for use in controlling the divergence of a simplified forward wing model. They found the optimal actuator size should span the entire wing. The required actuator’s thickness for divergence control decreases with increase airspeed due to the effectiveness softening of the wing in the presence of air loads. A uniform rectangle actuator is more efficient for controlling divergence; however, a linear shaped actuator is more efficient from the standpoint of control effectiveness per actuator weight. Liang et al. [16,17] presented optimization solution based on actuator power factor. Many researchers also investigated the actuator dimension and other optimization. Ip and Tse [18] presented the optimal position and orientation of a piezoelectric patch actuator for improving the controllability of isotropic plates. They found the optimal patch positions coincide exactly with anti-nodes of vibration modes for simply supported plates. They suggest aligning the patch (0°) is already a good choice; the improvement by varying orientation is insignificant (less than 1%). Main et al. [19] investigated the optimal embedding location and thickness of PZT pair in order to get maximum bending effect from actuator. Nam et al. [20] calculated the best geometry (placement, thickness, width and length) of piezo-actuators using numerical method. Seeley and Chattopadhyay [21] showed optimal thickness for box beams with a bimorph actuator configuration. Kim and Jones [22] investigated the effect of piezo-actuator thickness on the active vibration control of a cantilever beam. They found there exists an optimal thickness for actuators in static excitation of beam by the piezo-actuators. It was shown that the optimal thickness is a strong function of the Young’s modulus ratio of the actuator/beam configuration, becoming thinner with stiffer piezo-actuators. They also studied the optimal thickness of actuator for dynamic case. In all these works, the PZT patch’s mass and stiffness are either assumed to be negligible, or considered but dealt with numeric methods. It is much more complicated, however, if the structural modifications by the presence of actuators are considered. Wang and Wang [8] attempted to consider the effects of PZT patches on the substructure, their model did include the actuator’s stiffness. Nevertheless, the model still ignored the inertia of PZT patches. Yang and Lee [23] proposed the stepped beam model to describe the actuator-beam structure, but they only studied the changes of natural frequencies and modal shape functions. They did not mention those changes would influence the control efforts. Researchers did realize the PZT patches change the structure at certain level. However, they may have simply ignored the effect, thinking an analytical solution including PZT patches’ inertia and stiffness may be too complicated even for simple substructures like beams and plates. Instead, numerical methods, such as finite element techniques, were used to circumvent the problem. Based on classical laminate theory (CLT), Tzou and Tseng [24], Hwang and Park [25], Lam et al. [26], Oh et al. [27], Liu et al. [28] developed finite element models for structures with PZT patches or layers. In order to improve the accuracy of finite element model, some researchers developed their models based on higher order approximation. Bhattacharya et al. [29] developed a finite element model for smart beams and plates based on the Raleigh-Ritz principle and the first order shear deformation theory. Peng et al. [30] introduced the third order laminate theory to their finite element model for the active position control as well as vibration control of composite beams with distributed piezoelectric sensors and actuators. Based on the third order shear theory, Zhou et al. [31] presented another development in the finite element models by introducing a three order potential field that describe the field distribution in the piezoelectric elements along the thickness direction. They found that the results from their model were significant different from other models for thick piezoelectric layers. Another branch of PZT applications in vibration control is algorithm implementation. Using available analytic actuation mechanism model or finite element method, many researchers were able to applied different strategies to control structures containing PZT as actuators. The popular algorithms are proportional velocity feedback control, LQG (linear quadratic Gaussian) control, IMSC (independent modal space control) and PPF (positive position feedback) control. Recently several researchers investigated the possibility using piezoelectric materials simultaneously as sensor and actuator in vibration suppression. PZT has also been used widely in passive vibration suppression and shape control purpose. But we will focus on the active vibration control in this study. Although there are many papers studying the PZT’s applications in various areas, little work was done about the modification of host structure by attaching the PZT patches. Previous researchers rarely notice this alteration could degrade the control efficiency substantially. In this paper, we will concentrate on the control ability that PZT patches can provide to the host structure. By comparing the efficiency estimates with and without consideration of structural alteration because of the presence of actuators, a new method is introduced to better predict the control ability of PZT patches. UNIFORM-BEAM MODEL When the model ignores the inertia and stiffness of PZT patches, we call it the uniform-beam model. As shown in Fig. 1, a cantilever beam attached with one pair of PZT patches. Using the distributed moment forcing model, the dynamic of beam with actuator can be described as: EIw Aw cw M ρ Λ ′′′′ ′′ + + = (1)