Elsevier

Journal of Sound and Vibration

Volume 412, 6 January 2018, Pages 1-16
Journal of Sound and Vibration

Fractional-order positive position feedback compensator for active vibration control of a smart composite plate

https://doi.org/10.1016/j.jsv.2017.09.009Get rights and content

Abstract

In this paper, Active Vibration Control (AVC) of a rectangular carbon fibre composite plate with free edges is presented. The plate is subjected to out-of-plane excitation by a modal vibration exciter and controlled by Macro Fibre Composite (MFC) transducers. Vibration measurements are performed by using a Laser Doppler Vibrometer (LDV) system. A fractional-order Positive Position Feedback (PPF) compensator is proposed, implemented and compared to the standard integer-order PPF. MFC actuator and sensor are positioned on the plate based on maximal modal strain criterion, so as to control the second natural mode of the plate. Both integer and fractional-order PPF allowed for the effective control of the second mode of vibration. However, the newly proposed fractional-order controller is found to be more efficient in achieving the same performance with less actuation voltage. Moreover, it shows promising performance in reducing spillover effect due to uncontrolled modes.

Introduction

In the past decades research on Active Vibration Control (AVC) has found increasing interest in control of flexible thin-walled structures, mainly made of new advanced materials such as carbon fibre composites. These type of composite structures combine high stiffness with good flexibility in achieving complex shapes, and are mostly used in automotive and aerospace applications where they are often subjected to undesirable vibrations. In the field of AVC, a new type of sensor and actuator have become popular by using the so-called smart materials, such as piezoelectric materials, shape-memory alloys and electroactive polymers. These materials are called smart because they are inherently capable of detecting or responding to changes in their environment, making them very suitable both for sensing and actuation purposes. Given their distributed nature, they can be easily mounted on different types of structures, thus making them smart structures.

Piezoelectric transducers are often selected as sensors and actuators for the active control of smart flexible structures because of their unique properties including low cost, low mass, ease of integration and wide frequency range of control. In 1996 NASA invented a specific type of transducer called Macro Fibre Composite (MFC) which provides high performance, durability and a very good flexibility, making these transducers a preferred option in the case they shall be adapted to different structure geometries. Piezoelectric transducers in general, when used as sensors, measure strain which is proportional to the physical displacement. In fact, control schemes specifically designed to use position as feedback signal have been extensively studied and applied in this context.

The main objective of the controller is to provide active damping to the structure (plant), which results in an attenuation of the resonance peak in the dynamic amplification. The dynamics of flexible structures have very interesting properties: because of their flexibility, they have a large number of elastic modes resulting in very high order transfer functions that are rather difficult to control. Controllers are designed to target specific vibration modes in a restricted bandwidth of interest, and the fact that transfer functions are of high order means that there are out-of-bandwidth modes which are neglected, but whose effect might influence the closed-loop response. The effect of the uncontrolled, or out-of-bandwidth modes, is known in the literature as spillover [1]. Another important aspect regarding the controller is that, apart from being able to reduce structural vibrations, it should ensure robustness and closed-loop stability for the controlled system. In this sense, careful positioning of sensors and actuators can have a great influence. The majority of the controllers studied in literature use a collocated configuration, where sensors and actuators are related to the same Degree of Freedom (DOF) of the structure. The phase of the open-loop collocated transfer function is always between 0° and 180°, meaning that poles and zeros interlace on the imaginary axis, where zeros and poles correspond to anti-resonances and resonances of the frequency response, respectively. Collocated systems have the property of being always closed-loop stable with respect to out-of-bandwidth dynamics [2] and that is why most of the research involves collocation.

One of the most popular collocated modal control schemes is Positive Position Feedback (PPF), which has been first proposed in 1985 by Goh and Caughey [3] to overcome the instability associated with finite actuator dynamics. This controller was applied for the first time in 1987 by Fanson [4] to experimentally suppress vibrations in large space structures. PPF is effective if tuned to suppress a chosen frequency and mode. It is a second order low-pass filter which rolls off quickly at high frequencies, making it very appealing against possible instability or performance losses due to out-of-bandwidth dynamics.

Applications of PPF with smart structures are also extensively studied in literature. Kwak and Heo [5] applied a Multi-Input Multi-Output (MIMO) PPF to control vibrations of a smart grid structure equipped with piezoelectric transducers. They proposed a new technique to control a higher number of modes than the number of actuators and sensors. Zippo et al. [6] applied PPF for active vibration control of a composite sandwich plate using MFC transducers. Ferrari and Amabili [7], as a continuation of the work of Zippo, applied non-collocated PPF both in Single-Input Single-Output (SISO) and MIMO.

Direct Velocity Feedback (DVF), Resonant Control (RC) and Integral Resonant Control (IRC) are also collocated control techniques which are popular in literature. However, the DVF and RC [8] use velocity and acceleration as their feedback, respectively and thus they present limitations when used with piezoelectric transducers. Moreover, compared to PPF, IRC is a position feedback control with less suppression of higher order modes due to its first order filter behaviour [9]. In the field of AVC in general, apart from the aforementioned methods, many other different control strategies have been applied for several types of applications. Fractional-order calculus has been found to be an effective tool in control (see e.g. Refs. [10], [11], [12], [13], [14], [15]), however, it has rarely found room in the context of AVC [16], [17] and little research is present [18], [19], [20].

Fractional-control has never been experimentally applied before in the field of AVC of smart structures, and it is for the first time presented in this paper. A fractional-order versions of the PPF compensator is proposed, and compared to the standard integer-order PPF. Controllers are then implemented to control the 2nd mode of a carbon fibre/epoxy composite plate equipped with MFC actuator and sensor. A similar setup to the one used by Alijani and Amabili for similar studies [21], [22] is built to test and control the plate with all edge free boundary conditions. These type of boundary conditions reduce the influence of temperature variations and other non-ideal boundary conditions which are generally associated with relevant changes in natural frequencies due to thermal stress. Therefore, completely free boundary conditions have been chosen to perform experimental modal analysis on the composite plate. In the next section, integer and fractional-order PPF compensators are elaborated and compared. In section 4 the experimental dynamics setup and complete AVC setup are described in detail. In section 5 results are presented, whereas final remarks and conclusions follow in section 6.

Section snippets

Direct velocity feedback

Direct Velocity Feedback of a 1-DOF system is defined as follows:ξ¨+2ζωξ˙+ω2ξ=ω2fwhere ξ, ω, ζ are modal coordinate, natural frequency and modal damping of the structure, respectively; f=gξ˙ is the modal control force and g is the feedback gain. Equation (1) can be rewritten in the following form:ξ¨+(2ζω+gω2)ξ˙+ω2ξ=0It can be noted that active damping in this case is achieved through direct velocity feedback signal with gain g. DVF does not prevent the occurrence of spillover effect, but

Fractional-order control

As stated by Monje et al. [24], ‘Fractional calculus can be defined as the generalization of classical calculus to orders of integration and differentiation not necessarily integer’. Fractional-order dynamic systems can be expressed by fractional-order transfer functions whose simplest form in Laplace domain is sα, where αR and s is the Laplace transform variable. The benefit of fractional-order transfer functions comes from the fact that they have magnitude and phase response which are not

Experimental setup

An experimental set up is built to conduct modal analysis and active vibration control of a rectangular carbon fibre/epoxy composite plate (see Fig. 8). This setup is part of the Engineering Dynamics Laboratory of the Department of Precision and Microsystems Engineering (PME) at Delft University of Technology (TU Delft).

The plate has been produced at the Delft Aerospace Structures and Materials Laboratory by vacuum infusion process (VIP), using 6 layers of 0–90° carbon fibres and epoxy resin

Implementation of AVC

The integer and fractional-order PPF controllers presented in section 3 are first implemented in a Simulink model which is then compiled and built in the dSPACE ControlDesk environment.

Controller transfer function is the same as the one of the controllers explained in section 3. The controller can be activated by means of the gain block ‘ON_OFF_switch’ which is loaded as a switch button in the dSPACE environment. ‘Gain_IN’ and ‘Gain_OUT’ represent hardware gains required from the dSPACE system.

Conclusions

In this paper active vibration control of a rectangular carbon fibre/epoxy composite plate with free edges has been presented. A novel fractional-order compensator based on Positive Position Feedback has been proposed, analysed, and successfully applied in practice for the first time.

Numerical simulation has shown the benefit of using fractional-order PPF with respect to reduction of spillover effect, compared to the standard integer-order PPF. Controllers have been tested in an experimental

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