3rd Ankara International Aerospace Conference

In this study the fractional controllers, which were realized by the different degrees of the Continued Fractional Expansion (CFE) method, were experimentally evaluated for the suppression of the vibrations of the second mode of a smart beam. The smart beam is equipped with PZT patches and these patches were used both as actuator and/or sensor. The control strategy was based on the fractional derivation of the measurement signal which was the displacement values and filtering that signal by using a filter which was designed to characterise the dynamic properties of the second mode of the smart beam. The experimental results showed that when the controller was realized with a higher fractional derivation degree, better vibration suppression was provided for the second mode. INTRODUCTION This study gives the analysis, design and applications of the fractional order differentiators for the active vibration control of the second mode of a smart beam. The fractional order differentiators are the examples of fractional order systems. The fractional order systems are described by the fractional order differential equations [1, 2]. Fractional order differentiators are used to compute the fractional order time derivative of the given signal [3-6]. Geometrical and physical interpretations of fractional order differentiators are widely discussed in literature [7-11]. Fractional order control systems have transfer functions with fractional derivatives s α and fractional integrals s -α where α€R. It is not very easy to compute the frequency and time domain behaviours of such fractional order transfer functions with available software packages. It is well known that the simulation programs have been prepared to deal with integer power only. Although, there are some recent works dealing with implementation of a controller using fractance device [12], this area also needs further studies since an electronic component to implement fractional order systems is not, recently, commercially available. Therefore, the problem of integer order approximations of fractional order functions becomes a very important one to be attempted. A fractional transfer function can be replaced with an integer order transfer function which has almost the same behaviours with the actual transfer function. There are various methods [13-20] for computing the integer order approximations of the fractional order operators such as s α or s -α . One of the most widely encountered approximations for fractional order systems is the CFE method [21]. 1 Assist. Prof. in the Department, Email: [email protected] 2 Assist. Prof. in the Department, Email: msahin@ metu.edu.tr 3 Prof. in the Department, Email: [email protected] Cem ONAT Department of Mechanical Engineering Inonu University Malatya, TURKEY and Department of Aerospace Engineering Middle East Technical University Ankara, TURKEY Melin SAHIN and Yavuz YAMAN Department of Aerospace Engineering Middle East Technical University Ankara, TURKEY AIAC-2011-1021 Onat, Sahin & Yaman 2 Ankara International Aerospace Conference A fractional order controller which was designed by using a fourth degree approach of CFE method was successfully applied for the active control of the first flexural mode vibrations of a smart beam [24]. A variety of different degree approaches of CFE method was also applied, again in order to suppress the first flexural mode vibrations of the same smart beam. The performance and the robustness characteristics were evaluated. It was shown that the increase in the approach value of CFE method significantly increased the performance of the developed controller [25]. In this study a fractional order controller, developed by using the CFE method, was designed and implemented for the suppression of the second flexural mode vibration of a smart beam. The first, second, third and fourth degree approaches of CFE method were studied together with an integer counterpart for the performance analysis of the controller. Experimentally obtained results were presented for the suppression of the free and forced vibrations SMART BEAM The smart beam studied was a cantilever aluminium beam with eight surface bonded Lead-ZirconateTitanate (PZT) patches and is shown in Figure 1. A thin isolation layer was placed between the aluminium beam and PZT patches hence each PZT patch might independently be employed as a sensor and/or an actuator [22]. Figure 2 and Figure 3 show the experimental setup used in the study and the frequency response of the smart beam covering the first two flexural resonance frequencies respectively. Figure 1: The smart beam used in the study The smart beam was then harmonically excited in a frequency range to cover the second resonance frequency (approx. at 41.25 Hz) with piezoelectric patches acting as actuators and the response of the smart beam was obtained from a different single piezoelectric sensor patch acting as a sensor in order to obtain the necessary experimental frequency response of the smart beam for the system identification. The mathematical model of the smart beam was obtained by processing the measured frequency response data. By using MATLAB’s “fitsys” command located in μ Analysis and Synthesis Toolbox the transfer function of the smart beam was determined [24]. MATLAB “fitsys” command builds a statespace model based on estimated transfer function. Transfer function of the smart beam is estimated within the frequency range between 30 Hz and 50 Hz. This frequency range included the second flexural mode (approx. at 41.25 Hz) of the smart beam. PZT patches Aluminium beam AIAC-2011-1021 Onat, Sahin & Yaman 3 Ankara International Aerospace Conference Figure 2: The experimental setup used in the study 5 10 15 20 25 30 35 40 45 -60 -40 -20 0 20 Frequency (Hz) A m p lit u d e ( V /V d B ) 5 10 15 20 25 30 35 40 45 -200 -100 0 100 Frequency (Hz) P h a s e ( D e g re e ) Figure 3: Frequency response of the smart beam covering the first two flexural modes AIAC-2011-1021 Onat, Sahin & Yaman 4 Ankara International Aerospace Conference The determined 6 th order transfer function of the smart beam is given in Equation 1 and Figure 4 shows the experimentally measured and analytically estimated transfer functions of the smart beam. 14 10 2 10 3 6 4 5 5 6 13 9 2 8 3 4 4 4 5 6 10 089 . 3 10 779 . 5 10 371 . 1 10 716 . 1 10 028 . 2 74 . 12 10 922 . 1 10 314 . 3 10 8.405 10 642 . 9 10 225 . 1 0.701 0.05947 ) (                                   s s s s s s s s s s s s s G (1) 30 32 34 36 38 40 42 44 46 48 50 -45 -40 -35 -30 -25 -20 -15 -10 Frequency (Hz) M a g n it u d e ( d B ) Measurement Analytical model Figure 4: Experimentally measured and analytically estimated transfer functions of the smart beam around its second mode CONTROL DESIGN METHOD In this study, the differential effect was included as the fractional one and the controller for active vibration suppression was synthesized in two steps. First, the fractional differential effect of the smart beam was derived from the measurement signal by using the fractional derivative effect s μ . In this study different approximations for s μ was considered by using first, second, third and fourth degree approach of CFE method. These approximations are given in Equations 2 to 5 in ascending order [25, 26].                       1 1 1 1 s s s (2)             2 3 8 2 2 3 2 3 8 2 2 3 2 2 2 2 2 2 2 2                                       s s s s s (3) AIAC-2011-1021 Onat, Sahin & Yaman 5 Ankara International Aerospace Conference                 6 11 6 54 27 6 3 54 27 6 3 6 11 6 6 11 6 54 27 6 3 54 27 6 3 6 11 6 2 3 2 3 2 2 3 3 2 3 2 3 2 3 2 2 3 3 2 3                                                                                      