Active Vibration Damping of Composite Beam using Smart Sensors and Actuators

Transcription

1 Active Vibration Damping of Composite Beam using Smart Sensors and Actuators G. Song 1 ; P. Z. Qiao 2 ; W. K. Binienda 3 ; and G. P. Zou 4 Abstract: This paper discusses active vibration control of an E-glass/epoxy-laminated composite beam using smart sensors and actuators. The smart sensors and actuators used in this study are piezoelectric ceramic patches. The composite beam is in a cantilevered configuration. Both theoretical and numerical finite-element analysis studies of the laminated composite beam are conducted to reveal the beam s fundamental modal frequencies and modal shapes. The results based on the theoretical predication and numerical simulation are then compared with those from experimental modal testing, and a good correlation is obtained. Utilizing results from the model analysis and experimental modal testing, two control algorithms, namely, positive position feedback control and strain rate feedback control, are designed. Both single-mode vibration suppression and multimode vibration suppression are studied. An experimental apparatus has been developed to implement the control algorithms. The apparatus consists of a voltage amplifier and a data acquisition and real-time control system, in addition to the composite beam with bonded piezoelectric ceramic sensors and actuators. Experiments show that the proposed controllers can achieve active vibration damping of the composite beam. DOI: / ASCE :3 97 CE Database keywords: Vibration; Damping; Beams; Composite materials; Sensors. Introduction By combining the advantage of composites and smart materials, smart composites or intelligent composites can be created. One reason for this activity is that it may be possible to create certain types of structures and systems capable of adapting to, or correcting for, changing operating conditions. The add-on advantage of incorporating these special types of materials into the structure is that the sensing and actuating mechanism becomes part of the structure by sensing and actuating directly Reddy In recent years the subject area of smart composites has experienced tremendous growth in terms of research and development. The purpose of the experiment is to examine the effectiveness of using smart sensors and actuators for active vibration control of a flexible composite beam. Piezoelectric materials will be used as both sensors and actuators in this research, since these materials have the advantages of high stiffness, light weight, low power consumption, and easy implementation. Piezoelectric materials are primary candidates for dynamic stability applications such as resonant vibrations, active vibration suppression, and dynamic 1 Assistant Professor, Dept. of Mechanical Engineering, Univ. of Akron, Akron, OH Assistant Professor, Dept. of Civil Engineering, Univ. of Akron, Akron, OH Professor, Dept. of Civil Engineering, Univ. of Akron, Akron, OH Research Scientist, Dept. of Civil Engineering, Univ. of Akron, Akron, OH Note. Discussion open until December 1, Separate discussions must be submitted for individual papers. To extend the closing date by one month, a written request must be filed with the ASCE Managing Editor. The manuscript for this paper was submitted for review and possible publication on February 5, 2002; approved on February 12, This paper is part of the Journal of Aerospace Engineering, Vol. 15, No. 3, July 1, ASCE, ISSN /2002/ /$8.00 $.50 per page. buckling due mainly to the fact that they tend to have bandwidths beyond the frequency range of structural and acoustic control applications. PZT lead zirconate titanate is a commonly used piezoelectric ceramic. In this research, the object for active vibration suppression is an E-glass/epoxy-composite laminated beam fabricated using the vacuum bagging process. The composite beam is set up in a cantilevered configuration. A modal analysis based on the Euler- Bernoulli beam theory is developed to obtain the natural frequencies of composite beams. Finite-element analysis FEA of the laminated composite beam is also performed to further reveal its fundamental modal frequencies and modal shapes. The theoretical and numerical predictions are then compared with the results from experimental modal testing. To achieve active vibration suppression of the flexible composite beam, two PZT patches are bonded on both sides of the beam near its clamped end as a sensor and an actuator, respectively. The complete vibration suppression system includes a digital data acquisition and real-time control system, and a voltage amplifier, in addition to the composite beam with PZT sensor and actuator. Two active vibration control methods, strain-rate feedback SRF and positive-position feedback PPF, are designed and implemented. PPF Goh and Caughey 1985; Fanson and Caughey 1990; Agrawal and Bang 1994; Song et al is applied by feeding the structural position coordinate directly to the compensator and the product of the compensator and a scalar gain positively back to the structure. PPF offers quick damping for a particular mode provided that the modal characteristics are known. PPF is also easy to implement. Song et al experimentally demonstrated that PPF is insensitive to a varying modal frequency. SRF control is used for active damping of a flexible space structure Newman Using SRF, the structural velocity coordinate is fed back to the compensator, and the compensator position coordinate multiplied by a negative gain is fed back to the structure. SRF has a wider active damping region and can stabilize more than one mode given a sufficient bandwidth. In this JOURNAL OF AEROSPACE ENGINEERING / JULY 2002 / 97

2 Table 1. Ply Material Properties Ply Nominal weight Thickness in. V f E 1 (10 6 psi) E 2 (10 6 psi) G 12 (10 6 psi) v 12 0 UD 1 oz/ft UD 1 oz/ft Table 2. Laminate Properties of in. Composite Beam Micro/macromechanics E x (10 6 psi) E y (10 6 psi) G xy (10 6 psi) v xy Table 3. Comparison of Natural Frequencies Mode Theoretical Hz Experiment Hz Finite element Hz fiber, and the resin or matrix is epoxy. The composite beam used in this study is cut from the laminated panel. The laminated panel consists of six comb layers of C1800 fabric a 0 /90 cross-ply stitched fabric, provided by Brunswick Technologies, Inc., Maine; there is a total of 12 layers (0 /90 ) 3 s in the laminated panel. The material properties of this symmetric cross-ply laminated panel are evaluated by a combined micro/ macromechanics approach Davalos et al. 1996; Luciano and Barbero 1994; Qiao Each layer is modeled as a homogeneous, linearly elastic, and generally orthotropic material, and to evaluate its properties the information provided by the material supplier is used to compute the fiber volume fraction (V f ) of each equivalent layer. Each layer is simulated as a unidirectional composite, and a micromechanics for composites with periodic microstructure Luciano and Barbero 1994 is used to compute plystiffness properties. The fiber volume fraction (V f ) and ply stiffnesses of the constituent layers are summarized in Table 1. Once the ply-stiffness properties for the laminated panel are computed through micromechanics formulas, the stiffness properties of the symmetric panel are computed from classical lamination theory Jones In particular, the laminate engineering properties Table 2 can be evaluated from the compliance matrix corresponding to the extensional stiffness matrix. Because of the lamination layup of cross-plies, the material properties of the laminated panel are orthotropic in nature. Since the thickness of a composite beam or panel is relatively thin beam dimensions are in., the predicted in-plane properties in Table 2 are later used in numerical modeling. Theoretical Modal Analysis Based on the Euler-Bernoulli beam theory, a theoretical modal analysis of composite beams is developed to predict the natural frequencies and modal shapes of the structures. Considering a composite beam structure with piezoelectric actuators, the dynamic equation is given as 4 w 2 w D 11 x 4 M t 2 f t u t (1) Fig. 1. Analytical deformed shapes of a Mode I and b Mode II vibrations where N D k 1 C 11 z 3 3 k z k 1, N M k 1 k h k (2) research, the SRF is designed to control the vibration of the first mode. Two PPF controllers in parallel are designed to control the vibration of both the first and second modes. Experimental results demonstrate that both the SRF and PPF methods are effective in actively increasing damping of the flexible composite beam with the PZT sensor and actuator. Material Modeling of Composite Beams A composite laminated panel is fabricated using the vacuum bagging process, and the reinforcement used is continuous E-glass in which C 11 material stiffness; k and h k density and thickness of the kth layer, respectively; w transverse displacement of the beam; f(t) external force vector; and u(t) input vector provided by the piezoelectric actuators. The transverse deformation of one-dimensional plate w can be decomposed into the modal summation as w x,t m 1 m t m x where m (t) and m (x) modal coordinate and modal shape of mode m, respectively. The modal shape for a cantilever beam can be written as Warburton / JOURNAL OF AEROSPACE ENGINEERING / JULY 2002

3 Fig. 2. Finite-element deformed shapes of a Mode I and b Mode II vibrations where x x m x cosh m L cos m L m x x sinh m L sin m L (3) m sin m sinh m (4) cos m cosh m where m 1.875,4.694,...,2m 1/2, and L beam length. For the natural frequency problem free vibration, no external force is applied; i.e., f(t) u(t) 0, and the modal coordinate m (t) is assumed as m t m e i mt (5) where m magnitude of modal coordinate and m mth-mode natural frequency. Substituting Eqs. 3, 4, and 5 and d 4 m (x)/dx 4 ( m ) 4 m (x) into Eq. 1, the natural frequency can be obtained as m 2 m D 11 L 2 (6) M Based on Eq. 6 and the material properties given in Table 2, the natural frequencies for the first two modes are given in Table 3, and their corresponding modal shapes are depicted in Fig. 1. Modal Analysis by Finite-Element Method A modal analysis using the commercial finite-element package ANSYS 1999 is also used to study the dynamic behavior of the composite beam. The orthotropic material properties given in Table 2 are used in the modeling, and the composite beam is simulated using 4-node shell elements SHELL 63. Based on the eigenvalue solution analysis available in ANSYS, the natural frequencies of the first several modes are obtained Table 3. The deformed shapes of the first and second modes are shown in Figs. 2 a and b, respectively. The comparison of natural frequencies among theoretical predictions, numerical modeling FE, and experimental testing is also shown in Table 3. As indicated in that table, a relatively close agreement within the three approaches is obtained, and it demonstrates that micro/macromechanics for material prediction, the theoretical free-vibration analysis using a unique shape function for a cantilever beam, and the finiteelement method for modal analysis can be effectively used to evaluate the dynamic behavior of composite beam structures. Experimental Setup Fig. 3. Experimental setup for active vibration control of flexible composite beam The experimental setup for active vibration control of the flexible composite beam is shown in Fig. 3. To better view the functionality of each component, a schematic of this setup is shown in Fig. 4. As shown in both Figs. 3 and 4, a cantilevered composite beam is used as the object for vibration control. The beam is the E-glass/epoxy-composite beam described in earlier sections. The beam is clamped such that its length is parallel to the supporting table below it. This allowed the bending to be strictly in the horizontal plane. The beam has a PZT sensor and a PZT actuator bonded on its face near its cantilevered base. The PZT sensor has a dimension of in. The PZT actuator has two stacks of two piezo wafers, and each wafer has a dimension of in. A dspace digital data acquisition and real-time control system is used for data collection and implementation of control algorithms. Using a TMS320C31 microprocessor, the dspace processes the data from the PZT sensor and JOURNAL OF AEROSPACE ENGINEERING / JULY 2002 / 99

4 Fig. 4. Schematic of experimental setup Fig. 5. Plot of 30 s time history of free vibration of composite beam Fig. 6. Power spectrum density plot using 30 s data Fig. 7. Comparison of power spectrum density plots using a first and b last 2 s data during free vibration 100 / JOURNAL OF AEROSPACE ENGINEERING / JULY 2002

5 Positive-Position Feedback Control For control of the flexible structures, the PPF control scheme is well suited for implementation utilizing the piezoelectric sensors and actuators. In PPF control, structural position information is fed to a compensator. The output of the compensator, magnified by a gain, is fed directly back to the structure. The equations describing PPF operation are given as 2 2 G 2 (7) Fig. 8. Plot of 30 s vibration of beam with strain-rate feedback control generates a control signal according to the control algorithm. The control signal is then amplified by the voltage amplifier and finally sent to the PZT actuator to suppress vibrations. Experimental Modal Testing For the initial testing, a free vibration of the composite beam excited by manual tapping is conducted. Data is recorded for 30 s. Fig. 5 shows the time history of the free vibration of the beam. It takes more than 30 s 50 s to be exact for the beam to settle down. Fig. 6 shows the power spectrum density PSD plot of this 30 s data and reveals the first four modal frequencies. It is clear that the first two modes at 2.6 and 15.9 Hz are dominant. Fig. 7 presents the PSD plots of the same signal for the first and last 2 s. During this 30 s period, the energy level at mode 2 dropped 70 db, while it dropped only 18 db at mode 1. This revealed that mode 1 at 2.6 Hz would be the primary target for active vibration suppression. Methods for Vibration Suppression Two vibration suppression methods positive-position feedback control and strain-rate feedback control are reviewed in this section. These two methods are implemented to suppress vibrations of the flexible beam. 2 c c 2 c 2 c (8) where modal coordinate describing the displacement of the structure; damping ratio of the structure; natural frequency of the structure; G feedback gain; compensator coordinate; c compensator damping ratio; and c natural frequency of the compensator. To achieve maximum damping, c should be closely matched. Also, any structural natural mode above c will experience increased stiffness with the PPF control. This often results in high structural modes being exited by the PPF control. Strain-Rate Feedback Control SRF control is achieved by feeding the structural velocity coordinate to the compensator. The compensator position coordinate is then fed back to the structure after a negative gain is applied. When using a PZT sensor and a PZT actuator, this is realized by feeding the derivative of the voltage from the sensor, which is proportional to the strain rate, to the input of the compensator and applying the negative compensator output voltage to the actuator. The equations of motion in modal coordinates are 2 2 G 2 (9) 2 c c c 2 c (10) where the variables are the same as those defined for the case of PPF in the previous section. As compared to the PPF, SRF has a much wider active damping frequency region, which gives a designer some flexibility. Selecting a precise compensator frequency for SRF is not as critical as for PPF. As long as the compensator frequency is greater than the structural frequency, a certain amount of damping will be provided. A limitation of SRF is that the magnitude of the transfer function in the active damping region becomes extremely small very quickly. Therefore, the amount of damping provided SRF over a certain frequency range is limited. Fig. 9. Comparison of power spectrum density plots using a first and b last 2 s data during strain-rate feedback control JOURNAL OF AEROSPACE ENGINEERING / JULY 2002 / 101

6 vibration to settle down. Energy at mode 1 has been brought down by 70 db Fig. 9 as compared with 18 db for the case of free damping. This experiment demonstrates the effectiveness of SRF control for active damping of a flexible composite beam with a PZT sensor and a PZT actuator. Fig. 10. Plot of 30 s vibration of beam with PPF-PPF control Experimental Results Experimental Procedures For each test, data are obtained for a time interval of 30 s after beam excitation. This allows ample time to measure damping effects. The experimental data are then processed to show effectiveness of the tested control algorithm. A fast Fourier transform FFT is performed in Matlab to provide a power spectral density PSD plot of the beam response. PSD gives a measure of signal energy level at different frequencies. A comparison of the ratios of the last-second modal energy level in db to the initial one provides an indication of the damping effectiveness on this particular mode. Also, a direct comparison of the modal energy level drop with that of an open-loop response can indicate the effectiveness of the control algorithm. Strain-Rate Feedback Control Results An SRF controller with a cut-off frequency at 4 Hz is designed and implemented. The SRF controller should bring positive damping to the first mode. Fig. 8 shows the time history of beam vibration with active vibration suppression. It took about 7.5 s with SRF control, versus 50 s for the case of free damping, for the Positive-Position Feedback Control Results A controller employing two positive-position feedback PPF-PPF controllers are designed and implemented. The first PPF controller targets the first mode while the second PPF controller targets the second mode. Fig. 10 shows the time history of beam vibration with active vibration suppression. It took about 13 s for the vibration to settle down with PPF-PPF control, versus 50 s for the case of free damping. Energy at mode 1 has been brought down by 74 db Fig. 11 as compared with 18 db for the case of free damping. This experiment demonstrates the effectiveness of positive-position feedback control for active damping of a flexible composite beam with a PZT sensor and actuator. For this particular composite beam, the experimental results indicate that the SRF is more effective than the PPF in providing overall damping: a 7.5 s settling time Fig. 8 versus a 13 s settling time Fig. 10. The reason SRF is more effective is that the energy level at the second mode in the case of SRF Fig. 9 is reduced more than that in the case of PPF Fig. 11. This is because the PPF control has the tendency to excite higher modes. Conclusions This paper discusses the active vibration control of an E-glass/ epoxy-composite beam using a PZT sensor and actuator. The composite beam is in a cantilevered configuration. A theoretical formulation based on the Euler-Bernoulli beam theory and a unique shape funcation for the cantilever beam boundary condition is developed to predict the natural frequencies and model shapes of the structure. A commercial FEM software is also used to model the laminated composite beam and reveal fundamental modal frequencies and modal shapes. Before the controller design, experimental modal testing of the composite beam is conducted. Based on modal analysis and testing results, two different active controllers, positive-position feedback control and strainrate feedback control, are designed. Both single-mode vibration suppression and multimode vibration suppression are studied. An experimental apparatus is developed to implement the control algorithms. The apparatus consists of a voltage amplifier and a data Fig. 11. Comparison of power spectrum density plots using a first and b last 2 s data during PPF-PPF control 102 / JOURNAL OF AEROSPACE ENGINEERING / JULY 2002

7 acquisition and real-time control system, in addition to the composite beam with bonded piezoelectric ceramic sensors and actuators. Experiments demonstrate that the piezoelectric sensors can accurately and effectively obtain the natural frequencies of the structures, and the proposed controllers can achieve active vibration damping of the composite beam with piezoelectric ceramic patches as both sensor and actuator. Acknowledgments The writers would like to thank Michael C. Uhrain IV for his assistance in fabrication of the laminated composite beam, and Jialai Wang for the modal analyses using ANSYS. This study was partially supported through a NASA collaborative research grant, an NSF CAREER grant, and a collaborative research program on smart composites by the College of Engineering at the University of Akron. References Agrawal, B. N., and Bang, H Adaptive structure for large precision antennas. Proc., 45th Congress of the Int. Astronautical Federation (Jerusalem), October. ANSYS User s Manual Swanson Analysis System, Inc., Houston, Pa. Davalos, J. F., Salim, H. A., Qiao, P. Z., Lopez-Anido, R., and Barbero, E. J Analysis and design of pultruded FRP shapes under bending. Composites, Part B, , Fanson, J. L., and Caughey, T. K Positive position feedback control for large space structure. AIAA J., 28 4, Goh, C. J., and Caughey, T. K On the stability problem caused by finite actuator dynamics in the collocated control of large space structure. Int. J. Control, 41 3, Jones, R. M Mechanics of composite materials, Taylor & Francis, Philadelphia. Luciano, R., and Barbero, E. J Formulas for the stiffness of composites with periodic microstructure. Int. J. Solids Struct., 31 21, Newman, S. M Active damping control of a flexible space structure using piezoelectric sensors and actuators. Master thesis, U.S. Naval Postgraduate School. Qiao, P. Z Design analysis and optimization of FRP structural beams. PhD dissertation, Dept. of Civil and Environmental Engineering, West Virginia Univ., Morgantown, W.V. Reddy, J. N On laminated composite plates with integrated sensors and actuators. Eng. Struct., 21, Song, G., Schmidt, S. P., and Agrawal, B. N Experimental study of vibration suppression of flexible structure using modular control patch. Proc., IEEE Aerospace Conf., Snowmass, Colo. Song, G., Schmidt, S. P., and Agrawal, B. N Active vibration suppression of a flexible structure using smart material and a modular control patch. Proc. Inst. Mech. Eng., 214, Part G, JOURNAL OF AEROSPACE ENGINEERING / JULY 2002 / 103