Smart actuator effectiveness improvement through modal analysis A. Joshi a and S. M. Khot b a Professor, Department of Aerospace Engineering, Indian Institute of Technology, Bombay. b Research Scholar, Department of Aerospace Engineering, Indian Institute of Technology, Bombay. The effectiveness of the overall smart structure depends to a great extent on the number and distribution of active material patches as well as controller design. Therefore, placement of actuators and sensors plays an important role in the design procedure of the smart structures; particularly it is one of the main problems in control of plate like structures. Actuators and sensors placement as well as their estimation of an optimal shapes are a very complex problems which have not yet been fully solved. The attempt is made here to study the problem of locating the piezoelectric patches for controlling multimode vibration of plate structure. The positions of the plate at which the mechanical strain/curvature is highest are the best locations for sensors and actuators. Therefore, process is based on the identification of high average strain/curvature points and nodal lines (zero strain) through modal analysis. The effect of addition of patches to the overall curvature distribution is investigated and an attempt is made to relocate the patches to improve their strain sensitivity. ANSYS (Multiphysics) software is used for the modal analysis. The results show that peak strain locations are affected by putting patches at these points 1. Introduction Structural vibration and static shape control depends mainly on sensor/actuator selection, their placement and also on control law design. Piezoelectric actuators have proven to be experimentally effective in controlling the structural vibration as these have significantly large band width. Therefore, piezoelectric patches are the most popular control actuators that are being employed in many vibration control problems. Control of vibration in thin plates using smart concept has received wide spread attention from many researchers, as thin plates are common structures employed in many engineering applications. In general, when the control of vibration is for lower modes the use of only 1-2 patches to achieve the objective is adequate. However, in case of acoustic excitation, plates can exhibit significantly higher vibration amplitudes even in their higher modes, which need to be controlled in order to improve overall fatigue life of the structural component. One of the major challenges in controlling higher mode of vibration is the choice of number of patches as well as their location, to maximize the vibration reduction, for a given input power, or to minimize the input power for a given amplitude level. The optimal location is obvious in case of static shape control, as the given structure contains one point of maximum strain energy. This is not applicable to vibration control, because dynamic response of the structure is due to the weighted contribution of several modes. The highest strain energy for a given mode may be found at more than one point, and the high strain energy locations may be different for different modes. Modal phase cancellation is one of the major problems with distributed form, which restricts the control capability for higher vibration modes of the structure. These problems indicate the necessity of using number of discrete patches and identifying their locations for the vibration control. By placing the discrete actuators at the point of maximum strain/curvature on control surfaces and controlling these actuators through voltages, higher vibration modes may be controlled. Therefore, motivation of the present study is to identify the maximum strain/curvature points for locating the discrete patches for controlling multimode vibration of a plate structures. Methods for optimal placement of actuators and sensors are relatively new and need more exploration [1]. The optimal location determination of an actuator/sensor for desired applications is complex and challenging. Crawley and et al. [2] suggests the location of high average strain for placement of actuators for control of vibration of first two modes of flexible beams. Devasia and et al. [3] incorporated the sizing of distributed piezoelectric actuator as an additional optimization parameter and formulated simultaneous placement and sizing methodologies for vibration suppression in uniform beam. Gawronski [4] addresses the problem of actuator and sensor placement using their notion of modal controllability and observability. Yong Li and et al. [5] formulated a new optimal design methodology for simultaneous optimization of the placement of actuators and feedback gains for vibration suppression of beams. The notion of spatial controllability and modal controllability are used by D. Halim and et al. [6], for optimal placement of collocated piezoelectric actuator sensor pairs on a thin flexible simply supported plate.
All these researchers have developed optimization algorithms for identifying the actuator patch locations for effective control. Young-Hum Lim [7] has proposed patch location identification based on modal analysis of structure for multimode control of structural elements. The approach and proposed patch configuration of reference [7] is verified in the present study by using commercial finite element software ANSYS, and a new patch configuration is suggested for controlling first six modes of clamped square plate structure. 2. Problem Formulations and Solution The control of a structural vibration in plate structure is more complex because of its two dimensional nature, particularly when it is necessary to control the higher vibration modes. Therefore multiple discrete sensors and actuators are used for the multimode vibration control of the plate. Many researchers used finite element method to analyze and design smart structures. Special finite elements have been developed to account for piezoelectric effect [8]. These elements have also become available in some of commercial finite element codes such as ANSYS [9] (Multiphysics, Mechanical). Utilization of these commercial finite element codes can be a first step towards developing smart structures for real life industrial applications. Therefore ANSYS software is used for the present study. 2.1 Host Plate The clamped square plate of size 35x35x.8 mm 3 is selected for the analysis. The Solid45 3-d solid elements are used to model the entire plate for analysis. The modal analysis of structure involving piezoelectric elements can be performed only by the reduced method (Householder method) in ANSYS. For comparative study of the frequencies and mode shapes of host plate and host plate with piezoelectric patches, analysis of both structures can be performed by same method. Therefore, reduced method (Householder method) is selected for the modal analysis of host structure. Appropriate mesh size is identified through frequency convergence of mode shapes by modal analysis of host plate for different mesh size. The frequencies corresponding to mode shapes of plate structure are listed in Table1 for different mesh size. The observation of data clearly shows that frequency starts converging from 6х6 mesh size; hence this mesh size is adopted for further analysis. Table 1 List of frequencies (Hz) and mode shapes for different mesh size Mode shapes Mesh size Modal order and the corresponding mode shape function provide a significant insight into the best actuator locations for vibration control. For enhancing the effectiveness of the actuators, it is necessary to place all of the actuators in the regions of maximum strain/curvature and away from areas of zero strain. Therefore mode shapes of the structure can be plotted and studied carefully for maximum strain/curvature points and as well as nodal lines (zero strain). The first six mode shapes of clamped plate structure (host plate) are shown in Figure 1. The first mode shape (fundamental mode) has an associated motion with no phase change across the plate surface. It has one maximum curvature point at the centre of the plate. Therefore, one patch can be mounted at the centre of the plate for controlling fundamental mode of vibration. The higher order modes are characterized by nodal lines through which relative phase of the displacement changes by 18. The second mode shape has nodal line (zero strain) along one of the diagonal of plate and deformation changes about this nodal line. In the third mode shape, nodal line is now along another diagonal. These two mode shapes have two maximum curvature points, where patches are to be located. The fourth mode shape has two nodal lines parallel to x and y axis and passes through centre of plate. This diagram clearly shows that there are
four maximum curvature points along diagonals of plate. In case of fifth mode shape nodal lines are along diagonals of plate structure. There are four maximum curvature points on the plate structure. Sixth mode shape of clamped square plate structure has circular nodal line about the centre of the plate. It has five points of maximum curvature one at the centre and four along diagonals, where actuators are to be located for controlling this mode of vibration. From this study it is clear that each set of predicted piezoelectric patch locations applies to only one mode; therefore it is essential to know which modes need to be controlled based on requirement. a) First Mode (Freq. = 74.64 Hz) b) Second Mode (Freq. = 153.4 Hz) c) Third Mode (Freq. = 153.5 Hz) d) Fourth Mode (Freq. = 228.79 Hz) e) Fifth Mode (Freq. = 275.92 Hz) f) Sixth Mode (Freq. = 276.99 Hz) Figure 1. Mode shapes of host plate with natural frequencies.
2.2 5-Patch Configuration The first six mode shapes of host plate structure are studied for identifying number of piezoelectric patches and their locations for controlling specific number of vibration modes. As discussed in earlier section mode shapes gives the idea of patch location for higher effectiveness. The centre of the plate face is one of the locations of the maximum curvature (highest strain) for the first mode; hence one patch can be placed at the centre of the plate. Nodal lines of modes second, third; fourth and fifth intersect the piezoelectric patch placed at the centre of the plate. Therefore, these modes may be uncontrollable by central patch. On the other hand nodal lines corresponding to modes first and sixth do not intersect the patch, so it is anticipated that these modes can be controlled by central patch. From the modal analysis study, it can be predicted that control of modes second, third and fourth improve if four more patches are arranged along the diagonals of the plate. Nodal lines of fifth mode shape intersect the five patches located on the surface of the plate; therefore fifth mode is uncontrollable. As a first case, five piezoelectric patches made of PZT-5H are selected for use as collocated sensors and actuators in bimorph configuration for control of multimode vibration of the plate. Four patches of 1mm x 1mm size are placed diagonally and one patch of 2mm x 2mm size is placed centrally. Square host plate model with five actuator and sensor configuration is shown in Figure 2. Results of this configuration, particularly natural frequencies and mode shapes are available in reference [7]; these can be used for comparison. The piezoelectric ceramic, PZT-5H material is selected as actuators/sensors to be mounted on plate for the analysis. The material properties and data are listed in appendix. Further, it is necessary to determine whether any patches placed for controlling specific modes of vibration are affected because of change of mode shapes due to mass and stiffness contribution 35mm from piezoelectric patches. This analysis is carried Top View out, so that relocation of those patches can be considered if necessary, for improving actuator effectiveness. For this purpose integrated structure with patch configurations are modeled for more.8mm detailed study of first six mode shapes. The Solid5 coupled field elements are used to End View model the piezoelectric actuators/sensors part, because these elements suitable for solid geometry and available in ANSYS. Solid45 elements are used to model solid geometry of host plate. Figure 2. Integrated plate with 5- patch configuration Modelling of integrated structure is carefully done. The mapped meshing with 6x6 mesh size is selected for the modal analysis of the integrated structure, same as that of host plate analysis. This analysis is performed to identify the effect of actuators/sensors mass and stiffness on the natural frequencies and mode shapes. Modal analysis of 5-patch configuration of integrated structure shows that, piezoelectric elements bonded to the host plate have significant influence on the plate natural frequencies and associated mode shapes. Fifth and sixth mode shapes of 5-patch integrated structure as compared with the corresponding mode shapes of the elastic plate without piezoelectric patches (host plate) are significantly changed. From the mode shapes of integrated structure, it is clear that the piezoelectric patch location for controlling vibration modes of first, fourth and fifth are appropriate. The five patches located on the plate surface are not on the maximum curvature points of second and third symmetrical modes; hence they can only provide some control effect in controlling these modes. For fifth mode, all patches are on the maximum curvature points due to change in mode shapes after mounting the piezoelectric patches; hence this mode is now controllable. Frequency of this mode reduces drastically as compared to other modes because of all the patches are on maximum curvature points. In case of sixth mode all patches placed are on nodal lines, therefore sixth mode is uncontrollable. Natural frequency of this mode only has increased as compared to corresponding mode shape of host plate. This discussion reveals that, there is a need of analyzing the effect of mass and stiffness of piezoelectric patch on the mode shape and natural frequencies of structural element. By 35mm
considering this effect, locations of piezoelectric patches as actuators/sensors are to be finalized for effective control of multimode vibration. 2.3 9-Patch Configuration As discussed in earlier section, all patches located on the plate for controlling multimode vibration are on nodal lines for fifth mode; hence this mode is uncontrollable based on the above 5-patch distribution. After careful study of this mode shape four more patches are added to 5-patch configuration. The patch positions are determined from a detailed study of this mode as well as other modes for overall control of first six modes. These positions are the high curvature/strain regions of first six modes, where patches are to be placed for effective control of structure. The nine piezoelectric patches of 2mm x 2mm size are selected, which are hardly covering 4% of the plate surface area. Optimum size of actuators can be investigated for improving actuators effectiveness at later stage. The locations of the piezoelectric patches are decided on the basis of range of maximum curvature/strain area for covering majority of mode shapes. Integrated structure with 9-patch configuration in bimorph form, as collocated sensors/actuators for controlling higher modes of vibrations, is shown in Figure 3. The 9-patch actuator locations are initially decided on the basis of modal analysis of host plate alone. Modal analysis of integrated structure with 9-patch configuration is performed to identify whether the patches located are in the appropriate areas. The mode shapes of integrated structure with 9-patch configuration are shown in Figure 4. First mode shape clearly shows that central patch mounted is on the appropriate place (maximum curvature point); hence this mode can be controlled effectively. For the second and third mode shapes four patches are nearer to the maximum curvature region, this arrangement is better as compared to 5- patch configuration. Fourth mode shape also clearly shows that four patches are at the appropriate place out of nine patches. Fifth mode shape diagram clearly indicates that five patches are on nodal lines as similar to 5-patch configuration. The additional four patches added to the 5-patch configuration for controlling this mode 35mm in 9-patch configuration are on the right locations. Top View The mode shape diagram of sixth mode shows that five patches out of nine are on the maximum curvature region. From the discussion it is clear that out of nine patches some patches are on nodal.8mm lines and remaining on the curvature regions. For more detailed study of mode shape diagrams, strain End View diagrams at surface of the plate can be plotted along pre identified paths of mode shape by using path operation feature available in ANSYS. These Figure 3. Integrated plate with 9-patch configuration plots help in determining the distances from plate reference point to maximum curvature points as well as checking whether patches located are either on maximum curvature points or not. Further, this study reveals that central patch is exactly at the maximum curvature point in case of first and sixth mode only. Four patches mounted diagonally are little away from maximum curvature/strain points towards outwards of the centre of plate in case of fourth mode and towards inwards of the centre of plate for sixth mode. If relocation is done of the same patches, it will be better for one mode and worst for other. Similarly, remaining four patches are also little away from maximum curvature points towards inward of the centre of plate in case of fifth mode, if these are relocated exactly for this mode it will be worst for second and third mode. From this discussion it is clear that locating patches for maximum controllability/observability of all modes with fixed number of patches is difficult. However, in practice all modes are not equally important for the system response; some are less important and while some more important. Hence, suitable weightages are to be given to all the required modes of interest. Therefore, optimization of patch locations based on modal cost objective function is considered to be essential. 35mm
In view of the fact that, the patches mounted on nodal lines are not useful in controlling the mode shape, the present study also reveals the information regarding patches which should be actuated while controlling a desired mode shape. This information may help in improving the actuator effectiveness for controlling the modes. a) First Mode (Freq. = 64.6Hz) b) Second Mode (Freq. = 133.76 Hz) c) Third Mode (Freq. = 133.77 Hz) d) Fourth Mode (Freq. = 197.62 Hz) e) Fifth Mode (Freq. = 244.35 Hz) f) Sixth Mode (Freq. = 247.58 Hz) Figure 4. Mode shapes of integrated structure with 9-patch configuration and natural frequencies.
3. Results and Discussion The study conducted in previous section provides information in terms of natural frequencies, mode shape deformation, strain variation etc. The results of 5-patch configurations, particularly mode shapes and corresponding natural frequencies are in close agreement with the results of reference [7] and also observations are similar. Modal analysis is performed for different cases like host plate alone, 5-patch configuration of integrated structure with and without adhesive and 9-patch configuration of integrated structure without adhesive. Natural frequencies of first six modes are listed for different configuration in Table 2 for the comparison. Table 2 Natural frequencies (Hz) of plate and plate with piezoelectric patches Mode shapes Host plate Plate with 5 patches (without adhesive) Plate with 5 patches (with adhesive) Plate with 9 patches (without adhesive) 1 74.12 69.74 69.37 64.6 2 151.42 148.56 148.28 133.76 3 151.42 148.56 148.28 133.77 4 223.88 218.82 218.4 197.62 5 272.4 25.21 248.27 244.35 6 273.29 279.83 279.94 247.58 Natural frequencies and mode shapes of plate with 5-patch configuration, particularly for fifth and sixth modes are significantly changed as compared with the mode shapes of the elastic plate without piezoelectric patches. In case of fifth mode of integrated structure with 5-patch configuration all five patches are on nodal lines. Natural frequency of this mode is reduced by 8.7% as compared to corresponding mode shape of host plate. All five patches of this configuration are on the maximum curvature regions for sixth mode shape. Only the natural frequency of this mode is increased by 2.4%. The effect of adhesive between the host plate and piezoelectric patches is negligible in terms of frequencies, but it is significant with reference to mode shapes. Particularly, in case of higher modes maximum strain values are considerably increased as compared with without adhesive. Integrated structure with 9-patch configuration s natural frequencies for all mode shapes are reducing. The reduction in natural frequencies from first mode to sixth mode is gradual, which varies from 15% to 1% respectively. This configuration gives appropriate patch distribution for the first six modes; hence first six modes of vibration can be effectively controlled. 4. Conclusion From the above discussion it can be concluded that, the bonded piezoelectric elements have significant influence on the plate natural frequencies and associate mode shapes. By considering the effect of addition of patches to the overall curvature distribution and relocating the patches the strain sensitivity of actuator can be improved. Proposed 9-patch configuration is more suitable in controlling all the first six modes of clamped square plate structure. This actuator distribution can be studied further for determining optimum size of patches for effective control of first six modes. More number of actuator/sensor patches are required for controlling multimode vibration of plate structure. The present study also reveals the information regarding patches which should be actuated while controlling a desired mode shape. 5. References [1] M. Sunar and S. S. Rao (1999) Recent advances in sensing and control of flexible structures via piezoelectric materials technology. Applied Mechanics Review, 52(1), 1-16. [2] Crawley E.F. and Javier (1987) Use of piezoelectric actuators as elements of intelligent structures. AIAA Journal, 1773-1785.
[3] Santosh Devasia, Tesfay Meressi, Brad Paden and Eduardo Bayo (1993) Piezoelectric actuator design for vibration suppression: placement and sizing. Journal of Guidance Control and Dynamics, 16(5), 859-864. [4] Gawronski W. K. (1997) Actuator and sensor placement for structural testing and control. Journal of Sound and Vibration, 28(1), 11-19. [5] Yong Li, Junjiro Onoda and Kenji Minesugi (22) Simultaneous optimization of piezoelectric actuator placement and feedback for vibration suppression. Acta Astronautica, 5(6), 335-341. [6] Dunant Halim, S. O. Reza Moheimani (23) An optimization approach to optimal placement of collocated piezoelectric actuators and sensors on a thin plate. Mechatronics, 13, 27-47. [7] Young-Hun Lim (23) Finite-element simulation of closed loop vibration control of a smart plate under transient loading. Smart Materials and Structures, 12, 272-286. [8] A. Benjeddou (2) Advances in piezoelectric finite element modeling of adaptive structural elements: a survey. Computers and Structures, 76, 347-363. [9] ANSYS Analysis guide, ANSYS Software (Release 5.4) ANSYS Inc., Canonsburg, PA, USA. [1] Ahmed K. Noor, Samuel L. Venneri, Donald B. Paul, Mark A. Hopkins (2) Structures technology for future aerospace systems. Computers and structures 74, 57-519. [11] A. Joshi (22) Multi-layered piezoelectric inserts as vibration control actuators. Journal of Sound and Vibration, 253(4), 917-925. Appendix Material properties of aluminum and PZT 5H are [56], Aluminum material data, Density of material = 28 Kg/m 3 Poisson s ratio =.32 Young s modulus = 6.8 x 1 1 N/m 2 PZT 5H Density = 75 Kg m -3 Elastic stiffness matrix at constant electric field, 12.6 7.95 12.6 8.41 8.41 11.7 1 2 C E = 1 Nm 2.33 2.3 2.3 Dielectric matrix at constant strain, b 1.53 1.3 8 1 = 1.53 1 Fm Piezoelectric strain matrix, h T 17 = 17 Cm 6.5 6.5 23.3 2