Experiments in active control of panel vibrations with spatially weighted objectives using multiple accelerometers

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1 Experiments in active control of panel vibrations with spatially weighted objectives using multiple accelerometers D. Halim a School of Mechanical Engineering, University of Adelaide SA, Australia 55 G. Barrault b Departamento de Engenharia Mecânica, UFSC Florianopolis, SC, Brasil B. Cazzolato c School of Mechanical Engineering, University of Adelaide SA, Australia 55 ABSTRACT The work is aimed to control the spatial vibration profile of a panel structure using multiple accelerometers that are distributed over the structure. The spatial vibration profile is estimated based on the measurements from the accelerometers and a spatially weighted vibration objective is utilised to emphasise controlling a particular panel region. Experiments on a rectangular panel structure are performed to test the effectiveness of the spatial control strategy with the use of the FX-LMS based adaptive control together with the filtering of the acceleration signals. 1 INTRODUCTION The work presented in this paper concentrates on the experimental implementation of active structural vibration control using spatially-weighted structural signals [1]. A spatial filtering is generated by using vibration signals from structural sensors, incorporating spatial interpolations to obtain vibration estimate of an entire structure. Research in structural vibration control has been intensive such as the one using multiple discrete sensors for generating modal/spatial filters [2, 3, 4] in which spatial interpolations were used (other researchers have also utilised spatial interpolations for structural vibration control [5, 6]). However, the previous control methods require a-priori structural information such as mass and stiffness distribution or structural modal properties, which are in contrast with the proposed method that uses vibration information directly from structural sensors. Other research utilises shaped piezoeletric films to generate spatial filters for structural vibration control [7] or for structural sound radiation control [8, 9, 1, 11]. However, the use of piezoelectric films for efficient spatial filtering generally requires accurate a-priori mode shapes and boundary conditions. It is therefore the aim of this work to provide an alternative method that depends less on a-priori structural dynamic information. a address: dunant.halim@adelaide.edu.au b address: guillaume@lva.ufsc.br c address: benjamin.cazzolato@adelaide.edu.au

2 2 VIBRATION CONTROL WITH SPATIALLY WEIGHTED SIGNALS In some cases, it may not be practical to use a model-based control approach for complex structures. Alternatively, non-model based control can be used where structural vibration information can be accessed from a number of structural sensors distributed over a structure. This work concentrates on the experimental attempts to control the spatial vibration profile of a structure, since the ability to spatially control structural vibration can be advantageous such as for controlling the associated sound radiation. To measure the spatial profile of a vibrating structure, spatial interpolations can be employed such as the one used in numerical finite element method [12, 13]. The principle of using spatial interpolations have also been utilised in other work such as in [2, 5]. The proposed spatial control method is briefly described as follows [1]: N structural sensors structure v i (x i, y i ) m th element y (m) y x (m) N o nodes at boundaries x Figure 1: An arbitrary flexible structure with N structural sensors and N o nodes at structural boundaries. The vibration level measured by the i th sensor at location (x i,y i ) is v i. The m th element has local coordinates of (x (m),y (m) ) and is constructed from 4 nodes. Suppose there are N structural discrete sensors distributed over an arbitrary structure, in which the i th sensor measures vibration v i at a particular location (x i, y i ) on a structure. In addition, N o nodes at structural boundary conditions may be included to improve the spatial interpolation of the entire structure. This approach can be particularly useful if it is known that there is minimal vibration at the boundary conditions. M elements/regions can then be constructed from N + N o nodes. For the m th element/region with local coordinates of (x (m), y (m) ), the vibration level v xy (m) at any location (x (m), y (m) ) within the element can be estimated from sensor measurements. If l sensors are used to construct nodes for m th element, v xy (m) R k 1 will be the estimated elemental vibration profile which are based from sensor measurements v [1]: v (m) xy (x(m), y (m), t) = H(x (m), y (m) ) v (m) (t) (1)

3 where v (m) consists of a group of l sensor measurements v i associated with the m th element, and H(x (m), y (m) ) is a k l interpolation function matrix. Transforming the local coordinates into the global coordinates using a linear transformation matrix, the vibration profile can be shown to be in the form of: v xy (x, y, t) = M(x, y) v (2) where v consists of sensor signals v i and the nodes at boundaries with minimal vibrations. 2.1 Spatial cost function Next, a spatial cost function to be used for control is described. A reference structural vibration profile may be included in the control formulation that reflects the desired vibration profile and is denoted by f xy (x, y, t). A spatial error function can then be defined by: e(x, y, t) = v xy (x, y, t) f xy (x, y, t). (3) A real symmetric spatial weighting matrix W(x, y) (W(x, y) > for all locations (x, y) R where R is the structural region of interest) can be chosen to emphasise regions that are more important to control. The following spatial cost function can be obtained [1]: J(t) = e(x, y, t) T W(x, y)e(x, y, t)dr with R = (v(t) v o (t)) T A (v(t) v o (t)) + c(t) (4) v o (t) = A 1 b(t) (5) and A = b(t) = c(t) = R R R M(x, y) T W(x, y) M(x, y) dr M(x, y) T W(x, y) f(x, y, t) dr f(x, y, t) T W(x, y)f(x, y, t) dr v o (t) T b(t). (6) The dimensions of A and b(t) can then be reduced from N +N o to N by removing the relevant rows and columns associated with the constraints at structural boundaries. Now, matrix A can be shown to be real and symmetric so it can be decomposed into: A = F T sp F sp. (7)

4 Incorporating Eq. (7), the cost function J in Eq. (4) can then be simply represented as: with J(t) = ( v sp (t) v o sp (t)) T ( vsp (t) v o sp (t)) + c(t) (8) v sp (t) = F sp v(t) v o sp(t) = F sp v o (t) = F sp A 1 b(t). (9) From the results above, it can be seen that v sp is a signal that represents the spatiallyweighted vibration over the entire structure. If required, v o sp can be used to describe the desired spatial vibration profile. However, the minimisation of spatial structural vibration is the main interest in this experimental work, thus f xy (x, y, t) = (x, y) R and consequently v o sp (t) =, c(t) =. The filter matrix F sp can be obtained using the eigenvalue decomposition of A: A = UVU T (1) where V is a diagonal matrix containing positive real valued eigenvalues and U is the matrix containing the associated eigenvectors. It is also possible to reduce the dimensions of the filter matrix by including only the dominant eigenvalues and the associated eigenvectors [1]. Thus, F sp = V 1/2 U T. (11) Thus, an active vibration control strategy can utilise the spatially-weighted structural signals v sp that are obtained from the linear filtering F sp of the sensor signals. The spatial signals v sp can be used as error signals in which an active control algorithm can be used to minimise a spatially-weighted cost function. For a feedforward control problem with a tonal excitation at frequency ω o, let the disturbance and control input signals to be d and u respectively. The vibration levels v measured by N structural sensors are: v(ω o ) = G vd (ω o )d(ω o ) + G vu (ω o )u(ω o ) (12) where G pq represents the transfer matrix from q to p. The cost function J is now: J = v H spv sp v sp = F sp G vd d + F sp G vu u (13) where (F) H is the Hermitian transpose of matrix F. In practice, an adaptive control algorithm can be used to minimise the cost function. If the steepest-descent algorithm [14] is used, the feedforward control input signal u at the (p + 1) th iteration can be shown to be: u(p + 1) = u(p + 1) 2µ ( G H vu FT sp F spg vu u(p) + G H vu FT sp F spg vd d(p) ) (14) where µ is the convergence adaptation factor.

5 3 EXPERIMENTAL IMPLEMENTATION: SPATIAL CONTROL OF A PLATE A simply-supported steel plate (4 mm x 35 mm x 2.8 mm) was used with N = 16 Analog Devices MEMS ADXL5 accelerometers distributed across the plate as shown in Fig. 2. Two LING V-2 electrodynamic shakers were used as disturbance and control sources. The properties of the plate and 4 lowest resonance frequencies are shown in Table 1 and Table 2 respectively. A modal filtering box was used to implement the spatial filtering based on matrix F sp and EZ-ANC II FX-LMS based adaptive feedforward controller was used to implement the active control by utilising the error signals obtained from the spatial signals v sp. In principle, vibration signals from the accelerometers were initially filtered to obtain spatial signals that represent the spatially-weighted vibration of the entire plate. The energy of the spatial signals can be minimised by employing the FX-LMS adaptation algorithm to generate necessary control actuation via the attached electrodynamic shaker. A Polytec PSV-4-3D laser scanning vibrometer was then used to obtain velocity measurements over the entire plate structure. Although the proposed control strategy can also be implemented for multiple disturbance/control sources and general broadband disturbance, the tonal disturbance case with a single control source was considered in the experiments to clearly demonstrate the effect of spatial control on the spatial vibration profile of the plate. In general, the broadband control implementation would require good control performances on each of vibration modes within the bandwidth of interest, as well as the satisfactory implementation of broadband (stochastic) adaptive control process accelerometers control actuator disturbance (a) experimental set-up (b) Locations of 16 accelerometers, disturbance and control actuator. Figure 2: Experimental set-up for a simply-supported plate structure. For the experiment, linear interpolations and rectangular elements/regions were used to obtain the matrix of interpolation functions Eq. (1). Each rectangular element thus consisted of l = 4 sensors at all 4 corners with the elemental dimensions of h (m) x and h (m) y (in x (m) and y (m) directions respectively). The interpolation matrix is

6 Table 1: Properties of the plate Plate Young s Modulus N/m 2 Plate Poisson s ratio.292 Plate density 78 kg/m 3 Table 2: Resonance frequencies of the first four modes of the plate No. Mode Frequency (Hz) 1 (1, 1) 92 2 (2, 1) (1, 2) (2, 2) 385 (see [1] for details): H(x (m), y (m) ) = { ( 1 ( x (m) h (m) x){ x (m) h (m) x ( x (m) { 1 ( )}{ ( )} y 1 (m) ( h )} (m) y 1 y (m) )}( h (m) y ) y (m) h (m) x)( ) h (m) y y (m) x (m) h (m) x h (m) y T. (15) Two normalised scalar spatial weighting functions W(x, y) > were chosen for the experiment as shown in Fig. 3 and F sp can be obtained from Eq. (7). Note that the two weighting functions have different structural regions of interest as reflected by high weighting values (a) reference function 1 (b) reference function 2 Figure 3: Two reference spatial weighting functions used in the experiment.

7 3.1 Results of the tonal experiment on the plate The first experiment concentrated on the first resonant mode (1,1) at 92 Hz. The vibration profile of the plate with and without control are shown in Fig. 4 for reference spatial weighting function 1. It can be seen that the vibration at the structural region of interest (at the lower left-hand-side (LHS) corner of the plate) has been reduced more than other regions as expected. The simulation result based on an idealised simply-supported plate is also shown in Fig. 4(c). It is interesting to note that the spatial vibration profiles obtained from the experiment and the idealised model are similar. At higher frequencies, however, more pronounced differences between the experiment and simulation can be expected. When the reference spatial weighting 2 was used as shown in Fig. 5, the control results obtained are different from those for reference 1 case. It can be seen that the region with minimal vibration, represented by the vibration nodal line, has been shifted closer to the lower right-hand-side (RHS) corner of the plate, which is the structural region of interest for reference function 2. As the consequence, the vibration level at the lower left-hand-side (LHS) corner has been increased compared to that of the previous case. It can be shown that the control performances can actually be improved further if more control actuators are used. Thus, it can be seen that the active control was able to change the spatial region of interest that is reflected by the spatial shape of the weighting function (the reference function 2, in this case). Experimental results for mode (2,1) at 213 Hz are shown in Figs. 6 and 7 for reference functions 1 and 2 respectively. It can be observed that the regions of low vibration actually occur along the diagonal profile lines over the plate. The results can be expected since for the reference 1 case, the diagonal line cut across the lower LHS region, which is the region of interest for vibration minimisation. Similarly, the reference 2 case results reflect the vibration minimisation that particularly includes the lower RHS region. Simulation results describe similar patterns of results for both control cases. Results for mode (1,2) at 261 Hz are shown in Figs. 8 and 9 for reference functions 1 and 2 respectively. Again the diagonal profile lines indicate the location where the vibration level is lower than that at other regions. It should be noted that these diagonal profile lines represent the modified nodal line of the plate. The implemented controller has actually modified the vibration shape of the plate at that particular frequency so that the structure has the minimal spatially weighted vibration. Simulation results also indicate a similar control behaviour. The final results on mode (2,2) at 385 Hz are shown in Figs. 1 and 11 for reference functions 1 and 2 respectively. Again, the nodal lines on the plate have been modified: for reference 1 case, there was a nodal line that occured at lower LHS of the plate. For reference 2 case, a nearly vertical nodal line occured closer to the RHS of the plate so that the vibration at the lower RHS can be supressed further. From the experiment results, the ability of a single control actuator to modify a spatial vibration profile is shown to be limited particularly for vibration modes with high spatial variations. Thus for higher frequency vibration control, an improved spatial vibration regulation can generally be expected by increasing the number of control actuators.

8 (a) no control - spatial vibration profile of mode (1,1) (b) with control - experiment (c) with control - simulation Figure 4: Experiment using the reference spatial weighting 1 at a frequency of 92 Hz (a) with control - experiment (b) with control - simulation Figure 5: Experiment using the reference spatial weighting 2 at a frequency of 92 Hz.

9 (a) no control - spatial vibration profile of mode (2,1) (b) with control - experiment (c) with control - simulation Figure 6: Experiment using the reference spatial weighting 1 at a frequency of 213 Hz (a) with control - experiment (b) with control - simulation Figure 7: Experiment using the reference spatial weighting 2 at a frequency of 213 Hz.

10 (a) no control - spatial vibration profile of mode (1,2) (b) with control - experiment (c) with control - simulation Figure 8: Experiment using the reference spatial weighting 1 at a frequency of 261 Hz (a) with control - experiment (b) with control - simulation Figure 9: Experiment using the reference spatial weighting 2 at a frequency of 261 Hz.

11 In addition, the number of sensors used can also limit the control performances since vibration modes with high spatial variations will require a more spatially dense sensor arrangement (a) with control - experiment (b) with control - simulation Figure 1: Experiment using the reference spatial weighting 1 at a frequency of 385 Hz (a) standard control (b) spatial control Figure 11: Experiment using the reference spatial weighting 2 at a frequency of 385 Hz. 4 CONCLUSIONS Experiments on a simply-supported plate structure has been shown, demonstrating the feasibility of implementing the proposed spatial control method. It is shown that the spatial control can be utilised to control vibration at particular structural regions which can also be useful such as for regulating the profile of the associated sound radiation. Since the vibration information is obtained directly from structural sensors, a dynamic model of the structure is not required which allows the control method to be used for complex structures.

12 ACKNOWLEDGEMENTS The authors would like to thank the Australian Research Council (ARC) for its financial support. REFERENCES [1] D. Halim and B. S. Cazzolato, A multiple-sensor method for control of structural vibration with spatial objectives, Journal of Sound and Vibration, 296, (26). [2] L. Meirovitch and H. Baruh, Control of self-adjoint distributed-parameter systems, Journal of Guidance, 5(1), 6 66 (1982). [3] L. Meirovitch and H. Baruh, The implementation of modal filters for control of structures, Journal of Guidance, 8(6), (1985). [4] L. Meirovitch, Some problems associated with the control of distributed structures, Journal of Optimization Theory and Applications, 54(1), 1 21 (1987). [5] G. A. Pajunen, P. S. Neelakanta, M. Gopinathan, and M. Arockaisamy, Distributed adaptive control of flexible structures, SPIE Proc North American Conf. on Smart Structures and Intelligent Systems, pp. 219:79 81 (1994). [6] M. Gopinathan, G. A. Pajunen, P. S. Neelakanta, and M. Arockaisamy, Recursive estimation of displacement and velocity in a cantilever beam using a measured set of distributed strain data, Journal of intelligent material systems and structures, 6, (1995). [7] S. Collins, D. W. Miller, and A. Von Flotow, Distributed sensors as spatial filters in active structural control, Journal of Sound and Vibration, 173(4), (1994). [8] A. Preumont, A. François, P. De Man, N. Loix, and K. Henrioulle, Distributed sensors with piezoelectric films in design of spatial filters for structural control, Journal of Sound and Vibration, 282, (25). [9] R. L. Clark and C. R. Fuller, Modal sensing of efficient acoustic radiators with polyvinylidene fluoride distributed sensors in active structural acoustic control approaches, Journal of Acoustical Society of America, 91(6), (1992). [1] S. D. Snyder and N. Tanaka, On feedforward active control of sound and vibration using vibration error signals, Journal of the Acoustical Society of America, 94(4), (1993). [11] F. Charette, A. Berry, and C. Guigou, Active control of sound radiation from a plate using a polyvinylidene fluoride volume displacement sensor, Journal of the Acoustical Society of America, 13(3), (1998). [12] K. J. Bathe and E. L. Wilson, Numerical Methods in Finite Element Analysis (Prentice Hall, Englewood Cliffs, New Jersey, 1976). [13] Y. K. Cheung and A. Y. T. Leung, Finite Element Methods in Dynamics (Science Press; Kluwer Academic Publishers, Beijing, New York; Dordrecht, Boston, 1991). [14] S. J. Elliot, Signal Processing for Active Control (Academic Press, 21).