In the second analysis, a hybrid actuated cantilevered sandwich beam is constructed with collocated shear and extension actuators and the beam configuration is presented in figure 9.
The two types of actuators are activated individually, as well as simultaneously to see the bending behaviour of sandwich beam. It is seen that a collective actuation effect made the beam to have more active stiffening effect in the clamped-free case (figure 10). The full- length extension type actuators can generate only blocking forces, if the edges are constrained, therefore the segmented piezoelectric wafers are considered to analyse the deflection behaviour of hybrid actuated sandwich beams for various boundary conditions.
Figure 11 shows one such analysis as an example with clamped-clamped conditions.
L La
Xa
Foam (2mm) AL (8mm) +10V
-20V
-10V
PZT-5H (2mm) PZT-5H (1mm)
Figure 9 Hybrid actuated sandwich beam
0.00 0.02 0.04 0.06 0.08 0.10
0.0 1.0x10-7 2.0x10-7 3.0x10-7 4.0x10-7 5.0x10-7
6.0x10-7 Present ABAQUS EAM
SAM HAM Actuator La= L
Transverse Deflection (m)
Axial Location (m)
Figure 10 Hybrid actuated sandwich C-F beam
0.00 0.02 0.04 0.06 0.08 0.10 -1.2x10-8
-8.0x10-9 -4.0x10-9 0.0 4.0x10-9
SAM
EAM Present ABAQUS Actuator L
a=0.01m Xa=0.05m
Transverse Deflection (m)
Axial Location (m)
Figure 11 Hybrid actuated C-C sandwich beam with segmented actuators 3.1.1 With Collocated Actuators
The segmented collocated actuators (EAM and SAM) are placed at different locations along the length of beam to see their efficiency in deflecting the sandwich beams. The Clamped- Free (C-F), Clamped-Clamped (C-C), Clamped-Hinged (C-H), and Hinged-Hinged (H-H) boundary effects are considered in the analysis. The actuators are activated individually and simultaneously by applying voltages ±10 to EAM and -20 to SAM. The electro-mechanically induced transverse displacements are estimated and the results are presented in figures 12 to 14.
It is seen that the boundary effects play a significant role on the piezoelectric actuation behaviour. The HAM captures the combined actuation effect (stiffening) of EAM and SAM in the case C-F due to the same deflection patterns. However, this trend is not followed in other cases. Interesting deflection patterns are observed with different boundary conditions;
which may be exploited for mode shape control applications. Further, the combination of EAM and SAM also can control more than one elastic mode by appropriately activating them (refer to deflection pattern). In general both actuators are efficient at locations, where the strains seem to be significant.
0.00 0.02 0.04 0.06 0.08 0.10
-4.0x10-8 0.0 4.0x10-8 8.0x10-8 1.2x10-7 1.6x10-7 2.0x10-7
BC: C-F La=0.02m Xa = 0.01m Xa = 0.03m Xa = 0.05m Xa = 0.07m Xa = 0.09m
Transverse Deflection (m)
Axial Location (m)
0.00 0.02 0.04 0.06 0.08 0.10
-4.0x10-8 -2.0x10-8 0.0 2.0x10-8 4.0x10-8 6.0x10-8
BC: C-H La=0.02m Xa = 0.01m Xa = 0.03m Xa = 0.05m Xa = 0.07m Xa = 0.09m
Transverse Deflection (m)
Axial Location (m)
12 (a) 12 (b)
0.00 0.02 0.04 0.06 0.08 0.10 -6.0x10-8
-4.0x10-8 -2.0x10-8 0.0
BC: H-H L
a=0.02m Xa = 0.01m Xa = 0.03m Xa = 0.05m
Transverse Deflection (m)
Axial Location (m)
0.00 0.02 0.04 0.06 0.08 0.10
-2.0x10-8 0.0 2.0x10-8
4.0x10-8 BC: C-C La=0.02m
Xa = 0.01m Xa = 0.03m Xa = 0.05m
Transverse Deflection (m)
Axial Location (m)
12 (c) 12 (d)
Figure 12 Bending behaviour of sandwich beams with collocated EAM/SAM -EAM active
0.00 0.02 0.04 0.06 0.08 0.10
0.0 2.0x10-8 4.0x10-8
BC: C-F La=0.02m Xa = 0.01m Xa = 0.03m Xa = 0.05m Xa = 0.07m Xa = 0.09m
Transverse Location (m)
Axial Location (m)
0.00 0.02 0.04 0.06 0.08 0.10
-2.0x10-8 0.0 2.0x10-8 4.0x10-8
BC: C-H L
a=0.02m X
a = 0.01m Xa = 0.03m X
a = 0.05m Xa = 0.07m X
a = 0.09m
Transverse Deflection (m)
Axial Location (m)
13 (a) 13 (b)
0.00 0.02 0.04 0.06 0.08 0.10
-1.0x10-8 0.0 1.0x10-8 2.0x10-8 3.0x10-8
BC: H-H La=0.02m Xa = 0.01m Xa = 0.03m Xa = 0.05m
Transverse Deflection (m)
Axial Location (m)
0.00 0.02 0.04 0.06 0.08 0.10
-1.0x10-8 -5.0x10-9 0.0 5.0x10-9 1.0x10-8 1.5x10-8 2.0x10-8
BC: C-C La=0.02m Xa = 0.01m Xa = 0.03m Xa = 0.05m
Transverse Deflection (m)
Axial Location (m)
13 (c) 13 (d)
Figure 13 Bending behaviour of sandwich beams with collocated EAM/SAM -SAM active
0.00 0.02 0.04 0.06 0.08 0.10 0.0
5.0x10-8 1.0x10-7 1.5x10-7 2.0x10-7 2.5x10-7
BC: C-F La=0.02m Xa = 0.01m Xa = 0.03m Xa = 0.05m Xa = 0.07m Xa = 0.09m
Transverse Deflection (m)
Axial Location (m)
0.00 0.02 0.04 0.06 0.08 0.10
-4.0x10-8 0.0 4.0x10-8
8.0x10-8 BC: C-H La=0.02m Xa = 0.01m Xa = 0.03m Xa = 0.05m Xa = 0.07m Xa = 0.09m
Transverse Deflection (m)
Axial Location (m)
14 (a) 14 (b)
0.00 0.02 0.04 0.06 0.08 0.10
-6.0x10-8 -4.0x10-8 -2.0x10-8 0.0 2.0x10-8
BC: H-H La=0.02m Xa = 0.01m Xa = 0.03m Xa = 0.05m Xa = 0.07m Xa = 0.09m
Transverse Deflection
Axial Location (m)
0.00 0.02 0.04 0.06 0.08 0.10
-6.0x10-8 -3.0x10-8 0.0 3.0x10-8 6.0x10-8
BC: C-C La=0.02m Xa = 0.01m Xa = 0.03m Xa = 0.05m Xa = 0.07m Xa = 0.09m
Transverse Deflection (m)
Axial Location (m)
14 (c) 14 (d)
Figure 14 Bending behaviour of sandwich beams with collocated EAM/SAM- both active 3.1.2 With Non-collocated Actuators
The active stiffening effect with hybrid actuation using non-collocated EAM and SAM has been studied (figure 15). Unlike in collocated EAM/SAM, the non-collocated configuration has generated more deflection not only in the C-F case, but in the C-C and C-H cases also (figure 16). However in the H-H case, the SAM influence appears to be predominant.
Therefore to achieve a cumulative actuation effort of EAM/SAM, they can be placed in a non-collocated fashion for better control action.
La
L
XSAM
Foam (2mm) AL (8mm) +10V
-20V
-10V
PZT-5H (2mm) PZT-5H (1mm)
XEAM
La
Figure 15 Hybrid actuated sandwich beams with non-collocated actuators
0.00 0.02 0.04 0.06 0.08 0.10 -5.0x10-8
0.0 5.0x10-8 1.0x10-7 1.5x10-7 2.0x10-7 2.5x10-7 L
a =0.02m X
EAM = 0.01m X
SAM = 0.05m C-F C-H
H-H C-C
Transverse Deflection (m)
Axial Location (m)
0.00 0.02 0.04 0.06 0.08 0.10
-4.0x10-8 -2.0x10-8 0.0 2.0x10-8 4.0x10-8 6.0x10-8 8.0x10-8 1.0x10-7
La= 0.02m XEAM = 0.01m & 0.09m XSAM =0.05m C-H H-H C-C
Transverse Deflection (m)
Axial Location (m)
16 (a) 16 (b)
Figure 16 Bending behaviour of sandwich beams with non-collocated EAM/SAM -both active
4 Distributed Active Vibration Control
The flexible structural system is a distributed parameter system that has a large number of degrees of freedom (Eigen modes). Also the dynamic behaviour of a structural system is a function of spatial and time variables. To control such a flexible system, it is necessary to use a number of actuators and sensors, spatially distributed so that they are sensitive to spatially distributed structural behaviour. Numerical models (Finite Element Method: FEM) are mostly employed to estimate the structural parameters, namely, stiffness, mass and damping (Rayleigh’s proportional damping) for the prediction of system response. The finite element model represents a structure in the form of a multi-degree of freedom system. Further it generates a set of second order simultaneous equations (Equation 36) that will define the dynamic equilibrium of the structural system. In order to design a viable control law for the control of elastic modes of the structure, the discrete numerical system model (FEM) must be condensed or reduced to a reasonable size.
Control design methods normally assume that a full state vector is available. However, in reality to construct the system state completely, a large number of sensors are needed;
otherwise an estimator must be designed to build the system state. In addition, when a large structural system is reduced (condensed) for use in control design, the un-modelled modes may some time destabilise the closed loop system. Therefore, in order to tolerate the model errors, it is desirable to optimise the robustness of controller i.e., tolerance to model errors and disturbance. The output feedback control is designed by taking a few selected modes only. Due to incomplete modelling of the structural system, when such a control is applied on real time structure experimentally, the actual damping ratio and free vibration frequency of each mode may not be the same as in the simulated model. Therefore, additional sensors and actuators are needed to increase the degrees of freedom of the control system to provide the necessary stability. Otherwise, a state observer or estimator is required to get the additional system states.
Independent Modal Space Control (IMSC) is a modal filter approach to transfer the responses of multi-degrees of freedom system into independent modal coordinates (single input and single output). Using the IMSC concept, the feedback control force is made
independent (as a function of modal coordinates) completely to decouple the structural modes in a feedback control environment. Since the complete control energy is driven to concentrate on a particular mode, spillover problem (exciting uncontrolled modes statically: residual energy) may be avoided. A structural system is controllable, if the installed actuators excite all the possible modes to be controlled. Similarly, the system is observable, if the installed sensors detect the motions of all the possible modes to be controlled.
The feedback control can compensate the external disturbances only in a limited frequency band that is known as control bandwidth. Thus, the control bandwidth is normally limited by the accuracy of the model. There is always some destabilisation of the flexible modes outside the control bandwidth i.e., the disturbance is actually amplified by the control system. Therefore, the control bandwidth must be sufficient enough to ensure better closed loop performance with the designed actuators and sensors configuration. Also, out of bandwidth correction (DC compliance) may be included to avoid influence of higher uncontrolled modes.