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
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.
4.1 Active Control Strategies
The multifunctional piezoelectric lamina as an actuator (fully active actuator) is able to supply mechanical power to the structural system to modify its dynamic response. The actuator lamina can be used to generate a secondary vibration response, which can reduce the total system response by destructive interference with the original response of the system due to primary source of vibration. Feedback controller requires actually no knowledge of the incoming disturbance to the system and acts to change the system response by changing the system resonance and damping. In this control, both the primary source and the secondary actuators (piezoelectric) influence the sensor signal.
4.1.1 Displacement Feedback (Proportional Control)
The sensor signal (charge) is used directly in a closed loop feedback system, which is proportional to the strain developed (due to deflection) in a piezoelectric lamina. The feedback voltage is obtained by amplifying the sensor signal by a feedback gain.
The displacement gain is ( )
( ) .
a d
s
amplitude of feedback voltage F amplitude of sensor voltage
I I
§ ·
¨ ¸
© ¹
In the displacement control, the feedback gain represents a true amplification ratio. Here the vibration suppression is achieved by generating actuation signal 1800 out of phase with respect to disturbance signal. Actually, the anti-phase actuation signal modifies the strain energy and actively stiffens the vibrating elastic system (see figure 17 a).
4.1.2 Velocity Feedback (Derivative Control)
The sensor signal is differentiated (charge rate), conditioned and then fedback into actuator lamina. The signal used in the velocity feedback is actually strain/unit-time (i.e., strain rate).
Since the elastic strain field is expressed in terms of displacements, the time derivative of the
displacement is velocity. The electric charge (q) denotes the system displacement, and therefore the charge rate (dq/dt = i: sensor current) is the system velocity.
The velocity gain is
( )
( ) .
a v
s
amplitude of feedback voltage F amplitude of sensor voltage rate
I I
§ ·
¨ ¸
© ¹
In the velocity feedback control, the actuation signal is in-phase with the system velocity (900 phase lead with system displacement) so as to introduce a dissipative force (active damping) into the system (refer to figure 17 b).
(a) Displacement feedback
Amplitude
Time
Sensor signal
Actuator signal amplified, 180o phase shifted
(b) Velocity feedback
Amplitude Time
Sensor signal (displacement) Actuator signal (velocity) System velocity
Figure 17 Active control concepts using feedback controller
4.2 State-Space Formulation
4.2.1 State-Space Models in Physical Domain
The dynamic equation of a smart structural system (Linear Time Invariant) is presented with feedback control as,
uu uu uu u m 0 a
M u C u K u b f b I
(38) where
Muu (n x n) is the mass matrix, Cuu (n x n) is the damping matrix, Kuu (n x n) is the stiffness matrix,
0 u a
b (n x a) K
I , is an actuator influence matrix with ‘a’ as number of actuators, bu is an (n x 1) matrix of influence functions associated with mechanical force vector fm, and, velocity system
the is u
, nt displaceme system
the is u
(ax1)isthecontrolinput vector.
, on accelerati system
the is u Ia
The block diagram (figure 18) shows the signal flow of an output feedback control system.
The sensor output can be related to system states (displacement, velocity) as follows:
s d
s v
s s
1
d v s us
c u , c u ,
where , are the voltage and voltage rate, respectively, and
c ,c KII K ,I I
I
I I
(39)
K
IIs is the capacitance of piezoelectric sensor lamina,K
Ius is the charge sensitivity matrix of sensor lamina.fd
s s,I I x
Ia
2 v s
1 d s
x c
x c
u B x A x I I
Plant
s v a
s d a
F F
I I
I I
Control input
Controller
Figure 18 Block diagram of control system (Physical domain)
It may be observed that the sensor patch senses the charge induced due to the nodal displacements associated with it and outputs as sensor voltage. The velocity information of the sensor patch further can be obtained by differentiating the charge signal using a differentiator (analog or digital). This can be realised by properly selecting the circuit capacitance (Cf) and resistance (Rf) for the charge amplifier such that the time periodW CfRf. The equivalent circuit makes the sensor output independent of crystal capacitance and resistance.
A state vector x is defined with two new states (x1: displacement, x2: velocity) as follows:
. x u x and x u x ; x u
x u 2 1 2
2 1
¿¾
½
¯®
¿¾
½
¯®
(40)
The second order dynamic equation (38) is reduced to two first order equations using equation (40) as,
1 2
1 1 1 1
2 uu uu 2 uu uu 1 uu 0 a uu u m
x x ,
x M C x
M K x
M b
I M b f .
(41)The state-space equation (41) can be expressed in matrix form as,
1 1 1 1
uu uu uu uu uu u uu 0
d 1
2 a
0 I 0 0
x(t) M K M C x(t) M (b ) M b u(t), f (t)
x (t)
x(t) x (t) , u(t) (t) ,