A cantilevered steel beam with full-length active lamina (EAM) on top and bottom is considered to study the pole placement technique and mode shape control (refer to section 4.5). The beam geometry is shown in figure 27 and the material data is taken in the analysis as in table 6.
Steel (0.00635 m) 1 2 3 4 5 6 7 8 9 10 11 12
Actuator Position
Sensor Position
PZT (254 P m )
1.0 m
Figure 27 Cantilever PZT/Steel/PZT Beam
The beam is discretized using 12 elements and dynamic analysis is carried out. The main objectives of active vibration control are to place the desired Eigenvalues (for first three modes) and control the tip motion of the cantilever beam to achieve desired mode shapes. In
the present control study model, two actuators and six sensors are employed. The following shape constraints are imposed: Deflection at the free end is to be 1/10th of the open loop system and the deflection at grid point 4 as in the open loop system to maintain a smooth deflected shape. The optimal positions of the actuators are selected by minimising the number of unstable modes.
The following states are assumed to be measured: displacements at grids 2, 6, 10 and velocities at grids 4, 8, 12. The measured states are taken as specified elements in equation 69.
The desired Eigenvalues of the closed loop system is chosen as
I R
d / r j/
/ (76)
where /I Zdis the damped natural frequency (rad/sec) and /R ]Zn, Zn is the undamped natural frequency (rad/sec).
Table 6 Material Data of the PZT/Steel/PZT Beam
Properties Steel PZT
E (GPa) 210.0 63.0
G (GPa) 80.8 24.2
ȝ 0.3 0.3
ȡ (Kg/m3) 7750.0 7600.0
d31(C/m2) - 11.28
N11, N33 (F/m) - 16.5x10-9
Geometric Properties: Length = 1000 mm, Width = 50.8 mm.
And the damping factor is,
) (
/ 2R 2I
R / /
/
] . (77)
The closed loop desired Eigenvalues are given below along with the open loop eigen values in parentheses.
) 25 . 622 j 0 ( 0 . 600 j 5 . 6
) 18 . 215 j 0 ( 0 . 200 j 5 . 2
) 19 . 34 j 0 ( 0 . 30 j 0 . 2
d 6 , 5 d 4 , 3 d 2 , 1
r r
/
r r
/
r r
/
(78)
0 2 4 6 8 10 -1
-0.5 0 0.5 1 1.5
Time (secs)
Amplitude
10 0
10 1
10 2
10 3
10 -150 4
-100 -50 0
Frequency (rad/sec)
Gain dB
10 0
10 1
10 2
10 3
10 4 -360
0 360
Frequency (rad/sec)
Phase deg
0 0.5 1 1.5 2
-0.6 -0.4 -0.2 0 0.2 0.4 0.6
Time (secs)
Amplitude
10 0
10 1
10 2
10 -100 3
-50 0
Frequency (rad/sec)
Gain dB
10 0
10 1
10 2
10 3 -360
0 360
Frequency (rad/sec)
Phase deg
Figure 28 Time and frequency responses of open loop and closed loop systems.
-1.00E+00 -8.00E-01 -6.00E-01 -4.00E-01 -2.00E-01 0.00E+00
0 0.2 0.4 0.6 0.8 1 1.2
Open Loop Closed Loop
Displacement
Length of the beam (m)
Figure 29 First mode shape of the smart cantilever beam
-1.00E+00 -5.00E-01 0.00E+00 5.00E-01 1.00E+00
0 0.2 0.4 0.6 0.8 1 1.2
Open Loop Closed Loop
Displacement
Length of the beam (m)
Figure 30 Second mode shape of the smart cantilever beam
-1.00E+00 -5.00E-01 0.00E+00 5.00E-01 1.00E+00
0 0.2 0.4 0.6 0.8 1 1.2
Open Loop Closed Loop
Displacement
Length of the beam (m)
Figure 31 Third mode shape of the smart cantilever beam
The real parts in these Eigenvalues are arrived at optimal values after considering the facts that the uncontrolled modes are stable and the developed control scheme could be successfully applied to place the poles. Figure 28 shows the time and frequency response analyses of open loop and closed loop systems (impulse function in MATLAB is used for this purpose). It may be observed that vibration levels of the controlled modes are significantly reduced. From table 7, it can be noticed how exactly the first three Eigenvalues are altered without affecting much of the other modes. The uncontrolled and controlled mode shapes of the first three elastic modes are depicted in figures 29 to 31. The controlled mode shapes are observed to be quite different from uncontrolled ones and this is achieved through the application of shape constraints.
It is worth noticing that the damping factors, actuators and sensors locations are found to be significant factors in order to arrive at a constant gain matrix (controller), which not only controls few elastic modes, but also keeps the system stable.
Table 7 Dynamic Characteristics of PZT/Steel/PZT Beam
Open loop Closed loop
Mode Damping (%) Frequency (rad/sec) Damping (%) Frequency (rad/sec)
1,2 0 ± 34.19 6.65 ± 30.06
3,4 0 ± 215.18 1.25 ± 200.02
5,6 0 ± 622.25 1.08 ± 600.04
7,8 0 ± 1289.51 2.22 ± 1214.9
9,10 0 ± 2302.89 0.90 ± 2243.4
Open loop Closed loop
Mode Damping (%) Frequency (rad/sec) Damping (%) Frequency (rad/sec)
11,12 0 ± 3824.29 0.95 ± 3847.0
13,14 0 ± 6127.07 0.51 ± 6113.9
15,16 0 ± 9494.63 0.45 ± 9528.0
17,18 0 ± 14,847.95 0.52 ± 14,850.0
19,20 0 ± 25,649.66 0.02 ± 25,653.0
21,22 0 ± 43,355.17 0.06 ± 43,358.0
23,24 0 ±130,891.75 0.15 ± 130,888.0
5 Conclusions
A hybrid piezoelectric actuation concept is developed using shear and extension actuators.
Finite element procedures are derived to characterise the hybrid actuation mode by combining shear and extension actuation constitutive models. Further, a two node sandwich beam element is developed using three layer theory and sub-laminate concept to analyse smart sandwich structures. Numerical results are presented to qualify the element to model shear actuation, extension actuation independently and simultaneously. Interesting results have been obtained to demonstrate the influence of active stiffening and active damping effects of shear actuators and extension actuators in the structural control of sandwich beams. Finally, a shape control concept is presented with an example to show the capability of distributed PZT actuators to place desired poles and modify their associated mode shapes.
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Editor: Peter L. Reece, pp. 159-176 © 2006 Nova Science Publishers, Inc.
Chapter 4