Utilizing the Finite Element Method to Analyze the Active Vibration
Suppression of Structural Systems
Brian P. Baillargeon, Application Engineer
ABAQUS East, LLC, Warwick, Rhode Island, 02886
Tel: 401-739-3637, Fax: 401-739-3302, Email: [email protected]
Introduction:
Finite element model:
Cantilever Beam:
A two dimension plane stress implicit dynamic ABAQUS model is
utilized to characterize the structural response of the cantilever
beam. The model consisted of 2754 CPS8R and CPS8RE
elements (8893 nodes).
The mathematic representation of the compensator is that of a
spring-mass-damper system. A spring-mass-damper system is
included in the model to act as a physical representation of the
compensator.
The two systems (beam and compensator) are coupled using user
subroutines intrinsic to ABAQUS. In the experiment, the strain at
the root of the beam is used as the sensor input to the control
system. The numerical model is consistent, in that the strain at the
root of the beam applies a force to the compensator. In turn, the
value of the compensator displacement, multiplied by a scalar gain,
is applied as an electric potential to the piezoelectric shear
actuators.
The use of piezoelectric materials in smart/adaptive structures has
been studied intensely for more than a decade. Engineering
applications using this technology have been proposed and
conceived experimentally, such as for active vibration suppression,
noise cancellation, and shape control.
In this work, a distinctly different piezoelectric transduction mode is
utilized where the electric field is applied perpendicular to the poling
direction. In this case, a piezoelectric rectangular element that is
poled in the longitudinal direction and subjected to an electric field E3
in the thickness direction will undergo shear deformation as shown in
Figure 1(b). This is referred to as the piezoelectric shear actuation
- V +
mechanism.
Steel Block
Piezostack
Beam
(Used for repeatable beam excitation)
Figure 3. Cantilever beam configuration
Results:
a)
Extension Actuator
- V +
- V +
Rigid Foam
Table 1. Natural frequency comparison
Mode
b)
Shear Actuator
1
2
3
4
Experimental Natural
Frequency (Hz)
123.0625
495.7125
1098.275
1538.200
Finite Element Natural
Frequency (Hz)
126.85
495.12
1094.6
1558.3
% Diff.
2.9858
0.11967
0.33574
1.2899
Figure 1. Piezoelectric extension and shear actuators
The finite element method is utilized to replicate experimental results
of the active vibration suppression of a cantilever beam with
embedded piezoelectric shear actuators. The commercial finite
element package ABAQUS is utilized for the numerical modeling.
The feedback control is implemented using FORTRAN subroutines
coupled to the ABAQUS analysis. It is shown that the numerical
model accurately predicts the influence of the feedback control
system on the dynamic response of the beam.
Figure 4. Experimental and finite element comparison of settling time of the
cantilever beam
Table 2. Results of the settling time comparison
Feedback control:
In this study, active vibration suppression is implemented using a
feedback control system. The controller design is that of the
Positive Position Feedback (PPF) algorithm. The structure and the
compensator are coupled using the following relationships:
2 n n 2 g c 2
2 c c c 2 c 2
Structure
Compensator
96.7% reduction in
Amplitude!!!
Figure 5. Cantilever beam tip acceleration with
various feedback control options
Conclusion:
The goal of this study was to accurately reproduce the observed experimental response of the vibration suppression of a
cantilever beam using the finite element method. The control algorithm utilized was Positive Position Feedback. It is
shown that the dynamic characteristics of the cantilever beam are accurately represented utilizing the finite element method
with the active vibration suppression active and inactive.
Acknowledgement:
I wish to acknowledge the support of the Maine Space Grant Consortium under the collaborative seed Grant No. EP-02-11.
Figure 2. Experimental implementation of PPF
control