Enhancement of buckling load of thin plates using
Piezoelectric actuators
R. Indira Priyadarshinia, C.Lakshmana Raob, S.M.Siva Kumarb
a) Master of Science by Research Student, Department of Applied Mechanics.
b) Professor from Department of Civil Engineering.
Indian Institute of Technology Madras, Chennai, India 600036.
[email protected], [email protected], [email protected]
Abstract
Experiments are carried out on an aluminium plate of 0.3 X 0.5 X 0.001 m to examine the effect of piezoelectric forces on the
buckling load of the plate. The experimental procedure is outlined and the load strain plots, obtained with and without control
are presented. The loading edges of the plate are simply supported and the other two edges are free. Initial experiments are
conducted on the plate without piezoelectric actuators, later the piezoelectric actuators are surface bonded on the plate. Due
to initial imperfections and load eccentricities, the initial buckling load of the plate is less than the critical buckling load of the
plate. The converse piezoelectric effect of the piezo actuators is used for applying to control the bending strains. Open loop
and closed loop experiments are conducted on the aluminium specimen. It is observed from the open loop experiments that
the load carrying capacity of the plate without control was around 4.32 kg where as plate with 200 volts passive control was
5.5 kg which is 9.09% increase in the buckling load. Closed loop experiments with 200 gain factor showed an enhancement of
22% in the load carrying capacity. The experimental results have clearly demonstrated the effect of piezoelectric force in
enhancing the buckling load of plate structures.
Introduction
Design of material systems and structures is based on their response to the applied load. Lightweight and low cost, space
vehicles are designed for high flexibility. Thus, it is highly desirable to control the deformations of structures when they are
subjected to compressive loads. Due to the remarkable advances in sensors and actuators, active systems have become the
cost effective solutions to most vibration, shape and buckling control problems. Thin plates of various shapes used in
application of aeronautical structures are often subjected to in-plane compressive loads. Under certain conditions, these loads
can result in buckling (elastic instability) of plates. One method of enhancing the buckling load of plates without increasing the
weight of the structure considerably is to use smart materials such as piezoelectric materials and shape memory alloys as
actuators.
In the recent past, the construction and operation of space structures have generated an interest in using smart materials like
piezoelectric materials (PZT), shape memory alloys (SMA) and polyvinylidene fluoride (PVDF) to form a smart structure.
Among all these materials, piezoelectric materials (PZT) offer a great promise in vibration, shape and buckling control. Since
piezoelectric materials have high actuation performance, great ease in control, fast response etc compared to other smart
materials, they are strongly proffered as actuator materials compared to other materials. The piezoelectric material generates
an electrical charge under the application of mechanical force or deformation. This is called as the direct piezoelectric effect.
Applying an electric field to these materials results in mechanical stresses or strains and this phenomenon is called as the
converse piezoelectric effect. With these two effects, the PZT materials can be used for sensing and actuation.
Buckling is a form failure that is caused by in-plane compressive loading. When these compressive loads are combined with
structural imperfections, load eccentricities or other out-of-phase loads, the structure may experience large lateral deflections,
structural instability and finally to failure. Elastic instability of the column and plate structure was complied exhaustively in the
monograph of Timoshenko and Gere (1961) [1]. Piezoelectric materials have been used by several researchers to achieve
vibration and buckling control. Bailey and Hubbard [2] presented active vibration control of cantilever beam using piezoelectric
polymer. The axial forces and the corresponding moments generated by actuating the PZT can be used to counteract the
forces acting on the plate and hence the deflections of the structure can be controlled. Meressi et al [3] have reported that
resistive strain and piezoelectric (PVDF) film may be used to actively stabilize the first buckling mode of a column with
proportional feedback. A new concept of follower force is proposed by Venkateswar Rao and Gajbir singh [4]. These authors
have pasted PZT patches on surfaces on either sides of the column, and the forces generated by actuating the PZT layers act
as the follower force on the column. The result shown by these authors indicate that the Euler’s buckling load can be
increased by a factor 3.5. Thomson and Loughlan [5] have conducted experiments on composite column strips and
demonstrated the enhancement of buckling load. A pair of PZT was surface bonded at the center of the column by these
authors. The PZT patches were actuated by them by applying voltage in the thickness direction. The results demonstrated that
buckling load was enhanced by 32.4% compared to uncontrolled column. Later Mini et al [6] conducted experiments on
aluminum column, they used a single PZT for actuation the results obtained shows an enhancement of buckling load by
11.7%.
The study on buckling control needs to be extended to two-dimensional structures like plates. Most of the researchers
concentrated on vibration control of plates. Very little literature is available with regard to buckling control and shape control of
plates. Chandrasekhara and Bhatia [7] have developed a finite element model for active buckling control of laminated
composite plates using piezoelectric materials. The results have demonstrated that uncontrolled fiber-reinforced composite
plate buckles at 23.4 seconds, and with sensors and actuators, the same plate buckles after 24.6 seconds. The application of
PZT patches as actuators is also extended in the field of shape control. Chien-Chang Lin et al [8] have developed a finite
element model on deflection control of plates using piezoelectric actuators. The results obtained from the model were
compared with the analytical solutions. Young-Hun Lim [9] formulated for a closed loop vibration control of plate with
piezoelectric patches, the piezoelectric patch used by these authors is made up of PZT-5H. V.Birman and S.Adali [10] have
developed a method to damp the vibrations of an orthotropic plate using piezoelectric stiffener-actuators. These authors have
used the concept of applying the in-phase and out of-phase voltage on the PZT patches, thus resulting in effective in plane
strains and effective moments. They bonded the piezoelectric patches on the opposite surfaces of the plate and placed
symmetrically with respect to the middle plane. With the same methodology, Sarp Adali et al [11] in 2005 have extended the
concept of vibration control to buckling control of composite plate. They have discussed the buckling load enhancement of a
fiber reinforced rectangular plate with initial imperfection. The above authors have developed a closed form solution for a
composite plate with PZT strips. It was observed from the literature that not much attention was paid toward experiments on
plates, especially on buckling control of plates. Hence, in this paper an attempt is made to conduct experiments on rectangular
plate with piezoelectric actuators pasted at discrete locations unlike strips. Initial experiments were conducted on a plate
without PZT actuators, later with PZT actuators and actuated at different voltages. For a specified deflection the increase in
the load carrying capacity of the plate with control and without control is observed from the load strain plots. The buckling load
enhancement at chosen deflection was calculated from these plots.
Experiment setup
The apparatus is exclusively designed and fabricated for loading slender columns and thin plate structures. The loading
apparatus consists of two platforms, one at the top (loading platform) and the other at bottom. The self weight of the loading
platform is 941gm. The plate is held in-between the two platforms using two holders. These holders are designed in such a
way that they ensure a simply supported boundary condition for the plate. The holder is fabricated out of aluminium, and the
weight of each holder is 304gm. The plate is fixed in-between the two holders as shown in the figure1 (a). An “S” type 50kg
load cell is placed on the bottom platform, such that it measures the load applied on the plate. The calibration constant for the
load cell is obtained by calibrating the load cell against standard loads. The plate with holders is held in-between the loading
platform and the load cell. In order to load the plate continuously water is drawn continuously from a source at a rate of
6.30895 g/s into a can placed on the loading platform. The loading platform is allowed to slide in the two vertical polished
supports. As the load progresses the loading platform slides down there by increasing the load on the plate. The load cell
placed at the bottom initially measured a weight of 1.9 kg indicated by the digital indicator. The initial load measured by the
load cell included the weight of the loading platform, weigh of the two holders, self weight of the plate, weight of the can and
the cables used for strain measurement and voltage supply for PZT actuators. The output of the load cell is within the range of
+2 to -2 volts. This analog output of the load cell is fed to an ADC card, where the digitized data is appropriately converted to
voltages and stored in an excel file.
An aluminium plate of dimension 0.3m X 0.5m X 0.001m was used for conducting experiments. The loading edges of the plate
are simply supported and the other two edges are free. The plate is anodized to prevent continuity of voltage within the plate.
Three piezoelectric actuators of dimensions 0.076m X 0.026m X 0.001m are surface bonded on the plate. The piezoelectric
actuators are coated with conducting material so as to facilitate easy connection to voltage flow. The piezoceramics chosen
belong to the class of SP-5H.
Table 1. Material properties of SP-5H
Piezoelectric Charge Constants
d33
( x 10-12 C/N)
d31
-12
( x 10
Elastic Constants, short circuit
550
C/N)
- 265
Piezoelectric Voltage Constants
g33
g31
( x 10-3 Vm/N)
-3
( x 10 Vm/N)
T
Relative Dielectric Constant, K
E
–12
21
2
s 33 (x 10
m /N)
Frequency Constants (Hz-m)
15
19
Np (planar mode disk)
1950
-9
Nt (thickness mode disk)
2000
3100
Mechanical Quality Factor, Qm
Dissipation factor, tan δ (low field)
0.020
Figure of Merit dh x gh ( x 10
Density, ρ (kg/m3)
7500
Curie temperature, Tc (oC)
3
(low signal)
sE11 (x 10 –12 m2/N)
-15
65
)
67
190
Strain gauges were used to measure the bending strain of the plate. A resistive strain gauge of 120 ± 2Ω, gauge factor of 2.0 ±
2% and gauge length of 10 mm is used. The bending strains of the plate are measured at two locations as shown in the figure
1 (b). At every location a pair of strain gauges is pasted on either sides of the plate. The four leads from the strain gauges are
connected to half bridge circuit of the strain gauge amplifier, where the net strains measured on the plate are amplified. From
the initial mode shape of the plate it was observed that the maximum deflection occurs at the center of the plate, accordingly
the arrangement of the piezoelectric actuators is optimized as shown in figure 1 (b).
Figure: 1 (a) Plate with holder
Figure: 1 (b) Plate with PZT and Strain gauge
Experiment procedure
The test procedure is outlined in Figure 2. The experiment was initially done without applying the voltage to the PZT. The
structure i.e. the aluminium plate mounted with the PZT and strain gauges was loaded continuously with water. The strains in
the plate were sensed by the strain gauges. The strain gauges were preferred to LVDT for sensing because the setting up of
the LVDT in position causes some initial deformations on the plate. A half bridge circuit with two strain gauges was employed
to measure bending the strains in the plate subjected to slowly varying load. These were connected to a strain gauge amplifier
which had the remaining Wheatstone bridge circuit as well as the power supply for voltage excitation. The voltage change
caused because of the change in the resistance of the strain gauges was amplified with the strain gauge amplifier. The
amplified voltage was varying between -2 volts to +2 volts. The analog voltage output from the strain gauge amplifier was fed
to the ADC card interfaced with the computer. The ADC card converted the analog voltage signal to digital values and stored it
in an excel file. The sampling was done at every 100 milliseconds.
The experiment was then done with applying a voltage of 200V to the PZT. The voltage was applied immediately after the
loading commenced. The maximum DC operating field of the PZT was 200 V/mm. The PZT was polarized along the thickness
direction so that a voltage V, applied across its thickness resulted in strain along its length and width. Piezoelectric materials
exhibit a linear relationship between the components of the electric-field vector and the induced strain tensor components.
However the linear relationship is only valid for a low electric field. Hence a maximum voltage of 200 V was applied in the
experiments. The piezoceramics were used in preference to PVDF polymer because the piezoceramics have a much higher
d31 coefficient and can efficiently convert electrical energy to mechanical energy when compared to the PVDF polymer.
Figure: 2 Block diagram of experimental procedure.
Open loop experiments: The test procedure for the open loop experiments is shown in the block diagram of Figure.2
connecting the various elements with a dotted arrow marks. A constant voltage is applied on PZT actuators passively. The
voltage was applied immediately after the loading commenced. The bending strains measured by the strain gauge are fed to a
strain gauge amplifier. The analog output from the amplifier is fed to ADC integrated with a computer which stores the digital
data in an excel file. There is no feed back system to control the voltages that are applied on the PZT strips. A constant
voltage of either 100 V or 200 V is applied continuously on the PZT strips.
Closed loop experiments: In closed loop experiments the voltage applied on the PZT strips is proportional to the bending
strains measured by the strain gauges. The experimental procedure is outlined in the Figure.2 connected by solid arrow
marks. As the load progresses the bending strains measured also increases. These strains are amplified using strain gauge
amplifier and fed to ADC. A code is developed to handle the data from the ADC that converts the data to proportional
command voltage. This command voltage from the computer is fed to DC amplifier. The amplified voltage is proportionally
applied on the piezoelectric actuators. Here the voltage applied on the PZT patches is controlled by the feed back from the
strains measured. The results from the open loop experiments and from closed loop experiments are compared.
Results and discussion
Open loop experiment results: The graph shown in the Figure 3, gives the results open loop experiments. The data from the
load cell and strain gauge amplifier are multiplied with appropriate calibration factor in order to convert strains into microns and
load to kilogram. The graph consists of three load strain curves which represents for plate without PZT actuators, with PZT
actuators without voltage and plate with PZT with 200 volts passive voltage. For the strains equal to 15 microns it was
observed that the plate without control was able to carry a load of 4.32 kg. The same plate when loaded with PZT actuators
but without voltage, had a load carrying capacity of 5.5 kg. Later by actuating the PZT patches with 200 volts passively there is
an increase in the load carrying capacity from 5.6 kg to 6.1 kg which is 8.92% increase in the buckling load. Since the
sampling rate is 0.1 second the data collected contains some noise, the graphs shown in Figure 3 are smoothened by
averaging the data over 20 samples.
Figure: 3 Load vs. lateral strain graph from Open Loop Experiments
Closed loop experiment results: The graph shown in the Figure 4. gives results from closed loop experiments. The closed
loop experiments are also continued on the same specimen which was used earlier in open loop experiments. Hence there
was an initial strain of 65 microns in the specimen. This strains measured initially is permanently induced in the plate due to
repeated experiments on the same specimen. Figure 4. shows the results from closed loop experiment, with gain factor of 200.
Gain factor is a constant with which the command voltage is multiplied to set the initial voltage applied on the PZT actuators.
For a strain of 100 microns the plate without voltage was able to carry a load of 3.5 kg where as plate with active control was
able to carry a load of 4.28 kg which indicates an increase of 22% in the buckling load.
Figure: 4 Load vs. lateral strain graph from Active Control Experiments with a Gain Factor of 200
Figure: 5 Load vs. lateral strain graph from Active Control Experiments with a Gain Factor of 400
The graph shown in Figure 5. gives results from active control experiments with a gain factor of 400. From the graph it is
observed that for 100 microns strain the load carrying capacity of the plate without control was 3.5 kg. The load carrying
capacity of the plate is increased from 3.5 kg to 4.33 kg with active control which is approximately 23.715% increase in the
load carrying capacity. It is also observed from the figure 4 and 5 that as the gain factor increases there is slight increase in
the buckling load. Since the control strategy adapted is proportional controller, the voltages applied on the piezoelectric
actuator increases proportional to the strains measured multiplied with gain factor.
Summary and Conclusions
This paper investigates the use of PZT actuators to control the buckling of plate structures. Open loop and closed loop
experiments were conducted on aluminium specimens. The open loop experiments clearly indicated that the piezoelectric
force could remove bending strains due to loading in an axially loaded plate. From the experience obtained from open loop
experiments, closed loop experiments are also conducted on plates with two different gain factor. It is observed from closed
loop experiments that the voltage required for controlling the bending strains is much higher compared to columns. The
enhancement in the buckling is higher in closed loop experiments when compared with open loop experiments. Since the
piezoelectric actuators are pasted only at the center, the first mode alone could be controlled in the current set of experiments.
This paper demonstrates that piezoelectric materials are good in enhancing the buckling load of plate structures without much
increase in the weight of the whole system. The experiment results have indicated a way to work with smart controls and
smart structures where the weight is a major constraint. Further studies will focus on optimizing the number of piezoelectric
actuators required for control and the location of the actuators in the plate.
Acknowledment
This study is been carried out as a part of an investigation of the project ‘Buckling Control of Shells using PZT
actuators a sopnsored project by IIT-Madras-ISRO cell. This support is kindly acknowledged. The technical assistance offered
by Mr. Santosh Kumar S and Mr. France K is also acknowledged.
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