In this paper, the authors present results on dynamic behavior analysis of the stiffened composite plate with piezoelectric patches under airflow by finite element method and experimental study. The first-order shear deformation plate theory and nine-noded isoparametric piezoelectric laminated plate finite element with five elastic degrees of freedom at each node and one electric degree of freedom per element per piezoelectric layer were used in the dynamic analysis of plates by finite element method. The modern equipment was used in the dynamic behaviors analysis of plates subjected to airflow load by experimental method. In this study, the results of the theoretical method have been compared with experimental studies.
The research and development of smart stiffened composite structures represent one of the most significant recent trends in the mechanics of structures, especially in the aerospace industry. These structures can provide significant advantages over traditional structures, notably active vibration control. In 1957 and 1959, the first investigations of piezoelectric structures were published by Haskins and Walsh [
In this paper, in order to have more studies about the dynamic responses of piezoelectric composite plates, the authors examine the problems with piezoelectric stiffened composite plates subjected to airflow by finite element and experimental method.
Consider isoparametric piezoelectric laminated stiffened plate with the general coordinate system (x, y, z), in which the x, y plane coincides with the neutral plane of the plate. The top surface and lower surface of the plate are bonded to the piezoelectric patches (actuator and sensor). The plate is subjected to the airflow load acting (Figure
Smart stiffened plate and coordinate system (a), and lamina details (b).
Consider a laminated composite plate with integrated sensors and actuators as shown in Figure
Upon transformation, the lamina piezoelectric equations can be expressed in terms of the stress, strains, and electric displacements in the plate coordinates as [
Equations (
The
The piezoelectric constants matrix [e] is unavailable and it can be expressed by the more commonly available piezoelectric strain constant matrix which is [d] as [
A smart piezoelectric composite plate element is considered with coordinates x, y along the in-plane direction and z along the thickness direction as shown in Figure
Smart piezoelectric composite plate element.
Using the Mindlin formulation, the displacements u, v, and w at a point (x,y,z) form the median surface and are expressed as functions of mid-plane displacements u0, v0, and w0 and independent rotations
The components of the strain vector corresponding to the displacement field (
In the finite formulation, the displacement field
Equation (
The dynamic equations of a finite smart laminated composite plate can be derived by using Hamilton’s principle [
The kinetic energy at the element level is defined as
The strain energy can be written as
The work done by the external forces
Substituting (
Substitute (
For the sensor layer, charge sensing is considered. With zero voltage, from (
The operation of the amplified control loop implies the actuating voltage is
From (
Substitute (
Equation (
Modeling of plate and stiffener element.
If we consider that the x-stiffener is attached to the lower side of the plate, conditions of displacement compatibility along their line of connection can be written as
The element stiffness and mass matrices are defined as follows [
The same as for x-stiffener, the element stiffness and mass matrices of the y-stiffener are defined as follows:
Based on the first-order theory, the aerodynamic pressure
The functions
Using finite element method, aerodynamic force vector can be described as
From (
Finally, the elemental equations of motion are assembled to obtain the open-loop global equation of motion of the overall stiffened composite plate with the PZT patches as follows:
The solution of nonlinear equation (
A rectangle cantilever laminated composite plate is assumed to be
The stiffened plate is subjected to the airflow in the positive x direction as shown in Figure
Figure
History of the plate at a critical airflow velocity
Figure
A rectangle cantilever laminated composite plate is assumed to be
The preparation of experimental model.
The real PZT model.
The diagram of experiment.
The diagram of an experiment is shown in Figure
The main experiments are shown herein: Measurement of the acceleration response of the structure at the measuring points arranged on the top of the plate corresponding to different voltage and frequency stimulation levels placed on two piezoelectric plates. Measurement of the first free vibration of the structure.
The signal generator has a function to generate randomly the sinusoidal alternating current with voltage V and frequency f placing on 2 piezoelectric plates attached to the shell to establish vibration of the structure. In this experiment, the signal generator is used by combining the speaker test oscillator and a linear amplifier as in Figure
The signal generator.
The speaker test oscillator
The linear amplifier
The airflow load is created through an open wind tunnel with capacity 11kW, the area of test cross section is 1000mm×1000mm, and airflow speed can be changed from 0 to 40 m/s (Figure
The open wind tunnel and the bracket.
The oscilloscope has a function which is capturing the acceleration response of the plate at the point of measurement, then displaying on screen and storing the data in the computer. The devices to measure the vibrations used in this experiment include accelerometer sensor (2 sensors), two piezoelectric data receivers HnB75B, oscilloscope display Tektronix TDS-1012, and a computer. The accelerometer sensors used in this experiment are type ACH01-02 (Figure
Two piezoelectric channels’ data acquisition HnB75B has a function to amplify the electrical signals from the sensor to the oscilloscope and the computer (Figure
Through the simulation program set up already in devices and computer, the vibrations of the plate at measurement points are displayed on the screen of Tektronix TDS-1012 machine with the vibration parameters at each oscillation cycle (Figure
The oscilloscopes.
The acceleration sensor
HnB75B data receiver
The oscilloscope display
In this experiment, the authors conduct three cases:
The acceleration-time and amplitude-frequency response at the measuring point of the above stiffened composite plate with PZT patches are shown in Figures
The acceleration response in one measurement with
The acceleration response and frequency spectrum with f = 13.672Hz, U = 10m/s,
We calculate the acceleration response for the above stiffened composite plate with PZT patches by our computer program. The acceleration response at the measuring point of the plate will be compared with those of experimental ones and is given in Figure
The maximum acceleration at U = 5m/s.
Method | | | Error [%] |
---|---|---|---|
Experiment | 0.050 | 45 | 11.8 |
FEM | 0.0559 | ||
| |||
Experiment | 0.061 | 22.5 | 10.7 |
FEM | 0.0675 | ||
| |||
Experiment | 0.072 | 0 | 10.1 |
FEM | 0.0793 |
The acceleration response of plate at the measuring position (U = 5m/s).
The nonlinear dynamics analysis of the piezoelectric stiffened composite plate subjected to airflow using the finite element and experimental method has been presented. In this paper, we have presented a nine-noded stiffened rectangular composite plate element with PZT patches for the nonlinear vibration analysis of the piezoelectric stiffened composite plates subjected to airflow. The critical velocity of the airflow is determined by numerical calculations. The finite element results compare well with experimental ones. It is recommended that the present formulation can be used to determine the characteristics of the vibration and stability in the analysis and design of the piezoelectric stiffened composite plate structures subjected to airflow applied to the flying instruments.
The data used to support the findings of this study are available from the corresponding author upon request.
The authors declare that they have no conflicts of interest.