Abstract
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.
1. Introduction
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 [1] and Toupin [2] who considered cylindrical and spherical shells, respectively. Tiersten’s [3] analysis was in linear vibrations of piezoelectric plates. Adelman and Stavsky [4] studied vibrations of composite cylinders by piezoceramic and metallic layers. V. Balamurugan, S. Narayanon and Rajan L.Wankhade, Kamal M.B [5, 6] studied consistency technique for vibration control of smart stiffened plates using distributed piezoelectric sensors and actuators subjected to cyclic loads. P.veera Sanjeeva Kumar, B.Chandra Mohana Reddy [7] have presented a paper on the vibration of smart composite laminate plates using higher order theory. Sangamesh B. Herakal, Sai Kumar Dathrika, P. Giriraj Goud, and P. Sravan [8] analyzed bending of the smart composite plate under thermal environment, and Kanjuro Makihara, Junjiro Onoda, and Kenji Minesugi [9] studied flutter of cantilevered plate wing with piezoelectric material layers by finite element method. In 2016, Zafer K and Zahit M [10] presented a paper on flutter analysis of a laminated composite plate with temperature dependent material properties. N. T. Chung, H. X. Luong, and N. T. T. Xuan [11] studied the dynamic stability of laminated composite plate with piezoelectric layers. N. N. Thuy and N. T. Chung [12] have presented a paper on the dynamic analysis of smart stiffened composite plates using higher order theory.
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.
2. Finite Element Formulation and the Governing Equations
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 1).
(a)
(b)
2.1. Piezoelectric Equations
Consider a laminated composite plate with integrated sensors and actuators as shown in Figure 1. It is assumed that each layer of the plate possesses a plane of elastic symmetry parallel to the x-y plane; the layer’s lamina constitutive equations coupling the direct and converse piezoelectric equations can be expressed as [13, 14]where are, respectively, the plane-stress reduced elastic constants, the piezoelectric constants, and the permittivity coefficients of the lamina in its material coordinate system; , , , are the stress, strain, electric field, and electric displacement matric components, respectively, to the material coordinate system. If the voltage is applied to the actuator in the thickness only, then , is the applied voltage across the ply, and is the thickness of the piezoelectric layer. The plane-stress reduced elastic constants are given as
Upon transformation, the lamina piezoelectric equations can be expressed in terms of the stress, strains, and electric displacements in the plate coordinates as [13, 14]
Equations (4) and (5) can also be written as
The are the components of the transformed lamina stiffness matrix which are defined as follows:where is the lamina orientation angle.
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 [13]where
2.2. Smart Piezoelectric Composite Plate Element Formulation
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 2.
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 and of the normal in xz and yz planes, respectively, as [11, 15, 16]
The components of the strain vector corresponding to the displacement field (10) are [7, 11]where are the linear and nonlinear strain vector, respectively, and defined as follows:
In the finite formulation, the displacement field and the electric potential over an element are related to the corresponding node values of the element and the electric charges of the piezo layer by the mean of the shape functions , , as follows [15]:
Equation (11) can be expressed aswhere and are linear and nonlinear strain matric, respectively, are the shape functions, and are the shape functions for the electric potential.
The dynamic equations of a finite smart laminated composite plate can be derived by using Hamilton’s principle [15, 17]:where is the kinetic energy, is the strain energy, and is the work done by the applied forces.
The kinetic energy at the element level is defined aswhere is the volume of the plate element.
The strain energy can be written as
The work done by the external forceswhere is the body force, is the surface area of the plate element, is the surface force, and is the concentrated load.
Substituting (17), (18), and (19) into (16), noting that the electric field vector , and using (6), (7), (13), (14), and (15), the dynamic matrix equations can be written as
Substitute (21) into (20) to obtainwhere , , , , , , which are, respectively, the element mass, linear mechanical stiffness, nonlinear mechanical stiffness, mechanical-electrical coupling, piezoelectric permittivity, and electrical-mechanical coupling matrix. The vector and are element external mechanical force vector and external voltage applied to the piezo layer, respectively, which are defined as follows:
For the sensor layer, charge sensing is considered. With zero voltage, from (21), the sensed voltage is given by [13, 16, 18]
The operation of the amplified control loop implies the actuating voltage iswhere and are the feedback control gains for displacement and velocity, respectively.
From (21), the charge in the actuator due to actuator strain in response to plate vibration modified by control system feedback is
Substitute (26) into (22) to yieldwhere are the mechanical-electrical coupling and pizoelectric permittivity stiffness matrices, respectively.
Equation (27) can be rewritten as follows:where is the element active damping matrix and is the element active stiffness matrix.
2.3. Formulation of Stiffener
2.3.1. Formulation of x-Stiffener
where x-axis is taken along the stiffener centerline and the z-axis is its upward normal. The plate and stiffener element are shown in Figure 3.
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 aswhere is the plate thickness and is the x-stiffener depth.
The element stiffness and mass matrices are defined as follows [15]:where is the strain-displacement relations matrix, is the stress-strain relations matrix, and is the element length. are the shape function matrices relating the primary variables , w, x, in terms of nodal unknowns, is the area moment of inertia related to the y-axis, and , with being density of layer.
2.3.2. Formulation of y-Stiffener
The same as for x-stiffener, the element stiffness and mass matrices of the y-stiffener are defined as follows:
2.4. Modeling the Effect of Aerodynamic Pressure and Motion Equations of the Smart Composite Plate-Stiffeners Element
Based on the first-order theory, the aerodynamic pressure and moment can be described as [19, 20]where is defined as the reduced frequency, is the circular frequency of osscillation of the airfoil, U is the mean wind velocity, B is the half-chord length of the airfoil or half-width of the plate, is the air density, and is the angle of attack.
The functions are defined as follows:where F(k) and G(k) are defined as
Using finite element method, aerodynamic force vector can be described aswhere and are the aerodynamic stiffness, damping matrices, and lift force vector, respectively:where is the element area and are the shape functions.
From (28), (31), (32), (33), (34), and (38), the governing equations of motion of the smart composite plate-stiffeners element subjecteced to aerodynamic without damping can be derived aswith ,,.
2.5. Governing Differential Equations for Total System
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:where , , , and are Rayleigh’s coefficients, to account for inherent structural damping.
The solution of nonlinear equation (43) is carried out using Newmark direct and Newton-Raphson iteration method [11, 21].
2.6. Numerical Applications
A rectangle cantilever laminated composite plate is assumed to be with total thickness of 4mm, length of 600mm, and width of 400mm with three stiffeners along each of the directions x and y. The geometrical dimension of the stiffener is height of 5mm and width of 10mm. The material properties for plate and stiffeners made of graphite/epoxy are E11 = 181GPa, E22 = E33 = 10.3GPa, E12 = 7.17GPa, = 0.35, = = 0.38, are = 1,600kg·m−3. Material properties for piezoelectric layer made of PZT-5A are d31 = d32 = -171×10−12m/V, d33 = 374×10−12m/V, d15 = d24 = -584×10−12m/V, G12 = 7.17GPa, G23 = 2.87GPa, G32 = 7.17GPa, = 0.3, = 7,600kg·m−3 and thickness = 0.15876mm, = 0.05, = 0.5, = 15. The effects of the excitation frequency and location of the actuators are presented through a parametric study to investigate the vibration shape of the composite plate activated by the surface bonded piezoelectric actuators. The iterative error of the load = 0.02% is chosen.
The stiffened plate is subjected to the airflow in the positive x direction as shown in Figure 1(a).
Figure 4 presents the time history response of the plate at critical airflow velocity and α = 0°.
Figure 7 shows the history of plate dynamics at α = 0° and critical airflow velocity = 30.5m/s. In this case, at t = 15s, tip displacement (point A in Figure 1(a)) reached 1.75cm. The plate begins to be instable and, at this time, piezoelectric voltage was 125V due to the piezoelectric effect.
3. Experimental Validation
3.1. Experimental Model
A rectangle cantilever laminated composite plate is assumed to be with total thickness of 4mm, length of 500mm, and width of 300mm with three stiffeners along each of the directions x and y (). The geometrical dimension of the stiffener is height of 8mm and width of 10mm (Figure 5). The piezoelectric slice is 200 mm by 90mm dimension and placed in the location as in Figure 6. The plate and stiffeners made of T300/2500 graphite/epoxy are produced by Torayca Co., Japan. The mean values and coefficients of variation of the experimentally determined material constants are given as E1 = 124.68GPa, E2 = 9.6GPa, G12 = 8.64GPa, G23 = 2.32GPa, and = 0.33.
3.2. The Diagram of Experiment
The diagram of an experiment is shown in Figure 7. Alternating current from the electric grid running through transmitter is linearly amplified to generate voltage V with frequency imposing on PZT slice. The PZT slices attaching on the shell are playing a role to arouse the vibration of the whole system. Through two acceleration sensors attached to the shell, the dynamic response of the shell at the location of the measurement points will be captured, displaying on the oscilloscope display and then storing in the computer.
The main experiments are shown herein:(i)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.(ii)Measurement of the first free vibration of the structure.
3.3. Measuring Devices
3.3.1. The Signal Generator
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 8. The speaker test oscillator is used in this experiment is Onsoku branch (Japan) model OG-422A. The specifications of the machine are as follows: the maximum output voltage: ± 200 V, the response frequency range (output): kHz, capacity compliant amplifier: 20 (W), compatible power supply: 100V, 115V, 220V and 240 V, frequency 50/60Hz.
(a) The speaker test oscillator
(b) The linear amplifier
3.3.2. Generating Load Equipment
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 9). The experimental model is placed on a bracket with a diameter 500 mm; this bracket can rotate with different angles compared to the wind direction.
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3.3.3. The Oscilloscopes
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 9(a)). The specifications of these accelerometer sensors include parameters: the sensitivity 10mV/g, frequency band 1,0Hz ÷ 20kHz, dynamic range: ± 250g, resolution 40μg, resonant frequency: > 35kHz, linearity: 0.1%, and the maximum pick: 1000g (g: gravity acceleration).
Two piezoelectric channels’ data acquisition HnB75B has a function to amplify the electrical signals from the sensor to the oscilloscope and the computer (Figure 9(b)). The specifications of this device include: number of channels - 2 simultaneous channels, voltage range measurement: 0 5V, accuracy: +/- 0.2%, sampling frequency: maximum 10 kHz, connection port: 1 com port, 2 AC ports.
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 10(c)).
(a) The acceleration sensor
(b) HnB75B data receiver
(c) The oscilloscope display
3.4. The Results of Measuring Acceleration Response Experiment
In this experiment, the authors conduct three cases: = 0°, = 22.5°, and = 45°.
For = 0°: The speed of the airflow U = 5m/s and 10 m/s; the voltage = 9.30V, and = 20. Hence the voltage level will be × = 9.30 V × 20 = 186.0V; sampling frequency = 1000 Hz; frequency of stimulation f = 6.944 Hz.
For = 22.5°: The speed of the airflow U = 5m/s and 10m/s; the voltage = 9.50V, and = 20. Hence the voltage level will be × = 9.50 V × 20 = 190.0V; sampling frequency = 1000 Hz; frequency of stimulation f = 6.981 Hz.
For = 45°: The speed of the airflow U = 5m/s and 10m/s; the voltage = 9.20V, and = 20. Hence the voltage level will be × = 9.20 V × 20 = 184.0 V; sampling frequency = 1000 Hz; frequency of stimulation f = 6.993 Hz.
The acceleration-time and amplitude-frequency response at the measuring point of the above stiffened composite plate with PZT patches are shown in Figures 11 and 12.
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 13 and Table 1.
4. Conclusion
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.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.