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Original Article | Open Access | Br. J. Arts Humanit., 5(5), 243-252 | doi: 10.34104/bjah.02302430252

Vibration Analysis of Two-dimensional Functionally Graded Plate with Piezoelectric Layers using the Classical Theory of Plates

Mohammad Sharafi*

Abstract

In this article, two-dimensional functional materials have been used using the power law in them, which is a good measure to obtain the properties of a composite material of metal and ceramic. At first, the equations of motion were obtained using Hamiltons method and solved by the GDQ method, and finally, the accuracy of the obtained answers was compared with the existing articles. In the following, the dynamic model of the sheet with two piezoelectric actuator layers at the top and bottom was investigated and the obtained equations were solved using the Ritz method.

Keywords

INTRODUCTION

Rectangular plates are one of the mechanical elements that are widely used in various industries such as oil and gas and petrochemical industries. For this reason, investigating the phenomenon of vibrations in this type of plates is a necessary subject, therefore, in this thesis; the vibrations of such plates under the effect of piezoelectric layers that actuate them have been investigated. Examining the vibrations of sheets leads to a series of differential equations. Numerical solution of differential equations is widely used in various fields of engineering. Common methods for solving these equations are FEM, GDQ and RITZ, which are among the numerical methods for solving equations with partial derivatives. In this article, the stability of the sheet with the boundary conditions cccc, ccss, ccsc, cscs, sss and csss has been investigated. The variable of the problem is the piezoelectric thickness and the aim is to find the natural frequency of the sheet along with the piezoelectric. Investigating the piezoelectric effect on plate vibrations and applying the GDQ method to rectangular plates made of two-dimensional FGM materials are new aspects of the work. Research objectives investigating the natural frequencies of two-dimensional FGM sheet under the effect of piezo-electric layers is one of the research objectives.

Hypotheses or research questions 1- The sheet is assumed to be made of two-dimensional graded material. 2- The power law is used to obtain the mechanical properties of the FGM sheet structure. 3- To obtain the stress and strain tensors in the main structure and the piezoelectric layer, the classic sheet model is used. 4- When using the piezoelectric layer, they are used in pairs at the top and bottom of the sheet (the layers are assumed as operators). 5- Poissons ratio is considered the same. Classical plate theory: In this theory, only strain is ignored and the displacement field inside the plate changes as a linear function of thickness. Hamiltons principle: Hamiltons principle is used as a reference to obtain the equations of motion of the sheet will be GDQ: It is a numerical method in the definition of the derivative and it is used to solve the problem numerically.

Review of Literature

In 2009, Liu (Zhao and Liew, 2009) investigated the free vibrations of a rectangular sheet made of graded functional materials using the Ritz method. The properties of the functional material are considered according to the variable tracking power model in the direction of the sheet thickness. The differential equations of motion of the problem are obtained based on the first-order deformation theory. Finally, the solution of the obtained equations has led to the solution of an eigenvalue problem that has been solved by the Ritz method. In 2012, Akbari and his colleagues (Alashti and Khorsand, 2012) analyzed the vibrations of a cylindrical shell made of graded functional materials based on three-dimensional elasticity with piezoelectric layers and taking into account dynamic and thermal loads, and the equations obtained by the differential quadratic method have been solved and compared with the finite difference method. Akhwan and his colleagues (Akhavan et al., 2009) in 2008 have investigated the exact solution of the buckling analysis of a rectangular sheet under load in an elastic bed. The analysis method is based on the theory of rectangular sheet taking into account the first order shear deformation effect. In reference number (Yas and Aragh, 2010) Yas and his colleagues in 2010 have analyzed the free vibration of a plate reinforced with continuous fiber in an elastic bed. The equations of motion based on three-dimensional elasticity have been solved using the differential quadratic method, and finally the frequencies of the system have been investigated by changing the volume of fiber and composite. In 2009, Nemat-Alla and his colleagues (Nemat-Alla et al., 2009) analyzed the plastic elastic analysis of functionally graded materials under thermal loading. The stress-strain relationships based on the elastic-plastic rule of functionally graded materials under thermal loading have been introduced in this model, and the effects of thermal load on functionally graded material and stress-strain changes in the material have been investigated. In 2009, Malekzadeh (Malekzadeh, 2009) analyzed the vibration of a thick plate with a graded function on an elastic bed. The formulation is based on three-dimensional elasticity and FGM relationships are assumed to be exponential. Shaaban and his colleagues (Sharma and Singh, 2011) in 2011 presented an analytical solution for the free vibration of a thick sheet of functionally graded material in an elastic bed with elastic edges. Governing motion equations based on the first order theory of shear deformation and have been solved numerically and the effects of different parameters have been evaluated. Hashemi (Hashemi et al., 2010) in 2010 presented the free vibrations of a rectangular sheet of functionally graded material using the first-order theory of shear deformation, and a new formula for shear correction of the sheet theory is presented in this paper. Hong et al. (Huang et al., 2011) in 2011 have investigated crack vibrations in rectangular thin sheet of functionally graded material. Hu and his colleagues (Thai and Choi, 2011) in 2011 have presented the vibration of a rectangular sheet of functionally graded material in an elastic bed based on the first-order theory of shear deformation. Nezahat and his colleagues (Shirazi et al., 2011) in 2011 investigated the control of active vibrations of a functional rectangular sheet using the fuzzy control method and comparing it with PID controllers and using classical theory and considering a series of local functions of equations obtained the motion and obtained the natural frequencies using the Fourier series.

In 2013, Sadek and his colleagues (Sadek et al., 2003) analyzed the vibrations of a sheet with simple supports based on the feedback control method, which was controlled by sensors and piezoelectric movements, and obtained the equations from the Greens function. The controls are based on displacement and speed feedbacks and solved the equations using a numerical method. In 2013, Rehmat Talebi and his colleagues (Talabi and Saidi, 2013) analyzed the vibrations of complete circular and mounted circular plates made of functional materials due to the effect of two layers of piezoelectric actuators on the top and bottom of the plate using the third order theory of shear and with a new method 5 The main equation of motion is obtained by considering Maxwells equation. Hashemi and his colleagues (Hosseini Hashemi, 2019) in this paper, the analysis and free vibrations of the targeted relatively thick rectangular plate (FG) with smart layers based on Mendelin theory and providing an exact closed-response solution are investigated. This structure consists of a FG sheet and two piezoelectric layers.

Shariayat and his colleagues (Shariayat, 2018) investigated the effects of using piezoelectric sensor and actuator layers on the vibrations of the quadrangular FGM sheet, studied free vibrations, forced vibrations and active control of transient vibrations in this regard. Graded functional material Sheets are structures whose initial shape is flat and their thickness is very small. The behavior of the sheets can be distinguished depending on the type of material and the structure of their formation. Graded functional materials are materials whose mechanical properties change gently and continuously. In recent years, with the development of electric motors, turbines and reactors, it seems necessary to use materials that have high thermal and mechanical resistance. And in this chapter, the introduction of the graded functional materials is discussed. Then, how to make and uses of this material will be discussed. This class of materials is introduced here. Among the main applications of these graded functional materials are the use of nuclear reactors (components of the inner wall of the reactor), use in chemical industries (membranes and catalysts) use in medical engineering (implantation of artificial teeth, bones or artificial organs) and mentioned other new technologies such as ceramic motors and coating against corrosion and heat.

Also, these materials can be widely used in making plates and shells of tanks, reactors, turbines and other components of machines exposed to high heat. Because these parts are highly prepared for failure due to thermal buckling. Another advantage of FG materials compared to layered composite materials is the lack of discontinuity at the junction of layers, because as mentioned in FGM materials, the combination of elements is continuous and gradual. Among the other advantages of graded functional materials, we can mention their use in the construction of thermal insulation coatings, which are widely used. Graded functional materials are composite materials that are microscopically inhomogeneous and their structural characteristics such as the type of distribution, the size of phases, gradually change from one surface to another, and this gradual change leads to a gradual change in their properties. These materials are generally made of a mixture of ceramic with metal or a combination of different metals (with different thermal coefficients).

The ceramic material has high temperature resistance due to its low thermal conductivity, and the metal material prevents breakage or cracking due to thermal stress due to its malleability. In the simplest FGMs, two different material components are continuously changed from one to the other. The transverse distri-bution of electric potential satisfies the Maxwell equation as well as the electrical boundary conditions for both closed and open circuit piezoelectric layers. (Farsangi and Saidi, 2012) In the micro-electro-mechanical systems (MEMS), Coulombs force plays a major role as a mechanism in sensing and actuation. MEMS devices due to their high sensitivity are widely used in different fields of science (Kazemi et al., 2019). The free vibration analysis of two different configurations of functionally graded material (FGM) plates and cylinders is proposed. The first configuration considers a one-layered FGM structure. The second one is a sandwich configuration with external classical skins and an internal FGM core. Low and high order frequencies are analyzed for thick and thin simply supported structures. Vibration modes are investigated to make a comparison between results obtained via the 2D numerical methods and those obtained by the means of the 3D exact solution (Brischetto et al., 2016; Akter et al., 2023).

METHODOLOGY

Table 1: specifications of one-dimensional functional grade material (AL/AL2O3).

Validation of two-dimensional functional sheet vibration with existing articles to validate the results of this research, we first run the written program in the special case of one-dimensional graded material and without piezoelectric. Then we compare the obtained results with references 7, 8 and 11. In these references, the function sheet one dimension is made of aluminum metal and aluminum oxide ceramic with the following specifications.

Table 2: Comparison of natural four-frequency with dimension of present research with existing articles for one-dimensional functional material SSSS

The results of this review are as follows

Table 2 Comparison of the main natural frequency with the dimension of the current research with existing articles for the one-dimensional functional material AL/AL2O3 for boundary conditions SSSS

Investigation of two-dimensional functional sheet vibration (without piezoelectric)

Table 3: Specifications of two-dimensional graded material.

Table 4: Checking the vibration of two-dimensional graded material with support conditions SSSS.

The resulting graphs from this table are as follows

According to the graph (1), it can be seen that with the increase of the ratio or the decrease of the thickness, the natural frequency of the two-dimensional functional sheet decreases. 2- According to the diagram (2), it can be seen that it increases with increasing (constant) frequency. 3- According to the graph (2), it can be seen that it increases with the increase (constant) of the frequency. This increase is faster for values less than one, and for values greater than one, this increase is slower. 4- According to the diagram (1) by comparing the two cases, it can be seen that the increase has a greater effect on the frequency increase than the increase. 5- According to the diagram (1), it can be seen that the frequency reduction is faster before and after this ratio, the frequency reduction speed becomes lower.

In the case of increasing the piezoelectric thickness ( ), the frequency decreases until the thickness and the frequency increases from this thickness. 2- In the case of ( ) increasing the piezoelectric thickness ( ), the frequency decreases ( ) to the thickness ( ) and the frequency increases from this thickness. 3- In the case of increasing the piezoelectric thickness, the frequency decreases to the thickness ( ) and the frequency increases ( ) from this thickness. 4- In the case of increasing the piezoelectric thickness, the frequency decreases to the thickness and the frequency increases from this thickness.

Table 5: Checking the vibration of two-dimensional graded material with SCSC support conditions of the main natural frequency with the dimension of the two-dimensional functional sheet with SCSC boundary conditions.

5- In the case of increasing the piezoelectric thickness ( ), the frequency decreases to the thickness ( ) and the frequency increases from this thickness.6- By increasing the thickness of the piezo ( ), all the graphs are closer to each other and finally one be. 7- As the effect of piezoelectric increases in the initial reduction of frequencies, it increases. For example, in the case of frequency reduction, it is equal to 5.3% and for the case of equal to 8.6% is. 8- As the effect of piezoelectric increases in the initial reduction of frequencies, it increases. For example, in the mode of reducing the frequency in the mode equal to 5.3% and for the mode equal to 5.8% is. 9- The effect of piezoelectric in the initial reduction of frequencies is greater than For example, in the mode of reducing the frequency in the mode equal to 5.8% and for the mode equal to 8.6% is 6-3-5- Investigating the vibration of two-dimensional graded material with CCCC support conditions along with piezoelectric.

The resulting graphs from this table are as follows

The graph resulting from this table is as follows

Comparing the vibration of two-dimensional graded material with two piezoelectric layers with SSSS, SCSC, SSSC, SSCC, SCCC and CCCC boundary conditions.

RESULTS

Fig. 8: Boundary condition in the state.

According to the diagram above, in a specific state (here for), the percent increase and decrease in frequency caused by changes in the piezoelectric thickness is almost constant and uniform. So that all six lines in the above diagram move parallel to each other. Investigating the causes of errors 1- The classical theory of sheets is only valid for thin sheets. In this theory, shear force and rotational inertia are zero. 2- To consider zero and to simplify the equations. 3- Using the numerical method (GDQ) to solve equations: the causes of errors in this part include two parts becomes: a: The smaller the number of points in the grid of the sheet, the greater the error. b: In this research, the method is used to apply the boundary conditions. In this method, we apply the boundary conditions regardless of the delta distance from the edges.

CONCLUSION

According to the tables and charts, it was observed that: For all boundary conditions and different power coefficients, natural frequency decreases with increasing ratio ( ). They increase with increasing and frequencies. It should be noted that the frequency changes depend on the arrangement of metals and ceramics in the two-dimensional graduated functional material. As expected, in a certain case, the thickness of the functional sheet and the power coefficients are ordered from higher to lower according to the boundary conditions as follows: from CCCC, CCCS, SCSC, CCSS, SSSC and SSSS. The effect of the presence of two piezoelectric layers on the top and bottom of the sheet in the (close-circuit) mode is such that with the increase in thickness, the piezoelectric frequencies decrease up to a certain thickness and then increase. The piezo effect in a certain state of plate thickness and power coefficients is completely similar for different boundary conditions. This means that the frequencies decrease and then increase at a certain rate.

ACKNOWLEDGEMENT

We are grateful to all the dear professors for providing their information regarding this research.

CONFLICTS OF INTEREST

Conflicts of interest are declared obviously in the manuscript. Authors also state separately that they have all read the manuscript and have no conflict of interest.

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Article Info:

Academic Editor

Dr. Sonjoy Bishwas, Executive, Universe Publishing Group (UniversePG), California, USA

Received

September 18, 2023

Accepted

October 9, 2023

Published

October 20, 2023

Article DOI: 10.34104/bjah.02302430252

Corresponding author

Mohammad Sharafi*

English Language Teaching, Faculty of Humanities, Sepidan Azad University, Tehran, Iran

Cite this article

Sharafi M. (2023). Vibration analysis of two-dimensional functionally graded plate with piezoelectric layers using the classical theory of plates, Br. J. Arts Humanit., 5(5), 243-252. https://doi.org/10.34104/bjah.02302430252