scholarly journals A Multi-Parameter Perturbation Solution for Functionally Graded Piezoelectric Cantilever Beams under Combined Loads

Materials ◽  
2018 ◽  
Vol 11 (7) ◽  
pp. 1222 ◽  
Author(s):  
Yongsheng Lian ◽  
Xiaoting He ◽  
Sijie Shi ◽  
Xue Li ◽  
Zhixin Yang ◽  
...  
Keyword(s):  
Cantilever Beam ◽  
Functionally Graded ◽  
Airy Stress Function ◽  
Parameter Perturbation ◽  
Combined Loads ◽  
Piezoelectric Cantilever ◽  
Basic Equations ◽  
Perturbation Parameters ◽  
Functionally Graded Piezoelectric ◽  
Parameter Perturbation Method

In this study, we use a multi-parameter perturbation method to solve the problem of a functionally graded piezoelectric cantilever beam under combined loads, in which three piezoelectric coefficients are selected as the perturbation parameters. First, we derive the two basic equations concerning the Airy stress function and electric potential function. By expanding the unknown Airy stress function and electric potential function with respect to three perturbation parameters, the two basic equations were decoupled, thus obtaining the corresponding multi-parameter perturbation solution under boundary conditions. From the solution obtained, we can see clearly how the piezoelectric effects influence the behavior of the functionally graded piezoelectric cantilever beam. Based on a numerical example, the variations of the elastic stresses and displacements as well as the electric displacements of the cantilever beam under different gradient exponents were shown. The results indicate that if the pure functionally graded cantilever beam without a piezoelectric effect is regarded as an unperturbed system, the functionally graded piezoelectric cantilever beam can be looked upon as a perturbed system, thus opening the possibilities for perturbation solving. Besides, the proposed multi-parameter perturbation method provides a new idea for solving similar nonlinear differential equations.

scholarly journals Application of Multi-Parameter Perturbation Method to Functionally-Graded, Thin, Circular Piezoelectric Plates

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 342 ◽  
Author(s):  
Xiao-Ting He ◽  
Zhi-Xin Yang ◽  
Yang-Hui Li ◽  
Xue Li ◽  
Jun-Yi Sun
Keyword(s):  
Perturbation Method ◽  
Electrical Potential ◽  
Functionally Graded ◽  
Parameter Perturbation ◽  
Graded Materials ◽  
Cylindrical Coordinate ◽  
Stress Functions ◽  
Perturbation Parameters ◽  
Parameter Perturbation Method ◽  
Piezoelectric Plates

In this study, a multi-parameter perturbation method is used for the solution of a functionally-graded, thin, circular piezoelectric plate. First, by assuming that elastic, piezoelectric, and dielectric coefficients of the functionally-graded materials vary in the form of the same exponential function, the basic equation expressed in terms of two stress functions and one electrical potential function are established in cylindrical coordinate system. Three piezoelectric coefficients are selected as perturbation parameters, and the established equations are solved by the multi-parameter perturbation method, thus obtaining up to first-order perturbation solutions. The validity of the perturbation solution obtained is verified by numerical simulations, based on layer-wise theory. The perturbation process indicates that adopting three piezoelectric coefficients as perturbation parameters follows the basic idea of perturbation theory—i.e., if the piezoelectricity may be regarded as a kind of introduced disturbance, the zero-order solution of the disturbance system corresponds exactly to the solution of functionally-graded plates without piezoelectricity. The result also indicates that the deformation magnitude of piezoelectric plates is smaller than that of plates without piezoelectricity, due to the well-known piezoelectric stiffening effect.

scholarly journals One-Dimensional Theoretical Solution and Two-Dimensional Numerical Simulation for Functionally-Graded Piezoelectric Cantilever Beams with Different Properties in Tension and Compression

Polymers ◽  
2019 ◽  
Vol 11 (11) ◽  
pp. 1728 ◽  
Author(s):  
Xiao-Ting ◽  
Zhi-Xin ◽  
Hong-Xia ◽  
Jun-Yi
Keyword(s):  
Numerical Simulation ◽  
Cantilever Beam ◽  
Functionally Graded ◽  
Theoretical Solution ◽  
Two Dimensional ◽  
One Dimensional ◽  
Analysis And Design ◽  
Piezoelectric Cantilever ◽  
Tension And Compression ◽  
Functionally Graded Piezoelectric

The existing studies indicate polymers will present obviously different properties in tension and compression (bimodular effect) which is generally ignored because of the complexity of the analysis. In this study, a functionally graded piezoelectric cantilever beam with bimodular effect was investigated via analytical and numerical methods, respectively, in which a one-dimensional theoretical solution was derived by neglecting some unimportant factors and a two-dimensional numerical simulation was performed based on the model of tension-compression subarea. A full comparison was made to show the rationality of one-dimensional theoretical solution and two-dimensional numerical simulation. The result indicates that the layered model of tension-compression subarea also makes it possible to use numerical technique to simulate the problem of functionally graded piezoelectric cantilever beam with bimodular effect. Besides, the modulus of elasticity E* and the bending stiffness D* proposed in the one-dimensional problem may succinctly describe the piezoelectric effect on the classical mechanical problem without electromechanical coupling, which shows the advantages of one-dimensional solution in engineering applications, especially in the analysis and design of energy harvesting/sensing/actuating devices made of piezoelectric polymers whose bimodular effect is relatively obvious.

Exact Electroelastic Field of a Functionally Graded Piezoelectric Cantilever Beam Subjected to Pure Body Force Loading

2010 ◽  
Author(s):  
A. A. Emami ◽  
R. Hashemi ◽  
M. H. Kargarnovin ◽  
R. Naghdabadi
Keyword(s):  
Differential Equations ◽  
Body Force ◽  
Functionally Graded ◽  
The Body ◽  
Cantilever Beams ◽  
Exponential Distributions ◽  
Partial Differential ◽  
Numerical Studies ◽  
Piezoelectric Cantilever ◽  
Functionally Graded Piezoelectric

The electroelastic response of functionally graded piezoelectric cantilever beams which includes the effect of body force is presented in this paper. The material properties such as elastic compliance, piezoelectric and dielectric impermeability are assumed to be graded with different indices in the thickness direction according to exponential distributions. Systems of fourth order inhomogeneous partial differential equations (PDEs) which are satisfied by the stress and induction functions and involve the body force terms are derived. Spectral forms for electrical and mechanical variables in the x-axis are employed to convert the partial differential governing equations and the associated boundary conditions into sets of ordinary differential equations, and the resulting equations are solved in a closed form manner. Subsequently, in numerical studies, the effects of the material property graded indices are examined upon the electroelastic response of FGP cantilever beams under pure body force loadings.

scholarly journals A Multi-Parameter Perturbation Solution and Experimental Verification for Bending Problem of Piezoelectric Cantilever Beams

Polymers ◽  
2019 ◽  
Vol 11 (12) ◽  
pp. 1934 ◽  
Author(s):  
Zhi-Xin Yang ◽  
Xiao-Ting He ◽  
Hong-Xia Jing ◽  
Jun-Yi Sun
Keyword(s):  
Cantilever Beam ◽  
Perturbation Solution ◽  
Experimental Results ◽  
Cantilever Beams ◽  
Parameter Perturbation ◽  
Analysis And Design ◽  
Piezoelectric Polymers ◽  
Piezoelectric Cantilever ◽  
Piezoelectric Structure ◽  
Coupling Characteristics

The existing studies indicate that the application of piezoelectric polymers is becoming more and more extensive, especially in the analysis and design of sensors or actuators, but the problems of piezoelectric structure are usually difficult to solve analytically due to the force–electric coupling characteristics. In this study, the bending problem of a piezoelectric cantilever beam was investigated via theoretical and experimental methods. First, the governing equations of the problem were established and non-dimensionalized. Three piezoelectric parameters were selected as perturbation parameters and the perturbation solution of the equations was finally obtained using a multi-parameter perturbation method. In addition, the relevant experiments of the piezoelectric cantilever beam were carried out, and the experimental results were in good agreement with the theoretical solutions. Based on the experimental results, the effect of piezoelectric properties on the bending deformation of piezoelectric cantilever beams was analyzed and discussed. The results indicated that the multi-parameter perturbation solution obtained in this study is effective and it may serve as a theoretical reference for the design of sensors or actuators made of piezoelectric polymers.

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