Flexoelectric Actuation and Vibration Control of Ring Shells

2015 ◽  
Author(s):  
Bolei Deng ◽  
Huiyu Li ◽  
Hornsen Tzou
Keyword(s):  
Electric Field ◽  
Vibration Control ◽  
Contact Point ◽  
Flexible Structures ◽  
Bending Moment ◽  
Delta Function ◽  
Elastic Ring ◽  
Dirac Delta ◽  
Probe Radius ◽  
Afm Probe

The converse flexoelectric effect that the gradient of polarization (or electric field) induces internal stress (or strain) can be utilized to control the vibration of flexible structures. This study focuses on the microscopic actuation behavior and effectiveness of a flexoelectric actuator patch on an elastic ring. An atomic force microscope (AFM) probe is placed on the upper surface of the patch to implement the inhomogeneous electric field inducing stresses inside the actuation patch. The flexoelectric membrane force and bending moment, in turn, actuate the ring vibration and its actuation effect is studied. Actuator’s influence in the transverse and circumferential directions is respectively evaluated. For the transverse direction, the gradient of the electric field decays quickly along the ring thickness, resulting in a nonuniform transverse distribution of the induced stress and such distribution is not influenced by the patch thickness. The flexoelectric induced circumferential membrane force and bending moment resembles the Dirac delta function at the AFM contact point. The influence of the actuator can be regarded as a drastic bending on the ring. To evaluate the actuation effect, dynamic response of controllable displacements of the elastic ring under flexoelectric actuation is analyzed by adjusting the geometric parameters, such as the thickness of flexoelectric patch, AFM probe radius, ring thickness and ring radius. This study represents a thorough understanding of the flexoelectric actuation behavior and serves as a foundation of the flexoelectricity based vibration control.

Flexoelectric Actuation and Vibration Control of Ring Shells

2017 ◽  
Vol 139 (3) ◽  
Author(s):  
Hornsen Tzou ◽  
Bolei Deng ◽  
Huiyu Li
Keyword(s):  
Electric Field ◽  
Electric Field Gradient ◽  
Field Gradient ◽  
Design Parameters ◽  
Internal Stresses ◽  
Control Force ◽  
Elastic Ring ◽  
Dirac Delta ◽  
Control Moment ◽  
Afm Probe

The converse flexoelectric effect, i.e., the polarization (or electric field) gradient-induced internal stress (or strain), can be utilized to actuate and control flexible structures. This study focuses on the microscopic actuation behavior and effectiveness of a flexoelectric actuator patch laminated on an elastic ring shell. An atomic force microscope (AFM) probe is placed on the upper surface of the flexoelectric patch to induce an inhomogeneous electric field resulting in internal stresses of the actuator patch. The flexoelectric stress-induced membrane control force and bending control moment regulate the ring vibration and their actuation mechanics, i.e., transverse and circumferential control actions, are, respectively, studied. For the transverse direction, the electric field gradient quickly decays along the ring thickness, resulting in a nonuniform transverse distribution of the induced stress, and this distribution profile is not influenced by the actuator thickness. The flexoelectric-induced circumferential membrane control force and bending control moment resemble the Dirac delta functions at the AFM contact point. The flexoelectric actuation can be regarded as a localized drastic bending to the ring. To evaluate the actuation effect, dynamic responses and controllable displacements of the elastic ring with flexoelectric actuations are analyzed with respect to design parameters, such as the flexoelectric patch thickness, AFM probe radius, ring thickness, and ring radius.

Analytical and Experimental Studies of Flexoelectric Beam Control

2016 ◽  
Author(s):  
Xufang Zhang ◽  
Huiyu Li ◽  
Hornsen Tzou
Keyword(s):  
Electric Field ◽  
Design Parameters ◽  
Analytical Prediction ◽  
Probe Radius ◽  
Flexoelectric Effect ◽  
Control Moment ◽  
Probe Location ◽  
Afm Probe ◽  
Fixed End ◽  
Tip Displacement

Flexoelectricity includes two effects: the direct flexoelectric effect and the converse flexoelectric effect, which can be respectively applied to flexoelectric sensors and actuators to monitor structural dynamic behaviors or to control structural vibrations. This study focuses on the converse flexoelectric effect and its application to dynamic control of cantilever beams analytically and experimentally. In the mathematical model, a conductive atomic force microscope (AFM) probe with an external voltage is used to generate an inhomogeneous electric field driving the flexoelectric beam. The electric field gradient leads to an actuation stress in the longitudinal direction due to the converse flexoelectric effect. The actuation stress results in a bending control moment to the flexoelectric beam since the stress in the thickness is inhomogeneous. In order to evaluate the actuation effect of the flexoelectric actuator, the flexoelectric induced tip displacement is evaluated when the mechanical force is assumed zero. With the induced control moment, vibration control of the cantilever beam is discussed and the control effect is evaluated. Flexoelectric control effects with different design parameters, such as AFM probe location, AFM probe radius and flexoelectric beam thickness, are evaluated. Analytical results show that the optimal AFM probe location for all beam modes is close to the fixed end. Besides, thinner AFM probe radius and thinner flexoelectric beam enhance the control effects. Laboratory experiments are also conducted with different probe locations to validate the analytical predictions. Experimental results show that the induced tip displacement decreases when the input location moves away from the fixed end, which is consistent with the analytical prediction. The studies provide design guidelines of flexoelectric actuations in engineering applications.

Dynamic Analysis of Sandwich Beams Filled with ER Elastomers

2011 ◽  
Vol 50-51 ◽  
pp. 843-848 ◽  
Author(s):  
Quan Bai ◽  
Ke Xiang Wei ◽  
Wen Ming Zhang
Keyword(s):  
Finite Element ◽  
Electric Field ◽  
Vibration Control ◽  
Natural Frequencies ◽  
Flexible Structures ◽  
Sandwich Beam ◽  
Element Model ◽  
Sandwich Beams ◽  
Simulation Results ◽  
Er Fluids

Considered electrorheological (ER) elastomers as the visco-elasticity material, a finite element model of a sandwich beam filled with ER elastomers was developed based on Hamilton’s principle and sandwich beam’s theory. Then its dynamic characteristics were analyzed. Simulation results show that natural frequencies of the sandwich beam increase and vibration amplitudes of the beam decrease as the intensity of applied electric field increases. Increased the thickness of the ER elastomers layer, natural frequencies of the beam decrease and loss factors increase. Those indicate that the dynamic characteristic of ER elastomers sandwich beams is similar as that of ER fluids beam, which can be used for vibration control of flexible structures by applied a electric field.

Comparison of Flexoelectric and Piezoelectric Actuating of Beams

2020 ◽  
Author(s):  
Mu Fan
Keyword(s):  
Electric Field ◽  
Electric Field Gradient ◽  
Vibration Control ◽  
Case Studies ◽  
Field Gradient ◽  
Patch Size ◽  
Flexoelectric Effect ◽  
Control Moment ◽  
Modal Force ◽  
Afm Probe

Abstract The flexoelectric and piezoelectric effect on the actuating of a cantilever beam are compared in this study to explore how the size-dependent effect could affect the application of the flexoelectric effect. An AFM (atomic force microscopy) probe is used to generate the electric field in the flexoelectric patch, significant electric field gradient is induced. The electric field, distribution of control moment, induced modal force and the vibration control efficiency in terms of transverse displacements are analyzed in case studies. Analytical results show that the control moment of flexoelectric effect highly concentrates at the location of the AFM probe due to the inhomogeneous electric field, which shrink the effect area of flexoelectric patch size. The distribution of the flexoelectric control moment is an impulse function and the distribution of the piezoelectric control moment is a step function, which results to the flexoelectric modal force strongly affected by the electric field gradient while the piezoelectric modal force highly depends on the patch size. For the flexoelectric actuating, decreasing the AFM probe radius can increase the electric field gradient and induce larger modal force. The thickness effect of flexoelectric patch depends on the electric field gradient and the control moment arm, and in the current study, increasing the patch size, the induced flexoelectric modal force increases slightly. Case studies on vibration control show that both the flexoelectric actuating and piezoelectric actuating could generate larger transverse tip displacement with increasing the patch size. This study proves that the flexoelectric actuating can provide effectively actuating and control effect to engineering structures when the size decreases.

Static Nano-Control of Cantilever Beams Using the Inverse Flexoelectric Effect

2011 ◽  
Author(s):  
S. D. Hu ◽  
H. Li ◽  
H. S. Tzou
Keyword(s):  
Electric Field ◽  
Cantilever Beam ◽  
Mechanical Strain ◽  
Effective Control ◽  
Bottom Surface ◽  
Control Force ◽  
Dirac Delta ◽  
Flexoelectric Effect ◽  
Afm Probe ◽  
Converse Piezoelectric Effect

Flexoelectricity, an electromechanical coupling effect, exhibits two opposite electromechanical properties. One is the direct flexoelectric effect that mechanical strain gradient induces an electric polarization (or electric field); the other is the inverse flexoelectric effect that polarization (or electric field) gradient induces internal stress (or strain). The later can serve as an actuation mechanism to control the static deformation of flexible structures. This study focuses on an application of the inverse flexoelectric effect to the static displacement control of a cantilever beam. The flexoelectric layer is covered with an electrode layer on the bottom surface and an AFM probe tip on the top surface in order to generate an inhomogeneous electric field when powered. The control force induced by the inverse flexoelectric effect is evaluated and its spatial distribution resembles a Dirac delta function. Therefore, a “buckling” characteristic happens at the contact point of the beam under the inverse flexoelectric control. The deflection results of the cantilever beam with respect to the AFM probe tip radius indicate that a smaller AFM probe tip achieves a more effective control effect. To evaluate the control effectiveness, the flexoelectric deflections are also compared with those resulting from the converse piezoelectric effect. It is evident that the inverse flexoelectric effect provides much better localized static deflection control of.flexible beams.

scholarly journals Using Fractal Calculus to Express Electric Potential and Electric Field in Terms of Staircase and Characteristic Functions

2020 ◽  
Vol 13 (1) ◽  
pp. 19-32
Author(s):  
Nasibeh Delfan ◽  
Amir Pishkoo ◽  
Mahdi Azhini ◽  
Maslina Darus
Keyword(s):  
Characteristic Function ◽  
Electric Field ◽  
Electric Potential ◽  
Discrete Distribution ◽  
Delta Function ◽  
Dirac Delta Function ◽  
Characteristic Functions ◽  
Dirac Delta ◽  
Fractal Calculus ◽  
And Function

The Dirac Delta function is usually used to express the discrete distribution of electric charges in electrostatic problems. The integration of the product of the Dirac Delta function and the Green functions can calculate the electric potential and the electric field. Using fractal calculus, characteristic function, $\chi_{C_{n}}(x)$, as an alternative for dirac delta function is used to describe Cantor set charge distribution which is typical example of a discrete set. In these cases we deal with $F^{\alpha}$-integration and $F^{\alpha}$-derivative of the product of characteristic function and function of staircase function, namely $f(S^{\alpha}_{C_{n}}(x))$, which lead to calculation of electric potential and electric field. Recently, a calculus based fractals, called F$^{\alpha}$-calculus, has been developed which involve F$^{\alpha}$-integral and F$^{\alpha}$-derivative, of orders $\alpha$, $0<\alpha<1$, where $\alpha$ is dimension of $F$. In F$^{\alpha}$-calculus the staircase function and characteristic function have special roles. Finally, using COMSOL Multiphysics software we solve ordinary Laplace's equation (not fractional) in the fractal region with Koch snowflake boundary which is non-differentiable fractal, and give their graphs for the three first iterations.

scholarly journals Using Fractal Calculus to Express Electric Potential and Electric Field in Terms of Staircase and Characteristic Functions

2020 ◽  
Vol 13 (1) ◽  
pp. 19-32
Author(s):  
Nasibeh Delfan ◽  
Amir Pishkoo ◽  
Mahdi Azhini ◽  
Maslina Darus
Keyword(s):  
Characteristic Function ◽  
Electric Field ◽  
Electric Potential ◽  
Discrete Distribution ◽  
Delta Function ◽  
Dirac Delta Function ◽  
Characteristic Functions ◽  
Dirac Delta ◽  
Fractal Calculus ◽  
And Function

The Dirac Delta function is usually used to express the discrete distribution of electric charges in electrostatic problems. The integration of the product of the Dirac Delta function and the Green functions can calculate the electric potential and the electric field. Using fractal calculus, characteristic function, $\chi_{C_{n}}(x)$, as an alternative for dirac delta function is used to describe Cantor set charge distribution which is typical example of a discrete set. In these cases we deal with $F^{\alpha}$-integration and $F^{\alpha}$-derivative of the product of characteristic function and function of staircase function, namely $f(S^{\alpha}_{C_{n}}(x))$, which lead to calculation of electric potential and electric field. Recently, a calculus based fractals, called F$^{\alpha}$-calculus, has been developed which involve F$^{\alpha}$-integral and F$^{\alpha}$-derivative, of orders $\alpha$, $0<\alpha<1$, where $\alpha$ is dimension of $F$. In F$^{\alpha}$-calculus the staircase function and characteristic function have special roles. Finally, using COMSOL Multiphysics software we solve ordinary Laplace's equation (not fractional) in the fractal region with Koch snowflake boundary which is non-differentiable fractal, and give their graphs for the three first iterations.

Dirac delta regularization

2020 ◽  
Author(s):  
Matheus Pereira Lobo
Keyword(s):  
Delta Function ◽  
Dirac Delta Function ◽  
Dirac Delta ◽  
Finite Result

I present a finite result for the Dirac delta "function."

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