Method for analyzing power generation performance of single and double crystal piezoelectric cantilever beam under different shapes

Abstract

The application provides a power generation performance analysis method for single and double wafer piezoelectric cantilever beams in different shapes, and belongs to the technical field of vibration power supply; solves the problem of low piezoelectric cantilever beam power generation efficiency of the existing single rectangular piezoelectric sheet; and comprises the following steps: S1: five piezoelectric sheets with the same length and thickness but different shapes are formulated; S2: stress of the five piezoelectric sheets with different shapes is analyzed; and S3: COMSOL simulation software is used to set material properties and sizes, and static force analysis, modal analysis and harmonic response analysis are performed on the five cantilever beams with different shapes; the application is applied to piezoelectric sheet power generation performance research.

Description

Technical Field

[0001] This invention provides a method for analyzing the power generation performance of single and dual crystal piezoelectric cantilever beams under different shapes, belonging to the field of vibration power supply technology. Background Technology

[0002] Existing battery-powered systems are increasingly polluting the environment, and subsequent treatment is quite troublesome. Therefore, the development of vibration-powered systems is needed. Current piezoelectric cantilever beams only utilize a single rectangular piezoelectric sheet for power generation, and their efficiency is not particularly high, usually requiring combination with electromagnetic, electrostatic, or solar power. Currently, single piezoelectric cantilever beams are simple and convenient, and suitable for low-to-medium frequency applications and situations with less demanding system performance requirements. Therefore, to investigate how the structure, shape, and connection method of different piezoelectric sheets specifically affect the performance of piezoelectric cantilever beams, a method for analyzing the power generation performance of single and bicrystalline piezoelectric cantilever beams under different shapes is proposed. Summary of the Invention

[0003] To address the problem of low power generation efficiency of existing piezoelectric cantilever beams using a single rectangular piezoelectric sheet, this invention proposes a method for analyzing the power generation performance of single and dual-crystal piezoelectric cantilever beams under different shapes.

[0004] To solve the above-mentioned technical problems, the technical solution adopted by the present invention is as follows: a method for analyzing the power generation performance of single and dual crystal piezoelectric cantilever beams under different shapes, comprising the following steps:

[0005] S1: Design five types of piezoelectric sheets with the same length and thickness but different shapes;

[0006] S2: Stress analysis of five different piezoelectric sheet shapes;

[0007] S3: Using COMSOL simulation software, set material properties and dimensions, and perform static analysis, modal analysis, and harmonic response analysis on five different shapes of cantilever beams.

[0008] The five different shapes of piezoelectric sheets in step S1 are: rectangle, right trapezoid, isosceles trapezoid, triangle, and inverted trapezoid. The length of the rectangle, right trapezoid, isosceles trapezoid, triangle, and inverted trapezoid is L in mm, the width of the bottom is w in mm, and the width of the top is w, 1 / 2w, 1 / 2w, 0, and 2w in mm, respectively.

[0009] The stress analysis steps for the five different piezoelectric sheet shapes in step S2 are as follows:

[0010] Based on the static equilibrium equations, a spatial coordinate system is established, where the bending moment of the piezoelectric element after applying a pressure P at a distance x is:

[0011] My(x) = -P(xL), where L is the length of the piezoelectric sheet of different shapes;

[0012] The maximum bending moment of the piezoelectric element after applying pressure P is: My(x) = P * L;

[0013] Let the range of action of the uniform load Q be B, then the bending moment at the center of action of the uniform load Q is:

[0014] My(x) = Q*B*L;

[0015] When the uniformly distributed load B = L across the full span has a moment of action of L / 2, the bending moment is: My(x) = Q * L * L / 2;

[0016] Based on the second kind of piezoelectric equation and its boundary conditions, the relationships between stress T and strain S of the piezoelectric element are as follows:

[0017]

[0018] Where T = cS, S = sT;

[0019] Strain S and electric field intensity E are independent variables, stress T and electric displacement D are dependent variables, and e represents the piezoelectric stress constant, which indicates the change in stress component caused by a unit change in electric field intensity under constant strain conditions, or the change in electric displacement component caused by a unit change in strain component under constant electric field conditions. The units are N / Vm or C / m. 2 ε is the dielectric constant obtained under constant strain, called the clamping dielectric constant; c is the stiffness coefficient obtained under constant electric field, called the short-circuit stiffness coefficient; s is the elastic compliance constant; i, j, and k are the unit vectors of the x, y, and z axes, respectively.

[0020] The strain energy and strain of a cantilever beam under pure bending are:

[0021]

[0022]

[0023] In the above formula: V is the volume, W is the strain energy, M is the mass, θ is the bending angle corresponding to the x direction, l is the length of the cantilever beam, I is the moment of inertia, X is the stress, and z is the deflection;

[0024] From the approximate differential equation of the deflection curve of the cantilever beam, we get:

[0025] In the above formula: E1I1 is the bending stiffness of the upper cantilever beam, and E2I2 is the bending stiffness of the lower cantilever beam;

[0026] Let I be the moment of inertia of the piezoelectric cantilever beam about the y-direction, then we get:

[0027]

[0028]

[0029]

[0030]

[0031] In the above formula: b k Let b be the thickness of the cantilever beam. p Let be the thickness of the piezoelectric element, z be the equivalent moment of inertia in the z-direction, k = E2 / E1, w be the width of the rectangle, x be the length of the cantilever beam, and the moments of inertia of the isosceles trapezoid and the right trapezoid are equal, both being I. 梯 .

[0032] The piezoelectric sheet simulated in step S3 includes monocrystalline piezoelectric sheets and bicrystalline piezoelectric sheets.

[0033] The static analysis of the piezoelectric cantilever beam in step S3 involves simulating the changes in stress, strain, displacement, and voltage of five different shapes of single-crystal and double-crystal cantilever beams in steady-state mode.

[0034] The modal analysis of the piezoelectric cantilever beam in step S3 involves simulation experiments on the first-order natural frequencies of five different shapes of monocrystalline and bicrystalline cantilever beams.

[0035] The harmonic response analysis of the piezoelectric cantilever beam in step S3 involves applying a sinusoidal load signal to five different shapes of single-crystal and double-crystal cantilever beams and observing the peak voltage output.

[0036] The beneficial effects of this invention compared to the prior art are as follows: The method for analyzing the power generation performance of single and double crystal piezoelectric cantilever beams under different shapes provided by this invention conducts theoretical analysis and simulation experiments on five different piezoelectric sheet shapes with the same length and thickness, analyzes the various shapes of piezoelectric sheets to observe the various performances of different piezoelectric sheets, and provides a good test basis for improving the power generation output efficiency of piezoelectric cantilever beams in the future. Attached Figure Description

[0037] The present invention will be further described below with reference to the accompanying drawings:

[0038] Figure 1 These are schematic diagrams of the structures of the five different piezoelectric sheets proposed in this invention;

[0039] Figure 2 These are three views of the piezoelectric element of the present invention;

[0040] Figure 3 The figures show the static analysis results of the monocrystalline cantilever beams of five different shapes according to the present invention.

[0041] Figure 4 The figures show the static analysis results of the bicrystalline cantilever beams under five different shapes according to the present invention.

[0042] Figure 5 The diagram shows the modal analysis results of the monocrystalline cantilever beams of five different shapes according to the present invention.

[0043] Figure 6 The diagram shows the modal analysis results of the bicrystalline cantilever beams under five different shapes according to the present invention.

[0044] Figure 7 The graph shows the output performance results of the single-wafer cantilever beam under five different shapes according to the present invention.

[0045] Figure 8 The figures show the output performance results of the bicrystalline cantilever beam under five different shapes according to the present invention. Detailed Implementation

[0046] like Figures 1 to 8 As shown, this invention provides a method for analyzing the power generation performance of single and double crystal piezoelectric cantilever beams under different shapes, studies the power generation efficiency of piezoelectric sheets of different shapes, and establishes five piezoelectric sheet shapes with the same length and thickness but different shapes, such as... Figure 1 As shown, the shapes are rectangles, right trapezoids, isosceles trapezoids, triangles, and inverted trapezoids, respectively. The length of the rectangle, right trapezoid, isosceles trapezoid, triangle, and inverted trapezoid is L, the width of the top is w, 1 / 2w, 1 / 2w, 0, and 2w, respectively, and the width of the bottom is w. All units are mm.

[0047] Then, the stress of the five piezoelectric sheets mentioned above is analyzed. Based on the static equilibrium equation, a spatial coordinate system is established, where the bending moment of the piezoelectric sheet after applying a pressure P at a distance x is:

[0048] My(x) = -P(xL), where L is the length of the piezoelectric sheet of different shapes;

[0049] The maximum bending moment of the piezoelectric element after applying pressure P is: My(x) = P * L;

[0050] Let the range of action of the uniform load Q be B, then the bending moment at the center of action of the uniform load Q is:

[0051] My(x) = Q*B*L;

[0052] When the uniformly distributed load B = L across the full span has a moment of action of L / 2, the bending moment is: My(x) = Q * L * L / 2;

[0053] Based on the second kind of piezoelectric equation and its boundary conditions, the relationships between stress T and strain S of the piezoelectric element are as follows:

[0054]

[0055] Where T = cS, S = sT;

[0056] Strain S and electric field intensity E are independent variables, stress T and electric displacement D are dependent variables, and e represents the piezoelectric stress constant, which indicates the change in stress component caused by a unit change in electric field intensity under constant strain conditions, or the change in electric displacement component caused by a unit change in strain component under constant electric field conditions. The units are N / Vm or C / m. 2 ε is the dielectric constant obtained under constant strain, called the clamping dielectric constant; c is the stiffness coefficient obtained under constant electric field, called the short-circuit stiffness coefficient; s is the elastic compliance constant; i, j, and k are the unit vectors of the x, y, and z axes, respectively.

[0057] The strain energy and strain of a cantilever beam under pure bending are:

[0058]

[0059]

[0060] In the above formula: V is the volume, W is the strain energy, M is the mass, θ is the bending angle corresponding to the x direction, l is the length of the cantilever beam, I is the moment of inertia, X is the stress, and z is the deflection;

[0061] From the approximate differential equation of the deflection curve of the cantilever beam, we get:

[0062] In the above formula: E1I1 is the bending stiffness of the upper cantilever beam, and E2I2 is the bending stiffness of the lower cantilever beam;

[0063] Let I be the moment of inertia of the piezoelectric cantilever beam about the y-direction, then we get:

[0064]

[0065]

[0066]

[0067]

[0068] In the above formula: b k Let b be the thickness of the cantilever beam. p Let be the thickness of the piezoelectric element, z be the equivalent moment of inertia in the z-direction, k = E2 / E1, w be the width of the rectangle, x be the length of the cantilever beam, and the moments of inertia of the isosceles trapezoid and the right trapezoid are equal, both being I. 梯 .

[0069] From the above formula, we can see that strain is inversely proportional to moment of inertia. Based on the moments of inertia of piezoelectric sheets of different shapes, the triangular piezoelectric sheet has the largest strain, followed by the isosceles trapezoid, right trapezoid, and inverted trapezoid, while the rectangular sheet has the smallest strain.

[0070] COMSOL simulation software is used for simulation experiments. COMSOL Multiphysics is a large-scale, advanced numerical simulation software widely used in scientific research and engineering calculations across various fields to simulate various physical processes in science and engineering. Based on the finite element method, it simulates real physical phenomena by solving partial differential equations. It uses mathematical methods to solve real-world physical phenomena. Solving multi-field problems in COMSOL is equivalent to solving systems of equations. Users can easily achieve direct coupling analysis of multiphysics fields by simply selecting or customizing partial differential equations from different disciplines and combining them arbitrarily. COMSOL has general modules, additional modules (electromagnetism, structural mechanics & acoustics, fluid flow & heat transfer, chemical engineering, multi-functional modules, interface products, etc.).

[0071] First, set the material properties and dimensions in the software. The settings for the material property parameters and dimensions are shown in Table 1 below:

[0072]

[0073] Table 1. Material property parameters, size and dimensions.

[0074] Five different cantilever beams were compared in the following three ways.

[0075] (1) Static analysis of piezoelectric cantilever beam

[0076] Static analysis studies the changes in stress, strain, displacement, and voltage of a structure. Simulation results only present the results under steady-state conditions. The static analysis results for a single-crystal cantilever beam and a bicrystalline cantilever beam are shown below. Figure 3 and Figure 4 As shown in the figure, under the same applied load, the triangular piezoelectric sheet exhibits the largest strain, followed by the isosceles trapezoid, right trapezoid, and inverted trapezoid, with the rectangular sheet showing the smallest strain. The right trapezoid has smaller deformation than the isosceles trapezoid because the analysis is performed using the center point of the cantilever beam's fixed end, while the upper base of the right trapezoid is to the left of the center point, resulting in smaller deformation. By comparing the stress distribution diagrams of single and bicrystalline sheets, it can be concluded that the strain of the bicrystalline sheet is twice that of the single-sheet sheet, consistent with the theoretical calculations above.

[0077] (2) Modal analysis of piezoelectric cantilever beam

[0078] Modal analysis refers to the analysis of the natural frequencies of a cantilever beam. Lower-order modes have relatively weak modal stiffness, and under the same magnitude of excitation, their responses will have a relatively larger weight. Theoretically, lower-order mode theory is also relatively mature. Therefore, the first-order natural frequencies are generally studied. Table 2 below shows the natural frequencies of single-crystal piezoelectric elements, and Table 3 shows the natural frequencies of bicrystalline piezoelectric elements.

[0079] rectangle right trapezoid isosceles trapezoid triangle Inverted trapezoid Natural frequency 17.173Hz 16.266Hz 16.664Hz 15.202Hz 21.497Hz

[0080] Table 2. Natural frequencies of single-crystal piezoelectric elements;

[0081] rectangle right trapezoid isosceles trapezoid triangle Inverted trapezoid Natural frequency 20.706Hz 19.478Hz 20.22Hz 17.793Hz 28.778Hz

[0082] Table 3. Natural frequencies of bicrystalline piezoelectric elements.

[0083] Depend on Figure 5 , Figure 6 As shown in Tables 2 and 3, the natural frequencies of piezoelectric cantilever beams with different shapes are not significantly different. However, for the same length and thickness, the narrower the width, the lower the natural frequency, and vice versa. Furthermore, the natural frequency of a single-crystal piezoelectric beam is always slightly lower than that of a double-crystal piezoelectric beam. This is because, for the same shape, the thickness of a double-crystal piezoelectric beam is twice that of a single-crystal piezoelectric beam; the greater the thickness, the higher the natural frequency. Among the five shapes of single and double piezoelectric beams, it is evident that the first-order natural frequency of the triangular piezoelectric beam is the smallest, indicating that the triangular piezoelectric beam is the easiest to reach a resonant state.

[0084] (3) Harmonic response analysis of piezoelectric cantilever beam

[0085] Harmonic response analysis of a piezoelectric cantilever beam involves applying a sinusoidal load signal to the structure and observing its voltage peak value. Figure 7 and Figure 8 The harmonic response analysis results for monocrystalline and bicrystalline piezoelectric cantilever beams are presented. Analysis of five different bicrystalline beam shapes reveals that, compared to monocrystalline beams, bicrystalline beams have a higher output voltage and a slightly higher natural frequency. This suggests that bicrystalline beams are a better option for future research on piezoelectric cantilever beams. Figure 7 and Figure 8 The results show that the output voltage of different piezoelectric elements is the largest for the triangle, followed by the smallest for the isosceles trapezoid, inverted trapezoid, rectangle, and right trapezoid.

[0086] This invention provides the optimal output voltage at the first-order natural frequency, from Figure 7 and Figure 8 It can be seen that the power generation performance of the piezoelectric cantilever beam is at its maximum voltage at the resonant frequency. Around the resonant frequency, its power generation performance curve drops rapidly, and the rate of decline gradually slows down. The strain at different frequencies also varies, mainly depending on the shape and width of the piezoelectric element.

[0087] Regarding the specific structure of this invention, it should be noted that the connection relationships between the various component modules used in this invention are definite and achievable. Except as specifically described in the embodiments, their specific connection relationships can bring about corresponding technical effects and solve the technical problems proposed by this invention without relying on the execution of corresponding software programs. The models of the components, modules, and specific components appearing in this invention, the connection methods between them, and the conventional usage methods and expected technical effects brought about by the above technical features, unless specifically described, are all publicly disclosed content in patents, journal articles, technical manuals, technical dictionaries, and textbooks that can be obtained by those skilled in the art before the application date, or belong to conventional technology, common knowledge, and other existing technologies in this field. There is no need to elaborate, which makes the technical solution provided in this case clear, complete, and achievable, and can reproduce or obtain corresponding physical products based on this technical means.

[0088] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some or all of the technical features; and these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A method for analyzing the power generation performance of single and double crystal piezoelectric cantilever beams under different shapes, characterized in that: Includes the following steps: S1: Design five types of piezoelectric sheets with the same length and thickness but different shapes; S2: The stress of five different piezoelectric sheets was analyzed, and the steps are as follows: Based on the static equilibrium equations, a spatial coordinate system is established, where the bending moment of the piezoelectric element after applying a pressure P at a distance x is: , where L is the length of the piezoelectric sheet of different shapes; The maximum bending moment of the piezoelectric element after applying pressure P is: ; Let the range of action of the uniform load Q be B, then the bending moment at the center of action of the uniform load Q is: ； When the uniformly distributed load B=L across the full span and the moment of action is L / 2, the bending moment is: ; Based on the second kind of piezoelectric equation and its boundary conditions, the relationships between stress T and strain S of the piezoelectric element are as follows: ； Where T = cS, S = sT; Strain S and electric field intensity E are independent variables, stress T and electric displacement D are dependent variables, and e represents the piezoelectric stress constant, which indicates the change in stress component caused by a unit change in electric field intensity under constant strain conditions, or the change in electric displacement component caused by a unit change in strain component under constant electric field conditions. The units are N / Vm or C / m. 2 ε is the dielectric constant obtained under constant strain, called the clamping dielectric constant; c is the stiffness coefficient obtained under constant electric field, called the short-circuit stiffness coefficient; s is the elastic compliance constant; i, j, and k are the unit vectors of the x, y, and z axes, respectively. The strain energy and strain of the cantilever beam under pure bending are: ； ； In the above formula: V is the volume, W is the strain energy, M is the mass, θ is the bending angle corresponding to the x-direction, l is the length of the cantilever beam, I is the moment of inertia, and X is the stress. For deflection; From the approximate differential equation of the deflection curve of the cantilever beam, we get: ; In the above formula: E1I1 is the bending stiffness of the upper cantilever beam, and E2I2 is the bending stiffness of the lower cantilever beam; Let I be the moment of inertia of the piezoelectric cantilever beam about the y-direction, then we get: ； ； ； ； In the above formula: b k Let b be the thickness of the cantilever beam. p Let be the thickness of the piezoelectric element, z be the deformation equivalent moment of inertia in the z-direction, k = E2 / E1, w be the width of the rectangle, and x be the length of the cantilever beam. The moments of inertia of the isosceles trapezoid and the right trapezoid are equal. ; S3: Using COMSOL simulation software, set material properties and dimensions, and perform static analysis, modal analysis, and harmonic response analysis on five different shapes of cantilever beams.

2. The method for analyzing the power generation performance of single and double crystal piezoelectric cantilever beams under different shapes according to claim 1, characterized in that: The five different shapes of piezoelectric sheets in step S1 are: rectangle, right trapezoid, isosceles trapezoid, triangle, and inverted trapezoid. The length of the rectangle, right trapezoid, isosceles trapezoid, triangle, and inverted trapezoid is L in mm, the width of the bottom is w in mm, and the width of the top is w, 1 / 2w, 1 / 2w, 0, and 2w in mm, respectively.

3. The method for analyzing the power generation performance of single and dual crystal piezoelectric cantilever beams under different shapes according to claim 2, characterized in that: The piezoelectric sheet simulated in step S3 includes monocrystalline piezoelectric sheets and bicrystalline piezoelectric sheets.

4. The method for analyzing the power generation performance of a single- or dual-crystal piezoelectric cantilever beam under different shapes according to claim 3, characterized in that: The static analysis of the piezoelectric cantilever beam in step S3 involves simulating the changes in stress, strain, displacement, and voltage of five different shapes of single-crystal and double-crystal cantilever beams in steady-state mode.

5. The method for analyzing the power generation performance of a single- or dual-crystal piezoelectric cantilever beam under different shapes according to claim 3, characterized in that: The modal analysis of the piezoelectric cantilever beam in step S3 involves simulation experiments on the first-order natural frequencies of five different shapes of monocrystalline and bicrystalline cantilever beams.

6. The method for analyzing the power generation performance of single and double crystal piezoelectric cantilever beams under different shapes according to claim 1, characterized in that: The harmonic response analysis of the piezoelectric cantilever beam in step S3 involves applying a sinusoidal load signal to five different shapes of single-crystal and double-crystal cantilever beams and observing the peak voltage output.

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