Optimal design of piezoelectric cantilever velocity sensor based on PVDF

Abstract

The response charge of piezoelectric speed sensors using a conventional rectangular cantilever is low, which also causes a low sensitivity in speed measurement. To improve the sensor sensitivity, a piezoelectric speed sensor based on a streamlined piezoelectric cantilever is employed in this paper. Furthermore, a theoretical optimization model of the sensor based on Bernstein polynomial equation is established, and a simulation optimization flow work is also proposed. With method of moving asymptotes (MMA) algorithm, more charge output can be obtained than before. The simulation results show that the optimized sensor can output a voltage of 416 mV and obtain a sensitivity of 52 mV/m⋅s⁻¹ when the input speed is 8 m/s. As compared with the values of 300 mV and 37.5 mV/m⋅s⁻¹ in the un-optimized case, the improvement in the sensor sensitivity is up to 38%, which confirms the effectiveness of the proposed method.

Keywords

Piezoelectric velocity sensor sensitivity cantilever

1. Introduction

With the development of science and technology, the industry with fluid background is more and more extensive. More and more equipment, instruments and devices are integrated with fluid detection technology. From the aircraft in the air, the submersible in the water, the detection robot on the land and the ventilation pipe in the underground mine, to the quantitative detection of the blood flow velocity and pressure in the human blood vessels, to the accurate monitoring of the flow velocity and flow of the traditional Chinese medicine in the operation, to the current detection of the fluid characteristics (such as flow, pressure, etc.) in the “lab on chip (LOC)” microfluidic devices,they are all connected with the flow velocity detectiontechnology. The technique is closely connected. The precise detection of flow velocity can ensure the effective work of equipment and devices, and help medical staff reduce the risks of patients’ treatment. Therefore, flow rate detection has important application value [1–7]. Especially in the field of national defense detection, in fact, the change of flow velocity and direction in complex fluid environment often contains all kinds of effective information with the target, which is more hidden than the traditional detection through sonar, optics, etc. At the same time, it also fully explains the research significance brought by the detection of fluid velocity or flow direction.

At present, the flow rate sensor which uses intelligent material as energy exchange element develops rapidly. According to the different energy exchange elements, it mainly includes thermal flow sensor [8,9], capacitive flow sensor [10,11], and piezoelectric flow sensor [12–14]. The response time of piezoelectric velocity sensor is short, which can reach millisecond level, and it is suitable for dynamic measurement [15], at the same time, the device is more suitable for miniaturization. Structurally speaking, the piezoelectric flow sensor mainly utilizes the beam structure as the elastic support unit. After combining the piezoelectric layer with the elastic layer, it adopts the cantilever, fixed support, array and other structures. At the same time, combined with the micro processing technology. The volume of the device is also developing towards the direction of miniaturization [16,17]. In terms of materials, piezoelectric materials play an important part in the sensitivity, stability, response time and error of the device as the sensitive layer to external stimulation. How to prepare higher piezoelectric constant (mainly piezoelectric voltage coefficient (G33, G31) or piezoelectric charge coefficient (d33, d31) has attracted scholars’ attention [18–23].

The optimal design of polyvinylidene fluoride (PVDF) piezoelectric cantilever sensor mainly focuses on the optimization of the size, that is, the sensitivity of the sensor is optimized when the shape is certain (generally rectangular and triangular), so the optimization of the size can not guarantee the optimal output result in theory. Therefore, this paper will analyze the influence of design variables on the sensitivity of the sensor to establish the optimal model of PVDF piezoelectric cantilever velocity sensor.

2. Bending response characteristics of PVDF piezoelectric cantilever

According to direct piezoelectric effect, the strain of piezoelectric elastomer is directly converted into the charge. Firstly, the aerodynamic force is simplified as a uniform load applied on the surface of the beam, and the change of strain and output voltage in the length direction of the cantilever beam is analyzed, as shown in Fig. 1. As shown in the figure, the cantilever beam has a length of 10 mm, a width of 1 mm and a thickness of 0.07 mm, and an electrode layer is applied on the upper and lower surfaces of the piezoelectric elastomer. The results show that the strain of piezoelectric beam near the root is the largest (deformation displacement is the smallest), and the strain of the free end is the smallest (deformation displacement is the largest). As a whole, the strain in the length direction changes greatly (as shown in Fig. 1(c)). At the same time, the output voltage of the piezoelectric beam near the root is the largest, while the output voltage of the free end is the smallest. As a whole, the voltage change in the length direction is obvious. In particular,the output voltage of the root position is about 0.6 V, and that of the free end is about 0.06 V (as shown in Fig. 1(d)). It can be seen that if the shape of piezoelectric beam is designed as a rectangle, the sensitivity of the sensor can not be effectively improved, that is, the strain near the free end is too small, resulting in less output charge, and the strain distribution can not be fully utilized to generate more response charge. In this paper, the shape optimization method is used to improve the sensitivity of the sensor and the accuracy of the velocity sensor.

Fig. 1.

Vibration response characteristics of piezoelectric cantilever under uniform load: (a) Calculation model; (b) model cross section; (c) relationship between position and strain (ϵ) in X direction (cantilever length direction); (d) relationship between position and response voltage (U) in X direction (cantilever length direction).

3. Sensitivity optimization design of flow sensor

3.1. Optimization model

Figure 2 shows the structure of PVDF cantilever velocity sensor unit. The root of the cantilever is connected with the fixed base plate. According to the reference [24], the aerodynamic force is equivalent to the uniformly distributed load (f _wind). The sensitivity S _v of the air flow sensor isdeveloped as follows: $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}ll@{}}Q=\displaystyle \sum q=\displaystyle \int D_{z}\text{d}A & \\[12.0pt] U={\displaystyle \frac{Q}{C_{\text{P}}}} & \quad 1\text{b}\\[12.0pt] C_{P}={\displaystyle \frac{{\varepsilon}{\varepsilon}_{0}A}{h_{P}}} & \quad 1\text{c}\\[12.0pt] S_{v}={\displaystyle \frac{\text{U}}{v}} & \quad 1\text{d}\end{array}\right. & & \displaystyle\end{eqnarray}$ (1) where, q is the total response charge, C _P is the capacitance of piezoelectric film, ϵ is the relative dielectric constant, ϵ₀ is the vacuum dielectric constant, h _p is the plate spacing, v is the air flow velocity. It can be seen from (1) that the more responsive charge the piezoelectric cantilever generates, the greater the sensorsensitivity is.

Considering the stability of the cantilever when it vibrates, the beam model with variable cross-section is configured as a symmetrical structure, as shown in Fig. 2(a). For a cantilever beam with any cross-section, under the condition of controlling the total area of piezoelectric material. The gradient curve B (x) is defined as the curve basis function of variable cross-section. By coordinating the parameters C ₀, C ₁, C ₂, C ₃, C ₄ in the basis function, the cantilever beam can generate the largest response charge under the air flow load f _wind, and the response voltage can become the largest at the same time (according to formula (1(c)), the equivalent capacitance remain unchanged). The initial parameters of the optimization model are shown in Table 1. From this, the following relation expressions can be derived: $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}ll@{}} & \hspace{36.0pt}f\text{ind }\hspace{12.0pt}C_{0},C_{1},C_{2},C_{3},C_{4},l\\[6.0pt] \max & f(C_{0},C_{1},C_{2},C_{3},C_{4},l)=Q(C_{0},C_{1},C_{2},C_{3},C_{4},l)\\ & \hspace{72.0pt}s.t\quad A=\displaystyle \int _{0}^{l}B(x)\text{d}x\\ & {\displaystyle \frac{\text{d}B(x)}{\text{d}x}}>0\\ & B(x)>0\\ & \hspace{72.0pt}0\lt l\leq 20\times 10^{-3}\\ & \hspace{84.0pt}\quad {\sigma}_{i}\leq [{\sigma}_{\text{i}}]\\ & \hspace{-24.0pt}B(x)=C_{0}(1-x)^{4}+C_{1}x(1-x)^{3}+C_{2}x^{2}(1-x)^{2}+C_{3}x^{3}(1-x)+C_{4}x^{4}.\end{array}\right. & & \displaystyle\end{eqnarray}$ (2)

3.2. Derivation of force electricity relationship of PVDF piezoelectric cantilever

For PVDF materials,the piezoelectriccoefficient D _ij is expressed by the matrix: $\begin{eqnarray}\displaystyle \left[\begin{array}{@{}c@{}}{\delta}_{1}\\ {\delta}_{2}\\ {\delta}_{3}\end{array}\right]=\left[\begin{array}{@{}cccccc@{}}0 & 0 & 0 & 0 & d_{15} & 0\\ 0 & 0 & 0 & d_{24} & 0 & 0\\ d_{31} & d_{32} & d_{33} & 0 & 0 & 0\end{array}\right]\left[\begin{array}{@{}c@{}}{\sigma}_{1}\\ {\sigma}_{2}\\ {\sigma}_{3}\\ {\tau}_{4}\\ {\tau}_{5}\\ {\tau}_{6}\end{array}\right]. & & \displaystyle\end{eqnarray}$ (3)

Among them, δ₁, δ₂ and δ₃ represent the induced charge density in X, y and Z planes respectively; σ₁, σ₂ and σ₃ represent the axial stress in X, y and Z planes respectively; τ₁, τ₂ and τ₃ represent the tangential stress in X, Y and Z planes respectively.

Table 1

Initial parameters of piezoelectric cantilever model

Initial parameter			Thickness
v/(m/s)	A/(m²)	l ₀ /(m)	Ag/(m)	PVDF/(m)	Ag/(m)	epoxy resin (EP)/(m)	PET/(m)
8	1.5 × 10⁻⁵	10	5 × 10⁻⁶	50 × 10⁻⁶	5 × 10⁻⁶	5 × 10⁻⁶	10 × 10⁻⁶

Fig. 2.

Structure of PVDF piezoelectric cantilever velocity sensor unit: (a) schematic; (b) size and parameters.

According to the cantilever beam with a thin film structure and its working principle, the cantilever beam bears the wind load f _wind from the outside in the z-axis direction. When the cantilever beam has small deflection deformation, it can be considered that the beam has pure bending deformation, that is, the normal stress σ₂, σ₃ and shear stress τ₄, τ₅ and τ₆ are ignored. The formula (3) is simplified as following: $\begin{eqnarray}\displaystyle {\delta}_{3}=d_{31}\times {\sigma}_{1} & & \displaystyle\end{eqnarray}$ (4)

It can be seen from formula (4) that as long as the stress distribution σ₁ in the PVDF layer in the composite layer is calculated and the two sides of formula (4) are integrated, the response charge q generated by the sensor unit under the force f _wind can be calculated. It is necessary to find out σ₁ first.

Referring to Fig. 2(b), at coordinate X, i.e. section A–A, the resulting bending moment is: $\begin{eqnarray}\displaystyle M(x)=F_{\text{wind}}\cdot x={\rho}_{air}\cdot v_{air}^{2}\cdot {\xi}\cdot \displaystyle \int xB(x)dx. & & \displaystyle\end{eqnarray}$ (5)

At the same time, at coordinate X, when the ordinate of the distance from the neutral axis is Z, stress produced by bending σ_X: $\begin{eqnarray}\displaystyle {\sigma}_{x}=E\cdot {\varepsilon}=E{\displaystyle \frac{z}{{\rho}}}={\displaystyle \frac{M(x)\cdot z}{I_{y}}}. & & \displaystyle\end{eqnarray}$ (6)

Where, E is the elastic modulus of the beam in the PVDF layer; 𝜌 is the curvature radius of the beam after bending; Z is the coordinate of the PVDF piezoelectric layer from the neutral plane Z at coordinate X; m (x) is the moment at the origin X; I _y is the moment of inertia of the section relative to the neutral axis at coordinate X. Therefore, as long as the position of the neutral plane Z ₀ and the inertia moment I _y of the cross section relative to the neutral axis are calculated, formula (6) can be substituted into formula (3) to calculate the response charge density in the cross section. Finally, the charge quantity Q generated under the action of the load f _wind can be obtained through integration.

According to the equivalent section method [25], at x, the width of each layer of the composite cantilever beam is equivalent to the width of Polyethylene terepthalate (PET) layer, and the height of the neutral layer from the bottom of the composite beam Z ^′: $\begin{eqnarray} \displaystyle z^{\prime }={\displaystyle \frac{\displaystyle \mathop{\sum }_{i=1}^{5}\left\{{\displaystyle \frac{E_{i}}{E_{j}}}\cdot (2B_{x})\cdot h_{i}\cdot \left[\displaystyle \mathop{\sum }_{j=1}^{i=1}\left(h_{j}+{\displaystyle \frac{h_{i}}{2}}\right)\right]\right\}}{\displaystyle\sum _{i=1}^{5}\left[{\displaystyle \frac{E_{i}}{E_{j}}}\cdot (2B_{x})\cdot h_{i}\right]}}. & & \displaystyle \end{eqnarray}$ (7)

Where, i = 1 − 5, representing PET layer, glue layer, Ag layer, PVDF layer and Ag layer respectively; B _x is half of the width of the cantilever at coordinate X; E _i is the elastic modulus of the first layer of the cantilever; h _i is the thickness of the first layer of the cantilever.

Equivalent second moment of inertia of composite beam after equivalent to pet beam I ^′: $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}I^{\prime }=\displaystyle \mathop{\sum }_{i}^{5}[I_{i}+A_{i}(z_{i}-z^{\prime })^{2}]\\ A_{i}={\displaystyle \frac{E_{i}}{E_{\text{PET}}}}\cdot B_{x}\cdot h_{i}\end{array}\right.. & & \displaystyle\end{eqnarray}$ (8)

Where, I _i is the equivalent second moment of inertia of the i-th layer relative to the Si layer; Z _i is the distance from the neutral plane of the i-th layer to the bottom of the composite beam. By substituting formula ((7)) and ((8)) into formula ((6)), we can get the average stress σ_x of PVDF piezoelectric layer when the composite cantilever beam is under the action of f _wind. $\begin{eqnarray} \displaystyle {\sigma}_{x}={\displaystyle \frac{[{\rho}_{air}\cdot v_{air}^{2}\cdot {\xi}\cdot \displaystyle \int xB(x)dx]\cdot (z_{\text{PVDF}}-z^{\prime })}{\displaystyle\sum_{i=1}^{5}[I_{i}+A_{i}\cdot (z_{i}-z^{\prime })^{2}]}}\cdot {\displaystyle \frac{E_{\text{PVDF}}}{E_{\text{PET}}}}. & & \displaystyle \end{eqnarray}$ (9)

By substituting formula ((9)) into piezoelectric equation ((4)) and integrating the whole piezoelectric layer with upper and lower electrodes, the total charge Q of composite cantilever under f _wind action can be obtained. $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}Q=\int _{0}^{l}\int _{-B_{x}}^{B_{x}}D_{3}dxdy=\int _{0}^{l}\int _{-B_{x}}^{B_{x}}d_{31}\cdot \frac{[{\rho}_{air}\cdot v_{air}^{2}\cdot {\xi}\cdot \int _{0}^{l}xB(x)dx]\cdot (z_{\text{PVDF}}-z^{\prime })}{\displaystyle \mathop{\sum }_{i=1}^{5}[I_{i}+A_{i}\cdot (z_{i}-z^{\prime })^{2}]}\cdot {\displaystyle \frac{E_{\text{PVDF}}}{E_{\text{PET}}}}dxdy\\[12.0pt] B(x)=C_{0}(1-x)^{4}+C_{1}x(1-x)^{3}+C_{2}x^{2}(1-x)^{2}+C_{3}x^{3}(1-x)+C_{4}x^{4}\\[6.0pt] z^{\prime }={\displaystyle \frac{\displaystyle \mathop{\sum }_{i=1}^{5}\left\{{\displaystyle \frac{E_{i}}{E_{j}}}\cdot (2B_{x})\cdot h_{i}\cdot \left[\displaystyle \mathop{\sum }_{j=1}^{i=1}\left(h_{j}+{\displaystyle \frac{h_{i}}{2}}\right)\right]\right\}}{\displaystyle \mathop{\sum }_{i=1}^{5}\left[{\displaystyle \frac{E_{i}}{E_{j}}}\cdot (2B_{x})\cdot h_{i}\right]}}A_{i}={\displaystyle \frac{E_{i}}{E_{\text{PET}}}}\cdot B_{x}\cdot h_{i}\end{array}\right.. & & \displaystyle\end{eqnarray}$ (10) Where, and are the intermediate variables introduced in the calculation of the charge quantity Q.

3.3. Solution of constraints

Due to the pure bending of PVDF piezoelectric cantilever beam, when the uniform load (f _wind) acts on the surface of the cantilever beam, the bending moment M (x) occurs at coordinate X. At this time, the stress distribution at the section is as shown in Fig. 3. The normal stress at any point is directly proportional to the distance from the point to the neutral axis, that is, along the height direction, the normal stress changes according to the linear rule.

Fig. 3.

Stress distribution of PVDF piezoelectric cantilever when bending.

According to the allowable stress condition, at the fixed end X = L, it is deduced from formula (9): $\begin{eqnarray}\displaystyle \left[\begin{array}{@{}c@{}}{\sigma}_{\text{PET}\max }\\ {\sigma}_{\text{EP}\max }\\ {\sigma}_{\text{Ag}-\text{down}\max }\\ {\sigma}_{\text{PVDF}\max }\\ {\sigma}_{\text{Ag}-\text{upper}\max }\end{array}\right]=\frac{F_{\text{wind}}\cdot l}{I^{\prime }\cdot E_{\text{PET}}}\left[\begin{array}{@{}c@{}}(\text{z}_{\text{PET}}-\text{z}^{\prime })\cdot E_{\text{PET}}\\ (z_{\text{EP }}-\text{z}^{\prime })\cdot E_{\text{Si}\text{O}_{2}}\\ (\text{z}_{\text{Ag}-\text{down}}-\text{z}^{\prime })\cdot E_{\text{Ag}-\text{down}}\\ (\text{z}_{\text{PVDF}}-\text{z}^{\prime })\cdot E_{\text{PVDF}}\\ (\text{z}_{\text{Ag}-\text{upper}}-\text{z}^{\prime })\cdot E_{\text{Ag}-\text{upper}}\end{array}\right]. & & \displaystyle\end{eqnarray}$ (11)

Where σ_imax is the maximum stress of layer.

By substituting formulas ((10)) and ((11)) into the initial optimization model ((2)), it can be concluded that: $\begin{eqnarray} \displaystyle \left\{ \begin{array}{@{} l@{}}\hspace{108.0pt}f\text{ind }\qquad C_{0},C_{1},C_{2},C_{3},C_{4},l\\ \text{subject}\left\{ \begin{array}{@{} l@{}}\max \qquad f(C_{0},C_{1},C_{2},C_{3},C_{4},l)=Q(C_{0},C_{1},C_{2},C_{3},C_{4},l)=\\ [6.0pt] \displaystyle \int _{0}^{l}\int _{-B_{x}}^{B_{x}}D_{3}dxdy=\int _{0}^{l}\int _{-B_{x}}^{B_{x}}d_{31}\cdot {\displaystyle \frac{[{\rho}_{air}\cdot v_{air}^{2}\cdot {\xi}\cdot \displaystyle\int _{0}^{l}xB(x)dx]\cdot (z_{\text{PVDF}}-z^{\prime })}{\displaystyle\sum_{i=1}^{5}[I_{i}+A_{i}\cdot (z_{i}-z^{\prime })^{2}]}}\\ [12.0pt] \hspace{36.0pt}\cdot {\displaystyle \frac{E_{\text{PVDF}}}{E_{\text{PET}}}}dxdy\\ [12.0pt] z^{\prime }={\displaystyle \frac{\displaystyle\sum_{i=1}^{5}\left\{{\displaystyle \frac{E_{i}}{E_{j}}}\cdot (2B_{x})\cdot h_{i}\cdot \left[\displaystyle\sum_{j=1}^{i=1}\left(h_{j}+{\displaystyle \frac{h_{i}}{2}}\right)\right]\right\}}{\displaystyle\sum_{i=1}^{5}\left[{\displaystyle \frac{E_{i}}{E_{j}}}\cdot (2B_{x})\cdot h_{i}\right]}}A_{i}={\displaystyle \frac{E_{i}}{E_{\text{PET}}}}\cdot B_{x}\cdot h_{i} \end{array} \right.\\ [28.0pt] \hspace{108.0pt}s.tA=\displaystyle \int _{0}^{l}B(x)\text{d}x\\ [6.0pt] \hspace{108.0pt}{\displaystyle \frac{\text{d}B(x)}{\text{d}x}}>0\\ [6.0pt] \hspace{36.0pt}B(x)>0\\ \hspace{72.0pt}0\lt l\leq 20\times 10^{-3}\\ [6.0pt] \hspace{48.0pt}B(x)=C_{0}(1-x)^{4}+C_{1}x(1-x)^{3}+C_{2}x^{2}(1-x)^{2}+C_{3}x^{3}(1-x)+C_{4}x^{4}\\ [6.0pt] \hspace{36.0pt}\left[ \begin{array}{@{} c@{}}k1(C_{0},C_{1},C_{2},C_{3},C_{4},l)\\ k2(C_{0},C_{1},C_{2},C_{3},C_{4},l)\\ k3(C_{0},C_{1},C_{2},C_{3},C_{4},l)\\ k4(C_{0},C_{1},C_{2},C_{3},C_{4},l)\\ k5(C_{0},C_{1},C_{2},C_{3},C_{4},l) \end{array} \right]=\left[ \begin{array}{@{} c@{}}{\sigma}_{\text{PET}\max }\\ {\sigma}_{\text{EP}\max }\\ {\sigma}_{\text{Ag}-\text{down}\max }\\ {\sigma}_{\text{PVDF}\max }\\ {\sigma}_{\text{Ag}-\text{upper}\max } \end{array} \right]-\left[ \begin{array}{@{} c@{}}\left[{\sigma}_{\text{PET}}\right]\\ [3.60004pt] \left[{\sigma}_{\text{EP}}\right]\\ [3.60004pt] \left[{\sigma}_{\text{Ag}-\text{down }}\right]\\ [3.60004pt] \left[{\sigma}_{\text{PVDF }}\right]\\ [3.60004pt] \left[{\sigma}_{\text{Ag}-\text{upper }}\right] \end{array} \right]\leq 0. \end{array} \right. & & \displaystyle \end{eqnarray}$ (12)

3.4. Solution of optimization model

The finite element software (FEM, COMSOL multiphysics 5.3 software) is used to solve the above optimization model. In order to optimize the shape of the cantilever beam, the moving asymptote MMA algorithm in the software is used. This method is first proposed by svanberg, which makes up for the limitation of linear programming or quadratic programming in traditional optimization algorithm [26,27], and uses a series of convex problems with separable independent variables to approach the original nonlinear optimization problem continuously. Therefore, it is more suitable to deal with topology optimization problems with complex objective functions and multiple constraints. Set the optimization tolerance as 1E-6 and the constraint penalty factor at 1000. Figure 4 shows the implementation process of the sensitivity optimization model of the flow sensor in the COMSOL platform.

Fig. 4.

Shows the implementation process of the optimization model under the COMSOL platform.

4. Analysis of velocity sensing performance of PVDF piezoelectric cantilever

The parameters of the model optimized by MMA are as follows: S _v = 52 mV/m⋅s⁻¹, C ₀ = 0, C ₁ = 3.0904, C ₂ = 8.6280, C ₃ = 3.6000, C ₄ = 0.9000, l = 8 mm. Thus, the overall dimension of the optimized cantilever beam can be obtained, as shown in Fig. 5.

Fig. 5.

Overall dimension of optimized PVDF piezoelectric cantilever.

When the PVDF piezoelectric cantilever beam before and after optimization is affected by the airflow of different flow rates, the relationship between the flow rate obtained by the steady-state solution and the response voltage value is shown in Fig. 6. It is found that the output response voltage of the optimized cantilever is higher than that of the optimized cantilever at the same piezoelectric material area and at different flow rates.

For example, at 8 m/s, the sensitivity of u = 416 mV after optimization is 52 mV/m⋅s⁻¹, while that of u = 300 mV before optimization is 37.5 mV/m⋅s⁻¹. This shows that the sensitivity of the cantilever to the velocity is enhanced. It can be seen from the analysis that when the total piezoelectric material area is fixed, although the length of the cantilever beam decreases, the width increases, and from the strain curve in the length direction (as shown in Fig. 7(a) and Fig. 7(b)), the optimized strain is particularly uniform and the value is large. According to the piezoelectric equation, the larger the strain is, the more charge the piezoelectric layer will be output, and then the larger the response voltage will be. To sum up, the sensitivity optimization model of piezoelectric cantilever based on Bernstein polynomial equation is solved by MMA algorithm, and more responsive charge output is obtained than that of piezoelectric micro cantilever with traditional shape, which effectively improves the sensitivity of piezoelectric cantilever to velocity.

Fig. 6.

Sensitivity calculation results of optimized PVDF piezoelectric cantilever beam to flow velocity.

Fig. 7.

Calculation results of optimized PVDF piezoelectric cantilever: (a) comparison of strain curve in length direction before and after optimization; (b) strain moire diagram of optimized cantilever.

5. Conclusion

By analyzing the influence of design variables on the sensitivity of the sensor, the optimal theoretical model of PVDF piezoelectric cantilever velocity sensor is established to effectively improve the sensitivity of the sensor. When the area and velocity of piezoelectric material are fixed, the output response voltage of the optimized cantilever is higher than that before optimization. For example, at 8 m/s, the sensitivity of u = 416 mV after optimization is 52 mV/m⋅s⁻¹, while that of u = 300 mV before optimization is 37.5 mV/m⋅s⁻¹, which is 38% higher. It shows that the sensitivity of the cantilever beam with this shape to the velocity increases. It can be seen from the analysis that the optimized strain distribution is more uniform and the strain value is larger; the larger the strain is, the more the output charge of the piezoelectric layer is, and then the greater the response voltage. Therefore, the sensitivity optimization model of PVDF piezoelectric cantilever is solved by MMA algorithm, and more charge output is obtained than before, which effectively improves the sensitivity of piezoelectric cantilever to flow velocity.

Footnotes

Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant No. 51675265), Ministry of Education Humanities and Social Sciences Youth Fund (18YJC760085) and the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD). The authors gratefully acknowledge these supports and reviewers who given valuable suggestions.

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