差压式电容传感器膜片的变形分析与优化设计

图1 受预张力与均布横向载荷作用下的夹支圆薄板示意图

Fig.1 Diagram of clamped circular thin plate subjected to pretension and uniform lateral load

根据弹性理论可得到:

(2)

\begin{array}{l} N_{r} = & \frac{E h}{1 - ν^{2}} (ε_{r} + ν ε_{t}) = \\ \frac{E h}{1 - ν^{2}} [\frac{d u}{d r} + \frac{1}{2} {(\frac{d ω}{d r})}^{2} + ν \frac{u}{r}] + N_{0} = \\ {\tilde{N}}_{r} + N_{0} \\ N_{t} = & \frac{E h}{1 - ν^{2}} (ε_{t} + ν ε_{r}) = \\ \frac{E h}{1 - ν^{2}} [\frac{u}{r} + ν \frac{d u}{d r} + \frac{ν}{2} {(\frac{d ω}{d r})}^{2}] + N_{0} = \\ {\tilde{N}}_{t} + N_{0} \end{array}\}

式中:N_r和N_t分别为单位长度的径向和切向薄膜张力; ${\tilde{N}}_{r}$ 和 ${\tilde{N}}_{t}$ 分别为由于板的挠曲变形而增加的径向和切向薄膜张力.

基于小转角的假设,通过对薄板内任一点r处的静力及力矩平衡分析,得到平衡方程为

(3)

\frac{d N_{r}}{d r} + \frac{N_{r} - N_{t}}{r} = 0

(4)

Q_{r} = - N_{r} \frac{d ω}{d r} - \frac{1}{r} \int_{0}^{r} q r d r

(5)

D (\frac{d^{3} ω}{d r^{3}} + \frac{1}{r} \frac{d^{2} ω}{d r^{2}} - \frac{1}{r^{2}} \frac{d ω}{d r}) = - Q_{r}

式中: $Q_{r} 为剪力; D = E h^{3} / [12 (1 - ν^{2})],$ 为板的抗弯刚度.

结合式(2)~(5)即可得到von Kármán圆板大挠度控制方程:

(6)

r \frac{d}{d r} [\frac{1}{r} \frac{d}{d r} (r^{2} N_{r})] + \frac{E h}{2} {(\frac{d ω}{d r})}^{2} = 0

(7)

D (\frac{d^{3} ω}{d r^{3}} + \frac{1}{r} \frac{d^{2} ω}{d r^{2}} - \frac{1}{r^{2}} \frac{d ω}{d r}) = N_{r} \frac{d ω}{d r} + \frac{q r}{2}

当r=a时,边界条件为

ω = \frac{d ω}{d r} = 0, r \frac{d N_{r}}{d r} + (1 - ν) (N_{r} - N_{0}) = 0

当r=0时,dω/dr和N_r 均为有限值.由于 $N_{r} = {\tilde{N}}_{r} + N_{0},$ 控制方程也可表示为

(8)

\begin{matrix} r \frac{d}{d r} [\frac{1}{r} \frac{d}{d r} (r^{2} {\tilde{N}}_{r})] + \frac{E h}{2} {(\frac{d ω}{d r})}^{2} = 0 \end{matrix}

(9)

D (\frac{d^{3} ω}{d r^{3}} + \frac{1}{r} \frac{d^{2} ω}{d r^{2}} - \frac{1}{r^{2}} \frac{d ω}{d r}) = ({\tilde{N}}_{r} + N_{0}) \frac{d ω}{d r} + \frac{q r}{2}

当圆板为小挠度即挠度相对于板厚为小量时,增加的薄膜张力 ${\tilde{N}}_{r}$ 也很小,计算时可以忽略不计,此时控制方程可以简化为

(10)

D (\frac{d^{3} ω}{d r^{3}} + \frac{1}{r} \frac{d^{2} ω}{d r^{2}} - \frac{1}{r^{2}} \frac{d ω}{d r}) = N_{0} \frac{d ω}{d r} + \frac{q r}{2}

文献[5]应用贝塞尔函数给出了式(10)的解为

(11)

\hat{ω} = \frac{q}{4 N_{0}} (a^{2} - r^{2}) - \frac{q a^{2}}{2 N_{0} k} \frac{I_{0} (k) - I_{0} (\frac{k r}{a})}{I_{1} (k)}

式中: $\hat{ω}$ 为挠度的简化解;k为一种预张力的无量纲量,且 $k^{2} = N_{0} a^{2} / D;$ I₀为修正的零阶贝塞尔函数;I₁为修正的一阶贝塞尔函数.

在小挠度的情况下,由式(11)可以看出各点的挠度 $\hat{ω}$ 是随着压力q线性增长的.但随着圆板的挠度继续增大,相比板厚不再为小量时,因挠曲变形而增加的薄膜应力 ${\tilde{N}}_{r}$ 不能忽略.

2 圆薄板变形分析与求解

引入以下无量纲参数:

\begin{array}{r} y = \frac{r^{2}}{a^{2}}, W = \sqrt{3 (1 - ν^{2})} \frac{ω}{h} \\ φ (y) = y \frac{d W}{d y}, Q = \frac{3 (1 - ν^{2}) \sqrt{3 (1 - ν^{2})} a^{4} q}{4 E h^{4}} \\ s (y) = 3 (1 - ν^{2}) \frac{a^{2} N_{r}}{E h^{3}} y \\ \tilde{s} (y) = 3 (1 - ν^{2}) \frac{a^{2} {\tilde{N}}_{r}}{E h^{3}} \\ S_{0} = 3 (1 - ν^{2}) \frac{a^{2} N_{0}}{E h^{3}} \end{array}

其中: y为半径的无量纲量;W为挠度的无量纲量; $φ$ 为板斜度角的无量纲量;Q为压力的无量纲量;s为径向薄膜张力的无量纲量; $\tilde{s}$ 为径向增加薄膜张力的无量纲量;S₀为预张力的无量纲量.控制方程式(6)和(7)无量纲化为

(12)

y^{2} \frac{d^{2} φ (y)}{d y^{2}} - φ (y) s (y) - y^{2} Q = 0

(13)

y^{2} \frac{d^{2} s (y)}{d y^{2}} + \frac{1}{2} φ^{2} (y) = 0

当y=0和y=1时,对应的边界条件分别为

(14)

\begin{array}{l} φ (0) = s (0) = 0 \\ φ (1) = 0, (μ - 1) s (1) + S_{0} = μ s' (1) \end{array}\}

式中: $μ = \frac{2}{1 - ν} .$

为了方便求解,需进一步将微分方程转化成积分方程,使用解微分方程常见的格林函数法^[15],得到等价的积分方程为

(15)

N_{1} (y) = φ (y) + \int_{0}^{1} K (y, ξ) \frac{1}{ξ^{2}} φ (ξ) s (ξ) d ξ + \frac{Q}{2} (y - y^{2}) = 0

(16)

N_{2} (y) = s (y) - \frac{1}{2} \int_{0}^{1} G (y, ξ) \frac{1}{ξ^{2}} φ^{2} (ξ) d ξ - S_{0} y = 0

式中: y和ξ为格林函数K(y,ξ)和G(y,ξ)的变量,0≤y,ξ≤1.函数表示为

\begin{array}{l} \begin{matrix} K (y, ξ) = \{\begin{array}{l} (- y + 1) ξ, & ξ \leq y \\ (- ξ + 1) y, & ξ > y \end{array} \end{matrix} \\ G (y, ξ) = \{\begin{array}{l} [(μ - 1) y + 1] ξ, & ξ \leq y \\ [(μ - 1) ξ + 1] y, & ξ > y \end{array} \end{array}

HAM^[16]是取得该方程近似解析解的一种有效方法.在已知Q和S₀的情况下,构造如下零阶变形方程:

(17)

(1 - p) (Φ (y; p) - φ_{0} (y)) = p c [Φ (y; p) + \int_{0}^{1} K (y, ξ) \frac{1}{ξ^{2}} Φ (ξ; p) Ψ (ξ; p) d ξ + \frac{Q}{2} (y - y^{2})]

(18)

(1 - p) (Ψ (y; p) - s_{0} (y)) = p c (Ψ (y; p) - \frac{1}{2} \int_{0}^{1} G (y, ξ) \frac{1}{ξ^{2}} Φ^{2} (ξ; p) d ξ - S_{0} y)

式中: p为嵌入变量;c为收敛控制参数; $φ_{0} (y)$ 和 $s_{0} (y)$ 为初始近似解.当p=0,可得 $Φ (y; 0) = φ_{0} (y)$ , $Ψ (y; 0) = s_{0} (y)$ ;当p=1,可得 $Φ (y; 1) = φ (y)$ , $Ψ (y; 1) = s (y)$ . $Φ (y; p)$ 和 $Ψ (y; p)$ 分别为函数 $φ$ (y)和s(y)的同伦形式.

关于变量 $p, 对 Φ (y; p) 和 Ψ (y; p)$ 进行麦克劳林级数展开,得到:

\begin{array}{l} \begin{matrix} Φ (y; p) = φ_{0} (y) + \overset{+ \infty}{\sum_{n = 1}} φ_{n} (y) p^{n} \end{matrix} \\ Ψ (y; p) = s_{0} (y) + \overset{+ \infty}{\sum_{n = 1}} s_{n} (y) p^{n} \end{array}

其中:

\begin{array}{l} φ_{n} (y) = \frac{1}{n!} {\frac{\partial^{n} Φ (y; p)}{\partial p^{n}}|}_{p = 0} \\ s_{n} (y) = \frac{1}{n!} {\frac{\partial^{n} Ψ (y; p)}{\partial p^{n}}|}_{p = 0} \end{array}

如果以上级数在p=1时收敛,则可以得到级数解:

(19)

φ (y) = φ_{0} (y) + \overset{+ \infty}{\sum_{n = 1}} φ_{n} (y)

(20)

s (y) = s_{0} (y) + \overset{+ \infty}{\sum_{n = 1}} s_{n} (y)

将零阶变形方程中p的同幂次项进行合并,得到n阶变形方程:

(21)

φ_{n} = λ_{n} φ_{n - 1} + c [φ_{n - 1} + \overset{n - 1}{\sum_{i = 0}} \int_{0}^{1} K (y, ξ) \frac{1}{ξ^{2}} φ_{i} (ξ) s_{n - 1 - i} (ξ) d ξ + (1 - λ_{n}) \frac{Q}{2} (y - y^{2})]

(22)

s_{n} = λ_{n} s_{n - 1} + c [s_{n - 1} - \frac{1}{2} \overset{n - 1}{\sum_{i = 0}} \int_{0}^{1} G (y, ξ) \frac{1}{ξ^{2}} φ_{i} (ξ) φ_{n - 1 - i} (ξ) d ξ - (1 - λ_{n}) S_{0} y]

由此,可以把式(19)和(20)的部分和作为式(15)和(16)的近似解析解:

\overset{`}{φ} (y) = \overset{m}{\sum_{n = 0}} φ_{n} (y), \overset{`}{s} (y) = \overset{m}{\sum_{n = 0}} s_{n} (y)

并且,可以将多项式函数作为解表达式的基函数,选择符合边界条件式(14)的初始近似解:

φ_{0} (y) = y^{2} - y, s_{0} (y) = S_{0} y

为提高求解效率,采用文献[16]中的迭代技术,将m阶的同伦近似解作为新的初始近似解进行M次迭代运算.借助MATLAB软件中的多项式系列函数进行迭代运算,可以获得良好的求解速度.另外收敛控制参数c的最优值选取主要依据控制方程式(15)和(16)的最小化平方余量残差:

E_{m i n} (c) = \int_{0}^{1} [(N_{1} {(y))}^{2} + (N_{2} {(y))}^{2}] d y

表1为一部分不同 $S_{0}$ 和Q范围内,残差E_min<10^-20的求解时间和参数设置,可以看出随着 $S_{0}$ 和Q的增大,收敛控制参数c越接近于0,方程的非线性不断增强.为了保证一定的求解精度,需要增加迭代的阶数m和次数M,求解时间也随之增长.

表1 部分不同 $S_{0} 和 Q$ 范围的求解参数与求解时间

Tab.1 Solution parameters and solution time in different $S_{0}$ and $Q$ ranges

S₀	Q	m	M	c	E_min	计算时间/s
[0, 40]	[0, 400]	2	110	-0.1	(0, 10^-20)	0.07
[0, 150]	[0, 1 000]	2	300	-0.04	(0, 10^-20)	0.11
[0, 700]	[0, 4 200]	4	700	-0.01	(0, 10^-20)	0.35
[0, 1 500]	[0, 19 000]	10	5000	-0.001	(0, 10^-20)	8.00

FEM通常可以提供高精度的数值解,使用ANSYS有限元软件对计算数据做进一步对比验证.选择SHELL181单元构建圆薄板的实体模型,板的半径a=15 mm,板厚h=0.04 mm.SHELL181采用Reissner-Mindlin理论, 考虑了板横向剪切变形的影响, 从薄板到厚板都具有很强的适用性.使用HAM和FEM计算了有预张力和无预张力的解析解和数值解,圆板直径方向上的挠度曲线如图2所示,其中,实线表示HAM解,星号表示FEM解,可以看出HAM的挠度曲线与FEM的挠度曲线吻合度非常高.另外表2展示了当a=15 mm,h=0.04 mm时,在载荷 q=100,10,1 kPa下的6组FEM和HAM的中心挠度数据对比,显示最大的中心挠度结果偏差仅为0.38%.表中:ω₀为中心挠度;结果偏差为FEM和HAM求解的挠度与FEM求解的挠度之比.由此对于夹支圆薄板的挠度变形问题,通过与FEM数值解的对比,验证了文中大挠度理论方程及近似解析解的有效性.

图2

图2 圆板直径方向上的FEM解与HAM解的挠度曲线对比图

Fig.2 Comparison of deflection curves between FEM and HAM solution in diameter direction of circular plate

表2 FEA解和HAM解的中心挠度对比

Tab.2 Comparison of center deflection between FEA and HAM solutions

q/kPa	N₀/ (N·m^-1)	ω₀/mm		结果偏差/%
q/kPa	N₀/ (N·m^-1)	FEM	HAM	结果偏差/%
100	20 000	0.257 76	0.257 60	0.06
100	0	0.644 18	0.642 88	0.20
10	2 000	0.192 92	0.192 45	0.24
10	0	0.295 53	0.295 38	0.05
1	200	0.109 05	0.108 64	0.38
1	0	0.132 99	0.132 96	0.02

3 挠度的非线性分析

由计算可知,预张力可以使挠度增长变慢并在一定范围内趋于线性,此时得到的简化线性解式(11)计算误差较小.随着压力q的增大,简化线性解的误差也越大,以中心挠度分析其非线性特性.

对解式(11)无量纲化为

(23)

\hat{W} = \frac{Q}{S_{0}} (1 - y) - \frac{Q}{S_{0}^{\frac{3}{2}}} \frac{I_{0} (2 \sqrt{S_{0}}) - I_{0} (2 \sqrt{S_{0} y})}{I_{1} (2 \sqrt{S_{0}})}

式中: $\hat{W}$ 为 $\hat{ω}$ 的无量纲化表达,且

\hat{W} = \sqrt{3 (1 - ν^{2})} \hat{ω} / h

相应的中心挠度为

(24)

\begin{matrix} {\hat{W}}_{0} = \frac{Q}{S_{0}} - \frac{Q}{S_{0}^{\frac{3}{2}}} \frac{I_{0} (2 \sqrt{S_{0}}) - I_{0} (0)}{I_{1} (2 \sqrt{S_{0}})} \end{matrix}

考虑中心挠度的线性偏差为

(25)

\begin{matrix} δ = ({\hat{W}}_{0} - W_{0}) / W_{0} \end{matrix}

由式(24)和(25)可得:

(26)

W_{0} = \frac{Q}{1 + δ} (\frac{1}{S_{0}} - \frac{1}{S_{0}^{\frac{3}{2}}} \frac{I_{0} (2 \sqrt{S_{0}}) - I_{0} (0)}{I_{1} (2 \sqrt{S_{0}})})

令 $F = \frac{1}{1 + δ} (\frac{1}{S_{0}} - \frac{1}{S_{0}^{\frac{3}{2}}} \frac{I_{0} (2 \sqrt{S_{0}}) - I_{0} (0)}{I_{1} (2 \sqrt{S_{0}})})$ ,则 $W_{0} = F Q$ .另外由 $W_{0} = - \int_{0}^{1} \frac{φ (ξ)}{ξ} d ξ$ 可得:

(27)

\begin{matrix} Q = - \frac{1}{F} \int_{0}^{1} \frac{φ (ξ)}{ξ} d ξ \end{matrix}

将式(27)带入式(15)可得:

(28)

φ (y) + \int_{0}^{1} K (y, ξ) \frac{1}{ξ^{2}} φ (ξ) s (ξ) d ξ - \frac{1}{2 F} (y - y^{2}) \int_{0}^{1} \frac{φ (ξ)}{ξ} d ξ = 0

使用HAM求解式(28)和(16),即可得到不同δ值下的解析解.

在δ=0.3%,1%,2%,3%,5%时,S₀与Q的对应关系如图3所示,其中关系曲线与横坐标分别相交于Q=0.519 2,0.903 6,1.289 9,1.594 9,2.097 9 的点上,表示为在无预张力作用下,达到相应线性偏差的无量纲压力值.各曲线的左边区域表明von Kármán方程求解得到的中心挠度线性偏差小于对应的值,同时也表明使用简化方程求解的挠度值具有相对可靠的精度.

图3

图3 不同中心挠度线性偏差下的S₀与Q的关系

Fig.3 S₀ versus Q at different center deflection linear deviations

板抵抗外力变形的能力由弯曲内力和薄膜张力提供,而薄膜张力是由于板的绷紧拉伸变形而产生的平行于板中面的内力,由初始薄膜力和因挠曲变形而增加的薄膜力组成.在小挠度的情况下,增加的薄膜力可以忽略,弯矩、横向载荷以及初始薄膜力形成力学平衡,得到简化方程式(10),此时挠度值与压力成线性关系.当挠度随压力继续增大,增加的薄膜力随着板的进一步拉伸也逐渐增大, 此时由弯矩、初始薄膜力和增加的薄膜力共同抵抗横向载荷,形成大挠度控制方程组,其中式(8)表明增加的薄膜力 ${\tilde{N}}_{r}$ 受挠度ω限制,而式(9)中的非线性项 ${\tilde{N}}_{r}$ dω/dr又反过来影响挠度方程的求解,导致挠度随压力的非线性增长.提高初始薄膜力,可以降低新增薄膜力在受力平衡时的占比,同时能提高板的刚度,减慢挠度的增长,进而限制新增薄膜力的增大,从而降低挠度解的非线性.图4(a)和4(b)分别为在无量纲化预张力S₀=0, 5, 30的情况下, 新增中心薄膜张力和中心挠度对线性偏差的影响.图中可以看出随着新增薄膜张力的变大,中心挠度的非线性不断增长,但是预张力越大,挠度的非线性增长越缓慢.

图4

图4 不同S₀下中心挠度线性偏差的变化

Fig.4 Variation of linear deviation of central deflection at different S₀ values

4 差压式电容传感器输出特性分析

为研究板的变形特性对电容式压力传感器输出特性的影响,利用差压式传感器的实例进行分析.图5所示传感器的固定电极结构为圆形平面电极,作为传感器中心电极的测量膜片半径为a=15 mm,膜片的厚度为h,固定电极平行于中心电极,其半径为b=10.5 mm,固定电极与中心电极的距离为 H=0.1 mm.

图5

图5 差压式电容传感器结构示意图

Fig.5 Structure diagram of differential pressure capacitance sensor

考虑差动结构中低压腔电极间的电容为C₁,高压腔电极间的电容为C₂,无压力时初始电容C₀=C₁=C₂,真空介电常数为ε₀,电介质的介电常数为ε_r,本文电介质按干燥空气进行计算且假设恒定,ε_r=1.000 53.则电容的计算公式分别为

(29)

C_{0} = ε_{0} ε_{r} \int_{0}^{b} \frac{2 π r d r}{H}

(30)

C_{1} = ε_{0} ε_{r} \int_{0}^{b} \frac{2 π r d r}{H - ω (r)}

(31)

C_{2} = ε_{0} ε_{r} \int_{0}^{b} \frac{2 π r d r}{H + ω (r)}

在差动电容结构中,将低压端和高压端电容的对比变化值 $(C_{1} - C_{2}) / (C_{1} + C_{2})$ 作为输出量,分析不同预张力下输出的非线性误差,同时考虑不同膜片厚度影响.以传感器量程500 Pa为例,考虑4种膜片厚度h=0.02,0.03,0.04,0.05 mm,如表3所示计算相应的无量纲压力值Q,并根据图3分别确定中心挠度线性偏差δ= 5%, 1% 时的无量纲预张力S₀,然后计算不同压力及预张力下的挠度解析解,再通过电容计算公式得到传感器低压端及高压端的电容变化情况.图6表示0.05 mm厚的膜片在中心挠度线性偏差δ= 5%,1%下,低压端和高压端的压力-电容特性曲线,图中显示随着预张力增大,低压端及高压端电容的线性度得到提升,但灵敏度下降.

表3 500 Pa量程在不同膜厚和不同挠度线性偏差下的 $Q 和 S_{0}$

Tab.3 $Q$ and $S_{0}$ of 500 Pa range at different membrane thicknesses and deflection linear deviations

h/ mm	q_max/ Pa	Q_max	δ= 5%		δ= 1%
h/ mm	q_max/ Pa	Q_max	S₀	N₀/ (N·m^-1)	S₀	N₀/ (N·m^-1)
0.02	500	1 372.32	289	489.3	514.6	871.3
0.03	500	271.08	93.3	533.1	168.5	962.9
0.04	500	85.77	40.6	549.9	74.9	1 014.5
0.05	500	35.13	20.5	542.3	39.1	1 034.4

图6

图6 不同挠度线性偏差下低压端和高压端压力-电容特性曲线(h=0.05 mm)

Fig.6 Pressure-capacitance characteristic curves of low and high pressure parts at different deflection linear deviations (h=0.05 mm)

利用得到图6的相同方法,除了可以得到其他膜厚下的压力-电容特性曲线,也可以得到压力与输出量 $(C_{1} - C_{2}) / (C_{1} + C_{2})$ 的特性曲线,使用最小二乘法对该特性曲线进行拟合,得到的拟合直线与原曲线的最大偏差除以满量程的输出量, 即为差压式传感器的非线性误差.各膜厚和挠度线性偏差下的灵敏度和输出非线性误差如表4所示,当δ= 5%时,传感器的非线性误差为0.90%左右,但当δ= 1%时,随着预张力的增大,传感器的非线性误差减小到0.17%左右,由此可知膜片中心挠度的非线性对差压式传感器输出的非线性影响非常大.随着非线性误差得到改善,灵敏度也随之降低,而根据表4中结果可知,降低膜厚可以弥补灵敏度上的损失,这表明在中心挠度线性偏差不变的情况下,减小膜厚可以提高板的挠度.

表4 不同膜厚和不同挠度线性偏差下的灵敏度和输出非线性误差

Tab.4 Sensitivity and output nonlinearity at different membrane thicknesses and different deflection linear deviations

h/mm	δ= 5%			δ= 1%
h/mm	低压端灵敏度/(pF·Pa^-1)	高压端灵敏度/(pF·Pa^-1)	非线性误差/%	低压端灵敏度/(pF·Pa^-1)	高压端灵敏度/(pF·Pa^-1)	非线性误差/%
0.02	0.039 4	-0.016 8	0.86	0.018 3	-0.011 3	0.15
0.03	0.030 8	-0.015 0	0.94	0.015 1	-0.010 0	0.17
0.04	0.025 8	-0.013 7	0.95	0.013 2	-0.009 1	0.18
0.05	0.022 4	-0.012 6	0.94	0.011 7	-0.008 4	0.18

5 测试结果

图7(a)为制作的差压式电容传感器,膜片材料为QBe2膜片,壳体材料为黄铜,使用机械拉伸结构获得预张力,考虑传感器性能要求及加工性,传感器的实际结构参数如表5所示. 表中:H₁为低压腔极板间距;H₂为高压腔极板间距.

图7

图7 传感器测试系统

Fig.7 Schematic diagram of sensor test system

表5 传感器实际结构参数

Tab.5 Actual structure parameters of sensor

传感器参数	数值	传感器参数	数值
a/mm	15	H₁/mm	0.32
h/mm	0.02	H₂/mm	0.34
b/mm	10.5	N₀/(N·m^-1)	840

利用高精度数字电容转换芯片PCap02实现对传感器电容数据的测量^[17].利用图7(b)所示的测试系统对制作的传感器进行加压测试,以每次20 Pa的步进由0加压至500 Pa,测试的结果具体见图8.对测试数据进行最小二乘直线拟合,得到非线性误差为0.58%,理论计算值的非线性误差为0.28%,由于测试设备精度与测试环境的影响,测试结果虽然与计算存在一定偏差,但仍能表明在预加适当初始张力的情况下,传感器具有较好的非线性性能.同时观察到测试值大于计算值,造成这类误差的原因可能与传感器结构的加工有关,或者受温度影响材料的性能参数产生一些偏离,但测试结果与理论分析的方向在一定程度上表现一致.

图8

DOI:10.6117/kmeps.2017.24.1.051 URL [本文引用: 1]

图8 计算及测试的传感器特性

Fig.8 Characteristics of calculated and tested sensors

6 结论

本文研究了夹支圆板在预张力和均布横向载荷联合作用下的挠曲变形问题,参照von Kármán方程推导了相关公式,应用HAM并借助计算机程序求解出了近似解析解,同时对比FEM的数值解,验证了理论公式及求解方法的有效性.此外,通过对简化线性解和精确非线性解的对比分析,发现新增薄膜张力是造成板挠度非线性变化的主要因素.并得到了不同中心挠度线性偏差下的预张力和横向均布载荷的关系:预张力越大,横向均布载荷越小,板挠度的线性偏差越小.

研究板挠度的非线性特性对掌握电容式压力传感器的输出特性具有重要的现实意义.利用板挠度的非线性理论结果,可以指导差压式电容传感器测量膜片的参数设计,并以一种差压式电容传感器的结构为例,按照中心挠度线性偏差δ在5%和1%的情况下,计算传感器的非线性误差.结果表明:δ=5%时的非线性误差是δ=1%时的5倍左右,预张力可以显著降低传感器输出的非线性误差,至于预张力增加而带来的灵敏度降低的问题,可以通过减小膜片的厚度来弥补.最后,结合差压式电容传感器的性能测试,验证了本文对膜片预张力和非线性进行计算分析的合理性和正确性.

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