李盼1,杨睿1,任宗金1,钱卫2
(1.大连理工大学 机械工程学院, 辽宁 大连 116024; 2. 大连理工大学 航空航天学院, 辽宁 大连 116024)
摘要:
针对大载荷、狭窄安装空间的全动舵面气动载荷测量风洞试验，设计一种四支点三向力压电传感器并联式天平.选择压电石英作为天平的力敏元件，讨论四支点三向力传感器在其中的空间布局及其测量原理，研究并联式天平的标定方法，并对天平进行静态标定、复合加载试验.应用基于标定矩阵的求解矩阵广义逆的静态解耦算法进行解耦，从传感器等效作用点偏移的角度，利用ANSYS有限元软件分析并联式压电天平误差产生机理.实验结果表明：该天平的最大非线性误差和重复性误差分别为1.352%、1.019%，最大向间干扰系数为2.865%，复合加载对竖直力的测量精度影响较小，而对弯矩影响较为明显; 天平静态标定指标均满足测试精度要求，但不同加载方向及其大小对传感器间距的影响各异，复合加载时，间距影响相互叠加，使天平测量精度降低，此天平并不适用于多向载荷测量.
关键词: 风洞天平 压电传感器 天平标定 解耦 等效作用点
DOI：10.11918/j.issn.0367-6234.201704095
分类号:TH823
文献标识码:A
基金项目:
Design and error analysis of a piezoelectric balance for the measurement of aerodynamic loads on an all-movable rudder
LI Pan1,YANG Rui1,REN Zongjin1,QIAN Wei2
(1.School of Mechanical Engineering, Dalian University of Technology, Dalian 116024, Liaoning, China; 2.School of Aeronautics and Astronautics, Dalian University of Technology, Dalian 116024, Liaoning, China)
Abstract:
A three-direction piezoelectric sensor parallel-type balance with four supporting points was designed for the dynamic load measurement of the all-movable rudder which withstands high-load and installed in narrow space. The piezoelectric quartz was selected to be the force sensing element, the spatial layout and measuring principle of this three-direction force sensor with four supporting points were discussed and the calibration method of parallel balance was studied. Static calibration and compound loading test were carried on this parallel balance. The generalized inverse static decoupling algorithm based on the solution matrix of calibration matrix was used to decouple. From the view of equivalent application points' offset, the error generation mechanism of parallel piezoelectric balance was analyzed using ANSYS finite element software. The experimental results showed that the maximum nonlinear error and repeatability error of the balance were 1.352% and 1.019% respectively. The largest interference coefficient was 2.865%. The influence of composite load on vertical force measurement accuracy was less, yet on the bending moment was obvious. The balance static calibration indexes satisfied the test precision requirements. However, due to the different effects of different load on the range interval between sensors and the superimposed impacts of the composite loading time interval, the balance measurement accuracy was reduced and not satisfied multi-directional load measurement.
Key words: wind tunnel balance piezoelectric sensor balance calibration decoupling equivalent action point
李盼, 杨睿, 任宗金, 钱卫. 全动舵面载荷测量的压电天平设计与误差分析[J]. 哈尔滨工业大学学报, 2018, 50(1): 181-185. DOI: 10.11918/j.issn.0367-6234.201704095.
LI Pan, YANG Rui, REN Zongjin, QIAN Wei. Design and error analysis of a piezoelectric balance for the measurement of aerodynamic loads on an all-movable rudder[J]. Journal of Harbin Institute of Technology, 2018, 50(1): 181-185. DOI: 10.11918/j.issn.0367-6234.201704095.
作者简介 李盼(1990—)，男，硕士研究生;
杨睿(1974—)，男，教授，硕士生导师 通信作者 杨睿，yangrui@dlut.edu.cn 文章历史 收稿日期: 2017-04-19
Contents -->Abstract Full text Figures/Tables PDF
全动舵面载荷测量的压电天平设计与误差分析
李盼1, 杨睿1
, 任宗金1, 钱卫2 1. 大连理工大学 机械工程学院, 辽宁 大连 116024;
2. 大连理工大学 航空航天学院, 辽宁 大连 116024
收稿日期: 2017-04-19
作者简介: 李盼(1990—)，男，硕士研究生;
杨睿(1974—)，男，教授，硕士生导师
通信作者: 杨睿，yangrui@dlut.edu.cn
摘要: 针对大载荷、狭窄安装空间的全动舵面气动载荷测量风洞试验，设计一种四支点三向力压电传感器并联式天平.选择压电石英作为天平的力敏元件，讨论四支点三向力传感器在其中的空间布局及其测量原理，研究并联式天平的标定方法，并对天平进行静态标定、复合加载试验.应用基于标定矩阵的求解矩阵广义逆的静态解耦算法进行解耦，从传感器等效作用点偏移的角度，利用ANSYS有限元软件分析并联式压电天平误差产生机理.实验结果表明：该天平的最大非线性误差和重复性误差分别为1.352%、1.019%，最大向间干扰系数为2.865%，复合加载对竖直力的测量精度影响较小，而对弯矩影响较为明显; 天平静态标定指标均满足测试精度要求，但不同加载方向及其大小对传感器间距的影响各异，复合加载时，间距影响相互叠加，使天平测量精度降低，此天平并不适用于多向载荷测量.
关键词: 风洞天平 压电传感器 天平标定 解耦 等效作用点
Design and error analysis of a piezoelectric balance for the measurement of aerodynamic loads on an all-movable rudder
LI Pan1, YANG Rui1
, REN Zongjin1, QIAN Wei2 1. School of Mechanical Engineering, Dalian University of Technology, Dalian 116024, Liaoning, China;
2. School of Aeronautics and Astronautics, Dalian University of Technology, Dalian 116024, Liaoning, China
Abstract: A three-direction piezoelectric sensor parallel-type balance with four supporting points was designed for the dynamic load measurement of the all-movable rudder which withstands high-load and installed in narrow space. The piezoelectric quartz was selected to be the force sensing element, the spatial layout and measuring principle of this three-direction force sensor with four supporting points were discussed and the calibration method of parallel balance was studied. Static calibration and compound loading test were carried on this parallel balance. The generalized inverse static decoupling algorithm based on the solution matrix of calibration matrix was used to decouple. From the view of equivalent application points' offset, the error generation mechanism of parallel piezoelectric balance was analyzed using ANSYS finite element software. The experimental results showed that the maximum nonlinear error and repeatability error of the balance were 1.352% and 1.019% respectively. The largest interference coefficient was 2.865%. The influence of composite load on vertical force measurement accuracy was less, yet on the bending moment was obvious. The balance static calibration indexes satisfied the test precision requirements. However, due to the different effects of different load on the range interval between sensors and the superimposed impacts of the composite loading time interval, the balance measurement accuracy was reduced and not satisfied multi-directional load measurement.
Key words: wind tunnel balance piezoelectric sensor balance calibration decoupling equivalent action point
全动舵面是飞机、导弹和航天飞行器的主要控制面[1].舵面的操纵力矩和受力状态关系着飞行器的正常飞行状态，以舵面天平为测力技术手段的风洞试验，是预测舵面气动特性的重要技术途径[2].
当前，风洞试验中应用最为广泛的是应变天平，但是其刚度低、动态特性差[3-4]，严重制约全动舵面气动载荷测量精度.在文献[5-7]中，研究人员利用多个压电传感器组合的方式测量空间六维力，这种天平刚度极高，并具有良好的静态和动态测试性能，适用于全动舵面气动载荷测量.
受风洞水平、飞行器整体尺寸等诸多因素的影响，目前普遍采用缩比模型进行风洞测力实验，大大缩小了风洞天平的设计空间[8].秦岚等[9-10]提出8点支撑结构的压电式六维力传感器，实现了传感器的微小型化; 李映君等[11-12]提出的并联式压电六维力传感器，实现了对大力值的测量.上述研究成果具有极高的刚度和测量精度，但不适宜用于大载荷、狭窄型安装空间的全动舵面气动载荷测量风洞试验.
本文研制了一种四支点三向力压电传感器并联式天平，用于风洞试验中全动舵面气动载荷测量.分析了天平的测量原理，完成了静态标定，应用基于标定矩阵的求解矩阵广义逆的静态解耦算法进行解耦，得出其性能指标，并进行了复合加载试验.从传感器等效作用点偏移的角度，分析了并联式压电天平误差产生机理，研究结果对并联式压电天平设计，尤其对大长宽比压电天平设计具有指导意义.
1 压电天平设计与分析 1.1 压电天平设计针对大载荷、狭窄安装空间的全动舵面气动载荷测量风洞试验，设计了一种四支点三向力压电传感器并联式天平，天平爆炸图如图 1所示.天平整体尺寸为200 mm×70 mm×55 mm，测力范围为0~6 kN.
Figure 1
1—预紧螺钉; 2—天平底座; 3—三向力传感器; 4—天平盖板图 1 天平爆炸图 Figure 1 Explosion diagram of balance 1.2 压电天平测量原理四支点三向力压电传感器并联式天平布局如图 2所示. 空间作用力F可等效到参考坐标系中，分解为6维力/力矩(FX、FY、FZ、MX、MY、MZ)，并转化到天平中每个三向力传感器上.
Figure 2
图 2 天平平面布局示意 Figure 2 Sketch map of balance 根据空间力/力矩平衡方程可得六维力/力矩的表达式[13]为
$\begin{align} &{{F}_{X}}={{F}_{{{X}_{1}}}}+{{F}_{{{X}_{2}}}}+{{F}_{{{X}_{3}}}}+{{F}_{{{X}_{4}}}}, \text{ } \\ &{{F}_{Y}}={{F}_{{{Y}_{1}}}}+{{F}_{{{Y}_{2}}}}+{{F}_{{{Y}_{3}}}}+{{F}_{{{Y}_{4}}}}, \text{ } \\ &{{F}_{Z}}={{F}_{{{Z}_{1}}}}+{{F}_{{{Z}_{2}}}}+{{F}_{{{Z}_{3}}}}+{{F}_{{{Z}_{4}}}}, \text{ } \\ &{{M}_{X}}=\left( h/2 \right)(-{{F}_{{{Z}_{1}}}}+{{F}_{{{Z}_{2}}}}+{{F}_{{{Z}_{3}}}}-{{F}_{{{Z}_{4}}}}), \text{ } \\ &{{M}_{Y}}=\left( l/2 \right)(-{{F}_{{{Z}_{1}}}}-{{F}_{{{Z}_{2}}}}+{{F}_{{{Z}_{3}}}}+{{F}_{{{Z}_{4}}}}), \text{ } \\ &{{M}_{Z}}=\left( h/2 \right)({{F}_{{{Z}_{1}}}}-{{F}_{{{Z}_{2}}}}-{{F}_{{{Z}_{3}}}}+{{F}_{{{Z}_{4}}}})+ \\ &\ \ \ \ \ \ \ \ \text{ }l/2({{F}_{{{Z}_{1}}}}+{{F}_{{{Z}_{2}}}}-{{F}_{{{Z}_{3}}}}-{{F}_{{{Z}_{4}}}}). \\ \end{align}$
式中：FXi、FYi、FZi (i=1~4)分别表示4个传感器的三向分力，l为相邻传感器X方向间距，h为相邻传感器Y方向间距.
2 解耦算法及标定 2.1 解耦算法无论在设计、制造和安装天平等方面做出多大的努力，向间耦合是不可避免的，并且十分复杂.基于求解矩阵广义逆的静态解耦算法原理简单, 运算量小, 解耦精度高[14-15].
设输入载荷F=(FX FY FZ MX MY MZ), 输出电压信号U=(UX UY UZ UMX UMY UMZ)，输入载荷与输出信号关系如下：
$\left\{ \begin{align} &{{U}_{X}}={{U}_{XX}}+{{U}_{YX}}+{{U}_{ZX}}+{{U}_{{{M}_{X}}X}}+{{U}_{{{M}_{Y}}X}}+{{U}_{{{M}_{Z}}X}}, \text{ } \\ &{{U}_{Y}}={{U}_{XY}}+{{U}_{YY}}+{{U}_{ZY}}+{{U}_{{{M}_{X}}Y}}+{{U}_{{{M}_{Y}}Y}}+{{U}_{{{M}_{Z}}Y}}, \\ &\text{ }{{U}_{Z}}={{U}_{XZ}}+{{U}_{YZ}}+{{U}_{ZZ}}+{{U}_{{{M}_{X}}Z}}+{{U}_{{{M}_{Y}}Z}}+{{U}_{{{M}_{Z}}Z}}, \text{ } \\ &{{U}_{{{M}_{X}}}}={{U}_{X{{M}_{X}}}}+{{U}_{Y{{M}_{X}}}}+{{U}_{Z{{M}_{X}}}}+{{U}_{{{M}_{X}}{{M}_{X}}}}+{{U}_{{{M}_{Y}}{{M}_{X}}}}+{{U}_{{{M}_{Z}}{{M}_{X}}}}, \text{ } \\ &{{U}_{{{M}_{Y}}}}={{U}_{X{{M}_{Y}}}}+{{U}_{Y{{M}_{Y}}}}+{{U}_{Z{{M}_{Y}}}}+{{U}_{{{M}_{X}}{{M}_{Y}}}}+{{U}_{{{M}_{Y}}{{M}_{Y}}}}+{{U}_{{{M}_{Z}}{{M}_{Y}}}}, \text{ } \\ &{{U}_{{{M}_{Z}}}}={{U}_{X{{M}_{Z}}}}+{{U}_{Y{{M}_{Z}}}}+{{U}_{Z{{M}_{Z}}}}+{{U}_{{{M}_{X}}{{M}_{Z}}}}+{{U}_{{{M}_{Y}}{{M}_{Z}}}}+{{U}_{{{M}_{Z}}{{M}_{Z}}}}. \\ \end{align} \right.$ (1)
式中：Uij表示i方向载荷对j(i≠j)方向输出电压信号的干扰量.
假设输出电压与输入载荷之间呈线性关系，通过最小二乘法可得，各方向输出的回归直线方程为
${{\mathit{\boldsymbol{U}}}_{ij}}={{a}_{ij}}{{\mathit{\boldsymbol{F}}}_{i}}(\text{或}{{\mathit{\boldsymbol{M}}}_{i}}).$
式中a为干扰系数.
式(1)可用矩阵表示为
$\left[{\begin{array}{*{20}{c}}{{U_X}}\\{{U_Y}}\\{{U_Z}}\\{{U_{{M_X}}}}\\{{U_{{M_Y}}}}\\{{U_{{M_Z}}}}\end{array}} \right]{\text{ = }}\left[{\begin{array}{*{20}{c}}{{a_{XX}}}&{{a_{YX}}}&{{a_{ZX}}}&{{a_{{M_X}X}}}&{{a_{{M_Y}X}}}&{{a_{{M_Z}X}}}\\{{a_{XY}}}&{{a_{YY}}}&{{a_{ZY}}}&{{a_{{M_X}Y}}}&{{a_{{M_Y}Y}}}&{{a_{{M_Z}Y}}}\\{{a_{XZ}}}&{{a_{YZ}}}&{{a_{ZZ}}}&{{a_{{M_X}Z}}}&{{a_{{M_Y}Z}}}&{{a_{{M_Z}Z}}}\\{{a_{X{M_X}}}}&{{a_{Y{M_X}}}}&{{a_{Z{M_X}}}}&{{a_{{M_X}{M_X}}}}&{{a_{{M_Y}{M_X}}}}&{{a_{{M_Z}{M_X}}}}\\{{a_{X{M_Y}}}}&{{a_{Y{M_Y}}}}&{{a_{Z{M_Y}}}}&{{a_{{M_X}{M_Y}}}}&{{a_{{M_Y}{M_Y}}}}&{{a_{{M_Z}{M_Y}}}}\\{{a_{X{M_Z}}}}&{{a_{Y{M_Z}}}}&{{a_{Z{M_Z}}}}&{{a_{{M_X}{M_Z}}}}&{{a_{{M_Y}{M_Z}}}}&{{a_{{M_Z}{M_Z}}}}\end{array}} \right]\left[{\begin{array}{*{20}{c}}{{F_X}}\\{{F_Y}}\\{{F_Z}}\\{{M_X}}\\{{M_Y}}\\{{M_Z}}\\\end{array}} \right],\\即\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mathit{\boldsymbol{U}} = \mathit{\boldsymbol{AF}} $ (2)
式中A为耦合矩阵.
在对天平进行解耦时，传感器输出的电压量为自变量，而作用在天平上的六维力/力矩分量是未知的，式(2)可变换为
$\mathit{\boldsymbol{F}} = {({\mathit{\boldsymbol{A}}^T}\mathit{\boldsymbol{A}})^ - }^1{\mathit{\boldsymbol{A}}^T}\mathit{\boldsymbol{U}} = \mathit{\boldsymbol{CU}}.$
式中，C为标定矩阵，又称解耦矩阵.
2.2 静态标定实验静态标定是指在不同静态力作用下对天平所呈现出的力与变形的规律进行测试，以便确立天平的静态特性.釆用逐级加载法[16]，分别施加6个单方向的力/力矩对天平进行标定，其具体标定系统示意图如图 3所示.
Figure 3
图 3 天平静态标定系统示意 Figure 3 Schematic diagram of static calibration system for balance 根据回归直线方程和标定曲线，对标定数据进行最小二乘法计算及解耦，性能指标如表 1所示，图 4为天平标定曲线.压电式风洞天平的标定矩阵为
$\mathit{\boldsymbol{C}} = \left[{\begin{array}{*{20}{c}}{-0.770{\text{ }}6}&{0.004{\text{ }}7}&{0.014{\text{ }}9}&{-0.028{\text{ }}7}&{0.150{\text{ }}2}&{0.058{\text{ }}9}\\{-0.054{\text{ }}6}&{0.789{\text{ }}7}&{0.014{\text{ }}4}&{0.020{\text{ }}9}&{ - 0.165{\text{ }}3}&{0.082{\text{ }}9}\\{ - 0.007{\text{ }}3}&{0.014{\text{ }}9}&{1.066{\text{ }}2}&{ - 0.010{\text{ }}5}&{ - 0.080{\text{ }}0}&{ - 0.010{\text{ }}8}\\{ - 0.000{\text{ }}2}&{0.000{\text{ }}3}&{0.001{\text{ }}2}&{0.914{\text{ }}2}&{ - 0.006{\text{ }}4}&{0.001{\text{ }}3}\\{ - 0.000{\text{ }}4}&{0.005{\text{ }}6}&{0.002{\text{ }}6}&{0.003{\text{ }}6}&{ - 0.961{\text{ }}0}&{0.000{\text{ }}4}\\{ - 0.010{\text{ }}1}&{0.000{\text{ }}3}&{0.001{\text{ }}8}&{ - 0.002{\text{ }}0}&{ - 0.004{\text{ }}0}&{0.931{\text{ }}9}\end{array}} \right]{\text{ }}{\text{.}}$
表 1
表 1 天平的主要性能指标 Table 1 Primary performance of the balance %六维力 非线性 重复性 向间干扰系数/%
FX FY Fz MX MY MZ
FX 1.068 0.644 0.153 0.789 0.009 0.037 0.107
FY 1.330 0.438 1.571 1.876 0.011 0.078 0.057
Fz 0.201 0.141 0.116 0.328 0.010 0.033 0.014
MX 1.293 0.635 2.739 1.625 0.828 2.095 1.478
MY 1.257 0.987 1.155 0.748 3.050 0.257 0.768
MZ 1.352 1.019 2.865 2.443 0.695 0.261 1.683
表 1 天平的主要性能指标 Table 1 Primary performance of the balance
Figure 4
图 4 天平静态标定曲线 Figure 4 Static calibration curve of balance 3 复合加载试验及误差分析 3.1 复合加载试验为验证天平的标定性能，同时为了更加真实的模拟风洞试验中天平所受气动载荷，有必要对天平进行静态复合加载试验，具体加载方式如表 2所示.
表 2
表 2 压电天平二分量复合加载 Table 2 Two component composite loading of balance 输入载荷 输出载荷 误差/% FZ/kN MX/(N·m) FZ/kN MX/(N·m) FZ MX
-3.000 50 -3.008 4 48.87 0.28 2.26
-3.000 100 -3.011 2 97.68 0.37 2.32
-3.000 150 -3.006 6 146.29 0.22 2.47
-2.000 100 -2.004 3 98.38 0.22 1.62
-4.000 100 -4.012 7 96.93 0.32 4.07
表 2 压电天平二分量复合加载 Table 2 Two component composite loading of balance
综合表 1、2可得，考虑加载精度、标定系统精度等因素，该天平测力精度与标定时相差无几; 但MX的误差普遍较大，并随着FZ的增大而增大，而MX变化对天平测量精度影响较小.
3.2 误差分析由天平测量原理可知，MX的变化与FZi、h直接相关.由于该天平宽度方向相邻传感器距离较近，为了简化计算，可将这两个传感器单独提取，组成两支点并联式天平，等效四支点并联式天平在载荷作用下进行有限元分析，传感器中心间距为50 mm，预紧力10 kN. 图 5分别为天平不受力、压力、弯矩情况下的应变云图.
Figure 5
图 5 天平受载时传感器应变云图 Figure 5 Load strain nephogram of balance 从图 5可以看出，载荷将使传感器的等效作用点发生偏移.这是由于传感器与上盖是面接触，外载作用下传感器应力分布不均，从而使传感器等效作用点发生偏移. 表 3为在不同加载方式下传感器等效作用点的偏移情况.
表 3
表 3 传感器等效作用点偏移情况 Table 3 Sensor offset 输入载荷 传感器等效作用点位置/mm 传感器等效作用点距离/mm 传感器间距改变百分比/% FZ/kN MX/(N·m) 1 2
0 0 25.2 74.8 49.6
0 100 22.7 71.7 49.0 1.21
-2.5 21.0 72.7 51.7 4.23
-5.0 18.8 74.2 55.4 11.69
-7.5 16.8 76.3 59.5 19.96
-2.5023.876.352.55.85
5022.374.552.25.24
1021.072.751.74.23
150 19.9 71.3 51.4 3.63
表 3 传感器等效作用点偏移情况 Table 3 Sensor offset
从表 3可得，在FZ的作用下两个传感器等效作用点偏移的方向相反，使得传感器等效作用点间距不断变化; 在MX作用下两个传感器等效作用点偏移的方向相同，且偏移量较小，使得传感器间距变化较小.
由两支点推导到四支点并联式天平，基于天平测量原理，力与传感器间距无关，而力矩大小与传感器间距呈线性关系，验证了复合加载试验中FZ的测量精度较高，而MX的误差普遍较大.
4 结论针对大载荷、狭窄安装空间的全动舵面气动载荷测量风洞试验，设计了一种四支点三向力压电传感器并联式天平.推导了天平的测量原理，完成了天平的静标定及解耦，并进行复合加载试验及误差分析，得出如下结论：
1) 静态标定实验表明，该天平解耦后的非线性误差和重复性误差均＜1.5%，最大向间干扰系数均不超过3%.由于宽度方向相邻传感器的间距较小，使得某些通道的向间干扰较大.
2) 复合加载试验及误差分析表明，FZ对传感器等效作用点间距具有极大的影响，并且随着FZ的变化，其对传感器等效作用点间距的影响也不同; MX对传感器等效作用点间距有一定的影响，且MX的变化对传感器等效作用点间距影响较小.复合加载时间距影响相互叠加，使得天平测量精度降低，此天平并不适用于多向载荷测量.这对多支点并联式天平(尤其对传感器紧凑布局的天平)设计具有指导意义.
本文仅对一种复合加载方式进行了分析，还需进一步研究其他方向对天平精度的影响规律，以期提高并联式压电天平的性能.
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