# Error Characteristic Analysis of Fluid Momentum Law Test Apparatus
## 4 Error Characteristic Analysis
Comprehensively considering the vibration characteristics of the sensor, it is important to simplify the deformation \(s_{Y}\) of the sensor strain gauge in the Y direction and accumulate the absolute value of each signal amplitude in the spectral characteristics of \(s_{Y}\), as the maximum value of \(s_{Y}\), that is, \(s_{Ymax}=sum(A_{0},\,A_{1},\,A_{2},\,A_{3})\). According to the two-way response characteristics of strain gauges and the theory of lateral error, the calculation method of periodic system error caused by the sensor lateral effect can be expressed as [14; 27]:  
\[\delta=\frac{\varepsilon^{\prime}-\varepsilon}{\varepsilon}\times 100\%=\frac{k \left\{\frac{\varepsilon_{y}}{\varepsilon_{x}}+\mu_{1}\right\}}{1-\mu_{1}k} \tag{6}\]  
In Formula (6), \(\delta\) represents relative error; \(k\) represents the transverse sensitivity coefficient; \(\varepsilon_{x}\) represents the strain in the X direction of the strain gauge; \(\varepsilon_{y}\) represents the strain in the Y direction of the strain gauge; \(\mu_{1}\) represents the Poisson's ratio of the strain gauge material. The deformation \(s_{Y}/s_{X}\) of the characteristic point C on the sensor is equivalent to \(\varepsilon_{x}/\varepsilon_{y}\). After the sensor is stabilized in the jet direction (X), \(s_{X}\) is a fixed value, and assumed as the strain gauge Poisson's ratio \(\mu_{1}=0.3\). Take a different value for \(k\) and estimating the error of the sensor under the different jets impacts the load, as shown in Table 4. It can be seen from Table 4 that the larger lateral sensitivity coefficient \(k\) of the sensor, the larger measurement error. Under the same sensor parameter conditions, the larger jet flow, the smaller relative error. Figure 10: Relative displacement changing spectrum graph of sensor characteristic point under impacts of different momentum jets in the Y direction. The experiment on the apparatus is measured on the condition to the same as numerical parameters. The test data is shown in Table 5, which is similar to the estimated error (Table 4). Comparing the sensor estimated error (Table 4) and measured error (Table 5), it shows that most of measured errors are in the area that is ranged by estimated error. The result shows that the difference between the estimated error and the measured error is small, and the error analysis and estimation methods used in this article are effective. The reasons for the difference between the estimated error and the measured error are as follows [14; 27; 28; 29; 30]:  
1. Ignoring the linear system error caused by the deformation of the jet direction base;
2. Error compensation inside the sensor;
3. Sensor measurement error caused by temperature and humidity;
4. The displacement response of point C cannot fully reflect the true change of the strain gauge;
5. The jet velocity changing caused by the pulsation of pump flow during the actual measurement;
6. The difference between the calculation model, material properties, etc. and the real system. Comprehensive analysis shows that the experimental system error consists of two parts, namely, the linear system error caused by the jet impact direction (X direction) base deformation and the periodic system error caused by the sensor's Y direction periodic vibration and lateral effect. The linear error caused by the deformation of the base increases with the increase of the jet flow. The system error can be reduced by increasing the stiffness of the base; the measurement error caused by the lateral effect decreases with the increase of the jet, and the measurement error can be reduced by sensor correction.