A novel device for measuring velocity of projectile

Abstract

A novel device used to measure the velocity of a projectile for the military has been developed. This set of equipment based on the principle of statistics is composed of five groups of sensors. These five groups of sensors are symmetrically arranged to ensure that the groups of sensors measure the same point velocity, and the average value of the groups of sensors is regarded as the final test result. Compared with the traditional test method for projectile velocity, this method reduces the uncertainty of measurement by increasing the measurement samples. In this paper the F-test, T-test and variance are used to prove that the measurement value of this device is an unbiased, consistent and a valid estimation of the true value. Experiments are carried out over a wide operational range from 40 m/s to 2000 m/s velocity and from 4-mm to 12.7-mm calibre. The results show that the measurement uncertainty of this device is less than 1‰ (k = 2). This device can be suggested as a prior calibration in order to estimate other devices for measuring projectile velocity.

Keywords

Laser screen photoelectric sensor uncertainty analysis velocity of projectile zone-block device

Introduction

The velocity of a projectile, as one of the exterior ballistic parameters, has been used extensively to monitor some important attributes of shooter and ammunition. Usually, velocity measurement of a projectile is categorized in three different methods, which are respectively radar, high-speed photography and zone-block. The radar measuring velocity is mainly based on the Doppler effect principle whereby the frequency difference between a transmitted wave and reflected wave is related to the velocity of the projectile (He and Li, 2009). Because high-speed photography recording frequency is fixed, the velocity of the projectile can be analysed by recording the frames and capturing the moving position of the object in the field of view. The zone-block method is widely used to measure the velocity of projectiles. As shown in Figure 1, the zone-block device consists of two targets. Each target with an induction area is perpendicular to the trajectory line, and when the projectile passes through this area, an induction signal will be generated. If the distance between two targets known as L, then by detecting the time difference Δt of these two induction signals, the velocity of an intermediate point between two targets can be calculated with Equation (1).

v = L / Δ t

(1)

Figure 1.

The principle of zone-block device measuring velocity.

According to the different principles of the induction area, zone-block can be divided into three types, including light screen, electromagnetic induction and switch. The main advantages of zone-block are that it is easy to design, has low maintenance cost and is inexpensive. However, there are also some disadvantages, such as the sensitivity not being high, it is susceptible to external stray light and shows poor vibration resistance. This study selects the light screen as zone-block where the laser is chosen as the light source to form a sensing area. The reason for choosing a laser as a light source is that the laser has the advantages of being stable, and having high energy and high sensitivity (Kang et al., 2011).

For velocity measurement, there is still a problem in there is no standard instrument directly calibrating the velocity measuring device, as the velocity is a calculated value (Wang et al., 2009). The most common method is to calibrate the zone-block by firing standard ammunition; the calibration of the ammunition fired is important for accuracy of measuring the projectile velocity device and determining the indication error, but there is a velocity error of 2–5% for standard ammunition because of inconsistency, so the calibration precision is difficult to guarantee. In this study, we hope not only to improve the test precision, but also hope that this set of equipment can be used as a standard to calibrate other zone-block devices.

Theory and method

Statistical theory and method

In this study, the measurement principle of the device is shown in Figure 2. Through the trajectory line, five groups of zone-block are symmetrically arranged at both sides of the monitored point. For each shot, five velocity values will be obtained.

Figure 2.

Measuring principle.

These five values can be calculated by Equation (2), where v is the velocity, j the range and Δt the time difference.

v_{i} = Li / Δ ti, i = 1, 2, 3, 4, 5

(2)

The average velocity of the five-group zone-block can be calculated by Equation (3), where n is the total number of the zone-blocks. In this study, n is equal to five.

\bar{v} = \frac{1}{n} \sum_{i = 1}^{n} v_{i}

(3)

The standard deviation of each group is evaluated with the Bessel formula, as shown in Equation (4).

s_{i} = \sqrt{\frac{\sum_{i = 1}^{n} (v_{i} - \bar{v})^{2}}{n - 1}}

(4)

The standard uncertainty of the average velocity can be evaluated by Equation (5).

σ_{\bar{v}} = \sqrt{\frac{\sum_{i = 1}^{n} (v_{i} - \bar{v})^{2}}{n (n - 1)}}

(5)

Based on the above analysis, the velocity measurement accuracy of the measured point can obviously be improved by increasing the number of zone-block devices, and when n = 1, this is the common used method.

Unbiased, consistent and valid test

It should be discussed whether the result calculated by Equation (3) can be regarded as an estimation for true value, in an unbiased, consistent and valid test.

If the mathematical expectation of the estimators is equal to itself, they is unbiased estimators. When the sample number is limited, whether the estimator is unbiased can be checked by the F-test. Adopting the F-test method, if the Equation (6) is true, it can be considered that the result of Equation (3) is unbiased compared with other groups. In Equation (6), m is sample size for each experiment, $\bar{v_{i}}$ and $\bar{v_{j}}$ are the average velocities obtained by i and j group zone-block, respectively, and α is a given significant level.

\frac{\frac{1}{m_{i}} \sum_{l = 1}^{m_{i}} {(V_{il} - {\bar{V}}_{i})}^{2}}{\frac{1}{m_{j}} \sum_{l = 1}^{m_{j}} {(V_{jl} - {\bar{V}}_{j})}^{2}} < F_{\frac{α}{2}} (m_{i} - 1, m_{j} - 1)

(6)

Howell (2012) and Chen et al. (2005) have used the T-test as a method of consistency estimator, so the method is used in this study. The T-test is used to infer the probability difference happening with the t-distribution theory, and then the difference of the average value is estimated. The projectile velocity measured by each zone-block obeys normal distribution. The consistency can be decided by Equation (7), wherein $\bar{v}$ represents the average velocity calculated with the Equation (3). In Equation (7), i is the serial number for each zone-block, $v_{i}$ is the measured valid velocity value, α is a given probability level, σ_v is standard deviation and n is the total number of the zone-block.

| v_{i} - \bar{v} | > \frac{σ_{\bar{v_{i}}}}{\sqrt{n}} \cdot t_{\frac{a}{2}} (n - 1)

(7)

The test results $\bar{v}$ can be considered more valid than $v_{i}$ if Equation (8) is true, where $D (v_{i})$ represents the variance of $v_{i}$ and $D (\bar{v})$ represents the variance of $\bar{v}$ .

D (\bar{v}) < D (v_{i})

(8)

Systems architecture

The system architecture consists of five groups of zone-blocks, the frame, the signal conditioning module and the data acquisition equipment, shown in Figure 3. The detailed architecture will be explained in this section.

Figure 3.

System architecture.

A zone-block, as shown in the Figure 4, is composed of two parts that are the light-emitting portion and the photosensitive part. The frame size of the zone-block is 1360×920 mm and the internal dimension of zone-block is 800×800 mm. In the light-emitting part, many lasers pass through the optical lens and the diaphragm, and form a light area of a thickness of 3 mm.

Figure 4.

Zone-block.

The performance of the light-emitting portion is shown in Figure 5. The photosensitive part comprises a plurality of photoelectric sensors, as shown in Figure 6 (Dabov et al., 2007). One piece of the photosensitive device has 16 photosensitive pixels. The signal processing circuit is composed of a photoelectric conversion, the amplifier circuit, a comparing circuit and a shaping circuit (Kejzlar and Fischer, 2003). The principle block diagram is shown in Figure 7. A projectile target occlusion causes a light changing flux and a photoelectric sensor will convert the changing flux into a current signal. The current signal is finally converted into a pulse signal output after the signal processing circuit (Garinei and Marsili, 2013).

Figure 5.

The performance of the light-emitting portion.

Figure 6.

Photosensitivity.

Figure 7.

Signal processing circuit block diagram.

The device uses an NI data acquisition system with a timer of 60 MHz frequency. The measurement accuracy of the timer depends on the frequency. LABVIEW provides special engineering tools to gain information from the data obtained by the hardware platform.

The framework is built of aluminium alloy with high strength and rigidity, as shown in Figure 8. The section dimensions are 80×80 mm. The outer frame section adopts a special link part and a fasteners link. The lead rail providing support and installation guide is mounted, the dimensions of which are 3980×1360×1620 mm (length×width×height).

Figure 8.

The framework built with aluminium alloy.

The conditioning signal is sampled by a data acquisition system, and the test data display, analysis and processing through the special test program written (Qiu, 2013).

Uncertainty analysis

As seen from Equation (1), the uncertainty of the velocity is determined by the distance L and the time difference Δt. Measuring the distance uncertainty is composed of five parts: 1) the error $u_{s 1}$ from measuring the two zone-blocks distance determined by a measuring tool; 2) the error $u_{s 2}$ caused by two zone-block being non-parallel; 3) the error $u_{s 3}$ caused by the screen and the horizontal plane being non-parallel, 4) the error $u_{s 4}$ introduced by the screen having uneven thickness, and 5) the error $u_{s 5}$ produced by the angle of incidence of the projectile expressed as $θ$ being non-zero.

The laser ranger used to measure the distance has an accuracy of 1 mm, so $u_{s 1}$ is equal to 1 mm. Fixing the zone-blocks ensures that the maximum distance of the four corners of a group of zone-block is 0.5 mm, so $u_{s 2}$ can be less than 0.25 mm. By adjusting the verticality of the zone-block plane with a laser level meter, $u_{s 3}$ can be limited to within 0.1 mm. According to the measurements of the target plane, the maximum error of $u_{s 4}$ is 0.2 mm. If shooting on the level, the angle of incidence $u_{s 4}$ may be less than 1, and if two zone-block distance L is 1500 mm, the maximum error of $u_{s 5}$ is 0.23 mm according to Equation (9).

u_{S 5} = \frac{L}{\cos θ} - L = L (\frac{1}{\cos θ} - 1)

(9)

From the above analysis, the composite error can be obtained by Equation (10) (Fei, 2010).

\begin{matrix} u_{s} & = \sqrt{u_{s 1}^{2} + u_{s 2}^{2} + u_{s 3}^{2} + u_{s 4}^{2} + u_{s 5}^{2}} \\ = \sqrt{1^{2} + 0 . 25^{2} + 0 . 1^{2} + 0 . 68^{2} + 0 . 23^{2}} = 1.26 mm \end{matrix}

(10)

The time difference Δt uncertainty is composed of two parts, including the uncertainty of the instrument measuring time and the uncertainty caused by the electrical characteristics differences from the signal processing circuit. As the time difference is measured by the data acquisition device with a sampling frequency of 60 MHz, the maximum measurement error is about 0.0167 µs, a measuring time uncertainty so small that can be neglected. The differences in electrical characteristics can cause the two timing signals from a group of zone-blocks to generate a phase difference.

Based on strictly screening the electronic components and ensuring consistency of each functional unit, the uncertainty $u_{t}$ brought by the electrical characteristics differences can be controlled for 0.1 μs.

The temperature, pressure and crosswind must influence the measurement result in measurement as a type B uncertainty, but the influence of the temperature and pressure is so small that it can be neglected in the measurement. When test is carried out indoors, it is not necessary to consider the influence of crosswinds because of the windless condition; the calibre of projectile is small and the body of projectile is based on the principle of reducing air resistance. The wind speed is negligible when the test is carried out outdoors, so the uncertainty caused by crosswind is much smaller than that caused by instruments and can be neglected.

According to Equation (11) of the combined standard uncertainty, the standard uncertainty $u_{v}$ of velocity can be obtained by Equation (12).

u_{c} = \sqrt{\sum_{i = 1}^{n} (\frac{\partial f}{\partial x_{i}}) u_{xi}^{2}}

(11)

u_{v} = \sqrt{\frac{u_{s}^{2}}{t^{2}} + \frac{L^{2}}{t^{4}} u_{t}^{2}}

(12)

If the zone-block distance L is regarded as 1500 mm and the maximum velocity of projectile v is 2000 m/s, the standard uncertainty of velocity can be calculated by Equation (12). So the uncertainty of a group of zone-block can be confirmed as less than 0.2% by Equation (12).

From the above analysis, for actual operations considering the uncertainty of various factors for measurement, the measurement uncertainty of a group of zone-blocks is better than 0.2%. If the measurement uncertainty of a group of zone-blocks is $u_{v}$ , then the uncertainty of the whole system can be calculated by Equation (13), where n denotes the total group number of zone-blocks.

u = \frac{u_{s}}{\sqrt{n}}

(13)

Taking the $u_{v}$ = 0.2% into Equation (13), it can be known that if the whole system uncertainty is less than 0.1%, then the value of n should not be less than 5.

Experimental investigation

Experimental

In order to check whether the system measurement uncertainty meets the design requirements, the experimental investigations are carried out indoors and outdoors. The indoor test objects are those weapons with different calibres of 5.56, 5.8 and 7.62 mm, and the velocity range is from 400 to 1000 m/s. Outdoor experiments are conducted on 12.7-mm and 30-mm-calibre weapons and the examination velocity range is from 1000 to 2000 m/s.

As shown in Figure 9, the system is assembled in the room with the requirements that each zone-block is perpendicular to the horizontal plane and trajectory, so the error is not more than ±1°. Regarding the measured point as the centre, the five groups of zone-blocks are arranged symmetrically to ensure that the groups of zone-blocks measure the velocity of the same point.

Figure 9.

The system assembled in the room with these requirements that each zone-block is perpendicular to the horizontal plane and trajectory.

Figures 10 and 11 for the shooting scene firing test requires each group to launch 10 rounds and in 20 minutes the shooting is over. Data acquisition and processing software programmed in the data acquisition system can give the velocity measured by each zone-block, the average velocity of the five group targets and its uncertainty.

Figure 10.

The 4-mm air gun weapons firing test are carried out indoors.

Figure 11.

The 5.56-mm rifle weapons firing test are carried out indoors.

The test results are shown in Tables 1 –7. The uncertainty (k = 2) of $\bar{x}$ is calculated with Equation (14), where $x_{i}$ is the velocity of a group zone-block and n is the total number of zone-blocks, here n = 5.

\begin{matrix} u (\bar{x}) = \frac{2}{\sqrt{n}} \cdot \sqrt{\frac{\sum_{i = 1}^{n} {(x_{i} - \bar{x})}^{2}}{n - 1}} \times 100 % \\ = \frac{2}{\sqrt{n}} \cdot \sqrt{\frac{n \sum_{i = 1}^{n} x_{i}^{2} - {(\sum_{i = 1}^{n} x_{i})}^{2}}{n (n - 1)}} \times 100 % \end{matrix}

(14)

Table 1.

Test results of the 4-mm air gun.

4-mm air gun	I	II	III	IV	V	$\bar{x}$	$u (x)$
Time	Velocity (m/s)	Velocity (m/s)	Velocity (m/s)	Velocity (m/s)	Velocity (m/s)	Average velocity (m/s)	Uncertainty
1st shot	140.3	140.2	140.1	140.2	140.2	140.2	0.025
2nd shot	161.8	161.7	161.6	161.8	161.7	161.7	0.027
3rd shot	134.4	134.3	134.2	134.3	134.3	134.3	0.027
4th shot	153.7	153.6	153.5	153.6	153.6	153.6	0.026
5th shot	120.6	120.6	120.5	120.5	120.6	120.6	0.029
6th shot	153.8	153.7	153.6	153.7	153.7	153.7	0.027
7th shot	147.1	147.0	146.9	147.1	147.0	147.0	0.034
8th shot	167.8	167.7	167.6	167.7	167.7	167.7	0.028
9th shot	150.2	150.1	150.0	150.1	150.1	150.1	0.027
10th shot	155.3	155.2	155.1	155.2	155.2	155.2	0.028

Table 2.

The results of 5.56-mm rifle.

5.56-mm rifle	I	II	III	IV	V	$\bar{x}$	$u (x)$
Time	Velocity (m/s)	Velocity (m/s)	Velocity (m/s)	Velocity (m/s)	Velocity (m/s)	Average velocity (m/s)	Uncertainty
1st shot	891.482	891.116	891.473	890.434	890.462	890.993	0.522
2nd shot	896.156	896.702	896.552	896.069	895.466	896.189	0.482
3rd shot	884.672	884.171	884.615	883.415	884.857	884.346	0.585
4th shot	891.482	891.599	891.207	890.728	890.462	891.096	0.490
5th shot	889.495	889.189	889.087	889.256	888.146	889.035	0.522
6th Shot	892.147	891.599	892.005	891.024	891.126	891.580	0.506
7th shot	894.370	894.022	894.139	893.094	893.792	893.883	0.488
8th shot	907.253	906.836	907.160	906.640	905.989	906.776	0.497
9th shot	887.954	888.229	887.767	887.203	887.486	887.728	0.402
10th shot	890.377	890.875	890.146	890.139	890.794	890.466	0.352

Table 3.

The results of 5.8-mm rifle.

5.8-mm rifle	I	II	III	IV	V	$\bar{x}$	$u (x)$
Time	Velocity (m/s)	Velocity (m/s)	Velocity (m/s)	Velocity (m/s)	Velocity (m/s)	Average velocity (m/s)	Uncertainty
1st shot	970.008	969.358	969.206	968.671	968.788	969.206	0.490
2nd shot	966.873	966.226	966.387	966.583	965.267	966.267	0.564
3rd shot	957.333	957.230	957.413	956.274	956.383	956.927	0.515
4th shot	962.209	961.988	962.963	962.777	962.545	962.496	0.372
5th shot	954.787	954.730	954.357	954.239	953.711	954.365	0.407
6th shot	967.394	966.794	966.699	966.583	965.657	966.625	0.579
7th shot	961.951	962.833	962.033	961.401	960.997	961.843	0.647
8th shot	972.636	972.510	971.726	971.119	971.150	971.828	0.665
9th shot	967.655	967.647	967.638	967.278	966.438	967.331	0.485
10th shot	966.873	966.510	967.638	966.583	966.047	966.730	0.544

Table 4.

The results of 7.62-mm pistol.

7.62-mm pistol	I	II	III	IV	V	$\bar{x}$	$u (x)$
Time	Velocity (m/s)	Velocity (m/s)	Velocity (m/s)	Velocity (m/s)	Velocity (m/s)	Average velocity (m/s)	Uncertainty
1st shot	436.474	436.224	436.242	435.910	435.734	436.117	0.602
2nd shot	449.987	449.761	449.962	449.607	449.586	449.781	0.377
3rd shot	435.891	435.646	435.860	435.486	435.337	435.644	0.489
4th shot	454.488	454.294	454.407	453.855	453.855	454.180	0.600
5th shot	428.452	428.162	428.490	428.139	427.932	428.235	0.488
6th shot	446.129	445.981	446.002	445.586	445.314	445.802	0.683
7th shot	445.575	445.377	445.404	444.996	444.817	445.234	0.633
8th shot	436.899	436.629	436.815	436.547	436.291	436.636	0.490
9th shot	434.783	434.668	434.719	434.291	434.151	434.522	0.581
10th shot	454.028	453.856	453.924	453.549	453.510	453.774	0.456

Table 5.

The results of 7.62-mm rifle.

7.62-mm rifle	I	II	III	IV	V	$\bar{x}$	$u (x)$
Time	Velocity (m/s)	Velocity (m/s)	Velocity (m/s)	Velocity (m/s)	Velocity (m/s)	Average velocity (m/s)	Uncertainty
1st shot	722.625	721.808	722.571	721.954	721.618	722.115	0.566
2nd shot	722.770	722.442	722.222	721.986	721.183	722.121	0.741
3rd shot	693.586	693.508	693.575	692.942	692.754	693.273	0.509
4th shot	715.852	715.062	715.140	714.475	714.499	715.006	0.706
5th shot	711.455	711.044	711.058	710.512	711.098	711.033	0.424
6th shot	716.567	716.619	716.339	715.997	715.569	716.218	0.547
7th shot	717.713	717.401	717.543	716.760	717.072	717.298	0.476
8th shot	728.047	727.715	728.380	727.617	727.993	727.950	0.370
9th shot	720.884	721.649	720.829	720.444	720.880	720.937	0.544
10th shot	703.370	702.991	703.529	702.717	702.321	702.986	0.622

Table 6.

The test results of 12.7-mm gun.

12.7-mm	I	II	III	IV	V	$\bar{x}$	$u (x)$
Time	Velocity (m/s)	Velocity (m/s)	Velocity (m/s)	Velocity (m/s)	Velocity (m/s)	Average velocity (m/s)	Uncertainty
1st shot	804.714	804.555	804.799	803.892	803.571	804.306	0.605
2nd shot	787.220	787.015	786.561	786.467	787.129	786.878	0.389
3rd shot	814.218	814.126	814.017	813.144	813.715	813.844	0.478
4th shot	810.353	810.111	810.701	809.466	810.398	810.206	0.512
5th shot	794.724	794.822	794.940	793.909	794.206	794.520	0.498
6th shot	832.559	832.911	831.708	832.042	832.170	832.278	0.502
7th shot	831.594	831.225	831.476	830.755	830.143	831.038	0.641
8th shot	837.617	837.583	836.838	836.449	837.724	837.242	0.605
9th shot	792.616	792.712	793.039	791.802	792.096	792.453	0.561
10th shot	830.245	829.965	830.089	829.472	829.277	829.810	0.448

Table 7.

Test results of the 30-mm cannon of 2000-m/s loading dose.

2000m/s	I	II	III	IV	V	$\bar{x}$	$u (x)$
Time	Velocity (m/s)	Velocity (m/s)	Velocity (m/s)	Velocity (m/s)	Velocity (m/s)	Average velocity (m/s)	Uncertainty
1st shot	2075.090	2076.748	2077.479	2075.384	2075.472	2076.035	0.443
2nd shot	2067.912	2070.465	2071.285	2067.559	2066.309	2068.706	0.903
3rd shot	1993.221	1996.111	1997.192	1993.323	1992.321	1994.434	0.940
4th shot	2029.190	2031.791	2031.973	2029.763	2030.280	2030.599	0.544
5th shot	2031.719	2034.434	2034.048	2032.065	2032.873	2033.028	0.524
6th shot	2034.831	2037.209	2038.630	2038.076	2033.566	2036.462	0.955
7th shot	2012.907	2015.461	2015.384	2013.790	2015.026	2014.514	0.498
8th shot	2057.002	2059.045	2058.155	2056.797	2057.404	2057.681	0.401
9th shot	2059.364	2061.759	2060.995	2059.477	2060.957	2060.510	0.454
10th shot	2016.641	2018.805	2018.789	2014.696	2017.751	2017.337	0.765

The velocity, average velocity and uncertainty with 4-mm calibre air gun firing are measured for each zone-block device.

From the test and calculation results in the tables, the uncertainty of the system can meet 1‰ (k = 2) requirements.

As shown in Figures 12 and 13, the 12.7-mm and 30-mm-calibre weapons firing tests are carried out outdoors.

Figure 12.

The 12.7-mm-calibre weapons firing test are carried out outdoors.

Figure 13.

The 30-mm-calibre weapons firing test are carried out outdoors.

Statistical analysis

In order to verify that the measurement result is unbiased, consistent and valid for this true velocity, the methods described above may be put to the test. The method of analysis is the same in each one, so here only the 5.8-mm test results are analysed. Table 3 is the test results of the 5.8-mm rifle. By the normalization of processing, Table 8 can be obtained.

Table 8.

Test results of the 5.8-mm rifle.

5.8-mm rifle	I	II	III	IV	V	$\bar{x}$
Time	Velocity (m/s)	Velocity (m/s)	Velocity (m/s)	Velocity (m/s)	Velocity (m/s)	Average velocity (m/s)
1st shot	1.000827	1.000157	1.000000	0.999448	0.999569	1
2nd shot	1.000627	0.999958	1.000124	1.000327	0.998965	1
3rd shot	1.000424	1.000317	1.000508	0.999318	0.999432	1
4th shot	0.999702	0.999472	1.000485	1.000292	1.000051	1
5th shot	1.000442	1.000382	0.999992	0.999868	0.999315	1
6th shot	1.000796	1.000175	1.000077	0.999957	0.998999	1
7th shot	1.000112	1.001029	1.000198	0.99954	0.99912	1
8th shot	1.000831	1.000702	0.999895	0.99927	0.999302	1
9th shot	1.000335	1.000327	1.000317	0.999945	0.999077	1
10th shot	1.000148	0.999772	1.000939	0.999848	0.999293	1

The average velocity standard deviation of each target is shown in Table 9, in which it can be seen that the standard deviation is not equal for each one. If the significance level $α$ is 0.1, the F hypothesis testing can be obtained for each projectile test result value, as shown in Table 10.

Table 9.

Standard error of the mean velocity for the targets.

5.8-mm rifle	Target I	Target II	Target III	Target IV	Target V
$\frac{1}{m} \sum_{l = 1}^{m_{i}} {(V_{il}^{g} - {\bar{V}}_{i}^{g})}^{2} / 10^{- 7}$	1.211	1.768	0.905	1.273	0.937

Table 10.

Data analysis results based on F hypothesis test for each target.

5.8-mm rifle	I–II	I–III	I–IV	I–V
$F_{0.05} (9, 9) = 3.179$	0.685	1.338	0.951	1.292
	II–I	II–III	II–IV	II–V
	1.460	1.954	1.389	1.887
	III–I	III–II	III–IV	III–V
	0.747	0.512	0.711	0.966
	IV–I	IV–II	IV–III	IV–V
	1.501	0.720	1.407	1.359
	V–I	V–II	V–III	V–IV
	0.774	0.530	1.035	0.736

From the above results can be seen, for different kinds of projectile data to calculate the F hypothesis testing results greater than the I–V target, the ratio between any two target standard deviation can be obtained, and therefore the target accuracy requirements.

With Equation (7), the data in Table 8 can be estimated. If the significance level $α$ is 0.5, the results of data processing are shown in Table 11.

Table 11.

T-test results.

5.8-mm rifle	Target I	Target II	Target III	Target IV	Target V
${\bar{V}}_{i}^{g}$	1.000424	1.000229	1.000254	1.000254	0.999781
$σ_{{\bar{V}}_{i}^{g}} / 10^{4}$	3.480	4.205	3.007	3.569	3.062
$\| {\bar{V}}_{i}^{g} - {\bar{V}}^{g} \| / 10^{4}$	4.243	2.290	2.534	2.188	6.878
$σ_{\frac{{\bar{V}}_{i}^{g}}{\sqrt{n}} \cdot t_{\frac{α}{2}} (n - 1) / 10^{4}}$	1.153	1.393	0.997	1.183	1.015

As shown in Table 11, the significance level $α = 0.5$ , $t_{0.25} (5 - 1) = 0.741$ , the calculation method according to the consistency and $| \bar{V_{i}^{g}} - \bar{V^{g}} |$ can be obtained in the I, II, III, IV and V zone-blocks, respectively, and the average velocity of the target. Compared with $(σ_{\bar{v} i} / \sqrt{n}) \cdot t_{\frac{a}{2}} (n - 1)$ , it can be seen that the $| \bar{V_{i}^{g}} - \bar{V^{g}} |$ is significantly greater than $(σ_{\bar{v} i} / \sqrt{n}) \cdot t_{\frac{a}{2}} (n - 1)$ , so the I, II, III, IV and V zone-blocks meet the requirement of consistency.

If Equation (8) is a criterion for the validity of estimated value, it is obviously for Equation (3) that $D (\bar{v})$ is less than $D (v_{i})$ .

In summary, by analysing the normalized firing data, the measured velocity in this study is unbiased, consistent and valid for the true velocity value.

Conclusions

This set of equipment based on the principle of statistics to design a projectile testing system, compared with the traditional velocity measurement system, is effective for improving test precision.

The equipment frame for the aluminium section is built and equipped with rubber wheels, a zone cutting device of the modular design, and has convenient disassembly, assembly and movement.

A sensor that comprises a laser-generating, photoelectric induction is a silicon photocell array sensor with higher strength and better stability, and can improve the test sensitivity, detect the target object diameter over 2 mm and detect a maximum speed range of 2000 m/s.

Because mostly a derived quantity for speed and not speed directly calibrated, and equipment has a high precision, so the system can be used not only as the projectile velocity detecting device, but also can be used for the calibration of other projectile velocity measurement equipment.

Footnotes

Conflict of interest

The authors declare that there is no conflict of interest.

Funding

This research received no specific grant from any funding agency in the public, commercial or not-for-profit sectors.

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