1 Introduction

Despite continuous advances in our ability to numerically simulate blast waves, the field of blast analytics still relies heavily on accumulating experimental data. The importance of this data cannot be overstated, particularly within the defence industry. Capturing blast waves, commonly in large arrays, requires a considerable investment in setup time. Such arrays include sensors, hundreds of meters of cabling (typical in significant explosive events), power generation units, amplifiers, and data loggers. This investment in time and resources often conflicts with industry priorities, resulting in smaller, inadequate arrays being used or none at all.

At present, data acquisition (DAQ) instrument manufacturers and research institutes have access to essentially the same processing technology. This study investigates the feasibility of leveraging commercially available, robust, low-cost processors commonly housed on integrated circuits (ICs). These ICs already include many of the required peripheral circuitry, such as power regulators, analog-to-digital converters (ADCs), and noise filtering. This study investigates the use of such an IC to accurately measure blast wave profiles (pressure vs. time) to be later used in a portable, meshed array of sensors. The use of these ICs as data loggers has previously been extensively investigated. Ying et al. [1] demonstrated a use case in which such an IC successfully measures blast waves generated by 10 g of TNT and continuous logging approach, employing a single ARM (advanced RISC machine) IC. Other authors [2,3,4] have also documented similar data logging use cases for these boards, citing ease of use, low cost, and high computing power as the primary reasons for adopting these prototyping ICs, more traditionally used for prototyping, thanks to their easy to access general purpose input/output (GPIO) pins. The benchmark scores of various processors are shown in Fig. 1.

Fig. 1
figure 1

The benchmark scores shown here were derived by executing linked list management and matrix multiplication operations on each processor, creating a standardized workload for evaluating the computational capabilities of each. A higher score indicates superior performance and can serve as a simplified metric for measuring performance. For example, the ARM Cortex M7 achieved a score of 2314, making it approximately 4.3 times faster than the ARM Cortex M4, which scored 536. These results highlight the M7 processor’s excellent suitability for data logging compared to the M4 processor [5]

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The Teensy 4.1 prototyping board with ARM Cortex M7 and dual ADC was selected as the most suitable evaluation board to capture 10-bit data at a theoretical un-averaged 2.66-Msps (megasamples per second) sampling rate, including bus cycle estimates and direct memory access (DMA) overhead. A sampling rate of 2.66 Msps would adequately capture the blast profile. Skotak et al. [6] demonstrated this in explosive shock tube measurements with blast waves moving slower than Mach 2, suggesting that sampling frequencies beyond 1 MHz provide only superfluous improvements in results. Moon et al. [7] compared 24-bit 204-ksps and 10-bit 12.5-Msps DAQ systems, concluding that both systems demonstrate similar performance in capturing blast waves despite their vastly different capabilities. Neither frequency nor resolution plays a superior role in capturing blast waves. Instead, a balance between these parameters is required. The Nyquist theorem states that a signal must be sampled at least twice the speed of the highest frequency components of that signal to avoid aliasing, which is often cited as a limitation in blast wave analysis. In reality, it is proposed that this has little relevance when considering a conventional Friedlander blast wave, which is not a repeating sinusoidal curve. Any aliasing that may occur shall only affect the signal noise generated by the explosive environment. This will result in only superficial variations of recorded pressures and impulses. In this study, aliasing shall be considered to have an identical effect on both experimental and commercial systems; this allows the investigator to negate the effect of such variables on metrics such as peak pressure, impulse, and duration. One exception to this, however, is the measurement of rise time, to be discussed later.

In this study, the evaluation of an ARM Cortex M7 processor to accurately measure and record hemispherical free-field blast waves was undertaken. These results were then compared against an industry-standard blast measurement DAQ system to evaluate the accuracy of the experimental system. Successful validation of this system shall lay the groundwork for a proposed large array of remote DAQ nodes capable of measuring full-scale blast events and generating a three-dimensional map of the blast wave with real-time data.

2 Measurement of hemispherical free-field blast waves

Fig. 2
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An ideal blast wave profile represented by the Friedlander waveform, which characterizes a blast wave as a sudden increase in pressure from gauge pressure, \(P_{\textrm{0}}\), to a peak pressure, \(P_{\textrm{max}}\), eventually decaying into a negative overpressure zone

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Conventional blast analytics involves the study of blast waves generated by the detonation of a high explosive. The result is the near-instantaneous generation of a shock front moving at supersonic velocity outwards from the point of initiation. As all the explosive material behind the shock front is consumed, the blast wave transitions gradually into a more stable far-field blast wave in air, characterized by Friedlander [8] as a near-instantaneous rise in ambient pressure to a peak pressure, \(P_{\textrm{max}}\), followed by the gradual decay into a negative overpressure zone until a minimum pressure, \(P_{\textrm{min}}\), is reached, before returning to ambient pressure, \(P_{\textrm{0}}\) (see Fig. 2).

The far field is defined in this work as the domain within which all combustion of explosive material has been completed and the shock front has stabilized into an expanding front; this is predominantly where overpressure blast wave measurements are taken. Recent research has identified three distinct ranges in shock front behavior based on scaled distances, Z, given by:

$$\begin{aligned} Z=\frac{R}{W^{\frac{1}{3}}} , \end{aligned}$$
(1)

where W (kg) represents the TNT (trinitrotoluene) equivalent mass and R (m) represents the distance from the explosive charge. In the extreme near field (\(\le \) 0.5 m/kg\(^{\mathrm {1/3}})\) and far field (\(\ge \) 2 m/kg\(^{\mathrm {1/3}})\), shock front data are consistent, whereas in the intermediate range (0.5–2 m/kg\(^{\mathrm {1/3}})\), there is more variability. Studies [9,10,11] using different methods, including ground detonations and image tracking, all support these findings. Although not universally accepted, it is generally agreed that the far field begins at roughly 1 m/kg\(^{\mathrm {1/3}}\) [12, 13]. For this reason, it was deemed prudent to maintain these experiments beyond the scaled distance of 2 m/kg\(^{\mathrm {1/3}}\). However, it is crucial to understand where this area is and the approximate blast wave characteristics to establish an optimal test setup. These blast wave characteristics are calculated using a set of well-established empirical formulae derived from experimental data [14, 15]. Post-experimentation, these formulae will also provide an additional means of validation of the measured blast wave.

The modified Friedlander equation describes the typical blast wave profile (see Fig. 2) as follows:

$$\begin{aligned} P\left( t \right) =P_{0}+P_{\textrm{pos}}\left( 1-\frac{t}{t_{\textrm{pos}}} \right) \textrm{e}^{-b\frac{t}{t_{\textrm{pos}}}} , \end{aligned}$$
(2)

where \(P_{0}\) is the ambient pressure, \(P_{\textrm{pos}}\) is the peak overpressure, b is the decay coefficient, and\( t_{\textrm{pos}}\) is the duration of positive overpressure. Equation (2) gives pressure for any point in time, t, from the arrival of the blast wave and represents a first-order approximation for blast wave profiles providing peak overpressure, duration, and impulse. Within far-field blast waves, these parameters may be successfully measured at sampling speeds as low as 10 kHz, often with no statistically significant differences found compared to samples taken at sampling speeds upwards of 1 MHz [6]. Empirical values established through experiments are well documented for blast waves [16]. In this study, empirical formulas are used for calculating various blast wave parameters. These formulas have been extensively discussed in the literature, including a review in reference [14], and are particularly applicable under near-perfect free-field conditions. These parameters include positive peak overpressure, \(P_{\textrm{pos}}\) (Pa), given by:

$$\begin{aligned} P_{\textrm{pos}}= \frac{808\left[ 1+\left( \frac{Z}{4.5} \right) ^{2} \right] }{\sqrt{\left[ 1+\frac{Z}{0.048} \right] ^{2}}.{\sqrt{\left[ 1+\frac{Z}{0.32} \right] ^{2}.}} {\sqrt{\left[ 1+\frac{Z}{1.35} \right] ^{2} }}}; \end{aligned}$$
(3)

duration of positive overpressure, \(t_{\textrm{pos}}\) (µs), given by:

$$\begin{aligned} t_{\textrm{pos}}= \frac{980\left[ 1+\left( \frac{Z}{0.54} \right) ^{10} \right] }{\left[ 1+\frac{Z}{0.02} \right] ^{3}.\left[ 1+\frac{Z}{0.74} \right] ^{6}.\left[ 1+\frac{Z}{6.9} \right] ^{2}} ; \end{aligned}$$
(4)

and positive impulse, \(I_{\textrm{pos}}\) (bar ms), given by:

$$\begin{aligned} I_{\textrm{pos}}= \frac{0.067\sqrt{1+\left( \frac{Z}{0.23} \right) ^{4}} }{Z^{2}\root 3 \of {1+\left[ \frac{Z}{1.55} \right] ^{3}}} . \end{aligned}$$
(5)

The equations (3)–(5) are based on the work of Kinney and Graham [17]. They show excellent correlation with experiments for the calculation of positive peak overpressure, \(P_{\textrm{pos}}\), positive pulse duration, \(t_{\textrm{pos}}\), and positive impulse, \(I_{\textrm{pos}}\), across a wide range of scaled distances. A correction factor of 1.8 for overpressure may be applied to account for blast wave reflection from the ground [14].

Fig. 3
figure 3

Experimental system and comparative commercial system hardware layout

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A final critical parameter to consider is rise time, \(t_{\textrm{r}}\), not accounted for in the modified Friedlander equation. It is required to accurately predict the viscoplastic effect of a shock wave on structures and human tissues. Therefore, accurate measurement of rise times plays a significant role in holistic blast analysis, where the rise time response to the natural period of porous bodies, such as buildings, vehicles, and personnel, is of vital importance [16, 18, 19]. Unlike \(P_{\textrm{pos}}\), \(t_{\textrm{pos}}\), and \(I_{\textrm{pos}}\), the rise time in the far-field blast wave conventionally requires significantly higher sampling speeds and is the one exception to the statement that blast waves are immune to the effects of aliasing. Little openly available empirical data have been published to calculate rise time based on scaled distance, the possible exception being that of the US Army [16] directed at determining structural loading on walls and roofs. Therefore, the evaluation of rise time accuracy shall only be made against the comparative commercial system and not empirical relationships. In addition, atmospheric conditions influence the propagation and attenuation of blast waves by refraction, scattering, and absorption [20, 21]. Modeling occasionally disagrees with measured data due to lack of empirical blast data.

3 Hardware

The Teensy 4.1 (T4.1) produced by PJRC was selected to evaluate the M7 processor. The T4.1 incorporates the ARM Cortex M7 processor, operating at 600 MHz with easily accessible general purpose input/output (GPIO) pins, two analog-to-digital converters (ADCs), 1024 kb tightly coupled random access memory (RAM), and a 4-bit secure digital input/output (SDIO), making it suitable for this DAQ use case.

Signal conditioning is handled by an Analog Devices™AD8429 instrument amplifier connected in a unipolar configuration and set at a suitable gain. A 1.2-V reference voltage output from the T4.1 is used as the instrument amplifier reference voltage. The ADC range is thus limited to a 2.1-V range. This limit is considered in the test setup discussed later to avoid saturation of the measurements. The system diagram is illustrated in the Appendix.

The commercial system used for comparison was a 2310 Vishnay signal conditioning amplifier and DAQ system capable of sampling at 10 MHz with 16-bit resolution. During testing, sampling rates were matched at 1 MHz. Figure 3 illustrates the hardware setup for the experimental and comparative commercial systems used in the tests.

Critical to pressure measurements using micro-electromechanical systems (MEMS) pressure transducers are the effects of sensor mounting, ionization, light, thermal transients, signal drift, cable resistance, and differences in dynamic range and frequency of the pressure transducer [22, 23]. By using identical piezoresistive Endevco pressure transducers in both systems (see Table 1), these effects were mitigated and assumed to be identical for both systems, as both transducers were mounted at an equal distance from the explosive. Both were connected to their respective DAQs by an equal length of identical shielded cable with near-identical internal resistance. All pressure transducers were calibrated against a Bourdon gauge according to BS 1780:1985.

Table 1 The Endevco 8530C-100 pressure transducer specifications
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Fig. 4
figure 4

Code workflow from signal monitoring to logging

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4 Software

The experimental system code was written in C\(++\) and built around the Pedvide/ADC and Greiman/SDFat libraries. The code workflow is illustrated in Fig. 4. Testing with the experimental DAQ system showed up to 0.42 ms of jitter at the start of the timer interrupt handler due to the SD card drivers initializing (A in Fig. 4). The jitter was only present upon starting the recording loop and did not interfere with readings recorded after this initial phase. Using interleaved ADC collection made it possible to achieve 1.8 Msps at 10 bits (B in Fig. 4). The ADC data were then written in binary format to the RAM and transferred to the SD card when full (C in Fig. 4). This cycle repeats indefinitely or until the SD is full. It can store 1.5 days of data at 1 Msps using a 128-GB SD card. The process flow for signal processing once a signal is parsed via the instrument amplifier is shown in Fig. 4.

5 Calibration

Calibration was conducted by connecting the commercial and experimental systems to an air source and Bourdon gauge. Transducer excitation was maintained at 10-V DC as per datasheet best practice. The commercial system was initially calibrated using the Bourdon gauge measuring between two known pressures and found to be within a tolerance of 0.3%.

The instrument amplifier output voltage was then trimmed to the lower end of the measurement bandwidth at atmospheric pressure. It was assumed that, with the exception of drift caused by thermal gradients, the atmospheric baseline would be maintained for both systems. No change in altitude would be noted between calibration and field testing locations. Both systems were then coupled to a pressure bar and the pressure increased to 372 kPa. Once this pressure was reached and maintained for 2 s, the pressure bar was purged. This was repeated ten times. The recorded values were averaged and are shown in Fig. 5. The relationship is near linear across the pressure range with a correlation of 0.99 at a 95% confidence interval.

The experimental DAQ system was connected to a nickel metal hydride (NIMH) battery with a 1-k\(\Omega \) current limiting resistor and 10-µF capacitor from the ADC port to ground. This was sampled at various bit rates to measure “internal” ADC variation. An inherent variance of 0.4 % at a nominal voltage of 1.3 V and a sampling speed of 1 Msps was measured. The test was repeated at multiple sample rates, and the results are summarized in Table 2.

Fig. 5
figure 5

Averaged pressure readings across ten calibration cycles. An average deviation of 0.04 kPa was recorded (note that commercial values are shifted by \(+\)8 kPa in order to be visible)

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Table 2 Standard deviation of constant voltage maintained for 2 s
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Fig. 6
figure 6

PE-9 charge shaped using a plastic mold. Material was removed from flat surface until desired mass is reached (± 1 g)

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6 Test setup

The TNT equivalence model provides predictions of blast wave characteristics. Equivalencies for most high explosives are commonly available in the literature. Equivalent TNT mass (\(W_{\textrm{TNT}})\) is given by:

$$\begin{aligned} \frac{W_{\textrm{TNT}}}{{\mathrm {kcal/g}}_{\textrm{TNT}}}=\frac{W_{\textrm{PE}9}}{{\mathrm {kcal/g}}_{\textrm{PE}9}}. \end{aligned}$$
(6)

Empirical formulae (1)–(6) provide boundaries for the test setup, ensuring that a suitable scaled distance and charge weight are used, preventing saturation of pressure transducers while still using the maximum allowable range of the measurement bandwidth.

PE-9 explosive charges of 100-g mass and density of 1.45 g/cm\(^{\textrm{3}}\) (see Fig. 6) were placed 1.4 m from the commercial and experimental sensors, as shown in Fig. 7. The tests produced a series of twenty pressure histories. The empirical blast characteristics obtained are shown in Table 3. Of importance is the scaled distance of 2.7 m/kg\(^{\mathrm {1/3}}\) at which measurements are taken. At this distance according to [9,10,11], the measurements are beyond the near-field region, i.e., outside of combustion and in a region in which the blast wave behaves more predictably. Only static pressure was measured by mounting the pressure sensor on a lollipop gauge ring and positioning the sensor surface side-on to the blast front. Static pressure would negate the effects of dynamic pressure associated with particle motion.

Before conducting the tests, six calibration shots were performed using only commercial DAQ systems at points S1 and replacing S2 with an equivalent commercial sensor (refer to Fig. 7). The purpose of these calibration shots was to evaluate the expected variation between sensors if only the commercial DAQ system were used. Measurements were taken at 10 Msps. Results are presented in Table 4. It is immediately apparent that the measured Spearman correlation and peak pressure are not in a linear relationship. These measurement differences are caused by environmental variations, such as topology and the proximity of each sensor to reflective surfaces, such as walls [24]. It can be concluded from these calibration shots that an averaged pressure history variation between the identical sensor and DAQ combinations in the same test measuring the same blast wave may be up to 8%. It is also noted that the measured peak pressures show variations up to 4 kPa (2.1%).

Fig. 7
figure 7

Test layout: PE9 explosive in the center, surrounded by four Endevco transducers. The green transducers are connected to the experimental DAQ system, while the red transducers are connected to the commercial DAQ system

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Table 3 Blast wave characteristics calculated using formulae (1)–(6)
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Table 4 Calibration shots of Endevco DAQ system using two commercial systems
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Fig. 8
figure 8

Commercial blast wave data exhibiting a typical Friedlander blast curve except for multiple reflected blast peaks captured in the curve

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7 Measurement results and discussion

Pressure versus time measurements were captured as 10-bit binary data and converted using MATLAB. Statistical analysis was performed using Excel and Statistica software. All measurements were evaluated using non-parametric correlation analysis. All p values < 0.05 were considered statistically significant, and the standard error of mean (SEM) was assessed for each test comparison. Only Spearman correlation was used to account for the monotonic relationship of the data. Figures 8 and 9 show all commercial and experimental results, respectively. Worth noting are two distinct reflected blast waves caused by the proximity of the test charge to the reflective walls of the test arena (see Fig. 10). This effect of the reflected wave on the blast profile in individual commercial DAQ results was mimicked in the experimental system.

Fig. 9
figure 9

Experimental blast wave data mimic the commercial pressure data, including the reflected blast peaks

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Fig. 10
figure 10

Blast wave experiments with 100-g shaped PE-9 explosives

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Fig. 11
figure 11

Averaged commercial and experimental test data show close correlation and near-identical feature detection

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Fig. 12
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Linear correlation between experimental and commercial DAQs shows a close correlation at lower pressures, with system deviation only slightly increasing toward higher pressures, and a Spearman correlation of 0.928

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As discussed earlier, the natural variation is expected to be up to 8% per the calibration. In addition, features of both systems, such as wave decay and reflected waves, were consistently similar. Time to peak pressure, reflected waves, and crossover from positive to negative pressure were within 0.2% of each other over the positive period. Figure 11 considers the averaged values for both commercial and experimental systems, showing more clearly the conformance of the experimental system to the commercial results within an average 7% variation. In this study, it is assumed that the commercial system is deemed to be 100% accurate. While 7% represents a significant averaged variation in data, it should be noted that the commonly used blast equations may exhibit large deviations between themselves in excess of 30% [25, 26].

Table 5 Summarized experimental results
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Figure 12, which shows a scatterplot of the averaged commercial sensor measurements versus the experimental sensor measurements, illustrates variations in pressures across the measured pressure range, demonstrating close conformance throughout the range. This is also seen in the small divergence between the experimental and commercial curve in Fig. 11. The calculated positive phase duration is notably shorter than the measured positive phase duration (0.74 ms vs. 1.15 ms, respectively), as the empirical formulae do not consider the effect of reflected waves.

Table 5 summarizes the measured results. Variation in measured peak pressures between sensors S1 and S2 during calibration showed an average 1.9% variance, while the variance measured during testing increased to 3.4% (the highest variation of 5% recorded in test 6). The variation in experimental DAQ results may be due to aliasing caused by the low sampling rate of 1 Msps. The same argument can be applied to the commercial system, although the variation appears to be smaller, likely because the system has undergone more refinement. However, assessing these refinements is beyond the scope of this article. It is noted that this variance may be negligible in many blast wave investigations, and even those evaluating the rate-dependent results of blast waves may consider a 5% variation negligible. In future, however, the experimental system may be assessed at a higher sampling rate, as per Table 2. The rise time was calculated as the time it took for the measured pressure to increase to more than 20 kPa and reach the maximum recorded pressure. Average rise times were measured to be 12.45 µs. The variation between commercial and experimental rise time never exceeded 3 µs; however, pressure increases as high as 12 kPa were recorded in 1 µs, contributing to the higher variations in peak pressure observed in the experimental system. A standard mean error was calculated for each test showing a low discrepancy of the samples’ mean by measuring the sample-to-sample variability.

8 Conclusion

Hemispherical blast waves generated by detonating 100-g PE-9 hemispherical charges have been studied experimentally. An experimental proof of concept for a low-cost DAQ system capable of recording far-field blast waves was successfully demonstrated. An average Spearman correlation coefficient r of 0.928 was recorded for this experimental system compared to the commercial DAQ conventionally used for similar testing, with a worstcase variation in a correlation coefficient of 0.87. This worstcase variation is 5% greater than the worstcase measured during calibration firings, where variation between commercial DAQ systems of up to \(r=0.92\) was recorded. It was concluded that there was a good correlation between the experimental and commercial systems.

Impulse, rise time, peak pressures, and blast wave duration are measured with sufficient fidelity to visualize the blast wave with an appropriate level of accuracy. This challenges the status quo, which seeks to record far-field blast waves at sampling speeds beyond 10 MHz with costly test hardware simply by employing more modern commercially available hardware. It may be concluded that the proposed experimental system provides valid measurements of far-field blast waves, albeit at lower sampling speeds than conventional test hardware. This experimental system would not replace commercial systems; instead it would serve to capture data when the time and budget to layout a commercial sensor array are unavailable.

As an additional benefit, the hardware lends itself to a wireless form factor. Future work will include the design of a stand-alone pressure transducer. This consists of a power source with the evaluated DAQ hardware for use in an array of pressure transducers placed by RTK GPS into rapidly deployable wireless blast transducer arrays.

9 Appendix: Schematic layout of the experimental system

figure a