Introduction

The assessment of blunt force trauma is an essential issue in forensic medicine. About 10% of acts of physical violence are committed with weapons [1]. Forensic scientists are therefore often required to give an expert opinion. In the matter of biomechanics, it is of common concern if a blow to the victim’s head with a suspected weapon can potentially cause a fatal injury, e.g. skull fractures. In such cases, it is also of interest to which extent such a blow is capable of inflicting the documented injury. Moreover, the courses of events given by witnesses, victims and offenders are reviewed in terms of plausibility regarding injury patterns and severity.

Contact forces can be computed using the conservation of momentum, which states that the total momentum prior to the contact equals the total momentum after the contact. Contact forces can be compared to fracture thresholds given in literature (e.g. [2]) allowing the assessment of injury and fatality risk. v 1 and v 2 are the velocities prior to the contact, v 1 * and v 2 * denote the velocities after the contact and m 1 and m 2 are the masses of the objects.

$$ {v}_1{m}_1+{v}_2{m}_2={v}_1^{\ast }{m}_1+{v}_2^{\ast }{m}_2 $$

A system of linear equations results from the incorporation of the formula for the coefficient of restitution k. The maximum contact force F max is yielded with the contact time ∆t (more details and a sample calculation can be found in the ESM):

$$ k=\frac{v_1^{\ast }-{v}_2^{\ast }}{v_2-{v}_1} $$
$$ {F}_{\mathrm{max}}=2\bullet \frac{m_2\left({v}_2^{\ast }-{v}_2\right)}{\Delta t}=2\bullet \frac{m_1\left({v}_1^{\ast }-{v}_1\right)}{\Delta t} $$

The following input parameters need to be estimated: the striking velocity v, the effective mass m acting upon contact, the contact time Δt and the coefficient of restitution k. The effective mass m of longish instruments can determined according to [3]. The contact time Δt and the coefficient of restitution k can be found in references as well [4,5,6,7]. However, all parameters are subject to uncertainties.

Striking velocities are crucial parameters. In current research, there have been studies on velocities of body parts causing bodily harm, showing stomping velocities up to 11.1 m/s [8, 9], head butt velocities up to 4.7 m/s [4] and fist punch velocities up to 12 m/s [10]. Furthermore, there have been studies on impact velocities of several blunt instruments. For example, there have been measurements showing impact velocities of baseball bats up to 31.8 m/s [11, 12], of beer steins up to 14.8 m/s [13, 14] and of aluminium rods up to 34.9 m/s [15]. Adamec et al. also measured impact velocities of aluminium rods. As the strikes in their study were limited to a maximum reaction force below 6 kN, their documented impact velocities cannot be used in terms of maximum striking velocities [3]. These studies on blunt instruments had a maximum volunteer count of 19.

The aim of our study was to measure maximum striking velocities of blunt instruments as an upper bound of impact velocities. With instrument lengths and masses in particular, the influence of the blunt instruments’ parameters on maximum striking velocities was further analysed. Several steel rods were therefore exemplarily tested as striking implements. Moreover, we also acquired a larger volunteer group of 51 in order to examine the influence of volunteer parameters on maximum striking velocities.

Materials and methods

In our experiments, we used five different steel rods as striking implements. Each one had a diameter of 3 cm and an evenly distributed mass. In order to examine the influence of the length and mass of the striking implement on maximum striking velocities, we chose three rods differing in length but not in mass as well as three rods differing in mass but not in length. One rod (no. 2) was used in both settings. In preparation for the measuring of striking velocities by means of infrared light, the rods were lacquered with matte black colour to prevent measurement disturbances due to possible reflections of infrared light by metal surfaces. The proximal end of each rod was wrapped with nonslip tape for secure handling. All rods are listed in Table 1 and photographed in Fig. 1.

Table 1 List of all rods
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Fig. 1
figure 1

Rods no. 1 to 5 from bottom to top

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A 5-cm-thick mat composed of all-weather running track and polyurethane foam rubber was used as a target. It showed only minor rebound on impact in order to keep any injury risks as low as possible. The mat was strapped onto a table with a height of 72 cm. Its position on the table could be adjusted to each volunteer’s preference.

Fifty-one volunteers were requested to one-handedly hit the nonmoving target with each rod from above as fast and as hard as possible. The resultant striking velocities were measured by a Qualisys motion capture system. Twelve Qualisys ProReflex Motion Capture Unit (MCU) 1000 high-speed cameras emitting infrared light were used. The emitted light was reflected by passive retroreflective spherical markers with a diameter of 20 mm. The sampling rate of the high-speed cameras was 500 Hz. One marker each was attached to the distal end of the rods (dr marker) and 19.5 cm distally from the proximal end of the rods (pr marker) leaving a handle. In this study, the maximum striking velocity was defined as the maximum velocity of the dr marker.

After calibrating the MCU cameras, motion capturing was enabled. The motion capturing time was set to 6 s for each strike resulting in image sequences with 3000 frames. During motion capturing time, volunteers had to perform the strikes starting from a neutral position. Without any prior instructions respecting striking technique, each volunteer performed two valid strikes with each implement. Strikes were valid when dr markers were missing less than ten frames per second in each measurement. If more than two strikes were valid, only the ones with the highest and lowest maximum striking velocity were further analysed.

By means of a triangulation algorithm, Qualisys Track Manager 2.10 software calculated the marker positions with a spatial resolution of less than/up to 1.4 mm (standard deviation in system calibration). A 3D data ASCII file was generated with tab-separated coordinates for each frame. Recordings were manually cut adjusting to the striking motions’ duration. Striking duration was defined as the time from the beginning of the backswing to the time of impact. Resultant maximum striking velocities were automatically calculated and analysed with a MATLAB R2015b script. Random noise was eliminated by applying an 11th order median filter. Statistical evaluation was done using IBM SPSS Statistics 24.

Before testing, experiment and safety instructions were given and each volunteer had to sign a written informed consent. To ensure the security of each volunteer and optimal motion capturing, volunteers had to wear formfitting nonreflecting clothes and footwear. In order to prevent any physical injuries, they also had to do warm-up as well as practise strikes. Each strike had to be performed one-handedly, but volunteers were free to choose their individually preferred striking technique.

By means of Qualisys Track Manager 2.10 software, three-dimensional visualisations of a striking performance are illustrated in Fig. 2.

Fig. 2
figure 2

Three-dimensional visualisation of a striking performance with rod no. 2 at marker frames a 1014, b 1172 and c 1259 of 3000. The marker trace of the dr marker follows a bold black trajectory whereas the pr marker trace follows a thin black trajectory. Anatomical landmarks of the body are represented by dark grey spheres and connected by grey bones

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The following terminology will be used only in arguments where confusion has to be expected otherwise. The index n references the volunteers and the index k refers to the experimental setup (identical to the rod number).

  • Set of volunteers: Ω = {ω 1,…, ω N }, N = 51

  • Set of experimental setups: M = {m 1,…, m K }, K = 5

  • Repetitions per (ω n , m k ): I = 2

  • Velocity curve of the ith strike of volunteer ω n in setup m k : v i (t, n, k)

  • Maximum velocity: v i, max(n, k): = max t v i (t, n, k)

  • Maximum striking velocity over both repetitions i = 1, 2 per volunteer n and setup k: v max = max i (v i, max(n, k))

  • Intra-individual ∆v max difference: ∆v max(n, k): = |v 1, max(n, k) − v 2, max(n, k)|

  • Mean intra-individual ∆v max difference: \( \widehat{\mu}\left(\Delta {v}_{\mathrm{max}}(k)\right)=\frac{1}{N}{\sum}_n\Delta {v}_{\mathrm{max}}\left(n,k\right) \)

  • Empirical standard deviation: \( \widehat{\sigma}\left(\Delta {v}_{\mathrm{max}}(k)\right):= \sqrt{\frac{1}{N-1}\sum \limits_n\Big(\Delta {v}_{\mathrm{max}}\left(n,k\right)-\widehat{\mu}{\left(\Delta {v}_{\mathrm{max}}(k)\right)}^2} \)

The ethical committee of the University Hospital Jena gave approval to conduct this study.

Results

Out of 52 volunteers, one volunteer had to be excluded from the test series due to complaints of his right shoulder and wrist during warm-up and practice striking. One volunteer did not perform strikes with rod no. 3. The majority of the volunteers were college students among which eight studied sports science. Relevant parameters of the volunteer sample are given in Table 2.

Table 2 Descriptive statistics of male (m, n = 29) and female (f, n = 22) volunteers
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The velocity curve of the dr marker in the course of a striking performance is demonstrated in Fig. 3. At the beginning of a strike, volunteers make a backswing with the rod and thereby accelerate the dr marker. After the backswing, the marker shortly decelerates as the volunteers prepare for the actual strike which then shows a sudden large increase of marker velocity. At the moment of impact, the acceleration abruptly stops and the marker velocity instantly drops.

Fig. 3
figure 3

Example of a dr marker velocity curve in the course of a striking performance of a female volunteer with rod no. 4. The vertical line marks the point in time where the maximum striking velocity of 20.27 m/s is reached (0.76 s after the start of the striking motion)

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Table 3 shows the descriptive statistics of maximum striking velocities v max for each rod and both male and female volunteers. Compared to female volunteers, male volunteers distinctly achieved higher striking velocities with each rod. Among all the rods with a mass of 1000 g, both male and female volunteers reached higher mean maximum striking velocities with increasing rod lengths. Regarding all the rods with a length of 65 cm, all volunteers obtained higher maximum striking velocities with decreasing rod masses. In general, the lowest striking velocities occurred in the shortest rod of 40 cm. About one-third faster, the mean highest striking velocities were achieved by using the lightest rod of 500 g.

Table 3 Descriptive statistics of maximum striking velocities v max for each rod
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Intra-individual variability is shown in Fig. 4 for strikes with rod no. 1 (Fig. 4 in the ESM contains plots for all rods). Black vertical bars mark the difference ∆v max(n, k) between the maximum striking velocities in both valid measurements of each volunteer and rod. The mean differences of maximum striking velocities \( \widehat{\mu}\left(\Delta {v}_{\mathrm{max}}(k)\right) \) were between 0.76 and 1.3 m/s with empirical standard deviations \( \widehat{\sigma}\left(\Delta {v}_{\mathrm{max}}(k)\right) \) between 0.55 and 1.01 m/s; see Table S1 in the ESM.

Fig. 4
figure 4

Intra-individual variability of maximum striking velocities v max for rod no. 1. The vertical bars show the difference between the two valid measurements of each volunteer that mark the top and bottom end of each bar. Volunteers are sorted by their highest achieved maximum striking velocity in both measurements regardless of their sex and ID in all diagrams

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Statistical analysis

Volunteer parameters as well as maximum striking velocities considering volunteer sex and different rod lengths and masses were tested for normal distribution. For nearly all variables, normal distributions can be assumed. According to both the Kolmogorov-Smirnov as well as the Shapiro-Wilk test, only age did not follow normal distribution. In male volunteers, the Kolmogorov-Smirnov test rejected normal distribution for body mass and average amount of physical exercise per week. Furthermore, the Shapiro-Wilk test rejected normal distribution of the BMI. Please see supplementary Table S2 (ESM) for all results of normal distribution testing.

Results of a two-sample t test comparing mean maximum striking velocities of male and female volunteers for each rod are given in Table 4 together with the p values of Levene’s tests on equality of variances. Levene test p values above 0.05 support the assumption of variance equality. With all t test p values below 0.05, there are significant differences between both sexes. Women achieved 4.95 to 7.27 m/s smaller mean maximum striking velocities than men.

Table 4 Two-sample t test comparing mean maximum striking velocities of male and female volunteers for each rod
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To examine the influence of rod parameters on maximum striking velocities, the paired-sample t test was performed comparing maximum striking velocity means between groups with different rod lengths and masses. According to Kolmogorov-Smirnov and Shapiro-Wilk tests, normal distribution for maximum striking velocity differences of every sample pair can be assumed (Table S3, ESM). With the exception of females using rod 2 (L65M1000) versus rod 3 (L90M1000), mean maximum striking velocities of rods differed highly significantly (Table 5).

Table 5 Paired-sample t test for all six parameter combination pairs
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Figure 5 shows bar graphs of mean maximum striking velocities for male and female volunteers depending on rod lengths and masses. In general, maximum striking velocities increased with longer as well as lighter rods. However, as an exception, female volunteers achieved about the same velocities with both 65- and 90-cm rods. Compared to the 40-cm rod, female volunteers achieved about 2.3 m/s higher velocities with both longer rods without significant differences between the 65- and 90-cm rod. Male volunteers achieved with the 65- and 90-cm rods about 4.0 and 5.0 m/s higher mean maximum striking velocities than with the 40-cm rod. In terms of rod mass, heavier rods resulted in lower maximum striking velocities in both sexes. With rod masses of 1000 and 1500 g, male volunteers reached about 4.8 and 7.8 m/s and female volunteers reached 4.2 and 6.4 m/s lower striking velocities than with a rod mass of 500 g, respectively.

Fig. 5
figure 5

Mean maximum striking velocities for both sexes, separately regarding rod lengths (left) and masses (right)

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In order to examine dependencies between maximum striking velocities and volunteer parameters, a linear regression model was applied. Maximum striking velocities were analysed as a function of body height, body mass, BMI and the average amount of physical exercise per week for each rod. Male and female volunteers were examined separately because the descriptive statistics showed that maximum striking velocities were generally higher for men. Resulting values of the linear regression model are summarised in Table 6. As required for a full regression analysis, residues follow normal distribution (see Table S4, ESM).

Table 6 Linear regression models for both male and female volunteers using volunteer parameters as independent variables and mean maximum striking velocities of each rod as dependent variables. Results leading to a rejection of the null hypothesis are marked in italic (p(a) < 0.05)
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The results of linear regression analysis with the volunteers’ body height, body mass and BMI as the independent variables did not indicate a systematic effect on maximum striking velocities. Only for the average amount of physical exercise per week did most cases show p values below 0.05 indicating a systematic relationship between the amount of exercise and maximum striking velocities. However, this does not apply to strikes with rods 2, 3 and 4 by male volunteers. Diagrams depicting corresponding scatter plots and regression lines are given in Fig. S1 in the ESM.

Discussion

The purpose of our study was to measure maximum striking velocities of blunt instruments. By means of a group of 51 volunteers and five steel rods of several lengths and masses, we analysed potentially influencing aspects such as volunteer and rod parameters.

Compared to data of offenders convicted of murder, manslaughter and bodily harm, our volunteers only partly corresponded. Out of all volunteers, 54.9% were aged 20 to 24 years and 31.4% were aged 25 to 29 years. According to literature, the largest group of offenders is aged between 30 and 40 years followed by the age group of 21 to 25 years. As about 90% of offenders are usually men, women were overrepresented in our volunteer group [16,17,18]. Regarding body lengths and masses, our volunteer sample corresponded well to the German population group aged between 20 and 30 years [19].

Concerning the influence of rod parameters on maximum striking velocities, the paired-sample t test showed that mean maximum striking velocities generally increased with higher rod lengths but smaller rod masses (Fig. 5). The movement during a strike with a rod-like implement consists of translatory and rotary motion. The rotary component of said implements’ motion during striking can be characterised by its angular velocity as well as by its tangential velocity of its end point (the dr marker in the case of our rods). The linear relationship between angular and tangential velocity for circular motion with the radius r of the trajectory circle of the end point is shown in Eq. (1) [20]. Bearing this in mind, the end point of the implement reaches higher tangential velocities than a point nearer to the axis of rotation as the radius increases but the angular velocity stays the same. Since the dr marker was fixed to the very distal end of the rod, its tangential velocity is supposed to be highest compared with those of all other material points of the rod.

$$ v=\omega r $$
(1)

where

v :

tangential velocity

ω :

angular velocity

r :

radius

Nevertheless, female volunteers did not achieve significantly higher maximum striking velocities with the 90-cm rod compared to the 65-cm rod. In rotary motion, the moment of inertia has to be considered as well. As shown in Eq. (2) [20], it depends on mass and its distribution. The moment of inertia increases with higher masses. Applying this relationship, the decreasing mean maximum striking velocities with increasing rod mass can be straightforwardly explained because heavier rods require more strength to move. However, the increase of tangential velocity with rod length is not straightforward just like that. The increase of movement radius suggests an increase of tangential velocity. At the same time, the moment of inertia that counteracts rotation increases. These counteracting effects could explain the slight increase in maximum striking velocity in male volunteers and the insignificant increase in maximum striking velocity in female volunteers (Fig. 5, Table 5) when comparing rods of 65 cm length with rods of 90 cm length. Since the skeletal muscle mass in women is usually smaller than in men [21], it is supposed to be even more difficult for them to reach higher striking velocities with longer implements.

$$ I={\sum}_i{m}_i\bullet {r}_i^2 $$
(2)

where

I :

moment of inertia about an axis of rotation

m i :

mass of particle i

r i :

orthogonal distance of particle i to axis of rotation

Further on, the influence of volunteer parameters on maximum striking velocities is to be discussed. In this work, we applied linear regression models to identify linear relationships. Besides, spurious correlation due to an influence of sex was excluded by separate regression analyses for male and female volunteers.

Results showed that a linear relationship can only be assumed between maximum striking velocities and the average amount of physical exercise for female volunteers striking with all rods and male volunteers striking with rods 1 (L65M1000) and 5 (L65M1500) (Fig. S1 in the ESM). It seems reasonable that higher fitness due to a regular and higher amount of physical exercise allows for higher striking velocities. Nevertheless, the average amount of physical exercise per week does not specify the sports type. For example, endurance training like running would be considered as less influential on maximum striking velocity compared to racket sports where a combination of movements similar to this experiment can be found. Therefore, it is quite remarkable that maximum striking velocities increase with the average amount of physical exercise per week regardless of sports type. However, maximum striking velocities that male volunteers achieved with rods 2 (L65M1000), 3 (L90M1000) and 4 (L65M0500) do not show a fitness level dependence (Table 6). This might indicate the expected influence of the type of physical exercise, which cannot be quantified based on volunteer interrogation within this study.

Surprisingly, there was no linear relationship between maximum striking velocities and body height, body mass or BMI. It is arguable that taller volunteers would have achieved higher striking velocities because their height would allow them to have a larger range of motion to accelerate the rods. Considering the relationship between angular and tangential velocity for rotary motion, it would have been sensible to measure the volunteers’ arm length along with their body height as well. In addition to the expectable higher range of motion, longer arms ought to extend the distance between the distal end of the rod and the axis of rotation. The experimental tests showed that our volunteers used highly individual striking techniques which can hardly be categorised into distinct motion patterns. Maximum striking velocities seem to depend more crucially on striking technique than on body parameters. Moreover, kinematics of effective striking or beating, e.g. in racket sports like tennis, are very complex as different interactions of various muscles and muscle groups are required [22,23,24]. In this context, club-level tennis players achieved lower serve speeds than elite tennis players [25].

Regarding BMI, Wong et al. [24] found that in tennis serve, the BMI significantly correlates with ball speed to the effect that volunteers with higher BMI achieved significantly higher ball speeds. In contrast with the study of Wong et al. [24] whose volunteers were all male elite tennis players with a probably similar muscle to fat ratio, the BMI in our case does probably not reflect physical fitness.

One important fact to be considered is that in our study, we analysed maximum striking velocities instead of impact velocities. Unlike maximum striking velocities, impact velocities represent the biomechanically relevant parameter for the assessment of blunt trauma. The impact velocity of an object can be described as the velocity of said object at the moment of impact with another one. Nevertheless, the maximum striking velocities measured within this study can be regarded as an upper bound for the impact velocity. Considering our experimental setup in which the volunteers were asked to hit the target as fast and as hard as possible, striking velocities naturally reached their maximum at the end of each striking process either shortly before or exactly in the moment of impact.

As this experiment was conducted under laboratory conditions, several limitations have to be considered. We used custom-built steel rods with, apart from different lengths and masses, the same properties in order to guarantee their comparability. We also limited the different rod lengths and masses to three each in respect of our volunteers’ time and strength capacities. A test on all sorts of rod-like implements in every possible size would have gone far beyond any viable and sensible study. However, in everyday life, the misuse of common items such as tools like hammers, rackets like baseball bats or golf clubs and general household items like broomsticks for an act of physical violence is much more likely than the use of a steel rod which is harder to come by. Compared to our rods, such items are often heterogeneously shaped and mass distributed. While the lengths and masses of our rods were sensibly chosen to cover a broad range of possible impact tools, misused tools may also have lengths and masses outranging the ones we tested.

Furthermore, the height of our target was unalterable and the target itself was passive and immobile. In real cases, bodily harm often occurs in physical conflicts in which both parties are actively moving and reacting to each other if not one of them being unconscious or otherwise immobilised.

Standardisations concerning the experimental setup may influence the applicability in real case work. Firstly, they were not free to grip the rods anywhere else but at the proximal end. For some volunteers, this was slightly uncomfortable especially with the very heavy or very long rods or when their hands were too large to fit well between the pr marker and the proximal end of the rod. Secondly, the room dimensions in which our volunteers performed the strikes were limited. Particularly tall volunteers sometimes had to restrain themselves in order not to accidentally destroy camera equipment or touch the ceiling with a rod whilst striking. Lastly, the volunteers performed the strikes under laboratory conditions and were not in a situation of drunken violence, heightened irritability and aggressiveness facilitating affect-based actions [17, 18, 26]. Exceeding the usual strength accessible to voluntary motor function, the human body is able to mobilise excessive strength in such alerting situations [27].

Because our study had limited resources and a limited number of volunteers, numerous disturbance variables in our experiments could not be controlled by stratification. Their influence on our measurement results can therefore be estimated by looking at our standard deviations. Potential disturbance variables can be, e.g. experiment day time, (time of) last ingestion or time of last sportive activities.

We measured maximum striking velocities of steel rods between 10.36 and 35.52 m/s, depending on volunteer sex as well as rod lengths and masses (see Table 3). These velocities exceed the considerably lower velocities achieved without using any implements. Head impact velocities in head butt scenarios reached up to 4.7 m/s [4], fist punches up to 12 m/s [10] and foot velocities in foot stomping up to 11.1 m/s [8]. Our velocities were comparable to those found in other studies examining velocities of different rod-like striking objects but exceeded those reported for beer steins (Table 7).

Table 7 Selection of studies using different blunt instruments for striking, beating or hitting
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A sample calculation using the results of this study as well as a comparison between calculated contact forces and fracture thresholds are presented in the electronic supplementary material (ESM).

Conclusions

In vertical strikes with steel rods, maximum striking velocities of 14.0 to 35.5 m/s can be found in men and of 10.4 to 28.3 m/s in women. In general, our results indicate higher maximum striking velocities with increasing implement lengths. Smaller rod masses generally result in higher maximum striking velocities.

Expert opinions need to incorporate case-specific circumstances especially with regard to length and mass of the striking implement and individual parameters of the accused. Our study provides the basis for controlling the following influence factors, which had not been considered in previous studies: gender, body mass and body height, BMI, fitness level and rod mass and length. It therefore represents the basis for more case individual expert opinions at court.