# Biodynamic responses of the human hand-arm system under nail gun shock vibration and differences between males and females

Maxime Vincent, Akira Engelbrecht, Marc-André Gaudreau, Mathias Legrand, Thomas Dupont, Pierre Marcotte

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Maxime Vincent, Akira Engelbrecht, Marc-André Gaudreau, Mathias Legrand, Thomas Dupont, et al. Biodynamic responses of the human hand-arm system under nail gun shock vibration and differences between males and females. 2025. hal-05044707

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# Biodynamic responses of the human hand-arm system under nail gun shock vibration and differences between males and females

Maxime Vincent<sup>a</sup>, Akira Engelbrecht<sup>b,c</sup>, Marc-André Gaudreau<sup>d</sup>, Mathias Legrand<sup>b</sup>, Thomas Dupont<sup>a</sup>, Pierre Marcotte<sup>a,b</sup>

<sup>a</sup>Department of Mechanical Engineering, École de technologie supérieure, Montréal, Canada

<sup>b</sup>Institut de Recherche Robert-Sauvé en Santé et en Sécurité du Travail, Montréal, Canada

<sup>c</sup>Department of Mechanical Engineering, McGill University, Montréal, Canada

<sup>d</sup>Department of Mechanical Engineering, Université du Québec à Trois-Rivières, Drummondville, Canada

## Abstract

Occupational exposure to vibration from hand tools poses significant health risks, potentially leading to chronic disorders affecting nerves, muscles, and joints in the hand-arm system (HAS). Despite extensive research, the biodynamic response of the HAS to real-world shock vibrations remains insufficiently characterized, limiting effective risk assessment and prevention strategies. This study aims to quantify vibration transmissibility along the HAS during realistic nail gun operation and examine potential sex-related differences. Thirty participants (15 males, 15 females) performed standardized nail gun tasks while vibration transmissibility was measured at the wrist, forearm, upper arm, and shoulder. Results show a consistent decrease in peak transmissibility frequency from the wrist to the shoulder, reflecting a shift in biodynamic response along the HAS. Experimental modal analysis identified distinct natural frequencies across the HAS, with statistical analyses confirming significant sex-related differences, particularly in lower-frequency modes. These differences remained significant after adjusting for age, hand length, palm width, and BMI, suggesting underlying sex-specific biodynamic and anthropometric traits. The importance of a sufficiently large and balanced sample for reliably detecting sex-related effects is demonstrated. The findings highlight the importance of evaluating vibration exposure under realistic conditions, as conventional shaker tests may not fully replicate the dynamics of shock vibrations. Notably, transmissibility increased at high frequencies and decreased below 10 Hz, likely due to differences in experimental conditions. Furthermore, the observed sex-related differences support the need to incorporate sex-specific considerations into occupational health guidelines, ergonomic tool design, and exposure standards to reduce vibration-induced health risks and improve worker safety.

**Keywords:** Biodynamics, Hand-arm vibration, Shock vibration, Hand tools, Sex differences
**ANCOVA** Analysis of Covariance

**ANOVA** Analysis of Variance

**BMI** Body Mass Index

**DPMI** Driving Point Mechanical Impedance

**HAS** Hand-Arm System

**HAVS** Hand-Arm Vibration Syndrome

**MAC** Modal Assurance Criterion

**MICE** Multiple Imputation by Chained Equations

**OMA** Operational Modal Analysis

**PSD** Power Spectral Density

**SD** Standard Deviation

**VIF** Variance Inflation Factor

**ÉTS** École de technologie supérieure

# 1. Introduction

The Hand-Arm Vibration Syndrome (HAVS) is an occupational health disorder caused by prolonged exposure to vibrations affecting the Hand-Arm System (HAS), mainly from the operation of hand-held power tools. The disorder manifests through a triad of symptoms affecting distinct bodily systems: vascular (Raynaud's phenomenon or vibration white finger), neurological (tingling and numbness), and musculoskeletal (muscle weakness, reduced grip strength, and joint pain) (Griffin, 1990; Heaver et al., 2011; Shen and House, 2017). Fortunately, exposure to HAS vibration from power tools can be reduced by improving ergonomic working conditions, using low vibration tools and implementing anti-vibration solutions to existing tools (de Kok et al., 2019; Marcotte et al., 2022). However, even in highly industrialized countries like Canada, the HAVS remains a significant challenge (Heaver et al., 2011; Thompson et al., 2011; Ziamanesh, 2023; Poole et al., 2019), where exposure to cold temperatures present an additional risk, especially for Raynaud's phenomenon and musculoskeletal disorders (de Kok et al., 2019; Flavahan, 2015; Farbu et al., 2022).

To address these challenges and enhance worker protection, it is crucial to clarify dose-response relationships between hand-arm vibration exposure and HAVS. Determining the biodynamic response of the human upper limb segments to vibration is needed to help understand the transmission of vibrations from power tools to the HAS. Unfortunately, most studies in that area have been conducted using simulated tool vibrations generated by electrodynamic shakers. While useful for controlled experiments, these setups are unable to replicate the high peak accelerations of shock vibration tools, limiting their relevance for shock vibration biodynamic studies. This is reflected in the ISO 10068:2012 standard (International Organization for Standardization, 2012) defining the biodynamic of the human HAS, which applicability is limited to low magnitude random and harmonic vibration. A deeper understanding of human biodynamics could also pave the way for innovative approaches, including the development of human-free test benches for vibration tool testing (Vincent et al., 2023; Marcotte et al., 2010), which would simplify measurement procedures, while reducing associated material costs and participant recruitment. These systems could replace human participants with mechanical analogues or dynamic manikins reproducing the dynamic of the HAS, while providing more representative and replicable measurements. Additionally, such advances can facilitate the creation of comprehensive models that integrate the dynamics of the HAS (Dong et al., 2021; International Organization for Standardization, 2012; Adewusi et al., 2012; Rakheja et al., 2002; Matthiesen et al., 2012), the tool (Matthiesen et al., 2012; Ahmadabadi et al., 2019), and the working material, allowing manufacturers to
design efficient tools minimizing workers exposure to harmful vibration (Pitts et al., 2017; Ko et al., 2011; Jain A. R. Tony and Alphin, 2023; Lindell, 2017).

Human biodynamic characterization is performed using either the transmissibility (Dong et al., 2021; Ziamanesh, 2023; Adewusi et al., 2010; Xu et al., 2009; Wu and Lin, 2023; Pyykkö et al., 1976; Reynolds and Angevine, 1977; Marchetti et al., 2017; Saha and Kalra, 2016; Kihlberg, 1995; Xu et al., 2017; Vergara et al., 2008) or the Driving Point Mechanical Impedance (DPMI) (Dong et al., 2021; Pitts et al., 2017; Saha and Kalra, 2016; Hida et al., 2021; Marcotte et al., 2005). Transmissibility is defined as vibrations transmitted to different segments of the HAS and can be expressed as

$$T_{s,i,j}(f) = \frac{a_{\text{HAS},s,j}(f)}{a_{\text{tool},i}(f)} \qquad (1)$$

where $a_{\text{HAS}}$ and $a_{\text{tool}}$ are the acceleration amplitudes on the HAS segment and on the tool, at frequency $f$. The index $i$ refers to the excitation axis, $j$ to the response axis, and $s$ to the different segments along the HAS (e.g., wrist, forearm, upper arm, and shoulder) and at the vibration source (e.g., the tool handle), as illustrated later in Figures 1 and 2. High transmissibility generally occurs when the tool's excitation frequency is near a natural frequency of the HAS. At these frequencies, the segments of the HAS can reach large amplitudes due to vibratory resonances. This increased motion of the upper limb segments can lead to a higher level of strain on the operator's HAS, which in turn could contribute to HAVS. Accordingly, manufacturers should take this transmissibility into account when designing tools to ensure they minimize vibration exposure and prioritize operator health and safety (International Organization for Standardization, 2001).

On the other side, DPMI is the overall vibration transmitted to the HAS and can be expressed as

$$DPMI(f) = \frac{F(f)}{v(f)} \qquad (2)$$

where $F$ represents the dynamic force transmitted from the tool handle to the hand, and $v$ denotes the velocity of the tool handle, both measured at the driving point between the hand and the tool handle, at frequency $f$. While DPMI measurements allow to evaluate the total energy that is transmitted to the HAS (Dong et al., 2021; Aldien et al., 2005; Dong et al., 2008; Champoux et al., 2015), these are particularly challenging since they require precise force measurement at the interface between the human hand and the source of vibration. Achieving this without disrupting the natural interaction between the hand and the tool handle or compromising the tool's ergonomics in real-world scenarios is a significant technical challenge. Furthermore, DPMI measurements are restricted to a single location at the driving point between the hand and the tool, providing no vibration information through the body or vibration propagation information along the HAS different substructures. In addition, transmissibility measurements, taken at multiple locations over the entire HAS, enable the application of modal analysis techniques. These techniques are essential for determining the intrinsic characteristics of the HAS, in the form of the HAS natural frequencies and associated damping properties (Ewins, 2000). However, classical modal analysis can be more complex as it requires knowledge of the excitation force. When the excitation force is unknown, Operational Modal Analysis (OMA) (Brincker and Ventura, 2015) can be used to extract the system's dynamic characteristics.
While most studies on HAS transmissibility have used instrumented handles attached to single-axis shakers, which may neither reproduce the acceleration amplitude nor the direction of excitation, particularly for shock vibrations (Adewusi et al., 2010; Pyykkö et al., 1976; Reynolds and Angevine, 1977; Marchetti et al., 2017; Kihlberg, 1995; Gurram et al., 1994), a few studies have directly measured transmissibility on the tool and segments of the HAS (Ziamanesh, 2023; Chand et al., 2013), reporting higher transmissibility than the ones observed with shakers.

Nail guns are a prime example of shock vibration tools, widely used in the construction industry for their efficiency and precision in achieving quick and accurate fastening. However, they expose operators to intense shocks, with ISO/TS 15694:2004 (International Organization for Standardization, 2004) *hF* weighted peak acceleration reaching up to 8000 m/s² (Vincent et al., 2023), and occurring at a rate as high as 12 000 shocks per day for typical daily use (Gaudreau et al., 2018). This combination of high peak acceleration and repetition may increase the risk of HAVS (Bovenzi et al., 1997; Starck, 1984; Krajnak et al., 2013; Pelmear et al., 1995; Raju et al., 2011; Poole and Cleveland, 2016; Zimmerman et al., 2020; Ando et al., 2005; Gaudreau et al., 2018). Indeed, there are currently concerns about the adequacy and accuracy of current standards and regulations in protecting workers from shock vibration, as these standards are primarily based on vibration exposure with relatively low peak acceleration (International Organization for Standardization, 2012; Council of the European Union, 2002; Lindell et al., 2023; Pitts et al., 2017). Moreover, standardized methods to determine nail guns vibration emission values, such as the ISO 28927-13:2022 (International Organization for Standardization, 2022), are based on the participation of only three trained humans to operate the tools, which may cause limitations in the representativity and accuracy of the results. Individual differences in HAS biodynamics naturally lead to variability in the measurements (Vincent et al., 2023; Xu et al., 2017; Chand et al., 2013), which can cause large variations in the results. These are also sensitive to the selection of the three operators, who may not represent a good sample of human HAS biodynamics (Dong et al., 2021; Ziamanesh, 2023; Vincent et al., 2023; Marcotte et al., 2010; Sörensson, 1998). Another concern is the limited number of HAS biodynamic studies involving female participants, since obtaining information about possible differences in HAS biodynamic between males and females is particularly important given the higher prevalence of musculoskeletal disorders among females when compared to males. In addition, their increasing representation in traditionally male-dominated sectors (de Kok et al., 2021, 2019) highlights the need to understand these differences to develop effective protective measures (Côté, 2012; Hooftman et al., 2009).

In the present study, vibration transmissibility along different HAS segments were measured on a cohort of 30 participants (15 males and 15 females) while operating a battery-powered nail gun under standardized conditions (International Organization for Standardization, 2022). The measurements were further analyzed using OMA to identify the characteristic natural frequencies of the HAS. The initial hypothesis is that a deeper understanding of vibration propagation along the HAS can provide valuable insights for preventing HAVS and improving the reliability of risk assessments for exposure to shock vibrations, including potential sex-based differences in response to such exposures. Moreover, the presented method could help establish novel risk assessment methods, particularly for shock vibration, where a consensus on evaluation methods has not yet
been established. While both sexes were equally represented in the study, the current
analysis is limited to examining frequency-dependent vibration transmissibility, natural
frequencies of the HAS, and statistical analysis of the differences in natural frequencies
between males and females.

This paper is structured as follows: Section 2 details the general methodology, including participant selection, the experimental setup and the accelerometer positioning, as well as data acquisition and analysis. Section 3 presents the results and discussion, covering the transmissibility responses, the extraction of the HAS natural frequencies and their statistical analysis. Section 4 concludes with a summary of the study's findings.

**2. Methodology**

2.1. *Experimental setup*

For the vibration measurements along the HAS, four triaxial accelerometers were placed on the participant's right HAS, as depicted in Figure 1: a PCB 356B21 accelerometer (500 g) at the wrist, and three PCB 356A22 accelerometers (50 g) at the forearm, upper arm and shoulder. The orientations of the accelerometers follow the coordinate system outlined in Figure 1.

<figure><img src="image_3.png" /><figcaption>Figure 1: Location and orientation of the accelerometers with red caps along the HAS of one participant with their support: $z_h$-axis [···]; $y_h$-axis [---]; $x_h$-axis [—].</figcaption></figure>

The accelerometers were mounted onto aluminum plates and secured to the HAS using Velcro straps, in a similar way as described in (Adewusi et al., 2010). To ensure consistent measurements, the cited approach was further refined by standardizing strap tightness and accelerometer location across all trials and participants. To this end, a 5.2 N force obtained by suspending a 0.53 kg mass was applied to tighten the straps on each segment of the HAS, and the accelerometers were placed directly on the bone structures. The tightening force was maximized to reduce skin measurement artifacts along rigid bone structures while ensuring comfort and safety for participants. A handheld lithium-ion battery-powered electric nail gun was used in this study. To achieve precise vibration measurements at the tool handle-HAS interface, a PCB 356B20 (5000 g) accelerometer
was rigidly mounted on the nail gun using a mounting stud (PCB 080A25 and 081A90) and epoxy paste. Its placement followed the accelerometer location requirements specified in ISO 28927-13:2022 (International Organization for Standardization, 2022) and was aligned with the basicentric coordinate system, as described in (International Organization for Standardization, 2001). The tests were carried out on the standardized nail gun setup (International Organization for Standardization, 2022), as illustrated in Figure 2, and consisting of a sand-filled test bench and a buried wooden white pine block.

<figure><img src="image_4.png" /><figcaption>Figure 2: Basicentric coordinates on the nail gun and location of the accelerometers on the tool marked by the red squares: $z_h$-axis [···]; $y_h$-axis [---]; $x_h$-axis [---].</figcaption></figure>

To ensure consistent nail placement and to minimize variability during measurements, each pine block was laser-engraved with circular targets, allowing participants to drive nails into nearly identical positions across all trials. A circular bubble level was glued on top of the nail gun to help the participants maintain the tool perpendicular to the wooden block. Participants were instructed to begin at the top-left corner of the pine block and drive five nails vertically in the first column, followed by another five nails in the adjacent column to complete the first series of ten nails. This process was then repeated five times to complete the measurement of 50 nails for each participant. The pine blocks used in the experiment measured 35 cm × 21 cm × 10 cm (length × width × height) and were dried prior to the measurement campaign to maintain a humidity level (12 ± 3) % consistent with the standard (International Organization for Standardization, 2022). Each of them was retained and labeled with the participant's number. The nails used were 3-1/4-inch (≈ 8.3 cm) framing nails. Participants were instructed to maintain a posture in accordance with the recommendations provided by the standard (International Organization for Standardization, 2022). The experimental setup, standardized posture and nail sequence are illustrated in Figure 3.
<figure><img src="image_5.png" /><figcaption>Figure 3: Posture adopted according to the ISO 28927-13:2022 standard (International Organization for Standardization, 2022) with a pre-marked target pine wooden block. Participants were asked to follow the nail sequence from the first to the tenth nail until the series is completed: $z_h$-axis [....]; $y_h$-axis [---]; $x_h$-axis [---].</figcaption></figure>

## 2.2. Participant selection and measurement procedure

A total of 30 volunteers participated in the study, with an equal distribution between males ($n = 15$) and females ($n = 15$). Most participants were recruited among student and employees from the École de technologie supérieure (ÉTS) and had no prior experience with operation of nail guns. During a measurement session, participants received comprehensive information about the objectives of the investigation, procedures, and safety measures. They were also informed that they could withdraw from the study at any time without prejudice, with no need to provide a reason. Participants were given the opportunity to review the information provided, ask any questions, and provide their informed consent by completing and signing the consent form. The session continued with questions about health issues to assess study eligibility, and participants could be deemed ineligible if their answers indicated potential health risks with using nail guns. Participants were introduced to the experimental setup in a semi-anechoic chamber and asked to test the protective equipment, including helmet-mounted earmuffs, safety glasses, and steel-toed caps. The experimenter demonstrated the proper operation of the nail gun and instructed participants on the correct posture to maintain. After that, the participant's ability to handle the weight of the nail gun (≈ 4 kg when fully loaded with nails and the battery) with their right hand and operate it safely was verified. Additionally, depending on the participants' height, a platform with a height of nine
centimeters was used to allow the correct posture throughout the experiment.

After training and once the participants felt ready, they were asked to drive 5 series of 10 nails into the standardized pine wooden block. According to the standard (International Organization for Standardization, 2022), the recommended time is 30 seconds, which is equivalent to driving one nail every three seconds. However, since the participants were not professional operators familiar with nail guns, the time limit was extended to 90 seconds. If the participant exceeded the time limit, the series would need to be repeated. Failure to meet the time limit again would result in the participant being deemed ineligible for the study. However, no participant results were deemed ineligible. Finally, anthropometric measurements were taken, as illustrated in Appendix A; they summarized separately for male and female participants in Table 1.

<table><caption>Table 1: Anthropometric and demographic measurements of male and female participants, presenting the range, mean, and SD for each variable.</caption><thead><tr><th rowspan="2">Variable</th><th colspan="2">Male (n = 15)</th><th colspan="2">Female (n = 15)</th></tr><tr><th>Range</th><th>Mean (SD)</th><th>Range</th><th>Mean (SD)</th></tr></thead><tbody><tr><td>Age (years)</td><td>21.0 - 52.0</td><td>29.8 (9.9)</td><td>20.0 - 29.0</td><td>23.4 (2.1)</td></tr><tr><td>Height (m)</td><td>1.65 - 1.93</td><td>1.84 (0.1)</td><td>1.58 - 1.82</td><td>1.70 (0.1)</td></tr><tr><td>Weight (kg)</td><td>69.0 - 109.8</td><td>85.2 (12.3)</td><td>52.3 - 76.7</td><td>63.5 (6.6)</td></tr><tr><td>Body mass index (kg/m<sup>2</sup>)</td><td>19.5 - 33.0</td><td>25.4 (3.9)</td><td>17.1 - 27.5</td><td>22.0 (3.1)</td></tr><tr><td>Hand length (cm)</td><td>16.5 - 20.5</td><td>18.7 (1.1)</td><td>16.0 - 18.5</td><td>17.5 (0.7)</td></tr><tr><td>Palm length (cm)</td><td>8.5 - 11.5</td><td>10.0 (0.7)</td><td>8.5 - 11.5</td><td>9.6 (0.7)</td></tr><tr><td>Palm width (cm)</td><td>8.0 - 10.0</td><td>9.0 (0.6)</td><td>7.5 - 8.5</td><td>8.0 (0.4)</td></tr><tr><td>Palm circumference (cm)</td><td>19.5 - 23.0</td><td>21.4 (1.0)</td><td>17.5 - 20.0</td><td>18.8 (0.8)</td></tr><tr><td>Wrist circumference (cm)</td><td>16.0 - 18.5</td><td>17.3 (0.7)</td><td>14.5 - 16.0</td><td>15.4 (0.4)</td></tr><tr><td>Forearm length (cm)</td><td>31.5 - 40.5</td><td>37.6 (2.2)</td><td>32.5 - 36.5</td><td>34.4 (1.2)</td></tr><tr><td>Forearm circumference (cm)</td><td>22.5 - 31.0</td><td>27.1 (2.1)</td><td>20.0 - 25.0</td><td>22.7 (1.3)</td></tr><tr><td>Upper arm circumference (cm)</td><td>24.5 - 32.0</td><td>28.9 (2.3)</td><td>21.5 - 30.0</td><td>24.9 (2.3)</td></tr><tr><td>Arm length (cm)</td><td>69.5 - 81.5</td><td>75.4 (3.1)</td><td>65.5 - 75.5</td><td>69.6 (3.4)</td></tr><tr><td>Shoulder circumference (cm)</td><td>26.5 - 36.0</td><td>32.2 (2.8)</td><td>21.5 - 33.0</td><td>27.4 (2.9)</td></tr></tbody></table>

## 2.3. Signal processing

All acceleration time signals were recorded using a Siemens Simcenter SCADAS Mobile system with a sampling frequency of 51.2 kHz. Each signal consisted of 10 distinct shocks, and only one second of useful signal was retained for each shock. To ensure the signal captured the entire shock events, 1/7 of a second was retained before the peak amplitude and 6/7 afterward.

For each participant, transmissibility samples were averaged from all 50 shocks (10 shocks per series × 5 series) using the $H_1$ frequency response estimator, which is appropriate when noise is primarily present in the output signals. The $H_1$ transfer function estimate is defined as the ratio between the cross-Power Spectral Density (PSD) of the output b (nail gun handle) and the input a (HAS segment), denoted as $P_{ba}$, and the PSD of the input, denoted as $P_{aa}$, both at frequency f. Mathematically, this reads

$$H_1(f) = \frac{P_{ba}(f)}{P_{aa}(f)} \qquad (3)$$
## 2.4. Operational Modal Analysis

The dynamic response of the HAS was further analyzed using the OMA Polymax algorithm (Elkafafy et al., 2014; Peeters et al., 2004, 2012), within Siemens Testlab 2021.1. The stability diagram was first used to identify and extract the natural frequencies and associated damping ratio of the HAS; however, only the natural frequencies are reported in the present study. To validate the mode order identification of the associated natural frequencies, the Modal Assurance Criterion (MAC) (Pastor et al., 2012; Allemang, 2003) was computed. The MAC quantifies the correlation between two modal vectors by comparing their spatial distributions, values close to 1 indicate a strong similarity, thus confirming consistent mode identification. First, the AutoMAC method confirmed minimal off-diagonal values for the first ten modes, indicating low spatial aliasing and well-identified modes. Beyond the tenth mode, distinguishing modes became challenging, as higher-order modes tend to become increasingly localized at the hand (Xu et al., 2009, 2015, 2017; Welcome et al., 2015), providing limited information on the global dynamic behavior of the HAS. Then, the CrossMAC method was used to compare modal vectors between different participants to identify similar modes across individuals. This approach helps establish mode correspondences despite variations in natural frequencies among participants. As explained earlier, only the first ten modes with CrossMAC values greater than or equal to 0.7 were retained (Martin et al., 2014).

## 2.5. Statistical analyses

The natural frequencies obtained from OMA were initially tested using one-way Analysis of Variance (ANOVA) with sex as a single factor, to determine whether significant differences in natural frequencies exist between males and females. Since modal analysis provides natural frequencies that are independent of each other, ANOVA were performed independently for each natural frequency. To assess the magnitude of observed differences, Hedges' *g*<sub>s</sub> was calculated for each mode (Lakens, 2013). This effect size metric is particularly suitable for small to moderate sample sizes and provides an unbiased estimate of the magnitude of group differences. Following the guidelines in (Lakens, 2013), effect sizes are interpreted as small if 0.2 ≤ *g* < 0.5, moderate if 0.5 ≤ *g* < 0.8, and large if *g* ≥ 0.8. Analysis of Covariance (ANCOVA) were also performed to adjust for the potential confounding of participants' anthropometric characteristics between sexes. The selection process of covariates began with an initial set of the 14 variables from Table 1. Each variable was individually tested using ANCOVA, and only covariates with a potential influence (*p* < 0.1) were initially retained for further analysis. Subsequently, covariates exhibiting a Pearson correlation coefficient *r* > 0.7 with any other retained covariate were excluded to avoid multicollinearity (Dormann et al., 2013). After these exclusions, all remaining covariates had *r* < 0.4 between them. The final set of covariates consists of age, hand length, palm width, and Body Mass Index (BMI). All four retained covariates had Variance Inflation Factor (VIF) values below 2.5, indicating a low level of multicollinearity (James et al., 2013), and confirming that the selected variables can be included without introducing redundancy.

All statistical analyses were conducted using RStudio for Windows, running R version 4.4.2 (2024-10-31 ucrt), with a significance level set at $\alpha$ = 0.05. To address missing data from occasional undetected mode occurring in some participants, the Multiple Imputation by Chained Equations (MICE) method was applied to manage missing
data (van Buuren and Groothuis-Oudshoorn, 2011). All variables were transformed (van Albada and Robinson, 2007) to ensure normal distributions. They were then verified to ensure they met the criteria for parametric testing, namely normality of the residuals using the Shapiro-Wilk test and homogeneity of variances using Levene's test.

# 3. Results and discussion

## 3.1. Vibration transmissibility

Vibration transmissibility and PSDs are all presented within the 5 - 500 Hz frequency range, based on two key considerations: (i) low coherence observed below 5 Hz, attributed to the free motion of the hand, and (ii) low transmissibility beyond 500 Hz, stabilizing around -40 dB. Furthermore, this frequency range aligns closely with results from previous studies (Adewusi et al., 2010; Xu et al., 2009; Wu and Lin, 2023; International Organization for Standardization, 2012; Kihlberg, 1995; Xu et al., 2017).

Figure 4 shows the mean acceleration PSDs at the tool handle, averaged over all 50 shocks from the 30 participants, along the $z_h$-, $y_h$-, and $x_h$-axes during nail gun operation. The shaded regions represent the ±1 standard deviation across participants, highlighting inter-subject differences.

<figure><img src="image_6.png" /><figcaption>Figure 4: Mean power spectral density and standard deviation envelope at the tool handle, averaged over 50 shocks from 30 participants during nail gun operation, along the $z_h$-, $y_h$-, and $x_h$-axes.</figcaption></figure>

The motion along the $z_h$-axis exhibited the highest magnitude, with three distinct spectral energy peaks concentrated in the frequency ranges of 50 - 100 Hz, 200 - 300 Hz, and 400 - 500 Hz. Additionally, high variability at low frequencies (< 10 Hz) is observed, which is assumed to be associated with the free motion of the hand. Next, the motion along $y_h$-axis shows a notable peak in the 350 - 400 Hz range, while relatively much lower spectral energy is observed below 200 Hz. Finally, the motion along $x_h$-axis exhibits negligible spectral energy across the entire frequency range. Since PSDs in the $x_h$- and
$y_h$-axes were generally much lower than in the $z_h$-axis over the entire frequency range, although the difference was less pronounced for the $y_h$-axis above 200 Hz, vibration transmissibility analysis was therefore limited to handle excitation along the $z_h$-axis.

### 3.1.1. Reproducibility and repeatability of vibration transmissibility

Measurement reproducibility was assessed by repeatedly removing and reattaching the Velcro straps securing the accelerometers to a participant's arm segments before each trial, simulating variations in sensor positioning. This procedure was repeated three times, with each trial consisting of 10 shocks, and all straps were fastened by the same operator to ensure consistency. The results are presented at the wrist for handle excitation along the $z_h$-axis in Figure 5, in terms of transmissibility and coherence along the $z_h$-, $y_h$-, and $x_h$-axes. The transmissibility values are expressed in dB, facilitating comparison between curves across different frequency ranges.

<figure><img src="image_7.png" /><figcaption>Figure 5: Comparison of vibration transmissibility (left) and coherence (right) for the reproducibility test at the wrist, considering responses along three different axes under $z_h$-axis handle excitation. The test was conducted on a single participant across three trials. Trial 1 [—], 2 [—], and 3 [—].</figcaption></figure>

For all three axes, the transmissibility and coherence curves exhibit very good reproducibility between 10 and 100 Hz, with increased variability at frequencies below 10 Hz and above 100 Hz. Within 10 – 100 Hz frequency range, the reproducibility variability in transmissibility remains small, with SD ranging from 0.1 to 1.2 dB for the $z_h$-axis, 0.2 to 2.1 dB for the $y_h$-axis, and 0.6 to 2.5 dB for the $x_h$-axis. At higher frequencies, low transmissibility are also related to low coherence, since low transmissibility corresponds to a weak output signal, thus providing a lower signal-to-noise ratio. In the $z_h$-axis, two noticeable drops appear in the 100 - 200 Hz and 300 - 400 Hz ranges, whereas in the other axes, a decline is observed only above 200 Hz. Beyond 200 Hz, coherence progressively declines, reaching near-zero around 300 Hz, which corresponds to the reduction in excitation energy observed in the PSD in Figure 4
between 200 and 300 Hz. At low frequencies below 10 Hz and for all three axes, coherence slightly decreases, likely due to the free movement of the hand. Despite minor variations between trials, these trends remain consistent, suggesting good measurement reproducibility under the tested conditions. This trend also aligns with the one reported in (Adewusi et al., 2010), where coherence in the $z_h$-axis remained near unity up to 200 Hz, but dropped sharply around 300 Hz, since higher frequency vibrations tend to remain localized to the hand (Pyykkö et al., 1976; Reynolds and Angevine, 1977; Aatola, 1989; Marcotte et al., 2005).

Measurement repeatability was also assessed for one participant over five measurement series, each consisting of 10 shocks, with the accelerometers remaining attached throughout the measurements in order to assess intra-subject variability. Again, the results are presented at the wrist for handle excitation along the $z_h$-axis in Figure 6, in the form of transmissibility and coherence along the the $z_h-$, $y_h-$, and $x_h$-axes. The transmissibility values are also expressed in dB.

<figure><img src="image_8.png" /><figcaption>Figure 6: Comparison of vibration transmissibility (left) and coherence (right) for the intra-subject variability test at the wrist, considering responses along three different axes under $z_h$-axis handle excitation. The test was conducted on one participant across five measurement series. Measure 1 [—], 2 [—], 3 [—], 4 [—], and 5 [—].</figcaption></figure>

Compared to the reproducibility test, the transmissibility and coherence curves exhibit consistent responses up to 300 Hz for all three axes. In the 10–100 Hz frequency range, the intra-subject variability in transmissibility remains low, with SD ranging from 0.3 to 0.9 dB for the $z_h$-axis, 0.5 to 1.4 dB for the $y_h$-axis, and 1.4 to 2.7 dB for the $x_h$-axis. The coherence results followed a trend similar to the reproducibility test but with two notable differences: the absence of the drop observed around 100 - 200 Hz for the $z_h$-axis and a more pronounced drop around 300 Hz for all three axis still due to the excitation signal drop (Figure 4) at this frequency. The absence of a drop in the 100 - 200 Hz range is likely due to lower variability for the repeatability test, where the
repositioning of the accelerometers is eliminated between each series of measurements. Additionally, along the $x_h$-axis, an anti-resonance appears slightly above 10 Hz, which was not clearly previously highlighted. Along both the $x_h$- and $y_h$-axes, a coherence drop is observed below 10 Hz. This behavior is not visible along the $z_h$-axis, which could be attributed to free motion of the hand in this direction by the participant. Despite minor variations between trials, these trends remain highly consistent, demonstrating great measurement repeatability under the tested conditions.

### 3.1.2. Inter-subject variability

In order to assess inter-subject variability among all the participants, the mean transmissibility, averaged over 50 shocks, was computed for each participant for the $z_h$-axis handle excitation. Those transmissibility, in dB, and the associated coherence for the responses along the $z_h$, $y_h$, and $x_h$-axes at the wrist are presented in Figure 7.

<figure><img src="image_9.png" /><figcaption>Figure 7: Comparison of vibration transmissibility (left) and coherence (right) of the vibration responses at the wrist among 30 participants under $z_h$-axis handle excitation.</figcaption></figure>

The most notable observation is the strong consistency in vibration transmissibility and coherence along the $z_h$-axis, particularly between 10–100 Hz, indicating relatively low inter-subject variability. This consistency is somewhat less evident along the $y_h$-axis and significantly diminished for the $x_h$-axis, where greater inter-subject variability is observed. The lower transmissibility and coherence observed along the $x_h$-axis can primarily be attributed to the wrist accelerometer axis alignment relative to the nail gun's movement plane; specifically, the $x_h$-axis at the wrist is oriented out-of-plane, making it inherently more susceptible to instability and less correlated vibration signals. Additionally, the coherence drop at 300 Hz, attributed to the decrease in PSD shown in Figure 4, is further confirmed as it is clearly visible for all subjects. Moreover, factors such as wrist posture and anatomical differences may amplify subject-dependent variations in the
$x_h$-axis, whereas the $y_h$-axis remains relatively stable due to its perpendicular alignment with $z_h$-axis.

### 3.1.3. Vibration transmissibility along the hand-arm system

Figure 8 illustrates mean vibration transmissibility, in linear scale, across the different segments of the HAS under vertical ($z_h$-axis) excitation, with a reference line at $T(f) = 1$ to facilitate interpretation. The linear frequency scale was chosen to enhance the visibility of low frequency dynamics, where most biodynamic responses typically occur, and to align with the approach used in previous studies (Adewusi et al., 2010; Xu et al., 2009; Pyykkö et al., 1976; Reynolds and Angevine, 1977; Marchetti et al., 2017; Kihlberg, 1995; Xu et al., 2017). The associated SD and coherence are presented for each segment and each axis response in Appendix B.

<figure><img src="image_10.png" /><figcaption>Figure 8: Mean vibration transmissibility of all participants at different segments of the HAS for $z_h$-axis handle excitation, for different response axes; wrist [---], forearm [···], upper arm [---], and shoulder [----].</figcaption></figure>

The vibrational response along the $z_h$-axis peaks at the wrist around 80 Hz ($T = 1.1$), with a larger response along the $y_h$-axis ($T = 1.4$), whereas the response along the $x_h$-axis remains much lower. The forearm exhibits a peak transmissibility at approximately 30 Hz ($T = 1.1$), with lower response along the other axes. Upper arm and shoulder responses follow a similar pattern but remain below unity. Notably, wrist vibrations persist up to 300 Hz, whereas the forearm, upper arm, and shoulder show lower response from approximately 100 Hz. Vibration amplitudes generally decrease as the measurement location moves farther from the excitation source, showing a progressive attenuation from the wrist to the shoulder. In addition to the decrease in amplitude, the frequency of peak transmissibility also shifts, with higher frequencies observed on the wrist and lower frequencies on the shoulder. These findings align with (Adewusi et al., 2010), confirming both reduced transmissibility and downward frequency shift from the wrist to the shoulder.
The upper arm and shoulder exhibit a similar transmissibility trend along the $z_h$-axis, with a peak around 25 Hz. This similarity is attributed to minimal relative motion between accelerometers due to shoulder-upper arm segment rigidity along the $z_h$-axis. In contrast, along the $x_h$-axis, the upper arm shows a slightly higher transmissibility than the shoulder around 20 Hz when the tool is used in the posture illustrated in Figure 3. This difference arises because the shoulder functions as a pivot joint, allowing the shoulder-upper arm segment to rotate in response to nail gun excitations, particularly through upper arm abduction.

### 3.1.4. Measured transmissibility: bibliographic analysis

Figure 9 shows comparisons between the current study and previous studies on measured vibration transmissibility. Wrist and forearm transmissibility are considered

<figure><img src="image_11.png" /><figcaption>Figure 9: Comparisons of the measured vibration transmissibility across different studies along the HAS and for the $z_h$-axis excitation. Mean transmissibility [—], Adewusi et al. (2010) (Bent) [—], Adewusi et al. (2010) (Extended) [—], Xu et al. (2009) [••••], Xu et al. (2017) [•••••], Pyykkö et al. (1976) [---], Reynolds and Angevine (1977) [---], Kihlberg (1995) [---] and Marchetti et al. (2017) [---].</figcaption></figure>

from (Adewusi et al., 2010; Xu et al., 2009; Pyykkö et al., 1976; Reynolds and Angevine, 1977; Marchetti et al., 2017; Kihlberg, 1995; Xu et al., 2017), while upper arm and shoulder are limited to (Adewusi et al., 2010; Xu et al., 2017). At low frequencies below 10 Hz, transmissibility found in the current study is lower compared to the other studies ($T \le 0.5$), where a shaker was used to excite the HAS. Using a shaker restricts hand movement to displacements along the $z_h$-axis, with no movements possible along the other axes, while the free-hand condition associated with nail gun operation allows movement in all axes. The restricted movement of the hand associated with shaker excitation results in low frequency unitary transmissibility at low frequencies due to a rigid body behavior, while some of that transmissibility can be transferred to the other axes with the free-hand associated to nail gun operation, resulting in lower transmissibility along
the $z_h$-axis. The transmissibility at the wrist, upper arm, and shoulder shifts towards higher frequencies compared to the reported studies, while the forearm generally follows the trend observed in the reported studies. In addition, transmissibility at the shoulder is higher than the reported studies above 20 Hz. These discrepancies in transmissibility may be attributed to differences in experimental conditions, particularly in posture, excitation characteristics, and participant sample, including sample size and the inclusion of female participants. Effect of posture on transmissibility, as reported by Adewusi et al. (2010), is shown in Figure 9, where difference in transmissibility between 10 and 100 Hz is observed between the bent arm and extended arm posture. Indeed, the posture adopted in the present study followed the nail gun standardized condition of the ISO 28927-13:2022 (International Organization for Standardization, 2022), where the reported studies mostly adopted the posture highlighted in ISO 10068:2012 (International Organization for Standardization, 2012). Regarding the excitation, the reported studies used broadband vibration from a shaker, while the present study employs shock excitation from a nail gun, which in turn could contributed to the higher transmissibility above 20 Hz at the shoulder, where posture effect seems more limited, as reported by Adewusi et al. (2010). Although studies using actual tools in operation have reported even higher transmissibility values (Ziamanesh, 2023; Chand et al., 2013), no major transmissibility level differences are observed in the results when compared to studies using shaker excitation.

## 3.2. Natural frequencies

As explained in Section 2.4, only the first ten natural frequencies of the HAS obtained from the OMA across the 30 participants were retained, the average ranging from 6.4 Hz for the first mode to 110.2 Hz for the tenth mode. For each participant, the mode order was identified using the MAC criterion relative to a reference mode set, which could vary across participants. The results, including the frequency range, mean values, and SD for each mode, are presented in Table 2, alongside corresponding values from other studies (Adewusi et al., 2014; Xu et al., 2009, 2015, 2017; Welcome et al., 2015). Additionally, the sample size *n* for each study is indicated, allowing for a direct comparison between the current dataset and previous studies.

The natural frequencies identified in the present study show strong overall agreement with previously reported values, despite some variations. Compared to Adewusi et al. (2014), the relative differences in mean natural frequencies ranged from 0.5 % (Modes 4 and 8) to 21.9 % (Mode 1). Despite these differences, the reported mean values fall within the measured ranges. A similar trend is observed when comparing the mean measured natural frequencies with those reported in (Xu et al., 2017; Welcome et al., 2015; Xu et al., 2009, 2015), as the values generally remain close to or within the reported ranges. For example, the first mode at 6.4 Hz is slightly below the typical range of 7 – 9 Hz, the second mode at 13.9 Hz slightly exceeds the 8 – 12 Hz interval, while the third (24.6 Hz) and fourth (38 Hz) modes fall well within the ranges of 16 – 30 Hz and 20 – 40 Hz, respectively. The frequency ranges observed in the present study were broader for certain modes compared to previous works, highlighting large inter-subject variability over the identified natural frequencies. This is probably related to the larger sample size, ($n = 30$), and to the inclusion of females participants, who may have different natural frequencies when compared to males (see Section 3.3). Indeed, most
<table><caption>Table 2: Natural frequencies for the first ten modes (in Hz), with sample sizes indicated for each study. Column (n = 6) corresponds to (Adewusi et al., 2014), column (n = 8) to (Xu et al., 2017), and column (n = 6 or 7) to (Welcome et al., 2015; Xu et al., 2009, 2015)</caption><thead><tr><th rowspan="2">Mode #</th><th colspan="2">(n = 30)</th><th>(n = 6)</th><th>(n = 8)</th><th>(n = 6 or 7)</th></tr><tr><th>Range</th><th>Mean (SD)</th><th>Mean</th><th>Range</th><th>Range</th></tr></thead><tbody><tr><td>1</td><td>4.4 - 9.1</td><td>6.4 (1.4)</td><td>7.8</td><td>7.0 - 9.0</td><td rowspan="2">8.0 - 12.0</td></tr><tr><td>2</td><td>9.8 - 17.9</td><td>13.9 (2.3)</td><td>12.3</td><td>7.0 - 12.0</td></tr><tr><td>3</td><td>19.2 - 31.3</td><td>24.6 (3.4)</td><td>27.8</td><td>—</td><td>16.0 - 30.0</td></tr><tr><td>4</td><td>32.2 - 41.4</td><td>38.0 (2.4)</td><td>38.2</td><td>—</td><td>20.0 - 40.0</td></tr><tr><td>5</td><td>43.1 - 52.5</td><td>48.1 (2.6)</td><td>45.7</td><td>—</td><td>—</td></tr><tr><td>6</td><td>51.0 - 59.9</td><td>55.8 (2.9)</td><td>55.2</td><td>—</td><td>—</td></tr><tr><td>7</td><td>60.5 - 69.9</td><td>65.3 (2.9)</td><td>63.0</td><td>—</td><td>—</td></tr><tr><td>8</td><td>73.2 - 82.1</td><td>78.9 (2.3)</td><td>78.5</td><td>—</td><td>—</td></tr><tr><td>9</td><td>93.8 - 103.4</td><td>99.6 (2.6)</td><td>97.8</td><td>—</td><td>—</td></tr><tr><td>10</td><td>102.7 - 117.0</td><td>110.2 (4.3)</td><td>109.1</td><td>—</td><td>—</td></tr></tbody></table>

prior studies focused exclusively on males (n = 6 in Adewusi et al. (2014), n = 6 or n = 7 in Welcome et al. (2015); Xu et al. (2009, 2015)), except for Xu et al. (2017), which also included two female participants (n = 8).

Some of those reported works have associated the first mode (8.0 - 12.0 Hz) with shoulder and upper arm vibration, resembling a rigid-body mode of the HAS (Adewusi et al., 2014; Xu et al., 2009, 2015, 2017; Welcome et al., 2015), and the third and fourth modes (16.0 - 30.0 Hz and 20.0 - 40.0 Hz) have been linked to wrist, forearm, and elbow motion, while higher modes progressively localize towards the hand (Adewusi et al., 2014; Xu et al., 2009, 2015, 2017; Welcome et al., 2015). As discussed in Section 2.4, distinguishing between modes beyond the tenth became increasingly difficult due to their strong spatial localization in the hand region. This observation aligns with previous interpretations, and our findings are consistent with this pattern.

### 3.3. Statistical analysis

To assess whether male and female participants exhibit distinct modal properties, as hypothesized in Section 3.2, and to determine whether the differences between the results of the previous study and the current study can be attributed to the inclusion of female participants, a one-way ANOVA was conducted on the first ten modes. The analysis was performed with a significance level of $\alpha = 0.05$ to test for significant differences in the mean natural frequencies between sexes. The results, presented in Table 3, provide the frequency range and the mean frequencies with SD for both sexes, along with the corresponding p-values. Significant values are highlighted: values with one asterisk (*) correspond to $p < 0.05$ and those with two asterisks (**) correspond to $p < 0.01$. Additionally, Hedges' effect size g between groups (denoted as $g_s$) is reported for each significant difference to assess the magnitude of those differences.

The ANOVA results indicate statistically significant differences between sexes for modes 1, 2, and 8, with Modes 1 and 2 showing larger effect sizes, and females exhibiting lower natural frequencies. These differences can be attributed to greater muscle tissue mass and larger hand dimensions in male participants, resulting in increased overall
<table><caption>Table 3: ANOVA *p*-values for sex differences in the first ten natural frequencies ($\alpha$ = 0.05). The table includes the range, mean, and SD (all in Hz) of natural frequencies obtained from OMA for both male and female participants.</caption><thead><tr><th rowspan="2">Mode #</th><th rowspan="2">p-values</th><th rowspan="2">g<sub>s</sub></th><th colspan="2">Male (n = 15)</th><th colspan="2">Female (n = 15)</th></tr><tr><th>Range</th><th>Mean (SD)</th><th>Range</th><th>Mean (SD)</th></tr></thead><tbody><tr><td>1</td><td><strong>0.005 (**)</strong></td><td>1.73</td><td>5.2 - 9.1</td><td>7.4 (1.3)</td><td>4.4 - 7.3</td><td>5.5 (0.8)</td></tr><tr><td>2</td><td><strong>0.006 (**)</strong></td><td>1.38</td><td>11.5 - 17.9</td><td>15.1 (1.9)</td><td>9.8 - 14.7</td><td>12.4 (1.9)</td></tr><tr><td>3</td><td>0.258</td><td>0.32</td><td>19.2 - 31.3</td><td>25.1 (3.4)</td><td>20.4 - 30.4</td><td>24.0 (3.3)</td></tr><tr><td>4</td><td>0.469</td><td>0.24</td><td>32.2 - 41.0</td><td>37.7 (2.7)</td><td>35.2 - 41.4</td><td>38.3 (2.2)</td></tr><tr><td>5</td><td>0.886</td><td>0.18</td><td>43.1 - 52.5</td><td>48.4 (2.9)</td><td>43.4 - 52.4</td><td>47.9 (2.5)</td></tr><tr><td>6</td><td>0.138</td><td>0.40</td><td>51.0 - 59.9</td><td>55.2 (3.4)</td><td>53.0 - 59.9</td><td>56.4 (2.4)</td></tr><tr><td>7</td><td>0.192</td><td>0.32</td><td>61.3 - 69.9</td><td>65.8 (2.6)</td><td>60.5 - 69.8</td><td>64.8 (3.3)</td></tr><tr><td>8</td><td><strong>0.049 (*)</strong></td><td>0.61</td><td>76.1 - 82.1</td><td>79.5 (2.0)</td><td>73.2 - 81.0</td><td>78.1 (2.4)</td></tr><tr><td>9</td><td>0.929</td><td>0.11</td><td>95.3 - 103.4</td><td>99.4 (2.7)</td><td>93.8 - 102.4</td><td>99.7 (2.7)</td></tr><tr><td>10</td><td>0.955</td><td>&lt; 0.01</td><td>104.1 - 117.0</td><td>110.2 (4.4)</td><td>102.7 - 116.8</td><td>110.2 (4.3)</td></tr></tbody></table>

$g_s$: Hedges' *g* effect size. (*) *p* < 0.05; (**) *p* < 0.01.

rigidity, particularly since the first two modes involve vibrations of the entire HAS (Xu et al., 2009, 2015, 2017; Welcome et al., 2015). Additionally, when differentiating the results by sex, it appears that the natural frequencies for male participants are closer to those reported by (Adewusi et al., 2014) for the first mode. This is particularly notable because the first mode initially exhibited the largest absolute difference compared to the previous study, reaching 21.9%. However, this difference decreases to 5.4% when considering only male participants. This observation suggests that the inclusion of female participants may explain the observed discrepancies between the results of the two studies, particularly for the first mode. Overall, the sample size of *n* = 30 (15 males, 15 females) appears to offer a good balance, as all moderate-to-large effect sizes (*g*<sub>s</sub> > 0.5) reached statistical significance (*p* < 0.05), suggesting that the study had sufficient statistical power.

To account for the potential influence of covariates on differences in natural frequencies between males and females, an ANCOVA was performed. The covariates selected (Age, Hand length, Palm width, and BMI) were chosen according to the criteria described in Section 2.5. The results are presented in Table 4, which shows the ANCOVA *p*-values for the effect of sex and each anthropometric variable on the first ten natural frequencies. The column corresponding to sex spans two rows to reflect its role as a categorical factor.

After adjusting for covariates, relatively small differences in *p*-values were observed between the ANOVA and ANCOVA results, respectively in Table 3 and Table 4. Effectively, sex continues to exhibit a significant influence on natural frequencies, with the most pronounced differences still observed in Mode 1 (*p* = 0.009) and Mode 2 (*p* = 0.003), but no more significance difference in Mode 8, the corresponding *p*-value just failing to reach significance (*p* = 0.052). Among the covariates, age emerges as a significant factor in two modes: Mode 4 (*p* = 0.004) and Mode 6 (*p* = 0.050). Hand length also appears to significantly influence natural frequencies, specifically in Mode 2 (*p* = 0.006). However, the inclusion of covariates had limited effects on the observed sex-related differences, suggesting that no single covariate acts as a clear explanatory
<table><caption>Table 4: ANCOVA *p*-values between sex and anthropometric variables for the first ten modes, with the sex column spanning two rows to indicate this categorical factor.</caption><thead><tr><th>Mode #</th><th>Sex</th><th>Age</th><th>Hand length</th><th>Palm width</th><th>Body mass index</th></tr></thead><tbody><tr><td>1</td><td><strong>0.009 (**)</strong></td><td>0.759</td><td>0.818</td><td>0.696</td><td>0.444</td></tr><tr><td>2</td><td><strong>0.003 (**)</strong></td><td>0.807</td><td><strong>0.006 (**)</strong></td><td>0.603</td><td>0.201</td></tr><tr><td>3</td><td>0.281</td><td>0.358</td><td>0.558</td><td>0.854</td><td>0.610</td></tr><tr><td>4</td><td>0.405</td><td><strong>0.004 (**)</strong></td><td>0.082</td><td>0.804</td><td>0.946</td></tr><tr><td>5</td><td>0.890</td><td>0.658</td><td>0.909</td><td>0.266</td><td>0.513</td></tr><tr><td>6</td><td>0.110</td><td><strong>0.050 (*)</strong></td><td>0.840</td><td>0.068</td><td>0.290</td></tr><tr><td>7</td><td>0.197</td><td>0.393</td><td>0.146</td><td>0.489</td><td>0.755</td></tr><tr><td>8</td><td>0.052</td><td>0.886</td><td>0.855</td><td>0.073</td><td>0.980</td></tr><tr><td>9</td><td>0.926</td><td>0.062</td><td>0.317</td><td>0.390</td><td>0.307</td></tr><tr><td>10</td><td>0.955</td><td>0.588</td><td>0.267</td><td>0.178</td><td>0.688</td></tr></tbody></table>

factor for male and female differences.

## 4. Conclusion

This study provided a detailed characterization of shock vibration transmissibility along the HAS during standardized nail gun operation in a balanced and relatively large sample of 30 participants (15 males, 15 females), which proved sufficient to detect sex-related differences and assess inter-subject variability with adequate statistical power. Triaxial accelerometers captured vibration at the tool handle and across four segments of the HAS, revealing that vibration was primarily transmitted along the $z_h$-axis, with peak transmissibility observed at the wrist around 80 Hz and progressively lower and lower-frequency responses toward the shoulder. This spatial pattern confirms previous interpretations of biodynamic behavior, where the HAS behaves as a rigid body at low frequencies and exhibits localized segmental responses at higher frequencies. Measurement reproducibility and repeatability were strong, particularly between 10–100 Hz, validating the experimental protocol. Comparisons with previous shaker-based studies showed lower transmissibility below 10 Hz, likely due to the free-hand condition and multi-directional tool dynamics inherent to nail gun operation.

Using OMA, the first ten natural frequencies were extracted, ranging from 6.4 to 110.2 Hz, and results were broadly consistent with values reported in the literature. However, the larger sample size and inclusion of female participants revealed wider inter-subject variability. Statistical analyses confirmed significant sex-related differences for Modes 1 and 2, with males exhibiting higher natural frequencies, likely due to greater hand dimensions and muscle mass, which increase system rigidity. These differences persisted after adjusting for covariates (age, hand length, palm width, and BMI), suggesting that sex has an independent influence on low-frequency dynamic response.

The results highlight the importance of studying biodynamic responses under realistic tool-use conditions, as traditional shaker-based setups may underestimate dynamic variability, especially in free-hand scenarios. This dataset provides a foundation for developing accurate dynamic models of the human-tool-material system and supports the design of better ergonomic tools and dynamic manikins that reflect realistic vibration
transmission through the HAS, while accounting for inter-subject variability. Future
work may extend this framework to other tool types and exposure scenarios to improve
assessment of vibration-related health risks.

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## A. Location of the sensors and measurements taken of the HAS

Figure A.1 shows the anthropometric measurements and sensor locations of the right HAS for all participants during the measurements. All anthropometric measurements
<figure><img src="image_12.png" /><figcaption>Figure A.1: Location of the anthropometric measurement and accelerometers: HAS [left] and hand [right].</figcaption></figure>

were obtained with the participants' arms as relaxed as possible and free from muscle contractions. To measure the lengths of the arm and forearm, participants were instructed to hold a pencil in their hand and stand with their back against a wall. The distance was measured from the wall to the pencil.

## B. Mean transmissibility and standard deviation along the HAS

Figures B.2 to B.5 present the mean transmissibility of all participants along with the SD envelope (mean ± 1 SD) respectively at the wrist, forearm, upper arm, and shoulder under handle excitation along the $z_h$-axis.
<figure><img src="image_13.png" /><figcaption>Figure B.2: Mean vibration transmissibility at the wrist.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>Figure B.3: Mean vibration transmissibility at the forearm.</figcaption></figure>
<figure><img src="image_15.png" /><figcaption>Figure B.4: Mean vibration transmissibility at the upper arm.</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>Figure B.5: Mean vibration transmissibility at the shoulder.</figcaption></figure>
*Author Statement.* The authors confirm that the work is in accordance with the Code of Ethics of the World Medical Association (Declaration of Helsinki (Association, 2013)) for experiments involving humans. The study protocol, including the collection and storage of data, was reviewed and approved by the ÉTS Research Ethics Committee.

*CRediT authorship contribution statement.*

**Maxime Vincent** Conceptualization, Methodology, Software, Validation, Writing – original draft, Writing – review & editing.

**Akira Engelbrecht** Software, Validation.

**Marc-André Gaudreau** Conceptualization, Methodology, Supervision, Resources, Funding acquisition, Writing – original draft, Review & editing.

**Mathias Legrand** Methodology, Writing – original draft, Writing – review & editing.

**Thomas Dupont** Conceptualization, Methodology, Supervision, Resources, Funding acquisition, Writing – original draft, Review & editing.

**Pierre Marcotte** Conceptualization, Methodology, Supervision, Resources, Funding acquisition, Writing – original draft, Writing – review & editing.

*Declaration of Generative AI and AI-assisted Technologies in the Writing Process.*
During the preparation of this work, the authors used GPT-4o (OpenAI, 2025), retrieved from https://chatgpt.com, in order to enhance the text's coherence, grammar, and syntax. After using this tool, the authors reviewed and edited the content as needed and take full responsibility for the content of the publication.

*Declaration of Competing Interest.* The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

*Funding.* The project was funded by the Institut de recherche Robert-Sauvé en santé et en sécurité du travail (project #2018-0024) and Mitacs Accelerate (project #IT 27303). This work also received financial support from the AIRS (air, intersectorialité, recherche respiratoire et sonore) research network.

*Acknowledgments.* Special thanks to Christian Larivière for his invaluable contributions on the statistical analysis and careful reading of the manuscript. We also thank Hakim Mecheri for his assistance with data transformation, and Serigne Ngagne Cisse for his support during the measurement process.