Current research status on the occupational hazards of hand-transmitted vibration: a case study in China

2.1. Literature search and selection

To search CNKI for studies on hand-transmitted vibration, HTV, HAVD, and occupational exposure in China. The categories of literature were limited to domestic and foreign standards, domestic core journals, and foreign literature by domestic authors; and the publication time was limited to 1983 to June 2024; the search terms in Chinese were “HTV”, “HAVD”, “occupational exposure”, and then the extensions such as vibration intensity and vibration frequency were selected as needed.

2.2. Correlation analysis

SPSS 27.0 software was used for correlation analysis. Since none of the data conformed to a normal distribution, Spearman correlation analysis was performed to evaluate the relationship between vibration intensity, fundamental frequency of each vibration tool, and prevalence. Statistical significance was defined as $P<0.05$.

3.1. Measurement of occupational hand-transmitted vibration

At present, there are two main evaluation systems for HTV, namely, the absorbed power method and the$$ISO 5349 system, which takes the root mean square of acceleration as the index.$$ISO 5349-1:2001 clearly specifies the methods for measuring and evaluating HTV [8]. Compared with the absorbed power method, the$$ISO 5349 system is more convenient to calculate and the evaluation system is more complete. Based on this standard, combined with the actual national production situation, China has promulgated the new national standards$$GB/T 14790.1-2009 [9] and$$GB/T 14790.2-2014 [10], which are suitable for our country. According to the$$ISO 5349-1:2001 evaluation system, when evaluating HTV, the evaluation is mainly based on the total vibration value ($a_{hv}$), which is the root-mean-square (RMS) value of the three-axial frequency-weighted acceleration [8]. The standard$$ISO 2631-1:1997 [1] considers that the human body, when subjected to vibration, has different sensitivities to vibrations of different frequencies;$$ISO 5349-1:2001 [8] mentions that the response of the human body to vibration depends strongly on the frequency of the vibration, and therefore the calculation of the acceleration during operation needs to be frequency-weighted. The root mean square value of acceleration needs to be calculated within the range of 6.3-1250 Hz specified by 1/3 octave, and the frequency weighting factor ($W_h$) shall comply with the provisions of$$GB/T 23716-2009 [11], as shown in Table 1.

Single-axis frequency weighting vibration acceleration root-mean-square value calculation formula shown in Eq. (1):

where $W_{hi}$ is the weighting factor for the $i$th 1/3-octave band; $a_{hi}$ is the RMS value of acceleration measured in the$i$th 1/3-octave band; and $a_{hw}$ is the RMS value of the weighted acceleration.

Then the frequency-weighted acceleration RMS values in the $x$-axis, $y$-axis, and $z$-axis directions are calculated using Eq. (1), and the total vibration value, i.e., the three-axis total frequency-weighted acceleration RMS value, is calculated based on Eq. (2). If the working day consists of several operations with different vibration states, the total vibration value $a_{hv}$ is calculated according to Eq. (3).$$ISO 5349-2:2001 [12] mentions that the triaxial weighted acceleration RMS value can be up to 1.7 times the maximum uniaxial weighted acceleration RMS value (generally 1.2 to 1.5 times, the text takes 1.35 times), therefore, in the actual measurement, can only be measured to the uniaxial, can be used to correct for this factor:

where $a_{hwx}$, $a_{hwy}$, $a_{hwz}$ are the RMS values of the frequency-weighted acceleration along the $x$, $y$, and $z$ axes, respectively, in m/s²:

where $T$ is the daily exposure duration to vibration with respect to $a_{hv}$, $n$ is the number of vibration states, $a_{hvi}$ is the total vibration value for the $i$th vibration state, and $T_i$ is the operating time for the$i$th vibration state.

ISO 5349-1:2001 specifies that the daily exposure value is derived from the total vibration value and the daily exposure duration to vibration [8]. Currently, different standards are used both domestically and internationally to express daily exposure value. Internationally, standards such as 2002/44/EC [13],$$ISO 5349-1:2001 [8], etc., express daily exposure value in terms of the 8 hours energy equivalent frequency-weighted acceleration, denoted as $[{a_{hv}}_{\left(8\right)}$, or $A\left(8\right)]$; while in our country, daily exposure value is expressed in terms of the 4 hours energy equivalent frequency-weighted acceleration, denoted as $[{a_{hv}}_{\left(4\right)}$, or $A\left(4\right)]$. According to ISO 5349-1:2001 [8], if the daily exposure duration to vibration is not 8 hours, it is converted to the $A\left(8\right)$ value using Eq. (4). According to$$GBZ 2.2-2007 [14], if the daily exposure duration to vibration is not$4$hours, it is converted to the $A\left(4\right)$ value using Eq. (5):

where $T$ is the daily exposure duration to vibration relative to $a_{hv}$, and $T_0$ in Eq. (4) and Eq. (5) represents the reference times of 8 hours and 4 hours, respectively.

Xiao et al. [15] suggested that if the exposure of the vibration worker is not 5 days and 8 hours per week, it should be converted to the daily exposure hours for a 5-day work week. Lin et al. [16] proposed that the daily exposure hours for a 6-day work week can be converted to the daily exposure hours for a 5-day work week using Eq. (6). Accordingly, if the weekly work schedule is not 5 days, the daily exposure can be converted to a 5-day work week daily exposure duration to vibration based on Eq. (7):

where $T$ represents the daily exposure duration to vibration for a 5-day work week, and $T_0$ represents the daily exposure duration to vibration for a 6-day work week:

where $T$ is the daily exposure duration to vibration relative to $a_{hv}$; $T_0$ in Eq. (4) and Eq. (5) represents the reference times of 8 hours and 4 hours, respectively.

There is no difference in nature between $A\left(4\right)$ and $A\left(8\right)$, and $A\left(4\right)$ can be converted to $A\left(8\right)$ by Eq. (8) according to$$ISO 5349-1:2001 [8]. However, when comprehensively evaluating all human exposures in the environment, $A\left(8\right)$ can be consistent with the time-weighted averaging method for evaluating other physical factors such as noise and chemical substances [15], [17], so it is recommended to use $A\left(8\right)$ for convenience:

where $a_{hv}$ is the total value of vibration used in the calculation of $A\left(4\right)$; $T_0$ is the reference time of 8 hours.

GB/T 14790.1-2009 [9] stipulates that the number of years of vibration exposure ($D_y$) of the vibration-exposed population and the $A\left(8\right)$ value of the 10 % VWF prevalence rate of the investigated population are shown in Table 2.

$D_y$ for a 10 % prevalence of white finger in the exposure population was calculated according to Eq. (9):

At present, domestic standards primarily follow the requirements of$$GBZ/T 189.9-2007 [18] and$$GBZ 2.2-2007 [14] to record the hand biomechanical coordinate system’s $x$, $y$, and $z$ axes vibration acceleration and the daily average exposure time to calculate $A\left(4\right)$. The measuring instrument adopted is a specialized HTV vibration meter with a weighting network. It records the weighted acceleration or weighted acceleration level. If the measuring instrument measures the vibration amplitude as a weighted acceleration level (dB), it can be converted to weighted acceleration using Eq. (10):

where $L_h$ is the acceleration level in$$dB;$a$ is the RMS value of the vibration acceleration; $a_0$ is the reference value of vibration acceleration, $a_0=$10^-6 m/s².

3.2. Occupational hand-transmitted vibration limits

The vibration dose is mainly affected by vibration intensity, vibration frequency and duration of vibration exposure. Zhou and Chen [19] calculated the cumulative vibration dose of each worker according to Eq. (11), and then obtained the relationship between the cumulative vibration dose of the worker and the cumulative incidence of white finger, and thus calculated that, at the level of protecting 90 per cent of the workers from the occurrence of white finger for 20 years, the limit value of $A\left(4\right)$ should not exceed 5.0 m/s², and the limit value of the standard of our country was precisely set in accordance with this:

where $D$ represents the cumulative vibration dose when white finger occurs, $Y$ is the number of years of vibration exposure, and $d$ is the number of working days with vibration exposure per year.

Exposure action value refers to the value that requires the control of risks arising from an employee’s exposure to vibration once exceeded; exposure limit value refers to the vibration exposure level that employees should not exceed. Currently, internationally recognized standards for hand-transmitted vibration limits include 2002/44/EC [13],$$ISO 5349-1:2001 [8], and$$ANSI S2.70:2006 [20]. Most of these standards set $A\left(8\right)$ daily exposure values at 2.5 m/s² as the action level and 5.0 m/s² as the limit value, which have been widely accepted by many countries. In China, the limit value is specified in$$GBZ 2.2-2007 [14], where $A\left(4\right)$ is considered to exceed the standard when the daily exposure value surpasses 5.0 m/s². According to Eq. (8), the $A\left(8\right)$ value in China exceeds the standard when the daily exposure value surpasses 3.5 m/s².

4.1. Distribution characteristics of hand-arm vibration disease

HAVD is a disease caused by prolonged HTV work, use of hand-held vibrating tools or contact with vibrated workpieces. At least 2 million workers in China are currently exposed to HTV tools, with prevalence rates ranging from 2.5 % to 82.5 % in different occupational groups [21], [22]. With the change of industrial structure in China, the number of occupational HAVD incidence due to HTV is on the rise [23], [24].

In terms of geographic distribution, the study of HAVD was mainly in the northern region in the past, but in recent years, the number of cases suffering from HAVD in subtropical areas in the south of China has also gradually increased [25]-[27]; outbreaks of HAVD have occurred in places such as Guangdong Province, and nearly 60 % of HAVD cases in occupational disease reports in recent years have come from Guangdong Province [2]; unlike the industry distribution characteristics of northern regions where HAVD is mostly found in the mining industry, HAVD in Guangdong Province is mostly found in industries such as sports equipment manufacturing and wooden furniture manufacturing [17], [24], [28], and is more obviously spatially clustered [29], [30]. In terms of industry distribution, HAVD is mainly concentrated in the construction, machining, and furniture manufacturing industries [31]. In terms of the distribution of work types, according to the domestic data, the work types in which HAVD occurs are mainly rock drillers, wind shovellers, tampers, chain sawyers, riveters, etc.

The author reviewed more than 290 pieces of literature related to HTV vibration, in which the relevant literature mentioning vibration tools, job types, positions and their vibration intensity [the text is evaluated by $A\left(4\right)$] is recorded in Table 3.

The vibration intensity in the table is $A\left(4\right)$, and $A\left(8\right)$ in the literature has been converted to $A\left(4\right)$ values according to Eq. (5); $MA\left(4\right)$ denotes the median of $A\left(4\right)$.

Data collation of Table 3 leads to Fig. 2, which shows that there are more domestic investigations of vibration tools such as rock drills, tamping machines, sanders, belt sanders, grinders, polishers, wind shovels, sanding machines, angle grinders, etc., and the number of literature involving grinding tools is significantly higher than that of the rest of the vibration tools; the number of literature on rock drills and grinding tools accounts for the vast majority of the literature. Among them, Fig. 3 shows the vibration tools, work types or positions with $A\left(4\right)$ exceeding the national limit of 5 m/s², which can be seen in the figure, the tools, work types or positions such as large machine grinding, iron grinding, and grinders have $A\left(4\right)$ significantly higher than the limit value of 5 m/s² for HTV in China; vibrating tools that cause occupational HAVD mostly occur in the mining industry as well as in the manufacturing industry, with rock drilling work types and positions dominating in the mining industry and grinding work types and positions dominating in the manufacturing industry; the distribution of $A\left(4\right)$ for rock drilling tools is more concentrated, indicating that there is a high degree of consistency in $A\left(4\right)$ for different types or models of rock drilling tools; the $A\left(4\right)$ of grinding tools is involved in all value intervals, with a wide distribution of values and a large span, reflecting the diversity of their types or models and functional differences. Fig. 4 shows the vibration tools with a higher number of investigations conducted in China.

Fig. 2The number of times vibration tools recording energy equivalent frequency-weighted acceleration have been mentioned in the literature

Fig. 3Vibration tools with A(4) values greater than 5 m/s2 and their corresponding A(4) values

4.2. The hazards of hand-arm vibration disease

HTV refers to the mechanical vibration or shock that acts directly on or is transmitted to the human arm when vibrating tools are used or vibrated workpieces are touched during the production process [58]. HTV-induced HAVD can cause serious damage to the health of workers, which is a disease that mainly consists of peripheral circulation disorders in the hand, nerve dysfunction in the arm, and bone, joint and muscle injuries, and it can lead to changes in the blood biochemical indexes and electrophysiological indexes reflecting the functioning of the nervous system of the workers [59]-[62]. Clinical symptoms initially include hand symptoms such as hand numbness, hand stiffness, hand swelling, sweaty hands, and neurological symptoms such as headache and memory loss [43], [63]. The main clinical manifestations of HAVD are VWF, also known as occupational Raynaud's phenomenon, caused by microcirculation damage at the end of the fingers, and osteoarticular-muscular injury of the arm, mainly including deformation of the finger joints and ‘eagle-claw hand’ [3].

Fig. 4Vibration tools with a higher number of investigations

a) Rock drill

b) Tamper machine

c) Bench grinder

d) Belt sander

GBZ 7-2014 [59] grades the diagnosis of HAVD, which is categorized as mild, moderate and severe HAVD based on different symptoms. The episodes of HAVD are characterized by transient and temporal phases, i.e., after being exposed to cold, numbness, swelling and pain will appear in the fingers in a short period of time, and will change from gray to pale, with distal to proximal progression and clear boundaries; the duration of the episodes is 5 to 30 min; the more severe the disease the more frequent the episodes are; after the episodes, they gradually change from pale, gray to flushed, and finally change to normal skin color, and the hand’s sensation gradually recovers, and pain may appear, burning, itching and other sensations [3], [64].

At present, the pathogenesis of HAVD has not been clarified, and both treatment and recovery are difficult; once the disease develops, it is difficult to recover, and even if the exposure to HTV operations is stopped, recovery remains difficult and disability is high without effective intervention or treatment, which seriously affects the physical and mental health of the patients and may jeopardize the society [6], [50], [59], [65]-[70].

In addition, HTV vibration often coexists with noise as a synergistic factor in noise-induced hearing damage, while noise can also exacerbate hazards to circulatory and neurological function, and the higher the noise intensity the more synergistic the two are, and the greater the impact on physical health [33], [50], [71].

It is evident that HTV can cause various symptoms. Due to the unclear pathogenic mechanisms and the lack of effective preventive and treatment measures, recovery is difficult. Therefore, early diagnosis of workers exposed to vibration, with the timely detection of pathogenic signs, plays a crucial role in preventing the onset of HAVD.

4.3. Diagnostic methods and limitations of hand-arm vibration disease

Currently, health examination signs are dominated by abnormalities in thresholds of heat and cold, pain, touch, and vibration sensation [3], [31], [45], [72].

According to$$GBZ 188-2014 [73] and$$GBZ 7-2014 [59], the examination of HAVD includes symptom inquiry, physical examination, and laboratory and other tests, and the routine physical examination, cold provocation test, nerve electromyography, and other objective examination results are mainly used as diagnostic criteria for health testing. Among them, routine blood test is compulsory, and cold provocation test, nerve electromyography, fingertip vibration sensation and fingertip temperature sensation are optional.

In recent years, in order to further study diagnostic methods with higher sensitivity and specificity, more suitable for the human body and stronger screening ability, many scholars in China have carried out in-depth research on the detection methods of finger systolic blood pressure [74]-[76], fingertip thermotactile perception [31], [66], [77], [78], and fingertip vibrotactile perception [45], [79]-[81], and so on.

Studies have shown that the cold provocation test is affected by factors such as room temperature, water temperature, and immersion time, resulting in low specificity and poor screening power [82]. Even in healthy populations, the abnormality rate of the cold provocation test is relatively high [83]. It is less effective in diagnosing white finger in subtropical regions [84], with poor specificity, difficulty in inducing VWF, poor sensitivity, and a high likelihood of missed diagnosis. Moreover, there is a lack of simple, stable, and effective laboratory diagnostic indicators. Neuromuscular electromyography has the advantages of convenience and high reproducibility, but it is an invasive examination. It cannot be used to detect fibers that conduct fast cold sensation and nociception, or those that conduct slow temperature sensation and nociception. Age and length of service are influencing factors for abnormalities in neuromuscular electromyography among vibration-exposed workers, and it is difficult to reflect early changes in the sensory nerves of the fingertips [2], [77], [85]. Finger systolic blood pressure, fingertip thermotactile perception, and fingertip vibrotactile perception are not clearly defined in terms of testing procedures and standards in$$GBZ 188-2014 [73], [86].

Therefore, the early identification of health damage among vibration-exposed workers is still challenging [2], [87], which may lead to delayed implementation of preventive and control measures.

4.4. Protective measures for hand-arm vibration disease

The early preventive measures for HAVD mainly include early detection, early diagnosis, and early treatment [88]. Among these, effective prevention and control of HAVD have become a research focus [5], [6]. Treatment methods primarily target peripheral circulatory disorders in the arms, neurological dysfunction, and osteoarticular injuries. Symptomatic treatments involve improving microcirculation, dilating blood vessels, and nourishing nerves through a combination of traditional Chinese and Western medicine, physical therapy, and exercise therapy. Additionally, attention should be paid to keeping both the hands and the body warm [3], [5].

In recent years, progress has been made in the treatment of HAVD, such as acupuncture [89] and warm needling therapy [90]. However, the damaged peripheral circulation and peripheral nerves have not fully recovered after treatment, and the therapeutic effects remain limited [5].

Mitigation measures for HTV in the workplace primarily involve early prevention strategies such as engineering controls and personal protection. Engineering controls include measures like controlling the vibration source, improving power tools and production processes, using elastic pads and supports for vibration reduction, isolating the hand from the tool with elastic components (e.g., effective shock absorbers, rings, handles), and reducing the weight of handheld vibrating tools [3], [91], [92]. Other measures include the mechanization and automation of operations [33]. Personal protective measures involve keeping warm to prevent cold exposure, wearing anti-vibration gloves, and reducing vibration exposure time [21].

Furthermore, different vibration tasks or tools have varying vibration accelerations and frequency spectra, leading to different risks and control measures [17], [93]. When tools wear out or age, issues such as loose screws can alter the amplitude and frequency of the vibration tools, while air or water leaks can increase the risk of hand temperature loss for vibration-exposed workers, making HAVD prevention more challenging. Therefore, another important aspect of preventing HAVD is the regular maintenance and inspection of vibration tools [16], [92].

Additionally, some companies have implemented feasible measures, such as strengthening health management and education, regularly conducting health examinations for vibration-exposed workers, rotating and reassigning workers to reduce exposure time, providing vibration-damping gloves, and taking effective actions for workers with abnormal health reports to protect their well-being [16], [17], [94], [95].

However, many factories and enterprises, due to considerations of cost, risk avoidance, and operational convenience, have not arranged for workers to undergo optional health checks [30]. Literature [96] found that the rate of promotion and education on vibration hazards and protection in most factories was only 28.82 %, with a regular inspection rate of just 15.72 %. Literature [37] surveyed 20 companies with riveting operations, but only2of them had organized and implemented occupational health checks related to HTV. Literature [17] revealed that only a few companies provided dedicated anti-vibration gloves. Literature [50] surveyed$10$machinery manufacturing companies and found that none of the workers were equipped with anti-vibration gloves, nor had they undergone occupational health checks related to hand-transmitted vibration.

The analysis of the above literature suggests that the HTV industry still lacks sufficient attention to the hazards of HTV, with weak emphasis on the provision and use of personal protective equipment, education and training on vibration hazards, and the regular health examinations of workers.

5.1. The impact of vibration tool intensity on prevalence

To better analyze the relationship between vibration intensity and the prevalence of VWF, literature mentioning vibration intensity and prevalence has been compiled as shown in Table 4. Based on Table 4, data fitting resulted in the curve shown in Fig. 4. The results of the correlation analysis showed that there was no significant correlation between $A\left(4\right)$ and the prevalence of VWF $(P>0.05)$.

Furthermore, it is easy to find that the relationship between vibration intensity and prevalence and the fitted curves of some vibration tools vary widely. This discrepancy is because the prevalence of HAVD is influenced not only by vibration intensity but also by factors such as workers’ years of service, environmental temperature, and the vibration frequency of the tools [25], [67]. For example, under identical production conditions, the prevalence of VWF for the $C_{6}B$ wind shovel is higher than the fitted curve, while the prevalence for the $D_9$ tamper is lower. This is due to differences in the inherent frequencies of these two tools and the hardness of the processed material, resulting in significant differences in frequency distribution, and the frequency-weighted vibration acceleration is more affected by low-frequency and less attenuated mid-frequency components of vibration intensity [53]. The lower prevalence of VWF for the $1/T-24$ and $1/SP-45$ rock drills compared to the fitted curve may be due to higher temperatures in the southern regions, as low temperatures are an inducing factor for HAVD, resulting in a lower prevalence [34]. Therefore, $A\left(4\right)$ alone is insufficient to reflect the extent of the health hazards of hand-transmitted vibration operations, and environmental factors such as temperature should also be considered.

Fig. 5Prevalence of HAVD-A(4) curve

5.2. The effect of the vibration frequency of tools on the prevalence of disease

Vibration frequency, intensity, and exposure duration are the primary factors affecting human health. Therefore, studying the vibration frequency of tools is also highly significant. Some researchers have focused on the handles of rice transplanters and the arms of exposed workers, finding that the palms poorly absorb low-frequency vibrations below 25 Hz in the left-right ($x$) and front-back ($y$) directions; in the up-down ($z$) direction, they poorly absorb mid-frequency vibrations around 100 Hz. As a result, within the 25 Hz and 100 Hz ranges, vibrations transmitted to the palms are prone to resonance; moreover, the primary cause of Raynaud’s phenomenon is the destruction of cells and tissues caused by the dissipation of vibration energy between 100 to 200 Hz acting directly on the subcutaneous tissue [42], [98]. Another study [99], [100] on portable fire extinguishers found that the larger the fundamental frequency among different blower-type extinguishers, the fewer dominant frequencies there are in the 0 to 400 Hz range, leading to a reduced impact of vibration on the human body. Additionally, research [51] on tractor steering wheels showed that high-frequency vibrations above 100 Hz are largely absorbed and dissipated when transmitted from the steering wheel to the back of the hand. Furthermore, low-frequency vibrations below 10 Hz are amplified when transmitted to the back of the hand, possibly due to resonance at around 5 Hz, which is close to the natural frequency of the human arm, leading to an increase in vibration transmission rate. Therefore, as vibrations are transmitted through the arm, high-frequency vibrations decay rapidly, and the vibrations are primarily concentrated in the low-frequency range. The study in [101] found that the vibration frequency spectrum of the handle of a mini-tiller’s handrail is primarily distributed between 20 and 200 Hz; it also noted that the hands, wrists, and elbows tend to amplify vibrations in the low-frequency range of 4 to 20 Hz, while vibrations at frequencies greater than 100 Hz rapidly attenuate, making the human body more sensitive to low-frequency vibrations. Another study [25], [78], [102], discovered that high-frequency vibration exposure is significant in golf club head manufacturing and rock drilling in mining operations, with a high incidence of hand symptoms reported. The research in [43] observed that during rock drilling, the first peak in acceleration occurs at 25 to 40 Hz, with the highest peak between 125 and 160 Hz; in grinding operations, acceleration increases with frequency, peaking between 500 and 800 Hz; consequently, the incidence of hand symptoms in these two vibration-exposed populations is high, correlating with the exposure frequency.

Based on the above analysis, vibrations below 25 Hz are absorbed poorly, and resonance is likely to occur around 25 Hz in the forearm; low-frequency vibrations below 10 Hz may cause resonance in the back of the hand, amplifying the vibration; the hands, wrists, and elbows amplify low-frequency vibrations in the 4 to 20 Hz range. Vibrations in the 100 to 200 Hz range pose significant hazards; high-frequency vibrations above 100 Hz are largely absorbed and dissipated during transmission. The larger the fundamental frequency, the fewer dominant frequencies in the 0 to 400 Hz range, which reduces the impact of vibration on the human body. Therefore, both low-frequency and high-frequency vibrations can potentially harm the human body and contribute to the development of VWF. The fundamental frequency is the most impactful frequency component on the human body. To facilitate the analysis of vibration tools, literature involving the fundamental frequency of vibration tools has been integrated, as shown in Table 5 and Fig. 5. The results of the correlation analysis showed a significant positive correlation between the fundamental frequency of vibration tools and prevalence $(r>0,\mathrm{}P<0.05)$. That is, as the fundamental frequency of the vibration tools increases, the prevalence also rises to some extent, which is consistent with the findings of [25], [43], [78], [102], indicating that working under high-frequency vibration is associated with an increased prevalence of hand-related diseases. However, due to the limited data, the relationship between the two cannot be definitively determined. The fundamental frequencies in the third reference data are quite similar. Therefore, for ease of observation, the average value is temporarily used.

Fig. 6Prevalence of HAVD-fundamental frequency curve