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Vochiţoiu, Haralambie; Unguraş, Lavinia Camelia; Andreica, Mădălin; Benea, Mărioara; Taschi, Evgheni; Darcy, Anton; Iliaş, Nicolae; Vasilescu, Gabriel

Mining Revue
, Volume 27 (1): 9 – Mar 1, 2021

/lp/de-gruyter/risk-of-exposure-for-workers-to-professional-vibrations-H6fOpk0oQP

- Publisher
- de Gruyter
- Copyright
- © 2021 Haralambie Vochiţoiu et al., published by Sciendo
- eISSN
- 2247-8590
- DOI
- 10.2478/minrv-2021-0007
- Publisher site
- See Article on Publisher Site

Revista Minelor – Mining Revue ISSN-L 1220-2053 / ISSN 2247-8590 vol. 27, issue 1 / 2021, pp. 52-60 RISK OF EXPOSURE FOR WORKERS TO PROFESSIONAL VIBRATIONS 1 2 3 Haralambie VOCHIȚOIU , Lavinia Camelia UNGURAȘ , Mădălin ANDREICA , 4 5 6 7 8* Mărioara BENEA , Evgheni TASCHI , Anton DARCY , Nicolae ILIAȘ , Gabriel VASILESCU University of Petrosani, Petrosani, Romania, hvochitoiu@gmail.com University of Petrosani, Petrosani, Romania, camelialaviniaunguras@gmail.com University of Petrosani, Petrosani, Romania, mady_andreica@gmail.com University of Petrosani, Petrosani, Romania, mbenea40@yahoo.com University of Petrosani, Petrosani, Romania, taschi@ge.com University of Petrosani, Petrosani, Romania, darcy.anton@gmail.com University of Petrosani, Petrosani, Romania, iliasnic@yahoo.com INCD INSEMEX Petrosani, Petrosani, Romania, dragos.vasilescu@insemex.ro DOI: 10.2478/minrv-2021-0007 Keywords: vibrations, mathematic model, exposure, risk Abstract: The study focused on the model of estimating the risk of exposure of workers to global occupational vibrations / with local action on the hand-arm system. In order to estimate the risk of exposure to occupational vibrations, we developed a generalized mathematical model for estimating the risk of exposure to mechanical vibrations transmitted to the whole body / with action on the hand-arm system. This model is based on the statistical function of probability with exponential decrease (Gumbel function), and its argument is expressed either by the values of the weighted acceleration parameter or in the form of exposure points. 1. Introduction European Directive 2002/44/EC was adopted at national level by GD 1876/2005 on minimum occupational safety and health requirements regarding the exposure of workers to the risks arising from vibration. The regulation sets ”the limit values for exposure to mechanical vibration (global and hand-arm)” and ”exposure values that trigger the action”, specifying at the same time, the obligations of employers regarding the determination and assessment of risks, the elaboration of measures to be adopted to reduce or avoid exposure, as well as the details regarding the provision of information and training of workers. Thus, for global mechanical vibrations / (hand-arm), the occupational daily exposure limit value, calculated over a reference period of 8 hours, must be 1.15 m/s²/(5 m/s ) and the value of the daily exposure from which the employer's action is triggered, calculated at a reference period of 8 hours, must be 0.5 m/s²/(2.5 m/s ) [1,2]. 2. Mathematical models associated with the risk of exposure to occupational vibrations In order to estimate the risk of exposure to occupational vibrations, we developed a generalized mathematical model for estimating the risk of exposure to mechanical vibrations transmitted to the whole body / with action on the hand-arm system. This model is based on the statistical function of probability with exponential decrease (Gumbel function), and its argument is expressed either by the values of the weighted acceleration parameter or in the form of exposure points. 3,4,5,6. To quantify the exposure to different values of mechanical acceleration (global and hand-arm) in the work process, the following analytical relationships were used: ∗ 2 𝑃 = [𝐴 (8)] 100 𝐸𝑖 𝑖 * st Corresponding author: Vasilescu Gabriel, Scientific researcher 1 degree, Ph.D. eng., National Institute for Research and Development in Mine Safety and Protection to Explosion INSEMEX Petroșani, Petroșani Romania (32-34 G-ral Vasile Milea Str., 332047 Petrosani, dragos.vasilescu@insemex.ro) 52 Revista Minelor – Mining Revue vol. 27, issue 1 / 2021 ISSN-L 1220-2053 / ISSN 2247-8590 pp. 52-60 𝐴 (8) ∗ 𝑖 ( ) 𝐴 8 = , for vibrations transmitted to the whole body 0.5 𝐴 (8) ∗ 𝑖 𝐴 (8) = , for vibrations transmitted to the hand-arm 2.5 𝑒𝑗 𝐴 (8) = , for vibrations transmitted to the whole body (1) 0.5 𝑇 (𝑘𝑎 ) = 𝛼 , for vibrations transmitted to the whole body 𝑤𝑖 𝑖 𝑒𝑗 (𝑘𝑎 ) = 𝛽 , for vibrations transmitted to the hand-arm ℎ 𝑖 𝑒𝑗 where: A(8) - daily exposure to mechanical vibration, (m/s ); P – exposure points that quantify the values of mechanical vibrations transmitted to the whole body or hand-arm system; k – multiplication factor (is equal to 1.4 for the x and y axes and 1.0 for the z axis); a – weighted acceleration in the case of vibrations wi transmitted to the whole body, (m/s ); a – the magnitude of the acceleration in the case of vibrations hvi transmitted to the hand-arm system, (m/s ); T – daily duration of exposure to vibrations transmitted to the ej whole body or hand-arm system, (hours); T – reference duration, (8 hours) After applying mathematical relations ka =f(T ) (in the case of global vibrations) and a =f(T ) (in case wi ej hvi ej of hand-arm vibrations), value grids are obtained kawi/ahvi corresponding to mechanical vibrations (global and hand-arm) for different levels of the acceleration parameter A (8), respectively: Weighted acceleration values * 2 2 ka (a ) or parameter A (8) which do not exceed the level of 0.5m/s (2.5m/s ), from which the employer's wi hvi i action is triggered, in case of exposure to global vibrations (hand-arm vibrations); Weighted acceleration * 2 2 values ka (a ) or of parameter A (8) within the range (0.5;1.15) m/s ((2.5;5) m/s ), at which the action of wi hvi i the employer is triggered, in case of exposure to global vibrations (hand-arm vibrations); Weighted * 2 acceleration values ka (a ) or of parameter A (8) exceeding the maximum permitted limit of 1.15 m/s (5 wi hvi i m/s ), in case of exposure to global vibrations (hand-arm vibrations). Following the application of mathematical relations P =f(ka ,T ) (in case of global vibrations) and Ei wi ej P =f(a ,T ) (in case of hand-arm vibrations), grids of the exposure points are obtained P corresponding to Ei hvi ej E mechanical vibrations (global and hand-arm) in the work process for different levels of the acceleration parameter A (8), respectively: Exposure point values not exceeding 100 points, which quantifies the values of the weighted acceleration ka (a ) or of parameter A (8) from which the action of the employer is triggered, wi hvi i in case of exposure to global vibrations (hand-arm vibrations); Exposure point values in the range (100;529) points, ((100;400) points) which quantifies the range of values of the weighted acceleration ka (a ) or of wi hvi parameter A (8) at which the action of the employer is triggered, in case of exposure to global vibrations (hand-arm vibrations); Exposure point values exceeding 529 points (400 points), which quantify the weighted acceleration values ka (a ) or of parameter A (8) exceeding the maximum permissible limit in the event of wi hvi i exposure to global vibration (hand-arm vibration). In the following is presented how to use the differential equation that quantifies the risk of exposure to different levels of global mechanical vibration / hand-arm, which may occur during professional activity: 𝑔 (𝑥 ) = 𝐺 (𝑥 ) (2) 𝑖 𝑖 𝑥 −𝑥 𝑥 −𝑥 0 0 𝑖 𝑥 −𝑥 𝑖 − 𝑖 0 − 𝑎 𝑎 ′ −𝑒 −𝑒 𝑔 (𝑥 ) = 𝐺 (𝑥 ) = (𝑒 ) = 𝑒 𝑒 (3) 𝑖 𝑖 which admits as a solution, the following distribution function G(x ): 𝑥 −𝑥 𝑖 0 −𝑒 𝐺 (𝑥 ) = 𝑒 (4) where: x - function variable that can be explicit in the form of values ka /a or of exposure points P g(x ) - i wi hvi Ei; i the probability density function of the variable values x ; G(x ) - the function of distribution (of probability) of i i values x ; µ - the average value of the variable values x ; σ - standard deviation of variable values x ; i – order i i i index of the variable x i. Based on the distribution function G (xi), the expression of the average objective risk of exposure to different values of mechanical acceleration (global and hand-arm) is determined. 2.1 Elaboration of the mathematical model specific to the risk of exposure to vibrations transmitted to the whole body To quantify the exposure to different values of mechanical acceleration transmitted to the whole body in the work process, we have the following analytical relations [6,7,8]: 𝑣𝑖 𝑤𝑖 Revista Minelor – Mining Revue vol. 27, issue 1 / 2021 ISSN-L 1220-2053 / ISSN 2247-8590 pp. 52-60 𝐴 (8) ∗ 𝑖 ( ) 𝐴 8 = (5) 0.5 𝑘𝑎 𝑒𝑗 𝑃 = ( ) 100 (6) 0.5 𝑇 (𝑘𝑎 ) = 𝛼 (7) 𝑤𝑖 𝑖 𝑒𝑗 where: A(8) – daily exposure to vibrations transmitted to the whole body, (m/s ); PE – exposure points that quantify the values of mechanical vibrations transmitted to the whole body; k – multiplication factor (is equal to 1.4 for the x and y axes and 1.0 for the z axis); a – weighted acceleration, (m/s ); T – daily duration of w ej exposure to vibrations transmitted to the whole body, (hours); T – reference duration, (8 hours). Following the application of mathematical relations ka =f(T ) the grid of values is obtained ka wi ej wi corresponding to the mechanical vibrations transmitted to the whole body generated in the work process for different levels of the parameter A(8). Following the application of mathematical relations P =f(ka ,T ), the grid of exposure points P is Ei wi ej E obtained corresponding to the mechanical vibrations transmitted to the whole body generated in the work process for different levels of parameter A(8) In the following is presented the differential equation that quantifies the risk of exposure to different levels of mechanical vibration transmitted to the whole body, which are generated in the work process: 𝑔 (𝑥 ) = 𝐺 (𝑥 ) , (8) 𝑖 𝑖 respectively √2 ln 𝑖 (𝑥 −𝜇 −𝜎 √2 ln 𝑖 ) √2 ln 𝑖 √2 ln 𝑖 √2 ln 𝑖 𝜎 − (𝑥 −𝜇 −𝜎 √2 ln 𝑖 ) − (𝑥 −𝜇 −𝜎 √2 ln 𝑖 ) ′ −𝑒 𝑗 𝑗 𝜎 𝜎 𝑔 (𝑥 ) = 𝐺 (𝑥 ) = (𝑒 ) = 𝑒 𝑒 (9) 𝑖 𝑖 which admits as a solution, the following distribution function G(ka ) wi √2 𝑙𝑛 𝑖 (𝑥 −𝜇 −𝜎 √2 𝑙𝑛 𝑖 ) 𝐺 (𝑥 ) = 𝑒 (10) where: x - the function variable that can be explained in the form of values ka or of exposure points P ; g(x ) i wi Ei i - the probability density function of the variable values x ; G(x ) - the function of distribution (of probability) i i of values x ; µ - the average value of the variable values x ; σ - standard deviation of variable values x ; i order i i i index of the variable x Thus, the explanation of the functions according to the parameter ka or of the points P are reproduced wi Ei below, respectively: 𝑔 (𝑘 𝑎 ) = 𝐺 (𝑘 𝑎 ) (11) 𝑤𝑖 𝑤𝑖 respectively √2 𝑙𝑛 𝑖 ( −𝜇 −𝜎 √2 𝑙𝑛 𝑖 ) 𝑤𝑖 √2 𝑙𝑛 𝑖 √2 𝑙𝑛 𝑖 √2 𝑖 𝜎 − ( −𝜇 −𝜎 √2 𝑖 ) − ( −𝜇 −𝜎 √2 𝑖 ) ′ −𝑒 𝑤𝑖 𝑤𝑖 𝜎 𝜎 𝑔 (𝑘 𝑎 ) = 𝐺 (𝑘 𝑎 ) = (𝑒 ) = 𝑒 𝑒 (12) which admits as a solution, the following distribution function G(ka ) wi √2 𝑙𝑛 𝑖 (𝑃 −𝜇 −𝜎 √2 𝑙𝑛 𝑖 ) 𝐸𝑖 −𝑒 𝐺 (𝑃 ) = 𝑒 (13) 𝐸𝑖 or 𝑔 (𝑃 ) = 𝐺 (𝑃 ) (14) 𝐸𝑖 𝐸𝑖 respectively √2 𝑙𝑛 𝑖 (𝑃 −𝜇 −𝜎 √2 𝑙𝑛 𝑖 ) √2 𝑙𝑛 𝑖 √2 𝑙𝑛 𝑖 √2 𝑖 𝜎 − (𝑃 −𝜇 −𝜎 √2 𝑖 ) − (𝑃 −𝜇 −𝜎 √2 𝑖 ) ′ −𝑒 𝜎 𝜎 𝑔 (𝑃 ) = 𝐺 (𝑃 ) = (𝑒 ) = 𝑒 𝑒 (15) which admits as a solution, the following distribution function G(P ) Ei √2 𝑙𝑛 𝑖 (𝑃 −𝜇 −𝜎 √2 𝑙𝑛 𝑖 ) 𝐸𝑖 −𝑒 𝐺 (𝑃 ) = 𝑒 (16) 𝐸𝑖 Based on the distribution function G(xi) the expression of the average objective risk of exposure to different values of the mechanical acceleration transmitted to the whole body is determined, respectively: 𝑅 (𝑥 ) = ∫ 𝑥 𝐺 (𝑥 )𝑑 𝑥 𝑓𝑜𝑟 𝑎 𝑠𝑛𝑢𝑜𝑢𝑜𝑛𝑡𝑖𝑐 𝑟𝑎𝑛𝑔𝑒 𝑙𝑢𝑒𝑣𝑎𝑠 𝑥 𝑖 𝑖 𝑖 𝑖 𝑖 𝑖 𝑅 (𝑥 ) = ∑ 𝑥 𝐺 (𝑥 ), 𝑓𝑜𝑟 𝑎 𝑖𝑠𝑟𝑒𝑒𝑐𝑑𝑡 𝑟𝑎𝑛𝑔𝑒 𝑙𝑢𝑒𝑣𝑎𝑠 𝑥 𝑖 𝑖 𝑖 𝑖 𝑖 𝑜𝑓 𝑜𝑓 𝐸𝑖 𝐸𝑖 𝐸𝑖 𝐸𝑖 𝑙𝑛 𝑙𝑛 𝑙𝑛 𝐸𝑖 𝑤𝑖 𝑤𝑖 𝑙𝑛 𝑘𝑎 𝑙𝑛 𝑘𝑎 𝑙𝑛 𝑘𝑎 𝑤𝑖 Revista Minelor – Mining Revue vol. 27, issue 1 / 2021 ISSN-L 1220-2053 / ISSN 2247-8590 pp. 52-60 This risk can be explained as follows: Depending on the parameter kawi: √2 ln 𝑖 (𝑃 −𝜇 −𝜎 √2 ln 𝑖 ) 𝐸𝑖 −𝑒 ( ) 𝑅 𝑘𝑎 = ∫ 𝑘𝑎 𝐺 (𝑘𝑎 )𝑑 𝑘𝑎 = ∫ 𝑘𝑎 𝑒 𝑑 𝑘𝑎 , 𝑖 𝑤𝑖 𝑤𝑖 𝑤𝑖 𝑤𝑖 𝑤𝑖 𝑤𝑖 𝑓𝑜𝑟 𝑎 𝑠𝑛𝑢𝑜𝑢𝑜𝑛𝑡𝑖𝑐 𝑟𝑎𝑛𝑔𝑒 𝑙𝑢𝑒𝑣𝑎𝑠 𝑘𝑎 𝑤𝑖 √2 ln 𝑖 (𝑃 −𝜇 −𝜎 √2 ln 𝑖 ) 𝐸𝑖 −𝑒 𝑅 (𝑘𝑎 ) = ∑ 𝑘𝑎 𝐺 (𝑘𝑎 ) = ∑ 𝑘𝑎 𝑒 , 𝑖 𝑤𝑖 𝑤𝑖 𝑤𝑖 𝑤𝑖 𝑓𝑜𝑟 𝑎 𝑖𝑠𝑟𝑒𝑒𝑐𝑑𝑡 𝑟𝑎𝑛𝑔𝑒 𝑙𝑢𝑒𝑣𝑎𝑠 𝑘𝑎 𝑤𝑖 Depending of the points P : Ei √2 ln 𝑖 (𝑃 −𝜇 −𝜎 √2 ln 𝑖 ) 𝐸𝑖 −𝑒 𝜎 ( ) 𝑅 𝑃 = ∫ 𝑃 𝐺 (𝑃 )𝑑 𝑃 = ∫ 𝑃 𝑒 𝑑 𝑃 , 𝑖 𝐸𝑖 𝐸𝑖 𝐸𝑖 𝐸𝑖 𝐸𝑖 𝐸𝑖 𝑓𝑜𝑟 𝑎 𝑠𝑛𝑢𝑜𝑢𝑜𝑛𝑡𝑖𝑐 𝑟𝑎𝑛𝑔𝑒 𝑙𝑢𝑒𝑣𝑎𝑠 𝑃 𝐸𝑖 √2 ln 𝑖 (𝑃 −𝜇 −𝜎 √2 ln 𝑖 ) 𝐸𝑖 −𝑒 𝑅 (𝑃 ) = ∑ 𝑃 𝐺 (𝑃 ) = ∑ 𝑃 𝑒 , 𝑖 𝐸𝑖 𝐸𝑖 𝐸𝑖 𝐸𝑖 𝑓𝑜𝑟 𝑎 𝑖𝑠𝑟𝑒𝑒𝑐𝑑𝑡 𝑟𝑎𝑛𝑔𝑒 𝑙𝑢𝑒𝑣𝑎𝑠 𝑃 𝐸𝑖 Considering the results obtained based on the application of the previous relations, we obtain the following grids to assess the risk of exposure to different values of mechanical acceleration transmitted to the whole body, in the work process, depending on its levels or the value of exposure points, thus (table 1): Table 1 Estimation the exposure risk for the Assess the risk of exposure to Values of mechanical acceleration different values of mechanical different values of mechanical transmitted to the whole body, kawi acceleration transmitted to the acceleration transmitted to the m/s whole body, whole body, in the work process kawi*G(kawi) k ≤ 0.5 k * G(k ≤ 0.5) Low awi awi awi Medium 0.5 k ≤ 1.15 k * G(0.5 k ≤ 1.15) awi awi awi High 1.15 k k * G(1.15 k ) awi awi awi Assess the risk of exposure to Estimation the exposure risk The resulting value of the exposure different values of mechanical depending the resulting value of the points acceleration transmitted to the exposure points whole body, in the work process P ≤ 100.00 P * G(P ≤ 100.00) Low Ei Ei Ei Medium 100.00 P ≤ 529.00 P * G(100.00 P ≤ 529.00) Ei Ei Ei High 529.00 P P * G(529.00 P ) Ei Ei Ei 2.2 Elaboration of the mathematical model specific to the risk of exposure to vibrations with action on the hand-arm system To quantify the exposure to different values of the mechanical acceleration transmitted to the hand-arm system in the work process, we have the following analytical relations [6,7,8]: 𝐴 (8) 𝑃 = [ ] 100, (17) 𝐸𝑖 2.5 𝑒𝑗 𝑃 = ( ) 100, (18) 𝐸𝑖 2.5 𝑇 ( ) 𝑎 = 𝛽 , (19) ℎ𝑣𝑖 𝑖 𝑒𝑗 where: A(8) – daily exposure to vibrations transmitted to the hand-arm system, (m/s ); P – exposure points that quantify the values of mechanical vibrations transmitted to the hand-arm system; a – vibration hv magnitude, (m/s ); T – daily duration of exposure to vibrations transmitted to the hand-arm system, (hours); ej T – reference duration, (8 hours). 𝑣𝑖 𝑜𝑓 𝑜𝑓 𝑜𝑓 𝑜𝑓 Revista Minelor – Mining Revue vol. 27, issue 1 / 2021 ISSN-L 1220-2053 / ISSN 2247-8590 pp. 52-60 Following the application of the mathematical relations a =f(T ) (provided in the last column of the table hvi ej above), the grid of ahvi values corresponding to the mechanical vibrations transmitted to the hand-arm system generated in the work process for different levels of parameter A(8) is obtained. Following the application of mathematical relations P =f(a ,T ), the grid of exposure points is obtained Ei hvi ej P corresponding to the mechanical vibrations transmitted to the hand-arm system generated in the work process for different levels of parameter A(8). Next, we present the differential equation that quantifies the risk of exposure to different levels of mechanical vibration transmitted to the hand-arm system that are generated in the work process: 𝑔 (𝑥 ) = 𝐺 (𝑥 ) , (20) 𝑖 𝑖 respectively √2 𝑙𝑛 𝑖 (𝑥 −𝜇 −𝜎 √2 𝑙𝑛 𝑖 ) √2 𝑙𝑛 𝑖 √2 𝑙 𝑛 𝑖 √2 𝑖 𝜎 − (𝑥 −𝜇 −𝜎 2 𝑖 ) − (𝑥 −𝜇 −𝜎 2 𝑖 ) ′ −𝑒 √ √ 𝑖 𝑖 𝜎 𝜎 𝑔 (𝑥 ) = 𝐺 (𝑥 ) = (𝑒 ) = 𝑒 𝑒 (21) 𝑖 𝑖 which admits as a solution, the following distribution function G(ka ) wi √2 𝑙𝑛 𝑖 (𝑥 −𝜇 −𝜎 √2 𝑙𝑛 𝑖 ) −𝑒 𝐺 (𝑥 ) = 𝑒 (22) where: x - the function variable that can be explained in the form of values ka or of exposed points P ; g(x ) i wi Ei i - the probability density function of the variable values x ; G(x ) - the function of distribution (of probability) i i of values x ; µ - the average value of the variable values x ; - standard deviation of variable values x ; i – i i i order index of the variable x Thus, explaining the functions according to the parameter a or the points P are reproduced in the hvi Ei following, respectively: 𝑔 (𝑎 ) = 𝐺 (𝑎 ) , (23) ℎ𝑣𝑖 ℎ𝑣𝑖 respectively ( ) √2 𝑙𝑛 𝑖 𝑎 −𝜇 −𝜎 √2 𝑙𝑛 𝑖 ℎ𝑣𝑖 √2 𝑙𝑛 𝑖 √2 𝑙𝑛 𝑖 √2 𝑖 𝜎 − (𝑎 −𝜇 −𝜎 √2 𝑖 ) − (𝑎 −𝜇 −𝜎 √2 𝑖 ) ′ −𝑒 ℎ𝑣𝑖 ℎ𝑣𝑖 𝜎 𝜎 𝑔 (𝑎 ) = 𝐺 (𝑎 ) = (𝑒 ) = 𝑒 𝑒 (24) ℎ𝑣𝑖 ℎ𝑣𝑖 which admits as a solution, the following distribution function G(a ) hvi √2 𝑙𝑛 𝑖 (𝑃 −𝜇 −𝜎 √2 𝑙𝑛 𝑖 ) 𝐸𝑖 −𝑒 𝐺 (𝑃 ) = 𝑒 , (25) 𝐸𝑖 or 𝑔 (𝑃 ) = 𝐺 (𝑃 ) , (26) 𝐸𝑖 𝐸𝑖 respectively √2 𝑙𝑛 𝑖 (𝑃 −𝜇 −𝜎 √2 𝑙𝑛 𝑖 ) 2 𝑙𝑛 𝑖 2 𝑙𝑛 𝑖 √ √ 2 𝑖 𝜎 − (𝑃 −𝜇 −𝜎 √2 𝑖 ) − (𝑃 −𝜇 −𝜎 √2 𝑖 ) ′ −𝑒 𝜎 𝜎 𝑔 (𝑃 ) = 𝐺 (𝑃 ) = (𝑒 ) = 𝑒 𝑒 (27) 𝐸𝑖 which admits as a solution, the following distribution function G(P ) Ei 2 𝑙𝑛 𝑖 (𝑃 −𝜇 −𝜎 2 𝑙𝑛 𝑖 ) √ √ 𝐸𝑖 −𝑒 𝐺 (𝑃 ) = 𝑒 , (28) 𝐸𝑖 Based on the distribution function G(x ) the expression of the average objective risk of exposure to different values of the mechanical acceleration transmitted to the hand-arm system is determined, respectively: 𝑅 (𝑥 ) = ∫ 𝑥 𝐺 (𝑥 )𝑑 𝑥 , 𝑓𝑜𝑟 𝑎 𝑠𝑛𝑢𝑜𝑢𝑜𝑛𝑡𝑖𝑐 𝑟𝑎𝑛𝑔𝑒 𝑙𝑢𝑒𝑣𝑎𝑠 𝑥 𝑖 𝑖 𝑖 𝑖 𝑖 𝑖 𝑅 (𝑥 ) = ∑ 𝑥 𝐺 (𝑥 ) , 𝑓𝑜𝑟 𝑎 𝑖𝑠𝑟𝑒𝑒𝑐𝑑𝑡 𝑟𝑎𝑛𝑔𝑒 𝑙𝑢𝑒𝑣𝑎𝑠 𝑥 𝑖 𝑖 𝑖 𝑖 𝑖 This risk can be explained in this way: Depending on the parameter a : hvi √2 ln 𝑖 (𝑘𝑎 −𝜇 −𝜎 √2 ln 𝑖 ) −𝑒 ( ) 𝑅 𝑎 = ∫ 𝑎 𝐺 (𝑎 )𝑑 𝑎 = ∫ 𝑎 𝑒 𝑑 𝑎 , 𝑖 ℎ𝑣𝑖 ℎ𝑣𝑖 ℎ𝑣𝑖 ℎ𝑣𝑖 ℎ𝑣𝑖 ℎ𝑣𝑖 𝑓𝑜𝑟 𝑎 𝑠𝑛𝑢𝑜𝑢𝑜𝑛𝑡𝑖𝑐 𝑟𝑎𝑛𝑔𝑒 𝑙𝑢𝑒𝑣𝑎𝑠 𝑎 ℎ𝑣𝑖 √2 ln 𝑖 (𝑎 −𝜇 −𝜎 √2 ln 𝑖 ) −𝑒 𝑅 (𝑎 ) = ∑ 𝑎 𝐺 (𝑎 ) = ∑ 𝑎 𝑒 , 𝑖 ℎ𝑣𝑖 ℎ𝑣𝑖 ℎ𝑣𝑖 ℎ𝑣𝑖 𝑓𝑜𝑟 𝑎 𝑖𝑠𝑟𝑒𝑒𝑐𝑑𝑡 𝑟𝑎𝑛𝑔𝑒 𝑙𝑢𝑒𝑣𝑎𝑠 𝑎 ℎ𝑣𝑖 𝑜𝑓 𝑣𝑖 𝑜𝑓 𝑣𝑖 𝑜𝑓 𝑜𝑓 𝐸𝑖 𝐸𝑖 𝐸𝑖 𝑙𝑛 𝑙𝑛 𝑙𝑛 𝐸𝑖 𝑙𝑛 𝑙𝑛 𝑙𝑛 𝑙𝑛 𝑙𝑛 𝑙𝑛 Revista Minelor – Mining Revue vol. 27, issue 1 / 2021 ISSN-L 1220-2053 / ISSN 2247-8590 pp. 52-60 Depending on the points P : Ei √2 ln 𝑖 (𝑃 −𝜇 −𝜎 √2 ln 𝑖 ) 𝐸𝑖 −𝑒 𝑅 (𝑃 ) = ∫ 𝑃 𝐺 (𝑃 )𝑑 𝑃 = ∫ 𝑃 𝑒 𝑑 𝑃 , 𝑖 𝐸𝑖 𝐸𝑖 𝐸𝑖 𝐸𝑖 𝐸𝑖 𝐸𝑖 𝑓𝑜𝑟 𝑎 𝑠𝑛𝑢𝑜𝑢𝑜𝑛𝑡𝑖𝑐 𝑟𝑎𝑛𝑔𝑒 𝑙𝑢𝑒𝑣𝑎𝑠 𝑃 𝐸𝑖 √2 ln 𝑖 (𝑃 −𝜇 −𝜎 √2 ln 𝑖 ) 𝐸𝑖 −𝑒 𝑅 (𝑃 ) = ∑ 𝑃 𝐺 (𝑃 ) = ∑ 𝑃 𝑒 , 𝑖 𝐸𝑖 𝐸𝑖 𝐸𝑖 𝐸𝑖 𝑓𝑜𝑟 𝑎 𝑖𝑠𝑟𝑒𝑒𝑐𝑑𝑡 𝑟𝑎𝑛𝑔𝑒 𝑙𝑢𝑒𝑣𝑎𝑠 𝑃 𝐸𝑖 Taking into account the results obtained on the basis of the application of the previous relations, the following grids for assessing the risk of exposure to different values of mechanical acceleration transmitted to the hand-arm system in the work process are obtained depending on its levels or value of exposure points, as follows (Table 2): Table 2. Estimation the exposure risk for the Assess the risk of exposure to Values of mechanical acceleration different values of mechanical different values of mechanical transmitted to the hand-arm, ahvi acceleration transmitted to hand- acceleration transmitted to the m/s arm, a *G(a ) hand-arm, in the work process hvi hvi a ≤ 2.5 a * G(a ≤ 2.5) Low hvi hvi hvi 2.5 a ≤ 5 a * G(2.5 a ≤ 5) Medium hvi hvi hvi High 5 a a * G(5 a ) hvi hvi hvi Assess the risk of exposure to Estimation the exposure risk The resulting value of the exposure different values of mechanical depending the resulting value of the points acceleration transmitted to the exposure points hand-arm, in the work process P ≤ 100.00 P * G(P ≤ 100.00) Low Ei Ei Ei Medium 100.00 PEi ≤ 400.00 PEi* G(100.00 PEi ≤ 400.00) High 400.00 P P * G(400.00 P ) Ei Ei Ei After applying mathematical relations kawi=f(Tej) the grid of values is obtained, kawi from table 3, corresponding to the mechanical vibrations transmitted to the whole body generated in the work process for different values of the parameter A(8). Following the application of mathematical relations P =f(k ,T ), the grid of exposure points from table Ei awi ej 4 is obtained, P , corresponding to the mechanical vibrations transmitted to the whole body generated in the work process for different values of parameter A(8). Following the application of mathematical relations a =f(T ) the grid of values is obtained ahvi hvi ej corresponding to the mechanical vibrations transmitted to the hand-arm system generated in the work process from table 5 for different values of parameter A(8). Following the application of mathematical relations P =f(a ,T ), the grid of exposure points is obtained Ei hvi ej P corresponding to the mechanical vibrations transmitted to the hand-arm system generated in the work process from table 6 for different values of parameter A(8). 3. Conclusions The minimum requirements for the protection of workers against the risks to their health and safety, generated or which may be generated by exposure to mechanical vibration, are laid down in national legislation by GD 1876/2005 which takes over the European Directive 2002/44 / EC, and establish responsibilities of employers to ensure the elimination or minimization of the risks generated by professional vibrations. In order to estimate and assess the risk of exposure to occupational vibrations, a generalized mathematical model for diagnosis and prognosis of exposure risk was designed, both for mechanical vibrations transmitted to the whole body and for mechanical vibrations with action on the hand system -arm. This model is based on the exponentially decreasing Gumbel distribution function, whose variables can be made explicit either by the values of the weighted acceleration parameter or in the form of exposure points. 𝑜𝑓 𝑜𝑓 𝑤𝑖 Revista Minelor – Mining Revue vol. 27, issue 1 / 2021 ISSN-L 1220-2053 / ISSN 2247-8590 pp. 52-60 Table 3. Grid of values G(ka ) calculated using the mathematical relationship associated with the distribution function wi Table 4. Grid of exposure points ka *G(ka ) calculated using the mathematical relationship associated wi wi with the objective average risk function ( ) Revista Minelor – Mining Revue vol. 27, issue 1 / 2021 ISSN-L 1220-2053 / ISSN 2247-8590 pp. 52-60 Table 5. Grid of values G(P ) calculated using the mathematical relationship associated with the distribution function Ei Table 6. Grid of values P *G(P ) calculated using the relationship associated Ei Ei with the objective average risk function (PEi) Revista Minelor – Mining Revue vol. 27, issue 1 / 2021 ISSN-L 1220-2053 / ISSN 2247-8590 pp. 52-60 By applying the analytical relations within the mathematical model, the databases specific to the value grids were obtained, which constitute the results of the statistical functions that quantify the risk of exposure to professional vibrations. Thus, the assessment of the risk of exposure to occupational vibrations can be achieved both by using the value grid that quantifies the objective average risk, based on weighted acceleration values and their associated probabilities, and by using the value grid obtained according to exposure point values and corresponding probabilities. References [1] * * *, 2005 EU Good Practice Guide on Hoole-Body Vibrations [2] * * *, 2005 EU Good Practice Guide on Hand and Arm Vibrations [3] Conte J.C., Rubio E., Garcia A.I., 2011 Occupational accidents model based on risk-injury affinity groups, Safety science, 49, 306 – 314 [4] Covello V.T., Merkhofer M.W., 1993 Risk Assessment Methods. Approaches for Assessing Health and Environmental Risks, Springer, Berlin-Heidelberg-New Springer, Berlin-Heidelberg-New York [5] Vanderhaegen, F., 1999 APRECIH: a human unreliability analysis method - Application to railway system [6] Haimes Y.Y., 2004 Risk Modeling Assessment, and Management, Second edition, John Wiley & Sons, Inc. Publication, U.S.A., ISBN 0-471- 48048-7. [7] Vătășescu, Mihail, Vătășescu, Mihaela, Vasilescu, G., Lemle, L.D., 2014 Advanced Research in the Field of Instruments for Use in the Probabilistic Study of Security, Proceedings of the International Conference of Numerical Analysis and Applied Mathematics (ICNAAM-2014) Book, Series: AIP Conference Proceedings Volume: 1648 Article Number: UNSP 680008 DOI: 10.1063/1.4912914 Published: 2015, International Conference on Numerical Analysis and Applied Mathematics (ICNAAM), Rhodes, Greece, ISSN: 0094- 243X, ISBN: 978-0-7354-1287-3, WOS:000355339704048. [8] Vasilescu G., 2008 Probabilistic calculation methods used in the diagnosis and forecasting of industrial risk (in romanian), INSEMEX Publishing (2008), Petrosani, Romania, ISBN 978-973-88753-2-6. This article is an open access article distributed under the Creative Commons BY SA 4.0 license. Authors retain all copyrights and agree to the terms of the above-mentioned CC BY SA 4.0 license.

Mining Revue – de Gruyter

**Published: ** Mar 1, 2021

**Keywords: **vibrations; mathematic model; exposure; risk

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