Jtam-A4.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 53, 1, pp. 3-13, Warsaw 2015 DOI: 10.15632/jtam-pl.53.1.3 A COMPARISON OF HUMAN PHYSICAL MODELS BASED ON THE DISTRIBUTION OF POWER IN A DYNAMIC STRUCTURE IN THE CASE OF HAND-ARM VIBRATIONS Marian Witalis Dobry, Tomasz Hermann Poznan University of Technology, Institute of Applied Mechanics, Poznań, Poland e-mail: marian.dobry@put.poznan.pl; tomasz.hermann@put.poznan.pl Themain aimof this study is to present an energy comparisonof twohumanphysicalmodels taking into account hand-armvibrations, which are based on the power distribution in their dynamic structure. The method used in the study takes advantage of a close relationship between thedynamics of the systemsand energy-relatedphenomena that occurwithin them. The energy comparison of the two human physical models required construction of energy models of aHuman-Tool systemandfinding their solutions.For this purpose, programshave been developed using theMATLAB/simulink software to simulate power distribution in the systems. The simulation revealed a discrepancy between the two models in terms of three types of powers and globally in the system as a whole. Keywords: hand-arm vibrations, biomechanical system, power distribution 1. Introduction Research on the impact of vibrations on the humanbody is carried out inmany research centers around the world. Many studies in this area have contributed to the expansion of knowledge about the impact of vibrations on the human body, in particular studies by Griffin (1990), Reynolds and Soedel (1972), Meltzer (1981) andmany others (Adewusi et al., 2012). One of the basic problems, which has frequently been addressed, is the construction of a suitable human biomechanical model. At present, several models are used for analysis which differ, first of all, in terms of the number of components and the way in which they are connected. In addition to the already existing systems, newmodels are being created to replace the previous ones and to better reflect the human response tomechanical vibrations. Choosing the rightmodel is, therefore, becoming increasingly difficult. Consequently, this study attempted to compare two latest human physical models that have been used in recent studies. A measure of model adequacy should reflect to what extent a given model correctly represents energy phenomena which occur in the dynamic structure and in the entire system during its operation. Themainaimof this study is to assess theadequacy of four conditional variants of thehuman physicalmodel for hand-armvibrations presented inDong et al. (2007). The newphysicalmodel has been verified by comparing its energy inputs with the values obtained bymeans of a model with three points of reduction, which is specified in ISO 10068 (ISO 10068:1998). The two human physical models have been tested with respect to the power distribution in the dynamic structure and the test results served as the basis for a comparative assessment of the investigated biomechanical systems. Assessment of the human physical models for hand-arm vibrations in terms of energy inputs involved solving a set of differential equations of motion constituting the mathematical model, which are derived from Lagrange’s equations of the second kind expressed as 4 M.W. Dobry, T. Hermann d dt (∂E ∂q̇j ) − ∂E ∂qj =Qj +QjP +QjR j=1,2, . . . ,s (1.1) where: E is the kinetic energy of the system, qj – generalized coordinates, q̇j – generalized velocity,Qj –active external forces,QjP –potential forces,QjR –non-potential forces dependent on the power of dissipation, s – number of degrees of freedom. In themodels in question, it is necessary to assumegeneralized coordinates to clearly describe the displacement. The following generalized coordinates have been adopted in the model with three points of reduction specified in the ISO 10068 standard (Fig. 2a): j=1⇒ q1 = z1(t) – displacementof thepoint of reductionArm-Shoulder (massm1), j=2⇒ q2 = z2(t) – displacement of the point of reduction Forearm-Elbow (massm2), j=3⇒ q3 = z3(t) – displacement of the point of reduction Tool-Hand (mass m3 andmN). In the combinedphysicalmodel of thehumanand the tool (Fig. 2b), the following generalized coordinates have been assumed: j=1⇒ q1 = z1(t) – displacement of the Upper Arm-Shoulder point of reduction (massm1), j=2⇒ q2 = z2(t) – displacement of the Palm-Wrist-Forearm point of reduction (massm2), j=3⇒ q3 = z3(t) – displacement of the Fingers point of reduction (mass m3), j=4⇒ q4 = z4(t) – displacement of the sum of masses m3,m4 andmN. The differential equations of motion have been solved using a simulation programdeveloped in the MATLAB/simulink environment. The use of an Elementary Processor of Energy Flow and Power Distribution (EPEFPD) made it possible to determine the power distribution in the biomechanical system, in particular changes in momentary values of the power of inertia, loss and elasticity. Individual powers of structural forces have been added to calculate the total power and determine the order of energy inputs introduced into the biological structure under the human physical models analyzed in the study. 2. Theoretical basis – The First Principle of Power Distribution in a Mechanical System The First Principle of Power Distribution in a Mechanical System can be expressed in the following way (Dobry, 1998, 2001, 2004, 2012): “The net input power introduced into the mechanical system (after subtracting the power of dissipation) is equal to the reflected power (accumulated or stored) in the system and the output power generated by the system.” This rule can be expressed in the following mathematical form (Dobry, 1998, 2001, 2004, 2012) Pin(t)−Plos(t)=Pref(t)+Pout(t) (2.1) where: — power of the resultant force – the drive input power introduced into the mechanical system (power input) Pin(t)=Win(t) ·vin(t) (2.2) A comparison of human physical models based on the distribution... 5 —power of dissipation equal to the sumof internal losses in the system and the power of inertia in the system Plos(t)=Pintlos(t)+R(t) ·vR(t) (2.3) — reflected power in the mechanical system equal to the sum of the power of inertia and the power of elasticity Pref(t)=B(t) ·vB(t)+S(t) ·vS(t) (2.4) — output power equal to the output power of a mechanical system Pout(t)=O(t) ·vout(t) (2.5) TheFirstPrinciple ofPowerDistribution in aMechanical System(FPPDMS) canbe represented graphically, as shown in Fig. 1. Fig. 1. A graphical representation of the First Principle of Power Distribution inMechanical System (FPPDMS) and the universal model of power distribution in mechanical systems, their subsystems, elements and in points of reduction (Dobry, 1998, 2001, 2004, 2012) 3. Methods of solving the problem Figure 2b shows the physicalmodelwhich combines the newmodel of the humanbodyproposed byDong et al. (2007)with themodel of the tool.The results obtainedbymeans of thenewmodel to analyze vibrations acting on the humanbody through the upper limbs (hand-armvibrations) are comparedwith those generated by themodel with three points of reduction (nine-degrees of freedom), specified in ISO 10068 (ISO 10068:1998). Thatmodel is then combined with the tool, as shown in Fig. 2a. The two combined models are discrete in the sense that their corresponding points of re- duction are connected by means of elastic and damping systems which model the elastic and damping properties of the human body. Mathematical models of the dynamic structures are derived using equation (1.1), assuming, for the sake of simplicity, only one main direction of vibration – along the z axis. The mathe- matical model of the Human-Tool system constructed on the basis of the human model with three points of reduction specified in the ISO 10068 standard (ISO 10068:1998) (Fig. 2a), has 6 M.W. Dobry, T. Hermann Fig. 2. Physical models of the Human-Tool system: (a) the model specified in ISO 10068 (ISO 10068:1998) combined with the model of the tool, (b) the new human physical model proposed by Dong et al. (2007) combined with the model of the tool the form j=1 m1z̈1+(c1+ c2)ż1+(k1+k2)z1− c2ż2−k2z2 =0 j=2 m2z̈2+(c2+ c3)ż2+(k2+k3)z2− c2ż1−k2z1− c3ż3−k3z3 =0 j=3 (m3+mN)z̈3+ c3ż3+k3z3− c3ż2−k3z2 =F(t) (3.1) Themathematical model of the second combined human-toolmodel (Fig. 2b) can bewritten as j=1 m1z̈1+(c1+ c2)ż1+(k1+k2)z1− c2ż2−k2z2 =0 j=2 m2z̈2+(c2+ c3+ c4)ż2+(k2+k3+k4)z2− c2ż1−k2z1− c3ż3 −k3z3− c4ż4−k4z4 =0 j=3 m3z̈3+(c3+ c5)ż3+(k3+k5)z3− c3ż2−k3z2− c5ż4−k5z4 =0 j=4 (m4+m5+mN)z̈4+(c4+ c5)ż4+(k4+k5)z4− c4ż2−k4z2 − c5ż3−k5z3 =F(t) (3.2) Differential equations of motion (3.1) and (3.2) are the basis of constructing energy models for the systems in question.After applying theFirstPrinciple ofPowerDistribution in aMechanical System (2.1), it is possible to switch from the conventional dynamic analysis of amplitudes to the energy analysis of power distribution. Equations (2.2)-(2.5) are used to construct an energymodel of theHuman-Tool systembased on the model specified in the ISO 10068 standard (3.1). In this case, the energy model has the following form j=1 m1z̈1ż1+(c1+ c2)ż 2 1 +(k1+k2)z1ż1−c2ż2ż1−k2z2ż1 =0 j=2 m2z̈2ż2+(c2+ c3)ż 2 2 +(k2+k3)z2ż2−c2ż1ż2−k2z1ż2− c3ż3ż2 −k3z3ż2 =0 j=3 (m3+mN)z̈3ż3+ c3ż 2 3 +k3z3ż3− c3ż2ż3−k3z2ż3 =F(t)ż3 (3.3) A comparison of human physical models based on the distribution... 7 Thus, the corresponding energy model of the Human-Tool system based on the new human physical model (3.2) can be written thus j=1 m1z̈1ż1+(c1+ c2)ż 2 1 +(k1+k2)z1ż1−c2ż2ż1−k2z2ż1 =0 j=2 m2z̈2ż2+(c2+ c3+ c4)ż 2 2 +(k2+k3+k4)z2ż2− c2ż1ż2−k2z1ż2 − c3ż3ż2−k3z3ż2− c4ż4ż2−k4z4ż2 =0 j=3 m3z̈3ż3+(c3+ c5)ż 2 3 +(k3+k5)z3ż3−c3ż2ż3−k3z2ż3− c5ż4ż3 −k5z4ż3 =0 j=4 (m4+m5+mN)z̈4ż4+(c4+ c5)ż 2 4 +(k4+k5)z4ż4− c4ż2ż4 −k4z2ż4− c5ż3ż4−k5z3ż4 =F(t)ż4 (3.4) The energy models of the two Human-Tool systems have been implemented in the MATLAB/simulink environment to determine changes in momentary values of the power of inertia, loss and elasticity. The simulation results have been used to assess the models in terms of energy flows in the system.Theyalso demonstrated differences between the newhumanmodel (Dong et al., 2007) and the currently usedmodel specified in the ISO 10068:1998 standard. 4. Energy comparison of biomechanical Human-Tool systems with studied human models Energy analysis has been carried out for a sinusoidal driving force F(t) with an amplitude of 200N at different frequencies f inHz. The analysis has been conducted for four frequency values: 16Hz, 30Hz, 60Hz and 90Hz, assuming the mass of the tool mN to be 6kg. In each case, the simulation time t has been set to 300 seconds to allow an acceptable average deviation of power values (less than 3%) for each of the models. Simulations in the MATLAB/simulink software have been implemented using integration time steps ranging from amaximum of 0.001 to a minimum of 0.0001 second. The integration procedure ode113 (Adams) with a tolerance of 0.001 has been used. Values of dynamic parameters used in the simulations have been set according to specifications in the ISO 10068 (ISO 10068:1998) standard and in Dong et al. (2007). The simulation has been performed assuming different values of forces exerted by the human operator on the tool (Table 1). Table 1.Variants of forces in the new human physical model by Dong et al. (2007) The force of Variant I (V1) Variant II (V2) Variant III (V3) Variant IV (V4) grip Fg [N] 50 15 30 50 pushingFp [N] – 35 45 50 In the case of the new human physical model, four sets of dynamic parameters are available (Table 2),which correspond to fourdifferent conditions of thehuman interactionwith thehandle of the tool. Each variant corresponds to a different value of the grip force of the hand Fg and the pushing force exerted on the handle Fp (Table 1). Table 3 shows values of the dynamic parameters for the model specified in the ISO 10068 standard (ISO 10068:1998). The dynamic analysis of this model takes into account, as alre- ady mentioned, only the main direction of hand-arm vibrations, namely parameters for the z direction. Figure 3 shows the effect of frequency of the driving impulses f on the percentage increase in the contribution of the three types of power in the new model compared to values obtained 8 M.W. Dobry, T. Hermann Table 2. Values of the dynamic parameters of the new human physical model for different conditions of the human interaction with the handle of the tool (Dong et al., 2007) Parameter Unit Variant I (V1) Variant II (V2) Variant III (V3) Variant IV (V4) m1 kg 5.854 6.099 6.505 5.863 m2 kg 1.324 0.850 0.977 1.248 m3 kg 0.083 0.084 0.080 0.083 m4 kg 0.025 0.029 0.031 0.029 m5 kg 0.013 0.011 0.012 0.013 k1 N/m 13740 17270 18830 16900 k2 N/m 2460 2420 1020 1700 k3 N/m 6790 3450 4030 4040 k4 N/m 26190 38680 48930 52490 k5 N/m 157120 56150 96310 143920 c1 N·s/m 107.07 152.87 163.76 169.7 c2 N·s/m 97.80 159.20 158.94 140.53 c3 N·s/m 39.03 25.26 28.97 35.47 c4 N·s/m 81.79 86.53 101.31 114.83 c5 N·s/m 127.98 74.73 99.87 124.59 Table 3. Values of the dynamic parameters of the human physical model specified in the ISO 10068 standard (ISO 10068:1998) Parameter Unit The direction of hand-arm vibration x y z m1 kg 3.0952 3.2462 2.9023 m2 kg 0.486 0.3565 0.6623 m3 kg 0.0267 0.0086 0.0299 k1 N/m 1565 6415 2495 k2 N/m 132 300 299400 k3 N/m 4368 27090 5335 c1 N·s/m 9.10 30.78 30.30 c2 N·s/m 18.93 51.75 380.6 c3 N·s/m 207.5 68 227.5 for the ISO10068-basedmodel with three-point reduction. The percentage increase between the models is given by the formula IP = PDONG(RMS),f PISO(RMS),f ·100% (4.1) where PDONG(RMS),f is the effective value of the power of inertia, loss or elasticity in the entire Human-Tool system obtained under the newmodel at specific frequency – power (RMS) in [W]: — power of inertia expressed in [W] PDONG INE,f = √ √ √ √ √ 1 t t ∫ 0 [m1z̈1ż1]2 dt+ √ √ √ √ √ 1 t t ∫ 0 [m2z̈2ż2]2 dt+ √ √ √ √ √ 1 t t ∫ 0 [m3z̈3ż3]2 dt + √ √ √ √ √ 1 t t ∫ 0 [(m4+m5+mN)z̈4ż4] 2 dt (4.2) A comparison of human physical models based on the distribution... 9 —power of loss expressed in [W] PDONG LOS,f = √ √ √ √ √ 1 t t ∫ 0 [(c1+ c2)ż 2 1] 2 dt+ √ √ √ √ √ 1 t t ∫ 0 [(c2+ c3+ c4)ż 2 2] 2 dt + √ √ √ √ √ 1 t t ∫ 0 [(c3+ c5)ż 2 3] 2 dt+ √ √ √ √ √ 1 t t ∫ 0 [(c4+ c5)ż 2 4] 2 dt (4.3) — power of elasticity expressed in [W] PDONG ELA,f = √ √ √ √ √ 1 t t ∫ 0 [(k1+k2)z1ż1] 2 dt+ √ √ √ √ √ 1 t t ∫ 0 [(k2+k3+k4)z2ż2] 2 dt + √ √ √ √ √ 1 t t ∫ 0 [(k3+k5)z3ż3]2 dt+ √ √ √ √ √ 1 t t ∫ 0 [(k4+k5)z4ż4]2 dt (4.4) PISO(RMS),f is the effective value of the power of inertia, loss or elasticity in the entire Human- -Tool system obtained under the ISO 10068-based model at specific frequency-power (RMS) in [W]: — power of inertia expressed in [W] PISOINE,f = √ √ √ √ √ 1 t t ∫ 0 [m1z̈1ż1]2 dt+ √ √ √ √ √ 1 t t ∫ 0 [m2z̈2ż2]2 dt+ √ √ √ √ √ 1 t t ∫ 0 [(m3+mN)z̈3ż3]2 dt (4.5) — power of loss expressed in [W] PISOLOS,f = √ √ √ √ √ 1 t t ∫ 0 [(c1+ c2)ż 2 1] 2 dt+ √ √ √ √ √ 1 t t ∫ 0 [(c2+ c3)ż 2 2] 2 dt+ √ √ √ √ √ 1 t t ∫ 0 [c3ż 2 3] 2 dt (4.6) — power of elasticity expressed in [W] PISOELA,f = √ √ √ √ √ 1 t t ∫ 0 [(k1+k2)z1ż1]2 dt+ √ √ √ √ √ 1 t t ∫ 0 [(k2+k3)z2ż2]2 dt+ √ √ √ √ √ 1 t t ∫ 0 [k3z3ż3]2 dt (4.7) The results presented in Fig. 3 indicate that the new model (Dong et al., 2007) with four degrees of freedom is not comparable to the model specified in the ISO 10068 standard. The biggest increments in the power of elasticity can be observed in variant I and IV, where, depen- ding on the frequency, the inconsistency between themodels ranges from 66% up to as much as 519%. In variant III, the difference between the models decreases, ranging from 34% to 373%. The highest degree of correspondence is exhibited in variant II, where the increase in the power of elasticity ranges from 1% to 222%. The situation is much better in the case of the contribution of other types of power. The difference between the models in terms of the power of dissipation ranges from 12% to 132%, whereas in the case of the power of inertia – from 0.03% to 20%. Assuming the relative error of 30% between themodels, it can be assumed that the results obtained for variant II are correct, but only at an operational frequency of 16Hz and 90Hz. Under more stringent compatibility 10 M.W. Dobry, T. Hermann Fig. 3. The influence of frequency f on the percentage increase in three types of power for the new model for different values of the grip force Fg and push force Fp in relation to the model specified in ISO 10068: (a) variant I, (b) variant II, (c) variant III, (d) variant IV criteria (the relative error of, say, 10%) none of the conditional variants of the tested model could be used at the operational frequencies of the Human-Tool system. Figure 4 shows the influence of the frequencyof driving impulses f on thepercentage increase in total power, which is the sum of the three types of power in the new model relative to the values obtained for the reference model specified in the ISO 10068 standard (ISO 10068:1998). The above relationship can be expressed by the formula IG = PDONGINE,f +PDONGLOS,f +PDONGELA,f PISOINE,f +PISOLOS,f +PISOELA,f ·100% (4.8) The comparison reveals that the highest degree of compliance between the new model and the model specified in the ISO 10068 standard, regardless of the frequency f, can be observed for variant II. In addition, for each conditional variant, the largest difference could be observed at the frequency f = 30Hz. The cause of such increments is the frequency f, which is similar to resonant frequencies of the subsystems (Table 4). The greatest degree of similarity (the smallest difference) has been recorded for the highest frequency (f =90Hz), where the increase between the models ranged from 4% to 21%. Assuming the relative error of 25%, it can be concluded that for each conditional variant the values obtained with the newmodel (Dong et al., 2007) are true, but only at the frequency f =90Hz. It is worth noting that the second conditional variant can also be used for research A comparison of human physical models based on the distribution... 11 Fig. 4. The influence of frequency f on the percentage increase in total power for the newmodel in relation to the model specified in ISO 10068 (ISO 10068:1998) purposes, assuming the same level of the relative error. The results for this variant are similar to those specified in the standard, but not in all operational frequencies of the systemHuman-Tool (the highest discrepancy at the frequency 30Hz). Table 4.Resonant frequencies of the humanmodels at each point of reduction Model used The new (tested) model ISO 10068 model Variant j=1 j=2 j=3 j=4 j=1 j=2 j=3 Resonant frequency of subsystem [Hz] I 8.37 26.04 223.7 27.73 51.33 107.96 4.73 II 9.04 36.44 134.1 19.94 III 8.79 37.41 178.2 24.67 IV 8.96 34.38 212.5 28.70 Figure 5 presents a comparison of the percentage share of the energy input at individual points of reduction on the basis of the amount of power which is the sum of three types of power, at different frequencies f. It can be expressed in the following equation L= PINE(RMS),j +PLOS(RMS),j +PELA(RMS),j ∑s j=1PINE(RMS),j + ∑s j=1PLOS(RMS),j + ∑s j=1PELA(RMS),j ·100% (4.9) wherePINE(RMS),j, PLOS(RMS),j and PELA(RMS),j – power of inertia, loss and elasticity, respecti- vely, at the point of reduction obtained with a particular model and its conditional variant at a specific frequency-power (RMS) in [W]. In order to compare the energy inputs at various points of reduction in both models, a theoretical value has been introduced, which is equal to the sum of powers at the points of reduction corresponding to the same part of the operator’s hand in the model. The values of power at the third point of reduction (z3) have been added to these obtained at the fourth point of reduction (z4) with the new model (Dong et al., 2007). The resulting sum of power at the theoretical point has been compared with a corresponding value obtained with the model specified in the ISO 10068 standard (ISO 10068:1998). The results shown in Fig. 5 indicate that the four sets of dynamic parameters corresponding to different conditional variants of the hand interacting with the handle of the tool proposed by the authors in Dong et al. (2007) do not affect the order of energy inputs at the points of reduction, but they do have an impact on energy input values at individual points of reduction. Based on the percentage share of the energy input at individual points of reduction, it is possible to establish the order of energy input exerted on the biological structure in the new human physical model in terms of the sum of the three kinds of power. Theorderof energy inputs in thenewphysicalhumanmodel is independentof the frequencyf and is as follows: first, the theoretical point (z3+z4) –67%-99%, second,Palm-Wrist-Forearm (z2) 12 M.W. Dobry, T. Hermann Fig. 5. The percentage change of the energy input at points of reduction in terms of the sum of three types of power for both models and their conditional variants at different frequencies: (a) f =16Hz, (b) f =30Hz, (c) f =60Hz, (d) f =90Hz – 1%-30% and third, Upper Arm-Shoulder (z1) – 0.02%-4%. In the model specified in the ISO 10068:1998 standard, the order of energy inputs at individual points of reduction depends on the frequency f. At f = 16Hz the highest energy input is recorded at the reduction point of Shoulder-Arm (z1) – 40%, this is followed by Forearm-Elbow (z2) – 34% and finally Tool- Hand (z3) – 26%. At higher frequencies the highest energy input is observed at the reduction point of Tool-Hand, which is, just like in the case of the new physical human model, the point nearest the tool. What is different, however, is the order of points – Fig. 5b and 5c. The same order of energy input values at individual points of reduction with the tested model has been observed only at f = 90Hz. At this frequency, the energy input at the points of reduction decreased the more points have been removed from the tool, in the following manner: 91% at thepointTool-Hand (z3), over 5%atForearm-Elbow(z2) and less than4%atArm-Shoulder (z1). 5. Conclusions The comparison of energy inputs revealed that the models in questions are not comparable to each other in terms of the power of elasticity and loss, since the levels of these powers in the models differ. The energy analysis shows a higher degree of similarity between the models in terms of the percentage share of total power in comparison with their individual components, that is the three types of power. Based on the results of power distribution, it is possible to formulate four conditional variants of the new human physicalmodel in terms of their similarity to the reference system, i.e. the model specified in ISO 10068:1998 (Table 5). Developing the newmodel in further research and assuming the relative error of 25%, it can be assumed that the values obtained for each conditional variant are correct, but only at the frequency f = 90Hz. It is worth noting that the second conditional variant can also be used A comparison of human physical models based on the distribution... 13 Table 5.The relative error between themodels (in terms of total power – Fig. 4) New (tested) Frequency model 16Hz 30Hz 60Hz 90Hz Variant II 2% 47% 14% 2% Variant III 22% 78% 31% 12% Variant I 43% 101% 36% 16% Variant IV 40% 104% 45% 20% for research purposes, keeping in mind its high level of discrepancy at the frequency 30Hz in comparison with themodel specified in ISO 10068:1998. The topic requires further research. The analysis presented in the study can be extended to include other characteristics of driving impulses, for example the real impulse of pulsed forces occurring duringwork with hitting impact tools. Further studies should also be conducted for a wider range of frequency f in order to examine the similarity of themodels at higher frequencies, i.e. above 90Hz. In order to confirm the adequacy of the models, they should be verified by energymeasure- ments in a laboratory. Such verification could be performed at the Laboratory of Dynamics and Ergonomics of the Human-Technical Object-Environment Metasystem at Poznan University of Technology. The research in this area will be continued. References 1. Adewusi S., Rakheja S., Marcotte P., 2012, Biomechanical models of the human hand- arm to simulate distributed biodynamic responses for different postures, International Journal of Industrial Ergonomics, 42, 2, 249-260 2. DobryM.W., 1998,Optimization of the Energy Flow in theHuman-Tool-Base System (inPolish), Rozprawahabilitacyjna, Seria “Rozprawy”nr 330, ISSN0551-6528,Wyd.PolitechnikiPoznańskiej, Poznań 3. DobryM.W., 2001,Energy diagnostics and assessment of dynamics ofmechanical and biomecha- tronics systems,Machine Dynamics Problems, 25, 35-54 4. Dobry M.W., 2004, Energy Diagnosis of Technical Systems (in Polish), Inżynieria Diagnostyki Maszyn, PolskieTowarzystwoDiagnostykiTechnicznej, InstytutTechnologii Eksploatacji,Warsza- wa, Bydgoszcz, Radom, 314-339 5. Dobry M.W., 2012,Fundamentals of Energy Diagnostics of Mechanical and Biomechanical Sys- tems (inPolish),WydawnictwoNaukowe InstytutuTechnologiiEksploatacji –PIB,Poznań-Radom 6. Dong J.H., Dong R.G., Rakheja S., Wu J., 2007, Predictions of the distributed biodynamic responses in thehand-armsystem,11th InternationalConference onHand-ArmVibration,Bologna, Italy 7. Griffin M.J., 1990,Handbook of Human Vibration, Academic Press, London 8. Meltzer G., 1981,A vibrationmodel for the human hand-arm-system,Studies in Environmental Science, 13, 210-221 9. Reynolds D.D., Soedel W., 1972, Dynamic response of the hand-arm system to a sinusoidal input, Journal of Sound and Vibration, 21, 3, 339-353 10. ISO10068:1998,Mechanical vibrationand shock –Free,mechanical impedance of the humanhand- arm system at the driving point Manuscript received November 13, 2013; accepted for print June 10, 2014