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An investigation into a synthetic vibration model for humans:
International Journal of Industrial Ergonomics 27 (2001) 219–232
An investigation into a synthetic vibration model for humans: An investigation into a mechanical vibration human model constructed according to the relations between the physical, psychological and physiological reactions of humans exposed to vibration Mitsunori Kuboa,*, Fumio Terauchia, Hiroyuki Aokia, Yoshiyuki Matsuokab a
Department of Design and Architecture, Faculty of Engineering, Chiba University, 1-33 Yayoi-Cho, Inage-Ku, Chiba 268-8522, Japan b Faculty of Science and Technology, Keio University, 3-14-1, Hiyoshi, Kohoku-ku, Yokohama, Kanagawa 223-8522, Japan Received 20 March 1999; accepted 15 May 2000
Abstract We aimed to develop a synthetic vibration model reproducing the relations between the physical, psychological and physiological reactions of the human body exposed to external vibrations. The synthetic vibration model consisted of a mechanical vibration model simulating the physical behaviour of the human body and multiple regression equations describing the above three relations. The mechanical vibration models formalised according to Lagrange’s equation of motion were employed. The experiment was carried out under conditions in which five people were exposed to external vibration that vertically vibrates at various frequencies. As a result, it was clear that there were resonance points showing remarkable shaking of the head, the chest and the abdomen in the frequency range 2–11 Hz. Moreover, it was indicated that the relations between the physical reactions and the resulting psychological and physiological reactions might be expressed in terms of multiple regression analysis. Finally, the simple vibration model of a person riding in an automobile was numerically constructed to reproduce the physical reactions of the human body, and then the psychological and physiological reactions were predicted. Relevance to industry The synthetic vibration model could facilitate comfort design in the field of industrial design in general and the automotive industry in particular. Using the vibration model in industrial fields will enable us to efficiently develop various products, whose design will take into consideration of human comfort. # 2001 Elsevier Science B.V. All rights reserved. Keywords: Human vibration model; Physical reaction; Physiological reaction; Psychological reaction; Numerical vibration simulation
1. Introduction
*Corresponding author. E-mail address: [email protected].
Many people are exposed to whole-body vibration in vehicles: cars, buses, trains, ships and airplanes, on a daily basis. In our previous paper,
0169-8141/01/$ - see front matter # 2001 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 9 - 8 1 4 1 ( 0 0 ) 0 0 0 5 2 - 4
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it was confirmed that whole-body vibration caused a subject discomfort, fatigue and physical pains (Liu et al., 1995). There are several reports describing how vibration interferes with people’s working efficiency, safety and health (Bogert, 1994; Mcleod and Griffin, 1995; Qassem et al., 1996). Therefore many researchers have concentrated their efforts on reducing the amount of vibration from products and vehicles. There are many reports describing the measurement of the transmissibility of the human body under vibration (Griffin, 1975; Randall et al., 1997; Matsumoto and Griffin, 1998). We have also measured the transmissibility of the whole body in sitting and lying posture exposed to vertical vibration (Liu et al., 1996). The results of these reports indicated the resonance of the human body depended on various factors: the posture, the materials of the given seat surface, vibration magnitude and frequency. The measurements of the transmissibility of the body under various vibrations are inefficient, laborious, tedious and expensive. On the other hand, there are a few computer-automated procedures used to predict the human body’s responses to vibration (Amirouche, 1987; Liu et al., 1996; Kitazaki and Griffin, 1997; Kubo et al., 1997; Yogananden et al., 1997; Wei and Griffin, 1998). It is difficult to accurately estimate the behaviour of the human body under vibration, because it is a complex active dynamic system. Further, it is most important to bear in mind that the complexity is not only due to physical characteristics but also due to psychological and physiological characteristics. However, no vibration model concerning the physiological and the psychological reactions of a person exposed to vibration environments has been found. Therefore, we considered that the construction of a vibration model that could reproduce the characteristics of the vibrating human riding on an automobile should be a research task. The vibration model should not only be able to reproduce the behaviour of the physical human body but also predict the physiological and psychological reactions. In constructing the vibration model, we would predict the characteristics of the three reactions (physical, physiological and psychological), and then formalize the relations between
them. In this paper the vibration model was constructed in accordance with the results of our research into the characteristics of the human exposed to a vertical sinusoidal wave force.
2. Methods 2.1. Assumption of the structure of the synthetic vibration model of human We assumed that the characteristics of the vibration of the human body might be explained by the following three reactions when the human body is exposed to some vibration environments. (i) A physical reaction expressed by the transmissibility of the vibrations of each part of the human body to a standard part, e.g. the area on the vibration table which can vibrate a person sitting on a rigid chair. (ii) A physiological reaction manifested in terms of blood pressure, heart rate, etc.; these reactions are generally termed the physiological indices. (iii) A psychological reaction as illustrated by manifestation of the different symptoms induced by vibration. The human vibration models that have been defined by many researchers are generally limited to numerical models that reproduce only the physical reaction. Therefore, we proposed a basic structure for a synthetic vibration model of human beings, shown in Fig. 1, which could indirectly reproduce the characteristics of the physiological and the psychological reactions as well as the physical reaction through multiple regression equations. Moreover, we assumed that there were some linear relations between the physical, physiological and psychological reactions, so that a multiple regression analysis could be applied to analyse these relations. In the synthetic vibration model, the physical reaction can be simulated by equations of motion formalized by using Lagrange’s equation, and the physiological and psychological reactions can be predicted by multiple regression equations defined through the multiple regression analysis. In the multiple
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Fig. 1. A synthetic vibration model for human beings.
regression equations, the physical reaction directly relates to explanatory variables to predict the physiological and psychological reaction. Also, it was assumed that any physical reaction was not affected by the physiological and psychological reactions (Fig. 1). 2.2. Experimental conditions The basic experimental device was an electromagnetic vibration table (IMV Co.) on which a sufficiently rigid chair was installed (Fig. 2). The subjects who were exposed to periodical vertical vibrations were five males aged from 22 to 29. They were exposed to whole-body vibration while sitting under the following vibration conditions. (i) the vibration stimulus : sinusoidal waves in a vertical direction. (ii) the vibrating frequencies : 2, 5, 8, 11, 14, 17, and 20 Hz. (iii) the effective value of acceleration : 0.69 m/s2. (iv) the time of exposure to vibration : 10 min. During the experiments, the air temperature was 24–268C and the air humidity was 40–50%. Further, the characteristics of the physical, physiological and psychological reactions of the subjects in the vibration experiments were measured according to the following three methods, respectively.
Fig. 2. Experimental device.
2.2.1. Measurement of the physical reaction The physical reaction was measured with the data collected by 10 accelerometers (KYOWA, ASV-2GA); Fig. 2 shows the points of installation of the accelerometers. The data were measured with a Fast Fourier Transform (FFT) analyzer (ONO SOKKI, CF360) to obtain the vibration transmissibility. The transmissibility was defined according to the transmission ratio of acceleration sensed by each accelerometer to the acceleration of the vibration table, as well as by the phase difference between each vibration mode and the table vibration mode. The transmission ratios and phase differences were calculated by the FFT. 2.2.2. Measurement of the physiological reaction The measurements of the heart rate, highest and lowest blood pressure, respiration rate and saliva secretion quantity were carried out according to a timetable (Fig. 3). The heart rate was measured with the CM5-inducement method (Nakamura et al., 1983) through an electrical amplifier for a living body (NIHON KOUDEN, AB-620G) and a data recorder (TEAC, RD-111 T). The blood pressures were measured on the artery of the left upper arm with a simple sphygmomanometer (OMURON, HEM-700). The respiration rate
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Fig. 3. Timetable for the vibration experiment.
was calculated by using the nostril-type respiration curve. The curve was measured with a thermister (NIHON KOUDEN, TR-712 T). The saliva quantity was measured according to a method in which the saliva was collected by dental cotton. The weight of the dental cotton was controlled so that it maintained a constant value. The weight of the cotton after saliva absorption was measured with a precise scale (SHIMAZU, AEL-200). All the physiological measured values were expressed in terms of the ratio of the measured values when the subjects were exposed to vibration, to the initial values before exposure to vibration. We assumed that the influence of vibration would be weak when the ratio was near to one.
Fig. 4. Questionnaire of a feeling of tiredness.
3. Results 2.2.3. Evaluation of the psychological reaction The psychological change was evaluated according to semantic differential (SD) method. In this paper, a feeling of tiredness was considered as a psychological reaction. The subjects were equipped with earplugs in order to block the sound of the vibration table. They were questioned on their level of comfort before and after the time when they were exposed to the vibration in order to make clear the characteristics of their psychological reaction to the vibration. The questionnaire employed the terms shown in Fig. 4. It was assumed that the feeling of tiredness consisted of physical symptoms (tiredness, yawning, sleepiness, tired eyes, and absent-mindedness), mental symptoms (irritation, loss of patience, distracted attention), and nervous symptoms (headache, backache, dizziness, nausea, and stiff shoulders).
3.1. Physical reaction Fig. 5 shows the changes of transmissibility of five parts (head, chest, abdomen, thigh, and lower leg) according to the vibration frequencies. Although the changes of the transmission ratios according to the frequency were present in all parts, it was especially apparent in the chest at 5 Hz. Additionally, it was verified that the phase differences were also governed by the frequency. As a result, it was confirmed that the physical reaction would be affected by the frequency of the vibration force. In Fig. 5, the solid lines show the changes of the transmission ratios and phase differences in the direction of U, along the back of the rigid chair (see the local coordinate system illustrated in Fig. 2).
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Fig. 5. Physical reaction: Transmission ratio and phase difference to the direction U.
3.2. Physiological reaction The vertical axis of Fig. 6a shows the ratio of the heart rate after being exposed to the vibration, to the rate before it. Thus, when the ratio is greater than unity, it was apparent that the heart rate had been increased by the vibration exposure. Similarly, the other ratios in Fig. 6 show the changes between before and after exposure. As a result, it was clear that the heart rate, the respiration rate and the highest blood pressure had been increased by the vibration exposure, and the saliva-secretion quantity had been decreased.
exception of the yawning and sleepiness, the feelings of tiredness increased according to the vibration frequency. 3.4. Relations between the physical, physiological and psychological reactions The three relations between the physiological reaction and the physical reaction, the psychological reaction and the physical reaction, and the psychological reaction and the physiological reaction were formalized according to the multiple regression analysis using the above-mentioned experimental results.
3.3. Psychological reaction The horizontal axis of Fig. 7 shows the transition from the feelings of tiredness before exposure to the vibration, to the feelings after it. On this axis, ‘0’ means no transition of the feelings of tiredness caused by the vibration exposure. The broken line shows the transition at 2 Hz, and the solid line shows it at 5 Hz. In general, with the
3.4.1. Relation between the physiological reaction and the physical reaction The physiological reaction was expressed in terms of the purpose variables in the multiple regression analysis, and the physical reaction was expressed in the explanation variables of that analysis. The relation between them was described by multiple regression equations to predict the
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Fig. 6. Physiological reaction.
physiological reaction against the given vibration transmission ratios: physical reaction. From Table 1, it was suggested that the physiological reaction might be induced through the vibration of the head, the chest, the abdomen, and lower leg at 5 Hz. Also, it was realized that the characteristic of the physiological reaction and the relation between the physiological reaction and the physical reaction alter according to the frequency of the vibration. The dynamic characteristic would be due to the fact that there are some differences between the resonance frequencies of the head, of the chest and of the abdomen corresponding to the external vibration.
As an example, the multiple regression equations representing the relation between the physiological reaction and the physical reaction at 5 Hz are Yheart rate ¼ 2:388 þ 0:054X1 þ 0:200X2 ÿ 1:351X5 , Yhighest blood pressure ¼ ÿ 0:812 þ 2:055X1 , Ylowest blood pressure ¼ 0:634 ÿ 0:088X1 ÿ 0:175X2 þ 0:750X5 , Yrespiration rate ¼ 2:550 ÿ 0:212X2 ÿ 0:083X3 ÿ 0:692X5 ,
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Ysaliva secretion quantity ¼ 0:159 ÿ 0:280X1 ÿ 0:423X3 þ 1:362X5 ,
ð1Þ
where X1 shows the vibration transmission ratio of the acceleration of the head to the acceleration of the vibration table (Fig. 2). Similarly, X2 shows the ratio of the chest, X3 shows the ratio of the abdomen, X4 shows the ratio of the thigh, and X5 shows the ratio of the lower leg. From the equations, we could deduce that the heart rate, respiration rate and highest blood
pressure would increase with exposure to the vibration at 5 Hz, while the saliva quantity would decrease. 3.4.2. Relation between the psychological reaction and the physical reaction The transmissibility (transmission ratios, phase differences) of five body parts were defined as the explanation variables, and the 13 adjectives used to estimate the feeling of tiredness shown by the subjects when they were exposed to vibration, were defined as the purpose variables in the multiple regression analysis. The adjectives are displayed in the first column of Table 2. As a result, it was clear that the vibrations of the abdomen, head and chest mainly cause tiredness at 2 Hz and 5 Hz. In particular, it was confirmed that there is a strong relation between the vibration of each body part and the psychological reaction at 5 Hz. For example, the multiple regression equations representing the psychological reaction at 5 Hz are Ytiredness ¼ ÿ 716:335 þ 254:412X2 þ 120:435X4 , Yyawning ¼ ÿ 108:989 ÿ 16:744X2 þ 114:285X5 , Ysleepiness ¼ 69:861 ÿ 69:960X1 , Ytired eyes ¼ ÿ 364:568 þ 234:968X1 þ 59:181X2 , Yabsentÿmindedness ¼ ÿ 268:326 þ 234:906X1 þ 243:698X3 ÿ 177:265X5 , Yirritation ¼ ÿ 34:528 þ 36:232X1 , Yloss of patience ¼ ÿ 248:247 ÿ 45:156X1 þ 218:584X4
Fig. 7. Psychological reaction: Transition of the feelings of tiredness.
ÿ 73:250X5 ,
Table 1 Multiple regression relation between the physiological reaction and the physical reaction at 5 Hz Physiological reaction Heart rate Highest blood pressure Lowest blood pressure Respiration rate Saliva secretion quantity
Constant
2.388 ÿ0.812 0.634 2.550 0.159
Each body part Head
Chest
Abdomen
Thigh
Lower leg
0.054 2.055 ÿ0.088 } ÿ0.280
0.200 } ÿ0.175 ÿ0.212 }
} } } ÿ0.083 ÿ0.423
} } } } }
ÿ1.351 } 0.750 ÿ0.692 1.362
R
R2
0.999 0.689 0.998 0.999 0.995
0.998 0.475 0.996 0.998 0.990
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Table 2 Multiple regression relation between the psychological reaction and the physical reaction at 5 Hz Psychological reaction
Constant
Tiredness Yawning Sleepiness Tired eyes Absent-mindedness Irritation Loss of patience Distracted attention Headache Bachache Dizziness Nausea Stiff shoulders
ÿ716.335 ÿ108.986 69.861 ÿ364.568 ÿ268.326 ÿ34.528 ÿ248.247 ÿ266.597 ÿ284.597 ÿ656.687 ÿ44.854 ÿ210.231 ÿ408.998
Each body part Head
Chest
Abdomen
Thigh
Lower leg
} } ÿ69.960 234.968 234.960 36.232 45.156 17.850 32.515 } 47.534 24.882 }
254.412 ÿ16.744 } 59.181 } } } 71.059 16.151 32.846 } 26.664 173.352
} } } } 243.698 } } } } 255.421 } 120.426 32.438
120.435 } } } } } 218.584 } } 232.391 } } }
} 114.285 } } ÿ177.265 } ÿ73.050 75.595 171.099 } } } }
Ydistracted attention ¼ ÿ 266:597 þ 17:850X1 þ 71:059X2 þ 75:595X5 , Yheadache ¼ ÿ 284:713 þ 32:515X1 þ 16:151X2 þ 171:099X5 , Ybackache ¼ ÿ 656:687 þ 32:846X2 þ 255:421X3 þ 232:391X4 , Ydizziness ¼ ÿ 44:854 þ 47:534X1 , Ynausea ¼ ÿ 210:231 þ 24:882X1 þ 26:664X2 þ 120:426X3 , Ystiff shoulders ¼ ÿ 408:998 þ 173:352X2 þ 32:438X3 ,
ð2Þ
where, in the same manner as the above section, each Xi shows the vibration transmission ratio of the acceleration of each part to the acceleration of the vibration table. 3.4.3. Relation between the psychological reaction and the physiological reaction In this relation, the 13 adjectives were employed as the purpose variables, and the above-mentioned physiological reaction was employed as the explanation variables. As shown in Table 3, it was confirmed that the psychological reaction had a
R
R2
0.985 0.939 0.694 0.957 0.973 0.718 0.995 0.999 0.999 0.998 0.763 0.996 0.954
0.969 0.883 0.481 0.916 0.946 0.515 0.989 0.998 0.998 0.997 0.582 0.992 0.910
relevance to some symptoms of the physiological reaction. The symptoms were tiredness, absentmindedness, irritation and loss of patience, distracted attention, dizziness and nausea. Furthermore, it was apparent that the relevance was altered by the frequency of the vibration imposed on the human body and that the degree of the relevance would be remarkable according to increases in the frequency. For example, it was obtained that the heart rate would have a relevance to [absent-mindedness, loss of patience, dizziness and nausea], and the highest blood pressure would have a relevance to [irritation, loss of patience, distracted attention, dizziness, and nausea]. Additionally, it might be an interesting fact that there was a difference between the relevance of the lowest blood pressure and to that of the highest. 3.5. Construction of a physical vibration model of the human body In the synthetic vibration model, to reproduce a human being’s behaviour when exposed to vibration, the prediction of the physical reaction (Xi ) has to be performed as well as the formulation of the relations between the physical and the physiological and psychological reactions. The physical reaction is predicted by a numerical physical human body vibration model, which is a component
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M. Kubo et al. / International Journal of Industrial Ergonomics 27 (2001) 219–232 Table 3 Multiple regression relation between the psychological reaction and the physiological reaction at 5 Hz Physiological reaction Heart rate Highest Psychological reaction
Constant
Blood pressure
Tiredness Absent-mindedness Irritation Loss of patience Distracted attention Dizziness Nausea
ÿ155.561 116.276 ÿ40.026 178.652 48.026 } ÿ47.125 81.813 355.759 } 61.789 ÿ116.276 ÿ375.681 136.861
} } 70.588 103.645 134.872 74.356 }
of the synthetic vibration model. Basically, in the synthetic vibration model, the numerical physical vibration model predicts the physical reaction (Xi ) according to a physical environment in which the physical model vibrates numerically. And the physiological and psychological reactions are estimated through the multiple regression equations (Eqs. (1) and (2)) with the physical reaction as the explanation variables in the multiple regression equations.
3.5.1. Assumption to simplify the human body In this paper we assumed that parts of the human body would only swing back and forth as well as move up and down. Because it was apparent that the human body would remain physically symmetry during exposure to vibration in a vertical direction. Thus, in the physical vibration model, to predict the physical reaction the transverse shaking of the human body is ignored. Therefore, we can assume that a twodimensional model projected on the central plane, which is a midsagittal plane, of the human body would simulate the realistic vibration behaviour of the human body. Additionally, to simplify the model of the human body further, the following conditions were assumed: (1) It was assumed that the human body consists of head, chest (from the upper point of the breastbone to the third lumbar vertebra), abdomen (from the third lumbar vertebra to the trochanteric point), thigh, and lower leg. Each
Lowest
R
R2
0.982 0.987 0.968 0.993 0.968 0.945 0.922
0.964 0.974 0.937 0.986 0.938 0.893 0.850
Respiration rate Saliva secretion quantity
ÿ36.154 45.185 ÿ175.640 } ÿ125.396 } } ÿ150.042 ÿ276.595 ÿ202.071 } } 206.626 }
} } } } } } }
part of the human body has a mass and a rotating inertia at the centre of gravity (Fig. 8). (2) The lower leg could be connected to the thigh and the thigh to the abdomen by a joint with an axis of rotation and generating a viscosity resistance moment. The resistance moment represents the passive resistance element of ligaments. The abdomen and chest are connected by a viscoelasticity element that consists of a spring and a damper, and the chest and head are connected in the same way. The viscoelasticity element could simulate lumber and cervical vertebrae. (3) The weight of the lower legs could be supported by the horizontal plane of the experimental chair and the surface of the vibration table, so that the weight of the lower legs has no effect on the pelvis. (4) Only portions of the back of head, the back and the lower pelvis are exposed to the external force of the vibration (Fig. 2). (5) So that the head, trunk (chest, abdomen) and pelvis would never slip on the surface of the chair, there is sufficient frictional force at each point of contact. Finally, we simplified the human body to a twodimensional vibration model consisting of masses, rigid links, springs and dampers with nine degrees of freedom (Fig. 8). 3.5.2. Formulation of the equation of motion for the simplified human vibration model In order to simplify the formulation of the equation of motion for the two-dimensional
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Fig. 8. Two-dimensional human vibration model.
vibration model, we further assumed the following: (1) Each part of the vibration model slightly vibrates around each static force equalizing position. (2) The righting moment of springs and the attenuating force of dampers are in proportion to the displacement and the velocity, respectively. (3) The saturation viscosity resistance moment is applied to the resistance moments between the lower leg and the thigh and between the thigh and the abdomen. Finally, the equation of motion consists of the coefficient matrices illustrating the effects of the masses, rigid links, springs and dampers. The equation also has nine degrees of freedom, which were 3 rotations and 6 translations, which did not perpendicularly intersect each other. Therefore, the equations were formulated with generalized coordinates according to the general process of Lagrange’s equation of motion. The equation of motion of the human body is ½M d2 x=dt2 þ ½Cfdx=dtg þ ½K fxg ¼ f f g, ð3Þ where {x} is generalized coordinates: fxg ¼ fy1 , y2 , y3 , B, Z, v1 , u1 , v2, u2 gT and, { f } is general-
ized forces:f f g ¼ ff1 , f2 , f3 , f4 , f5 , f6 , f7 , f8, f9 gT . Each fi corresponds to each generalized coordinate in the equation of motion. [M ], [C] and [K] -coefficient matrices are symmetric positive matrices that have nine degrees of freedom. [M] consists of mi , lij , Ii and ai , [C] consists of ci , lij , Ii and ai , and [K] consists of ki , lij , Ii and ai . In this paper, ki was the spring constant, and ci was the damping coefficient. Furthermore, to quantitatively define the unknown constants: ki , ci included in the coefficient matrices, the transmission function, numerically calculated with the solution of the equation of motion, was compared to the experimental transmission function measured and calculated by the accelerometers and the FFTanalyzer. The coefficient matrices were controlled according to a general optimum design method so that the numerical transmission function would coincide with the experimental transmission function. The damping matrix [C] corresponds to velocity and [M] and [K] correspond to acceleration and displacement, respectively, so that the phase differences between the generalized coordinates {x} of each part of the body are induced. Therefore, {x} would be complex numbers in general, so that the transmission functions would be described with complex numbers. As a result,
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the spring and damping coefficients that were calculated through identifying the numerical transmission function with the experimental transmission function, would be complex numbers (Table 4). Finally, the equation of motion could simulate the physical vibration behaviour of the human body and could predict the physical reaction (transmission function: Xi ), which would be substituted for the explanation variables in the multiple regression equations.
4. Discussion 4.1. Investigation into the validity of the physical vibration model of the human body In order to investigate the validity of the physical vibration model, the experimental transmission functions were compared with the numerical transmission functions predicted by the above-mentioned physical vibration model. Both transmission functions relate to the direction U (see Fig. 2). The numerical transmission functions were calculated by solving the equation of motion with time history response analysis, in which the spring constants and damping coeffi-
cients identified in the above section were applied to the equation (see Table 4). And, the spring and damping coefficients at the frequencies 3, 4, 6, 7, 9 and 10 Hz were calculated by using interpolation polynomials with the known spring and damping coefficients shown in Table 4. Fig. 9 shows that the numerical transmissibility (broken lines) almost agrees with the experimental transmissibility (solid lines) in all the frequency range. Moreover, the discrepancy between the numerical results and experimental results in the range of 2–11 Hz was relatively low within the range of 0.14 and 12.27%. In particular, in the range lower than 8 Hz, the relative error was less than 10%, while at 11 Hz, the relative error was within 10.61 and 12.72%. The disparity between the numerical results and the experimental results is inevitable for the following reasons: (1) The precision of estimating the spring constants and damping coefficients was affected by the measurement precision of the curve fit, the window function and the number of averages involved in the FFT. (2) As a simplification, it was assumed that the human body consisted of 16 simplified geometrical elements: the elements being an ellipsoid of gyration, a truncated cone, a cylinder
Table 4 Identified spring constants (Unit: 10 N/m) and damping coefficients (Unit: N s/m) 2 Hz Real part
5 Hz Imaginary part
Real part
8 Hz Imaginary part
Real part
11 Hz Imaginary part
Real part
Imaginary part
k1 k2 k3 k4 k5 k6
1040.6 866.6 419.4 309.6 610.8 589.7
694.4 61.4 47.4 149.1 274.1 118.7
1261.7 452.2 457.6 665.7 948.5 338.0
839.0 22.5 44.0 148.0 66.2 209.6
1307.8 710.1 286.9 367.4 936.6 546.8
686.1 116.1 116.9 10.6 181.9 244.8
1233.2 842.0 567.9 765.2 1550.9 565.3
60.5 36.5 144.6 40.6 118.9 462.8
k7 c1 c2 c3 c4 c5 c6 c7
401.2 783.4 364.4 164.3 170.3 146.7 250.3 224.6
173.0 ÿ724.7 ÿ96.1 ÿ74.2 ÿ301.1 ÿ169.4 ÿ252.1 ÿ117.6
239.5 754.1 469.8 168.0 181.2 457.8 256.9 182.8
62.2 ÿ496.0 ÿ32.1 ÿ89.0 ÿ61.2 ÿ180.5 ÿ209.4 ÿ53.6
295.5 664.7 861.0 214.8 110.7 541.3 388.7 164.2
103.2 ÿ288.3 ÿ107.3 ÿ129.7 ÿ37.3 ÿ223.5 ÿ161.1 ÿ62.1
269.4 636.6 312.3 222.7 227.7 485.9 356.6 242.1
128.8 ÿ479.7 ÿ46.5 ÿ109.9 ÿ50.9 ÿ223.8 ÿ301.1 ÿ103.3
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Fig. 9. Comparison between the measured- and the predicted-transmission ratio.
and a rectangular prism. Moreover, it was assumed that all the elements were homogeneous substances and that the densities were equal to the mean density of the human body. The precision of the mass, the centre of gravity and the moment of inertia also contribute to the discrepancy. (3) From a different viewpoint, a real human body consists of bone, internal organs, muscles and fat. Therefore, the human body can be regarded as an elastic body performing complicated motions while resonance is occurring. When the human body is exposed to lowfrequency vibration, we consider that the weight of the human body consists of only rigid body masses, because the relative elastic displacements of the human body are sufficiently infinitesimal. However, when the body is exposed to higher frequency vibration, it is difficult to consider that the weight of the human body consists of rigid body masses, because the nonlinearity of muscles and fat would become stronger in relation to the increase in frequency.
4.2. An example simulating the vibration characteristics of the human body by using the vibration model The human vibration model was installed on a concentrated frame of two-dimensional automobile vibration model (Nishiyama, 1993) to simulate the vibration behaviour of a human body riding in an automobile (Fig. 10). We predict the unknown psychological and physiological reactions of a person riding the two-dimensional automobile vibrating at a given frequency, by using the above-mentioned multiple regression equation representing the relations between the psychological and physical reactions, and between the physiological and physical reactions. As an example, a simulation of the vibration in the human– automobile system exposed to a perpendicular sine-wave force of 5 Hz was performed. This frequency violently shakes the head and the chest. In this case, it was assumed that the concentrated frame of the automobile (Fig. 10) remain horizontal and vibrate throughout at 5 Hz. Therefore, the spring constants and damping coefficients
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Table 5 Ratios between the reactions before and after exposure to vibration
Fig. 10. A human vibration model riding in an automobile vibration model.
calculated through the above-mentioned identification of the transmission functions at 5 Hz were applied to the equation of motion of the human vibration model riding the automobile. At first, the transmission ratios of the vibration of each body part to the table were estimated through the simulation of the vibration system. And, the physiological and psychological reactions to the vibration were easily predicted by using the above-mentioned multiple regression equations with the transmission ratios as the explanation variables. For example, from Table 5, we can read the change relating to tiredness as a magnification of 2.70. This change ratio illustrates that the quantified tiredness after exposure to the vibration is 2.70 times higher than the tiredness before it. As a result, it was suggested that the simple human-automobile vibration model could realistically predict the physical, physiological and psychological reactions of a person riding in an automobile. Additionally, we could modify the spring constants and damping coefficients according to the predicted vibration behaviour, so that the vibrations inducing disagreeable impressions could be reduced.
Reactions
Ratios
Psychological reaction Tiredness Yawning Sleepiness Tired eyes Absent-mindedness Irritation Loss of patience Distracted attention Headache Backache Dizziness Nausea Stiff shoulder
2.70 1.01 0.54 2.71 5.47 9.25 6.70 6.00 5.52 13.95 13.62 6.98 7.69
Physiological reaction Heart rate Highest blood pressure Lowest blood pressure Breathing rate Saliva quantity
1.16 1.26 1.12 1.12 1.16
5. Conclusions In this paper, the possibility of a synthetic vibration model that could enable us to numerically estimate the synthetic behaviour of a person exposed to vibration was investigated. The synthetic behaviour would be expressed by synthesizing the characteristics of the physiological and psychological reactions to the vibration, as well as those of the physical reaction. The vibration model consists of two basic equations: *
*
the multiple regression equations formalized by multiple regression analysis according to experimental data; the equation of motion formalized by using Lagrange’s equation of motion.
The equation of motion of the human body was expressed in terms of masses, springs, dampers, links and rotating inertia. These parameters were also defined according to the physical reaction measured in the vibration experiments with the FFT-analyzer. As an example, the constructed
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vibration model simulated the vibration behaviour of the human body in a sitting posture at 5 Hz. As a result of the simulation, it was demonstrated that the physical reaction could be adequately estimated through the equation of motion of the mechanical human model, and that the physiological and psychological reactions could be numerically predicted by using the multiple regression equations in which the physical reaction (transmission ratios) was substituted in place of the explanation variables. Additionally, it was suggested that this vibration system could be the optimum design for a comfortable ride, in terms of the psychological and physiological reactions.
References Amirouche, F.M.L., 1987. Modeling of human reactions to whole-body vibration. Journal of Biomechanical Engineering 109 (3), 210–217. Bogert, A.J., 1994. Analysis and simulation of mechanical loads on the human musculoskeletal system. Exercise and Sport Sciences Reviews 22, 23–51. Griffin, M.J., 1975. Vertical vibration of seated subject, effect of posture, vibration level, and frequency. Aviation Space and Environmental Medicine 46 (3), 269–276. Kitazaki, S., Griffin, M.J., 1997. A model analysis of wholebody vertical vibration, using a finite element model of the human body. Journal of Sound and Vibration 200 (1), 83–103.
Kubo, M., Terauchi, F., Aoki, H., Suziki, T., 1997. Development of the vibration model of human. Proceeding of Korea– Japan Joint Design Symposium. pp. 125–128. Liu, J.Z., Kubo, M., Aoki, H., Liu, N., Kou, P.H., Suzuki, T., 1995. A study on the difference of human sensation evaluation to whole body vibration in sitting and lying postures. Journal of Physiological Anthropology 14 (5), 219–226. Liu, J.Z., Kubo, M., Aoki, H., Terauchi, F., 1996. The transfer function of human body on vertical sinusoidal vibration. Japanese Journal of Ergonomics 32 (1), 29–38. Matsumoto, Y., Griffin, M.J., 1998. Dynamic response of the standing human body exposed to vertical vibration. Journal of Sound and Vibration 212 (1), 85–108. Mcleod, R.W., Griffin, M.J., 1995. Mechanical vibration included interference with manual control performance. Ergonomics 38, 1431–1444. Nakamura, R., Saito, H., Kitahara, K., 1983. Undougaku Zissyu: Isiyaku Syuppan Kabusikigaisya, Tokyo, 182pp. Nishiyama, S., 1993. Development of simulation system on vehicle–occupant dynamic interaction. Transactions of the Japan Society of Mechanical Engineers Seriese-C 59 (568), 1004–1012. Qassem, W., Al-Nashash, H., Zabin, A., Othman, M., 1996. ECG response of the human body subjected to vibration. Journal of Medical Engineering & Technlogy 20 (1), 2–10. Randall, J.M., Mattehews, R.T., Stiles, M.A., 1997. Resonance frequencies of standing humans. Ergonomics 40 (9), 879–886. Wei, L., Griffin, M.J., 1998. Mathematical models for the apparent mass of the seated human body. Journal of Sound and Vibration 212 (5), 855–874. Yogananden, N., Kumaresan, S., Voo, L., Pintar, A., 1997. Finite-element model of the human lower cervical spine: parametric analysis of the C4–C6. Journal of Biomechanical Engineering 119 (1), 87–92.
An investigation into a synthetic vibration model for humans: An investigation into a mechanical vibration human model constructed according to the relations between the physical, psychological and physiological reactions of humans exposed to vibration Mitsunori Kuboa,*, Fumio Terauchia, Hiroyuki Aokia, Yoshiyuki Matsuokab a
Department of Design and Architecture, Faculty of Engineering, Chiba University, 1-33 Yayoi-Cho, Inage-Ku, Chiba 268-8522, Japan b Faculty of Science and Technology, Keio University, 3-14-1, Hiyoshi, Kohoku-ku, Yokohama, Kanagawa 223-8522, Japan Received 20 March 1999; accepted 15 May 2000
Abstract We aimed to develop a synthetic vibration model reproducing the relations between the physical, psychological and physiological reactions of the human body exposed to external vibrations. The synthetic vibration model consisted of a mechanical vibration model simulating the physical behaviour of the human body and multiple regression equations describing the above three relations. The mechanical vibration models formalised according to Lagrange’s equation of motion were employed. The experiment was carried out under conditions in which five people were exposed to external vibration that vertically vibrates at various frequencies. As a result, it was clear that there were resonance points showing remarkable shaking of the head, the chest and the abdomen in the frequency range 2–11 Hz. Moreover, it was indicated that the relations between the physical reactions and the resulting psychological and physiological reactions might be expressed in terms of multiple regression analysis. Finally, the simple vibration model of a person riding in an automobile was numerically constructed to reproduce the physical reactions of the human body, and then the psychological and physiological reactions were predicted. Relevance to industry The synthetic vibration model could facilitate comfort design in the field of industrial design in general and the automotive industry in particular. Using the vibration model in industrial fields will enable us to efficiently develop various products, whose design will take into consideration of human comfort. # 2001 Elsevier Science B.V. All rights reserved. Keywords: Human vibration model; Physical reaction; Physiological reaction; Psychological reaction; Numerical vibration simulation
1. Introduction
*Corresponding author. E-mail address: [email protected].
Many people are exposed to whole-body vibration in vehicles: cars, buses, trains, ships and airplanes, on a daily basis. In our previous paper,
0169-8141/01/$ - see front matter # 2001 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 9 - 8 1 4 1 ( 0 0 ) 0 0 0 5 2 - 4
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it was confirmed that whole-body vibration caused a subject discomfort, fatigue and physical pains (Liu et al., 1995). There are several reports describing how vibration interferes with people’s working efficiency, safety and health (Bogert, 1994; Mcleod and Griffin, 1995; Qassem et al., 1996). Therefore many researchers have concentrated their efforts on reducing the amount of vibration from products and vehicles. There are many reports describing the measurement of the transmissibility of the human body under vibration (Griffin, 1975; Randall et al., 1997; Matsumoto and Griffin, 1998). We have also measured the transmissibility of the whole body in sitting and lying posture exposed to vertical vibration (Liu et al., 1996). The results of these reports indicated the resonance of the human body depended on various factors: the posture, the materials of the given seat surface, vibration magnitude and frequency. The measurements of the transmissibility of the body under various vibrations are inefficient, laborious, tedious and expensive. On the other hand, there are a few computer-automated procedures used to predict the human body’s responses to vibration (Amirouche, 1987; Liu et al., 1996; Kitazaki and Griffin, 1997; Kubo et al., 1997; Yogananden et al., 1997; Wei and Griffin, 1998). It is difficult to accurately estimate the behaviour of the human body under vibration, because it is a complex active dynamic system. Further, it is most important to bear in mind that the complexity is not only due to physical characteristics but also due to psychological and physiological characteristics. However, no vibration model concerning the physiological and the psychological reactions of a person exposed to vibration environments has been found. Therefore, we considered that the construction of a vibration model that could reproduce the characteristics of the vibrating human riding on an automobile should be a research task. The vibration model should not only be able to reproduce the behaviour of the physical human body but also predict the physiological and psychological reactions. In constructing the vibration model, we would predict the characteristics of the three reactions (physical, physiological and psychological), and then formalize the relations between
them. In this paper the vibration model was constructed in accordance with the results of our research into the characteristics of the human exposed to a vertical sinusoidal wave force.
2. Methods 2.1. Assumption of the structure of the synthetic vibration model of human We assumed that the characteristics of the vibration of the human body might be explained by the following three reactions when the human body is exposed to some vibration environments. (i) A physical reaction expressed by the transmissibility of the vibrations of each part of the human body to a standard part, e.g. the area on the vibration table which can vibrate a person sitting on a rigid chair. (ii) A physiological reaction manifested in terms of blood pressure, heart rate, etc.; these reactions are generally termed the physiological indices. (iii) A psychological reaction as illustrated by manifestation of the different symptoms induced by vibration. The human vibration models that have been defined by many researchers are generally limited to numerical models that reproduce only the physical reaction. Therefore, we proposed a basic structure for a synthetic vibration model of human beings, shown in Fig. 1, which could indirectly reproduce the characteristics of the physiological and the psychological reactions as well as the physical reaction through multiple regression equations. Moreover, we assumed that there were some linear relations between the physical, physiological and psychological reactions, so that a multiple regression analysis could be applied to analyse these relations. In the synthetic vibration model, the physical reaction can be simulated by equations of motion formalized by using Lagrange’s equation, and the physiological and psychological reactions can be predicted by multiple regression equations defined through the multiple regression analysis. In the multiple
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Fig. 1. A synthetic vibration model for human beings.
regression equations, the physical reaction directly relates to explanatory variables to predict the physiological and psychological reaction. Also, it was assumed that any physical reaction was not affected by the physiological and psychological reactions (Fig. 1). 2.2. Experimental conditions The basic experimental device was an electromagnetic vibration table (IMV Co.) on which a sufficiently rigid chair was installed (Fig. 2). The subjects who were exposed to periodical vertical vibrations were five males aged from 22 to 29. They were exposed to whole-body vibration while sitting under the following vibration conditions. (i) the vibration stimulus : sinusoidal waves in a vertical direction. (ii) the vibrating frequencies : 2, 5, 8, 11, 14, 17, and 20 Hz. (iii) the effective value of acceleration : 0.69 m/s2. (iv) the time of exposure to vibration : 10 min. During the experiments, the air temperature was 24–268C and the air humidity was 40–50%. Further, the characteristics of the physical, physiological and psychological reactions of the subjects in the vibration experiments were measured according to the following three methods, respectively.
Fig. 2. Experimental device.
2.2.1. Measurement of the physical reaction The physical reaction was measured with the data collected by 10 accelerometers (KYOWA, ASV-2GA); Fig. 2 shows the points of installation of the accelerometers. The data were measured with a Fast Fourier Transform (FFT) analyzer (ONO SOKKI, CF360) to obtain the vibration transmissibility. The transmissibility was defined according to the transmission ratio of acceleration sensed by each accelerometer to the acceleration of the vibration table, as well as by the phase difference between each vibration mode and the table vibration mode. The transmission ratios and phase differences were calculated by the FFT. 2.2.2. Measurement of the physiological reaction The measurements of the heart rate, highest and lowest blood pressure, respiration rate and saliva secretion quantity were carried out according to a timetable (Fig. 3). The heart rate was measured with the CM5-inducement method (Nakamura et al., 1983) through an electrical amplifier for a living body (NIHON KOUDEN, AB-620G) and a data recorder (TEAC, RD-111 T). The blood pressures were measured on the artery of the left upper arm with a simple sphygmomanometer (OMURON, HEM-700). The respiration rate
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Fig. 3. Timetable for the vibration experiment.
was calculated by using the nostril-type respiration curve. The curve was measured with a thermister (NIHON KOUDEN, TR-712 T). The saliva quantity was measured according to a method in which the saliva was collected by dental cotton. The weight of the dental cotton was controlled so that it maintained a constant value. The weight of the cotton after saliva absorption was measured with a precise scale (SHIMAZU, AEL-200). All the physiological measured values were expressed in terms of the ratio of the measured values when the subjects were exposed to vibration, to the initial values before exposure to vibration. We assumed that the influence of vibration would be weak when the ratio was near to one.
Fig. 4. Questionnaire of a feeling of tiredness.
3. Results 2.2.3. Evaluation of the psychological reaction The psychological change was evaluated according to semantic differential (SD) method. In this paper, a feeling of tiredness was considered as a psychological reaction. The subjects were equipped with earplugs in order to block the sound of the vibration table. They were questioned on their level of comfort before and after the time when they were exposed to the vibration in order to make clear the characteristics of their psychological reaction to the vibration. The questionnaire employed the terms shown in Fig. 4. It was assumed that the feeling of tiredness consisted of physical symptoms (tiredness, yawning, sleepiness, tired eyes, and absent-mindedness), mental symptoms (irritation, loss of patience, distracted attention), and nervous symptoms (headache, backache, dizziness, nausea, and stiff shoulders).
3.1. Physical reaction Fig. 5 shows the changes of transmissibility of five parts (head, chest, abdomen, thigh, and lower leg) according to the vibration frequencies. Although the changes of the transmission ratios according to the frequency were present in all parts, it was especially apparent in the chest at 5 Hz. Additionally, it was verified that the phase differences were also governed by the frequency. As a result, it was confirmed that the physical reaction would be affected by the frequency of the vibration force. In Fig. 5, the solid lines show the changes of the transmission ratios and phase differences in the direction of U, along the back of the rigid chair (see the local coordinate system illustrated in Fig. 2).
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Fig. 5. Physical reaction: Transmission ratio and phase difference to the direction U.
3.2. Physiological reaction The vertical axis of Fig. 6a shows the ratio of the heart rate after being exposed to the vibration, to the rate before it. Thus, when the ratio is greater than unity, it was apparent that the heart rate had been increased by the vibration exposure. Similarly, the other ratios in Fig. 6 show the changes between before and after exposure. As a result, it was clear that the heart rate, the respiration rate and the highest blood pressure had been increased by the vibration exposure, and the saliva-secretion quantity had been decreased.
exception of the yawning and sleepiness, the feelings of tiredness increased according to the vibration frequency. 3.4. Relations between the physical, physiological and psychological reactions The three relations between the physiological reaction and the physical reaction, the psychological reaction and the physical reaction, and the psychological reaction and the physiological reaction were formalized according to the multiple regression analysis using the above-mentioned experimental results.
3.3. Psychological reaction The horizontal axis of Fig. 7 shows the transition from the feelings of tiredness before exposure to the vibration, to the feelings after it. On this axis, ‘0’ means no transition of the feelings of tiredness caused by the vibration exposure. The broken line shows the transition at 2 Hz, and the solid line shows it at 5 Hz. In general, with the
3.4.1. Relation between the physiological reaction and the physical reaction The physiological reaction was expressed in terms of the purpose variables in the multiple regression analysis, and the physical reaction was expressed in the explanation variables of that analysis. The relation between them was described by multiple regression equations to predict the
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Fig. 6. Physiological reaction.
physiological reaction against the given vibration transmission ratios: physical reaction. From Table 1, it was suggested that the physiological reaction might be induced through the vibration of the head, the chest, the abdomen, and lower leg at 5 Hz. Also, it was realized that the characteristic of the physiological reaction and the relation between the physiological reaction and the physical reaction alter according to the frequency of the vibration. The dynamic characteristic would be due to the fact that there are some differences between the resonance frequencies of the head, of the chest and of the abdomen corresponding to the external vibration.
As an example, the multiple regression equations representing the relation between the physiological reaction and the physical reaction at 5 Hz are Yheart rate ¼ 2:388 þ 0:054X1 þ 0:200X2 ÿ 1:351X5 , Yhighest blood pressure ¼ ÿ 0:812 þ 2:055X1 , Ylowest blood pressure ¼ 0:634 ÿ 0:088X1 ÿ 0:175X2 þ 0:750X5 , Yrespiration rate ¼ 2:550 ÿ 0:212X2 ÿ 0:083X3 ÿ 0:692X5 ,
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Ysaliva secretion quantity ¼ 0:159 ÿ 0:280X1 ÿ 0:423X3 þ 1:362X5 ,
ð1Þ
where X1 shows the vibration transmission ratio of the acceleration of the head to the acceleration of the vibration table (Fig. 2). Similarly, X2 shows the ratio of the chest, X3 shows the ratio of the abdomen, X4 shows the ratio of the thigh, and X5 shows the ratio of the lower leg. From the equations, we could deduce that the heart rate, respiration rate and highest blood
pressure would increase with exposure to the vibration at 5 Hz, while the saliva quantity would decrease. 3.4.2. Relation between the psychological reaction and the physical reaction The transmissibility (transmission ratios, phase differences) of five body parts were defined as the explanation variables, and the 13 adjectives used to estimate the feeling of tiredness shown by the subjects when they were exposed to vibration, were defined as the purpose variables in the multiple regression analysis. The adjectives are displayed in the first column of Table 2. As a result, it was clear that the vibrations of the abdomen, head and chest mainly cause tiredness at 2 Hz and 5 Hz. In particular, it was confirmed that there is a strong relation between the vibration of each body part and the psychological reaction at 5 Hz. For example, the multiple regression equations representing the psychological reaction at 5 Hz are Ytiredness ¼ ÿ 716:335 þ 254:412X2 þ 120:435X4 , Yyawning ¼ ÿ 108:989 ÿ 16:744X2 þ 114:285X5 , Ysleepiness ¼ 69:861 ÿ 69:960X1 , Ytired eyes ¼ ÿ 364:568 þ 234:968X1 þ 59:181X2 , Yabsentÿmindedness ¼ ÿ 268:326 þ 234:906X1 þ 243:698X3 ÿ 177:265X5 , Yirritation ¼ ÿ 34:528 þ 36:232X1 , Yloss of patience ¼ ÿ 248:247 ÿ 45:156X1 þ 218:584X4
Fig. 7. Psychological reaction: Transition of the feelings of tiredness.
ÿ 73:250X5 ,
Table 1 Multiple regression relation between the physiological reaction and the physical reaction at 5 Hz Physiological reaction Heart rate Highest blood pressure Lowest blood pressure Respiration rate Saliva secretion quantity
Constant
2.388 ÿ0.812 0.634 2.550 0.159
Each body part Head
Chest
Abdomen
Thigh
Lower leg
0.054 2.055 ÿ0.088 } ÿ0.280
0.200 } ÿ0.175 ÿ0.212 }
} } } ÿ0.083 ÿ0.423
} } } } }
ÿ1.351 } 0.750 ÿ0.692 1.362
R
R2
0.999 0.689 0.998 0.999 0.995
0.998 0.475 0.996 0.998 0.990
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Table 2 Multiple regression relation between the psychological reaction and the physical reaction at 5 Hz Psychological reaction
Constant
Tiredness Yawning Sleepiness Tired eyes Absent-mindedness Irritation Loss of patience Distracted attention Headache Bachache Dizziness Nausea Stiff shoulders
ÿ716.335 ÿ108.986 69.861 ÿ364.568 ÿ268.326 ÿ34.528 ÿ248.247 ÿ266.597 ÿ284.597 ÿ656.687 ÿ44.854 ÿ210.231 ÿ408.998
Each body part Head
Chest
Abdomen
Thigh
Lower leg
} } ÿ69.960 234.968 234.960 36.232 45.156 17.850 32.515 } 47.534 24.882 }
254.412 ÿ16.744 } 59.181 } } } 71.059 16.151 32.846 } 26.664 173.352
} } } } 243.698 } } } } 255.421 } 120.426 32.438
120.435 } } } } } 218.584 } } 232.391 } } }
} 114.285 } } ÿ177.265 } ÿ73.050 75.595 171.099 } } } }
Ydistracted attention ¼ ÿ 266:597 þ 17:850X1 þ 71:059X2 þ 75:595X5 , Yheadache ¼ ÿ 284:713 þ 32:515X1 þ 16:151X2 þ 171:099X5 , Ybackache ¼ ÿ 656:687 þ 32:846X2 þ 255:421X3 þ 232:391X4 , Ydizziness ¼ ÿ 44:854 þ 47:534X1 , Ynausea ¼ ÿ 210:231 þ 24:882X1 þ 26:664X2 þ 120:426X3 , Ystiff shoulders ¼ ÿ 408:998 þ 173:352X2 þ 32:438X3 ,
ð2Þ
where, in the same manner as the above section, each Xi shows the vibration transmission ratio of the acceleration of each part to the acceleration of the vibration table. 3.4.3. Relation between the psychological reaction and the physiological reaction In this relation, the 13 adjectives were employed as the purpose variables, and the above-mentioned physiological reaction was employed as the explanation variables. As shown in Table 3, it was confirmed that the psychological reaction had a
R
R2
0.985 0.939 0.694 0.957 0.973 0.718 0.995 0.999 0.999 0.998 0.763 0.996 0.954
0.969 0.883 0.481 0.916 0.946 0.515 0.989 0.998 0.998 0.997 0.582 0.992 0.910
relevance to some symptoms of the physiological reaction. The symptoms were tiredness, absentmindedness, irritation and loss of patience, distracted attention, dizziness and nausea. Furthermore, it was apparent that the relevance was altered by the frequency of the vibration imposed on the human body and that the degree of the relevance would be remarkable according to increases in the frequency. For example, it was obtained that the heart rate would have a relevance to [absent-mindedness, loss of patience, dizziness and nausea], and the highest blood pressure would have a relevance to [irritation, loss of patience, distracted attention, dizziness, and nausea]. Additionally, it might be an interesting fact that there was a difference between the relevance of the lowest blood pressure and to that of the highest. 3.5. Construction of a physical vibration model of the human body In the synthetic vibration model, to reproduce a human being’s behaviour when exposed to vibration, the prediction of the physical reaction (Xi ) has to be performed as well as the formulation of the relations between the physical and the physiological and psychological reactions. The physical reaction is predicted by a numerical physical human body vibration model, which is a component
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M. Kubo et al. / International Journal of Industrial Ergonomics 27 (2001) 219–232 Table 3 Multiple regression relation between the psychological reaction and the physiological reaction at 5 Hz Physiological reaction Heart rate Highest Psychological reaction
Constant
Blood pressure
Tiredness Absent-mindedness Irritation Loss of patience Distracted attention Dizziness Nausea
ÿ155.561 116.276 ÿ40.026 178.652 48.026 } ÿ47.125 81.813 355.759 } 61.789 ÿ116.276 ÿ375.681 136.861
} } 70.588 103.645 134.872 74.356 }
of the synthetic vibration model. Basically, in the synthetic vibration model, the numerical physical vibration model predicts the physical reaction (Xi ) according to a physical environment in which the physical model vibrates numerically. And the physiological and psychological reactions are estimated through the multiple regression equations (Eqs. (1) and (2)) with the physical reaction as the explanation variables in the multiple regression equations.
3.5.1. Assumption to simplify the human body In this paper we assumed that parts of the human body would only swing back and forth as well as move up and down. Because it was apparent that the human body would remain physically symmetry during exposure to vibration in a vertical direction. Thus, in the physical vibration model, to predict the physical reaction the transverse shaking of the human body is ignored. Therefore, we can assume that a twodimensional model projected on the central plane, which is a midsagittal plane, of the human body would simulate the realistic vibration behaviour of the human body. Additionally, to simplify the model of the human body further, the following conditions were assumed: (1) It was assumed that the human body consists of head, chest (from the upper point of the breastbone to the third lumbar vertebra), abdomen (from the third lumbar vertebra to the trochanteric point), thigh, and lower leg. Each
Lowest
R
R2
0.982 0.987 0.968 0.993 0.968 0.945 0.922
0.964 0.974 0.937 0.986 0.938 0.893 0.850
Respiration rate Saliva secretion quantity
ÿ36.154 45.185 ÿ175.640 } ÿ125.396 } } ÿ150.042 ÿ276.595 ÿ202.071 } } 206.626 }
} } } } } } }
part of the human body has a mass and a rotating inertia at the centre of gravity (Fig. 8). (2) The lower leg could be connected to the thigh and the thigh to the abdomen by a joint with an axis of rotation and generating a viscosity resistance moment. The resistance moment represents the passive resistance element of ligaments. The abdomen and chest are connected by a viscoelasticity element that consists of a spring and a damper, and the chest and head are connected in the same way. The viscoelasticity element could simulate lumber and cervical vertebrae. (3) The weight of the lower legs could be supported by the horizontal plane of the experimental chair and the surface of the vibration table, so that the weight of the lower legs has no effect on the pelvis. (4) Only portions of the back of head, the back and the lower pelvis are exposed to the external force of the vibration (Fig. 2). (5) So that the head, trunk (chest, abdomen) and pelvis would never slip on the surface of the chair, there is sufficient frictional force at each point of contact. Finally, we simplified the human body to a twodimensional vibration model consisting of masses, rigid links, springs and dampers with nine degrees of freedom (Fig. 8). 3.5.2. Formulation of the equation of motion for the simplified human vibration model In order to simplify the formulation of the equation of motion for the two-dimensional
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Fig. 8. Two-dimensional human vibration model.
vibration model, we further assumed the following: (1) Each part of the vibration model slightly vibrates around each static force equalizing position. (2) The righting moment of springs and the attenuating force of dampers are in proportion to the displacement and the velocity, respectively. (3) The saturation viscosity resistance moment is applied to the resistance moments between the lower leg and the thigh and between the thigh and the abdomen. Finally, the equation of motion consists of the coefficient matrices illustrating the effects of the masses, rigid links, springs and dampers. The equation also has nine degrees of freedom, which were 3 rotations and 6 translations, which did not perpendicularly intersect each other. Therefore, the equations were formulated with generalized coordinates according to the general process of Lagrange’s equation of motion. The equation of motion of the human body is ½M d2 x=dt2 þ ½Cfdx=dtg þ ½K fxg ¼ f f g, ð3Þ where {x} is generalized coordinates: fxg ¼ fy1 , y2 , y3 , B, Z, v1 , u1 , v2, u2 gT and, { f } is general-
ized forces:f f g ¼ ff1 , f2 , f3 , f4 , f5 , f6 , f7 , f8, f9 gT . Each fi corresponds to each generalized coordinate in the equation of motion. [M ], [C] and [K] -coefficient matrices are symmetric positive matrices that have nine degrees of freedom. [M] consists of mi , lij , Ii and ai , [C] consists of ci , lij , Ii and ai , and [K] consists of ki , lij , Ii and ai . In this paper, ki was the spring constant, and ci was the damping coefficient. Furthermore, to quantitatively define the unknown constants: ki , ci included in the coefficient matrices, the transmission function, numerically calculated with the solution of the equation of motion, was compared to the experimental transmission function measured and calculated by the accelerometers and the FFTanalyzer. The coefficient matrices were controlled according to a general optimum design method so that the numerical transmission function would coincide with the experimental transmission function. The damping matrix [C] corresponds to velocity and [M] and [K] correspond to acceleration and displacement, respectively, so that the phase differences between the generalized coordinates {x} of each part of the body are induced. Therefore, {x} would be complex numbers in general, so that the transmission functions would be described with complex numbers. As a result,
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the spring and damping coefficients that were calculated through identifying the numerical transmission function with the experimental transmission function, would be complex numbers (Table 4). Finally, the equation of motion could simulate the physical vibration behaviour of the human body and could predict the physical reaction (transmission function: Xi ), which would be substituted for the explanation variables in the multiple regression equations.
4. Discussion 4.1. Investigation into the validity of the physical vibration model of the human body In order to investigate the validity of the physical vibration model, the experimental transmission functions were compared with the numerical transmission functions predicted by the above-mentioned physical vibration model. Both transmission functions relate to the direction U (see Fig. 2). The numerical transmission functions were calculated by solving the equation of motion with time history response analysis, in which the spring constants and damping coeffi-
cients identified in the above section were applied to the equation (see Table 4). And, the spring and damping coefficients at the frequencies 3, 4, 6, 7, 9 and 10 Hz were calculated by using interpolation polynomials with the known spring and damping coefficients shown in Table 4. Fig. 9 shows that the numerical transmissibility (broken lines) almost agrees with the experimental transmissibility (solid lines) in all the frequency range. Moreover, the discrepancy between the numerical results and experimental results in the range of 2–11 Hz was relatively low within the range of 0.14 and 12.27%. In particular, in the range lower than 8 Hz, the relative error was less than 10%, while at 11 Hz, the relative error was within 10.61 and 12.72%. The disparity between the numerical results and the experimental results is inevitable for the following reasons: (1) The precision of estimating the spring constants and damping coefficients was affected by the measurement precision of the curve fit, the window function and the number of averages involved in the FFT. (2) As a simplification, it was assumed that the human body consisted of 16 simplified geometrical elements: the elements being an ellipsoid of gyration, a truncated cone, a cylinder
Table 4 Identified spring constants (Unit: 10 N/m) and damping coefficients (Unit: N s/m) 2 Hz Real part
5 Hz Imaginary part
Real part
8 Hz Imaginary part
Real part
11 Hz Imaginary part
Real part
Imaginary part
k1 k2 k3 k4 k5 k6
1040.6 866.6 419.4 309.6 610.8 589.7
694.4 61.4 47.4 149.1 274.1 118.7
1261.7 452.2 457.6 665.7 948.5 338.0
839.0 22.5 44.0 148.0 66.2 209.6
1307.8 710.1 286.9 367.4 936.6 546.8
686.1 116.1 116.9 10.6 181.9 244.8
1233.2 842.0 567.9 765.2 1550.9 565.3
60.5 36.5 144.6 40.6 118.9 462.8
k7 c1 c2 c3 c4 c5 c6 c7
401.2 783.4 364.4 164.3 170.3 146.7 250.3 224.6
173.0 ÿ724.7 ÿ96.1 ÿ74.2 ÿ301.1 ÿ169.4 ÿ252.1 ÿ117.6
239.5 754.1 469.8 168.0 181.2 457.8 256.9 182.8
62.2 ÿ496.0 ÿ32.1 ÿ89.0 ÿ61.2 ÿ180.5 ÿ209.4 ÿ53.6
295.5 664.7 861.0 214.8 110.7 541.3 388.7 164.2
103.2 ÿ288.3 ÿ107.3 ÿ129.7 ÿ37.3 ÿ223.5 ÿ161.1 ÿ62.1
269.4 636.6 312.3 222.7 227.7 485.9 356.6 242.1
128.8 ÿ479.7 ÿ46.5 ÿ109.9 ÿ50.9 ÿ223.8 ÿ301.1 ÿ103.3
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Fig. 9. Comparison between the measured- and the predicted-transmission ratio.
and a rectangular prism. Moreover, it was assumed that all the elements were homogeneous substances and that the densities were equal to the mean density of the human body. The precision of the mass, the centre of gravity and the moment of inertia also contribute to the discrepancy. (3) From a different viewpoint, a real human body consists of bone, internal organs, muscles and fat. Therefore, the human body can be regarded as an elastic body performing complicated motions while resonance is occurring. When the human body is exposed to lowfrequency vibration, we consider that the weight of the human body consists of only rigid body masses, because the relative elastic displacements of the human body are sufficiently infinitesimal. However, when the body is exposed to higher frequency vibration, it is difficult to consider that the weight of the human body consists of rigid body masses, because the nonlinearity of muscles and fat would become stronger in relation to the increase in frequency.
4.2. An example simulating the vibration characteristics of the human body by using the vibration model The human vibration model was installed on a concentrated frame of two-dimensional automobile vibration model (Nishiyama, 1993) to simulate the vibration behaviour of a human body riding in an automobile (Fig. 10). We predict the unknown psychological and physiological reactions of a person riding the two-dimensional automobile vibrating at a given frequency, by using the above-mentioned multiple regression equation representing the relations between the psychological and physical reactions, and between the physiological and physical reactions. As an example, a simulation of the vibration in the human– automobile system exposed to a perpendicular sine-wave force of 5 Hz was performed. This frequency violently shakes the head and the chest. In this case, it was assumed that the concentrated frame of the automobile (Fig. 10) remain horizontal and vibrate throughout at 5 Hz. Therefore, the spring constants and damping coefficients
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Table 5 Ratios between the reactions before and after exposure to vibration
Fig. 10. A human vibration model riding in an automobile vibration model.
calculated through the above-mentioned identification of the transmission functions at 5 Hz were applied to the equation of motion of the human vibration model riding the automobile. At first, the transmission ratios of the vibration of each body part to the table were estimated through the simulation of the vibration system. And, the physiological and psychological reactions to the vibration were easily predicted by using the above-mentioned multiple regression equations with the transmission ratios as the explanation variables. For example, from Table 5, we can read the change relating to tiredness as a magnification of 2.70. This change ratio illustrates that the quantified tiredness after exposure to the vibration is 2.70 times higher than the tiredness before it. As a result, it was suggested that the simple human-automobile vibration model could realistically predict the physical, physiological and psychological reactions of a person riding in an automobile. Additionally, we could modify the spring constants and damping coefficients according to the predicted vibration behaviour, so that the vibrations inducing disagreeable impressions could be reduced.
Reactions
Ratios
Psychological reaction Tiredness Yawning Sleepiness Tired eyes Absent-mindedness Irritation Loss of patience Distracted attention Headache Backache Dizziness Nausea Stiff shoulder
2.70 1.01 0.54 2.71 5.47 9.25 6.70 6.00 5.52 13.95 13.62 6.98 7.69
Physiological reaction Heart rate Highest blood pressure Lowest blood pressure Breathing rate Saliva quantity
1.16 1.26 1.12 1.12 1.16
5. Conclusions In this paper, the possibility of a synthetic vibration model that could enable us to numerically estimate the synthetic behaviour of a person exposed to vibration was investigated. The synthetic behaviour would be expressed by synthesizing the characteristics of the physiological and psychological reactions to the vibration, as well as those of the physical reaction. The vibration model consists of two basic equations: *
*
the multiple regression equations formalized by multiple regression analysis according to experimental data; the equation of motion formalized by using Lagrange’s equation of motion.
The equation of motion of the human body was expressed in terms of masses, springs, dampers, links and rotating inertia. These parameters were also defined according to the physical reaction measured in the vibration experiments with the FFT-analyzer. As an example, the constructed
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vibration model simulated the vibration behaviour of the human body in a sitting posture at 5 Hz. As a result of the simulation, it was demonstrated that the physical reaction could be adequately estimated through the equation of motion of the mechanical human model, and that the physiological and psychological reactions could be numerically predicted by using the multiple regression equations in which the physical reaction (transmission ratios) was substituted in place of the explanation variables. Additionally, it was suggested that this vibration system could be the optimum design for a comfortable ride, in terms of the psychological and physiological reactions.
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