an investigation into a synthetic vibration model for humans:: an investigation into a mechanical...

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 betweenthe physical, psychological and physiologicalreactions of humans exposed to vibration

Mitsunori Kuboa,*, Fumio Terauchia, Hiroyuki Aokia, Yoshiyuki Matsuokab

aDepartment of Design and Architecture, Faculty of Engineering, Chiba University, 1-33 Yayoi-Cho, Inage-Ku, Chiba 268-8522, JapanbFaculty 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 andphysiological 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 equationsdescribing the above three relations. The mechanical vibration models formalised according to Lagrange’s equation ofmotion were employed. The experiment was carried out under conditions in which five people were exposed to externalvibration 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 2–11Hz. Moreover, it wasindicated that the relations between the physical reactions and the resulting psychological and physiological reactionsmight 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 thepsychological 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 variousproducts, 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

Many people are exposed to whole-body vibra-tion in vehicles: cars, buses, trains, ships andairplanes, on a daily basis. In our previous paper,

*Corresponding author.

E-mail address: [email protected].

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

it was confirmed that whole-body vibration causeda subject discomfort, fatigue and physical pains(Liu et al., 1995). There are several reportsdescribing how vibration interferes with people’sworking efficiency, safety and health (Bogert,1994; Mcleod and Griffin, 1995; Qassem et al.,1996). Therefore many researchers have concen-trated their efforts on reducing the amount ofvibration from products and vehicles. There aremany reports describing the measurement of thetransmissibility of the human body under vibra-tion (Griffin, 1975; Randall et al., 1997; Matsu-moto and Griffin, 1998). We have also measuredthe transmissibility of the whole body in sittingand lying posture exposed to vertical vibration(Liu et al., 1996). The results of these reportsindicated the resonance of the human bodydepended on various factors: the posture, thematerials of the given seat surface, vibrationmagnitude and frequency. The measurements ofthe transmissibility of the body under variousvibrations are inefficient, laborious, tedious andexpensive. On the other hand, there are a fewcomputer-automated procedures used to predictthe human body’s responses to vibration (Amir-ouche, 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 accuratelyestimate the behaviour of the human body undervibration, because it is a complex active dynamicsystem. Further, it is most important to bear inmind that the complexity is not only due tophysical characteristics but also due to psycholo-gical and physiological characteristics. However,no vibration model concerning the physiologicaland the psychological reactions of a personexposed to vibration environments has beenfound. Therefore, we considered that the construc-tion of a vibration model that could reproduce thecharacteristics of the vibrating human riding on anautomobile should be a research task. The vibra-tion model should not only be able to reproducethe behaviour of the physical human body but alsopredict the physiological and psychological reac-tions. In constructing the vibration model, wewould predict the characteristics of the threereactions (physical, physiological and psychologi-cal), and then formalize the relations between

them. In this paper the vibration model wasconstructed in accordance with the results of ourresearch into the characteristics of the humanexposed to a vertical sinusoidal wave force.

2. Methods

2.1. Assumption of the structure of the syntheticvibration model of human

We assumed that the characteristics of thevibration of the human body might be explainedby the following three reactions when the humanbody is exposed to some vibration environments.

(i) A physical reaction expressed by the transmis-sibility of the vibrations of each part of thehuman body to a standard part, e.g. the areaon the vibration table which can vibrate aperson sitting on a rigid chair.

(ii) A physiological reaction manifested in termsof blood pressure, heart rate, etc.; thesereactions are generally termed the physiologi-cal indices.

(iii) A psychological reaction as illustrated bymanifestation of the different symptoms in-duced by vibration.

The human vibration models that have beendefined by many researchers are generally limitedto numerical models that reproduce only thephysical reaction. Therefore, we proposed a basicstructure for a synthetic vibration model of humanbeings, shown in Fig. 1, which could indirectlyreproduce the characteristics of the physiologicaland the psychological reactions as well as thephysical reaction through multiple regressionequations. Moreover, we assumed that there weresome linear relations between the physical, phy-siological and psychological reactions, so that amultiple regression analysis could be applied toanalyse these relations. In the synthetic vibrationmodel, the physical reaction can be simulatedby equations of motion formalized by usingLagrange’s equation, and the physiological andpsychological reactions can be predicted by multi-ple regression equations defined through themultiple regression analysis. In the multiple

M. Kubo et al. / International Journal of Industrial Ergonomics 27 (2001) 219–232220

regression equations, the physical reaction directlyrelates to explanatory variables to predict thephysiological and psychological reaction. Also, itwas assumed that any physical reaction was notaffected by the physiological and psychologicalreactions (Fig. 1).

2.2. Experimental conditions

The basic experimental device was an electro-magnetic vibration table (IMV Co.) on which asufficiently rigid chair was installed (Fig. 2). Thesubjects who were exposed to periodical verticalvibrations were five males aged from 22 to 29.They were exposed to whole-body vibration whilesitting under the following vibration conditions.

(i) the vibration stimulus : sinusoidal waves in avertical direction.

(ii) the vibrating frequencies : 2, 5, 8, 11, 14, 17,and 20Hz.

(iii) the effective value of acceleration : 0.69m/s2.(iv) the time of exposure to vibration : 10min.

During the experiments, the air temperaturewas 24–268C and the air humidity was40–50%.

Further, the characteristics of the physical,physiological and psychological reactions of thesubjects in the vibration experiments were mea-sured according to the following three methods,respectively.

2.2.1. Measurement of the physical reactionThe physical reaction was measured with the

data collected by 10 accelerometers (KYOWA,ASV-2GA); Fig. 2 shows the points of installationof the accelerometers. The data were measuredwith a Fast Fourier Transform (FFT) analyzer(ONO SOKKI, CF360) to obtain the vibrationtransmissibility. The transmissibility was definedaccording to the transmission ratio of accelerationsensed by each accelerometer to the accelerationof the vibration table, as well as by the phasedifference between each vibration mode and thetable vibration mode. The transmission ratios andphase differences were calculated by the FFT.

2.2.2. Measurement of the physiological reactionThe measurements of the heart rate, highest and

lowest blood pressure, respiration rate and salivasecretion quantity were carried out according to atimetable (Fig. 3). The heart rate was measuredwith the CM5-inducement method (Nakamuraet al., 1983) through an electrical amplifier fora living body (NIHON KOUDEN, AB-620G) anda data recorder (TEAC, RD-111T). The bloodpressures were measured on the artery of the leftupper arm with a simple sphygmomanometer(OMURON, HEM-700). The respiration rate

Fig. 1. A synthetic vibration model for human beings.

Fig. 2. Experimental device.

M. Kubo et al. / International Journal of Industrial Ergonomics 27 (2001) 219–232 221

was calculated by using the nostril-type respirationcurve. The curve was measured with a thermister(NIHON KOUDEN, TR-712T). The saliva quan-tity was measured according to a method in whichthe saliva was collected by dental cotton. Theweight of the dental cotton was controlled so thatit maintained a constant value. The weight of thecotton after saliva absorption was measured with aprecise scale (SHIMAZU, AEL-200). All thephysiological measured values were expressed interms of the ratio of the measured values when thesubjects were exposed to vibration, to the initialvalues before exposure to vibration. We assumedthat the influence of vibration would be weakwhen the ratio was near to one.

2.2.3. Evaluation of the psychological reactionThe psychological change was evaluated accord-

ing to semantic differential (SD) method. In thispaper, a feeling of tiredness was considered asa psychological reaction. The subjects were equip-ped with earplugs in order to block the sound ofthe vibration table. They were questioned on theirlevel of comfort before and after the time whenthey were exposed to the vibration in order tomake clear the characteristics of their psychologi-cal reaction to the vibration. The questionnaireemployed the terms shown in Fig. 4. It wasassumed that the feeling of tiredness consistedof physical symptoms (tiredness, yawning, sleepi-ness, tired eyes, and absent-mindedness), mentalsymptoms (irritation, loss of patience, distractedattention), and nervous symptoms (headache,backache, dizziness, nausea, and stiff shoulders).

3. Results

3.1. Physical reaction

Fig. 5 shows the changes of transmissibility offive parts (head, chest, abdomen, thigh, and lowerleg) according to the vibration frequencies.Although the changes of the transmission ratiosaccording to the frequency were present in allparts, it was especially apparent in the chest at5Hz. Additionally, it was verified that the phasedifferences were also governed by the frequency.As a result, it was confirmed that the physicalreaction would be affected by the frequency of thevibration force. In Fig. 5, the solid lines show thechanges of the transmission ratios and phasedifferences in the direction of U, along the backof the rigid chair (see the local coordinate systemillustrated in Fig. 2).

Fig. 3. Timetable for the vibration experiment.

Fig. 4. Questionnaire of a feeling of tiredness.


3.2. Physiological reaction

The vertical axis of Fig. 6a shows the ratio of theheart rate after being exposed to the vibration, tothe rate before it. Thus, when the ratio is greaterthan unity, it was apparent that the heart rate hadbeen increased by the vibration exposure. Simi-larly, the other ratios in Fig. 6 show the changesbetween before and after exposure. As a result, itwas clear that the heart rate, the respiration rateand the highest blood pressure had been increasedby the vibration exposure, and the saliva-secretionquantity had been decreased.

3.3. Psychological reaction

The horizontal axis of Fig. 7 shows the transi-tion from the feelings of tiredness before exposureto the vibration, to the feelings after it. On thisaxis, ‘0’ means no transition of the feelingsof tiredness caused by the vibration exposure.The broken line shows the transition at 2Hz, andthe solid line shows it at 5Hz. In general, with the

exception of the yawning and sleepiness, thefeelings of tiredness increased according to thevibration frequency.

3.4. Relations between the physical, physiologicaland psychological reactions

The three relations between the physiologicalreaction and the physical reaction, the psycholo-gical reaction and the physical reaction, and thepsychological reaction and the physiological reac-tion were formalized according to the multipleregression analysis using the above-mentionedexperimental results.

3.4.1. Relation between the physiological reactionand the physical reaction

The physiological reaction was expressed interms of the purpose variables in the multipleregression analysis, and the physical reaction wasexpressed in the explanation variables of thatanalysis. The relation between them was describedby multiple regression equations to predict the

Fig. 5. Physical reaction: Transmission ratio and phase difference to the direction U.


physiological reaction against the given vibrationtransmission ratios: physical reaction. FromTable 1, it was suggested that the physiologicalreaction might be induced through the vibration ofthe head, the chest, the abdomen, and lower leg at5Hz. Also, it was realized that the characteristic ofthe physiological reaction and the relation betweenthe physiological reaction and the physical reac-tion alter according to the frequency of thevibration. The dynamic characteristic would bedue to the fact that there are some differencesbetween the resonance frequencies of the head, ofthe chest and of the abdomen corresponding to theexternal vibration.

As an example, the multiple regression equa-tions representing the relation between the phy-siological reaction and the physical reaction at5Hz are

Yheart rate ¼ 2:388þ 0:054X1 þ 0:200X2 ÿ 1:351X5,

Yhighest blood pressure ¼ ÿ 0:812þ 2:055X1,

Ylowest blood pressure ¼ 0:634ÿ 0:088X1 ÿ 0:175X2

þ 0:750X5,

Yrespiration rate ¼ 2:550ÿ 0:212X2 ÿ 0:083X3

ÿ 0:692X5,

Fig. 6. Physiological reaction.


Ysaliva secretion quantity ¼ 0:159ÿ 0:280X1 ÿ 0:423X3

þ 1:362X5, ð1Þ

where X1 shows the vibration transmission ratio ofthe acceleration of the head to the acceleration ofthe vibration table (Fig. 2). Similarly, X2 shows theratio of the chest, X3 shows the ratio of theabdomen, 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 thevibration at 5Hz, while the saliva quantity woulddecrease.

3.4.2. Relation between the psychological reactionand the physical reaction

The transmissibility (transmission ratios, phasedifferences) of five body parts were defined as theexplanation variables, and the 13 adjectives usedto estimate the feeling of tiredness shown by thesubjects when they were exposed to vibration, weredefined as the purpose variables in the multipleregression analysis. The adjectives are displayed inthe first column of Table 2. As a result, it was clearthat the vibrations of the abdomen, head and chestmainly cause tiredness at 2Hz and 5Hz. Inparticular, it was confirmed that there is a strongrelation between the vibration of each body partand the psychological reaction at 5Hz. Forexample, the multiple regression equations repre-senting the psychological reaction at 5Hz are

Ytiredness ¼ ÿ 716:335þ 254:412X2 þ 120:435X4,

Yyawning ¼ ÿ 108:989ÿ 16:744X2 þ 114:285X5,

Ysleepiness ¼ 69:861ÿ 69:960X1,

Ytired eyes ¼ ÿ 364:568þ 234:968X1 þ 59:181X2,

Yabsentÿmindedness ¼ ÿ 268:326þ 234:906X1

þ 243:698X3 ÿ 177:265X5,

Yirritation ¼ ÿ 34:528þ 36:232X1,

Yloss of patience ¼ ÿ 248:247ÿ 45:156X1 þ 218:584X4

ÿ 73:250X5,Fig. 7. Psychological reaction: Transition of the feelings of

tiredness.

Table 1

Multiple regression relation between the physiological reaction and the physical reaction at 5Hz

Physiological

reaction

Constant Each body part R R2

Head Chest Abdomen Thigh Lower leg

Heart rate 2.388 0.054 0.200 } } ÿ1.351 0.999 0.998

Highest blood pressure ÿ0.812 2.055 } } } } 0.689 0.475

Lowest blood pressure 0.634 ÿ0.088 ÿ0.175 } } 0.750 0.998 0.996

Respiration rate 2.550 } ÿ0.212 ÿ0.083 } ÿ0.692 0.999 0.998

Saliva secretion quantity 0.159 ÿ0.280 } ÿ0.423 } 1.362 0.995 0.990


Ydistracted attention ¼ ÿ 266:597þ 17:850X1

þ 71:059X2 þ 75:595X5,

Yheadache ¼ ÿ 284:713þ 32:515X1 þ 16:151X2

þ 171:099X5,

Ybackache ¼ ÿ 656:687þ 32:846X2 þ 255:421X3

þ 232:391X4,

Ydizziness ¼ ÿ 44:854þ 47:534X1,

Ynausea ¼ ÿ 210:231þ 24:882X1 þ 26:664X2

þ 120:426X3,

Ystiff shoulders ¼ ÿ 408:998þ 173:352X2

þ 32:438X3, ð2Þwhere, in the same manner as the above section,each Xi shows the vibration transmission ratio ofthe acceleration of each part to the acceleration ofthe vibration table.

3.4.3. Relation between the psychological reactionand the physiological reaction

In this relation, the 13 adjectives were employedas the purpose variables, and the above-mentionedphysiological reaction was employed as the ex-planation variables. As shown in Table 3, it wasconfirmed that the psychological reaction had a

relevance to some symptoms of the physiologicalreaction. The symptoms were tiredness, absent-mindedness, irritation and loss of patience,distracted attention, dizziness and nausea.Furthermore, it was apparent that the relevancewas altered by the frequency of the vibrationimposed on the human body and that the degree ofthe relevance would be remarkable according toincreases in the frequency. For example, it wasobtained that the heart rate would have arelevance to [absent-mindedness, loss of patience,dizziness and nausea], and the highest bloodpressure would have a relevance to [irritation, lossof patience, distracted attention, dizziness, andnausea]. Additionally, it might be an interestingfact that there was a difference between therelevance of the lowest blood pressure and to thatof the highest.

3.5. Construction of a physical vibration model ofthe human body

In the synthetic vibration model, to reproduce ahuman being’s behaviour when exposed to vibra-tion, the prediction of the physical reaction (Xi)has to be performed as well as the formulation ofthe relations between the physical and the physio-logical and psychological reactions. The physicalreaction is predicted by a numerical physical hu-man body vibration model, which is a component

Table 2

Multiple regression relation between the psychological reaction and the physical reaction at 5Hz

Psychological

reaction

Constant Each body part R R2

Head Chest Abdomen Thigh Lower leg

Tiredness ÿ716.335 } 254.412 } 120.435 } 0.985 0.969

Yawning ÿ108.986 } ÿ16.744 } } 114.285 0.939 0.883

Sleepiness 69.861 ÿ69.960 } } } } 0.694 0.481

Tired eyes ÿ364.568 234.968 59.181 } } } 0.957 0.916

Absent-mindedness ÿ268.326 234.960 } 243.698 } ÿ177.265 0.973 0.946

Irritation ÿ34.528 36.232 } } } } 0.718 0.515

Loss of patience ÿ248.247 45.156 } } 218.584 ÿ73.050 0.995 0.989

Distracted attention ÿ266.597 17.850 71.059 } } 75.595 0.999 0.998

Headache ÿ284.597 32.515 16.151 } } 171.099 0.999 0.998

Bachache ÿ656.687 } 32.846 255.421 232.391 } 0.998 0.997

Dizziness ÿ44.854 47.534 } } } } 0.763 0.582

Nausea ÿ210.231 24.882 26.664 120.426 } } 0.996 0.992

Stiff shoulders ÿ408.998 } 173.352 32.438 } } 0.954 0.910


of the synthetic vibration model. Basically, in thesynthetic vibration model, the numerical physicalvibration model predicts the physical reaction(Xi) according to a physical environment inwhich the physical model vibrates numerically.And the physiological and psychological reactionsare estimated through the multiple regressionequations (Eqs. (1) and (2)) with the physical re-action as the explanation variables in the multipleregression equations.

3.5.1. Assumption to simplify the human bodyIn this paper we assumed that parts of the

human body would only swing back and forth aswell as move up and down. Because it wasapparent that the human body would remainphysically symmetry during exposure to vibrationin a vertical direction. Thus, in the physicalvibration model, to predict the physical reactionthe transverse shaking of the human body isignored. Therefore, we can assume that a two-dimensional model projected on the central plane,which is a midsagittal plane, of the human bodywould simulate the realistic vibration behaviour ofthe human body. Additionally, to simplify themodel of the human body further, the followingconditions were assumed:

(1) It was assumed that the human body consistsof head, chest (from the upper point of thebreastbone to the third lumbar vertebra), abdo-men (from the third lumbar vertebra to thetrochanteric point), thigh, and lower leg. Each

part of the human body has a mass and a rotatinginertia at the centre of gravity (Fig. 8).

(2) The lower leg could be connected to the thighand the thigh to the abdomen by a joint with anaxis of rotation and generating a viscosity resis-tance moment. The resistance moment representsthe passive resistance element of ligaments. Theabdomen and chest are connected by a viscoelas-ticity element that consists of a spring and adamper, and the chest and head are connected inthe same way. The viscoelasticity element couldsimulate lumber and cervical vertebrae.

(3) The weight of the lower legs could besupported by the horizontal plane of the experi-mental chair and the surface of the vibration table,so that the weight of the lower legs has no effect onthe pelvis.

(4) Only portions of the back of head, the backand the lower pelvis are exposed to the externalforce of the vibration (Fig. 2).

(5) So that the head, trunk (chest, abdomen) andpelvis would never slip on the surface of the chair,there is sufficient frictional force at each point ofcontact.

Finally, we simplified the human body to a two-dimensional vibration model consisting of masses,rigid links, springs and dampers with nine degreesof freedom (Fig. 8).

3.5.2. Formulation of the equation of motion for thesimplified human vibration model

In order to simplify the formulation of theequation of motion for the two-dimensional

Table 3

Multiple regression relation between the psychological reaction and the physiological reaction at 5Hz

Physiological reaction R R2

Heart rate Highest Lowest Respiration rate Saliva secretion

quantity

Psychological reaction Constant Blood pressure

Tiredness ÿ155.561 116.276 } ÿ36.154 45.185 } 0.982 0.964

Absent-mindedness ÿ40.026 178.652 } ÿ175.640 } } 0.987 0.974

Irritation 48.026 } 70.588 ÿ125.396 } } 0.968 0.937

Loss of patience ÿ47.125 81.813 103.645 } ÿ150.042 } 0.993 0.986

Distracted attention 355.759 } 134.872 ÿ276.595 ÿ202.071 } 0.968 0.938

Dizziness 61.789 ÿ116.276 74.356 } } } 0.945 0.893

Nausea ÿ375.681 136.861 } 206.626 } } 0.922 0.850


vibration model, we further assumed the follow-ing:

(1) Each part of the vibration model slightlyvibrates around each static force equalizing posi-tion.

(2) The righting moment of springs and theattenuating force of dampers are in proportion tothe displacement and the velocity, respectively.

(3) The saturation viscosity resistance moment isapplied to the resistance moments between thelower leg and the thigh and between the thigh andthe abdomen.

Finally, the equation of motion consists ofthe coefficient matrices illustrating the effectsof the masses, rigid links, springs and dampers.The equation also has nine degrees of freedom,which were 3 rotations and 6 translations, whichdid not perpendicularly intersect each other.Therefore, the equations were formulated withgeneralized coordinates according to the generalprocess of Lagrange’s equation of motion. Theequation of motion of the human body is

M½ � d2x=dt2�

þ C½ � dx=dtf g þ K½ � xf g ¼ ff g, ð3Þ

where {x} is generalized coordinates: fxg ¼ fy1,y2, y3, B, Z, v1, u1, v2, u2gTand, { f } is general-

ized forces:f f g ¼ ff1, f2, f3, f4, f5, f6, f7, f8,f9gT. Each fi corresponds to each generalizedcoordinate in the equation of motion. [M ], [C]and [K ] -coefficient matrices are symmetric positivematrices that have nine degrees of freedom. [M]consists of mi, l ji , Ii and ai, [C] consists of ci, l ji ,Ii and ai, and [K ] consists of ki, l ji , Ii and ai. Inthis paper, ki was the spring constant, and ci wasthe damping coefficient. Furthermore, to quantita-tively define the unknown constants: ki, ci includedin the coefficient matrices, the transmission func-tion, numerically calculated with the solution ofthe equation of motion, was compared to theexperimental transmission function measured andcalculated by the accelerometers and the FFT-analyzer. The coefficient matrices were controlledaccording to a general optimum design method sothat the numerical transmission function wouldcoincide with the experimental transmission func-tion. The damping matrix [C] corresponds tovelocity and [M] and [K ] correspond to accelera-tion and displacement, respectively, so that thephase differences between the generalized coordi-nates {x} of each part of the body are induced.Therefore, {x} would be complex numbers ingeneral, so that the transmission functions wouldbe described with complex numbers. As a result,

Fig. 8. Two-dimensional human vibration model.


the spring and damping coefficients that werecalculated through identifying the numericaltransmission function with the experimental trans-mission function, would be complex numbers(Table 4). Finally, the equation of motion couldsimulate the physical vibration behaviour of thehuman body and could predict the physicalreaction (transmission function: Xi), which wouldbe substituted for the explanation variables in themultiple regression equations.

4. Discussion

4.1. Investigation into the validity of the physicalvibration model of the human body

In order to investigate the validity of thephysical vibration model, the experimental trans-mission functions were compared with thenumerical transmission functions predicted bythe above-mentioned physical vibration model.Both transmission functions relate to the directionU (see Fig. 2). The numerical transmission func-tions were calculated by solving the equation ofmotion with time history response analysis, inwhich the spring constants and damping coeffi-

cients identified in the above section were appliedto the equation (see Table 4). And, the spring anddamping coefficients at the frequencies 3, 4, 6, 7, 9and 10Hz were calculated by using interpolationpolynomials with the known spring and dampingcoefficients shown in Table 4. Fig. 9 shows that thenumerical transmissibility (broken lines) almostagrees with the experimental transmissibility (solidlines) in all the frequency range. Moreover, thediscrepancy between the numerical results andexperimental results in the range of 2–11Hz wasrelatively low within the range of 0.14 and 12.27%.In particular, in the range lower than 8Hz, therelative error was less than 10%, while at 11Hz,the relative error was within 10.61 and 12.72%.The disparity between the numerical results andthe experimental results is inevitable for thefollowing reasons:

(1) The precision of estimating the springconstants and damping coefficients was affectedby the measurement precision of the curve fit, thewindow function and the number of averagesinvolved in the FFT.

(2) As a simplification, it was assumed thatthe human body consisted of 16 simplifiedgeometrical elements: the elements being anellipsoid of gyration, a truncated cone, a cylinder

Table 4

Identified spring constants (Unit: � 10N/m) and damping coefficients (Unit: N � s/m)

2Hz 5Hz 8Hz 11Hz

Real part Imaginary part Real part Imaginary part Real part Imaginary part Real part Imaginary part

k1 1040.6 694.4 1261.7 839.0 1307.8 686.1 1233.2 60.5

k2 866.6 61.4 452.2 22.5 710.1 116.1 842.0 36.5

k3 419.4 47.4 457.6 44.0 286.9 116.9 567.9 144.6

k4 309.6 149.1 665.7 148.0 367.4 10.6 765.2 40.6

k5 610.8 274.1 948.5 66.2 936.6 181.9 1550.9 118.9

k6 589.7 118.7 338.0 209.6 546.8 244.8 565.3 462.8

k7 401.2 173.0 239.5 62.2 295.5 103.2 269.4 128.8

c1 783.4 ÿ724.7 754.1 ÿ496.0 664.7 ÿ288.3 636.6 ÿ479.7c2 364.4 ÿ96.1 469.8 ÿ32.1 861.0 ÿ107.3 312.3 ÿ46.5c3 164.3 ÿ74.2 168.0 ÿ89.0 214.8 ÿ129.7 222.7 ÿ109.9c4 170.3 ÿ301.1 181.2 ÿ61.2 110.7 ÿ37.3 227.7 ÿ50.9c5 146.7 ÿ169.4 457.8 ÿ180.5 541.3 ÿ223.5 485.9 ÿ223.8c6 250.3 ÿ252.1 256.9 ÿ209.4 388.7 ÿ161.1 356.6 ÿ301.1c7 224.6 ÿ117.6 182.8 ÿ53.6 164.2 ÿ62.1 242.1 ÿ103.3


and a rectangular prism. Moreover, it wasassumed that all the elements were homo-geneous substances and that the densities wereequal to the mean density of the human body.The precision of the mass, the centre of gravity andthe moment of inertia also contribute to thediscrepancy.

(3) From a different viewpoint, a real humanbody consists of bone, internal organs, musclesand fat. Therefore, the human body can beregarded as an elastic body performing compli-cated motions while resonance is occurring.When the human body is exposed to low-frequency vibration, we consider that theweight of the human body consists of onlyrigid body masses, because the relative elasticdisplacements of the human body are suffici-ently infinitesimal. However, when the bodyis exposed to higher frequency vibration, it isdifficult to consider that the weight of thehuman body consists of rigid body masses,because the nonlinearity of muscles and fat wouldbecome stronger in relation to the increase infrequency.

4.2. An example simulating the vibrationcharacteristics of the human body by usingthe vibration model

The human vibration model was installed on aconcentrated frame of two-dimensional automo-bile vibration model (Nishiyama, 1993) to simulatethe vibration behaviour of a human body riding inan automobile (Fig. 10). We predict the unknownpsychological and physiological reactions of aperson riding the two-dimensional automobilevibrating at a given frequency, by using theabove-mentioned multiple regression equationrepresenting the relations between the psychologi-cal and physical reactions, and between thephysiological and physical reactions. As an exam-ple, a simulation of the vibration in the human–automobile system exposed to a perpendicularsine-wave force of 5Hz was performed. Thisfrequency violently shakes the head and the chest.In this case, it was assumed that the concentratedframe of the automobile (Fig. 10) remain horizon-tal and vibrate throughout at 5Hz. Therefore,the spring constants and damping coefficients

Fig. 9. Comparison between the measured- and the predicted-transmission ratio.


calculated through the above-mentioned identifica-tion of the transmission functions at 5Hz wereapplied to the equation of motion of the humanvibration model riding the automobile. At first, thetransmission ratios of the vibration of each bodypart to the table were estimated through the simu-lation of the vibration system. And, the physiolo-gical and psychological reactions to the vibrationwere easily predicted by using the above-mentionedmultiple regression equations with the transmissionratios as the explanation variables. For example,from Table 5, we can read the change relating totiredness as a magnification of 2.70. This changeratio illustrates that the quantified tiredness afterexposure to the vibration is 2.70 times higherthan the tiredness before it. As a result, it was sug-gested that the simple human-automobile vibrationmodel could realistically predict the physical,physiological and psychological reactions of aperson riding in an automobile. Additionally, wecould modify the spring constants and dampingcoefficients according to the predicted vibrationbehaviour, so that the vibrations inducing dis-agreeable impressions could be reduced.

5. Conclusions

In this paper, the possibility of a syntheticvibration model that could enable us to numeri-cally estimate the synthetic behaviour of a personexposed to vibration was investigated. The syn-thetic behaviour would be expressed by synthesiz-ing the characteristics of the physiological andpsychological reactions to the vibration, as well asthose of the physical reaction. The vibration modelconsists of two basic equations:

* the multiple regression equations formalized bymultiple regression analysis according to ex-perimental data;

* the equation of motion formalized by usingLagrange’s equation of motion.

The equation of motion of the human body wasexpressed in terms of masses, springs, dampers,links and rotating inertia. These parameters werealso defined according to the physical reactionmeasured in the vibration experiments with theFFT-analyzer. As an example, the constructed

Fig. 10. A human vibration model riding in an automobile

vibration model.

Table 5

Ratios between the reactions before and after exposure to

vibration

Reactions Ratios

Psychological reaction

Tiredness 2.70

Yawning 1.01

Sleepiness 0.54

Tired eyes 2.71

Absent-mindedness 5.47

Irritation 9.25

Loss of patience 6.70

Distracted attention 6.00

Headache 5.52

Backache 13.95

Dizziness 13.62

Nausea 6.98

Stiff shoulder 7.69

Physiological reaction

Heart rate 1.16

Highest blood pressure 1.26

Lowest blood pressure 1.12

Breathing rate 1.12

Saliva quantity 1.16


vibration model simulated the vibration behaviourof the human body in a sitting posture at 5Hz.As a result of the simulation, it was demon-strated that the physical reaction could beadequately estimated through the equation ofmotion of the mechanical human model, and thatthe physiological and psychological reactionscould be numerically predicted by using themultiple regression equations in which the physicalreaction (transmission ratios) was substituted inplace of the explanation variables. Additionally, itwas suggested that this vibration system could bethe optimum design for a comfortable ride, interms of the psychological and physiologicalreactions.

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 whole-

body 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. Develop-

ment 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. Reso-

nance 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...

Documents