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T h r e e - d i m e n s i o n a l b i o m e c h a n i c a lm o d e l fo r s i m u l a t in g t h e r e s p o n s e o ft h e h u m a n b o d y to v ib r a t io n s t re s sM . F r i t z

Inst i tu t f f i r Arb ei tsph ysio lo gie an der Universi t~t Dor tm und, ArdeystraGe 67, D-4413 9 Dortm und,G e r m a n yA b s t r a c t m Several investigations have revealed that long-term exposure to whole-bodyvibrations can induce low back pain. In analogy to materials handling, the health risk canbe assessed if the forces transmitted in the spine during vibration are k n o w n . To estimatethe forces a biomechanical model has been developed in which the human trunk, neck,head and arms are represented by t6 rigid bodies. An additional body simulates thevibrating seat. The bodies are connected by visco-elastic joint elements, and 56 forceelements imitate the trunk and neck muscles. The motion equations are derived by meansof the dynamics of systems of rigid bodies, and the motions are simulated in threedirections. The frequency-response functions between the accelerations of the seat andthe head satisfactorily correspond to data reported in the literature. The spine forces arecomposed of a static part, due to body posture, and a vibration-induced part. The relationbetween the oscillating parts of the forces transmitted from seat to pelvis and the spineforces are afso described by frequency-response functions. To assess the health risk thesimulated spine force s must be compared with the strength of the spine, bearing in mindthat this is dependent on the number of load cycles.

K e y w o r d s m Frequency-response function, Health r is ks, Rigid bo dies, Simulation, Spineforces, Strength of the spineMed. Biol. En g. Com put., 1998, 36, 686-692

N o m e n c l a t u r ex , y , z coord inates accord ing to the basicen t r ic system ofISO 2631-1accelera t ion of the exci tat ionf r eq u en cy o f t h e ex c i t a t io nv ec to r o f t h e g en era t i sed co o rd in a tes o f t h e mu l t i -b o d y s y s t e m/ ] seco n d t ime d er iv a t i v e o f qA an d B mat r i ces o f t h e eq u a t io n o f mo t ionK st i f fness ma tr ixHo . ,: y f r eq u en cy - r esp o n se fu n c t io n b e tween acce l e r a ti o n sH f~ y f r eq u en cy - r esp o n se fu n c t io n b e tween t ime d ep en -d en t fo rces(second su bscr ip t ind icates the d i rect ion of the input s ignal,th i rd subscr ip t ind icates the d i rect ion of the ou tpu t s ignal)

a

fq

1 I n t r o d u c t i o nDRIVERS OF ear th - mo ving m achine ry , t ractors o r t rucks a reex p o sed t o sev ere wh o le -b o d y v ib ra t i o n . Un d er v ib ra t i o ns t r es s t h e h u man p er fo rman ce can b e r ed u ced an d ac tu a lC o r r e s p o n d e n c e s h o u l d b e a d d r e s s e d t o D r . - I n g M , F r i tz ; e ma i l :f r i t z @ a r b - p h y s . u n i - d o r t mu n d . d eF i r s t rece i ved 18 Decem ber 1997 and i n f i na l fo rm 18 March 19989 IFMBE:1998

wel l -b e in g may b e a f f ec t ed b y d i s t u rb ed v eg e t a t i v e an dorgan ic funct ions . F inal ly , ma ny years o f v ibrat ion exposuremay cont r ibu te to low back pain , d isorders o r in ju r ies asdescr ibed in var ious rev iew ar t ic les (SEIDEL and HEIDE,1986; WIKSTROM e t a l . , 1994; WILDER and POP E, 1996).Th e sy mp to ms d o n o t d i f fe r f ro m lo w b ack p a in cau sed b y ag eor const i tu t ion . How ever , they oc cur at an ear l ier age or are moresevere in people ex posed to v ibrat ion (DuPuIs , 1986) .Occ upat ional act iv i t ies o f ma ter ials handl ing can resu l t inlo w b ack p a in an d d i so rd er s (ANDERSSON,1981; LUTTMANNe t a l . , 1988) . Obv iously there i s a relat ion between the heal thr i sk and the fo rces t ransmit ted in the sp ine which can bec o m p u t e d b y m e a n s o f b io m e c h a n i c a l m o d e ls (JA,GER andLUTTMANN, 1992). Thus i t should also be possib le to assessthe v ibrat ion- induced heal th r i sk by comput ing the sp inefo rces cau sed b y wh o le -b o d y v ib ra t i o n . To d o t h i s , a mo d e lmust be developed which imi tates a s i t t ing dr iver . Severalmo d e l s cap a b l e o f u n d erg o in g v ib ra to ry mo t io n h av e b eendescr ibed in the l i teratu re. The f i rst model , consis t ing of amass , sp r i n g an d d amp er , was d ev e lo p ed b y DIECKM ANN(1 9 5 7 ) . He s imu la t ed t h e imp ed an ce an d t h e f r eq u en cy -r esp o n se fu n c t i o n o f th e h u m an b o d y i n t h e v e r t i ca l z -d i r ec -t ion . A model which enab les one to assess roughly the sp ineforces i s the f ive m ass mo del o f MERTENS (1976). Mo re detai l so f t h e h u man b o d y were im i t a t ed i n th e mo d e l o f QAS SEM e ta l. (1994). Ho we ver , in th is model there i s no coupl ingb e tween t h e h o r i zo n t a l an d v er t i ca l mo t io n s o f t h e mo d e lma sses and the v ibrat ion prope r t ies do no t f it the data g ive n inli terature.

6 8 6 M e d i c a l & B i o lo g i ca l E n g i n e e r i n g & C o m p u t i n g N o v e m b e r 1 9 9 8

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A b io mech an i ca l mo d e l wh ich en ab l es t h e co mp u ta t i o n o fthe fo rces in the sp ine wa s desc r ibed by LEO and GOLDSMITH(1 9 9 i ) . I n t hi s mo d e l , t he h u m an h ead /n eck / t o r so sy s t em wasimi tated by 10 r ig id bodies connected by v isco-elas t ic ele-men t s r ep resen t i n g t h e m o t io n seg men t s o f t h e sp ine . Tosimulate the human responses to shock Luo and GOLDSMITH(1991) t ransmit ted the exci tat ion d i rect ly to the pelv is . Th isapproach i s d isadvantageous fo r s imulat ing the v ibrat ion-induced responses because data about the v ibrat ion s t ress areg iv en b y acce l e r a t i o n s i g n a l s measu red b e tween t h e sea tsurface and the bu t tocks accord ing to ISO 2631-1 . Therefore,FRITZ (1996 , 1997a) added an eleven th body to the modelwhich represen ts the v ibrat ing seat . A model which includes adetai led imi tat ion of the lum bar sp ine and the abdom en wasdevelo ped by BUCK (1997). Beca use i t en tai led som e expenseto s imulate the v ibrat ion response of the human b ody wi th th isc o m p l e x m o d e l , B U C K e t a l . (1997) s impl i f ied the model .Ho w ev er , t h e v ib ra t i o n p ro p er t i e s o f t hi s mo d i f i ed mo d e lco incided w i th the p roper t ies o f the human bo dy on ly at lowfrequencies .Th e p u rp o se o f t h e p resen t s t u d y i s to imp ro v e t h e b io m e-chanical m odel o f FRITZ (1996 , 1997a) . In the Fr i tz mo del , nobody represen ted the arms al though the arms can in f luence thev ib ra t io n - in d u ced mo v em en t s o f th e h u man b o d y . Co n se-quent ly , the improved model p resen ted here includes armsand the hands ho ld ing a v ibrat ing s teer ing wheel . Fur thermore,the mot ion range should include the fo re-and-af t and lateraldirections as well as the vertical direction.

2 M e t h o d2 .1 V i b r at o r y b i o m e c h a n i c a l m o d e l

Although the human t runk , neck and head are f lex ib le anddeformable bodies , in the model they can be imi tated by r ig idbodies . The t runk and n eck w ere d iv ided by f ict i tious cu ts in tosmal l body segments . The sect ional p lanes run th rough thein terver tebral d isks (F ig . 1 ). So f ive r ig id bodies represen t thepelv is , the abdomen wi th the lumbar sp ine, and the thorax .Four bodies represen t the shoulder and the neck , and one body

"6e~

P,

r ep resen ts t h e h ead. Th ese b o d i es a r e co n n ec t ed b y ' h i n g es ' a tthose po in ts where the sp ine pushes th rough the sect ionalplane s (Fig. 2). In the sense of WITTENSURG (1977), the'h inges ' enab le no t on ly ro tat ional bu t also t ranslat ionalmot ion s s imi lar to the mot ion segm ents o f the sp ine. Thisimpl ies that fo rces and to rques are t ransmit ted in the 'h inges . 'The forces and to rques resu l t f rom the deform at ions o f thes imu la t ed mo t io n seg men t s .In t he h u man b o d y g ro ss mo v em en t s o f th e sp in e a r econt ro l led and res t r icted by the musc les o f the t runk andn eck . Un d er wh o le -b o d y v ib ra t i o n t h e mu sc l es p e r fo rm th i stask by rhy thmical act ivat ion on ly at low frequencies( f < 8 H z , F REUND, 1 9 83 ; S E tDEL e t a l . , 1986; andf < 10Hz, SEROUSSI e t a L , 1989) . At h igher f requencies ,an d esp ec i a l l y u n d er r an d o m v ib ra t i o n , t h e b ack mu sc l eshave an e nhanced s teady-s tate act iv i ty (WIKSTROM, 1993)an d a r e s t r e t ch ed b y t h e mo v e men t s o f th e b o d y . In t h emodel the muscles are s imulated by in ternal fo rce elements .The i r fo rces are g iven by the s t rain o f the elem ents and theforce d i rect ions are dependent on the at tachm ent po in t o f thee l emen t s o n t h e mo v in g b o d i es . S in ce a lmo s t n o m u sc l e h as as t raigh t l ine o f act ion , the fo rce elem ents are t igh tenedb e tween two b o d i es an d t u rn ed a ro u n d a t an i n t e rmed ia t ebod y (Fig . 2 ) . So 56 force elements , nam ely 26 eleme nts in theab d o min a l an d l u mb ar r eg io n an d 3 0 e l emen t s i n t h e n eck -shoulder , are bu i l t symm etr ical ly to the media n p lane in themo d e l .The arms are each represen ted by th ree r ig id bodies . The'upper arm' r ig id bodies are l inked to the neck-shoulder bodyby bal l -and-so cket jo in ts , wherea s the bodies represen t ing'u p p er a rm, ' ' f o r ea rm ' an d ' h an d ' a r e co n n ec t ed t o eachother by un iversal jo in ts . As in the hand-arm mo del o fFRITZ (1991) the ro tat ions in the jo in ts are res t r icted byin ternal to rques s imulat ing the arm muscles . The relat ionsh ipbetw een the to rques and the ro tat ions i s that o f a to rs ion spr ingco mb in ed wi th a d amp er . Th e b o d ies r ep resen t i n g ' h an d ' a r et ied by v iscoelas t ic fo rce elements to the v ibrat ing s teer ingwheel . In th is way we avoid the s i tuat ion where the arms andthe s teer ing wheel fo rm a closed k inemat ic chain , whichwould lead to mechanical difficult ies.F inal ly , the r ig id body 'pelv is ' i s connected wi th thev ibrat ing seat by spr ings and dampers in the th ree coord inated i rect ions . These spr ings and dampers should imi tate theelas t ici ty o f the bu t tocks (Tab le 1 ). For the ver t ical spr inga n d d a m p e r a p r i o r i v a lu es were d e r i v ed f ro m th e d a t a o fBROMAN e t a l . (1996) and PRIVITZER and BELYTSCHKO(1980). The value s were im prove d by f i t ting the mod el

m N % ze (o)

Y

F i g . 1 Schem at i c o f t he t runk , neck , head and arm. T he dashedhor i zon ta l l i nes i nd i ca t e t he pos i t i on o f n ine cu t s ( adop tedf rom F R IT Z , 1997b , bas i cen t r i c coord ina te sys t em accord ingto I SO 2631- 1 )

F i g . 2 T wo r ig id bod ies o f the b iomechan ica l m ode l t he connec t ing' h inge , ' and a f o r ce e l emen t im i ta t ing a back m usc l e ;rc~ = rad ius vec tor f ro m the i ner t ia l sys t em to t he cen t re o fg r a v i ~ ; m k a k = f o r c e o f t h e t r a ns l a ti o n al a c c e l e r a ti o n ;Okd) k = to rque o f t he ro ta t iona l acce l era t ion

Medical & Biological Engineer ing & Com put ing No vem ber 1998 687

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Table 1 Springs and dampers between v ibrat ing seat (body k = 1)and pe lv i s ( k = 2)Direction of vibration x y zspring, N /m 21000 25000 53000damper, Ns /m 650 490 900

prope r t i e s t o t he v ib r a t i on p rope r t i e s o f t he hum an bod y . I n t heve r t i ca l z -d i r ec t i on t he sp r i ng and dam per a r e no rm a l ly com -p r e s s e d b y t h e w e i g h t o f t h e u p p e r b o d i e s . H o w e v e r , i f t h edow nw ard s ea t acce l e r a t i on i s g r ea t e r t han t he acce l e r a t i ondue t o g r av i t y t he ve r t i ca l f o r ce be tw een sea t and pe lv i sbecom es g r ea t e r t han ze ro and t hus i t i s equ iva l en t t o a t ens i l efo r ce m ean ing t ha t t he d r i ve r i s pu l l ed dow nw ards by t he s ea t .B ecause t h i s i s no t r ea l i s t i c t he connec t i on be tw een t he s ea tand t he pe lv i s i s cu t i n t h i s ca se .

2 .2 E q u a t i o n o f m o t io nT h e e q u a t i o n s o f m o t i o n w h i c h d e s c ri b e t h e r e l a t io n s h i pb e t w e e n t h e f o r c e s a n d t o r q u e s a n d t h e b o d y m o t i o n s i n t h em o d e l w e r e d e r i v e d f r o m t h e p r in c i p le o f d 'A l e m b e r t . T h i s

p r inc ip l e im p l i e s t ha t the sum o f t he v ir t ua l w ork w h ich r e su l t sf rom ex t e rna l f o r ce s i nduced by phys i ca l e f f ec t s and f romin t e rna l f o r ce s equa l s ze ro . The i n t e rna l f o r ce s o f t he m od e la r e t he fo r ce s i n t he ' h i n ges ' and i n t he fo r ce e l em en t s . I nseve ra l s t eps (desc r i bed by WITTENBURQ, 1977) t he equa t i ono f m o t io n can be s im p l i f ied t o t he d i f f e r en ti a l equa t i on :A ~ = B ( I)

w h e r e / I i s t h e a c c e l e r a t i o n v e c t o r o f t h e g e n e r a l i s e d c o o r d i -n a t e s q . T h e e l e m e n t s o f t h e m a t r i c e s A a n d B a r e g i v e n b y t h eg e o m e t r i c a l q u a n t i t i e s f t h e m o d e l , t h e q u a n t i t i e s f m o t i o n ,a n d b y t h e f o r c e s a n d t o r q u e s . T h e t w o m a t ri c e s w e r ec o m p u t e d b y m e a n s o f t h e s o f t w a r e M E S A V E R D E , V e r s i o n4 . 2, w h i c h w a s d e v e l o p e d f o r s i m u l a t i ng t h e m o t i o n s o f mu l t i -b o d y s y s te m s .2.3 M e c h a n i c a l p r o p e r ti e s o f m o d e l e l e m e n ts

B efo re t he i n t eg ra t i on o f t he m o t ion eqn . 1 , t he r e l a t i onsh ipb e t w e e n t h e f o r c e s a n d t o r q u e s a n d t h e d e f o r m a t i o n s i n t h e' h i n g e s ' h a d t o b e m a t h e m a t i c a l l y d e s c r i b e d . B e c a u s e t h e' h i n g e s ' b e t w e e n t h e t e n b o d i e s o f th e t r u n k, n e c k , a n d h e a dshou ld s im u la t e t he m o t ion s egm en t s o f t he sp ine , t he co r r e -spond ing r e l a t i onsh ip w as d e r ived f rom the o r i g ina l s t if f nes sm a t r i x K o f PA N JA B Ie t a l . ( 1 9 7 6 ) a n d t h e d a m p i n g f a c t o r s o fM A R K O LF (1971) . I t w as a s sum ed t ha t t he r e i s a d i f f e r encebe tw een t he s t i f f nes s a t pos i t i ve and nega t i ve de fo rm a t ions .F o r e x a m p l e , a n e g a t i v e d e f o r m a t i o n i s g i v e n b y a c o m p r e s -s i on o f t he i n t e rve r teb ra l d i sk o r by a backw ard sh i f t o f t hec ran i a l ve r t eb rae i n r e l a t i on t o t he cauda l ve r t eb rae .The sym m et r i ca l s t i f f nes s m a t r i x K o fPA N JA B I e t a l . ( 1 9 7 6 )f o r n e g a t i v e d e f o r m a t i o n s i s g i v e n b e l o w . U n i t s f o r t h e v a l u e sa r e N , m , and r ad .

1 0 0 00 0 0 - 2 0 0 0 0 0 - 1 5 6 0 6 00 20 110000 0 1640 0K = - 2 0 0 0 0 0 1 2 4 00 0 0 0 - 7 3 0 0 [i

0 1640 0 197 0 2 f1 5 6 0 0 - 7 3 0 0 1 86 00 620 0 2 0 145The c ros s - sec t i ona l a r ea s o f t he ve r t eb rae and t he i n t e r -ve r t eb ra l d i sks va ry f rom the pe lv i s t o t he head and t hus t he

s t i f fnes s o f t he sp ine i s no t un i fo rm . Th e s t i f f nes s and t hed a m p i n g m a t r ic e s o f e a c h ' h i n g e ' w e r e th e r e f o re m u l t i p l ie d b ya c o r r e c t i o n f a c t o r w h i c h c o r r e s p o n d s t o th e d i m i n u t i o n o f t h e

c ros s - sec t i ona l a r ea s a s g iven by Y A M A D A (1973) . Fo r t hen ine "h inges ' be tw een t he pe lv i s (k = 2 ) and t he head (k = 1 I )t he f ac to r s a r e :A k = 2 .9 , 2 .54, 2 .36, 2 .12, 1 .11, 0 .88, 0 .69, 0 .50

and 0 . 52 fo r k = 3 , 4 . . . . . 11I n s i m u l a ti n g t h e t r u n k a n d n e c k m u s c l e s , t h e f o r c e s o f th e 5 6f o r c e e le m e n t s s h o u l d b e c o m p o s e d o f a c o n s t a n t a n d a t im e -va r i ab l e fo r ce . I nduced by t he w e igh t f o r ce s o f t he bod i e s , t hem ode l s t a r t s t o v ib r a t e a t i t s na tu r a l f r equenc i e s . The cons t an tfo r ce pa r t s w e re ca l cu l a t ed i n such a w ay t ha t t he m ode lim i t a t ed an e r ec t s i t t i ng pos i t i on w hen t he w e igh t i nducedv i b r a ti o n s w e r e d a m p e d o u t . A s m e n t i o n e d a b o v e , t h e m u s c l e sr e spond t o v ib r a t i on w i th a rhy thm ica l ac t i va t i on on ly a t l owf requenc i e s . The re fo re , t he t im e-va r i ab l e fo r ce pa r t s w e redepend en t on t he s t r e t ch o f t he fo r ce e l em en t s r e su l t i ngf rom the ac tua l v ib r a t i on s t r e s s . The fo r ces w ere com pu tedb y m e a n s o f t h e e q u a t io n o f M E R R IL L e t a l . (1984) , w h ichdesc r ibes t he r e l a ti onsh ip be tw een t he s t r e t ch o f a m usc l e andi ts f o r ce , and by a m u l t i p l i ca t i on w i th t he co r r e sp ond ing c ros s -sec t i ona l a r ea s . These a r ea s and t he r ad ius v ec to r s o f t hea t t a c h m e n t p o i n ts o f t h e m u s c l e s w e r e t a k e n f r o m t h e s t u d y o fL uo and G O LD SM ITH (1991) .In a s se s s ing t he sp r i ng and dam per cons t an t s f o r t he fo r ce sb e t w e e n h a n d a n d s t e e r i n g w h e e l , a t t e n t i o n w a s p a i d t o t h ed i s t ance be tw een t hese bod i e s , w h ich shou ld be sm a l l t h rough-ou t t he v ib r a t i on . Th i s w as ach i eved us ing t he va lues g iven i nTab l e 2 . I n add i t ion , t he v ib r a t i on i nduc ed ro t a t i ons o f t heu p p e r a r m s a n d t h e f o r e a r m s s h o u l d b e s m a l l , w h i c h w a sm an aged b y t he i n te rna l t o rques . The co ns t an t s o f t he t o r s i onsp r ings and dam per s a r e l i s t ed i n Tab l e 2 w i th r ega rd t o t hef ac t t ha t t he i ne rt i a l o r i en t a t i on o f the t o rque s a r e pa r a l l e l t othe bas i cen t r i c coo rd ina t e sy s t em . I f no va lue i s g iven i n Tab l e2 a ro t a t i on a round t h i s ax i s i s b locked .T h e r e m a i n i n g a n a t o m i c a l , p h y s i o l o g i c a l a n d m e c h a n i c a ld a t a o f t h e d i f fe r e n t e le m e n t s o f t h e m o d e l w e r e d e r i v e d f r o mLU O and G O LD SM ITH (1991) and da t a fo r the a rm s f romD EM PSTER (1955) . F ina l l y , t he d a t a w ere f i tt ed t o a hu m a nb o d y o f h e i g h t 1 . 73 m a n d a b o d y m a s s 7 4 k g .2 .4 T e s t v i b r a t i on

To a s se s s t he v ib r a t i on p rope r t i e s o f t he m od e l i n t he x - , y -and t he z -d i r ec ti ons , r e spec t i ve ly , by m eans o f one s im u la t i on ,t h e s e a t v i b ra t i o n w a s c o m p o s e d o f 8 0 s i n u s o i d al v i b r a t i o n sw i th d i f f e r en t f r equenc i e sna( t ) = y ~ a i s i n ( 2 n f t ) (2 )i=1

T h e f r e q u e n c ie s o f th e s e v i b r a t io n s w e r ef / = ( 6 00 + i . 4 0 0 ) / 1 0 2 4 f o r i = 1 , 2 . . . . . 8 0

Table 2 Spr ings and dampers fo r for ce be tween hand and s t ee r ingwhee l and for in te rnal t orques in jo in t s o f t he arm s imulat ing themusc le sF o r C e h a n d spr ing, N /m 50000damper, Ns /m 400Torques around the axes x y zShoulder spring, Nm /rad 90.0 5.0 0damper, Nm s/rad 2.5 0.1 0Elbow spr ing, Nm /rad 0 0 bdamper, Nm s/rad 0 0 bWrist spring, Nm /rad b 600 160damper, Nm s/rad b 8.0 3.0b = blocked, i.e. rotation not possible

688 M ed ica l & B io log ica l Eng ineer ing & C om put ing N ove mb er 1998

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a n d t h u s t h e y c o v e r e d t h e f r e q u e n c y r a n g e f r o m I t o 3 2 H z .The am pl i t ude o f each v ib r a t i on pa r t w as a i = 0 . 2 m s -2 . Fo rthe z -d i r ec t i on t h i s y i e lded t o a w e igh t ed acce l e r a t i ona ~ = 0 . 7 8 m s - 2 w h i c h c o r r e s p o n d e d t o a f a ir l y u n c o m f o r t a -b l e s ea t v ib r a t i on ( ISO 2631-1 ) .

3 R e s u l t s3 .1 M o t i o n s o f r i g i d b o d i e s

T h e f r e q u e n c y d e p e n d e n t t r an s f e r pr o p e r ti e s o f t h e m o d e lc o n c e r n i n g t h e a c c e l e r a t i o n s o f th e s e a t a n d t h e h e a d w e r ef i t ted to da ta giv en in the l i tera ture (COERMA NN, 1962;D IECK M A N N , 1957 ; H A G EN A , 1985 ; M ER TEN S, 1978 o rPA D D A N and G R IFFIN , 1988a , b ) b y m od i fy ing t he sp r ingsa n d d a m p e r s b e t w e e n s e a t a n d p e l v i s a n d t h e c o r r e c t i o nfac to r s A k . The i n f l uence o f t he pa ram e te r s o f t he a rm s w ass m a ll . S o t h e b e s t a p p r o x i m a t i o n o f t h e m o d e l t o h u m a nv i b r a t i o n w a s f i n a l l y a c h i e v e d b y t h e m o d e l p a r a m e t e r sl i s t ed i n Tab l e s 1 and 2 .I n F i g . 3 t h e f r e q u e n c y - r e s p o n s e f u n c t i o n H a = b e t w e e n t h eve r t i ca l s ea t a cce l e r a t i on and t he s im u la t ed head acce l e r a t i on

i s p l o t te d . T h e m a g n i t u d e o f t he f u n c t i o n r e a ch e s t w oex t r em es , nam e ly b e tw e en 4 and 5 H z and nea r 29 I4_z . I n t helow e r pa r t o f F ig . 3 t he cou r se o f the phase ang l e i s show n int h e r a n g e f r o m 1 8 0 ~ to - 1 8 0 ~ w h i c h i s n o r m a l l y u s e d i nf r equency ana lys i s . The phase cou r se has i t s s t eepes t s l opea r o u n d 5 a n d 2 9 H z . T h e s e p r o p e r t i e s o f t he m a g n i t u d e a n d t h ep h a s e s h i f t p o i n t to t w o r e s o n a n c e f r e q u en c i e s o f t he m o d e l i nthe z -d i r ec t i on . U nde r ve r t i ca l v ib r a t i on t he hum an body a l som oves i n t he ho r i zon t a l x -d i r ec t i on , s im i l a r t o a w h ip s t i ck .H o w e v e r , t h e h o r i z o n t a l h e a d m o v e m e n t i s m u c h s m a l l e r t h a nt h e v e r t i c a l m o v e m e n t w h i c h b e c o m e s c l e a r f r o m t h e f r e -q u e n c y - r e s p o n s e f u n c t i o n H ~ :x be tw e en t he exc i t a t i on an d t hehor i zon t a l a cce l e r a t i on o f t he ' he ad ' (F ig . 3 ).U n d e r v i b r a t i o n i n x - d i r e c ti o n t h e h e a d m o v e s i n t h e x - a n dz - d i r e c t i o n a n d t h e c o r r e s p o n d i n g f r e q u e n c y - r e s p o n s e f u n c -t i ons H o~,= and H a= a r e show n in F ig . 4 . F rom the m ag n i tudeso f th e f u n c t i o n s i t c a n b e s e e n t h a t a t l o w f r e q u e n c i e s ( < 3 H z )t h e h o r i z o n t a l h e a d m o v e m e n t s a r e g r e a t e r t h a n t h e v e r t i c a l

E

2.0-1.5-1.0-0.5.0.0 i i i5 10 1'5 20 2'5 30frequency,Hz1 : / I i I I I

o 1

- 9 0 . j

Fig. 3 F r e q u e n c y - r e s p o n s e f u n c t i o n s b e t w e e n t h e a c c e l e r a t io n o fs e a t i n z -d i r e c t i o n a n d a c c e l e r a t io n s o f t h e h e a d i n z - a n d x -d i re c ti o n s. " - - - - = z - d i r e c t io n ; . . . . = x - d i r e c t io n

Fig. 4

2.01.5

~t~ 1.00 . 5 : '.0 . 0

0 5 1 0 1 5 20 25 3 0frequency,Hz18090oo,5 _

-90-180

I I I I I I

F r e q u e n c y - r es p o n s e f u n c t i o n s b e t w e e n t h e a c c e le r a t io n o ft h e s e a t i n x - d i r e c t i o n a n d t h e a c c e l e r a t i o n s o f t h e h e a d i n x -a n d z -d i r e c t i o n ; - - - - = x - d i r e c t io n ; . . . . ---- - d i r e c t i o n

m ov em e n t s . B u t a t t he ve r t ica l r e sonance f r eque ncy t h is r a t ioi s re v e r s e d . A b o v e 1 0 H z t h e m a g n i t u d e s o f b o th f u n c t i o n s a r es m a l l, w h i c h m e a n s t h a t t h e b o d y ' h e a d ' s c a r c e l y m o v e s u n d e rthose v ib r a t i ons . I n t he r ange o f l ow f r equenc i e s t he phases h i ft s i n b o t h d i r e ct i o n s v a r y r a p i d l y b e t w e e n 1 80 ~ a n d - 1 8 0 ~w h e r e a s a b o v e I 0 H z t h e y g r a d u a l l y d e c r e a s e w i t h i n c r e a s i n gf r e q u e n c y .U n d e r v i b r a t i o n i n t h e l a te r a l y - d i r e c t i o n th e h e a d m o v e s n o ton ly i n t h i s d i r ec t i on bu t a l so i n t he x - and z -d i r ec t i ons .The re fo re , t h r ee f r equency- r e sponse func t i ons ( H ay y , H ay = ,

and H ayx) a r e p lo t t ed i n F ig . 5 . The d i f f e r ences be tw een t he

Fig. 5

2 . 0

1.5 L.00.50.0 ~ . , . . . . ' 120 25 30

frequency,Hz1809Ooo

f ~

- 1 8 0

F r e q u e n c y - r e sp o n s e f u n c t i o n s b e t w e e n t h e a cc e l e ra t i o n o ft h e s e a t i n y - d i r e c t i o n a n d t h e a c c e l e r a t io n s o f t h e h e a d i n y ,x - a n d z - d i r e c t io n : - - - - = y - d i r e c t i o n ; - - = x - d i r e c t i o n ;. . . . . z - d i r e c t io n ; ( f o r c l e a r n e s s t h e p h a s e is p l o t t e d o n l yi n t h e y - d i r e c t i o n )

Med i ca l & B i o log i ca l E ng i neer i ng & C om put i ng N ov em ber 1998 689

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magnitudes show that the head movements in the y-directionare much greater than in the two other directions at frequen-cies below 5 Hz. At higher frequencies the head movementsare small. Because the phase shifts in the three directionsrepeatedly cross the whole given phase range only the phaseshift in the y-direction is shown in Fig. 5.

3.2 F o r c e s i n t h e l u m b a r s p i n eThe model was developed in order to compute the spineforces and assess health risk. The greatest risk is given for thelumbar spine transmitting the vibration from the pelvis to theupper body. In Fig. 6 the frequency-response functions H azand H azx between the vertical seat acceleration and the accel-erations of the vertebra L3 are shown. The maximal magni-tudes are reached between 4 and 5 Hz, and in the x-directionthis extreme is even greater than the corresponding extreme ofthe head.During still erect sitting the compressive force (z-direction)transmitted from seat to pelvis (called seat-pelvis force) was

- 483 N, whereas the compressive force acting on vertebra L3was -5 30N . The difference was caused by the lumbarmuscles being active during sitting. The shear force (x-direc-tion) on the vertebra was 14N. Under vibration the staticforces are superimposed by oscillating parts. Frequencydependent relations exist between these parts of the seat-pelvis force and the spine forces, which can be expressed byfrequency-response functions. For example, for a verticalexcitation these functions are shown in Fig. 7 for the com-pressive (H#=) and the shear force (Hfzx). According to themagnitudes given, the oscillating forces on the vertebra,especially the shear force, are lower than the seat-pelvisforce. Only at around 27 Hz is the compressive force greaterthan this force. The phase shift o f the compressive force issmall over the whole frequency range. The phase shift of theshear force decreases with increasing frequency.

2 . 0 -

1.S-

o 1 . O -E 0 . 5 -

0.00

j , , , .m ~ - . .

i i i l i i5 1 0 1 5 2 0 2 5 3 0

f r e q u e n c y , H z

18 0

9 0 -o

o -

- 9 0 -

- 1 8 0

I . I I I I I! :

Fig. 6 F r e q u e n c y - r e s p o n s e f u n c t i o n s b e t w e e n t h e a c ce l e r at i on o fthe s ea t i n z - d ir ec t ion and the acce l er a tions o f t he ver t ebr aL z in z - a n d x - di re c ti o n; - - - - = z - d i r e c t i o n ; . . . . = x -direct ion

6 9 0

Fig. 7

1 . 5 -

, a . , 1 . 0 -"E

E 0 . 5 -F" - - - ' - -

. - . . . . . ~,~ 9 . . . . . . . . . . . . . - ,0 . 0 ~ , , ,0 5 1 0 1 5 2 0 2 5 3 0

~ e q u e n c y , H z

1 8 0 I I I I I t. -

9 Oo ~ - . ' .

0 '=

- .- 1 8 0

F r e q u e n c y - re s p o n s e f i m c t i o n s b e t w e e n t h e c o m p r e s s iv e f o r c es ea t - pe l v is and co mpr es s i ve and s hear f or c e ac t ing onver t ebr a Ls f or v ibr a t ion exc i ta t ion in the z -d i r ec t ion; - -- - = c o m p r es s iv e f o r c e ; . . . . . s h e a r f o r c e

4 D i s c u s s i o n4 .1 M o t i o n s o f r i g i d b o d i e s

The presented frequency-response functions of the model donot agree in every detail with curves described in ISO 7962 or inthe literature. First, in the magnitude of function H a= (Fig. 3) thesecond extreme is not given in ISO 7962. This extreme isdependent upon the vibration level at high frequencies. Addi-tional simulations with higher accelerations (ai) showed that thesecond extreme decreased with accelerat ion and shifted to lowerfrequencies. On the other hand, the missing extreme in the curveof lS O 7962 may result from artefacts due to the fixation of theaccelerometers on the head as discussed by FRITZ (1996,1997a). PADDAN and GRIFFIN (1988a) simultaneously mea-sured head accelerations in three directions. In the x-directiontheir transmissibility curves generally coincide with the magni-tude of H azx . The transmissibility in the y-direction is notreflected by a frequency-response function of the model becausethe model is symmetrical to the xz-plane and the excitation takesplace in this plane.In a second study, PADDAN and GRIFFIN (1988b) exposedtheir subjects to horizontal vibrations in the frequency range0.2 to 16Hz. Unfortunately, they plotted transmissibilitycurves computed for each subject and each trial and not thecurves of mean values. Thus, the comparison of these curveswith the corresponding frequency-response functions of ourmodel can only lead to the statement that the frequencydependent tendency of the different curves coincides. Acomparison of the phase shifts may be confusing because inthe present study the phase angles are given in the rangebetween 180~ and - 180 ~ whereas PADDAN and GRIFFIN(1988b) extended the range from 400 ~ to -1 00 0 ~ However,a phase o f -36 0 ~ can be interpreted as a phase o f 0 ~ meaningthat between input and output signal no time or phase shiftexists. In this case, the phase curves o f PADDAN and GRIFFIN(1988b) tend to coincide with the simulated curves.

Conditioned by the springs and dampers between seat andpelvis, the imitation pelvis cannot rotate round the lateral axis.According to BROMAN t a l . (1996) the pelvis must be allowedM e d i c a l & B i o l o g ic a l E n g i n e e r i n g & C o m p u t i n g N o v e m b e r 1 9 9 8

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t o ro t a t e i n o rde r t o ge t su f f ic i en t con fo rm i ty be tw ee n t hev i b r a ti o n p r o p e r ti e s o f a m o d e l a n d o f t h e h u m a n b o d y . D u r i n ge rec t s i t t ing w i thou t a back re s t , t he ro t a t i on o f t he pe lv i s i sr educed by t he g lu t ea l m usc l e s connec t i ng t he pe lv i s w i th t hel egs . I n t he b iom echan i ca l m ode l t he l egs w ere no t r ep re -sen t ed . The re fo re , t he e f f ec t o f t he g lu t eal m usc l e s cou ld beim i t a t ed on ly by a t o r s i on sp r ing be tw een sea t and pe lv i s .H ow eve r , s im p ly no su f f i c i en t r e f e r ence i s ava i l ab l e i n o rde rto a s se s s t he s t r eng th o f the t o r s i on sp r ing a s s t a t ed byBROMAN e t a l . (1996) .In con t r a s t t o t he m od e l o f FR ITZ (1996 , 1 997a ) t he p r e sen tm o d e l i n c l u d es b o t h a rm s . B y c o m p a r i n g t h e f r e q u e n c y -r e sponse func t i ons o f t he tw o m ode l s i n the z -d i r ec t ion , i tc an be s een t ha t t he a rm s do no t i n f l uence t he v ib r a t i onp rope r t i e s o f t he m ode l . Th e sm a l l d i f f e r ences be tw een t hefunc t i ons r e su l t f r om the im prov ed m ode l pa r am e te r s . I n the x -d i r ec t i on t he m ove m e n t s o f t he bod i e s rep re sen t i ng t r unk ,n e c k , a n d h e a d a r e d i m i n i s h e d b y t h e a r m s . A n e n h a n c e m e n to f th e s e m o v e m e n t s b y t h e v i b r a t io n a p p l i e d t o th e h a n d s o ft he m ode l cou ld no t be de t ec t ed . Th i s m eans t ha t t he hand-a rmsys t em o f t he m ode l a t t enua t e s t he v ib r a ti on i n a s im i l a r w a yto t he r e su l t s o f PO PE e t a l . (1997) .

4 .2 F o r c e s i n th e l u m b a r s p i n eIn t he ca se o f v ib r a t i on - induced l ow back pa in , i t is o fi n t e re s t i f r e sonances a l so ex i s t i n t he v ib r a t i on r e sponses o ft he l um bar ve r t eb rae , f o r exam ple L3 . PA N JA B I e t a l . ( 1 9 8 6 )

a n d P O P E e t a l . (1989) d i r ec t l y m easu red t he acce l e r a t i ons o ft he ve r t eb ra L3 unde r ve r t i ca l s i nuso ida l v ib r a t i on and im pac t ,r e spec t i ve ly . B ased on t he i r m easu rem en t s t hey com pu ted t hef r equenc y- r e spon se func t i on fo r the v ib r a ti ons o f ve r t eb ra L3 .T h e s i m u l a t e d f u n c t i o n H azz (F ig . 6 ) i s som ew here be tw eenthese ' m easu red ' cu rves . The d i f f e r ences m ay r e su l t f r om thedev i a t i ons i n t he exc i t a t i ons , nam e ly s i nuso ida l , im pac t andb r o a d b a n d v i b r a t i o n . H o w e v e r , t w o c o n s p i c u o u s p r o p e r t i e scan be de r ived f rom the f r equency- r e sponse func t i ons . A sw i th t o t he head v ib r a t i ons , a f i r s t r e sonance ex i s t s nea r 5 H z .POPE e t a l . (1989) sugges t ed t ha t t h i s r e sonance i s due t o t heb io log i ca l subsys t em s be tw een t he ve r t eb ra L3 and t he s ea t ,e spec i a l l y t o t he com pre s s ion o f t he bu t t ocks t is sue . C om ple -m en ta ry t o t h is , the m e asu rem en t s o f PA N JA B I e t a l . (1986)show ed no d i f f e r ences be tw een t he acce l e r a t i ons a t ve r t eb raeLI an d L3 . I n t he m o de l t he f i rs t r e sonance i s due t o t he sp r i ngsa n d d a m p e r s b e t w e e n s e a t a n d p e l v i s . A s e c o n d r e s o n a n c en e a r 3 0 H z c a n b e d e r i v e d f r o m t h e f u n c ti o n H a = a n d t h ec u r v e s o f P O P E e t a l . ( 1 9 8 9 ) . A c c o r d i n g t o t h e m o d e l t h es e c o n d r e s o n a n c e i s d u e t o t h e e n h a n c e d m o v e m e n t s o f t h euppe r body and t he head a t h igh f r equenc i e s .PANJABI e t a l . ( 1 9 8 6 ) c a l c u l a t e d t h e m a g n i t u d e s o f t h ef r equen cy- r e spo nse fun c t i on o f ve r t eb ra L3 a l so i n t he x -d i r ec t i on . They r epo r t ed t ha t p robab ly no r e sonance ex i s t sw i t h i n t h e i n v e st i g a te d f r e q u e n c y r a n g e a n d th i s st a t e m e n topp osed t he func t i on H a= (F ig . 6 ). H ow eve r , one o f t he ir f i vecu rves show s a d i s t i nc t r e sonance co r r e l a t i ng w i th t he m agn i -t u d e o f Ha~ . In 1958 D IEC K M A N N had a l r eady r eg i s t e r ed t hem ax im a l acce l e r a t i on o f t he h ip unde r ho r i zon t a l v ib r a t i on a t af r eque ncy o f 2 H z . I t i s l i ke ly t ha t t he v ib r a t i on r e sponse o f t hev e r t e b r a L 3 re a c h e s a m a x i m u m a t l o w f r e q u e n c y e v e n i f t h ehor i zon t a l m ovem en t i s i nduced by a ve r t i ca l v ib r a t i on .The l um bar sp ine fo r ces p r e sen t ed i n t h i s s t udy fo r an e r ec ts i t t i n g p e r s o n c o i n c i d e w i t h f o r c e s c o m p u t e d b y m e a n s o fo the r m ode l s de sc r i bed i n t he l i t e r a tu r e . U nde r v ib r a t i on t heosc i l l a t i ng fo r ce pa r t s a r e l ow er t han t he o sc i l l a t i ng pa r t o f t hesea t -pe lv i s f o r ce (F ig . 7 ) . H ow ev e r , an excep t i on ex i s t s a round27 H z w here t he co m p res s ive sp ine fo r ce is g r ea t e r t han t hesea t -pe lv i s f o r ce . B u t t h i s does no t m ean t ha t t he h ighes tcom pres s ive sp ine fo r ces a r e a lw ays r eached a t h igh v ib r a t i onMedical & Biological Engineer ing & Comput ing

f r equenc i e s . Fu r the rm ore , i t c an be de r ived f rom the im pe-d a n c e o f t h e h u m a n b o d y ( I S O 5 9 8 2 ) t h a t t he s e a t - p e lv i s f o r c edec rease s w i th i nc r eas ing f r equenc i e s . Thus t he h ighes t sp inefo rces have t o be expec t ed a t f r equenc i e s l ow er t han 10 H z .To a s se s s t he v ib r a t i on i nduced hea l t h r i sk , t he sp ine fo r cesh a v e t o b e c o m p a r e d w i t h t h e c o m p r e s s i v e s t r e n g t h o f t h esp ine , a s m os t i n fo rm a t ion ex i s t s abou t t h i s s t r eng th . B u t i t i sno t su f f i c i en t t o com pare on ly t he peak fo r ces w i th t heu l t im a t e com pres s ive s t r eng th . A s w i th t echn i ca l m a t e r i a l st h e s t r e n g th o f th e s p i n e d e c r e a s es w i t h t h e n u m b e r o f l o a dcyc l e s . So SEID EL e t a l . (1995) sugges t ed a endurance l im i tw h ic h am o un t s t o 20% o f t he u l t im a t e s t r eng th . Fu r the r,acco rd ing t o t he G ood m a n r e l a t i on (F ISC H ER , 1976) , t hes t a t ic f o r ce and t he am pl i t ude o f the o sc i l l a t ing pa r t have t ob e c o m p a r e d w i t h t h e e n d u r a n c e l im i t. I f t h e c o m p a r i s o nind i ca t e s an ove r load t he ac tua l v ib r a t i on s t r e s s ha s t o ber educed , e . g . by im pro ved sea t s , o r i n t he ca se o f m an i f e s t edlow back pa in t he ove r load i nd i ca t e s t ha t t he pa in i s obv ious lyv ib ra t i on i nduced .

5 C o n c l u s i o nB y us ing t he p roposed m ode l , o r i t s f r equency- r e sponsefunc t i ons , t he fo r ce s i n t he l um bar sp ine can be com pu ted fo rd i f f e r en t v ib r a t i on s t r e s se s . B y com par ing t he fo r ce s w i th t heenduran ce s t r eng th o f t he sp ine , a va l i d a s se s sm e n t o f t heheal th r i sk i s poss ible .

R e f e r e n c e sANDERSSON, G. B . J. (1981): 'Epidemiologic aspects of low backpain in industry, ' S p in e , 6 , pp . 53 -60BROMAN, H., P OPE , M., a nd HANSSON,T. (1996): 'A mathematicalmodel for the impact response of the seated sub ject, ' Med. Eng.Ph ys . , 18, pp. 410-419BUCK, B. (1997): 'Ein M odell ftir das Sch wingun gsverhalten dessitzenden Menschen mit detaillierter Abbildung der WirbelsS,ul eund Muskulatur im Lendenbereich. ' Thesis, TechnischeHochschule D armstadt, Shaker VerlagBUCK, B., PANKOKE,S., and W6 LFEL, H. P. (199 7): 'Lateralsymm e-trisches Modell der Lendenwirbels/iule zur Berechnung dyna-mischer Bandscheibenkr/ifte (Schlul3bericht). ' Schriftenreihe derBundesa nstalt fiir Arbeitsschutz und A rbeitsmedizin, -Forschu ng-,Fb 770COERMANN,R. (1962): 'Th e mechanical impedance of the humanbody in sitting and standing position at low frequencies, ' H u m a nFa c to rs , 4, pp. 227-253DEMPSTER, W. T. (1955): 'Spa ce requirem ents of the seated opera-tor. ' WADC Technical report no. 5 5 -1 5 9DIECKMANN, D. (1957): 'Einflul3 vertikaler mecha nischer Schw in-gungen auf den Menschen, ' In t . Z. angew. Physio l . , 16, pp. 519-

56 4DIECKMANN,D. (1958 ): 'Einflul3 horizontaler mechanischer Schwin-gungen anfde n M enschen,' In t . Z. angew. Physio l . , 17, pp. 83 - 100DuPUIS, H. (1986): 'Ganz-K 6rper-Schw ingungen und W irbels~iule-Stellungnahme (1), ' Arb . Med. Soz. Med. P r?i. Me d., 21, pp. 186-1 87FISCHER, G. (1976): 'T heo ry of equipment design' in HARRIS, C. M. ,and CREDE, C. E. (Eds.): 'Sh ock and vibration handbook, 2nd edn.(McGraw-Hil l Book Company, New York), pp. 42-1-4 2-3 6FREUND, H. J. (1983): 'Mo tor unit and muscle activity in voluntarymotor co nt rol ,' Ph ys io l . Rev . , 63, pp. 387-428FRITZ, M. (1991): 'An improved biomechanical model fo r simulatingthe strain of the hand-arm system un der vibration stress, ' J.Bio mech . , 24, pp. 1165-1171FRITZ, M. (19 96): 'AbscNitzun g der in d er Wirbels~iule bei G anz-K6rper-S chwingu ngen wirkenden Kr~ifte mit Hilfe eines biome-chanischen Modells, ' Z Arb. Wiss., 50(22 NF) , pp. 174-179FRITZ, M. (1997a): 'Estimation of spine forces under w hole-bodyvibration by means of a biomechanical model and transfer func-tions, ' Aviat . Space Environ. M ed. , 68 , pp . 512 -51 9N o v e m b e r 1 9 9 8 6 9 1

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FRITZ, M. (1997b): 'Ermittlung der in der Wirbels/iule bei Ganz-Krrper-Schwingungen wirkenden Kr mit Hilfe eines biome-chanischen Modells' in VDI-Gesselschaft Entwicklung, Konstruk-Lion, Vertrieb (Eds.) : 'Schwingungen am Arbeitsplatz und in derUmwelt, VDI-Berichte 1345' (VDI Verlag, Dfisseldorf), pp. 21 7-224HAGENA, F. W. (1985): 'Zur Biomechanik der Wirbe lsgu le--Untersuchungen zum D/impfungsverhalten der Wirbels~iuledurchBeschleunigungsmessungen an gesunden Probanden und autop-tisch gewonnenen LWS-Prgparaten.' Thesis for the habilitation,Ludwig Maximilian University of MunchJ,~.GER, M., and LUTTMANN,A. (1992): 'The load on the lumbar spineduring asymmetrical bi-manual materials handling,' Ergonomics,35, pp. 783-805LUO, Z., and GOLDSMITH, W. (1991) : 'Reaction of human head]neck/torso system to shock,' J. Biomech., 24, pp. 499-510LUTTMANN, A., J.~GER, M., LAURIG, W., and SCHLEGEL, K. F.(1988): 'Orthopaedic disease among transport workers,' Int.Arch. Occup. Environ. Health, 61, pp. 197-205MARKOLF, K. L. (1971): Stiffness and damping characteristics of thethroacolumbar spine' in 'Proceedings of Workshop on Bioengi-neering Approaches to Problems of the Spine' (Department ofHealth, Education, and Welfare, California), pp. 87-143MERRILL, T., GOLDSMITH, W., and DENG, Y. C. (1984): 'Three-dimensional response of a lumped parameter head-neck model due

to impact on impulsive loading,' J. Biomechanics, 17, pp. 81-95MERTENS, H. (1976) : 'Das dynamische Verhalten des sitzendenMenschen unter BeschleunigungseinfluB.' PdD thesis, RWTHAachenMERTENS, H. (1978): 'Nonlinear behavior of sitting humans underincreasing gravity,' Aviat. Space Environ. Med., 49, pp. 287-298MESA VERDE, Version 4.2, IPG-Ingenieurgemeinschaft, Prof. Dr.-Ing, Gnadler, KarlsruhePADDAN, G. S., and GRIFFIN, M. J. (1988a): 'The transmission oftranslational seat vibration to the head -- I. Vertical seat vibration,'J. Biomech., 21, pp. 191-198PADDAN, G. S., and GRIFFIN, M. J. (1988b): 'The transmission oftranslational seat vibration to the h ead- -I I. Horizontal seat vibra-tion,' J. Biomech., 21, pp. 199-206PANJABI,M. M., BRAND, R. A., and WHITE,A. A. (1976): 'Three-dimensional flexibility and stiffness properties of the humanthoracic spine,' J. Biomech., 9, pp. 185-192PANJABI, M. M., ANDERSSON,G. B. J., JORNEUS, L., HULT, E., andMATTSSON, L. (1986): "In vivo measurements of spinal columnvibrations,' J. Bone Joint Surg., 68-A, pp. 695-702POPE, M. H., BROMAN, H., and HANSSON, T. (1989): 'The dynamicresponse of a subject seated on various cushions,' Ergonomics, 32,pp. 1155-1166POPE, M. H,. MAGNUSSON,M., and HANSSON, T. (1997): 'The upperextremity attenuates intermediate frequency vibrations' J. Bio-mech., 30, pp. 103-108PRIVITZER, E., and BELYTSCHKO,E. (1980): 'Impedance of a three-dimensional head-spine model,' Math. Model., 1, pp. 189-209

QASSEM, W., OTHMAN, M. O., and ABDUL-MAJEED, S. (1994): 'Theeffects of vertical and horizontal vibrations on the human body,'Med. Eng. Phys., 16, pp. 151-162SEIDEL, H., BLOTHNER,R., and HINZ,B. (1986): 'Effects of sinusoi-dal whole-body vibration on the lumbar spine: The stress-strainrelationship,' Int. Arch. Occup. Environ. Health, 57, pp. 207-223SEIDEL, H., and HEIDE,R. (1986): 'Long-term effects of whole-bodyvibration on the lumbar spine: a critical survey of the literature,'Int. Arch. Occup. Environ. Health, 58, pp. 1-26SEIDEL, H., BLOTHNER, R., HINZ, B., and SCHUST, M. (1995):'Belastung der Lendenwirbels/iule durch stoBhalt ige Ganz-krrperschwingungen,' Schriftenreihe der Bundesanstalt fftrArbeitsmedizin, Forschung FB 01 HK 061, BerlinSEROUSSI, R., WILD ER, D., and POPE, M. (1989): 'Trunk muscleelectromyography and whole body vibration,' J. Biomech., 23, pp.219-229WIKSTROM,B. O. (1993): 'Effects from twisted posture and whole-body vibration during driving,' Int..I. Industrial Ergonomics, 12,pp. 61-75WIKSTROM, B. O., KJELLBERG, A., and LANDSTROM, U. (1994):'Health effects of long-term occupational exposure to whole-bodyvibration: A review,' Int. J. Industrial Ergonomics, 14, pp. 273-292WILDER, D. G., and POPE, M. H. (1996): 'Epidemiological andaetiological aspects of low back pain in vibration

envi ronm ents --an update,' Clin. Biomech., 11, pp. 61-73WITTENBURG, J. (1977): 'Dynamics of systems of rigid bodies,'(LeitfMen der angewandten Mathematik und Mechanik, Bd. 33)Teubner, StuttgartYAMADA, H. (1973): 'Strength of biological materials' (R.E. KriegerPubl. Co., Huntington, New York)ISO 2631-1: 'Mechanical vibration and shoc k--Gu ide to the evalua-tion of human exposure to whole-body vibrat ion --Part 1: Generalrequirements' (1993)ISO 5982: 'Vibration and shock--Mechanical driving point impe-dance of the human body' (1981)ISO 7962: 'Mechanical vibration and shock--Mechanical transmis-sibility of the human body in the z-direction' (1987)Author's biography

MARTIN FRITZ was born in Hamm, Germany, in1949. He received his Dipl.-Ing. and Dr.-Ing.Degrees in Mechanical Engineering in 1974 and1979, respectively, from the Ruhr-UniversitgtBochum. From 1974 to 1982 he was with theAbteilung ftir Funktionelle Morphologie, Ruhr-Universitgt Bochum. Since 1984, he has beenworking at the Institut ftir Arbeitsphysiologie ander Universit~it Dortmund. His main researchinterests include the biomechanics of the musculoskeletal system,the simulation of rapid human movements and vibration inducedhealth risk.

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