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Determination of muscle, ligament and articular forces at the knee during a simulate skating thrust Halliwell, Albert A. 1977

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DETERMINATION OF MUSCLE, LIGAMENT AND ARTICULAR FORCES AT THE KNEE DURING A SIMULATED SKATING THRUST by ALBERT A. HALLIWELL B . E n g . , C l a r k s o n C o l l e g e o f Techno logy , I963 M . E n g . , M c G i l l U n i v e r s i t y , I967 B . P . E . , U n i v e r s i t y o f B r i t i s h Co lumbia , 197^ A THESIS SUBMITTED IN PARTIAL 'FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF PHYSICAL EDUCATION i n THE FACULTY OF GRADUATE STUDIES (Schoo l o f P h y s i c a l E d u c a t i o n arid R e c r e a t i o n ) We accept t h i s t h e s i s as conforming to the r e q u i r e d s tandard THE UNIVERSITY OF BRITISH COLUMBIA September, 1977 A l b e r t A l e x a n d e r H a l l i w e l l , 1977 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the requirements fo r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e fo r reference and study. I f u r t h e r agree t h a t permiss ion fo r e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . It i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l ga in s h a l l not be a l lowed without my w r i t ten pe rm i ss i on . Department of P h y s i c a l Educat ion and Recrea t ion The U n i v e r s i t y of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date September 22nd. 1Q77 ABSTRACT A number o f i n v e s t i g a t o r s have determined the j o i n t f o r c e s a c t i n g at the h i p and knee f o r normal human l o c o -m o t i o n as r e l a t e d to the d e s i g n o f p r o s t h e t i c d e v i c e s . T h i s r e s e a r c h has been extended to a l l o w the c a l c u l a t i o n or e s t i -m a t i o n o f the muscu lar and l igamentous f o r c e s o p e r a t i n g at the knee j o i n t f o r normal w a l k i n g . The c u r r e n t s tudy expanded upon the p a s t r e s e a r c h to e v a l u a t e the magnitude and tempora l sequence o f . t h e musc le , l i g a m e n t and a r t i c u l a r f o r c e s a c t i n g a t the knee j o i n t f o r a s i m u l a t e d s k a t i n g t h r u s t . A s k i l l e d i c e hockey p l a y e r was f i l m e d i n two r e f e r -ence p l a n e s w h i l e making a s k a t i n g t h r u s t from a l a b o r a t o r y f o r c e p l a t f o r m . The c i n e f i l m d a t a was s y n c h r o n i z e d w i t h the f o r c e p l a t e output to a l l o w c a l c u l a t i o n o f the o r t h o -g o n a l f o r c e s and moments imposed on the knee j o i n t . The o r t h o g o n a l f o r c e system was determined from a knowledge o f the i n e r t i a l , g r a v i t a t i o n a l and r e a c t i o n f o r c e s a c t i n g on the lower l i m b d u r i n g the s k a t i n g t h r u s t . The musc le , l i g a m e n t and j o i n t f o r c e s were determined from equat ions d e r i v e d from the c o n d i t i o n s o f j o i n t e q u i l i b r i u m . The e q u a t i o n s o f e q u i l i b r i u m were i n d e t e r m i n a t e and had to be reduced by making assumptions from e l e c t r o m y o g r a p h i c r e c o r d s to allow solution. Forces were calculated for a si m p l i f i e d muscle and ligament system which included the hamstrings, quadriceps and gastrocnemius muscle groups, the c o l l a t e r a l ligaments and the cruciate ligaments of the knee j o i n t . In addition, the a r t i c u l a r j o i n t force, joint torque and centre of pressure of the joi n t force were determined. Results of the inv e s t i g a t i o n revealed that the magni-tude of the muscle, ligament and j o i n t forces developed i n a skating thrust were considerably greater than respective forces exerted during l e v e l walking while the temporal sequence of the skating forces was comparable to walking upstairs. The quadriceps muscle group exerted the greatest c o n t r a c t i l e force while the gastrocnemius and hamstrings groups developed much smaller forces. The largest ligament forces were developed i n the c o l l a t e r a l ligaments and the posterior cruciate ligament to maintain s t a b i l i t y of the jo i n t . The knee jo i n t i s subject to the combined effects of a j o i n t force six times body weight and a large joint torque superimposed upon each other during the skating thrust and th i s fact i s considered important when discussing the cause of menisci knee i n j u r i e s . i v TABLE OF CONTENTS CHAPTER Page I . INTRODUCTION 1 I I . REVIEW OF LITERATURE 7 I I I . METHODS AND PROCEDURES , 19 Anatomy o f the Knee J o i n t 19 Concepts o f A n a l y s i s 27 Anthropometry 35 T e s t i n g P r o c e d u r e s 39 A n a l y s i s ^9 I V . RESULTS AND DISCUSSION 73 R e s u l t s 73 D i s c u s s i o n 89 V. SUMMARY AND CONCLUSIONS 106 REFERENCES 110 APPENDICES 117 V LIST OF TABLES TABLE Page 1. A n t h r o p o m e t r i c Data 37 2 . A n t h r o p o m e t r i c Data 7k 3. Maximum M u s c l e , Ligament and. J o i n t F o r c e s f o r the S i m u l a t e d S k a t i n g T h r u s t i . . . . 90 k. The Maximum Musc le and Ligament F o r c e s o f V a r i o u s A c t i v i t i e s . . . . ; 102 5. Maximum J o i n t F o r c e s and Torques o f V a r i o u s A c t i v i t i e s lO J^-v i LIST OF "FIGURES FIGURE Page 3 .01 Right Knee J o i n t - Bone S t r u c t u r e '. 20 3 .02 Superior Surface - T i b i a 22 3 .03 Ligaments of Knee J o i n t 22 3.04 Ligaments of Knee J o i n t 24 3 .05 P a t e l l a r Ligament 24 3 .06 Muscles C o n t r o l l i n g Knee J o i n t 26 3 .07 T i b i a l Reference Axes 28 3.08 T i b i a l Condyles 28 3 .09 Reference Axes - Femur, T i b i a 3 ° 3 .10 E r r o r i n Assumed -Centre of Ro t a t i o n of Femur i n 9 0 ° F l e x i o n 31 3 .11 P e l v i c Reference Axes 32 3 .12 S i m p l i f i e d Muscle and Ligament System A c t i n g at Knee J o i n t 35 3 .13 Angle of P a t e l l a r Ligament R e l a t i v e to Angle of Knee F l e x i o n 38 3.14 L o c a t i o n of Reference Markers 41 3 .15 Experimental Set-Up 43 3 .16 Force Record of Force P l a t e Expressed as S i x Measured V a r i a b l e s 44 3 .17 Experimental Set-Up Showing Test Subject on Force P l a t f o r m 47 3 .18 A n a l y s i s of Data 50 3 .19 E x t e r n a l Force System at Knee Expressed i n Terms of T i b i a l Reference Axes of Right Knee 52 3 .20 Forces A c t i n g on Lower Limb I n c l u d i n g Reaction Forces, G r a v i t y Forces and A c c e l e r a t i o n Forces 53 3 .21 Muscle Force i n Quadriceps to Balance Moment + Mxk " 58 3 .22 Muscle Force i n Hamstrings to Balance Moment - Mxk , 58 3 .23 Muscle Force i n Gastrocnemius to Balance Moment - Mxk 62 3 .24 Force i n P o s t e r i o r C r u c i a t e and A n t e r i o r C r u c i a t e Ligaments. . . . 64 3 .25 C o l l a t e r a l Ligament Forces 67 3 .26 Compressive J o i n t Force, Rz 69 4 . 0 1 Force P l a t e Output i n Form of Os c i l l o s c o p e Trace - T r i a l No. 4 76 v i i 4 . 0 2 E lec tromyogram - S k a t i n g T h r u s t on Ice 77 4 . 0 3 I l l u s t r a t e d Sequence o f S k a t i n g T h r u s t 78 4 . 0 4 Limb L i n e a r A c c e l e r a t i o n s 80 4 . 0 5 Limb A n g u l a r A c c e l e r a t i o n s 81 4 . 0 6 R e s o l v e d F o r c e Components a t Knee T r i a l No. 4 . . 83 4 . 0 7 Musc le and Ligament F o r c e s . f o r T r i a l No. 4 85 4 . 0 8 Musc le and Ligament F o r c e s 86 4 . 0 9 J o i n t F o r c e s A c t i n g at Knee 88 4 . 1 0 C e n t r e o f P r e s s u r e on Condy le s o f Knee 99 v i i i NOTATIONS Reference Axes X g , Yg , Zg G r i d r e f e r e n c e axes , o r i g i n a t c e n t r e o f f o r c e p l a t e , p o i n t p . X s , Y s , Zs Reference axes o f t i b i a , o r i g i n a t c e n t r e o f t i b i a l c o n d y l e s , p o i n t k. X f , Y f , Z f Reference axes o f femur, o r i g i n a t c e n t r e o f f e m o r a l c o n d y l e s , p o i n t f c . Xp, Yp, Zp Reference axes o f p e l v i s , o r i g i n a t c e n t r e o f f e m o r a l head, p o i n t f h . C o - o r d i n a t e s The c o - o r d i n a t e s o f a p o i n t , p , r e l a t i v e to a se t o f r e f e r e n c e axes , f o r example, the g r i d r e f e r e n c e axes, are expressed as Xgp, Ygp, and Zgp. P o i n t s a Ankle j o i n t c e n t r e . k Knee j o i n t c e n t r e a t c e n t r e o f s u p e r i o r s u r f a c e o f t i b i a , f c Centre o f f emora l c o n d y l e s . f h Centre o f f e m o r a l head o f h i p j o i n t . Musc le and Ligament Attachments h I n s e r t i o n o f h a m s t r i n g s , g I n s e r t i o n o f g a s t r o c n e m i u s . i x q I n s e r t i o n o f p a t e l l a r l i g a m e n t ( q u a d r i c e p s f e m o r i s ) . 1 L a t e r a l c o l l a t e r a l attachment to f i b u l a . m M e d i a l c o l l a t e r a l attachment to t i b i a . a A n t e r i o r c r u c i a t e attachment to t i b i a . p P o s t e r i o r c r u c i a t e attachment to t i b i a . ph O r i g i n o f h a m s t r i n g s . !•'.-.. f g O r i g i n o f g a s t r o c n e m i u s . f l L a t e r a l c o l l a t e r a l attachment to femur. fm M e d i a l c o l l a t e r a l attachment to femur. f a A n t e r i o r c r u c i a t e attachment to femur. fp P o s t e r i o r c r u c i a t e attachment to femur. F o r c e A c t i o n s F x i , F y i , F z i Components o f f o r c e a c t i n g i n d i r e c t i o n s o f X , Y , a n d Z r e f e r e n c e axes r e s p e c t i v e l y . M x i , M y i , M z i Components o f moment a c t i n g i n X, Y and Z r e f e r e n c e axes r e s p e c t i v e l y . Rx, Ry, Rz Components o f j o i n t f o r c e a c t i n g a t the knee j o i n t . Pa , Pp, Pm F o r c e a c t i o n s i n a n t e r i o r c r u c i a t e , p o s t e r i o r and PI c r u c i a t e , m e d i a l c o l l a t e r a l and l a t e r a l c o l -l a t e r a l l i g a m e n t s r e s p e c t i v e l y . Pq, Ph and F o r c e a c t i o n s i n q u a d r i c e p s f e m o r i s , ham-Pg s t r i n g s and gas trocnemius muscle groups r e s p e c t i v e l y . X Fxm, Fym, Fz.m Components of force acting i n muscle groups i n the Xs, Ys and Zs directions respectively. Fxcr, Fycr Components of force acting i n the cruciate and Fzcr ligaments i n the Xs, Ys and Zs directions respectively. Fxcol, Fycol Components of force acting i n the c o l l a t e r a l and Fzcol ligaments i n the Xs, Ys and Zs directions respectively. Symbols , ] g Acceleration of gravity, bw Body weight of subject, wi Weight of body segment i . I i Moment of i n e r t i a of segment i about axis perpendicular to the long axis of the segment and passing through the centre of mass of the segment. t Thigh segment. s Shank segment. f Foot segment. Ri Radius of segment or joint i . L i Length of body segment i . r i Radius of gyration of segment i about the axis perpen-dicul a r to the long axis of the segment and passing ^ through the centre of mass of the segment. LFO Left foot o f f during skating thrust. PO Push o f f of ri g h t foot i n skating thrust. x i ACKNOWLEDGEMENTS The author i s i n d e b t e d to s e v e r a l people f o r t h e i r guidance and a s s i s t a n c e i n c o m p l e t i n g t h i s i n v e s t i g a t i o n . To these i n d i v i d u a l s I would l i k e to express my s i n c e r e a p p r e c i a t i o n . D r . Ted Rhodes, my committee cha irman, p r o v i d e d b o t h i n s p i r a t i o n a l and monetary a s s i s t a n c e towards the r e s e a r c h s tudy and made h i m s e l f a v a i l a b l e f o r many hours o f h e l p f u l c o n s u l t a t i o n . To the o t h e r members o f my t h e s i s committee, D r . Merv O l s o n , D r . Bob Hindmarch and D r . D o r i s M i l l e r , I extend a s p e c i a l thanks . I n p a r t i c u l a r , to D r . M i l l e r who s u p e r v i s e d the t e s t i n g a t the U n i v e r s i t y o f Washington and gave i n v a l u a b l e a d v i c e r e g a r d i n g c inematography t e c h n i q u e s . A f u r t h e r thanks i s expressed to the t e s t s u b j e c t , Grant C u m b e r b i r c h , f o r h i s p a r t i c i p a t i o n i n the s tudy , to T e r r y S c h u l t z and Bruce Goldsmid f o r t h e i r h e l p i n the d a t a c o l l e c t i o n and to Mr . Hsu f o r h i s t e c h n i c a l a d v i c e and development o f t e s t i n g equipment. The i n v e s t i g a t o r i s a l s o i n d e b t e d to the a s s i s t a n c e o f M r . Frank Maurer o f the I n s t i t u t e o f Animal Resource E c o l o g y at U . B . C . i n making a v a i l a b l e f i l m r e d u c t i o n e q u i p -ment. F i n a l l y , the author g r a t e f u l l y a p p r e c i a t e s the a i d o f D r . .James M o r r i s o n who p r o v i d e d a copy o f h i s d o c t o r a l t h e s i s . x i i The author i s further indebted to Elizabeth Orne f o r the typing of t h i s manuscript. CHAPTER I INTRODUCTION The s tudy o f human l o c o m o t i o n over the p a s t s e v e r a l decades has "been approached from two separate d i r e c t i o n s . D i l l m a n (1970) and o t h e r s who were i n t e r e s t e d i n u n d e r -s t a n d i n g accompl i shed a t h l e t i c performance s t u d i e d the t i m e -d i s p l a c e m e n t o f the lower l i m b s u t i l i z i n g c i n e m a t o g r a p h i c t e c h n i q u e s . D i l l m a n was s p e c i f i c a l l y concerned w i t h the a c t i o n o f the lower l i m b d u r i n g the r e c o v e r y phase o f r u n n i n g and determined l i m b v e l o c i t y and a c c e l e r a t i o n d a t a which a l l owed an i n depth a n a l y s i s o f the t empora l sequence o f r u n n i n g . A second group o f r e s e a r c h e r s were more i n t e r e s t e d i n examining human l o c o m o t i o n from a b i o m e c h a n i c a l and "bio-e n g i n e e r i n g p o i n t o f v iew. B r e s l e r ( 1 9 5 0 ) , M o r r i s o n (1968) and P a u l (1964) de termined the f o r c e s deve loped at the j o i n t s o f the lower l i m b s d u r i n g w a l k i n g . These f o r c e - t i m e a n a l y s e s were i n i t i a t e d to s tudy the b i o m e d i c a l problems o f persons f i t t e d w i t h p r o s t h e t i c d e v i c e s such as a r t i f i c i a l l i m b s and rep lacement h i p j o i n t s . W i t h the a i d o f s o p h i s t i -c a t e d f o r c e measur ing d e v i c e s t h a t a c c u r a t e l y r e c o r d r e a c t i o n f o r c e s , these r e s e a r c h e r s c a l c u l a t e d the j o i n t f o r c e s a c t i n g a t the a n k l e , knee and h i p j o i n t s d u r i n g the w a l k i n g c y c l e from h e e l s t r i k e to toe o f f . These s t u d i e s i n d i c a t e d t h a t - 2 -the j o i n t f o r c e s imposed at the knee and h i p d u r i n g normal l e v e l w a l k i n g were equa l to t h r e e to f o u r t imes the body-weight . M o r r i s o n (1970) extended the s tudy o f human l o c o -m o t i o n to i n c l u d e the c a l c u l a t i o n o f muscle and l i g a m e n t f o r c e s deve loped at the knee j o i n t f o r l e v e l w a l k i n g , i n -c l i n e d w a l k i n g and s t a i r c l i m b i n g . I n a r e c e n t s tudy , Chao (1973) measured the v e r t i c a l r e a c t i o n f o r c e , the l a t e r a l and p o s t e r i o r r e a c t i o n f o r c e s and the t w i s t i n g moment o f the p u s h i n g f o o t o f a hockey p l a y e r wear ing s k a t e s . These r e a c t i o n f o r c e s were r e c o r d e d f o r a s i m u l a t e d e n e r g e t i c push o f f a c t i o n w i t h a f o r c e p l a t e . The p r e s e n t s tudy was des igned to extend the work o f Chao u t i l i z i n g the methodology o f M o r r i s o n ( I 9 6 8 ) to examine the magnitude o f the j o i n t , muscle and l i g a m e n t f o r c e s i m -posed a t the knee d u r i n g a s i m u l a t e d s k a t i n g t h r u s t . Because o f the p o w e r f u l na ture o f the s k a t i n g t h r u s t i t i s c o n s i d e r e d t h a t the j o i n t and l i g a m e n t f o r c e s deve loped at the knee are q u i t e s i g n i f i c a n t compared to the r e s p e c t i v e f o r c e s o f normal l o c o m o t i o n . The i n v e s t i g a t i o n i s t h e r e f o r e a f a c t f i n d i n g and d e s c r i p t i v e s tudy o f a s p e c i f i c a t h l e t i c movement, the s k a t i n g t h r u s t . A knowledge o f the f o r c e s a c t i n g d u r i n g t h i s manoeuvre are i m p o r t a n t i n terms o f the f u n c t i o n a l anatomy o f the knee j o i n t and c o u l d a s s i s t the p r a c t i t i o n e r or t h e r a p i s t i n judgements c o n c e r n i n g the s p e c i f i c f o r c e s a f f e c t i n g the s t a b i l i t y o f the knee d u r i n g the game o f i c e hockey . - 3 -S i g n i f i c a n c e o f the Study To d a t e , no d a t a has been p u b l i s h e d r e g a r d i n g the magnitude o f lower l i m b j o i n t f o r c e s i n c l u d i n g m u s c l e , l i g a -ment and a r t i c u l a r f o r c e s deve loped d u r i n g an exer ted a t h -l e t i c movement. A l s o , the methodology o f d e t e r m i n i n g the above f o r c e s from a knowledge o f f o r c e p l a t e r e a c t i o n f o r c e s ( M o r r i s o n , I968) was deve loped f o r l o c o m o t i o n i n a s i n g l e p l a n e , i . e . , the a n t e r i o r - p o s t e r i o r p l a n e . The c u r r e n t s tudy has extended the methodology so t h a t i t i s more g e n e r a l and can be a p p l i e d to a t h l e t i c s k i l l s such as s k a t i n g which have more than one p lane o f movement. Statement o f the Problem The purpose o f the i n v e s t i g a t i o n i s to determine the a r t i c u l a r and l i gamentous f o r c e s imposed a t the knee j o i n t d u r i n g a s i m u l a t e d s k a t i n g t h r u s t by an i c e hockey p l a y e r . D e f i n i t i o n o f Terms The f o l l o w i n g l i s t o f d e f i n i t i o n o f terms i s e s s e n t i a l to a l l o w i n t e r p r e t a t i o n o f the somewhat t e c h n i c a l t e x t o f the r e p o r t . A r t i c u l a r f o r c e s - the compress ive and shear f o r c e s a c t i n g at the a r t i c u l a r s u r f a c e s o f a j o i n t . F o r the knee , they r e p r e s e n t the f o r c e s imposed on the c o n d y l e s o f the t i b i a and the femur o f the j o i n t . Ligamentous f o r c e s - the t e n s i l e f o r c e s a c t i n g i n the l i g a -merits t h a t suppor t a j o i n t . F o r the knee, these f o r c e s a c t i n the c o l l a t e r a l and c r u c i a t e l i g a m e n t s . T ime-space c o - o r d i n a t e s - the l o c a t i o n o f a body segment or a p o i n t o f t h a t segment w i t h t ime as d e f i n e d by the x, y and z c o - o r d i n a t e s . F o r the lower l i m b these c o - o r d i n a t e s r e p -r e s e n t d i s p l a c e m e n t o f the c e n t r e o f mass o f the f o o t and shank and the d i s p l a c e m e n t o f the j o i n t c e n t r e s o f t h e ' . h i p , knee and a n k l e . R e a c t i o n f o r c e s - the equa l and o p p o s i t e f o r c e s e x e r t e d on a body by a r e s i s t i n g body or s u r f a c e a t the p o i n t o f c o n t a c t w i t h t h a t body or s u r f a c e . I n t h i s s tudy the r e a c t i o n f o r c e s are exer ted on the skate b lade by the f o r c e p l a t e s u r f a c e and r e c o r d e d by the f o r c e p l a t f o r m . I n e r t i a l f o r c e s - the p l a n a r f o r c e s deve loped by the l i n e a r a c c e l e r a t i o n o f a body segment as determined by Newtonian p r i n c i p l e s o f mass t imes a c c e l e r a t i o n . The i n e r t i a l f o r c e s o f the s tudy r e p r e s e n t the p r o d u c t o f the l i n e a r a c c e l e r a t i o n o f the c e n t r e o f mass o f the f o o t or shank and the r e s p e c t i v e mass o f the segment. I n e r t i a l t o r q u e s - the moment determined from the p r o d u c t o f the a n g u l a r a c c e l e r a t i o n o f the f o o t and shank and the r e s -p e c t i v e moment o f i n e r t i a o f the segment about the x, y and z axes . - 5 -S k a t i n g t h r u s t - s k a t i n g has t h r e e components: the t h r u s t , the weight s h i f t to the g l i d e l e g and the g l i d e on t h a t l e g . A s k a t i n g t h r u s t r e p r e s e n t s the e x t e n s i o n o f the t h r u s t l e g to d r i v e the s k a t e r f o r w a r d . L i m i t a t i o n s The knee j o i n t f o r c e s c a l c u l a t e d i n the s tudy were f o r a s i m u l a t e d s k a t i n g t h r u s t . The skate b lade to f o r c e p l a t e c o n t a c t does not e x h i b i t the same f r i c t i o n a l c h a r a c -t e r i s t i c s o f the a c t u a l skate b l a d e to i c e c o n t a c t o f the normal s k a t i n g environment . T h e r e f o r e , the c a l c u l a t e d j o i n t f o r c e s r e p r e s e n t those f o r c e s c o r r e s p o n d i n g to the r e a c t i o n f o r c e s as r e c o r d e d i n the l a b o r a t o r y by the f o r c e p l a t f o r m measurement t e c h n i q u e . To deve lop the r e q u i r e d f r i c t i o n a l r e s i s t a n c e f o r c e at the s u r f a c e o f the f o r c e p l a t f o r m the s k a t e r was r e q u i r e d to wear s p e c i a l l y p r e p a r e d r u b b e r skate b lade c o v e r i n g s . T h i s procedure i n t r o d u c e d f u r t h e r e r r o r i n t o the l a b o r a t o r y e s t imate o f the a c t u a l r e a c t i o n f o r c e s o f a s k a t i n g t h r u s t executed on i c e . The r e s u l t s o f the s tudy are l i m i t e d to the p a r t i -c u l a r s k a t i n g t h r u s t o f one s k i l l e d hockey p l a y e r . The n a t u r e o f the e x p e r i m e n t a l s e t - u p r e q u i r e d t h a t the s k a t i n g t h r u s t be executed from a s t a r t i n g s t a t i o n a r y p o s i t i o n . D e l i m i t a t i o n s and Assumptions The f o l l o w i n g assumptions were made f o r the purpose - 6 -o f f o r c e a n a l y s i s o f the knee j o i n t f o r c e s ( M o r r i s o n , 1968) : ( i ) R o t a t i o n at the knee i s c o n t r o l l e d by the f o r c e s i n t h r e e major muscle groups , the q u a d r i c e p s f e m o r i s , h a m s t r i n g s and gas trocnemius groups . The muscles o f each group are c o n s i d e r e d to have -conf luent l i n e s o f a c t i o n and t h e • d i r e c t i o n o f the r e s u l t a n t f o r c e i n each group i s assumed to be c o n s t a n t w i t h i n -c r e a s i n g f o r c e , ( i i ) The l i n e o f a c t i o n o f a muscle group or l i g a m e n t i s assumed to be c o i n c i d e n t w i t h the l i n e j o i n i n g i t s o r i g i n and i n s e r t i o n , ( i i i ) The a x i s o f r o t a t i o n o f the femur i s assumed to be c o n s t a n t r e l a t i v e to the c o - o r d i n a t e axes o f the t i b i a . ( i v ) The e f f e c t s o f f r i c t i o n a t the a r t i c u l a r s u r f a c e o f the knee j o i n t i s n e g l e c t e d f o r the a n t e r i o r -p o s t e r i o r movement o f the j o i n t . To a l l o w a n a l y s i s o f the e x t e r n a l f o r c e s imposed on the knee j o i n t the s i m p l i f i e d muscle and l i g a m e n t system shown i n F i g u r e 3-12 was adopted . - 7 -CHAPTER I I REVIEW OF LITERATURE I n t r o d u c t i o n The j o i n t s o f the lower l i m b s p r o v i d e the s t r u c t u r a l means f o r human l o c o m o t i o n i n a l l a c t i v i t i e s . Wi th the a i d o f muscu lar a c t i o n s the human body can p e r f o r m a m u l t i t u d e o f c o - o r d i n a t e d l o c o m o t i v e movements through the a r t i c u l a t i n g j o i n t s . However these movements and a c t i v i t i e s n e c e s s a r i l y s u b j e c t the j o i n t s and the s u p p o r t i n g l i gamentous s t r u c t u r e s to f o r c e s t h a t must be e q u i l i b r a t e d . The na ture o f a t h l e t i c s i n c r e a s e s the e f f e c t o f these f o r c e s on the j o i n t s o f the lower l i m b and s p e c i f i c a l l y the knee j o i n t as p o i n t e d out by Hughston, What ley and D o d e l i n (1961) and K l e i n ( 1 9 6 3 ) . The p r e v a l e n c e o f knee i n j u r i e s has been documented by s e v e r a l i n v e s t i g a t o r s such as Bender (1964) and S m i l l i e (1970) w h i l e the f u n c t i o n o f the knee j o i n t i n v a r i o u s a c t i v i t i e s o f l o c o -mot ion has been s t u d i e d and d e t a i l e d by M o r r i s o n ( 1 9 6 9 ) . Locomot ion S t u d i e s W a l k i n g i s the most common o f human l o c o m o t i v e a c t -i v i t i e s and was i n i t i a l l y i n v e s t i g a t e d by E l f t m a n (193&\ 1939) who examined the e x t e r n a l f o r c e s e x e r t e d , the energy . changes i n the l e g and the f u n c t i o n o f the muscles i n - 8 -w a l k i n g . E l f t m a n employed an e x p e r i m e n t a l t e c h n i q u e t h a t determined ground r e a c t i o n f o r c e s from the c a l i b r a t e d d e f l e c -t i o n s o f a f o r c e p l a t e supported by s t i f f s p r i n g s . B r e s l e r and F r a n k e l (1950) measured the o r t h o g o n a l r e a c t i o n f o r c e s and moments o f w a l k i n g u s i n g a f o r c e p l a t e based on the f o r c e - t i m e measur ing p r o p e r t i e s o f s t r a i n gauges and des igned by Cunningham and Brown ( 1 9 5 2 ) . The r e a c t i o n f o r c e s and moments t o g e t h e r w i t h c i n e f i l m r e c o r d s o f the p o s i t i o n o f the l e g i n space a l l owed B r e s l e r and F r a n k e l to e v a l u a t e the j o i n t f o r c e s t r a n s m i t t e d a t the a n k l e , knee and h i p d u r i n g l e v e l w a l k i n g . Cinematography has p l a y e d an i m p o r t a n t r o l e i n the s tudy o f l o c o m o t i o n . M u r r a y , Drought and K o r y (196^-) i n v e s t i g a t e d the d i s p l a c e m e n t o f the lower l i m b s i n the w a l k i n g p a t t e r n s o f normal men w h i l e W i n t e r (197^) has s t u d i e d the k i n e m a t i c s o f normal l o c o m o t i o n from TV d a t a . The development o f e l ec tromyography as a r e s e a r c h t o o l by r e s e a r c h e r s such as Basmaj ian ( I 9 6 2 ) has a l l o w e d i n -v e s t i g a t o r s , to s tudy the f u n c t i o n o f musc les and muscle groups i n l o c o m o t i o n . Joseph and N i g h t i n g a l e (1952) and Houtz and Walsh (1959) a n a l y z e d the f u n c t i o n o f the muscles o f the lower e x t r e m i t y d u r i n g weight b e a r i n g and normal p o s t u r e u s i n g e l e c t r o m y o g r a p h i c t e c h n i q u e s . The p h a s i c a c t i v i t y o f the muscles o f the l e g was s t u d i e d by C l o s e and Todd ( 1959) w h i l e L i n g e (I96I) examined the s p e c i f i c behav-i o u r of. the q u a d r i c e p s muscle group d u r i n g w a l k i n g from e lec tromyograms. A l t h o u g h e l ec tromyography has been - 9 -accepted, as a means o f measur ing muscle a c t i v i t y the r e l a t i o n between the a c t i o n p o t e n t i a l s o f an e lectromyogram and muscle t e n s i o n as s t u d i e d by L i p p o l d (1952) and o t h e r s i s not w e l l e s t a b l i s h e d . B i o m e c h a n i c a l A n a l y s e s o f J o i n t F o r c e s The o r i g i n a l s t u d i e s o f j o i n t f o r c e s were c o n f i n e d to the h i p j o i n t as r e l a t e d to the d e s i g n o f endo-pros theses f o r the h i p . D i r e c t d e t e r m i n a t i o n o f h i p j o i n t f o r c e was conducted by R y d e l l (1965 , 1966) who. f i t t e d two o f h i s p a -t i e n t s w i t h h i p - j o i n t p r o t h e s e s i n s t r u m e n t e d w i t h s t r a i n gauges. The l a r g e s t f o r c e a c t i n g on the f emora l head was 4 . 3 3 t imes the body weight d u r i n g r u n n i n g w h i l e l e v e l w a l k i n g f o r c e s reached v a l u e s o f 3-3 t imes the body we ight . R a d c l i f f e (1962) and Cunningham and Brown (1952) have both i n v e s t i g a t e d the b iomechanics o f below knee p r o s t h e s e s u s i n g i n s t r u m e n t e d p r o s t h e t i c d e v i c e s . Other r e s e a r c h e r s such as W i l l i a m s and . Evenson ( 1 9 6 8 ) , Denham (1959) and Inman (19^7) de termined j o i n t f o r c e s d i r e c t l y from t e s t s on d i s s e c t e d l i m b s w i t h muscles and l i g a m e n t s removed. Dynamometers were used to e s t imate the v a l u e o f h i p j o i n t f o r c e f o r the s t a n d i n g p o s i -t i o n . M c L e i s h and C h a r n l e y (1970) de termined h i p j o i n t f o r c e s i n the one - l egged s t a n c e . The most d e f i n i t i v e a n a l y s i s o f r e a c t i o n f o r c e s a t the h i p j o i n t , d u r i n g w a l k i n g has been the r e c e n t work o f P a u l (1964, I 9 6 6 , 1967, 1971 ) . P a u l determined the v a r i a t i o n - 10 -w i t h t ime o f the magnitude o f the h i p j o i n t f o r c e d u r i n g w a l k i n g from c i n e f i l m r e c o r d s o f l e g segments and the ground r e a c t i o n f o r c e s on the f o o t measured w i t h a f o r c e p l a t e . P a u l found maximum h i p j o i n t f o r c e s r a n g i n g from 3 . 3 9 to k.k6 t imes body we ight . M o r r i s o n (1968, 1970) extended P a u l ' s s t u d i e s to an e n g i n e e r i n g a n a l y s i s o f the f o r c e s t r a n s m i t t e d between the f emora l and t i b i a l c o n d y l e s o f the knee j o i n t d u r i n g v a r i o u s w a l k i n g a c t i v i t i e s . M a x i -mum knee j o i n t f o r c e s measured v a r i e d between 2 . 0 6 and ^ . 0 t imes body weight w i t h an average v a l u e o f 3 « 0 3 - M o r r i s o n a l s o i n t e g r a t e d f o r c e p l a t e , c i n e m a t o g r a p h i c and e l e c t r o -myographic r e c o r d s to e s t imate the t e n s i o n s deve loped i n muscle groups a c t i n g a c r o s s the knee j o i n t i n c l u d i n g the p h a s i c r e l a t i o n o f these f o r c e s d u r i n g w a l k i n g . By a d o p t i n g a s i m p l i f i e d model o f the knee j o i n t M o r r i s o n expressed the mechanics o f the j o i n t i n mathemat ica l terms and was a b l e to e v a l u a t e the l i gamentous f o r c e s from the e x p e r i m e n t a l d a t a . The g r e a t e s t muscle f o r c e c a l c u l a t e d was ^ 0 5 l b a c t i n g i n the q u a d r i c e p s f e m o r i s group and the maximum l i g a m e n t f o r c e determined was 1^8 l b i n the l a t e r a l c o l l a t e r a l l i g a m e n t . . M o r r i s o n ' s s t u d i e s p r o v i d e d the f i r s t d a t a p u b l i s h e d w i t h r e s p e c t to the magnitude o f the t e n s i l e f o r c e s a c t i n g i n the l i gamentous s t r u c t u r e s o f a j o i n t d u r i n g a c t i v i t y . Recent B i o m e c h a n i c a l A n a l y s e s The advent o f computer t e c h n o l o g y l e d to new methods o f - 11 -e v a l u a t i n g l o c o m o t i o n d a t a from f i l m . B e c k e t t and Chang (1968) de termined the k i n e m a t i c s o f g a i t w h i l e D i l l m a n (1970) made a s p e c i f i c k i n e m a t i c a n a l y s i s o f the r e c o v e r y l e g d u r i n g s p r i n t r u n n i n g . These s t u d i e s made use o f computer a n a l y s i s to reduce c i n e f i l m d a t a to c a l c u l a t e the t r a n s l a t i o n a l and a n g u l a r a c c e l e r a t i o n s o f the lower l i m b segments. T h o r n t o n -Trump and Daher (1975) used a n t h r o p o m e t r i c d a t a i n c l u d i n g l i m b segment weight and mass moments o f i n e r t i a to p r e d i c t the moments a c t i n g a t the h i p and knee and the f l o o r r e a c -t i o n f o r c e s from g a i t d a t a . The r e a c t i o n f o r c e s and knee moment d a t a were then a p p l i e d to the problem o f the d e s i g n o f an e x t e r n a l p r o s t h e t i c p o l y c e n t r i c knee j o i n t . A number o f mathemat ica l models have been deve loped f o r a n a l y s i s o f human body dynamics . C h a f f i n (1969) r e p r e s -ented the body as a s e r i e s o f seven s o l i d l i n k s a r t i c u l a t i n g at the a n k l e s , knees , h i p s , s h o u l d e r s , elbows and w r i s t s to deve lop a computer i zed model f o r s e v e r a l body movements. The model was d e s i g n e d to s tudy body movements o f l i f t i n g and c a r r y i n g but the assumptions made l i m i t e d the models p r a c -t i c a l a p p l i c a t i o n s . S e i r e g and A r v i k a r ( 1973i 1975) d e v e l -oped a mathemat i ca l model to s i m u l a t e the muscu lar a c t i o n s o f the lower e x t r e m e t i e s f o r d i f f e r e n t s t a t i c p o s t u r e s . They used a n a t o m i c a l d a t a f o r the p o i n t s o f o r i g i n and i n s e r t i o n o f musc les to approximate the l i n e s o f a c t i o n o f the mus-c u l a r t e n s i l e f o r c e s . S i n c e the mathemat ica l f o r m u l a t i o n i s s t a t i c a l l y i n d e t e r m i n a t e a l i n e a r programming method based on - 12 -a se lected, m i n i m i z i n g c r i t e r i a was used to determine the muscle l o a d s h a r i n g and c o r r e s p o n d i n g h i p , knee and ankle j o i n t r e a c t i o n s . In a subsequent s tudy S e i r e g and A r v i k a r extended t h e i r p r e v i o u s work to the a n a l y s i s o f muscu lar l o a d s h a r i n g and j o i n t r e a c t i o n s f o r q u a s i - s t a t i c w a l k i n g p a t t e r n s . Dynamic l i m b d a t a i n c l u d i n g i n e r t i a f o r c e s and moments were not i n c l u d e d i n the a n a l y s i s and the ground to f o o t r e a c t i o n f o r c e was assumed to v a r y l i n e a r l y from zero to the body we ight . S e i r e g and. A r ' v i k a r ' s c a l c u l a t e d muscle f o r c e s agreed c l o s e l y w i t h M o r r i s o n ' s (1969) r e s u l t s but t h e i r j o i n t f o r c e s were c o n s i d e r a b l y l a r g e r . Maximum j o i n t r e a c t i o n f o r c e s d u r i n g the c o n s i d e r e d q u a s i - s t a t i c w a l k i n g c y c l e were 7-1 and 5-2 t imes the body weight f o r h i p , knee and a n k l e j o i n t r e s p e c t i v e l y . Other r e s e a r c h e r s such as Chao and -Kwam (1973) have employed o p t i m i z a t i o n p r i n c i p l e s to e v a l u a t e the a p p l i e d moments i n the l e g d u r i n g w a l k i n g w h i l e Penrod , Dary and S i n g h (197^) used a s i m i l a r method to e s t imate tendon f o r c e . Mechanics o f the Knee J o i n t The most comprehensive s tudy o f the movements o f the knee j o i n t and the f u n c t i o n o f the l i g a m e n t s c o n t r o l l i n g these movements was made by B r a n t i g a n and V o s h e l l ( 1 9 ^ 1 ) . T e s t s o f f u n c t i o n and mot ion were made on a p p r o x i m a t e l y 100 f r e s h and p r e s e r v e d knee j o i n t s s t r i p p e d o f a l l p a r t s except the l i g a m e n t s . I n d i v i d u a l l i g a m e n t s and combinat ions o f - 13 -l i g a m e n t s were cu t and the f u n c t i o n o f the i n t a c t l i g a m e n t s and mot ion o f the j o i n t was o b s e r v e d . A s imple v i c e ap-p a r a t u s was used to measure the degree o f l a t e r a l mot ion and r o t a t i o n o f the t i b i a on the femur. The r e s u l t s i n d i c a t e d t h a t t h e r e i s a c l o s e i n t e r r e l a t i o n s h i p o f the l i g a m e n t s i n m a i n t a i n i n g j o i n t s t a b i l i t y which can be summarized as f o l l o w s : (1) L a t e r a l mot ion i n e x t e n s i o n i s c o n t r o l l e d by the c a p s u l e , c o l l a t e r a l l i g a m e n t s and c r u c i a t e l i g a m e n t s ; i n f l e x i o n by the same s t r u c t u r e s minus the l a t e r a l c o l l a t e r a l . (2) R o t a r y mot ion i n e x t e n s i o n i s c o n t r o l l e d by the c a p -s u l e , c o l l a t e r a l l i g a m e n t s and c r u c i a t e l i g a m e n t s ; i n f l e x i o n by the same s t r u c t u r e s minus the l a t e r a l c o l l a t e r a l . (3) Forward g l i d i n g o f the t i b i a on the femur i s c o n -t r o l l e d by the a n t e r i o r c r u c i a t e l i g a m e n t . (4) Backward g l i d i n g o f the t i b i a on the femur i s c o n -t r o l l e d by the p o s t e r i o r c r u c i a t e l i g a m e n t . (5) L a t e r a l g l i d i n g o f the t i b i a on the femur i s c o n -t r o l l e d by the t i b i a l i n t e r - c o n d y l o i d eminence and the f emora l c o n d y l e s w i t h a i d o f a l l the l i g a m e n t s . (6) H y p e r e x t e n s i o n i s c o n t r o l l e d by both the c o l l a t e r a l l i g a m e n t s , bo th c r u c i a t e l i g a m e n t s , both m e n i s c i , the p o s t e r i o r aspec t o f the a r t i c u l a r c a p s u l e , the o b l i q u e p o p l i t e a l l i g a m e n t and the a r c h i t e c t u r e - 11+ -o f the f emora l c o n d y l e s . (7) H y p e r f l e x i o n i s c o n t r o l l e d by both c r u c i a t e l i g a -ments, both m e n i s c i , the f emora l attachment o f the p o s t e r i o r a spec t o f the c a p s u l e , the f emora l attachment' o f both heads o f the gas trocnemius and the c o n d y l e s o f the femur and t i b i a . Recent s t u d i e s have examined the l e n g t h o f the l i g a -ments f o r d i f f e r e n t knee p o s i t i o n s . Edwards, L a f f e r t y and Lange (1970) a t t a c h e d gauges to the l i g a m e n t s o f a d i s s e c t e d j o i n t to measure the t e n s i o n s a t d i f f e r e n t ang les o f f l e x i o n and thereby e s t imate l i g a m e n t l e n g t h . Wang, Walker and W o l f (1973) de termined the l i g a m e n t l e n g t h p a t t e r n s f o r 12 s p e c i -mens f o r v a r i o u s ang le s o f f l e x i o n and degrees o f r o t a t i o n . The j o i n t c a p s u l e , the f o u r major l i g a m e n t s and the q u a d r i -ceps muscle t o g e t h e r w i t h p a t e l l a attachment were r e t a i n e d i n the t e s t i n g . The l e n g t h o f the l i g a m e n t s was determined from r a d i o g r a p h i c measurement o f l o n g meta l p i n s i n s e r t e d through d r i l l h o l e s i n the femur and t i b i a so t h a t the ends o f the p i n s l o c a t e d the r e s p e c t i v e l i g a m e n t a t tachments . R e s u l t s showed t h a t the c o l l a t e r a l s shortened about 20 p e r -cent i n l e n g t h i n f l e x i o n w h i l e the c r u c i a t e s a c t i o n was r e c i p r o c a l w i t h the a n t e r i o r c r u c i a t e l e n g t h e n i n g and the p o s t e r i o r c r u c i a t e s h o r t e n i n g d u r i n g f l e x i o n . The b iomechanics o f normal and abnormal knee j o i n t s was examined by E n g i n and Korde (197*0 • The- i n v e s t i g a t i o n was concerned w i t h an e x p e r i m e n t a l and t h e o r e t i c a l s tudy o f - 15 -the changes i n knee j o i n t mechanics i n d e g e n e r a t i v e j o i n t d i s e a s e . E x p e r i m e n t a l t e s t s were conducted to determine the s t r a i n a t the s t r a i n gauged c r o s s - s e c t i o n s o f a femur and t i b i a specimen f o r normal and abnormal c o n f i g u r a t i o n . S o f t t i s s u e s were d i s s e c t e d away l e a v i n g o n l y the r e m a i n i n g l i g a -ments o f the j o i n t to m a i n t a i n the j o i n t i n t e g r i t y . F o r normal j o i n t c o n f i g u r a t i o n the l a t e r a l condy le c o n t a c t f o r c e was g r e a t e r than the med ia l condy le c o n t a c t f o r c e w i t h a v a r u s d e f o r m i t y i n c r e a s i n g the med ia l c o n t a c t f o r c e and a v a l g u s d e f o r m i t y i n c r e a s i n g the l a t e r a l c o n t a c t f o r c e . The t e n s i o n s i n the c o l l a t e r a l l i g a m e n t s were found to be depen-dent on the degree o f knee a b n o r m a l i t y and not on magnitude o f a x i a l l o a d . Ket t l ekamp and Chao (1972) found s i m i l a r v a r i a t i o n i n the f o r c e d i s t r i b u t i o n w i t h v a l g u s and varus d e f o r m i t y o f the lower l i m b . E x p e r i m e n t a l T e s t i n g The s t a b i l i t y o f the knee and the e f f e c t o f v a r i o u s types o f l o a d i n g on the j o i n t has been s t u d i e d by s e v e r a l r e s e a r c h e r s . K l e i n (1962) des igned an i n s t r u m e n t f o r t e s -t i n g the m e d i a l and l a t e r a l c o l l a t e r a l l i g a m e n t s t a b i l i t y o f the knee . The d e v i c e was a t t a c h e d to the t h i g h and lower l e g o f a s u b j e c t w i t h c u f f s and a m a c h i n i s t ' s d i a l a t t a c h e d above the p i v o t p o i n t measured the': d e v i a t i o n o f the lower l e g as a t e s t e r adducted or abducted the lower l e g . In a l a t e r a r t i c l e , K l e i n ( 1 9 6 4 ) , s t a t e d t h a t the a b i l i t y o f the - 16 -t e s t to demonstrate s t a b i l i t y o r i n s t a b i l i t y o f the l i g a m e n t s o f the j o i n t was dependent upon the e x p e r i e n c e o f the t e s t e r . D e g r a d a t i o n o f the c a r t i l a g e i n the knee j o i n t has been the s u b j e c t o f r e c e n t r e s e a r c h . R a d i n (1973) s u b j e c t e d the knees o f a d u l t r a b b i t s to d a i l y i n t e r v a l s o f impulse l o a d i n g to induce c a r t i l a g e d e s t r u c t i o n s i m i l a r to c l i n i c a l l y observed d e g e n e r a t i o n i n human knees . R a d i n ' s s t u d i e s and those o f Simon, R a d i n and P a u l (1972) i n v e s t i g a t e d the e f -f e c t s o f suddenly a p p l i e d l o a d s or impact l o a d i n g . S e i r e g and G e r a t h (1975) i n v e s t i g a t e d i n v i v o , the e f f e c t s o f h i g h speed r u b b i n g o f the p a t e l l a j o i n t o f r a t s s u b j e c t e d to a c o n s t a n t s t a t i c compress ive l o a d . The r e s u l t s showed t h a t c a r t i l a g e damage i n c l u d e s an i n c r e a s e o f the s k i n temper-a t u r e o f the j o i n t , and a change i n s u r f a c e roughness , the c e l l u l a r s t r u c t u r e and m i n e r a l content o f the c a r t i l a g e . The s t r e n g t h o f l i g a m e n t s and tendons has been i n v e s -t i g a t e d i n an imal s by T i p t o n et a l . (1967 , 197*+) who measured the j u n c t i o n s t r e n g t h and l i gamentous s t r e n g t h i n r a t s and by V i i d i k (1966 , 1967) who determined the t e n s i l e s t r e n g t h o f i s o l a t e d r a b b i t t endons . A rev i ew o f l i t e r a t u r e on t h i s s u b j e c t has not r e v e a l e d any s t u d i e s r e l a t i n g to the s t r e n g t h o f the l i g a m e n t s i n man a l t h o u g h much l i t e r a t u r e has been p u b l i s h e d w i t h r e s p e c t to the s u r g i c a l t reatment o f l i g a m e n t i n j u r i e s . (O'Donoghue, 1950) - 17 -E x e r c i s e and the Knee J o i n t S e v e r a l s t u d i e s have "been made c o n c e r n i n g the e f f e c t s o f e x e r c i s e on the s t r e n g t h o f l i g a m e n t s o f the knee j o i n t . A g a i n , these i n v e s t i g a t i o n s have been c o n f i n e d to animal s t u d i e s . Adams ( 1 9 6 6 ) , Rasch e t a l . ( 1 9 6 7 ) , T i p t o n e t a l . (1967, 1975) and Zuckerman and S t u l l (1969 , 1973) a l l found t h a t the s t r e n g t h o f knee l i g a m e n t s i n r a t s i n c r e a s e d a f t e r a t r a i n i n g program o f r u n n i n g . T i p t o n et a l . (1970) made the same o b s e r v a t i o n c o n c e r n i n g the s t r e n g t h o f the m e d i a l c o l l a t e r a l l i g a m e n t o f dogs and V i i d i k (1968) found an i n -crease i n s t r e n g t h o f the a n t e r i o r c r u c i a t e l i g a m e n t i n r a b b i t s a f t e r t r a i n i n g . A l t h o u g h the v a l u e o f e x e r c i s e i n s t r e n g t h e n i n g l i g a -ments i s w e l l documented, K a r p o v i c h (1970) and K l e i n (1971) f e l t the e f f e c t o f deep knee bends or deep squat e x e r c i s e s was d e t r i m e n t a l to the l i g a m e n t s o f the knee. K l e i n and a s s o c i a t e s conducted e x t e n s i v e s t u d i e s o f such e x e r c i s e s and found t h a t f o o t b a l l p l a y e r s and weight l i f t e r s who i n c l u d e d deep squat e x e r c i s e s i n t h e i r t r a i n i n g and c o m p e t i t i o n were s u b j e c t to l i g a m e n t and knee i n s t a b i l i t y due to abnormal s t r e t c h i n g . S i m i l a r i l y , a s tudy o f p a r a t r o o p e r s , who used squat jumps i n t r a i n i n g , showed a h i g h i n c i d e n c e o f knee j o i n t i n s t a b i l i t y , e s p e c i a l l y i n the m e d i a l and l a t e r a l l i g a m e n t s . - 18 -Summary The rev i ew o f l i t e r a t u r e revealed, t h a t the biomechan-i c s ' o f the knee j o i n t ( B r a n t i g a n and V o s h e l l , 1941) are so i n v o l v e d to make a thorough a n a l y s i s o f the f o r c e s a c t i n g a t the j o i n t and w i t h i n the s t r u c t u r e s o f the j o i n t v e r y d i f -f i c u l t . The b i o m e c h a n i c a l a n a l y s e s to date ( P a u l , 1964, 1966; M o r r i s o n , 1968, 1970; S e i r e g and A r v i k a r , 1973. 1975) have t h e r e f o r e n e c e s s a r i l y i n v o l v e d p r o c e d u r e s which r e q u i r e d r e d u c t i o n o f i n d e t e r m i n a n c y through assumptions o f m i n i m i -z i n g p r o c e d u r e s and l i n e a r programming. There i s a l s o a p a u c i t y _ o f r e s e a r c h w i t h r e g a r d s the s t r e n g t h o f the l i g a m e n t s o f the knee i n man a l t h o u g h some t e s t i n g methods have been deve loped ( K l e i n , 1962) to e v a l u a t e the s t a b i l i t y o f the knee o f t e s t s u b j e c t s . . S i m i l a r s t a t e -ments can be made about the f u n c t i o n and p h y s i c a l p r o p e r t i e s o f the c a r t i l a g e s t r u c t u r e o f the knee. Human l o c o m o t i o n appears to have been t h o r o u g h l y i n -v e s t i g a t e d i n terms o f k i n e m a t i c s , g a i t p a t t e r n s , p r o s t h e t i c d e v i c e s , j o i n t f o r c e s and muscle invo lvement but t h e r e remains a ' l a c k o f i n f o r m a t i o n d i r e c t l y r e l a t e d to the f o r c e s and s t r a i n s imposed on the j o i n t s o f the lower e x t r e m i t i e s o f the "body i n the more dynamic l o c o m o t i v e a c t i v i t i e s o f a t h l e t i c s . - 19 -CHAPTER I I I METHODS AND PROCEDURES Anatomy o f the Knee J o i n t I n t r o d u c t i o n - • The knee j o i n t i s a s y n o v i a l c o n d y l a r h inge j o i n t t h a t a l l o w s f l e x i o n and e x t e n s i o n i n the s a g i t t a l p lane w h i l e r e s t r i c t i n g v a l g u s and v a r u s r o t a t i o n about the l o n g a x i s . The p o s i t i o n and s t r u c t u r e o f the knee makes i t an i m p o r t a n t j o i n t i n both weight b e a r i n g and l o c o m o t i o n . The s t a b i l i t y and e q u i l i b r i u m o f the knee i s m a i n t a i n e d by the s u p p o r t i n g s t r u c t u r e o f muscles and l i g a m e n t s c r o s s i n g the j o i n t . The s y n o v i a l c a p s u l e and the m e n i s c i deepened a r t i -c u l a t i n g s u r f a c e s between the femur and t i b i a a l s o a i d i n j o i n t s t a b i l i t y . R e l a t e d knee s t r u c t u r e s are the f i b u l a and the p a t e l l a . Bone s t r u c t u r e The femur and t i b i a are the weight b e a r i n g bones o f the l e g and t h e r e f o r e are d i r e c t l y a s s o c i a t e d w i t h j o i n t f u n c t i o n w h i l e the non-weight b e a r i n g f i b u l a and p a t e l l a are o n l y i n d i r e c t l y i n v o l v e d w i t h knee f u n c t i o n . ( F i g u r e 3-01) The knee j o i n t has three a r t i c u l a t i o n s . The lower end o f the femur has two c y l i n d r i c a l shaped c o n d y l e s which FIG. 3.01 RIGHT KNEE JO I NT - BONE STRUCTURE - 21 -t r a n s m i t the "body weight to the c i r c u l a r concave c o n d y l e s o f the upper s u r f a c e o f the t i b i a . ( F i g . 3 -02) These are r e f e r -r e d to as the med ia l and l a t e r a l c o n d y l a r a r t i c u l a t i o n s and are s e p a r a t e d by a n o n - a r t i c u l a t i n g i n t e r c o n d y l a r n o t c h . The l a r g e sesamoid p a t e l l a bone i s embedded i n the q u a d r i -ceps tendon and a r t i c u l a t e s w i t h the a n t e r i o r s u r f a c e ' o f the f emora l c o n d y l e s to form the p a t e l l a r a r t i c u l a t i o n . ( F i g . 3 .01 ) The f i b u l a , the l a t e r a l bone o f the l e g , i s not a f u n c t i o n i n g p a r t o f the j o i n t i t s e l f but r a t h e r p r o v i d e s s i t e s f o r muscle and l i g a m e n t a t tachments . Ligamentous S t r u c t u r e The s t a b i l i t y o f the knee j o i n t i s c o n t r o l l e d by two se t s o f l i g a m e n t s ; the a n t e r i o r and p o s t e r i o r c r u c i a t e s and the m e d i a l and l a t e r a l c o l l a t e r a l s . ( F i g . 3-03) The s t r o n g c r u c i a t e s c r o s s each o t h e r i n the middle o f the j o i n t as shown i n F i g . 3-0*1- T h e i r main f u n c t i o n i s to l i m i t a n t e r i o r and p o s t e r i o r d i s l o c a t i o n o f the j o i n t but they a l s o check h y p e r e x t e n s i o n , r o t a t i o n about the l o n g a x i s and s i d e to s i d e movement o f the j o i n t . The c o l l a t e r a l l i g a m e n t s p r e v e n t a b d u c t i o n and adduc-t i o n o f the j o i n t w h i l e a l l o w i n g a wide range o f f l e x i o n and e x t e n s i o n i n the s a g i t t a l p l a n e . The m e d i a l c o l l a t e r a l has a s u p e r f i c i a l p a r t and a deep p a r t w i t h the deep f i b r e s o f l i g a m e n t a d h e r i n g to the med ia l meniscus ( F i g . 3• 0*4-). The - 2 2 -FIG. 3.02 SUPERIOR SURFACE -TIBIA FIG. 3.03 L IGAMENTS OF KNEE JOINT - 23 -l a t e r a l c o l l a t e r a l i s a s i n g u l a r c o r d - l i k e l i gament w i t h no adherence to the l a t e r a l meniscus . The c o l l a t e r a l s are s l a c k i n f l e x i o n but t a u t i n e x t e n s i o n as the d i s t a n c e b e t -ween the j o i n t s u r f a c e s i n c r e a s e s . The c o l l a t e r a l l i g a m e n t s a l o n g w i t h the c r u c i a t e s and i n t e r c o n d y l a r notch r e s t r i c t r o t a t i o n o f the femur r e l a t i v e to the t i b i a . The s t r o n g p a t e l l a r l i g a m e n t i s an e x t e n s i o n o f the q u a d r i c e p s tendon and i t p r o v i d e s a n t e r i o r j o i n t s t a b i l i t y through i t s ' at tachment to the t i b i a l t u b e r o s i t y o f the t i b i a . ( F i g . 3-05) In a d d i t i o n to the main l i g a m e n t s o f the knee , there are two minor l i g a m e n t s whose f u n c t i o n i s not w e l l d e f i n e d . These are the o b l i q u e p o p l i t e a l l i gament and the a r c u a t e p o p l i t e a l l i g a m e n t s i t u a t e d on the p o s t e r i o r s ide o f the j o i n t . M e n i s c i and A s s o c i a t e d S t r u c t u r e s The m e n i s c i deepen the concave c o n d y l e s o f the t i b i a to a i d the a r t i c u l a t i o n o f the f emora l c o n d y l e s on the t i b i a l s u r f a c e s . The m e n i s c i , which are s e m i - l u n a r shaped, have a wedge l i k e c r o s s - s e c t i o n w i t h i n t e r i o r "borders t h a t t a p e r to t h i n edges. ( F i g . 3 -03) The l a t e r a l meniscus i s s m a l l e r i n d i a m e t e r , t h i c k e r about the p e r i p h e r y and more c i r c u l a r t h a n the m e d i a l meniscus but covers a l a r g e r p o r t i o n o f the a r t i -c u l a r s u r f a c e as i t i s w i d e r . ( F i g . J.Ok) The a n t e r i o r margins o f the two m e n i s c i are con t in u ou s w i t h the t r a n s v e r s e - 2^ -ANTERIOR CRUCIATE T R A N S V E R S E L I G A M E N T CORONARY L I G A M E N T FIG, 3.04 L IGAMENTS OF K N E E JO INT TEN DON OF QUADRICEPS PATELLA PAT E L L A R L I G A M E N T T I B I A L T U B E R C L E FIG, 3.05 P A T E L L A R L I G A M E N T - 25 -l i g a m e n t w h i l e the p e r i p h e r y o f the m e n i s c i are connected, to the t i b i a by a f i b r o u s p o r t i o n o f the a r t i c u l a r c a p s u l e known as the c o r o n a r y l i gament ( F i g . J.Ok). The m e d i a l meniscus i s a t t a c h e d t i g h t l y compared to the l a t e r a l meni s -cus a c c o u n t i n g f o r a d i f f e r e n c e . i n m o b i l i t y . Musc le System The muscle groups t h a t c o n t r o l the movements o f the knee j o i n t can be d i v i d e d i n t o e x t e n s o r s and f l e x o r s . ( F i g . 3 . 0 6 ) The main ex tensors form the s t r o n g q u a d r i c e p s f e m o r i s group and i n c l u d e r e c t u s f e m o r i s , v a s t u s l a t e r a l i s , v a s t u s m e d i a l i s and v a s t u s i n t e r m e d i u s muscles ( F i g . 3 - ° 6 ) which j o i n to form the q u a d r i c e p s tendon. Some f i b r e s o f t h i s tendon i n s e r t i n t o the p a t e l l a w h i l e o t h e r s pass over the p a t e l l a to b l e n d w i t h the p a t e l l a r l i g a m e n t . The main f l e x o r s o f the knee form the h a m s t r i n g group. These i n c l u d e the b i c e p s f e m o r i s , the semimembranosus and the semi tendinosus ( F i g . 3 - 0 6 ) . These musc les c r o s s the j o i n t p o s t e r i o r l y to a t t a c h to the s u r f a c e s o f the f i b u l a and the t i b i a . The two headed gas trocnemius muscle forms the g r e a t e r p a r t o f the c a l f and i s a weak f l e x o r o f the knee . The f i b r e s o f the gas trocnemius and the deeper so l eus muscle u n i t e to form the tendo c a l c a n e u s which a t t a c h e s to the c a l c a n e u s bone o f the f o o t . The s m a l l p l a n t a r i s musc le , the FIG. 3.06 MUSCLES CONTROLLING KNEE JOINT 1. B I C E P S LONG HEAD 5. GRACILIS 9 . VASTUS LATE RA LIS 2 . B I C E P S SHORT HEAD 6. GASTROCNEMIUS MED. HEAD 10. VASTUS INTERME DI U S 3 . SEMITE NDi .NOSUS 7. GASTROCNEMIUS LAT . HEAD 11. VASTUS MEDIALIS 4 . SEMI M E M B R A N O S U S 8 , SARTORl US 12. RECTUS FEMORIS 13. PLANTARIS 14. POPLITEAL FOSSA - 27 -so l eus and the gas trocnemius are the main p l a n t a r f l e x o r s o f the f o o t ( F i g . 3 . 0 6 ) . Other weak f l e x o r s o f the knee are the l o n g s a r t o r i u s musc le , the t h i n f l a t g r a c i l i s musc le , the p l a n t a r i s and the t r i a n g u l a r p o p l i t e u s muscle ( F i g . 3 . 0 6 ) . The s a r t o r i u s and g r a c i l i s a s s i s t m e d i a l r o t a t i o n o f the t i b i a i n f l e x i o n w h i l e the p o p l i t e u s a i d s i n u n l o c k i n g the j o i n t a t the onset o f f l e x i o n . Concepts o f A n a l y s i s Reference Axes o f Knee To a n a l y z e the f o r c e s a c t i n g a t the knee j o i n t , M o r r i s o n ' s (1973) r e f e r e n c e axes were adopted . These X s , Ys, Zs axes are r e f e r e n c e d w i t h r e s p e c t to the t i b i a and have t h e i r o r i g i n a t the assumed j o i n t c e n t r e o f the knee . ( F i g . 3 .O7) The h o r i z o n t a l Xs a x i s c o i n c i d e s w i t h the m e d i a l -l a t e r a l a x i s o f the t i b i a l condy le s u r f a c e s , the a n t e r i o r -p o s t e r i o r Ys a x i s c o i n c i d e s w i t h the-mid-raxis , d f the i n t e r -c o n d y l a r n o t c h w h i l e the Zs a x i s r e p r e s e n t s the v e r t i c a l a x i s o f the t i b i a . ( F i g . 3 .08) Axes o f Femur and P e l v i s In o r d e r to c a l c u l a t e the muscle and l i g a m e n t a t t a c h -ments to the femur and p e l v i s , two a d d i t i o n a l r e f e r e n c e axes are r e q u i r e d . M o r r i s o n ' s (1973) f e m o r a l r e f e r e n c e axes X f , Y f , Z f - 28 -FIG. 3.07 T I B I A L R E F E R E N C E A X E S I CONTACT AREA t Y s F I G . 3 . 0 8 T I B I A L C O N D Y L E S - 29 -were adopted ( F i g . 3 - 0 9 ) . The X f a x i s c o i n c i d e s w i t h the c e n t r e l i n e o f the f emora l c o n d y l e s and i s p a r a l l e l to the Xs a x i s o f the t i b i a and the Y f a x i s i s p a r a l l e l to the Ys a x i s .of the t i b i a . The Z f a x i s i s c o i n c i d e n t w i t h the Zs a x i s o f the t i b i a but o n l y i n 1 8 0 ° e x t e n s i o n . The i n t e r s e c t i o n o f the c e n t r e l i n e o f the f emora l c o n d y l e s and the Zs a x i s de termines the o r i g i n o f the fem-o r a l r e f e r e n c e axes . T h i s , o r i g i n i s not f i x e d i n f l e x i o n r e l a t i v e to the t i b i a l axes ( F i g . 3 * 1 ° ) d u e to the c h a r a c t e r o f j o i n t a r t i c u l a t i o n d i s c u s s e d below. The r e f e r e n c e axes o f the p e l v i s , Xp , Yp , Zp were chosen w i t h o r i g i n a t the c e n t r e o f the acetabulum ( F i g . 3 - 1 1 ) . The Xp a x i s i s p a r a l l e l to the l i n e j o i n i n g the a n t e r -i o r s u p e r i o r sp ines o f the h i p bones, the Yp axis , l i e s i n the s a g i t t a l p l a n e and the Zp a x i s i s p e r p e n d i c u l a r . Movements o f the Knee J o i n t . The main movements o f the knee j o i n t are f l e x i o n and e x t e n s i o n through about 135 degrees . These movements are governed by the shape o f the j o i n t a r t i c u l a r s u r f a c e s and the r e s i s t a n c e o f the l i g a m e n t s and musc les t h a t c r o s s the j o i n t . F o r the g r e a t e r p a r t o f the j o i n t ' s range o f mot ion , the c y l i n d r i c a l f emora l c o n d y l e s s l i d e on the t i b i a w i t h a f i x e d l i n e c o n t a c t on the concave t i b i a l c o n d y l e s . However, i n the l a s t 10 to 20 degrees o f e x t e n s i o n the f emora l c o n -d y l e s r o l l forward on the t i b i a l s u r f a c e . S i m i l a r i l y the FIG. 3.09 REFERENCE AXES - FEMUR ; TIBIA - 31 -9 0 ° FLEXION i - c c FIG. 3.11 PELV IC R E F E R E N C E AXES - 33 -f i r s t 10 to 20 degrees o f f l e x i o n i n v o l v e a backward r o l l i n g o f the f e m o r a l c o n d y l e s . The l i n e c o n t a c t o f the j o i n t s u r -f a c e s i s assumed to be c o i n c i d e n t w i t h the Xs a x i s o f the • t i b i a ( F i g . 3-°8). The r o l l i n g o f the f emora l c o n d y l e s causes an a n t e r i o r or p o s t e r i o r d i s p l a c e m e n t o f t h i s c o n t a c t l i n e d u r i n g a r t i c u l a t i o n and i n t r o d u c e s an e r r o r i n the above as sumpt ion . The f i n a l 10 to 15 degrees o f knee e x t e n s i o n are accompanied by an outward r o t a t i o n o f the t i b i a r e l a t i v e to the femur. T h i s movement i s known as " l o c k i n g " o f the j o i n t and a s i m i l a r " u n l o c k i n g " a c t i o n o c c u r s a t the onset o f f l e x i o n . The l o c k i n g and u n l o c k i n g a c t i o n s c o i n c i d e w i t h the r o l l i n g phase o f j o i n t movement. I n f u l l e x t e n s i o n , r o t a t i o n o f the j o i n t about the l o n g a x i s i s r e s t r i c t e d but up to 25 degrees o f r o t a t i o n can o c c u r i n f l e x i o n as the l i g a m e n t s a c r o s s the j o i n t b e g i n to s l a c k . A l s o some l a t e r a l mot ion a t t h e ; j j o i n t s u r f a c e s i s a l l o w a b l e between 30 and 50 degrees f l e x i o n . Musc le and Ligament System In o r d e r to complete an a n a l y s i s o f the f o r c e s imposed a t the knee , a s i m p l i f i e d muscle and l i g a m e n t system a f t e r M o r r i s o n (1970:3*0 was adopted . I n t h i s s i m p l i f i e d anatomi -c a l model , the four , main l i g a m e n t s o f the j o i n t , the c r u c i a t e s and the c o l l a t e r a l s , are assumed to have i n d i v i d u a l f o r c e a c t i o n s . The muscles t h a t c r o s s the j o i n t were d i v i d e d i n t o - 3^ -t h r e e s epara te groups so t h a t the muscles o f each group a c t s y n c h r o n o u s l y . The complete system i s d e f i n e d as f o l l o w s : ( F i g . 3 . 12 ) (1) H a m s t r i n g s ; i n c l u d i n g b i c e p s f e m o r i s , s e m i t e n d i n -osus and semimembranosus. (2) G a s t r o c n e m i u s ; i n c l u d i n g l a t e r a l and m e d i a l heads o f gas trocnemius and p l a n t a r i s . (3) ' Q u a d r i c e p s f e m o r i s ; i n c l u d i n g r e c t u s f e m o r i s , v a s t u s m e d i a l i s v a s t u s . i n t e r m e d i u s " a n d . v a s t u s l a t e r a l i s . (4) C r u d i a t e L igaments ; a n t e r i o r and p o s t e r i o r . (5) C o l l a t e r a l L igaments ; m e d i a l and l a t e r a l . The f o u r muscles o f the knee not accounted f o r i n the above model , t e n s o r f a s c i a l a t a e , g r a c i l i s , s a r t o r i u s and p o p l i t e u s are assumed to be o f i n s i g n i f i c a n t importance f o r t h i s s tudy and are t h e r e f o r e exc luded from the a n a l y s i s . In a c t u a l f a c t , the t e n s o r f a s c i a l a t a e a c t s w i t h the q u a d r i c e p s group , g r a c i l i s and s a r t o r i u s a i d the hamstr ings group and p o p l i t e u s i s a s s o c i a t e d w i t h the " u n l o c k i n g " a c t i o n o f the knee . Anthropometry I n t r o d u c t i o n I n o r d e r to c a l c u l a t e the l i n e s o f f o r c e a c t i o n o f the muscle groups and l i g a m e n t s at the knee,, the c o - o r d i n a t e s o f t h e i r s k e l e t a l a t tachments are r e q u i r e d . The p r e s e n t a n a l y s i s u t i l i z e s M o r r i s o n ' s (1970) a n t h r o -- 35 -FIG. 3.12 S IMPLIFIED MUSCLE AND LIGAMENT  SYSTEM ACTING AT KNEE JOINT 1. HAMSTRINGS 4 . GRUCIATE LIGAMENTS 2. GASTROCNEMIUS 5 . COLLATERAL LIGAMENTS 3. QUADRICEPS FEMORIS - 36 -pometr i c measurements to determine the c o - o r d i n a t e s o f a t -tachment f o r the t e s t s u b j e c t . Measurements M o r r i s o n (1970) de termined the c o - o r d i n a t e s o f muscle and l i g a m e n t attachment f o r a d i s s e c t e d male l i m b . These measurements, which were made i n i n c h e s r e l a t i v e to the t i b i a l (s) and f emora l ( f ) axes o f the amputated l i m b , are g i v e n i n Tab le No. 1. Musc le o r i g i n c o - o r d i n a t e s f o r the p e l v i s r e p r e s e n t measurements r e l a t i v e to M o r r i s o n ' s p e l v i c (p) a x i s w i t h c e n t r e at the a n t e r i o r s u p e r i o r sp ine o f the r i g h t h i p bone ( F i g . "},!!). F o r the hamstr ings group a common musc le o r i g i n was assumed f o r the b i c e p s f e m o r i s , • semimembranosus and semi tendinosus musc l e s . A l s o , s i n c e the gas trocnemius muscle group has a m e d i a l and l a t e r a l head, an e q u i v a l e n t common i n s e r t i o n on the femur was assumed so t h a t i t s l i n e o f a c t i o n passed through the tendo c a l c a n e u s to the m i d - p o i n t o f the p o s t e r i o r s u r f a c e o f the f emora l c o n d y l e s . S i n c e the l i n e o f a c t i o n o f the q u a d r i c e p s f e m o r i s muscle group i s con t inuous w i t h the l i n e o f a c t i o n o f the p a t e l l a r l i g a m e n t , M o r r i s o n (1970) deve loped a r e l a t i o n s h i p between the angle o f the p a t e l l a r l i g a m e n t and the angle o f knee f l e x i o n ( F i g . 3 - 1 3 ) -t h e t a q = 0 . 3 1 x 10"^ (ph i ) - 8 .4 x 1 0 " 3 ( p h i ) 2 + O . 37 x 1 0 ~ 2 (ph i ) + 15 . . 3 . 0 1 - 37 -TABLE NO.1 ANTHROPOMETR IC DATA Xsq 0.0 Xsl 1.65 Xf fa 0.15 Ysq -1.3 Ysl 1.0 Y f fa 0.4 Zsq -1.55 Zsl -1.1 Zffa 0-2 Xsh 0.0 Xffl 1.6 Xsp 0.0 Ysh 1.0 Yffl 0.2 Ysp 0.65 Zsh - 1 0 5 Zffl 0.25 Zsp -0 .25 Xph -2.24 X sm - 0 . 85 Xffp -0.3 Yph 4 .35 Ysm 0.3 Yffp - 0 3 Zph -3 .95 Zsm 2.35 Zffp -0.2 Xsg 0.0 Xf fm -1.8 Xpfh -1.1 Ysg 2.0 Yffm 0.1 Ypfh 2.0 Zsg -16.0 Zffm 0.65 Zpfh -2.4 Xsfg 0.0 X sa -0.15 XI 0.72 Ysfg 1.1 Ysa -0.4 Zsfc 0.8 Zsfg 1.1 Zsa 0.15 K N E E S C A L I N G D I M E N S I O N S PELVIC S C A L I N G D I M E N S I O N S X Y Z X Y Z 3.5 3.35 16 8.7 4.3.5 5.0 - 38 -PHI - D E G R E E S FIG. 3.13 ANGLE O F P A T E L L A R L IGAMENT RELAT IVE TO A N G L E OF KNEE F L E X I O N THETA cj, - A N G L E B E T W E E N L INE OF PATELLAR L IGAMENT AND LONG A X I S OF TIBIA ( Z s ) PHI - A N G L E OF FLEXION OF KNEE JOINT - 39 -where, t h e t a q = ang le between the l i n e o f p a t e l l a r l i g a m e n t and the v e r t i c a l a x i s o f the t i b i a ( Z s ) . ph i = angle o f f l e x i o n o f the knee j o i n t S c a l i n g F a c t o r s The c o - o r d i n a t e s o f muscle and l i gament attachment f o r the t e s t s u b j e c t were o b t a i n e d by u s i n g M o r r i s o n ' s a n t h r o -pometr i c measurements o f T a b l e No. 1 and a p p r o p r i a t e s c a l i n g f a c t o r s . To determine the s c a l i n g f a c t o r s f o r the t e s t sub-j e c t , the knee and p e l v i c s c a l i n g d imens ions o f the t e s t s u b j e c t were compared to the c o r r e s p o n d i n g measurements from the amputated l i m b and s k e l e t o n o f M o r r i s o n ' s (1970) s tudy . The s c a l i n g d imens ions are l i s t e d i n T a b l e No. 2 o f the r e s u l t s s e c t i o n a l o n g w i t h a d d i t i o n a l a n t h r o p o m e t r i c d a t a f o r the t e s t s u b j e c t w h i l e the s c a l i n g f a c t o r c a l c u l a t i o n s are g i v e n i n Appendix A . T e s t i n g P r o c e d u r e s I n t r o d u c t i o n The t e s t s u b j e c t f o r the s tudy was an e x p e r i e n c e d and accompl i shed i c e hockey p l a y e r . The t e s t i n g was conducted i n l a b o r a t o r i e s o f the U n i v e r s i t y o f Washington and the U n i v e r -s i t y o f B r i t i s h C o l u m b i a . S y n c h r o n i z e d f i l m r e c o r d s and o s c i l l o s c o p e f o r c e r e c o r d s were o b t a i n e d w i t h the f o r c e p l a t e apparatus a t the U n i v e r s i t y o f Washington f o r a s imu-. I s o -l a t e d s k a t i n g s t r i d e . E l e c t r o m y o g r a p h t e s t s were completed at the U n i v e r s i t y o f B r i t i s h Co lumbia to compare the p h a s i c muscle a c t i v i t y o f a s i m u l a t e d s k a t i n g s t r i d e i n the l a b o r a -t o r y w i t h an a c t u a l o n - i c e s k a t i n g s t r i d e . The e l e c t r o m y o -graph d a t a was n e c e s s a r y to a l l o w f o r a s o l u t i o n o f the i n d e t e r m i n a t e e q u a t i o n s o f the f o r c e system. Cinematography P r o c e d u r e s The t e s t s u b j e c t was f i l m e d i n the s a g i t t a l and f r o n -t a l p l a n e s w h i l e e x e c u t i n g a s i m u l a t e d s k a t i n g s t r i d e . To determine l i m b movement, the t e s t s u b j e c t wore swimming t r u n k s and white c i r c u l a r r e f e r e n c e markers , 0 . 7 5 i n c h e s i n d i a m e t e r , were p l a c e d on the lower r i g h t l i m b and p e l v i s . The l o c a t i o n s o f the r e f e r e n c e \ i m a r k e r s , shown i n F i g . 3 - 1 ^ were as f o l l o w s : (1) a t the h i p j o i n t c e n t r e over the f emora l head o f the acetabulum a r t i c u l a t i o n on the l a t e r a l and f r o n t a l s i d e s o f the h i p (2) a t the knee j o i n t c e n t r e over the l a t e r a l e p i c o n -d y l e o f the femur and a t the c o r r e s p o n d i n g f r o n t a l l o c a t i o n (3) a t the ank le j o i n t c e n t r e on the f r o n t and l a t e r a l s i d e s (4) a t the c e n t r e o f g r a v i t y o f the shank on the f r o n t and l a t e r a l s i d e s (see Appendix A) (5) a t the c e n t r e o f g r a v i t y o f the f o o t and skate on FIG. 3.14 LOCATION OF R E F E R E N C E MARKERS - k2 -the f r o n t and l a t e r a l s i d e s (see Appendix A) Two 16 mm m o t i o n p i c t u r e cameras d e t a i l e d i n Appendix B, were used to r e c o r d the s k a t i n g s t r i d e at a f i l m i n g speed o f 6^ frames per second. The f i l m r e c o r d s and the r e f e r e n c e markers . allowed.' l o c a t i o n i n t ime o f the X, Y, Z c o - o r d i n a t e s o f the j o i n t c e n t r e s and o f the c e n t r e s o f g r a v i t y o f the lower r i g h t l i m b . The e x p e r i m e n t a l s e t - u p w i t h the l o c a t i o n o f the cameras i s shown i n F i g . 3 -15-F o r c e Measurements A f o r c e a n a l y s i s o f a movement d u r i n g any a c t i v i t y i n v o l v e s the use o f p r e c i s e and s o p h i s t i c a t e d measur ing equipment. A KISTLER p i e z o - e l e c t r i c f o r c e p l a t e w i t h a s i x -component f o r c e measur ing system (Appendix B) was u t i l i z e d to determine the r e a c t i o n f o r c e s deve loped d u r i n g the s i m u l a t e d s k a t i n g s t r i d e . The s i x - c h a n n e l output o f the f o r c e p l a t e was s t o r e d as c o n t i n u o u s t r a c e s on three o s c i l l o s c o p e s . A 35 mm camera w i t h s u i t a b l e o s c i l l o s c o p e attachment was then used to photograph these t r a c e s f o r l a t e r d i g i t i z a t i o n . The f o r c e r e c o r d o b t a i n e d from the f o r c e p l a t e output i s expres -sed as the t h r e e o r t h o g o n a l f o r c e components (Fxp, Fyp and Fzp) a t the p l a t e , the moment about the Z - a x i s (Mzp) and the c o - o r d i n a t e s o f the p o i n t o f a p p l i c a t i o n (Ax, Ay) o f the f o r c e ( F i g . 3 . 1 6 ) . The s y n c h r o n i z a t i o n o f the f o r c e p l a t e o s c i l l o s c o p e - 4 3 -FIG. 3.15 EXPER IMENTAL S E T - U P - 44 -FIG.3,16 FORCE RECORD OF FORCE PLATE E X P R E S S E D AS SIX  M E A S U R E D VAR IABLES F x p , F Y p , F z p - O R T H O G O N A L FORCE COMPONENTS OF FORCE , F p M Z P — MOMENT ABOUT VERTICAL AXIS A x , Ay - POINT OF APPLICATION OF FORCE , Fp - 45 -t r a c e s w i t h the 16 mm f i l m record, was a c h i e v e d w i t h a f l a s h source as shown i n F i g . J.15. The f l a s h source was p l a c e d i n the f i e l d o f the 16 mm cameras and was connected i n s e r i e s to a c o n t r o l sw i t c h and the e x t e r n a l t r i g g e r source o f the o s c i l l o s c o p e . C l o s i n g the s w i t c h i n i t i a t e d the o s c i l l o s -cope t r a c e and i l l u m i n a t e d the f l a s h s i m u l t a n e o u s l y thereby s y n c h r o n i z i n g the f o r c e p l a t e output w i t h the f i l m r e c o r d . E l e c t r o m y o g r a p h y To compare the muscle a c t i v i t y o f a s i m u l a t e d s k a t i n g s t r i d e to an a c t u a l s k a t i n g t h r u s t , e l ec tromyograph t e s t s were conducted^ both i n the l a b o r a t o r y and on i c e . T h e r e f o r e , independent e l ec t romyograph t e s t s were conducted a t the e x e r c i s e p h y s i o l o g y l a b o r a t o r y o f the U n i v e r s i t y o f B r i t i s h Co lumbia and on i c e at the T h u n d e r b i r d W i n t e r S p o r t s C e n t r e . S u r f a c e e l e c t r o d e s were a t t a c h e d over the b e l l y o f the r e c t u s f e m o r i s , b i c e p s f e m o r i s and gas trocnemius musc les to r e c o r d the e l e c t r i c a l a c t i v i t y o f the q u a d r i c e p s f e m o r i s , hamstr ings and gas trocnemius muscle groups r e s p e c t i v e l y . A two-channe l Sanborn Recorder (Appendix B) was used to mon i tor the e l e c t r i c a l s i g n a l s . The e l e c t r o m y o g r a p h r e c o r d s were s y n c h r o n i z e d w i t h 16 mm f i l m r e c o r d s by means o f a f l a s h source i n the f i e l d o f the camera t r i g g e r e d s i m u l t a n e o u s l y w i t h the Sanborn Rec -o r d e r . S i n c e the muscle groups were connected to the two-channe l r e c o r d e r i n p a i r s , b e f o r e each t e s t the output - 46 -s i g n a l o f one muscle group was checked f o r i n t e r f e r e n c e from the output s i g n a l o f the o t h e r moni tored muscle group . S i m u l a t e d S k a t i n g T h r u s t The normal s k a t i n g s t r i d e has t h r e e d i s t i n c t movements which can be d e s c r i b e d as f o l l o w s : (1) the t h r u s t e x t e n s i o n by the p u s h i n g l e g , (2) the weight s h i f t to the g l i d e l e g and (3) the g l i d e on the s u p p o r t i n g g l i d e l e g . The p r e s e n t s tudy i s concerned w i t h an a n a l y s i s o f the f o r c e s a c t i n g at the knee d u r i n g the t h r u s t e x t e n s i o n phase o f a s i m u l a t e d s k a t i n g s t r i d e executed i n a l a b o r a t o r y env ironment . The t e s t s u b j e c t performed the s k a t i n g t h r u s t from a s t a t i o n a r y p o s i t i o n w i t h the r i g h t skate b lade p l a c e d at the c e n t r e o f a f o r c e p l a t e and p o i n t i n g i n the p o s i t i v e Y -d i r e c t i o n . ( F i g . 3«1?)« The s u b j e c t i n i t i a t e d movement by s h i f t i n g the body weight from the r e a r l e f t f o o t to the r i g h t f o o t and t h e n d r i v i n g o f f the r i g h t f o o t to l a n d on the l e f t f o o t ' in f r o n t o f the f o r c e p l a t e . To a l l o w the s k a t e r to deve lop the r e q u i r e d t h r u s t i n g f o r c e on the f o r c e p l a t e , r u b b e r skate b lade guards were p l a c e d on -bo th o f the s u b j e c t ' s s k a t e s . The t e s t s u b j e c t performed the s k a t i n g t h r u s t i n the l a b o r a t o r y u s i n g a hockey s t i c k to d u p l i c a t e the arm a c t i o n o f a hockey p l a y e r ' s s t r i d e . - kl -FIG.3.17 EXPERIMENTAL SET-UP SHOWING TEST S U B J E C T ON FORCE PLATFORM - 48 -Data R e d u c t i o n F o r each t e s t t r i a l , the c o - o r d i n a t e s f o r the lower l i m b were o b t a i n e d from the deve loped 16 mm f i l m and the f o r c e p l a t e output was o b t a i n e d from the deve loped 35 mm photographs o f the r e s p e c t i v e o s c i l l o s c o p e t r a c e s . A Vanguard M o t i o n A n a l y z e r o f the I n s t i t u t e o f Animal Resource E c o l o g y ( I . A . R . E . ) was used to d i g i t i z e the f i l m d a t a . F o r each frame o f the 16 mm f i l m d e s c r i b i n g the s k a t i n g s t r i d e , the c o - o r d i n a t e s o f each l i m b r e f e r e n c e marker r e l a t i v e to an o r i g i n l o c a t e d a t the c e n t r e o f the f o r c e p l a t e were measured. The s k a t i n g s t r i d e was a d e q u a t e l y r e p r e s e n t e d by 50 frames o f the 16 mm f i l m . The X, Y and Z c o - o r d i n a t e r e a d i n g s were made to the n e a r e s t 0 .001 i n c h w i t h r e s p e c t to the s c a l e o f the Vanguard p r o j e c t e d image. The a c t u a l r e a -d i n g s were o b t a i n e d by m u l t i p l y i n g the Vanguard measurements by an a p p r o p r i a t e s c a l i n g f a c t o r , the s c a l e determined from the f i l m image o f a f i v e f o o t l e n g t h s u r v e y o r ' s range p o l e h e l d a t the c e n t r e o f the f o r c e p l a t e . (Appendix C) The s y n c h r o n i z e d f o r c e p l a t e r e c o r d s on 35 mm f i l m were a l s o d i g i t i z e d w i t h the Vanguard M o t i o n A n a l y z e r by d i v -i d i n g the t r a c e s i n t o 50 equa l h o r i z o n t a l increments c o r r e s -ponding to the 50 frames o f the 16 mm f i l m . The a c t u a l f o r c e r e a d i n g s were then determined by a p p l y i n g the s u i t a b l e f o r c e p l a t e and o s c i l l o s c o p e c a l i b r a t i o n f a c t o r s (Appendix C) to the v e r t i c a l c o - o r d i n a t e measure o f the Vanguard image. The f o r c e p l a t e r e a d i n g s were taken f o r each frame o f f i l m at a f i l m speed o f 64 f r a m e s / s e c or f o r every 0 .16 sec t ime i n t e r v a l . - 49 -The e l e c t r o m y o g r a p h i c output was examined w i t h r e s p e c t to the t ime i n t e r v a l used to d e s c r i b e the s k a t i n g s t r i d e (50 frames or 0.80 sec) to determine the p h a s i c a c t i v i t y o f the muscle g r o u p s i n v o l v e d . The p e r i o d o f a c t i v i t y f o r each muscle group was d e f i n e d r e l a t i v e to the f i l m frames . A g e n e r a l computer program, DIFRN, was w r i t t e n to c a l c u l a t e the e x t e r n a l f o r c e system a c t i n g a t the knee and the c o r r e s p o n d i n g muscle and l igament , f o r c e s f o r each frame o f f i l m a n a l y z e d . Input d a t a i n c l u d e d l i m b c o - o r d i n a t e s , ' j o i n t c o - o r d i n a t e s , f o r c e p l a t e r e a c t i o n s , a n t h r o p o m e t r i c measurements and e l ec t romyograph i n f o r m a t i o n . A n a l y s i s I n t r o d u c t i o n The a n a l y s i s o f the f i l m and f o r c e p l a t e d a t a to d e t -ermine the musc le , l i g a m e n t and j o i n t f o r c e s a c t i n g at the knee i s o u t l i n e d i n the f low c h a r t o f F i g . 3 .18 . The f o l -l o w i n g t e x t summarizes the s teps i n v o l v e d i n the a n a l y s i s w h i l e a more d e t a i l e d d e s c r i p t i o n i s g i v e n i n the A p p e n d i c e s . Limb and J o i n t C o - o r d i n a t e s The c o - o r d i n a t e s o f the j o i n t s and mass c e n t r e s o f the lower l i m b were determined from the l o c a t i o n o f the c o r r e s -pond ing r e f e r e n c e markers i n the f r o n t a l and s a g i t t a l p l a n e s . The a c t u a l c o - o r d i n a t e s were c o r r e c t e d f o r the d i s t a n c e o f the r e f e r e n c e markers from the r e s p e c t i v e j o i n t c e n t r e or - 50 -F I L M RECORD FORCE PLATE RECORD ORTHOGONAL CO-ORDS OF JOINTS & LIMBS L IMB DISPLACEMENT LIMB PARAMETERS REACTION FORCES ON SKATE ANGULAR CO-ORDINATES ANGULAR DISPLACEMENT LINEAR ACC'N OF LIMBS ANGULAR ACC'N OF LIMBS FORCE SYSTEM AT KNEE RESOLVED FORCES AT KNEE ANTHROPOMETRIC MEASUREMENTS ORTHOGONAL CO-ORD MUSCLES,LIGAMENTS ELECTROMYOGRAM RECORD LIGAMENT AND MUSCLE FORCES F I G . 3.18 A N A L Y S I S O F D A T A - 51 -c e n t r e o f mass and f o r the r o t a t i o n o f the l i m b . (Appen-d i x D) The a n g u l a r c o - o r d i n a t e s o f the axes o f the t h i g h , the t i b i a l axes and the axes o f the f o o t were a l s o d e t e r -mined. Limb A c c e l e r a t i o n The d i s p l a c e m e n t s i n time o f the c e n t r e s o f mass o f the lower l i m b were determined from the l i m b c o - o r d i n a t e s f o r each frame o f f i l m a n a l y z e d . A n i n e r p o i n t n u m e r i c a l d i f -f e r e n t i a t i o n t echn ique deve loped by Lanczos (1957) was then used to c a l c u l a t e the l i n e a r a c c e l e r a t i o n o f the mass c e n t r e s o f the shank and f o o t from the d i s p l a c e m e n t d a t a . L i n e a r a c c e l e r a t i o n s were c a l c u l a t e d f o r the r e s p e c t i v e X, Y and Z d i r e c t i o n s . S i m i l a r i l y the a n g u l a r a c c e l e r a t i o n s o f the shank and f o o t about the X, Y and Z axes were determined from the c o r r e s p o n d i n g a n g u l a r c o - o r d i n a t e s and a n g u l a r d i s p l a c e m e n t s . The n u m e r i c a l d i f f e r e n t i a t i o n t e c h n i q u e which i s based on the t h e o r y o f f i n i t e d i f f e r e n c e s i s d e s c r i b e d i n Appendix H . E x t e r n a l F o r c e System at the Knee The s i x component e x t e r n a l f o r c e system a c t i n g a t the knee j o i n t c e n t r e , F i g . 3 .19 r e p r e s e n t s the o r t h o g o n a l sum-mat ion o f a l l moments and f o r c e s a c t i n g on the lower l i m b , F i g . 3 ' 2 0 . The f o r c e s a c t i n g on the. l i m b are': (a) the f o r c e p l a t e r e a c t i o n s - 52 -Zs FIG. 3.19 EXTERNAL FORCE SYSTEM  AT KNEE E X P R E S S E D IN  T E R M S OF T IB IAL REFERENCE A X E S OF RIGHT KNEE FIG.3.20 FORCES ACTING ON LOWER LIMB INCLUDING REACTION FORCES, GRAVITY FORCES AND ACCELERAT ION FORCES - 54 -(b) the g r a v i t a t i o n a l f o r c e s and (c) the i n e r t i a l f o r c e s o f the f o o t and shank. The g r a v i t a t i o n a l f o r c e s r e p r e s e n t the weight o f the f o o t and shank w h i l e the i n e r t i a l f o r c e s are due to the a c c e l e r a t i o n o f the f o o t and shank masses. The mass o f the s u b j e c t ' s shank and f o o t was de termined from Dempster ' s (1955) percentage f i g u r e s and the moments o f i n e r t i a o f the shank and f o o t were c a l c u l a t e d from Braune and F i s c h e r ' s (1889) c o e f f i c i e n t s and C ^ . (Appendix A) By c o n s i d e r i n g the lower l i m b as a f r e e body and by summing the f o r c e s and moments a c t i n g on the l i m b ( F i g . 3.20) i n terms o f the knee r e f e r e n c e axes , s i x equat ions e x p r e s s i n g the e q u i l i b r i u m c o n d i t i o n s o f the lower l i m b are d e r i v e d . Fxk = Fxp + — • Xs + — • X f 3.02a g g Fyk = Fyp + ~ • Ys + ^  • Y f . . . 3.02b o o Fzk = Fzp - p ( g + Zs) - | ^ ( g + Z f ) 3.02c o o Mxk = Fzp(Yk - Ay) - Fyp(Zk) - — • Y f ( Z k - Z f ) - | £ • Ys(Zk - Zs) - |£(g + Z f ) ( Y k - Yf ) - | ^ ( g + Z s ) ( Y k - Ys) + I s x • Osx + I f x - - Qfx 3.02d Myk = - Fzp(Xk - Ax) + Fxp(Zk) + ^ • X f ( Z k - Z f ) - 55 -+ ^ . Xs (Zk - Zs) + | ^ ( g + Z f ) ( X k - X f ) + | ^ ( g + Z s ) ( X k - Xs) - I sy • 9sy - I f y • Qfy 3 . 0 2 e Mzk = Mzp + Fyp(Xk - Ax) - Fxp(Yk - Ay) - ^ • X f ( Y k - Yf ) - ^ • Xs(Yk - Ys) g g + — • Ys(Xk - Xs) + — • Yf (Xk - X f ) g g - I f z • Qfz - I s z • 9sz 3 . 0 2 f The c a l c u l a t e d f o r c e s (Fxk, Fyk , Fzk) and moments (Mxk, Myk, Mzk) r e p r e s e n t the complete e x t e r n a l f o r c e system a c t i n g at the knee j o i n t c e n t r e , k, i n terms o f r e f e r e n c e axes a t the knee p e r p e n d i c u l a r to the f o r c e p l a t e s u r f a c e . To determine the a s s o c i a t e d musc le , l i g a m e n t and j o i n t f o r c e s , the f o r c e system at the knee must he r e s o l v e d w i t h r e s p e c t . to the t i b i a l axes X s , Y s , Z s . See Appendix G. Musc le and Ligament C o - o r d i n a t e s F o r the t e s t s u b j e c t , the c o - o r d i n a t e s o f muscle and l i g a m e n t at tachment to the t i b i a , femur and p e l v i s were ob-t a i n e d by m u l t i p l y i n g M o r r i s o n ' s a n t h r o p o m e t r i c measurements o f T a b l e No. 1 by the a p p r o p r i a t e s c a l i n g f a c t o r . (Appendix A) The c o - o r d i n a t e s o f attachment were determined r e l a t i v e to the t i b i a l axes f o r each frame o f f i l m d e s c r i b i n g the - 56 -s k a t i n g t h r u s t and. were c o r r e c t e d f o r r o t a t i o n s o f the lower l i m b . These c a l c u l a t i o n s are d e s c r i b e d i n Appendix E . Musc le and Ligament F o r c e s In o r d e r to determine the muscle and l i g a m e n t f o r c e s , the l i n e s o f f o r c e a c t i o n o f the r e s p e c t i v e muscle groups and l i g a m e n t s were c a l c u l a t e d from the c o - o r d i n a t e s o f muscle and l i g a m e n t at tachment . iTh-e l i n e s o f f o r c e a c t i o n were expressed as the ang les o f the l i n e o f a c t i o n r e l a t i v e to the t i b i a l axes and w i t h c o r r e c t i o n s f o r r o t a t i o n o f the lower l i m b . These c a l c u l a t i o n s are -g iven i n Appendix F . The l i n e o f a c t i o n o f the q u a d r i c e p s f e m o r i s muscle group was c a l c u l a t e d from e q u a t i o n 3 . 01 where the angle c a l c u l a t e d r e p r e s e n t s the angle o f the p a t e l l a r l i g a m e n t w i t h r e s p e c t to the l o n g a x i s o f the t i b i a . A knowledge o f the l i n e s o f a c t i o n o f the muscles and l i g a m e n t s a l l o w e d d e t e r m i n a t i o n o f the f o r c e s a c t i n g i n these t i s s u e s from the known e q u i l i b r i u m c o n d i t i o n s t h a t e x i s t at the knee. The e x t e r n a l force- system a c t i n g at the knee i s b a l a n c e d by (a) the f o r c e a c t i n g between the a r t i c u l a t i n g s u r f a c e s o f the femur and t i b i a ( condy le s ) and (b) the f o r c e s deve loped i n the muscle groups and l i g a -ment s t r u c t u r e s o f the knee . S i n c e the number o f muscle groups (3) and l i g a m e n t s t r u c t u r e s (4) i s g r e a t e r than the number o f a v a i l a b l e e q u i -- 57 -i b r i u m e q u a t i o n s , the a n a l y s i s i s e s s e n t i a l l y i n d e t e r m i n a t e . T h e r e f o r e a number o f assumptions based on the known f u n c -t i o n a l anatomy o f the knee and the determined p h a s i c a c t i v i t y o f the muscle groups (EMG r e c o r d ) were made to o b t a i n a com-p l e t e s o l u t i o n . F o l l o w i n g M o r r i s o n ' s (1970) p r o c e d u r e , each a c t i o n o f the e x t e r n a l f o r c e system, F i g . 3<19» was c o n s i d -ered s e p a r a t e l y i n terms o f the p a r t i c u l a r muscle or l i g a m e n t a c t i o n i n v o l v e d . F o r c e s i n Muscle. Groups The moment a c t i o n , Mxk, i n the a n t e r i o r - p o s t e r i o r p l a n e o f the t i b i a l axes i s e q u i l i b r a t e d by muscle a c t i o n i n the muscle groups i n t h a t p l a n e , i . e . , (1) h a m s t r i n g s group (2) gas trocnemius group (3) q u a d r i c e p s f e m o r i s group A p o s i t i v e Mxk corresponds to a f o r c e a c t i o n on the lower l i m b t h a t tends to f l e x the knee and e q u i l i b r i u m i s o b t a i n e d by muscle f o r c e i n the q u a d r i c e p s group, the ex ten- .. s o r s o f the lower l i m b . EMG r e c o r d s , F i g . 4 . 0 2 , show t h a t both q u a d r i c e p s and gas trocnemius groups were a c t i v e d u r i n g t h i s p e r i o d w h i l e hamstr ings were i n a c t i v e . To reduce the i n d e t e r m i n a n c y o f t h i s s i t u a t i o n the a n t a g o n i s t i c e f f e c t o f • the gas trocnemius group on e q u i l i b r i u m c o n d i t i o n s at the knee-was assumed to be s m a l l . The f o r c e a c t i n g i n the q u a d r i c e p s f e m o r i s group , P q , was t h e r e f o r e determined from F i g . 3 .21 - 58 -FIG.3.21 M U S C L E FORCE IN Q U A D R I C E P S TO B A L A N C E M O M E N T + Mxk FIG.3.22 MUSCLE FORCE IN H A M S T R I N G S TO BALANCE MOMENT - Mxk - 59 -and e q u i l i b r i u m e q u a t i o n 3 . 0 3 as , Pq = Mxk/ (Cos 9q • Ysq + s i n 9q • Zsq) 3 . 0 3 where, 9q = angle between l i n e o f a c t i o n o f p a t e l l a r l i g a m e n t and l o n g a x i s o f t i b i a . X s q , Ysq = c o - o r d i n a t e s o f p a t e l l a r l i g a m e n t at tachment to t i b i a . S i n c e the h a m s t r i n g s are i n a c t i v e and the a n t a g o n i s t i c a c t i o n o f the gas trocnemius f o r c e on the knee i s d i s c o u n t e d f o r t h i s phase Ph = Pg = 0 3 . 0 4 The h o r i z o n t a l component o f muscle f o r c e , Fym, and the v e r t i c a l component o f muscle f o r c e , Fzm, are t h e n Fym = Pq • s i n 9q 3 . 0 5 a Fzm = Pq - cos 9q 3 . 0 5 b A n e g a t i v e Mxk r e p r e s e n t s an e x t e n s i o n f o r c e a c t i o n oh the lower l i m b which i s e q u i l i b r a t e d by muscle f o r c e a c t i o n i n the hamstr ings group a n d / o r the gas trocnemius group, the f l e x o r s o f the lower l i m b . The moment, Mxk, was n e g a t i v e f o r the f i r s t h a l f o f the s k a t i n g t h r u s t , F i g . k.06 and f o r the f i n a l p u s h - o f f o f the t h r u s t . EMG r e c o r d s show - 60 -t h a t the h a m s t r i n g s were a c t i v e f o r the f i r s t p o r t i o n o f the t h r u s t w h i l e the g a s t r o c n e m i u s group was a c t i v e f o r the f i n a l p o r t i o n . The q u a d r i c e p s muscle group, w h i c h has an antagon-i s t i c e f f e c t r e l a t i v e to b o t h the h a m s t r i n g . a n d g a s t r o c n e m i u s groups, was a l s o i n t e r m i t t e n t l y a c t i v e d u r i n g t h e s e p e r i o d s , F i g . 4 . 0 2 . I n o r d e r t o o b t a i n a s i n g u l a r s o l u t i o n f o r the f o r c e s a c t i n g i n the h a m s t r i n g and g a s t r o c n e m i u s groups, the a n t a g o n i s t i c a c t i o n o f the quadriceps- was i g n o r e d . F o r the f i r s t phase o f the s k a t i n g t h r u s t , the f o r c e a c t i n g i n the h a m s t r i n g s group, Ph, was determined from F i g . 3 . 2 2 and e q u i l i b r i u m e q u a t i o n 3.06. Ph = Mxk/(cos 9 x h • Ysh + s i n Qxh • Zsh) 3 . 0 6 where, 9 x h = a n g l e between l i n e o f a c t i o n o f the h a m s t r i n g s muscle group and l o n g a x i s o f the t i b i a Ysh, Zsh = c o - o r d i n a t e s o f h a m s t r i n g attachment t o the t i b i a S i n c e the q u a d r i c e p s a n t a g o n i s t i c f o r c e i s not i n -c l u d e d i n the c a l c u l a t i o n , the f o r c e i n the h a m s t r i n g s group r e p r e s e n t s a l o w e r l i m i t value, where i n Pg = Pq = 0 3 . 0 7 and the h o r i z o n t a l and v e r t i c a l components o f muscle f o r c e are Fym = -Ph • s i n 9 x h 3 . 0 8 a - 61 -Fzm = Ph • cos Qxh 3 .08b F o r t he f i n a l p u s h - o f f phase o f the s k a t i n g t h r u s t , the f o r c e a c t i n g i n the g a s t r o c n e m i u s group, Pg, was d e t e r -mined from F i g . 3-23 and e q u i l i b r i u m e q u a t i o n 3 ' 0 9 so t h a t , Pg = Mxk/(cos Qxg • Ysg + s i n 9xg • Zsg) 3 .09 where, 9xg = angl e between l i n e o f a c t i o n o f the g a s t r o c n e m i u s group and the l o n g a x i s o f the t i b i a Xsg, Zsg = co-ordinate.s o f g a s t r o c n e m i u s mu,scle attachment t o the t i b i a . Once a g a i n the c a l c u l a t e d v a l u e o f g a s t r o c n e m i u s muscle f o r c e i s l o w e r l i m i t and Ph = Pq = 0 3 - 10 and t he v e r t i c a l and h o r i z o n t a l muscle components a re Fym = -Pg • s i n 9xg 3 - H a Fzm = Pg • cos 9xg 3 .11b F o r c e s i n C r u c i a t e Ligaments A n t e r i o r - p o s t e r i o r f o r c e a c t i o n i n the y - d i r e c t i o n a t the j o i n t i s e q u i l i b r a t e d by t e n s i o n i n the c r u c i a t e l i g a -ments as no t e d by B r a n t i g a n and V o s h e l l (19^1) and S t e i n d l e r ( 1 9 5 5 ) ' I t was f u r t h e r assumed t h a t a f o r c e a c t i o n d i r e c t e d FIG.3.23 MUSCLE FORCE IN GASTROCNEMIUS TO B A L A N C E MOMENT - Mxk f o r w a r d s d e v e l o p s t e n s i o n o n l y i n the a n t e r i o r - c r u c i a t e and a backwards f o r c e causes t e n s i o n o n l y i n the p o s t e r i o r c r u c i a t e . See F i g . 3.24 a f t e r M o r r i s o n (1970). The e q u i l i b r i u m e q u a t i o n f o r the a n t e r i o r - p o s t e r i o r f o r c e a c t i o n s i s Fyk + Fym = Pa • cos 9xa + Pp • cos 9xp 3.12 where, Fyk = y-component o f e x t e r n a l f o r c e system Fym = y-component o f muscle f o r c e Pa = f o r c e a c t i n g i n the a n t e r i o r c r u c i a t e l i g a m e n t Pp = f o r c e a c t i n g i n the p o s t e r i o r c r u c i a t e l i g a m e n t 9xa. = a n g l e between l i n e o f a c t i o n o f a n t e r i o r c r u c i a t e and t i b i a l a x i s 9xp = a n g l e between l i n e o f a c t i o n o f p o s t e r i o r c r u c i a t e and t i b i a l a x i s F o r a f o r w a r d d i r e c t e d f o r c e a c t i o n , the f o r c e , Pa, d e v e l o p e d i n the a n t e r i o r c r u c i a t e l i g a m e n t was c a l c u l a t e d from e q u a t i o n 3'12 by s e t t i n g Pp = 0 .3.13a Pa = -(Fyk + Fym)/cos 9xa 3-13b The h o r i z o n t a l and v e r t i c a l components o f the a n t e r -i o r c r u c i a t e l i g a m e n t f o r c e r e l a t i v e to the t i b i a l axes are - 64 -(a) FORCE IN ANTERIOR CRUCIATE FOR (Fyk+Fym) IN FORWARD DIRECTION Fzcr ( Fyk+ Fym) Ys (b) FORCE IN POSTERIOR CRUCIATE FOR  (Fyk+Fym) IN BACKWARD DIRECTION FIG. 3.24 FORCE IN POSTERIOR CRUCIATE AND ANTERIOR CRUCIATE L IGAMENTS - 65 -F y c r = Pa • cos 9xa 3 . l 4 a F z c r = Pa • s i n 9xa 3 . l 4 b F o r a backward d i r e c t e d f o r c e a c t i o n a t the knee, the f o r c e , Pp, deve l o p e d i n the p o s t e r i o r c r u c i a t e l i g a m e n t was c a l c u l a t e d from e q u a t i o n 3-12 by s e t t i n g Pa = 0 3 . 1 5 a Pp = - (Fyk + Fym)/cos 9xp 3-15b The h o r i z o n t a l and v e r t i c a l components o f the pos-t e r i o r c r u c i a t e l i g a m e n t f o r c e are F y c r = Pp • cos 9xp 3 . l 6 a F z c r = Pp • s i n 9xp 3 - l 6 b F o r c e s i n C o l l a t e r a l Ligaments and J o i n t F o r c e The moment, Myk, tends t o bend the knee l a t e r a l l y and m e d i a l l y i n the f r o n t a l p l a n e . M o r r i s o n (1970) i n t e r p r e t s the e q u i l i b r i u m c o n d i t i o n s a t the knee f o r the s e f o r c e a c t i o n s as f o l l o w s . When t h e r e i s a comp r e s s i v e f o r c e , Rz, between the f e m o r a l and t i b i a l c o n d y l e s , the f o r c e a c t i o n o f Myk s h i f t s t he c e n t r e o f p r e s s u r e o f t h i s f o r c e m e d i a l l y o r l a t e r a l l y depending on the d i r e c t i o n o f Myk. I f t h e s h i f t o f weight i s s u f f i c i e n t t o reduce the p r e s s u r e a t the p e r i p h e r y o f the c o n d y l e t o zero t h e n any - 66 -f u r t h e r i n c r e a s e i n Myk must he b a l a n c e d by a t e n s i o n i n the c o l l a t e r a l l i g a m e n t a t t h a t s i d e o f the j o i n t . ( F i g . 3 . 25 ) M o r r i s o n (1970) d e t e r m i n e d a l i m i t i n g v a l v e , X I , f o r the s h i f t o f the c e n t r e o f p r e s s u r e t h a t r e d u c e s the p r e s s u r e a t the p e r i p h e r y o f the c o n d y l e b e a r i n g a r e a to z e r o . See F i g . 3 . 1 3 -E s s e n t i a l l y t h e r e a re two e q u i l i b r i u m e q u a t i o n s t h a t d e s c r i b e the f o r c e a c t i o n o f moment, Myk. Rz = Fzk + Fzm + F z c r + Pm • cos Gym • cos 9xm + P I • cos 9 y l • cos 9x1 3 . 17 0 = Myk - P I • cos 9 y l • X s l + P I • s i n Qyl • Z s l + Pm • cos 9ym • Xsm + Pm • s i n 9ym • Zsm + Rz • Xo 3 .18 where, Fzk = Z-component o f e x t e r n a l f o r c e system Fzm = Z-component o f muscle f o r c e F z c r = Z-component o f c r u c i a t e l i g a m e n t f o r c e Pm = f o r c e a c t i n g i n m e d i a l l i g a m e n t P I = f o r c e a c t i n g i n l a t e r a l l i g a m e n t Rz = c o m p r e s s i v e f o r c e between t i b i a l and f e m o r a l c o n d y l e s Xo = o f f s e t o f c e n t r e o f p r e s s u r e from j o i n t c e n t r e These two e q u a t i o n s are i n d e t e r m i n a n t i n t h a t t h e y c o n t a i n f o u r unknowns, Rz, Xo, Pm and P I . However, a s o l u -t i o n can be o b t a i n e d by r e d u c i n g the number o f unknowns.to - 67 -(a) FORCE IN LATERAL COLLATERAL DUE  TO ADDUCT ION FORCE - Myk (b) FORCE IN MEDIAL COLLATERAL DUE TO ABDUCTION FORCE + Myk FIG. 3.25 C O L L A T E R A L L IGAMENT FORCES - 68 -two by c o n s i d e r i n g s p e c i f i c l o a d i n g c o n d i t i o n s . C o n d i t i o n 1: P r e s s u r e e x i s t s between the c o n d y l e s o f the j o i n t so t h a t Fzk + Fzm + F z c r > 0 Pm = P I = 0 F o r t h e s e l o a d i n g c o n d i t i o n s , F i g . 3 -26 , Rz i s c a l -c u l a t e d d i r e c t l y from e q u a t i o n 3-17 Rz = Fzk + Fzm + F z c r 3 .19 and Xo can be e v a l u a t e d by s u b s t i t u t i o n f o r Rz i n e q u a t i o n 3.18 Xo = - Mzk/(Fzk + Fzm + F z c r ) 3 .20 C o n d i t i o n 2 : The c a l c u l a t e d v a l u e o f Xo i s g r e a t e r t h a n the l i m i -t i n g v a l u e o f XI and Myk a c t s i n a p o s i t i v e d i r e c t i o n m e d i a l l y . F o r t h e s e l o a d i n g c o n d i t i o n s , F i g . 3-25>a s o l u t i o n i s o b t a i n e d f o r Rz and P I by s u b s t i t u t i n g Pm = 0 and Xo = XI i n t o e q u a t i o n s 3 .17 and 3 .18 so t h a t , P I = (Myk + [Fzk + Fzm + F z c r J • X I ) / (cos 9 y l • X s l - s i n 9 y l • Z s l - cos 9 y l • X l ) 3 .21 - 69 -FIG. 3.26 COMPRESS I VE JOINT FORCE ; Rz - 70 -Rz = Fzk + Fzm + F z c r + P I • cos 9 y l • cos 9x1 3 .22 T h e r e f o r e f o r a p o s i t i v e Myk, such t h a t Xo >X1, a t e n s i l e f o r c e , P I , a c t s i n the l a t e r a l l i g a m e n t and the c e n t r e o f p r e s s u r e o f the. c o m p r e s s i v e f o r c e between the j o i n t con-d y l e s i s s h i f t e d a d i s t a n c e XI m e d i a l l y from the j o i n t c e n t r e . C o n d i t i o n 3-The c a l c u l a t e d v a l u e o f Xo i s g r e a t e r t h a n X I and Myk a c t s i n a n e g a t i v e d i r e c t i o n l a t e r a l l y . F o r t h e s e l o a d i n g c o n d i t i o n s , F i g . 3 - 2 5 . a s o l u t i o n i s o b t a i n e d f o r Rz and Pm by s u b s t i t u t i n g P I = 0 and Xo = XI i n t o e q u a t i o n s 3 .17 and 3 .18 so t h a t Pm = (Myk + [F.zm + Fzk + F z c r ] • X l ) / ( c o s 9ym • Xsm + s i n 9ym • Z s m - cos 9ym • X I ) 3-23 Rz = Fzk + Fzm + F z c r + Pm • cos 9ym • cos 9xm 3.24 T h e r e f o r e , f o r a n e g a t i v e Myk such t h a t Xo >X1, a t e n s i l e f o r c e , Pm, a c t s i n the m e d i a l l i g a m e n t and the c e n t r e o f p r e s s u r e o f the c o m p r e s s i v e f o r c e a t the j o i n t i s d i s -p l a c e d a d i s t a n c e , X I , l a t e r a l l y from the j o i n t c e n t r e . Shear F o r c e and Torque The shear f o r c e a t the j o i n t i n the m e d i a l - l a t e r a l d i r e c t i o n i s d e t e r m i n e d by summing the X component o f the e x t e r n a l f o r c e , the muscle f o r c e and the l i g a m e n t f o r c e s . - 71 -Rx = Fxk + Fxm + F x c o l 3-25 where F x c o l i s the summation o f the X components o f Pm, the m e d i a l l i g a m e n t f o r c e . F x c o l = PI • s i n 9 y l + Pm • s i n 9ym 3-26 The t o r q u e a c t i n g on the j o i n t i s e q u a l to the sum-m a t i o n o f the t w i s t i n g f o r c e s a c t i n g on the j o i n t . Mz = Mzk + F x l c • Y s l + Fxmc • Ysm + F y l c • X s l + Fymc • Xsm 3-27 where, F x l c = x-component o f l a t e r a l l i g a m e n t f o r c e F y l c = y-component o f l a t e r a l l i g a m e n t f o r c e Fxmc = x-component m e d i a l l i g a m e n t f o r c e Fymc = y-component o f m e d i a l l i g a m e n t f o r c e The complete s o l u t i o n f o r the muscle, l i g a m e n t and j o i n t f o r c e s a t the knee i n c l u d e s , Rz - co m p r e s s i v e f o r c e between j o i n t c o n d y l e s Zo - c e n t r e o f p r e s s u r e o f comp r e s s i v e f o r c e w i t h r e s p e c t to the knee j o i n t c e n t r e Rx - shear f o r c e on j o i n t Ph - f o r c e a c t i n g i n h a m s t r i n g muscle group Pg - f o r c e a c t i n g i n g a s t r o c n e m i u s muscle group Pq - f o r c e a c t i n g i n q u a d r i c e p s muscle group - 72 -Pa - force acting i n anterior cruciate ligament Pp - force acting i n posterior cruciate ligament Pm - force acting i n medial c o l l a t e r a l ligament PI - force acting i n l a t e r a l c o l l a t e r a l ligament M z - torque on joint - 73 -CHAPTER IV RESULTS AND DISCUSSION R e s u l t s F o r c e p l a t e o u t p u t and l i m b d i s p l a c e m e n t d a t a were c o l l e c t e d f o r a t o t a l o f 15 t r i a l s . From th e s e d a t a the l i n e a r and a n g u l a r a c c e l e r a t i o n s o f the l o w e r l i m b were d e t -ermined, t h e e x t e r n a l f o r c e system a c t i n g a t the knee was c a l c u l a t e d and f i n a l l y the m u s c l e , l i g a m e n t and j o i n t f o r c e s were c a l c u l a t e d f o r each t r i a l . These r e s u l t s a re r e v i e w e d i n the f o l l o w i n g s e c t i o n w i t h r e f e r e n c e t o t r i a l No. 4 w h i c h i s c o n s i d e r e d t o be r e p -r e s e n t a t i v e o f the 15 t r i a l s made. A n t h r o p o m e t r i c Measurements The body parameters f o r the t e s t s u b j e c t are l i s t e d i n T a b l e No. 2 . The c a l c u l a t i o n s o f t h e s e a n t h r o p o m e t r i c measures are shown i n Appendix A. I n o r d e r t o s c a l e the b a s i c c o - o r d i n a t e s o f muscle and l i g a m e n t attachments o f T a b l e No. 1 t o the t e s t s u b j e c t , the t e s t s u b j e c t ' s knee and p e l v i c s c a l i n g d i m e n s i o n s were o b t a i n e d . These measurements are g i v e n i n Table No. 2 a l o n g w i t h the a d d i t i o n a l a n t h r o p o m e t r i c d a t a from the t e s t s u b j e c t . A l l measurements are i n i n c h e s and r e p r e s e n t the average - 7^ -TABLE NO.2 ANTHROPOMETRIC DATA Foot Shank Lf 10.50 Ls 16.50 Rf 2.07 Rs 2.31 r fx 1.45 rsx 4.15 r fy 3.15 r sy 4.15 r f z 3.15 rsz . 1.62 Ifx 0.15 I sx 3.92 Ify 0.71 Isy 3.92 Ifz 0.71 Isz 0.60 Wf 2.37 Ws 7.43 Knee Ankle Rxk 1.90 Ra 2.02 Ryk 2.44 Knee Scaling Dimensions Pelvic X 3.78 Y 3.64 Z 16.16 X 8.2 - 75 -v a l u e o f a t l e a s t f i v e i ndependent measuring t r i a l s . The knee and p e l v i c d i m e n s i o n s were' used to c a l c u l a t e the s c a l i n g f a c t o r s f o r the s u b j e c t . The s c a l i n g f a c t o r s are l i s t e d i n Appendix A w i t h c a l c u l a t i o n s . F o r c e P l a t e R e s u l t s The f o r c e p l a t e o u t p u t i n the form o f s i x s e p a r a t e o s c i l l o s c o p e t r a c e s i s shown i n F i g . 4 . 0 1 f o r a t r i a l No. 4 . These t r a c e s r e p r e s e n t the t h r e e o r t h o g o n a l p l a t e r e a c t i o n f o r c e s , the t o r q u e about a v e r t i c a l a x i s t h r o u g h the p o i n t o f a p p l i c a t i o n o f the r e s u l t a n t f o r c e o f the p l a t e and the c o - o r d i n a t e s o f the p o i n t o f a p p l i c a t i o n o f the r e s u l t a n t f o r c e , F i g . 3 . 1 6 . The v e r t i c a l a m p l i t u d e o f each t r a c e was measured w i t h the Vanguard M o t i o n A n a l y z e r a t i n t e r v a l s c o r r e s p o n d i n g to the 50 frames o f the s y n c h r o n i z e d c i n e f i l m o f the s k a t i n g t h r u s t . The a c t u a l v a l u e s o f the f o r c e s , t o r q u e and c o - o r d i n a t e s o f p o i n t o f a p p l i c a t i o n were o b t a i n e d by a p p l y i n g the a p p r o p r i a t e c a l i b r a t i o n f a c t o r s to the measured t r a c e v a l u e s . See Appendix 'G. E l e c t r o m y o g r a p h R e s u l t s F i g . 4 . 0 2 shows r e p r e s e n t a t i v e electromyograms f o r the t h r e e muscle groups i n s t r u m e n t e d and i n d i c a t e s the p h a s i c a c t i v i t y o f t h e s e muscle groups d u r i n g the s k a t i n g t h r u s t . The q u a d r i c e p s muscle group was a c t i v e t h r o u g h o u t the e n t i r e c y c l e , the h a m s t r i n g s were a c t i v e f o r the f i r s t p o r t i o n o f the t h r u s t and the g a s t r o c n e m i u s group showed a c t i v i t y d u r i n g - 76 -Fy 3 20C )N/( ii v. Fx| ) 10( DN/< jiv Fzc ) 50 ON/ ciiv. Mz D 2N m/c Jiv. Ax l Ay —f 2.5 0.2 sec FIG.4.01 FORCE PLATE OUTPUT IN FORM OF OSCILLOSCOPE TRACE-TRIAL NO.4 Hamstr ings Gast rocnemius Quadr iceps Femoris -N3 TIME - SECONDS FIG.4.02 E L E C T R O M Y O G R A M - SKATING THRUST ON ICE FRAMES d) 0. L FO WEIGHT SHIFT 0.2 L 0.4 L_ SUPPORT PHASE 0.6 0-5 CO SECONDS FIG. 4.03 I LLUSTRATED SEQUENCE OF SKATING THRUST - 79 -the l a t e r phase. The e l e c t r o m y o g r a p h r e s u l t s f o r a s i m u l a t e d s k a t i n g t h r u s t i n the l a b o r a t o r y were c l o s e l y r e l a t e d t o on-i c e e l e c t r o m y o g r a p h r e s u l t s i n d i c a t i n g comparable muscle a c t i v i t y . L i n e a r Limb A c c e l e r a t i o n s The l i n e a r a c c e l e r a t i o n s o f the shank and f o o t f o r the X, Y and Z d i r e c t i o n s a re p l o t t e d i n F i g . 4 .04 . As shown, the l i n e a r a c c e l e r a t i o n s o f the shank and f o o t a re • s m a l l u n t i l push o f f . The d r a m a t i c p o s i t i v e i n c r e a s e i n the v a l u e s o f AXS, AYS and AZS a t push o f f c o r r e s p o n d to an abr u p t i n w a r d , f o r w a r d and upward a c c e l e r a t i o n o f the shank. The f o o t a c c e l e r a t e s upward and backward i n p l a n t a r f l e x i o n a t push o f f as i n d i c a t e d by a p o s i t i v e AZF and a n e g a t i v e AYF. The i n w a r d a c c e l e r a t i o n o f the f o o t a t push o f f as i n d i c a t e d by the p o s i t i v e i n c r e a s e i n AXF i s accompanied by an outward r o t a t i o n o f the shank. A n g u l a r Limb A c c e l e r a t i o n s The a n g u l a r a c c e l e r a t i o n s o f the shank and f o o t about the X, Y and Z axes are p l o t t e d i n F i g . 4 . 0 5 . S i m i l a r t o the l i n e a r l i m b a c c e l e r a t i o n s , t h e a n g u l a r a c c e l e r a t i o n s are m i n i m a l f o r the f i r s t p o r t i o n o f the s k a t i n g t h r u s t but i n c r e a s e r a p i d l y d u r i n g push o f f . The l a r g e n e g a t i v e v a l u e s o f AASZ and AAFZ a t push o f f r e p r e s e n t a sharp outward r o t a t i o n o f the shank and f o o t about the v e r t i c a l a x i s w h i l e the p o s i t i v e i n c r e a s e i n AASY and AAFY a t push o f f - 80 -FIG.4.04 LIMB LINEAR ACCELERAT ION S -•81 -0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 .i 1 i i i i i i i TIME, seconds F I G . 4 . 0 5 L I M B A N G U L A R A C C E L E R A T I O N S - 82. -i n d i c a t e an inward r o t a t i o n o f the shank and f o o t about the a x i s o f p r o g r e s s i o n . The f o o t r o t a t e s forward i n p l a n t a r f l e x i o n at push o f f as shown by the p o s i t i v e i n c r e a s e i n AAFX w h i l e the shank a l s o appears to have some forward r o t -a t i o n as i n d i c a t e d by the p o s i t i v e v a l u e o f AASX at push o f f . E x t e r n a l F o r c e System at Knee The v a r i a t i o n s i n the e x t e r n a l f o r c e system d u r i n g one s k a t i n g s t r i d e i n terms o f the three f o r c e s and t h r e e moments a c t i n g a t the knee are p l o t t e d i n F i g . 4 . 0 6 . The l e n g t h o f the support phase i n the s k a t i n g s t r i d e as i n d i -c a t e d on the t ime base o f F i g . 4 . 0 3 i s a p p r o x i m a t e l y 0 . 4 seconds from the time the s k a t e r ' s l e f t f o o t l i f t e d o f f the f o r c e p l a t e (LFO) u n t i l the f i n a l push o f f .(P0) w i t h the r i g h t f o o t . The commencement o f the support phase i s c o i n -c i d e n t w i t h a p o s i t i v e i n c r e a s e i n the v e r t i c a l f o r c e Fzk and a change o f s i g n o f the moment Mxk i n the p lane o f p r o g -r e s s i o n from n e g a t i v e to p o s i t i v e . A n e g a t i v e Mxk tends to extend the knee w h i l e a . p o s i t i v e Mxk tends to f l e x the knee j o i n t . Moment Myk i s i n i t i a l l y p o s i t i v e and then n e g a t i v e d u r i n g the suppor t phase , a p o s i t i v e Myk r e p r e s e n t i n g a v a l g u s e f f e c t on' the knee j o i n t and a n e g a t i v e Myk a varus e f f e c t . The torque on the j o i n t about the l o n g a x i s o f the l i m b , Mzk, i s p o s i t i v e f o r the e n t i r e support phase i n d i c a -t i n g a c o n s t a n t c o u n t e r - c l o c k w i s e torque on the j o i n t . AT KNEE - TR IAL NO.4 - 84 -The f o r c e Fzk, t h r o u g h the l o n g a x i s o f the j o i n t , remains p o s i t i v e upward f o r the d u r a t i o n o f the su p p o r t phase but the shear f o r c e Fxk i s i n i t i a l l y n e g a t i v e i n a l a t e r a l d i r e c t i o n and t h e n becomes p o s i t i v e m e d i a l l y . The a n t e r i o r - p o s t e r i o r f o r c e , Fyk, i s n e g a t i v e and t h e r e f o r e d i r -e c t e d p o s t e r i o r l y f o r the d u r a t i o n o f the s k a t i n g t h r u s t . M uscle and Ligament F o r c e s M u s c l e and l i g a m e n t f o r c e s c o r r e s p o n d i n g to the ex-t e r n a l f o r c e system o f F i g . 4 . 0 6 and the electromyograms o f F i g . 4 . 0 2 are p l o t t e d i n F i g . 4 . 0 7 f o r one complete s k a t i n g t h r u s t . Muscle and l i g a m e n t f o r c e s f o r the 15 t r i a l s o f the stu d y are superimposed i n F i g . 4 .08 to i n d i c a t e the v a r i a t i o n i n t he r e s u l t s o b t a i n e d w h i l e maximum f o r c e s developed d u r i n g each o f the 15 t r i a l s are g i v e n i n Table No.'3-Peak f o r c e s d e t e r m i n e d i n the q u a d r i c e p s f e m o r i s , h a m s t r i n g s and g a s t r o c n e m i u s muscle groups were 665, 273 and 118 pounds r e s p e c t i v e l y . As i n d i c a t e d i n F i g . 4 . 0 8 , peak q u a d r i c e p s f o r c e s o c c u r r e d d u r i n g the su p p o r t phase as the s k a t e r extended the s u p p o r t l e g :to d r i v e f o r w a r d whereas maximum h a m s t r i n g s f o r c e s were r e c o r d e d i m m e d i a t e l y b e f o r e the commencement o f the su p p o r t phase. The g a s t r o c n e m i u s muscle, a c t i v e o n l y a t push o f f , developed c o n s i d e r a b l y s m a l l e r muscle f o r c e s . Maximum f o r c e s a c t i n g i n the l i g a m e n t s were r e s p e c t -i v e l y 240 and 62 pounds i n the p o s t e r i o r and a n t e r i o r c r u -- 85 -FIG.4.07 M U S C L E A N D L I G A M E N T F O R C E S FOR TRIAL NO.4 - 86 -FIG. 4 .08. M U S C L E AND L IGAMENT FORCES - 8? -c i a t e s and 289 and 258 pounds i n the m e d i a l and l a t e r a l c o l -l a t e r a l s . The p o s t e r i o r c r u c i a t e l i g a m e n t was ten s e f o r the d u r a t i o n o f the s k a t i n g t h r u s t w i t h peak f o r c e s o c c u r r i n g a t the onset o f t h e su p p o r t phase w h i l e the a n t e r i o r c r u c i a t e s were s l a c k u n t i l push o f f . The maximum m e d i a l c o l l a t e r a l f o r c e s were r e c o r d e d a t push o f f as opposed to peak l a t e r a l c o l l a t e r a l f o r c e s which o c c u r r e d d u r i n g the i n i t i a l p o r t i o n o f the s u p p o r t phase as the s k a t e r moved f o r w a r d and l a t e r -a l l y . I t i s i n t e r e s t i n g t o observe t h a t peak h a m s t r i n g s muscle f o r c e s were c o i n c i d e n t s e q u e n t i a l l y w i t h maximum pos-t e r i o r c r u c i a t e l i g a m e n t f o r c e s and maximum g a s t r o c n e m i u s muscle f o r c e s were c o i n c i d e n t w i t h maximum m e d i a l c o l l a t e r a l l i g a m e n t f o r c e s . On the o t h e r hand, a t t h a t time o f peak q u a d r i c e p s f o r c e , the c o l l a t e r a l l i g a m e n t s were s l a c k , no f o r c e was r e c o r d e d i n the a n t e r i o r c r u c i a t e l i g a m e n t and o n l y m i n i m a l f o r c e s were i n d i c a t e d i n the p o s t e r i o r c r u c i a t e l i g a m e n t . A r t i c u l a r F o r c e s The magnitude o f the normal j o i n t f o r c e Rz a c t i n g a t the knee d u r i n g the s k a t i n g t h r u s t i s p l o t t e d i n F i g . 4 . 0 9 . The maximum v a l u e s o f Rz v a r i e d between 666 pounds and 888 pounds f o r the 15 t e s t t r i a l s w i t h the peak v a l u e o f Rz c o i n c i d e n t i n time w i t h the maximum q u a d r i c e p s f o r c e r e c -orded. The maximum v a l u e s o f Rz deve l o p e d r e p r e s e n t j o i n t f o r c e s 4 . 2 to 5-6 t i m e s the 165 pound body w e i g h t o f the -.88 -FIG.4.09 JOINT FORCES A C T I N G AT K N E E - 89 -test subject. The v a r i a t i o n of the medial-lateral shear force Rx on the a r t i c u l a r surfaces of the joi n t i s shown i n Fig'. 4 . 0 9 . The shear force acts medially during the support phase of the skating thrust but i s l a t e r a l at push off. The maximum medial and l a t e r a l shear forces recorded were 96 and 57 pounds respectively. The anterior-posterior force, Ry, on the knee j o i n t i s assumed to be equilibrated "by the forces developed i n the cruciate ligaments, either the posterior cruciate or the ant-e r i o r cruciate dependent upon the d i r e c t i o n of the force, Ry. Fig. 3.24. The maximum values of Mz, the torque on the j o i n t about the long axis of the t i b i a , were developed during the support phase of the skating thrust and were coincident with peak quadriceps muscle forces. A plot of Mz i s given i n F i g . 4 . 0 9 for T r i a l No. 4 with peak values of torque for each test t r i a l given i n Table No. 3- The maximum values of torque on the joi n t varied from 455 to 735 pound-inches and were directed medially inward. Discussion The summary of re s u l t s as shown i n Fig. 4 . 0 8 and as tabulated i n Table No. 3 indicates that although the c y c l i c pattern of muscle and ligament forces was quite consistent from t r i a l to t r i a l , there was considerable v a r i a t i o n i n TABLE NO. 3 MAXIMUM MUSCLE LIGAMENT AND JOINT FORCES FOR THE SIMULATED SKATING THRUST TR Ph Pq Pg Pp Pa Pm PI Rz Rx Mz 2 193-6? 618.40 118.62 165.01 6.95 2.69 36.39 843.45 54.01 486. ?8. 3 272.92 613.89 20.31 239.51 20 .64 37 .04 131.11 822.28 -50.95 573-90 4 144.35 6OI.38 24.55 140.86 25.03 159.69 30.89 802.06 64.54 566.34 5 100.94 506.99 50.69 111.34 21.52 17.80 202.11 707.34 61.83 521.60 6 116.42 518.75 60.81 103.25 8.30 59 .48 69.95 721.24 -50-39 413 :81 7 30.75 633-39 57-51 73.01 4.99 86.01 15.05 855.18 68 .84 510.59 9 20.54 453.08 24.19 90 .42 17 .44 207 .42 102.61 666 t 44 82.97 455.37 10 87.43 639 .42 106.15 75-56 4. 74 178.71 51.06 872.22 64.53 497.96 11 74.06 545.16 29.10 86.76 9.. 28 79-13 52.68 767.69 -56.87 522.78 12 95-98 582.18 18.86 74.61 14.96 158.15 56.57 803.88 96.09 610.95 13 151.36 540.24 24.43 131.60 12.47 221.60 172.85 765.80 -61.27 590.76 14 40 .08 665.52 21.59 43.99 29.99 145.26 109.83 878.11 -55-52 735.62 15 78.89 660.43 25.14 74.99 61.76 164.33 194.21 888.29 47.63 675.50 16 130.25 663.77 43.11 139.83 17.92 288.88 258.81 885.78 96- 74 700.47 17 111 .01 625.56 25.34 95-57 12 .81 41.50 191-53 852.17 -56.16 674.23 muscle, ligament and j o i n t forces = pounds j o i n t torque = pound-inches - 91 -the peak f o r c e s r e c o r d e d . The v a r i a n c e i n the r e s u l t s can he l a r g e l y a t t r i b u t e d to the f a c t t h a t i t was e x t r e m e l y d i f -f i c u l t f o r the t e s t s u b j e c t to e x a c t l y d u p l i c a t e the s i m u l a t e d s k a t i n g t h r u s t f o r each t r i a l . M o r r i s o n (1970) a l s o found some v a r i a t i o n i n the muscle and j o i n t f o r c e s o f a s i n g l e t e s t s u b j e c t when t e s t e d t w i c e f o r s i m p l e l e v e l w a l k i n g . Due t o the l i m i t a t i o n s o f the a n a l y s i s , the antagon-i s t i c muscle a c t i o n s ( F i g . 4 . 0 2 ) c o u l d not be c o n s i d e r e d and t h e r e f o r e the v a l u e s o f muscle f o r c e c a l c u l a t e d are the m i n i -mum r e q u i r e d f o r e q u i l i b r i u m o f the e x t e r n a l f o r c e system. These minimum v a l u e s are a l s o r e f l e c t e d i n t h e v a l u e s o f l i g a m e n t f o r c e s and a r t i c u l a r f o r c e s . M u scle F o r c e s -The f o r c e s d e v e l o p e d i n the t h r e e muscle groups can be e x p l a i n e d w i t h r e f e r e n c e t o the a c t i o n s o f the s k a t i n g t h r u s t and the assumptions made i n the a n a l y s i s . The i n i t i a l move-ment o f the s k a t i n g t h r u s t i s a s h i f t o f w e i g h t from the r e a r f o o t t o the f o r w a r d s u p p o r t o r t h r u s t i n g f o o t . S i n c e the s u p p o r t f o o t i s a n t e r i o r to the knee a t t h i s p o i n t , F i g . 4 . 0 3 , the v e r t i c a l f o r c e , Fzp, on the s u p p o r t f o o t , F i g . 3 -20 , produces a n e g a t i v e •'.•moment Mxk on the knee t h a t tends t o extend the j o i n t . T h i s moment a c t i o n i s r e s i s t e d by f o r c e a c t i o n i n the h a m s t r i n g s , t o s t a b i l i z e the knee j o i n t as the f o r w a r d l e g assumes the body w e i g h t . The electromyogram o f F i g . 4 . 0 2 shows t h a t b o t h q u a d r i c e p s and h a m s t r i n g s muscle - 92 -groups a re a c t i v e d u r i n g t h i s phase but the magnitude o f q u a d r i c e p s f o r c e c o u l d n o t be e v a l u a t e d due t o the i n d e t e r -minancy o f the a n a l y s i s . S i n c e the f o r c e a c t i o n o f the q u a d r i c e p s i s a n t a g o n i s t i c t o the f o r c e a c t i o n o f the ham-s t r i n g s , t he v a l u e o f h a m s t r i n g s f o r c e c a l c u l a t e d i s by d e f i n i t i o n t he minimum muscle f o r c e r e q u i r e d t o b a l a n c e the e x t e r n a l moment, Mxk. The a c t u a l f o r c e a c t i n g i n the ham-s t r i n g s i s e q u a l t o the sum o f (a) the f o r c e a c t i o n r e q u i r e d t o r e s i s t t he e x t e r n a l moment, Mxk, and (b) the f o r c e a c t i o n r e q u i r e d t o b a l a n c e the antagon-i s t i c f o r c e a c t i o n o f the q u a d r i c e p s on the j o i n t . S i n c e the e l e c t r i c a l a c t i v i t y (EMG) o f the muscle groups was r e c o r d e d w i t h s u r f a c e e l e c t r o d e s , i n d i v i d u a l muscle a c t i v i t y w i t h i n a muscle group c o u l d n o t be a c c u r a t e l y d e f i n e d . I t i s p o s s i b l e t h a t the q u a d r i c e p s a c t i v i t y r e c -orded f o r t h i s phase was due to c o n t r a c t i o n p f the b i a r t i -c u l a r r e c t u s f e m o r i s muscle, the o n l y member o f the q u a d r i -ceps group t h a t c r o s s e s the h i p j o i n t and t h a t can t h e r e f o r e f l e x the h i p as w e l l as extend the. knee ( B a s m a j i a n , 1970 ) . T h i s d e d u c t i o n agrees w i t h the a n a l y s i s o f P a u l (1966) o f the f u n c t i o n o f the h i p i n . w a l k i n g . As the s k a t e r d r i v e s f o r w a r d and the knee moves ahead o f the s u p p o r t f o o t , F i g . 4.03> the j o i n t i s s u b j e c t t o a p o s i t i v e Mxk due t o the v e r t i c a l f o r c e , Fzp, and the p o s t -e r i o r f o r c e , Fyp, on the f o o t , F i g . 3 -20 . S i n c e a p o s i t i v e - 93 -Mxk t e n d s t o f l e x t he knee, e q u i l i b r i u m i s o b t a i n e d by f o r c e a c t i o n i n the q u a d r i c e p s f e m o r i s . A l t h o u g h electromyogram r e c o r d s , F i g . 4 . 0 2 , i n d i c a t e t h a t b o t h q u a d r i c e p s and g a s t r o -cnemius muscle groups a re a c t i v e i n the su p p o r t phase, the f o r c e a c t i o n o f the g a s t r o c n e m i u s muscle i s not c o n s i d e r e d to a f f e c t t h e knee. B e i n g a b i a r t i c u l a r muscle, the g a s t r o -cnemius can b o t h f l e x t he knee and produce p l a n t a r f l e x i o n o f the a n k l e . However, the f i b r e s o f the g a s t r o c n e m i u s a re so s h o r t t h a t t h i s muscle, cannot f l e x the knee and p l a n t a r f l e x the a n k l e a t the same time because o f t h e s l a c k i n the muscle. ( B a s m a j i a n , 1970 ) . I t i s t h e r e f o r e assumed t h a t i n the s u p p o r t phase o f the s k a t i n g t h r u s t , the g a s t r o c n e m i u s r e s i s t s d o r s i - f l e x i o n as the shank moves f o r w a r d over the f o o t and does n ot c o n t r i b u t e t o f o r c e a c t i o n s a t the knee. At push o f f , moment Mxk becomes n e g a t i v e due to the a n t e r i o r f o r c e Fyp on the f o o t , F i g . 3*20, and the i n c r e a s e d i n e r t i a l f o r c e s o f the shank and f o o t as the l o w e r l i m b a c c e l e r a t e s f o r w a r d , F i g . 4 . 0 4 . S i n c e the h a m s t r i n g s a re i n a c t i v e a t push o f f and the f o r c e a c t i o n i n the q u a d r i c e p s f e m o r i s i s assumed t o i m p a r t the f o r w a r d s a c c e l e r a t i o n t o the l e g , the n e g a t i v e e x t e n s i o n moment, Myk, devel o p e d a t push o f f must be e q u i l i b r a t e d by f o r c e a c t i o n i n the g a s t r o -cnemius muscle. As the g a s t r o c n e m i u s cannot f l e x t he knee and p l a n t a r f l e x the a n k l e a t the same time p l a n t a r f l e x i o n o f t he a n k l e a t push o f f must be due to f o r c e a c t i o n o f the s o l e u s muscle, the g r e a t p l a n t a r f l e x o r o f the a n k l e and a - 94 -member o f the t r i c e p s surae muscle group o f the l o w e r l e g . Ligament F o r c e s The f o r c e s c a l c u l a t e d , i n the l i g a m e n t s must be con-sidered, i n c o n j u n c t i o n , w i t h the l i m i t a t i o n s o f the s i m p l i f i e d muscle and l i g a m e n t system o f F i g . 3-12 and the e q u a t i o n s used t o o b t a i n e q u i l i b r i u m c o n d i t i o n s , S e c t i o n 3-The a n t e r i o r - p o s t e r i o r f o r c e s a c t i n g on the j o i n t were assumed t o be b a l a n c e d by f o r c e s d eveloped i n the c r u c i -a t e l i g a m e n t s . F o r the d u r a t i o n o f the s k a t i n g t h r u s t , e x c e p t i n g push o f f , t h e r e i s a n e g a t i v e p o s t e r i o r f o r c e , Fyk, imposed on the knee, F i g . 4 . 0 6 . E q u i l i b r i u m c o n d i t i o n s a t t h e j o i n t a r e m a i n t a i n e d by a t e n s i l e f o r c e d e v e l o p e d i n the p o s t e r i o r c r u c i a t e , F i g . 3 ' 2 4 . The h o r i z o n t a l component o f the h a m s t r i n g s muscle f o r c e a l s o imposes a p o s t e r i o r f o r c e on t h e . j o i n t w h i c h must be b a l a n c e d by an i n c r e a s e d t e n s i l e ' f o r c e i n the p o s t e r i o r c r u c i a t e . T h i s e x p l a i n s the c o i n c i -dence o f peak p o s t e r i o r c r u c i a t e l i g a m e n t f o r c e and maximum h a m s t r i n g s muscle f o r c e as mentioned i n the r e s u l t s s e c t i o n ; i l l u s t r a t e d i n F i g . 4 . 0 8 , and summarized i n Table No. 3 ' As the s k a t e r t h r u s t s f o r w a r d t o push o f f , the p o s t e r i o r f o r c e , Fyk, on the j o i n t d e c r e a s e s w h i l e the h o r i z o n t a l component o f the q u a d r i c e p s f o r c e on the j o i n t becomes a n t e r i o r as the h i p j o i n t moves ahead o f the knee, F i g . 4 . 0 3 . The . r e s u l t i s a s m a l l a n t e r i o r f o r c e on the knee which i s b a l a n c e d by a s m a l l t e n s i l e f o r c e a c t i n g i n the a n t e r i o r c r u c i a t e . I n a l l - 95 -t e s t s , the p o s t e r i o r c r u c i a t e c a r r i e d the g r e a t e r f o r c e , a maximum f o r c e o f 240 pounds compared to 62 pounds i n the a n t e r i o r c r u c i a t e . The f o r c e o f f r i c t i o n a c t i n g a t t h e a r t -i c u l a r s u r f a c e s o f the j o i n t was not c o n s i d e r e d when c a l -c u l a t i n g the c r u c i a t e f o r c e s . S i n c e the f e m o r a l c o n d y l e s r o t a t e backward on t h e t i b i a d u r i n g the s k a t i n g t h r u s t , t h e f o r c e o f f r i c t i o n on the j o i n t i s d i r e c t e d p o s t e r i o r l y and would t h e r e f o r e i n c r e a s e the t e n s i l e f o r c e i n the p o s t e r i o r c r u c i a t e . Assuming a v a l u e o f 0.02 f o r the c o e f f i c i e n t o f f r i c t i o n i n the i j o i n t ( R y d e l l , 1966), the maximum f o r c e o f f r i c t i o n would be i n the o r d e r o f 18 pounds a c t i n g a t the p o i n t i n the s k a t i n g t h r u s t when t h e normal j o i n t f o r c e i s maximum. M e d i a l - l a t e r a l s t a b i l i t y o f the knee i s m a i n t a i n e d by t e n s i l e f o r c e s a c t i n g i n the c o l l a t e r a l l i g a m e n t s o f the j o i n t . The v a l g u s - v a r u s moment, Myk, i s r e s i s t e d by a f o r c e i n e i t h e r the m e d i a l c o l l a t e r a l o r l a t e r a l c o l l a t e r a l when-ever the c e n t r e o f p r e s s u r e o f the normal j o i n t f o r c e , Rz, f a l l s o u t s i d e the l i m i t i n g v a l u e Xo, as shown i n F i g . 3-26. As t h e s k a t e r moves f o r w a r d on t o t h e s u p p o r t l e g , F i g . 4.03> the knee j o i n t i s l a t e r a l w i t h r e s p e c t t o the v e r t i c a l f o r c e , Fzp, on the f o o t , F i g . 3-20, and t h i s d e v e l o p s a p o s i t i v e moment, Myk, on the j o i n t . T h i s p o s i t i v e moment tends to s e p a r a t e the l a t e r a l c o n d y l e s o f the j o i n t and e q u i l i b r i u m i s a c h i e v e d t h r o u g h a t e n s i l e f o r c e i n the l a t e r a l c o l l a t e r a l . The magnitude o f the l a t e r a l c o l l a t e r a l f o r c e i s d i r e c t l y , _ 9 6 -r e l a t e d t o t h e r e l a t i v e a n g l e o f the shank t o the v e r t i c a l . T h i s a c c o u n t s f o r the wide range i n t e n s i l e f o r c e s c a l c u l a t e d i n t he l a t e r a l c o l l a t e r a l as r e p r e s e n t e d i n Ta b l e No. J. Minimum v a l u e s c o r r e s p o n d t o u p r i g h t a n g l e s o f the shank w h i l e maximum v a l u e s a c t when t h e knee i s l a t e r a l a s i g n i f i ^ c a n t degree w i t h r e s p e c t t o the f o o t . A p p r o a c h i n g push o f f i n t he s k a t i n g t h r u s t , the knee moves m e d i a l l y and i n s i d e the v e r t i c a l f o r c e , Fzp, on the f o o t , F i g . 3*20, and i s sub-j e c t t o a n e g a t i v e moment, Myk. T h i s n e g a t i v e Myk tends t o s e p a r a t e t h e m e d i a l c o n d y l e s o f the j o i n t and i s r e s i s t e d by a t e n s i l e f o r c e i n the m e d i a l c o l l a t e r a l . A g a i n the mag-n i t u d e o f the f o r c e i n the- m e d i a l c o l l a t e r a l i s dependent on the a n g l e o f the shank w i t h maximum v a l u e s a c t i n g when the knee i s m e d i a l a maximum amount w i t h r e s p e c t t o the f o o t . T a b l e No. 3 i n d i c a t e s t h a t when a l a r g e m e d i a l c o l l a t e r a l f o r c e i s de v e l o p e d d u r i n g a s k a t i n g t h r u s t , the f o r c e i n the l a t e r a l c o l l a t e r a l i s m i n i m a l and v i c e v e r s a . P a r t o f the v a l g u s - v a r u s moment, Myk, may be r e s i s t e d by the c r u c i a t e s . B r a n t i g a n and V o s h e l l (1941) demonstrated t h a t when the knee i s i n f l e x i o n some c r u c i a t e l i g a m e n t t e n -s i o n i s r e q u i r e d t o a s s i s t m e d i a l - l a t e r a l i n s t a b i l i t y o f the j o i n t due t o a s l a c k e n i n g o f the l a t e r a l c o l l a t e r a l . How-eve r , F i g . 4 . 0 6 i n d i c a t e s t h a t moment Myk i s n e g a t i v e f o r the f l e x i o n p o r t i o n o f the s k a t i n g t h r u s t and no f o r c e a c t s i n t he c o l l a t e r a l s ' d u r i n g t h i s phase, F i g . 4.08. On the o t h e r hand some o f the f o r c e a t t r i b u t e d t o the l a t e r a l c o l -- 97 -l a t e r a l l i g a m e n t i s p r o b a b l y c a r r i e d as t e n s i o n i n the i l i o -t i b i a l t r a c t . A l s o , McCloy (1959) has s t a t e d t h a t i t i s p o s s i b l e t h a t a p o r t i o n o f the f o r c e s a s c r i b e d t o the c o l -l a t e r a l l i g a m e n t s may be t a k e n by the q u a d r i c e p s muscle group s i n c e the v a s t u s m e d i a l i s and v a s t u s l a t e r a l i s b l e n d p o s t e r i -o r l y w i t h the c o l l a t e r a l s t h r o u g h t h e i r i n s e r t i o n i n t h e r e c t i n a c u l u m p a t e l a e and i t s ' a s s o c i a t i o n w i t h the c a p s u l a r l i g a m e n t o f the j o i n t . M o r r i s o n (1970) f u r t h e r s u g g e s t s t h a t the v a l g u s - v a r u s moment i s p a r t l y e q u i l i b r a t e d hy d i f f e r e n -t i a l f o r c e a c t i o n o f the l a t e r a l and m e d i a l h a m s t r i n g s o r the l a t e r a l and m e d i a l heads o f the g a s t r o c n e m i u s . These s t a t e -ments a re not e w o r t h y i f one a c c e p t s the t h e o r y o f s e v e r a l a u t h o r s such as S m i l l i e (1951) t h a t t e n s i o n r e c e p t o r s i n the a r t i c u l a r l i g a m e n t s , when a c t i v i a t e d , produce a r e f l e x con-t r a c t i o n i n t h e a s s o c i a t e d muscle groups. T h i s t h e o r y has not been p r o v e n and i n f a c t i s c o n t r a d i c t e d by S t e n e r ' s (1959) e x p e r i m e n t s f o r the m e d i a l c o l l a t e r a l l i g a m e n t o f the knee. A r t i c u l a r F o r c e s The j o i n t f o r c e , Rz, which a c t s normal t o the a r t i -c u l a r s u r f a c e o f the knee j o i n t a t t a i n s a peak v a l u e a t a p o i n t i n the su p p o r t phase o f the s k a t i n g t h r u s t , F i g . 4 . 0 9 , when the sum o f the.normal component o f the q u a d r i c e p s muscle f o r c e p l u s the e x t e r n a l f o r c e , Fzk, i s maximal, E q u a t i o n 3 - 1 7 . The maximum v a l u e o f Rz devel o p e d i n the - 98 -s k a t i n g t h r u s t was 888 pounds o r 5-60 t i m e s the body w e i g h t o f the t e s t s u b j e c t . T e s t r e s u l t s i n d i c a t e t h a t d u r i n g the sup p o r t phase o f the s k a t i n g t h r u s t t h e c e n t r e o f p r e s s u r e o f j o i n t f o r c e , Rz, i s p o s i t i o n e d i n i t i a l l y over the m e d i a l c o n d y l e s but t h e n s h i f t s t o the l a t e r a l c o n d y l e s a t push o f f , F i g . 4 . 1 0 . S i n c e the m e d i a l c o n d y l e has a l a r g e r a n a t o m i c a l b e a r i n g a r e a com-par e d t o the l a t e r a l c o n d y l e , F i g . 3 . 0 2 , i t f o l l o w s t h a t the comp r e s s i v e s t r e s s imposed a t t h e m e d i a l a r t i c u l a r s u r f a c e o f t h e j o i n t i s s m a l l e r t h a n the s t r e s s o f the l a t e r a l a r t i -c u l a r s u r f a c e . A l s o , as M o r r i s o n (1970) has note d , the s t r e s -ses a c t i n g i n the s h a f t o f the t i b i a would be l o w e r on the m e d i a l s i d e because the m e d i a l c o n d y l e overhangs the s h a f t o f the t i b i a l e s s t h a n the l a t e r a l c o n d y l e . ' The s i d e o r . s h e a r f o r c e , Rx, on the j o i n t i s r e l a -t i v e l y s m a l l , a p p r o x i m a t e l y 0 .10 t i m e s the magnitude o f the normal a r t i c u l a r f o r c e , Rz, F i g . 4 . 0 9 . The m e d i a l o r l a t e r a l movement o f the femur on the t i b i a i s c o n s i d e r e d t o be r e s -i s t e d by t h e t i b i a l i n t e r c o n d y l a r eminence, by a f r i c t i o n f o r c e between the c o n d y l e s and by t e n s i o n i n the l i g a m e n t s t r u c t u r e s ( B r a n t i g a n and V o s h e l l , 1941) . The femur tends to move l a t e r a l l y on the t i b i a d u r i n g the s u p p o r t phase o f the s k a t i n g t h r u s t and m e d i a l l y a t push o f f . The magnitude and d i r e c t i o n o f the a n t e r i o r - p o s t e r i o r f o r c e , Ry, on the j o i n t i s r e f l e c t e d by the f o r c e s a c t i n g i n t he c r u c i a t e l i g a m e n t s . F o r the g r e a t e r p o r t i o n o f the - 100 -s k a t i n g t h r u s t t he femur tends t o g l i d e f o r w a r d on the t i b i a and i s c o n t r o l l e d by a t e n s i o n d e v e l o p e d i n the p o s t e r i o r c r u c i a t e l i g a m e n t , F i g . 4 . 0 8 . At push o f f the femur i s f o r c e d backward r e l a t i v e to the t i b i a and t h i s m o t i o n i s r e s i s t e d by t e n s i o n i n the a n t e r i o r c r u c i a t e . As s t a t e d p r e v i o u s l y , t he e f f e c t s o f f r i c t i o n i n the j o i n t i n the a n t e r i o r - p o s t e r i o r d i r e c t i o n were n e g l e c t e d . J o i n t Torque F i g . 4 . 0 9 i n d i c a t e s a p o s i t i v e i n w a r d t o r q u e , Mz, ,v a c t i n g on the knee d u r i n g the s k a t i n g t h r u s t . T h i s t o r q u e produces m e d i a l r o t a t i o n o f the t i b i a on the femur which i s r e s i s t e d by the c o l l a t e r a l l i g a m e n t s . ( S t e i n d l e r , 1955) A c c o r d i n g t o M o r r i s o n (1970) an i n w a r d t o r q u e on the j o i n t i s r e s i s t e d by the o b l i q u e p o s t e r i o r f i b r e s o f the m e d i a l c o l l a t e r a l l i g a m e n t w h i l e B r a n t i g a n and V o s h e l l (1941) dem-o n s t r a t e d on i n t a c t knee j o i n t s t h a t m e d i a l r o t a t i o n i s b a l a n c e d by t e n s i o n i n a t i g h t e n i n g l a t e r a l c o l l a t e r a l l i g a -ment and t h e c r u c i a t e s as t h e y t w i s t on th e m s e l v e s . I t i s t h e r e f o r e a p p arent t h a t moment a c t i o n , Mz, about the l o n g a x i s o f the t i b i a must be b a l a n c e d by f o r c e a c t i o n s i n the a r t i c u l a r l i g a m e n t s and w i l l a l t e r the d i s t r i b u t i o n o f the l i g a m e n t f o r c e s as c a l c u l a t e d i n the a n a l y s i s . I t i s a l s o suggested by c e r t a i n a u t h o r s ( S t e i n d l e r , 1965; R a d i n and P a u l , 1970) t h a t t he m e n i s c i and t e n s o r f a s c i a l a t a e a i d i n r e s i s t i n g t he e f f e c t s o f a t o r q u e on the knee j o i n t . - 101 -F o r c e s a t the Knee i n V a r i o u s A c t i v i t i e s The p r e c e d i n g r e s u l t s r e v e a l l i t t l e c o n c e r n i n g the r e l a t i v e s t r a i n imposed on the l i g a m e n t s and muscles o f the knee s i n c e the a c t u a l t e n s i l e s t r e n g t h and s t r e s s - s t r a i n c h a r a c t e r i s t i c o f human l i g a m e n t and muscle t i s s u e has not been e x p e r i m e n t a l l y d etermined t o t h i s d a t e . C o n s i d e r a b l e r e s e a r c h has been done by T i p t o n and a s s o c i a t e s (1967, 1975) and Zuckerman (1969 . 1973) on the s t r e n g t h o f l i g a m e n t s from a n i m a l s but i t i s d i f f i c u l t i f n o t i m p o s s i b l e t o r e l a t e t h e s e r e s u l t s t o the s t r e n g t h o f human l i g a m e n t o u s t i s s u e . There-f o r e we are f o r c e d t o e v a l u a t e the s i g n i f i c a n c e o f the p r e s -ent s t u d y "by comparison w i t h r e s u l t s o f s i m i l a r s t u d i e s f o r o t h e r a c t i v i t i e s . ( M o r r i s o n , 1969) The mean maximum muscle and l i g a m e n t f o r c e s c a l c u l a t e d f o r the s k a t i n g t h r u s t and f o r v a r i o u s w a l k i n g a c t i v i t i e s are l i s t e d i n Table No. 4 . The peak s k a t i n g f o r c e s r e p r e s e n t the mean o f f i v e t r i a l s on a s i n g l e s u b j e c t w h i l e the f o r c e s f o r w a l k i n g r e p r e s e n t the mean from t h r e e t e s t s u b j e c t s . Upon e x a m i n a t i o n o f the t a b u l a t e d r e s u l t s , i t i s apparent t h a t the maximum f o r c e s d e v e l o p e d i n the c o l l a t e r a l l i g a m e n t s , i n p a r t i c u l a r the m e d i a l c o l l a t e r a l , are s i g n i f i c a n t l y g r e a t e r d u r i n g a s k a t i n g t h r u s t as compared t o l e v e l o r i n -c l i n e d w a l k i n g . The a n t e r i o r c r u c i a t e f o r c e f o r the s k a t i n g t h r u s t i s comparable to t h a t d e v e l o p e d i n l e v e l w a l k i n g but o n l y 0 . 3 t i m e s the t e n s i l e f o r c e a c t i n g i n the a n t e r i o r c r u c i a t e when w a l k i n g down an i n c l i n e d ramp. The f o r c e a c t i n g i n the p o s t e r i o r c r u c i a t e d u r i n g the s k a t i n g t h r u s t i s - 102 -TABLE NO. 4 THE MAXIMUM MUSCLE AND LIGAME'NT FORCES OF VARIOUS ACTIVITIES A c t i v i t y Muscle F o r c e , l b s Ligament F o r c e , l b s H Q G pc ac mc 1c L e v e l W a l k i n g 309 191 262 79 38 17 50 W a l k 1 ng\" Up V R amp 240 176 335 144 15 16 158 W a l k i n g Down Ramp 189 430 - 59 100 19 62 W a l k i n g Up S t a i r s 177 433 79 273 6 9 156 W a l k i n g Down S t a i r s 88 380 155 101 21 19 . 80 S k a t i n g T h r u s t 179 652 79 164 32 212 204 PC . p o s t e r i o r c r u c i a t e Q q u a d r i c e p s ac a n t e r i o r c r u c i a t e G g a s t r o c n e m i u s mc m e d i a l c o l l a t e r a l l c l a t e r a l c o l l a t e r a l - 103 -t w i c e as g r e a t as t h e f o r c e f o r l e v e l w a l k i n g but l e s s t h a n the p o s t e r i o r c r u c i a t e f o r c e c a l c u l a t e d when w a l k i n g up s t a i r s . The peak muscle f o r c e f o r the s k a t i n g t h r u s t , 652 pounds i n the q u a d r i c e p s , i s double the maximum muscle f o r c e r e c o r d e d i n l e v e l w a l k i n g , 309 pounds i n t h e h a m s t r i n g s . A d i r e c t c o m parison o f a c t i v i t i e s s u g g e s t s t h a t the s k a t i n g t h r u s t i s comparable to w a l k i n g u p s t a i r s i n t h a t the q u a d r i c e p s muscle group e x e r t s the maximum f o r c e i n each ca s e , w h i l e t h e f o r c e s a c t i n g i n t h e g a s t r o c n e m i u s and ham-s t r i n g s muscle groups a r e much s m a l l e r . However, the l a r g e f o r c e s d e v e l o p e d i n the q u a d r i c e p s muscle group and the c o l l a t e r a l l i g a m e n t s d u r i n g a s k a t i n g t h r u s t makes i t unique when compared t o e i t h e r s t a i r o r ramp w a l k i n g . Peak a r t i c u l a r f o r c e s i n c l u d i n g j o i n t t o r q u e a c t i n g a t the knee f o r the s k a t i n g t h r u s t and the v a r i o u s w a l k i n g a c t i v i t i e s s t u d i e d by M o r r i s o n (1969) are l i s t e d i n Table No. 5- A g a i n t h e s e r e s u l t s r e p r e s e n t t h e maximum mean o f f i v e t r i a l s on a s i n g l e s u b j e c t f o r the s k a t i n g t h r u s t and the maximum mean from t h r e e t e s t s u b j e c t s f o r the w a l k i n g a c t i v i t i e s . From T a b l e No. '5 i"t i s e v i d e n t t h a t the normal o r v e r t i c a l j o i n t f o r c e , Rz, i s s i g n i f i c a n t l y l a r g e r f o r s k a t i n g , 5-^8 t i m e s body weight compared to 4 . 2 5 t i m e s body w e i g h t f o r w a l k i n g u p s t a i r s and 3 .40 t i m e s body weight f o r l e v e l w a l k i n g . P r o b a b l y t h e s e v e r e s t f o r c e imposed on t h e knee j o i n t d u r i n g the s k a t i n g t h r u s t i s the j o i n t t o r q u e . I n the s k a t i n g - 104 -TABLE NO. 5 MAXIMUM JOINT FORCES AND TORQUES OF VARIOUS ACTIVITIES A c t i v i t y Max. J o i n t F o r c e Mz Rz/bw Rx/bw l b - i n s L e v e l W a l k i n g 3.40 0 .26 239 W a l k i n g Up Ramp 3-97 - -W a l k i n g Down Ramp 3-95 - -W a l k i n g Up S t a i r s 4 . 2 5 0 .89 -W a l k i n g Down S t a i r s 3 .83 - -S k a t i n g T h r u s t " 5.48 0 . 5 1 696 Rz/bw Rx/bw Mz. . . R a t i o normal j o i n t f o r c e t o body weight R a t i o s hear f o r c e t o body w e i g h t J o i n t t o r q u e - 10'5 -t h r u s t a mean maximum t o r q u e o f 696 pound-inches was deve l o p e d compared t o a maximum o f 239 pound-inches f o r l e v e l w a l k i n g . T h i s l a r g e t o r q u e on the j o i n t has s e r i o u s i m p l i c a t i o n s when i t i s c o n s i d e r e d t h a t the maximum v a l u e s o f t o r q u e and v e r t i c a l j o i n t f o r c e a c t a t the same p o i n t i n time d u r i n g the s k a t i n g t h r u s t , F i g . 4 . 0 9 . - 106 -CHAPTER V SUMMARY AND CONCLUSIONS The knee i s p r o b a b l y s u b j e c t to more s t r e s s and. s t r a i n t h a n any o t h e r j o i n t o f the body. The s t a b i l i t y o f the knee i s m a i n t a i n e d by the l i g a m e n t o u s s t r u c t u r e o f the j o i n t , the a r t i c u l a r c a r t i l a g e and the bone a r c h i t e c t u r e o f the t i b i a and femur. D u r i n g an a t h l e t i c movement such as the s k a t i n g t h r u s t , c o n t r a c t i l e muscle f o r c e s p l u s e x t e r n a l f o r c e s o f r e a c t i o n impose a c o n s i d e r a b l e s t r e s s on the knee. These f o r c e s are n e c e s s a r i l y b a l a n c e d by the a r t i c u l a r s u r f a c e s o f the j o i n t and by the t e n s i l e f o r c e s d e v e l o p e d i n the l i g a m e n t s t o p r e v e n t r e l a t i v e d i s p l a c e m e n t s o f the j o i n t . The p r e s e n t s t u d y was d e s i g n e d to determine the magni-tude and t e m p o r a l sequence o f muscle, l i g a m e n t and a r t i c u l a r f o r c e s a c t i n g a t the knee j o i n t d u r i n g a s i m u l a t e d s k a t i n g t h r u s t . One male t e s t s u b j e c t , an a c c o m p l i s h e d hockey p l a y e r , made 15 s i m u l a t e d s k a t i n g t h r u s t s . The r e a c t i o n f o r c e s and p o i n t o f a p p l i c a t i o n o f t h e s e f o r c e s d u r i n g the t h r u s t were r e c o r d e d w i t h the K i s t l e r f o r c e p l a t e a t the l a b o r a t o r y o f the U n i v e r s i t y o f Washington. A s y n c h r o n i z e d c i n e f i l m r e c o r d o f each t r i a l was o b t a i n e d from 16 mm f i l m i n b o t h the f r o n t a l and l a t e r a l p l a n e s . E l e c t r o m y o g r a p h i c ' d a t a f o r the t h r e e main muscle groups ( h a m s t r i n g s , g a s t r o c n e m i u s and q u a d r i c e p s ) - 107 -a c t i v e d u r i n g the s k a t i n g t h r u s t was o b t a i n e d from t e s t s conducted o n - i c e and o f f - i c e a t the U n i v e r s i t y o f B r i t i s h Columbia. A n t h r o p o m e t r i c d a t a a f t e r Dempster ( 1 9 5 5 ) > Braune and F i s c h e r ( I 8 8 9 ) and M o r r i s o n ( 1 9 6 8 ) a l l o w e d c a l c u l a t i o n o f g r a v i t a t i o n a l l i m b f o r c e s w h i l e a c c e l e r a t i o n d a t a d e t e r m i n e d from the d i g i t i z e d f i l m r e c o r d s was used t o c a l c u l a t e l i m b i n e r t i a l f o r c e s . U s i n g D'Alembert's p r i n c i p l e o f e q u i l i -b r ium o f b o d i e s i n mo t i o n , the j o i n t f o r c e s and moments a t the knee were de t e r m i n e d from t h e f o r c e p l a t e r e a c t i o n f o r c e s , l i m b g r a v i t a t i o n a l f o r c e s and l i m b i n e r t i a l f o r c e s . . The f o r c e s i n each muscle group and l i g a m e n t were de t e r m i n e d from the c a l c u l a t e d e x t e r n a l f o r c e system a c t i n g a t the knee and the r e q u i r e d s t a b i l i t y c o n d i t i o n s o f the j o i n t . R e s u l t s showed t h a t f o r c e a c t i o n d e v e l o p e d i n i t i a l l y i n the h a m s t r i n g s t o s t a b i l i z e t h e j o i n t as the s k a t e r s h i f t e d h i s w e i g h t t o the s u p p o r t l e g . As the s u p p o r t l e g assumed the body w e i g h t , p o w e r f u l f o r c e a c t i o n d e v e l o p e d i n the q u a d r i c e p s muscle t o extend the knee and d r i v e the s k a t e r f o r w a r d p a s t the s u p p o r t f o o t . The g a s t r o c n e m i u s e x e r t e d a s m a l l f o r c e t o p l a n t a r f l e x o r extend the a n k l e f o r the. f i n a l push o f f t h a t i s c h a r a c t e r i s t i c o f a l l a c c o m p l i s h e d s k a t e r s . The p o s t e r i o r c r u c i a t e l i g a m e n t developed t e n s i o n i n o r d e r t o r e s i s t t he a n t e r i o r d i s p l a c e m e n t o f the femur r e l a t i v e t o the t i b i a d u r i n g the t h r u s t r e a c h i n g a maximum f o r c e as t h e s k a t e r s h i f t e d h i s w e i g h t t o the s u p p o r t f o o t . The f o r c e s d eveloped i n the a n t e r i o r c r u c i a t e were r e l a t i v e l y - 108 -s m a l l and u n i m p o r t a n t i n the s k a t i n g t h r u s t . T e n s i l e f o r c e d eveloped i n the c o l l a t e r a l l i g a m e n t s was r e q u i r e d t o a s s u r e m e d i o - l a t e r a l s t a b i l i t y w i t h the peak l a t e r a l c o l l a t e r a l f o r c e s d e v e l o p e d a t the onset o f the s u p p o r t phase and maxi-mum m e d i a l c o l l a t e r a l f o r c e s d e v e l o p e d d u r i n g the f i n a l e x t e n s i o n o f the l e g t h r o u g h push o f f . A l t h o u g h the c u r r e n t i n v e s t i g a t i o n cannot comment on the r e l a t i v e s t a b i l i t y o f the knee i t can be s t a t e d t h a t the j o i n t i s most s u s c e p t i b l e t o i n j u r y i n s k a t i n g when the ex-t e r n a l f o r c e s and t o r q u e s o f a t h l e t i c c o m p e t i t i o n a r e super-imposed on the c r i t i c a l v a l u e s o f l i g a m e n t and a r t i c u l a r f o r c e s d e v e l o p e d by t h e t h r u s t i t s e l f . I t t h e r e f o r e appears t h a t the l a t e r a l c o l l a t e r a l l i g a m e n t would be most suscep-t i b l e t o a t e n s i l e s t r a i n a t the onset o f the s u p p o r t phase •while the m e d i a l c o l l a t e r a l i s most v u l n e r a b l e t o i n j u r y as the s k a t e r d r i v e s f o r w a r d a t push o f f . A l s o , the c o i n c i -dence o f a maximum j o i n t f o r c e , s i x t i m e s body w e i g h t , and a j o i n t t o r q u e a p p r o a c h i n g 700 pound-inches d u r i n g the su p p o r t phase p l a c e d a c o n s i d e r a b l e s t r e s s oh the a r t i c u l a r s u r f a c e s o f the j o i n t and t h e ; j o i n t m e n i s c i . Damage to the m e n i s c i o f the knee i n the form o f t e a r i n g i s a common i n j u r y i n the game o f i c e hockey. The magnitude o f the muscle, l i g a m e n t and j o i n t f o r c e s d e v e l o p e d i n the s k a t i n g t h r u s t were s i g n i f i c a n t l y g r e a t e r t h a n r e s p e c t i v e f o r c e s e x e r t e d d u r i n g l e v e l w a l k i n g '.while the c y c l i c p a t t e r n o f the s k a t i n g f o r c e s was comparable - 109 -to w a l k i n g u p s t a i r s i n d i c a t i n g a t e m p o r a l s i m i l a r i t y i n t h e s e a c t i v i t i e s . C o n c l u s i o n s The f o l l o w i n g s t a t e m e n t s are made w i t h r e s p e c t to the b i o m e c h a n i c s o f the s k a t i n g t h r u s t . (1) The q u a d r i c e p s are the most i m p o r t a n t muscle group i n the s k a t i n g t h r u s t d e v e l o p i n g c o n t r a c t i l e f o r c e s o f t h e o r d e r o f 700 pounds when e x t e n d i n g t h e knee j o i n t . (2) The h a m s t r i n g s and g a s t r o c n e m i u s muscle groups e x e r t f o r c e s l e s s t h a n 200 pounds and 100 pounds r e s p e c t i v e l y t o s t a b i l i z e the knee d u r i n g the weight s h i f t and push o f f phases o f the s k a t i n g t h r u s t . (3) The c o l l a t e r a l l i g a m e n t s and the p o s t e r i o r c r u c i a t e are i m p o r t a n t i n m a i n t a i n i n g s t a b i l i t y o f the j o i n t and t e n s i l e f o r c e s i n e x c e s s o f 250 pounds a c t i n t h e s e s t r u c t u r e s . (4) The knee j o i n t i s s u b j e c t to the combined e f f e c t s , o f a j o i n t f o r c e s i x t i m e s body w e i g h t and a j o i n t t o r q u e a p p r o a c h i n g 700 pound-inches superimposed upon each o t h e r d u r i n g the s u p p o r t phase o f the s k a t i n g t h r u s t . (5) M e d i o - l a t e r a l j o i n t d i s p l a c e m e n t i s not c r i t i c a l s i n c e s hear f o r c e s l e s s t h a n 0 . 50 t i m e s body weight are imposed d u r i n g the s k a t i n g t h r u s t . - 110 -\ R E F E R E N C E S - I l l -REFERENCES Adams, A. 1 9 6 6 . E f f e c t s o f e x e r c i s e upon l i g a m e n t s t r e n g t h . R e s e a r c h Q u a r t e r l y . 3 7 : 1 6 3 . Andrews, J . G., Chao, F. Y., J o h n s t o n , R. C. and S t a u f f e r , R. N. 1 9 7 2 . A g e n e r a l method f o r a c c u r a t e k i n e m a t i c a n a l y s i s o f b i o m e c h a n i c a l systems i n motion. P r o c s . Orth. Res. Soc. 2.9. Basmajian, J . V. 1 9 6 2 . 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M e d i c i n e and S c i e n c e i n S p o r t s . - 117 -APPENDICES - 118 -APPENDIX A ANTHROPOMETRIC MEASUREMENTS - 119 -APPENDIX A ANTHROPOMETRIC MEASUREMENTS 1. Body Segment Parameters The f o l l o w i n g body segment parameters r e p r e s e n t the average o f f i v e i ndependent measurements. Segment C i r c u m f e r e n c e , i n R a d i u s , i n Length, i n shank 14. 50 2 . 31 1 6 . 5 0 f o o t 13. 30 2 . 0 7 1 0 . 5 0 a n k l e 1 2 . 8 0 2 . 0 2 -knee 1 5 . 3 0 2 . 4 3 -r a d i u s o f segment = c i r c u m f e r e n c e o f l i m b 2 it« The l e n g t h o f a segment was measured from j o i n t c e n t r e t o j o i n t c e n t r e and the c i r c u m f e r e n c e o f a segment was measured a t the, j o i n t c e n t r e and c e n t r e o f g r a v i t y o f the segment. 2. C a l c u l a t i o n o f Segment Weights Dempster (1955) e x p r e s s e d the we i g h t o f each "body segment as a p e r c e n t a g e o f the t o t a l body weight, based on measurements from l i v i n g male s u b j e c t s o f v a r i o u s body t y p e s and age. Te s t s u b j e c t body w e i g h t = 165 l b s Segment - 120 -Dempster fo Weight l b s t h i g h 9 . 6 5 1 6 . 0 0 shank 4 . 5 0 7 . 4 3 f o o t 1 .40 2 . 3 1 segment w e i g h t = Dempster fo x body weight 3. Ce n t r e o f G r a v i t y o f Body Segments Dempster de t e r m i n e d the d i s t a n c e o f the mass c e n t r e o f each body segment from the p r o x i m a l end e x p r e s s e d as a p e r c e n t o f the l e n g t h o f the segment Segment L e n g t h , i n Dempster fo C e n t r e G r a v i t y , i n shank 1 6 . 5 0 . 4 3 3 7 - 1 5 f o o t 1 0 . 5 0 . 4 2 9 ^ . 5 0 d i s t a n c e o f e.g. from p r o x i m a l end o f segment = Dempster fo x segment l e n g t h I n the case o f the f o o t , the c e n t r e o f g r a v i t y l i e s a t the i n t e r s e c t i o n o f a v e r t i c a l l i n e 4 . 5 0 i n c h e s p r o x i m a l to the heel- and a l i n e j o i n i n g the a n k l e a x i s to the b a l l o f the f o o t . 4 . Moment o f I n e r t i a o f Segment The mass moment o f i n e r t i a ( I ) o f a body segment i s e q u a l to the p r o d u c t o f the segment's mass and the r a d i u s o f - 121 -g y r a t i o n ( r ) o f the segment. Braune and. F i s c h e r ( I 8 8 9 ) determined c o - e f f i c i e n t s t o a l l o w c a l c u l a t i o n o f the r a d i u s o f g y r a t i o n o f a segment. Segment L e n g t h , i n Diameter, i n r x r y r z shank 1 6 . 5 4 , 6 2 0 . 2 5 O .35 4 . 1 3 ^.13 1-62 f o o t 1 0 . 5 4.14 0 . 3 0 O .35 1 .45 3 .15 3 . 15 r a d i u s o f g y r a t i o n ( r ) i s e x p r e s s e d i n i n c h e s C ^ i s the F i s c h e r c o e f f i c i e n t f o r the r a d i u s o f g y r a t i o n f o r r o t a t i o n about t h e a x i s t h r o u g h t h e mass c e n t r e and p e r p e n -d i c u l a r t o the l o n g i t u d i n a l a x i s o f the segment r a d i u s o f g y r a t i o n = x l e n g t h o f segment F o r r o t a t i o n o f the segment about the l o n g i t u d i n a l a x i s , F i s c h e r used the c o e f f i c i e n t C^ and the d i a m e t e r o f t h e seg-ment to c a l c u l a t e the r a d i u s o f g y r a t i o n r a d i u s o f g y r a t i o n = x di a m e t e r o f segment Segment Weight, l b s I x I y I z shank 7-^3 3-92 3-92 0 . 6 0 f o o t 2 . 3 1 0 . 1 5 0 . 7 1 O .71 moment o f i n e r t i a ( I ) = mass o f segment x ( r a d i u s o f g y r a t i o n ) p and i s e x p r e s s e d i n s l u g - i n - 122 -The c o e f f i c i e n t s o f F i s c h e r and Dempster are t a k e n from a r e v i e w o f body segment parameters by D r i l l i s , C o n t i n i and B l u e s t e i n (1964) 5. A n t h r o p o m e t r i c S c a l i n g F a c t o r s S c a l i n g f a c t o r s were r e q u i r e d t o o b t a i n the t e s t s u b j e c t ' s c o - o r d i n a t e s o f muscle and l i g a m e n t attachment from t h e . . r e s p e c t i v e b a s i c c o - o r d i n a t e s o f M o r r i s o n as l i s t e d i n T a ble No. 1 o f the t e x t . The dime n s i o n s used to o b t a i n the s c a l i n g f a c t o r s f o r the knee a r e , X - the b r e a d t h o f the f e m o r a l c o n d y l e s measured from the l a t e r a l e p i c o n d y l e t o the m e d i a l c o n d y l e Y - the depth o f the f e m o r a l c o n d y l e s measured from the an-t e r i o r s u r f a c e o f the l a t e r a l c o n d y l e t o the p o s t e r i o r s u r f a c e o f the m e d i a l c o n d y l e . Z - the l e n g t h o f the t i b i a measured from the c e n t r e o f the knee j o i n t t o the c e n t r e o f the a n k l e j o i n t . F o r the p e l v i s a common s c a l i n g d i m e n s i o n was used f o r male s u b j e c t s . X - the d i s t a n c e between the a n t e r i o r s u p e r i o r i l i a c s p i n e s o f the p e l v i s T h e r e f o r e , a common s c a l i n g f a c t o r i s c a l c u l a t e d f o r the p e l v i s w hich i s a p p l i e d t o the t h r e e c o - o r d i n a t e s o f - 123 -muscle attachment a t the p e l v i s . J o i n t C o - o r d i n a t e S c a l i n g Dimension S c a l i n g F a c t o r B a s i c S u b j e c t knee X 3 .50 3-78 1 . 07 knee Y 3 .55 3.64 1 . 03 knee Z Z 16 .00 16.16 1.01 p e l v i s X 8.70 8. 20 0.94 s c a l i n g d i m e n s i o n s e x p r e s s e d i n i n c h e s s c a l i n g f a c t o r = d i m e n s i o n from t e s t s u b j e c t b a s i c d i m e n s i o n from d i s s e c t i o n The c o - o r d i n a t e o f a muscle o r l i g a m e n t attachment to the s k e l e t o n o f the t e s t s u b j e c t i s t h e n c o n s i d e r e d t o be the b a s i c d i m e n s i o n as measured from d i s s e c t i o n m u l t i p l i e d by the a p p r o p r i a t e s c a l i n g f a c t o r . F o r example the Xs Co-o r d i n a t e o f the p a t e l l a r l i g a m e n t attachment t o the t i b i a , Xsq, i s c a l c u l a t e d as f o l l o w s Xsq = (Xs c o - o r d i n a t e measured) x ( s c a l i n g f a c t o r i n d i s s e c t i o n ) - 124 -APPENDIX B INSTRUMENTATION \ - 125 -APPENDIX B INSTRUMENTATION 1. F i l m i n g Equipment , Two cameras were r e q u i r e d t o o b t a i n c i n e m a t o g r a p h i c d a t a i n the two p l a n e s o f movement. 16 mm Locam: motor d r i v e n a d j u s t a b l e f i l m speed i n t e r n a l p u l s e l i g h t 16 mm B o l e x : s p r i n g l o a d e d s e t f i l m speeds " Kodak 4X R e v e r s a l 320 ASA f i l m w i t h double edge p e r -f o r a t i o n s was used w i t h b oth cameras. F i l m i n g speed was 64 f r a m e s / s e c . A 35 mm T e k t r o n i x camera and a 35 mm Nikkormat camera were u t i l i z e d t o r e c o r d the s t o r e d o s c i l l o s c o p e t r a c e s o f the f o r c e p l a t e o u t p u t . 2 . D a t a R e d u c t i o n The Vanguard M o t i o n A n a l y z e r o f the I n s t i t u t e o f A n i m a l R e s o u r s e s (I.A.R.E.) c a p a b l e o f r e a d i n g s to the n e a r e s t 0 .001 i n c h was used to reduce cinema and f o r c e p l a t e d a t a . - 126 -3. E l e c t r o m y o g r a p h y B i p o l a r e l e c t r o d e s p l a c e d on the b e l l y o f two muscle groups were connected to s e p a r a t e c h a n n e l s o f a two-channel Sanborn R e c o r d e r . T h i s a l l o w e d the e l e c t r i c a l o u t p u t o f two muscle groups t o be m o n i t o r e d s i m u l t a n e o u s l y as a t r a c e on paper. paper speed - 100 mm/sec g a i n - same f o r b o t h c h a n n e l s 4. F o r c e M e a s u r i n g P l a t f o r m The multi-component measuring p l a t f o r m Type 9261A d e s i g n e d by the K i s t l e r I n s t r u m e n t s i s a p i e z o e l e c t r i c t r a n s -d ucer which measures any f o r c e a p p l i e d to i t i n t h r e e o r t h o -g o n a l components. I n a d d i t i o n the moment a p p l i e d to the p l a t f o r m and t h e c o - o r d i n a t e s o f the p o i n t o f a p p l i c a t i o n o f the f o r c e are r e c o r d e d . Four q u a r t z f o r c e - m e a s u r i n g - e l e m e n t s are f i t t e d t o the c o r n e r s o f the p l a t f o r m w h i c h has a h i g h r i g i d i t y a l l o w i n g o p e r a t i o n w i t h a minimum a measuring d i s p l a c e m e n t as w e l l as a wide f r e q u e n c y range. The e l e c t r i c a l c h arges y i e l d e d by the p l a t f o r m are s t r i c t l y p r o p o r t i o n a l to the measurands; by means o f charge a m p l i f i e r s t h e y are c o n v e r t e d i n t o a n a l o g dc v o l t a g e s , which t h e n can be r e c o r d e d , i n d i c a t e d o r o t h e r w i s e p r o c e s s e d as r e q u i r e d . I n our case t h e f o r c e p l a t e o u t p u t was s t o r e d as s i x s e p a r a t e s t o r a g e t r a c e s on t h r e e d u a l c h a n n e l T e k t r o n i x o s c i l l o s c o p e s . - 127 -K1STLER FORCE P L A T F O R M B A S E D  ON P I E Z O E L E C T R I C E F F E C T OF O U A R T Z CRYSTALS - 128 -I ( APPENDIX C DATA REDUCTION - 129 -APPENDIX C DATA REDUCTION 1. Cinema S c a l i n g F a c t o r s A s u r v e y o r ' s range p o l e was u t i l i z e d as the r e f e r e n c e image i n d e t e r m i n a t i o n o f the X and Y s c a l i n g f a c t o r s f o r r e d u c t i o n o f c i n e f i l m d a t a . C o - o r d i n a t e Image, i n A c t u a l , i n S c a l i n g F a c t o r X 6 .396 72 .00 10 ,38 Y 5 .592 72 .00 12 .88 The c a l c u l a t e d s c a l i n g f a c t o r s r e p r e s e n t the average o f t e n measurements s c a l i n g f a c t o r = a c t u a l l e n g t h o f range p o l e image l e n g t h o f range p o l e 2 . F o r c e P l a t e Data R e d u c t i o n The s y n c h r o n i z e d f o r c e p l a t e t r a c e s , r e c o r d e d on § 5 mm f i l m , were d i g i t i z e d by h o r i z o n t a l l y i n c r e m e n t i n g the t r a c e s i n t o 50 d i v i s i o n s c o r r e s p o n d i n g t o the 50 frames o f the 16 mm f i l m r e c o r d t h a t d e s c r i b e d t h e s k a t i n g t h r u s t . The a c t u a l v a l u e s o f the o u t p u t components were d e t e r m i n e d hy a p p l y i n g the f o r c e p l a t e and o s c i l l o s c o p e c a l i b r a t i o n f a c t o r s t o each o f the v e r t i c a l c o - o r d i n a t e s d e f i n e d by the - 130 -50 h o r i z o n t a l d i v i s i o n s . The c a l i b r a t i o n f a c t o r s f o r the ou t p u t v a r i a b l e s i n terms o f measurand u n i t s p e r v e r t i c a l d i v i s i o n a r e , V a r i a b l e F o r c e P l a t e O s c i l l o s c o p e M.U-./div. Fx 100 N / V X 2 V / d i v . = 200, N/div. Fy 100 N / V X 1 V / d i v . = 100 N / d i v . Fz 100 N / V X 5 V / d i v . = 500 N / d i v . Mz 100 Nm/V X 0 . 0 2V/div. = 2 Nm/div. Ax .5 Cm/V X 0 . 5 V / d i v . = 2 . 5 cm/div. Ay 5 Cm/V X 0 . 5 V / d i v . = 2 . 5 cm/div. i : N (Newton) = 0.2248 lbs 1 Nm (Newton-meter) = (0.2248 x 39-37) l b - i n s 1 cm (centimeter) = O .3937 i n Fx = (200) (0.2248) = 44.96 lb/div,. Fy = (100)(0.2248) = 22.48 lb / d i v . Fz = (500)(0.2248) =112.40 l b / d i v . Mz = (2)(0.2248 x 39-37) = 15-50 l b - i n / d i v . Ax = (2.5) (0.,3937) = 0.984 in/div. Ay = (2.5)(0.3937) = 0.984 i n / d i v - 131 -APPENDIX D LIMB AND JOINT CO-ORDINATES - 132 -APPENDIX D LIMB AND JOINT CO-ORDINATES 1. O r t h o g o n a l C o - o r d i n a t e s The l i m b and j o i n t c o - o r d i n a t e s were determined w i t h r e s p e c t t o the c e n t r e a x i s o f the f o r c e p l a t f o r m and f o r the f o l l o w i n g s i g n c o n v e n t i o n . ( i n i n c h e s ) X - p o s i t i v e m e d i a l t o c e n t r e o f p l a t f o r m Y - p o s i t i v e f o r w a r d o f c e n t r e o f p l a t f o r m Z - p o s i t i v e above p l a t f o r m The o r t h o g o n a l c o - o r d i n a t e s o f the l i m b s and j o i n t s are shown i n F i g . DI f o r the knee ahead o f and m e d i a l t o the c e n t r e a x i s o f the f o r c e p l a t f o r m . Ygf i Xgf i FIG. DI LIMB AND JOINT CO-ORDINATES - 133 -The o r t h o g o n a l c o - o r d i n a t e s o f F i g . DI r e p r e s e n t the c o - o r d i n a t e s o f t h e j o i n t c e n t r e o f the knee and a n k l e and the mass c e n t r e o f the shank and f o o t . These c o - o r d i n a t e s were de t e r m i n e d by c o r r e c t i n g the c o - o r d i n a t e s o f l i m b and j o i n t s u r f a c e markers f o r l i m b angle and l i m b r a d i u s . 2. A n g u l a r Limb C o - o r d i n a t e s The a n g u l a r l i m b c o - o r d i n a t e s e x p r e s s i n g l i m b a n g l e s were determined from the c o - o r d i n a t e s o f the j o i n t s u r f a c e markers. The a n g u l a r c o - o r d i n a t e s f o r r o t a t i o n o f the shank and f o o t w i t h r e s p e c t t o the X a x i s i s shown i n F i g . D2. ( Yk,Zk) ( Yt.Zt) DZS ( Ya,Za) DYS= Yk-Ya DZS= Zk-Za DYF= Yt -Ya DZF = Za-Zt FIG. D2 ANGULAR LIMB CO-ORDINATES 9 X S = a r c t a n DYS 9 X F = a r c t a n 'DYF DZS DZF - 134 -(Xk,Zk) (Xa,Za) DZS DXS=Xk-Xa DZS = Zk-Za DXF = X a - X t DZF= Za-Zt (Xt,Zt) FIG. D3 ANGULAR LIMB CO-ORDINATES 9 Y S = a r c t a n 9 Y F = a r c t a n ^|| The a n g u l a r c o - o r d i n a t e o f the f o o t about i t s l o n g i -t u d i n a l a x i s , 9 Y F , was d i r e c t l y d e t ermined as shown above i n F i g . D 3 . The a n g u l a r c o - o r d i n a t e o f the shank about i t s l o n g i t u d i n a l a x i s , 9 Z S , was c a l c u l a t e d from a knowledge o f the r a d i u s o f the shank and the l o c a t i o n o f the mass c e n t r e marker o f the shank as shown i n F i g . D4. A n g u l a r c o - o r d i n a t e 9 Z F r e p r e s e n t s the r o t a t i o n o f the f o o t the v e r t i c a l a x i s and i s c a l c u l a t e d as 9 Z F = a r c t a n The a n g u l a r r o t a t i o n o f the t h i g h about the X and Y 135 -axes were c a l c u l a t e d as f o l l o w s GXT = a r c t a n 9 Y T = a r c t a n DYT DZT DXT DZT DYS/2 Rs - r a d i u s o f s h a n k Y S A - a n t e r i o r c o - o r d i n a t e of s h a n k Y S P - p o s t e r i o r c o - o r d i n a t e o f s h a n k Y S - c o - o r d i n a t e o f s h a n k m a r k e r D Y S = Y S A - Y S P DS= Y S - Y S P FIG. D4 ROTATION OF SHANK ABOUT LONG AXIS where the a n g u l a r r o t a t i o n o f the shank 9 Z S i s c a l c u l a t e d f o l l o w s n ( 7 C . DYS/2 - DS 9 Z S = a r c s m 3. Limb and J o i n t C e n t r e C o - o r d i n a t e s The YZ p l a n e was chosen as the- r e f e r e n c e measuring p l a n e i n c a l c u l a t i o n o f the c o - o r d i n a t e s o f j o i n t c e n t r e s and mass c e n t r e s . 'Z' C o - o r d i n a t e - I36 -S i d e View, YZ P l a n e F r o n t View, XZ P l a n e FIG. D5 'Z' CO-ORDINATE OF KNEE AND ANKLE A'Zk =. Rxk • s i n 9 Y T Zk = Zkyz - Rxk • S i n 9 Y T WHERE Zk r e p r e s e n t s the a c t u a l c o - o r d i n a t e and h e i g h t o f the knee j o i n t c e n t r e and Rxk i s the r a d i u s o f the knee j o i n t i n X d i r e c t i o n . A Z a = Ra • s i n 9 Y S Za = Zayz - Ra • 9 Y S where Z<a. r e p r e s e n t s the a c t u a l c o - o r d i n a t e and h e i g h t o f the - 137 -ankle j o i n t c e n t r e and Ra i s the r a d i u s o f the ankle j o i n t . The v e r t i c a l c o - o r d i n a t e o f the shank mass c e n t r e i s c a l c u -l a t e d s i m i l a r i l y . A Zs = Rs • s i n 9YS .v Zs = Zsyz - Rs • s i n 9YS. " S ide View, YZ Plane F r o n t View, XZ Plane F I G . D6 'Z* CO-ORDINATE OF FOOT A.Zf = Rf • s i n 9YF Z f = Z f y z - Rf • s i n 9YF where Rf i s the w id th o f the f o o t a t the mass c e n t r e o f the f o o t . ' Y ' C o - o r d i n a t e The Y c o - o r d i n a t e o f the j o i n t c e n t r e s and l i m b mass - 138 -c e n t r e s i s determined by c o r r e c t i n g the Y c o - o r d i n a t e o f the s u r f a c e marker measured i n the YZ p l a n e f o r r o t a t i o n o f the l i m b about i t s l o n g a x i s , - F i g . D ? . Y k y z S i d e View, YZ P l a n e V e r t i c a l View, XY P l a n e FIG. D7 'Y' CO-ORDINATE OF KNEE AND SHANK AYs = Rs • s i n 9ZS Ys = Ysyz + Rs • s i n OZS where Ys i s the a c t u a l c o - o r d i n a t e o f the mass c e n t r e o f the shank and Rs i s the r a d i u s o f the shank a t the mass c e n t r e . The Yf. c o - o r d i n a t e o f the knee i s determined s i m i l -a r i l y from shank r o t a t i o n as - 139 -A Yk = Rxk • s i n 9ZS Yk = Ykyz + Rxk • s i n 9ZS where Rxk r e p r e s e n t s the c a l c u l a t e d , r a d i u s o f the knee a t the j o i n t marker i n X d i r e c t i o n . The Y c o - o r d i n a t e o f the a n k l e j o i n t c e n t r e and the f o o t mass c e n t r e was c a l c u l a t e d from the 9ZF. f o o t r o t a t i o n a p p l i e d t o t h e YZ c o - o r d i n a t e o f the r e s p e c t i v e s u r f a c e markers shown i n F i g . D6 S O t h a t Ya = Yayz + Ra • s i n 9ZF Y f = Y f y z + Rf • s i n 9ZF ' 'X' C o - o r d i n a t e The X c o - o r d i n a t e o f the knee j o i n t c e n t r e and shank mass c e n t r e were d e t e r m i n e d from the X c o - o r d i n a t e o f the s u r f a c e markers measured i n the XZ p l a n e and the shank r o t -a t i o n . Xk = Xkxz + Ryk • s i n 9ZS 1 Xs = Xsxz + Rs • s i n 9ZS w h i l e the X c o - o r d i n a t e o f the a n k l e j o i n t c e n t r e and the f o o t mass c e n t r e was c a l c u l a t e d from the r o t a t i o n o f the f o o t about i t s v e r t i c a l a x i s , 9ZF Xa = Xaxz + Ra • s i n 9ZF X f -= Y f x z + Rf • s i n 9ZF - 140 -APPENDIX E MUSCLE AND LIGAMENT CO-ORDINATES V - -APPENDIX E MUSCLE AND LIGAMENT CO-ORDINATES The c o - o r d i n a t e s o f a l l muscle and l i g a m e n t a t t a c h -ments were determined i n terms o f the same axes -- the g r i d axes w i t h o r i g i n a t t h e c e n t r e o f the f o r c e p l a t f o r m . 1.• M e d i a l C o l l a t e r a l Ligament Z s Y s exs Z s m ' FIG. E l MEDIAL COLLATERAL ATTACHMENT TO TIBIA IN YZ PLANE FOR ROTATION 9XS t a n 9xm = Ysm 2 2 Rmyz = Ysm + Zsm' Zsm 9xm 1 = 9xm + 9xs - 142 -Zsm' = Rrayz • cos 9xm' Ysm' = Rmyz • s i n 9xm 1 f o r r o t a t i o n i n the XZ p l a n e (9YS) and. r o t a t i o n i n the XY p l a n e (9ZS) the c o - o r d i n a t e s o f attachment reduce t o Zsm = Zsm* - cos 9YS Ysm = Ysm' - cos 9ZS X s m ' FIG. E2 MEDIAL COLLATERAL ATTACHMENT TO TIBIA IN XZ PLANE FOR ROTATION 9YS t a n 9ym = Rmxz = Xsm 2 + Zsm 2 9ym' = 9ym + 9;YS ^  Xsm' = Rmxz • s i n Gym' f o r r o t a t i o n o f t i b i a i n XY p l a n e (9ZS) the c o - o r d i n a t e o f - 143 -attachment r e d u c e s t o Xsm = Xsm' - cos 9ZS'- : The c o - o r d i n a t e s o f m e d i a l l i g a m e n t attachment t o the t i b i a w i t h r e s p e c t t o t h e g r i d axes a re t h e r e f o r e Xgm = Xgk + Rmxz • s i n 9ym• . cos 9ZS Ygm = Ygk - Rmyz • s i n 9xm' • cos 9ZS Zgm = Zgk - Rmyz • cos y9xm' • cos 9YS where Xgk, Ygk, Zgk are the g r i d c o - o r d i n a t e s o f the t i b i a l axes o f the knee FIG. E3 MEDIAL COLLATERAL ATTACHMENT TO FEMUR IN YZ PLANE FOR ROTATION 9XT - 144 -t a n 9xfm = Zffm Y f f m R f myz2 = Y f f m 2 + Z f f m 2 9xfm' * 9xfm - 9XT Zffm' = Rfmyz • s i n 9xfm' Yffm' = Rfmyz • cos 9xfm' f o r r o t a t i o n o f femur i n XZ p l a n e (9YT) and r o t a t i o n i n XY p l a n e (9ZT) t h e c o - o r d i n a t e s o f attachment reduce t o Zffm = Zffm' . cos 9YT Yffm =.' Yffm' . cos. 9ZT FIG. E4 u MEDIAL COLLATERAL ATTACHMENT TO FEMUR IN XZ PLANE FOR ROTATION 9YT . n „ Zffm t a n 9yfm = Rfmxz 2 = X f f m 2 + Z f f m 2 9yfm' = 9yfm - 9YT Xffm' = Rfmxz • cos 9yfm' - 145 -c o r r e c t e d f o r r o t a t i o n o f the femur i n the XY p l a n e (9ZT) X f f m = Xffm» • cos 9ZT The c o - o r d i n a t e s o f m e d i a l l i g a m e n t attachment t o t h e femur w i t h r e s p e c t t o the g r i d axes are t h e r e f o r e Xgfm = X g f + Rfmxz • cos 9yfm' • cos 9ZT Ygfm = Ygf - Rfmyz • cos 9xfm' • cos 9ZT . Zgfm = Zgf + Rfmyz • s i n i O x f m ' • cos 9YT where Xgf, Y g f , Zgf, are t h e g r i d c o - o r d i n a t e s o f t h e f e m o r a l axes o f the knee The c o - o r d i n a t e s o f l a t e r a l l i g a m e n t attachment to the f i b u l a and femur are c a l c u l a t e d s i m i l a r i l y . 2. L a t e r a l C o l l a t e r a l Ligament Z s l FIG. E5 LATERAL COLLATERAL ATTACHMENT TO FIBULA IN YZ PLANE FOR ROTATION 9XS - 146 -t a n 9x1 = | g R l y z ^ = Y s l ^ + Z s l ^ 9x1' = 9x1 + 9XS Z s l ' = R l y z • cos 9x1' Y s l • = R l y z • s i n 9x1' c o r r e c t e d f o r r o t a t i o n o f the t i b i a i n the XZ p l a n e (9YS) and the XY p l a n e (9ZS) the c o - o r d i n a t e s reduce to Z s l = Z s l ' • cos 9YS Y s l = Y s l ' • cos 9ZS FIG. E6 LATERAL COLLATERAL ATTACHMENT TO .FIBULA IN XZ PLANE FOR ROTATION 9YS t a n 9 y l = ||± R l x z 2 = X s l 2 + Z s l 2 9 y l ' = 9 y l - 9;YS - 14? -X s l ' = R l x z • s i n 9 y l ' c o r r e c t e d f o r r o t a t i o n o f the t i b i a i n the XY p l a n e (9ZS) X s l = X s l ' • cos 9ZS The c o - o r d i n a t e s o f the l a t e r a l l i g a m e n t attachment to t he f i b u l a w i t h r e s p e c t t o the axes a re t h e r e f o r e X g l = Xgk - R l x z • s i n 9 y l ' • cos 9ZS Y g l = Ygk - R l y z - • s i n 9x1' • cos 9ZS Z g l = Zgk - R l y z • cos 9x1' • cos 9YS Z f Y f Z f f I Y f f I FIG. E 7 LATERAL COLLATERAL ATTACHMENT TO FEMUR IN YZ PLANE FOR ROTATION 9XT t a n Q x f l = | R f l z y 2 = Y f f l 2 + Z f f l 2 9 x f l ' = 9 x f l - 9XT Y f f l ' = R f l y z • cos 9 x f l ' Z f f l ' = R f l y z . s i n 9 x f 1 ' c o r r e c t e d f o r r o t a t i o n o f the femur i n the XZ p l a n e (9YT) and r o t a t i o n i n the XY p l a n e (9ZT) the c o - o r d i n a t e s reduce t Z f f l = Z f f l ' • cos 9YT Y f f l = Y f f l ' • cos 9ZT ff i ' FIG. E8 LATERAL COLLATERAL ATTACHMENT TO FEMUR IN XZ? PLANE FOR ROTATION 9YT t a n 9 y f l = R f l x z 2 = X f f l 2 + Z f f l 2 9 y f l ' = 9 y f l + 9YT X f f l = R f l x z • cos 9 y f l ' - 14,9 -c o r r e c t e d , f o r r o t a t i o n o f femur i n XY p l a n e (9ZT) the c o r d i n a t e r e d u c e s to X f f l = X f f l ' • cos 9ZT The c o - o r d i n a t e s o f l a t e r a l l i g a m e n t attachment the femur w i t h r e s p e c t t o the g r i d axes are X g f l = X g f - R f l x z • cos 9 y f l ' • cos 9ZT Y g f l = Ygf - R f l y z • cos 9 x f 1 ' • cos 9ZT Z g f l = Z g f + R f l y z • s i n 9 x f 1 ' • cos 9YT 3- A n t e r i o r C r u c i a t e Ligament FIG. E9 ANTERIOR CRUCIATE ATTACHMENT TO TIBIA IN YZ PLANE FOR ROTATION 9XS - 150 -t a n Oxa = f | f R a y z 2 = Y s a 2 + Z s a 2 Qxa' = 9xa - 9XS Z s a ' = Rayz • s i n 9xa' Y s a ' = Rayz • cos 9xa' c o r r e c t e d f o r r o t a t i o n o f t i b i a i n XZ p lane (9YS) and the XY p lane (9ZS) Z s a = Z s a ' • cos 9YS Ysa = Y s a ' • cos 9ZS Z s X s a ' F I G . E10 ANTERIOR CRUCIATE ATTACHMENT TO TIBIA IN XZ PLANE FOR ROTATION 9YS t a n 9ya = R a x z 2 = X s a 2 + Z s a 2 A s a 9ya' = 9ya + 9YS X s a ' = Raxz • cos 9ya' c o r r e c t e d f o r r o t a t i o n o f the t i b i a i n the XY p lane (9ZS) - 151 -Xsa = Xsa' • cos 9ZS The c o - o r d i n a t e s o f a n t e r i o r c r u c i a t e l i g a m e n t a t t a c h -ment to the t i b i a w i t h r e s p e c t t o the g r i d axes are t h e r e f o r e Xga = Xgk + Raxz • cos 9ya' • cos 9ZS Yga = Ygk + Rayz • cos 9xa' * cos 9ZS Zga = Zgk + Rayz " s i n 9xa' • cos 9YS Yf e x f a ' =k Z f f a ' FIG. E l l ANTERIOR CRUCIATE ATTACHMENT TO FEMUR IN YZ PLANE FOR ROTATION 9XT t a n 9 s f a Z f f a Y f f a R f a y z2 = Y f f a 2 + Z f f a 2 9 x f a ' = 9 x f a - 9XT Z f f a ' = R f a y z • s i n 9 x f a ' Y f f a ' = Rf a y z • cos 9 x f a ' - 152 -corrected f o r r o t a t i o n of the femur i n the XZ plane (QYT) and r o t a t i o n i n the XY plane (9ZT) Yf f a = Yffa' • cos 9ZT Z f f a = Z f f a ' • cos 9YT Xf FIG. E12 ANTERIOR CRUCIATE ATTACHMENT TO FEMUR IN XZ PLANE FOR ROTATION 9YT tan 9yfa = |||| Rfaxz 2 = X f f a 2 + Z f f a 9yfa' = 9yfa + 9YT Xffa' = Rfaxz ; cos Qyfa' corrected f o r r o t a t i o n of the femur i n the XY plane (9ZT) Xffa = Xffa' • cos 9ZT The co-ordinates of anterior cruciate ligament - 153 -attachment t o the femur w i t h r e s p e c t t o the g r i d , axes X g f a = X g f - R f a x z • cos Qyfa' • cos 9 Z T Y g f a = Y g f - R f a y z • cos 9 x f a ' • cos 9 Z T Z g f a = Z g f + Rf a y z • s i n Q x f a ' • c o s ' , 9 Y T 4. P o s t e r i o r C r u c i a t e Ligament FIG. E 1 3 POSTERIOR CRUCIATE ATTACHMENT TO TIBIA IN YZ PLANE FOR ROTATION 9 X S t a n 9 x p ' = 9 x p - - 9 X S " R p y z 2 = Y s p 2 + Z s p 2 Zsp' = Rpyz • s i n 9 x p ' Ysp' = Rpyz • cos 9 x p ' - 15^ -c o r r e c t e d f o r r o t a t i o n o f the t i b i a i n the XZ p l a n e (GYS) and r o t a t i o n i n the XY p l a n e (9ZS) Ysp = Ysp' • cos 9ZS Zsp = Zsp' • s i n 9YS FIG. E l 4 POSTERIOR CRUCIATE ATTACHMENT TO TIBIA IN XZ PLANE FOR ROTATION 9YS Rpxz = Zsp 9yp' = 9YS Xsp' = Zsp • s i n 9YS c o r r e c t e d f o r r o t a t i o n o f the t i b i a i n the XY p l a n e (9ZS) Xsp = Xsp' • cos 9ZS The c o - o r d i n a t e s o f p o s t e r i o r c r u c i a t e l i g a m e n t - 155 -attachment to the t i b i a with respect the grid, axes Xgp = Xgk + Zsp • 9YS • cos 9ZS Ygp = Ygk - Rpyz • cos 9xp' •• cos 9ZS Zgp = Zgk - Rpyz • s i n 9xp' • cos 9YS Oxfp' = 9 x f p - 9XT Zffp* = Rfpyz • s i n 9 x f p ' Yffp' = Rfpyz • cos 9xfp' corrected for r o t a t i o n of the femur i n the XZ plane (9YT) and the XY plane (9ZT) - 156 -Y f f p = Y f f p ' • cos 9ZT Z f f p = Z f f p ' • cos 9YT FIG. E 1 6 POSTERIOR CRUCIATE ATTACHMENT TO FEMUR IN XZ PLANE FOR ROTATION 9YT t a n 9 y f p = R f p x z 2 = X f f p 2 + Z f f p 2 9xfp* = 9 y f p + 9YT X f f p ' = Rfpx z • cos Oyfp* c o r r e c t e d f o r r o t a t i o n o f the femur i n the XY p l a n e (9ZT) X f f p = X f f p ' •• cos 9ZT The c o - o r d i n a t e s o f p o s t e r i o r c r u c i a t e l i g a m e n t a t t a c h -ment t o the femur w i t h r e s p e c t t o the g r i d axes are Xgfp = Xgf + Rfpxz • Ygfp = Ygf + Rfpyz • Zgfp = Zgf - Rfpyz • - 157. -cos Oyfp 1 . cos 9 Z T cos Qxfp' • cos 9 Z T s i n Qxfp* " cos 9 Y T - 158 -MUSCLE CO-ORDINATES The c o - o r d i n a t e s o f the h a m s t r i n g and g a s t r o c n e m i u s muscle attachments were c a l c u l a t e d t o a l l o w d e t e r m i n a t i o n o f r e s p e c t i v e l i n e s o f f o r c e a c t i o n f o r t h e s e muscle groups. The l i n e o f f o r c e a c t i o n f o r the q u a d r i c e p s group was determined from e q u a t i o n 3-01 o f s e c t i o n No. 3 . 1. Ha m s t r i n g s M u s c l e Group FIG. E 1 7 HAMSTRINGS ATTACHMENT TO TIBIA IN YZ PLANE FOR ROTATION 9XS t a n 9xh = ||| R h y z 2 = Y s h 2 + Z s h 2 9xh' = 9xh + 9XS - 159 -Ysh' = Rhyz ' s i n 9xh' Zsh' = Rhyz • cos 9xh c o r r e c t e d f o r r o t a t i o n i n the XZ p l a n e (9YS) and f o r r o t a -t i o n i n the XY p l a n e (9ZS) Ysh = Ysh' • cos 9YS X s h ' FIG. E18 HAMSTRINGS ATTACHMENT TO TIBIA IN XZ PLANE FOR ROTATION 9YS Rhxz = Zsh 9yh' = 9YS-Xsh' = Zsh • s i n 9YS c o r r e c t e d f o r r o t a t i o n o f the t i b i a i n the XY p l a n e (9ZS) the c o - o r d i n a t e r e d u c e s t o Xsh = Xsh' • cos 0ZS The c o - o r d i n a t e s o f h a m s t r i n g attachment to the t i b i a - 160 -with respect to the grid, axes are Xgh = Xgk + Zsh • s i n 9YS . cos 9ZS Ygh = Ygk - Rhyz • s i n 9xh' • cos 9ZS Zgh = Zgk - Rhyz • cos 9xh' • cos 9YS The hamstrings being a b i a r t i c u l a r muscle group cross the knee j o i n t and. hip j o i n t to i n s e r t i o n i n the p e l v i s . To determine the co-ordinates of attachment to the pe l v i s with respect to the g r i d axes, the basic co-ordinates with respect to Morrison's pelvic o r i g i n at the anterior c ilia's) spine (p) are transposed to the femoral head o r i g i n (fh). The co-ordinates of hamstring muscle group attachment to p e l v i s with respect to the g r i d axes are Xgph = Xgfh + (Xph - Xpfh) Xgph = Ygfh - (Yph - Ypfh) Zgph = Zgfh - (Zph - Zpfh) The co-ordinates of hamstrings attachment to the pel v i s represent a common o r i g i n of i n s e r t i o n for the muscles which make up the hamstrings muscle group. The calculations for hamstrings attachment to the - 161 -Xfhh FIG. E 1 9 HAMSTRINGS ATTACHMENT TO PELVIS IN XZ PLANE WITH RESPECT TO FEMORAL HEAD Z p Yp Yfhh z Yph -Ypfh FIG. E'20 HAMSTRINGS ATTACHMENT TO PELVIS IN YZ PLANE WITH RESPECT TO FEMORAL HEAD - 1-6-2 -p e l v i s have not been c o r r e c t e d f o r r o t a t i o n s o f the p e l v i s , 9XP, 9YP and 9ZP, as these r o t a t i o n s c o u l d not be determined from the s u r f a c e markers used . However these r o t a t i o n s are assumed s m a l l . 2. Gas trocnemius Musc le Group Ysfg Ysfg' F I G . E21 GASTROCNEMIUS ATTACHMENT TO FEMUR IN YZ PLANE FOR ROTATION 9XT t a n 9xfg = | | | | , R f g y z 2 = Y s f g 2 + Z s f g 2 9xfg ' = 9xfg - 9XT Y s f g ' = Rfgyz • cos 9xfg Z s f g ' = Rfgyz • s i n 9xfg* c o r r e c t e d f o r r o t a t i o n i n the XZ p l a n e (9YT) and r o t a t i o n i n - 163 -the XY p l a n e (QZT) Y s f g = Y s f g ' • cos QZT A s f g = Z s f g ' • cos 9YT X s f g ' FIG. E22 GASTROCNEMIUS ATTACHMENT TO FEMUR IN XZ PLANE FOR ROTATION 9YT Rfgxz = Z s f g 9 y f g ' = 9YT X s f g ' = Z s f g • s i n 9YT c o r r e c t e d , f o r r o t a t i o n o f the femur i n the XY p l a n e (9ZT) X s f g = X s f g * cos 9ZT The c o - o r d i n a t e s o f g a s t r o c n e m i u s muscle group attachment to the femur w i t h r e s p e c t t o the g r i d axes - 164 -X g f g = Xgk + Z s f g • s i n 9YT . cos 9ZT Y g f g = Ygk - Rfgyz • cos 9 x f g ' • cos 9ZT Z g f g = Zgk + Rgxz • s i n 9 x f g ' . cos 9YT FIG. E 2 3 GASTROCNEMIUS ATTACHMENT TO TIBIA IN YZ PLANE FOR ROTATION 9XS t a n oxg = I § £ & Zsg 9xg' = 9xg + 9XS R g y z 2 = Y s g 2 + Z s g 2 Ysg' = Rgyz • s i n 9xg' Zsg' = Rgyz • cos 9xg' c o r r e c t e d f o r r o t a t i o n o f the t i b i a i n the XY p l a n e (9ZS) - 166 -and the p l a n e (9YS) Ysg = Ysg' • cos 9ZS Zsg = Zsg' • cos 0YS Xs Zsg Zs FIG. E24 GASTROCNEMIUS ATTACHMENT TO TIBIA IN XZ PLANE FOR ROTATION 9YS Rgxz = Zsg 9xg' = 9YS Xsg' = Zsg • s i n 9YS c o r r e c t e d f o r r o t a t i o n o f the t i b i a i n the XY p l a n e (9ZS) Xsg = Xsg' • cos 9ZS The c o - o r d i n a t e s o f g a s t r o c n e m i u s attachment to the - 166 -t i b i a with respect to the grid, axes are Xgg = Xgk + Zsg . s i n 9YS . cos 9ZS Ygg = Ygk - Rgyz • s i n 9 x g ' • cos 9ZS Zgg = Zgk - Rgyz • cos 9 x g ' • cos 9YS - 16? -APPENDIX F MUSCLE AND LIGAMENT FORCES - 168 -APPENDIX F MUSCLE AND LIGAMENT FORCES 1• Qu a d r i c e p s Muscle Group Z s Y s q t Y s FIG. F l FORCE ACTION OF QUADRICEPS MUSCLE GROUP IN THE YZ PLANE Mxk = Pq • cos 9q • Ysq + Pq • s i n 9q • Zsq Pq = Mxk / ( c o s 9q • Ysq + s i n 9q • Zsq) where 9q i s the angl e o f the q u a d r i c e p s t o the Zs a x i s i n the YsZs p l a n e and Mxk i s t h e moment on t h e knee i n the YsZs p l a n e \ - 169' -F I G . F2 ANGLE OF SHANK IN YsZs PLANE 9XS (+/ ARCTAN (DYS ' /DZS ' ) - 1?0 -Y s i w sin(GYS- ©YT)- DZT F I G . F3 ANGLE OF THIGH IN YsZs PLANE 9XT .= ARCTAN (DYT"/DZT") - 17-1 -9q = 0 .31 x 10"^ (phi) - 8 .4 x 1 0 " 3 ( p h i ) 2 + O.37 x 1 0 " 2 (phi) + 15 Phi = 9XS + 9XT where 9XS and 9XT are c a l c u l a t e d i n the YsZs plane as i n F i g . F2 and F3 2. Hamstrings Muscle Group - 172 -sin G y h - D Z H 1 FIG. F5 ANGLE OF HAMSTRINGS IN YsZs PLANE 9X = ARCTAN (DYH"/l)ZH") - 173 -Mxk = Ph • cos 9xh . Ysh + Ph • s i n 9xh • Zsh Ph = Mxk / (cos 9xh • Ysh + s i n 9xh • Zsh) 9xh = 9x + 9XS. i s the angl e o f the h a m s t r i n g s t o the Zs a x i s i n t he YsZs p l a n e w i t h 9x c a l c u l a t e d as i n F i g . F5 and 9XS as i n F i g . F2 3. Gastrocnemius Muscle Group Zs tan 9x= DYG/DZG Ys e x s ©xg - e x s - e x DZG = Zg fg -Zgg DYG=Ygfg-Ygg FIG. F6 FORCE ACTION.OF GASTROCNEMIUS MUSCLE GROUP IN THE YZ PLANE - 17k -Mxk = Pq • s i n 9xg • Zsg + Pq • cos 9xg • Ysg Pg = Mxk / ( s i n 9xg • Zsg + Fg . cos 9xg • Ysg) 9xg = 9XS - 9X i s the angle of the gastrocnemius to the Zs axis i n the YsZs plane with 9X calculated as i n Fig. F7 and 9XS from F i g . F2 - 175 -FIG. F7 ANGLE OF GASTROCNEMIUS IN YsZs PLANE 9X = ARCTAN (DYG'/DZG') - 1 7 6 -LIGAMENT FORCES 1. A n t e r i o r C r u c i a t e Ligament DYA: Yga-Ygfa DZA:Zgfa - Z g a t a n G x : DZA/DYA FIG. F8 FORCE ACTION OF ANTERIOR CRUCIATE LIGAMENT IN YZ PLANE Fym + Fyk = Pa • cos 9xa Pa = (Fym + Fyk) / cos 9xa 9xa - OX - 9XS i s the a n g l e o f the a n t e r i o r c r u c i a t e t o the Ys a x i s i n t h e YsZs p l a n e w i t h 9X c a l c u l a t e d as i n F i g . F9 and 9XS from F i g . F2 - 177 -Zs' 9YS Gyfa = Qyfa + GYS D Z F A ' = Rfaxz- s'mGyfa' Gya'= 6ya - GYS D Z A ' = Raxz- sin 9ya' DZFA DZA = DZFA +Zsfc.cosGXS- DZA* Xs DYFA Gza - GZS - Gza DYA = Raxy cos Gza 0zfa'= GZS - Q z f a ' DYA DYFA'= Rfaxy • cosGzfa' D Y A = DYA + D Y F A ' I Z S fC • si n 6XS FIG. F 9 ANGLE OF ANTERIOR CRUCIATE IN YsZs PLANE 9X = ARCTAN (DZA"/DYA") - 178' -2. P o s t e r i o r C r u c i a t e Ligament DYP= Ygf p - Ygp D 2 A = Zgfp - Zgp tan G\x = DZP/DYP FIG. F10 FORCE ACTION OF POSTERIOR CRUCIATE LIGAMENT IN YZ PLANE Fym + Fyk = Pp • cos 9xp Pp = (Fym + Fyk) / cos Gxp 9xp = 9X + 9XS i s the a n g l e o f the p o s t e r i o r c r u c i a t e t o the Ys a x i s i n the YsZs p l a n e w i t h 9X c a l c u l a t e d as i n F i g . F l l and 9XS from F i g . F2 - 179 -eyfp = BYS- eyfp DZFP'=Rfpxz-sineyfp' tan Gyfp = DZFP/DYFP DZP =Zsp DZP"= DZP'+Zsfc -coseXS-DZFP' eztp ' DYP"= DYP+ DYFP+ Zsfc-sin 0 X S FIG. F l l ANGLE OF POSTERIOR CRUCIATE IN YsZs PLANE, 9X = ARCTAN (DZP"/DYP")" - 180 -3- L a t e r a l C o l l a t e r a l Ligament DXL FIG. F 1 2 FORCE ACTION OF LATERAL COLLATERAL LIGAMENT.IN XZ PLANE Rz = Fzk + Fzm + F z c r P I = (Myk + Rz • X I ) / (cos 9yl • X s l - s i n Qyl • Z s l - cos 9yl • Y l ) 9yT = 9YS -• 9Y i s the a n g l e o f the l a t e r a l c o l l a t e r a l t o the Zs a x i s i n the YsZs p l a n e w i t h 9Y c a l c u l a t e d as i n F i g . F 1 3 and 9YS from F i g . F 2 / - 181 -DZ FL 9x1 = GXS + Gx l DZL = R l y z - s i n G x l 9 x f l = G X S t 9 x f l DZFL'= R f l y z - C O S 9 x f l ' DZL = DZL +• DZ FL +• Zsf c • cos G Y S 9 z l = GZS - Q z l DXL = Rlxy sin Gz l G z f l = 6 z f l - 0 Z S D X F L = Rf I xy c o s G z f l ' v DXL = DXL - DXFL t Z s f c - s i n G Y S FIG. F 1 3 ANGLE OF LATERAL COLLATERAL IN XsZs PLANE, Qy = ARCTAN (DXL"/DZL") - 182 -4. M e d i a l C o l l a t e r a l Ligament DXM = Xgm - Xgfm DZM = Zgf m - Z g m t a n e y = DXM/DZM FIG. F14 FORCE ACTION OF MEDIAL COLLATERAL LIGAMENT IN XZ PLANE Rz = FZK + FZM + FZCR Pm = (Myk + Rz • X I ) / (cos 9ym • Xsm + s i n 9ym • Zsm - cos 9ym • X I ) 9ym = 9YS - 9Y i s the a n g l e o f the m e d i a l c o l l a t e r a l t o the Zs a x i s i n the XsZs p l a n e w i t h 9Y c a l c u l a t e d as i n F i g . F 1 5 and 9YS from F i g . F2 - 183 -FIG. F 1 5 ANGLE OF MEDIAL COLLATERAL IN XsZs PLANE, 9Y = ARCTAN (DXM"/DZM") - 184 -APPENDIX G RESOLUTION OF EXTERNAL FORCE SYSTEM - 185 -APPENDIX G RESOLUTION OF EXTERNAL FORCE SYSTEM XY P l a n e Fxk = Fxk-cos9ZS + Fyk-si n9ZS Fyk'= Fyk-cosGZS - Fxk-sinGZS Mxk'= Mxk cos GZS + Myk-s inQZS Myk'= M y k - c o s 9 Z S - M x k - s i n e Z S XZ P l a n e Fxk'= Fxk-cosGYS - F z k - s i n ©YS Fz k' = Fzk-cosGYS + Fxk-s in ©YS Mxk' = Mxk-cosGYS - Mzk-sinQYS Myk' - Mzk-cos9YS + Mxk -s in9YS GYS \ YZ P l a n e - 186 -Ys Myk \ ^ Fyk= F y k . c o s e x S - F z k - s i n G X S Fzk' : Fzk-.cos GXS + Fykis inGXS Myk'= M y k c o s G X S - M z k - s i n G X S M z k ' r Mzk-cos GXS + M y k - s i n G X S E q u a t i o n s o f R e s o l v e d E x t e r n a l F o r c e System Fxk" = Fxk • cos 9ZS • cos 9YS + Fyk • s i n 9ZS - Fzk • s i n 9YS Fyk' = Fyk • cos 9ZS • cos 9XS - Fxk • s i n 9ZS - Fzk . s i n 9XS Fzk' = Fzk • cos 9YS • cos 9XS + Fyk • s i n 9XS + Fxk • s i n 9YS Mxk' = Mxk • cos 9ZS • cos 9YS + Myk • s i n 9ZS - Mzk • s i n 9YS Myk' = Myk • cos 9ZS • cos 9XS - Mxk • s i n 9ZS - Mzk • s i n 9XS Mzk' = Mzk • cos GXS • cos 9YS + Mxk • s i n 9YS + Myk • s i n 9XS - 187 -APPENDIX H LIMB ACCELERATIONS - 188 -APPENDIX H LIMB ACCELERATIONS 1. Li n e a r Limb A c c e l e r a t i o n A numerical d i f f e r e n t i a t i o n technique based on f i n i t e d i f f e r e n c e s (Lanczos, 1957) was;used to determine the l i n e a r limb a c c e l e r a t i o n s of the foot and. shank. The l i n e a r ac-c e l e r a t i o n s of the foo t and shank i n the X, Y and Z d i r e c -t i o n s were c a l c u l a t e d from the r e s p e c t i v e displacements of the centre of mass of the foot and shank. Therefore the l i n e a r a c c e l e r a t i o n s are c a l c u l a t e d as f o l l o w s : Xn = 4Xn - 4 + 4Xn - 3 + Xn - 2 - 4Xn - 1 - lOXn - 4Xn + 1 + Xn + 2 + 4Xn + 3 + 4Xn + 4 / 1 0 0 t 2 X n - displacement of centre of mass of segment i n X d i r e c t i o n at time, n X - a c c e l e r a t i o n of centre of mass i n X d i r e c t i o n at time, n n n - p o i n t i n time t - time i n t e r v a l of displacement from n to n + 1 S i m i l a r equations give the a c c e l e r a t i o n of centre of mass of a segment i n the Y and Z d i r e c t i o n s - 189 -Yn = 4Yn - 4 + 4Yn - 3 + Yn - 2 - 4Yn - 1 - lOYn-- 4Yn + 1 + Yn + 2 + 4Yn + 3 + 4Yn + 4 / 1 0 0 t 2 Zn = 4Zn - 4 + 4Zn - 3 + Zn - 2 - 4Zn - 1 - lOZn., - 4Zn +,'.1 + Zn +• 2 + 4Zn + 3 + 4Zn +.4 / 1 0 0 t 2 3- A n g u l a r Limb A c c e l e r a t i o n The a n g u l a r a c c e l e r a t i o n s o f the f o o t and. shank were determined, from t h e r e s p e c t i v e a n g u l a r d i s p l a c e m e n t s o f t h e f o o t and shank. The a n g u l a r a c c e l e r a t i o n s are c a l c u l a t e d about the X, Y and Z axes p a s s i n g t h r o u g h the c e n t r e o f mass o f t h e segment as f o l l o w s : 9x = 49Xn - 4 + 49Xn - 3 + 9Xn - 2 - 49Xn - 1 - 109Xn - 49Xn + 1 + 9Xn + 2 + 49Xn + 3 + 49Xn + 4 / 1 0 0 t 2 9xn - a n g u l a r d i s p l a c e m e n t o f segment about the X a x i s a t ,time, n 9x - a n g u l a r a c c e l e r a t i o n o f segment about the X a x i s a t t i m e , n n - p o i n t i n time t - t i m e i n t e r v a l o f d i s p l a c e m e n t from n t o n + 1 S i m i l a r e q u a t i o n s g i v e the a n g u l a r a c c e l e r a t i o n o f the segment about t h e Y and Z axes 9Yn = 49Yn - k + 49Yn - 3 + 9Yn - 2 - 49Yn - 1 - 109Yn - 49Yn + 1 + 9Yn + 2 + 49Yn + 3 + 49Yn + 4 / 1 0 0 t 2 - 19P -9Zn = 49Zn - 4 + 49Zn - 3 + 9Zn - 2 - 49Zn - 1 - 109Zn - 49Zn + 1 + 9Zn + 2 + 49Zn + 3 + 49Zn + 4 / 100t 2 

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