'''''''''''' THE MECHANICS OF THE DOUBLE-sBA'CK S‘OM'ERSAULT Them {hr the Dane of. M. A. MICHIGAN STATE UNIVERSITY Yoshiro Hatano 19:62 m..*. ‘1. “ “=2 win-1.1m (Hf THESIS LIBRARY Michigan State University :6'57wm.nnmw BUREAU OF SOUSKT'I'Jif‘xL RICE"? COLLEGE OF ELUCKI. N MlCi-HGAN Sit-ATE; U!\ll\.r'E-_"..3IEY EAjI' LAN‘ANG, MsCr-HuMN r'vr ._, ' fin 1': ‘7‘ m m 17-1 x ' — ~1fl‘r' r‘ r-v' ‘( ‘ '"m id; machaNiCb OF in; DOUEiE-Bacn oOhEkth-l By Yoshiro Hatano AN ABSTRACT OF A THTSIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF ARTS Department of Health, Physical Education, and -ecre zkppiwaveci st<.«,7 AB TRACT THE NECHANLCS OF THE DOUBLE-BACK SOMEESRUTT by Yoshiro Hatano Statement of the Problem To analyze the mechanics of the double-back somersault in mat—tumbling, utilizing a mathematical and geometrical analysis of motion pictures of this trick. Methodology This study was undertaken to analyze the mechanics of the double—back somersault, using the cinematographic tech— niques as described by Cureton. The subjects were selected from double-back somersaulters in the United States. Each frame of the pictures of the performance was drawn on paper. 5 the body rotation Ho The movement of the center of rrav ty an’ L ‘4 were mathematically and geometrically analyzed. Conclusions Subject to the limitations inherent in cinemato- graphic analysis and other limitations given for this study, the following conclusions have been drawn. 1. :n the subjects studied. it would be impossible to perform a double-back somersault without a preceding tack-handspring. Yoshiro Hatano The body angle of take—off should be less than 90 degrees and the initial flight angle of the center of gravity should be approximately 80 degrees. The body tuck during the latter half of the flight in all subjects was found to be more than that of the first half. The angular velocity of body rotation should be greater in the latter half of the stunt than in the first. The first somersault in all subjects required 55 per cent or more of the flight time.’ The height attained by the center of gravity in flight should be at least lOOcm above the center of gravity at take-off. THE MECHANTCS OF THE DOVBLE-EACK SOMERSAULT By Yoshiro Hatano A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF ARTS Department of Health, Physical Education, and Recreation 1962 ACKNOWLEDGEMENT The author would like to express his gratitude to Dr. Wayne D. Van Huss for his kindest help and advice throughout this study. Appreciation is also extended to Mr. George Szypula, the subjects, and Mr. Henry Cole, for their kind help and cooperation. TABZE OE CONTEFTS ‘ C) O-r—4 r 11 '15 'U H} [I] 'U :1: L‘\ I r. :. :NTPODUCTION TO THE PROBLEM. . . . . . 1 Purpose of the Study . . . . . . . 2 Need for the Study. . . . . . . . 2 Limitation of the Study . . . . . . 2 v- .‘ .L- wefinition of Terms . . . . . . . 2 ~— EEVIEW OF THE RELATED _--E8ATUQE . . . . 3 :::. METHODOLOGY . . . . . . . . . . . 6 Analytical Procedure . . . . . . . 6 IV. MECHANICAL ANALYSIS . . . . . . . . 9 Analysis of Flight. . . . . . . . 2O 0 V. EJNHAEY, CONCLUS:ONS, AND FECONKENDATZONS . LU i—-‘ U) F4 Summary . . . . . . . . . . . U) H Conclusions . . . . . . . . . . Recommendations. . . . . . . . . 32 ‘APE‘EIJD: XEAD O O O O I O O I O O O O O O BFZ APPENDIX A-—Definition of Terms . . . . . . 3: APPENDIX B--Quantitative Data of Body Rotation . A3 LIST OF TABLES TABLE PAGE I. Subjects and Films Used for the Study. . . 6 1:. Kinetic Elements of Double—back Somersault . 10 :12. Comparison of Kinetic Elements of Some Tricks . . . . . . . . . . . . ll IV. Elements at the Time of Maximum Body Rotation. . . . . . . . . . . . 21 V. Degree of Body Rotation and Time Proportion. 23 FIGURE 1-1. 1—3. 1-H. k,“ -4. EZST OF FIGURES Illustration of Locuses of Some Parts of the Body in Executing the Double—back Some‘sault: Mitakis Illustration of Double-back Somersault: Mitakis. Illustration of Locuses of Some Partszof the Body in Executing Double—back Somersault: Johnson. Illustration of Double—back Somersault: Johnson. Force and Velocity at the Kick Comparative Locuses of the Center of Gravity of Double—back Somersault Angular Velocity of Body Rotation. Changes of Hip and Knee Plexions as a Function of Time:_ Hyman Changes of Hip and Knee Flexions as a Function of Time: Holmes. Changes of Hip and Knee Flexions as a Function of Time: Mitchell Changes of Hip and Knee Flexions as a Function of Time: Johnson "0 13 14 14 16 19 26 26 27 F _.. I .. 'J TEE \Jl 'U x. . LI] Changes of Hip and Knee Flexions as a Function of Time: Mitakis . . . . . 27 Two Patterns of Tucking . . . . . . . 29 The Body Rotation and the Maximum Hip Flexion. . . . . . . . O {U \JL; Pelation Between Hip Flexion and Degree of 30 Forces at the Take—off . . . . . . . A2 Body Rotation. . . . . . Hip Al:\nr~lgle. . . C . C . C O . C . IC Knee Angle . . . . . . . . . . . A2 Angle of Body Rotation . . . . . . . A2 C'APTEE I .L INTRODUCTION TO THE PROBLEM It is natural for a gymnast to long for more skillful tricks. The double—back somersault is considered to be one of the most difficult tricks in the field of mat— tumbling. Approximately ten gymnasts have ever successfully completed this difficult stunt. All of the double somersaults have been executed in the United States. Tn international gymnastics competition, however, the United States representatives are not the best in the world. In fact, at the Home Olympics in 1960, the U. S. Gymnastic Team finished fifth, even though the team was reported to be the best ever assem led from the U. S. The popularization of gymnastics in the world in recent years has made it easy to exchange ideas of skills and performances of this sport from one country to another. The U. S. however, is dominating the field of tumbling as seen in the double-back somersault. As a foreigner to this country and as a gymnast, the writer had started his interest in this particular stunt soon after coming to the United States in 1960. Observing the double-back somersaults, it was noticed that some performers do not tuck as one would expect. Mechanically, it was thought that double-back somersaulters would tuck soon after the take-off and remain tucked until just prior to landing, for the shorter the radius, the faster the rotation of the body. The present study was undertaken to satisfy the writer's curiosity regarding the mechanics of the stunt and to supply further information necessary for adequate coaching of the stunt. Purpose of the Study To analyze the mechanics of the double—back somersault in mat-tumbling. Need for the Study The double-back somersault is an American speciality, and there is a definite need for the study among the gymnasts in the world. Very little literature is available on the double—back somersault. limitation of the Study The number of subjects was limited to five; Steve Johnson, Michigan State University, Lewis Hyman, University of Michigan, Hal Holmes, University of Illinois, Steve 'Hitakis, University of California in Los Angeles, and Rusty Mitchell, Southern Illinois University. The study is also limited by the usual limitations-of cinemato- graphic analysis in the accuracy of angles and dimensions. Definition Of Terms The definition of terms is presented in Appendix A. C AFTER I: REVIEW OF RELATED :ITEEATURE There is little research literature available concerning this stunt. Host of the material available is opinionated and nonquantitative. Mitakis pointed out, by a cinematographic study, that at the moment of body contact with the spotter's arms, the tumbler has completed slightly more than one and a quarter somersaults in executing the double-back somersault} Szypula describes the following items as the training techniques for a back-double; (1) to lift the somersault, (2) to throw the arms up and back hard without traveling back, (3) to bring the knees up quickly after the spring, and (u) to maintain a tight tuck through the trick.2 Brodeur and Johnson found that Johnson's single—back sonersault is in flight more than one second and that the 3 center of gravity is lifted a height of 7.3 feet. 1.. , . m s. Kitakis, ”Spotting the Back Double," lhe Modern GanaSt9 1:24-25, DecembeP, 1956' 23. Szypula, Tumbling and Balancing for All (Dubuque, lowa: Wm. C. Brown Company, 1957), pp. 149-150- (a i 3.. Brodeur and S. Johnson, HStudy of Back S mersault,‘ The Kodern Gymnast, 4:19, February, 1962. Austin found, in executing a double—back somersault, that the first somersault takes more than SO per cent of the whole flight time, and that in the second somersault, the performer tucks more than in the first one.“ Holmes believes the following techniques are the impor- tant mechanics of this stunt: (1) the body angle is sharp at take—off; (2) the body is lifted up soon after the take— off; (3) if the body is lifted enough, it is possible to perform a double—back somersault with less tuck of the body; (4) after the hands are lifted up at the take-off, the hands should grab the knees; and (5) the head is kept back until the mat is seen prior to the landing.5 Johnson's recommendations for the mechanics of a double-back somersault are as follows: (1) first possess the ability to perform a high back somersault; (2) the head should be kept up at the take-off, then kept back and straight at the landing; (3) late tuck; and (A) the hands should be kept up at the take-off, then grab the legs below knee at both sides at the tuck and the hands are to be placed in front at the finish.6 Hyman's recommendations for satisfactory performance Of a double-back somersault are as follows: (1) a sharp I ' ° /N ‘ - ' ' 4J. m. Austin, ”cinematographic AnalySis of the -. .1 1 g u . . ,_ . Double Backward somersault' (unpublished haSter's theSis, Department of Physical Education, University of Illinois, 1953 Pinterview with H. Holmes, March 1, 1962. 6 Interview with S. Johnson, March 1, 1962. 1 body angle of take—off; (2) hands kept up at the take—off, then later outside the knees to tuck the body; and (3) the head should be laid back and chest be thrown back right after the take—off.7 In reviewing the literature concerning the mechanics of the double-back somersault, it seems that the high flight, and a continuous tuck with hands grabbing the lower body are major keys in executing the stunt. The problem is, how does a performer fly and tuck mechanically. :shikawa's research on the factor of height of the athletes in various sports was introduced by Miyahata. In this study, lshikawa points out that the center of gravity of a gymnast is placed approximately 56 per cent of his normal height from the feet. 7 Interview with L. Hyman, March 1, 1962. ‘T. Kiyahata, Shintai Undo no Kagaku (Kinesiology) (Tokyo: Gakugei Shuppan Sha, 1960), p. 151. CHAPTER 7T7 METHODOLOGY This study was undertaken to analyze the mechanics of the double—back somersault, using the cinematographic tech- niques as described by Cureton.l The subjects were selected from double-back somersaulters in the United States. A 16mm camera, with maximum speed of 64 frames per second, and an 8mm camera with constant speed of 24 frames per second were used for taking pictures. The camera was set perpendicularly to the mat from 25 feet to 40 feet apart from it. The sub— jects, description of the films used for this study,and pertinent data on subjects where available are presented in Table I. Analytical Procedure Each frame was projected on the wall and drawn on paper. The center of gravity of the performer of each frame was estimated and placed on the projected picture. The movements of the whole body and each segment were observed by these pictures. The movements of the center of gravity at the preceding movement, take—Off, flight, and the landing 1T. Cureton, ”Elementary Principles and Techniques of Cinematographic Analysis as Aids in Athletic Pesearch, lesearch Quarterly, lO:3- 24, May, 1939. .m-xcpflg :o maco :oflpaEHfixopdasa * m.eg m.\a ms a.ms \.m\s an em .2; : zaass e m.ea m.em a. ...ms .~.mgs as ea .::: n gazalw a m.x\ H.mwl Om ©.H© «u.m©a 3% :m .:E_»w m,e:nxw x In m.mofi mm m.3@ :.:m~ O: :m .55 m :omzzow Q I: m.moa mm ©.m© :.:m~ ma :m .55 b cows:0m m :1 un :1 I: I: w id .55 m fimozooflg : Eon.wg m so mu m.nm \.m\1 mm m: .55 mg seen: m -- *@ mtg -- -- *c.mm .. so .2; as sweepsa r In Eom.mofi npmmm axm.QQ :c4 3mg .pu O: ocoxxo .55 c_ :On£:o% fl dish >p~>jpw eve uzaflcfi QLEHQ: ooCdme; Lidia pacing ;5fim .0”...H a mmcwp.ficmw ago .H¢+w~ficu do pzmwc: Finn...» man :0. 33m: .2; L... :z .1. 0.90”... Eb... m Maggi U.) were measured and graphed. The changes of hip angle, knee angle, body angle of rotation, and angular velocity of body rotation were also measured and graphed. Mathematical and kinetic calculations of each performance were made upon the measurement of the center of gravity movement. The metric system was used exclusively in the analysis. CHAPTER IV MECHANICAL ANALYSIS This study was undertaken in order to analyze the mechanics of flight and the body rotation of double—back somersault through cinematographical study. First, the mechanics of flight were studied by following the path of the center of gravity, where the center of gravity was located 56 per cent of the gymnast's normal standing height. During the trick, however, the center of gravity may be located in space outside of the performer somewhere near his stomach, e.g., when in tuck position. Secondly, the mechanics of body rotation were analyzed by considering the tuck angle and the speed of rotation of the body as a whole throughout the flight. Analysis of Flight Kinetic Analysis of the flight was made on four of the subjects, e.g., Johnson, Mitakis, Hyman, and Holmes. Many of the kinetic elements of these subjects found in Table I: are the same for each subject. The time of flight is roughly one second, and the time of take—off is approximately one—tenth of a second. The former is almost the same as that in Austin's study, but the latter, longer than his. The distance of the flight -- 0H.:H m©.HH Hm.@ HHHHHH Ho HHH>eHm mceHHV an mommH mmmHH ommm Amomm\Eov HOHx mnH Hm :oHHHHHHooo< In mw mw omw AmonoUVHOHx oLH Hm moHoH Ho ons< -- o m.HH- om HHQHHHHH HamEm>oE HHH>me Ho Hchmo mcHeoomHo Ho m .HH:< -- mm: :mm mmm Homm\eHv HHH>mHm Ho Hchmo Ho HHHooHo> mcHewoeHm HH: Hem com mmm AHHV HHHHHH Ho moHOH HHHHHHH mmH mam mHH mam Homn\eHv HHH>eH Ho Hszmo Ho HHHOOH®> HHHHHCH HHOHHoHooLe cow Hm» MHw Aomm\Eov HHH>mHm Ho Hchmo Ho HHHOOHm> HmHHHZH m.®w mw m.m> oo.xw HHH>mHm Ho Hchoo mo mHmzw HHHHHH ooH HHH mmH mmH AEHV HHH>HHH Ho HHHHHH Ho H: 1 maH Hmw mmm HHm Asov HHH>eHm Ho HHHHto Ho HHHH Hts H.tH Ht: my mm H» :m “gov HHo-mer Hm HHHsmHH Ho HtHsmo Ho HHHHtm ow m3 Ow om> HHOumwa Ho onzm Hoom mom mam mom @mm HEHV HHHHHH Ho mocmHmH: mmo wmo. wmo. moo. m:\: mmH\HH mmH\HH wm\m H.0mmv HHo-mer Ho HEHH om». mwo.H >30.H mmw. :m\mH :m\mo e©\Hw mm\oH H.0mmv HHHHHH Ho HEHH mmEHom memHHz comczcm cmEHm chmEmHm QHHocHx EM SimmmEOm m0 HcmHHm HHH>MHG HLMHHH HHH>mHm mHCHEmHm HwHHHCH Ho QEHB HoyHoHcmm Ho Ho wocmHmHQ Mo Hchoo)Ho mHmc< HzmHHm HgmHmm HmeguH: HmXOHmH mEOm ho mBszmgd OHDMZHK mo Zommmmhm mo Lapcmo wLp mo mSoom .OH «QOHmeSU pmwfl map pqmoxm “mmampm ma >L®>m campa 1M Locus of back of the Locus of the knee Locus of the toe head " \ 1”, 0 \ 0‘. .Q\ ’I .. (c.. I, .‘...H " '-:i~:.o o . ’ ‘\ \ l .0 .0 'I ‘ l.\\ O Q C ‘ Q. " . \ O.‘.. ’ C \‘ 0" ‘ “-” \\ '0'. ‘ \ . . x l «x 4“ \ x\\ ‘ \ O ' , 2 \ ’ . ) Mat Figure 1—3. Illustrations of Double-Back Somersault: Johnson Locus of the center of gravity is Projected every 12 frames, ' except the last duration, a . 10 frames. Mat V690 Figure 1—H. Illustration of Double-Back Somersault: Johnson #4 k)“ ranges between 2080m and 245 cm, and the lift of the center of gravity, lOOcm to lQMcm. The movement of the center of gravity in the very first moment of the flight in the picture gives the initial velocity and the flight angle of the center of gravity of each performance. Also, the following formula gives the initial velocity, when the initial flight angle of the center of gravity and the distance of the flight are deter- mined from the picture. - 2 . R = 2V0 Slnecose R = distance of the flight E g = gravity V0 = initial velocity 6 = initial flight angle of the center of gravity The four initial velocities and the flight angles of the center of gravity are the same for each performer, as are the theoretical and actual initial velocities. The velocity of the bddy movement before the kick is taken from the movement of the center of gravity in the last moment of the backhand—spring. This motion of the center of gravity is inclined at an angle to the mat. Hyman's ‘0 - center of gravity movement before the kick has an acclivity of-9 degrees, Johnson‘s, an declivity of 17.5 degrees, and Mitakis, no noticable slope. The acceleration which the center of gravity receives at the moment of kick is given as the change of speed and direction of the center of gravity. As seen in Figure 2, the initial velocity (V0) is given as a result of the 16 2—2: Johnson 985 /sec. \ 2-1: A Model \ .970 68“ 53Mcm/se , 17-50 Preceding Velocity 79.5 ‘\ 2-4: Mitakis 2 -3: Hyman F I Figure 2. Force and Velocity at the Kick (subject traveling left to right). l7 preceding velocity and the velocity just after the kick. Since the preceding velocity and the initial velocity are obtained from an analysis of the pictures, the velocity of kick can be stated mathematically and geometrically. After obtaining the velocity of kick, the force and the acceler- ation at the moment of kick are found and are Listed in Table II. Johnson has the largest initial velocity and the flight angle of the center of gravity. These are 767 cm/sec. and 79.50, respectively. Holmes' are the next largest with 723em/sec. and 78.50. Mitakis has the smallest flight angle of the center of gravity with 750, yet his initial velocity, 693cm/sec. is bigger than Hyman's 613cm/sec. Hyman's center of gravity takes off from an angle of 77.50. The force at the initial stage of the flight ranges \Q (”IQ between 355kg and 59 k It is interesting to note that the flight angle of the center of gravity and the angle of the force at the kick are close in each subject, these angles being approximately 770 and 670, respectively. It may be concluded, although limited to an examination of four subjects, that the trick is not possible if these angles are not the same. The acceleration which the center of gravity receives 2 and at the moment of kick ranges between 6380cm/sec lBQOQcm/secg, which are equivalent to 6.51 and 14.19 times the gravity of the earth. Although the time interval is roughly one-tenth of a second, it is found that at the 18 moment of kick, the whole body feels as if it weighed six to fourteen times what it would normally weigh. The comparison of the take-off angle, time of flight, the maximum height of the center of gravity, distance of the flight, and the projection velocity in the double- back somersault, broad jump, and high jump is seen in Table IV. Holmes has the best ability in the vertical jump among the subjects, as was shown in Table I. His vertical jump from the stand (or squat) position, without any pre- ceding run or any kind, produces only 389.80m/see of initial velocity which is less than any of the initial velocities of the double—back somersault done in this study. The calculation is as follows: (A 4 I! [\D m C”) l - initial velocity gravity of the earth _ height of the vertical jump H U) 09 I V =-/ 2 x 930 x 77.5 = 389.8cm/sec. Holmes' flight angle of his center of gravity is 78.50. If he had jumped with this angle of take-off from the squat position, with this vertical element, the initial velocity of this jump would be approximately MOOcm/sec., as seen below: 380 a v' = - “'L cosYd.5 = MOOcm/sec. This is about 55.3 per cent of the initial velocity of his 9 .l. pfizmmpmeom xommuoapsoa mo th>MLm mo popcoo ecu mo mmmSooq o>HpmmeEoo .m epsmflb conspmflm 0mm Ecoom one seoofi or o p B p r lbl / / a 50 z / OOH / r I / to to I 0. \Q I z .. a.“ / ... \s I no \ o.‘\ H I / oo . o.\ m. / on \o.\ y W :a / .0 es Qua U. 9 o \ n4, / k. \ .. x / o. o. \ / [b \\\~. \ / . t. x / f7 \ ... x // .3. I Ill. \ .... \ Eo / 00.0.5.0..06000 \\ ll CON / \ // \ / .\\ // \ I \\ l/ \ WQCHHOEH cocoa... ’n'll“ swat». nll II E .l Omm mflxmpflz lllll comczom 2O double-back somersault. Therefore, the preceding back- handspring is needed in order to supply sufficient velocity for execution of the double-back somersault. Analysis of Body Rotation The energy of body rotation comes originally from leaning the body back right after the take-off. The initial rotation, due to the little tuck, is slow. It increases, however, and at the fastest moment of rotation, the tuck of the body reaches to some 60 per cent of the standing height, as seen in Table IV. Keeping the head back helps the rotation backward. The hands, which, at the moment of take-off, are generally kept up, so that the center of gravity is placed as high as possible, grab some part of the legs below the knees so that the performer may keep tucking during the fast rotation. However, from the 8mm. and 16mm. photographs, with a maximum shutter speed of 64 frames per second, it was not possible to analyze the head and the hand movements into meaningful detail. Therefore, this study will be mostly concerned with the hip and knee flexion and the body rotation. As seen in Figure A, Johnson has an exceptional way of body rotation, i.e., just prior to the landing, his speed Of body rotation reaches the maximum, while the other sub— jects have maximum speeds of body rotation at an earlier stage of the latter half of the flight. Among these maximum rotational velocities, Eolmes' is [\D }__J fastest with 13630/sec.,i.e., 24 rao ians /sec The slower two, Johnson and Iitakis can he explained by their having longer flight time, than the other three. Holmes has the shortest rotational radius with 61.1 per cent of the standing height, at the time of the maximum angular velocity of body rotation (Table :v). Hyman has an exceptionally small degree of tuck with 74.4 per cent at the time of maximum rotation. He should be using more energy for the body rotation than any one else, since he tucks less all through the flight. His flight parabola is low and long, and he leans his body back more than the others right after take—off. These facts are consistent in understanding his method of performance. Actually he believes in this method according to the inter- view. TABI E IV ELEMENTS AT THE TI”E OF MAXIMUM BODY ROTATION Degree Energy of Maximum Angular Velocity of Body Angular Name of Body Rotation Tuck Rotation Johnson 836o/sec. l5. 7rad. /sec) 65.Z% 330. 5kg Hyman 984 17.3) 75. 317. 5 Mitakis 896 15.7) 66.2 309. 8 Holmes 136E 24.0) 61.1 475.3 Mitchell 1200 (21.2) 64.3 -- At the time of maximum rotation speed, Holmes' rotational energy is considerably large, due to the large rotational velocity. This rotational energy is given by the following formula: 22 E = l/2mVP2r2 E : energy of angular rotation :.®© moehwoo Com UmmeOL econ Haas: mean egmwae ac acme tea mm 1-- mm as --- engage macs: no ezc esm pzmflam mo oco :omzomn oopmoom zoom mmfi Ame woe msa oma ermwfie mo oco Ugo :\m Coozoon ompmpom aoom Ham smm 6mm saw oom :\m ace m\a aceseen emneaca seem smfl sow me sea owe m\H 6cm :\H semsncn emnsncs seem awe :62 mm mm we :\H was eec-eeme smasher ecesacw seem mmo -u- mmo mam --- enmafin cases to est as newsmacs seem coo one How :mm mmo enmwae to are he scaesnes seem ass mam om: Ham mes sewage no s\m as coarsest seem 6mm mam cam Ame mam enmefle ac m\a cm ceanmact seem mm mm mm om mo reggae to :\H em coarsest seem mm- ma- mm- mm- egg- oec-cxmn as ceaeeeca ac cflmcm eeem Hfimcooflz moEHom mflxopflz somcgow geezm oEHE ZOHBmOmOmm mEHH DZ< ZOHB mdm< fl FL 24 though his range of percentages of the time of whole flight for the first somersault is shorter and smaller than that of this study. Johnson's 76.6 per cent of the whole flight time for the first somersault is quite high, but it is understandable when one realizes that his body rotation is faster at the last moment of the flight. All the values of this item are the same for each subject and, therefore, indicate that the key to the body rotation of double—back somersault exists in the second half of the flight. The tuck of the body consists of hip and knee flexion besides the head and hands movements. These results are seen in Figure 5 and Appendix B. The hip and the knee are usually not flexed much at the take-off, but soon after starting the flight, both hip and knee start to flex. They remain tucked until the recovery just prior to the landing. The degree of the hip and the knee flexion can be considered to be proportional to the degree of tucking of the performer in each moment. There are two types of tucking among the subjects. Johnson, Mitakis, and Holmes stop increasing their tuck by the end of the first third of the flight. During the second third of the flight, the tuck is kept about in the same degree with a slight increase. The maximum tuck comes approximately at the two—thirds mark of the flight. Hyman's and Mitchell's tucks have more distinctive first peak. After the first peak, there is a depression, then the inclining comes more obviously. 25 Johnson ..... Mitakis ,7 ..........Hyman -—---—-Holmes 12000/7 ,r"‘\\ ___ ___.watchell sec . {:T?:$§k 96o - . 720 i 430 i 240 J O I T Y Y 1/4 1/2 3/4 1 Time in Seconds Figure 4. Angular Velocity of Body Rotation 1200i 100 80 1 6O 4 g 40 1 w 2 <11 20 . o s .- fi , l/4 1/2 3/4 Time in Seconds Figure 5-1. Changes of Hip and Knee Flexions as a Function of Time: Hyman 1200‘ 100 - 80 « 0) r6?) ’ a 9 6o 40 . hip angle 20 ~ \ —————— knee angle ~.. O . , i 1/4 1/2 3/4 Time in Seconds Figure 5—2. Changes of Hip and Knee Flexions as a Function of Time: Holmes 12oQ 100. 80 . CD '61“, 60 l G <1: 40 . hip angle 204 ----- knee angle 0 Fr“ I ‘ I l/4 1/2 3/4 Time in Seconds Figure 5-3. Changes of Hip and Knee Flexions as a Function _of Time: Mitchell 26 120Q lOO ‘ hip angle l ______ knee angle ‘ I ' V 1/4 1/2 3/4 H1 Time in Seconds Figure 5—4. Changes of Hip and Knee Flexions as a Function of Time: Johnson Figure C—5 2 Changes of Hip and Knee Flexions as a Function of Time: Mitakis '\ ’ .4 2 \ .,/ These two patterns of tucking are modeled in Figure 6. Thus, the maximum of the body tuck is always at the latter half of the flight. According to the result, this Hmore tuck in the latter half” is considered to be an important key of this stunt. There is a consistent relation between the hip angle and the angle of body rotation in each subject, as seen in Figure 8. The maximum hip flexion of all the subjects occurs when the body rotation is between 400 degrees and 520 degrees. This is the time when the head is going down in the second rotation, as seen in Figure 7. Presumably, it is comparatively easy to increase tucking when the upper body is going down and the lower body is being pulled down. This is understandable, from other gymnastic tricks, too. Hyman's tuck is much different from other performers. His hip flexion comes much later than knee flexion at the first third of the flight. His hip is flexed only about 70 degrees when the body rotates 90 degrees, while all other subjects flex their hips more than 90 degrees at the same instant of the body rotation. Consequently, his ”late tuck” is obvious and the Ffifgfmmsa distinctively positive degree of incline as seen in Figure 5-1. His hip flexion is netrly always less than knee flexion, while Holmes, Mitakis, and Mitchell flex the hip more than the knee, and Johnson's hip flexion is about the same as knee flexion. All the subjects flex knees more than hip before they reach to the point of ,E—J I‘D \ { , \ 4‘ ' x P a v a 2 .L 0 E P] 2 P E 1 9—4 LN. v o a I o G 1 9 . 9 a) 1 g 4 Cl) 1 b1) ' C1 ' 53:: 1 l L 1 l ' t I I * 1/3 2/3 end 1/3 2/3 end Time of Flight Time of Flight Figure 6. Two Patterns of Tucking Figure 7. The Maximum Hip Flexion Comes When the Body Beaches to this Position commom ocm cowxoam oflm Cmozuom cowpmamx .m ohsmflg newcsnem seem to coflmeom xoom mo oepmcfl cog. com com. co: com com cog \/ o . N 3 s X z I _ o . _ r...- meEfiom ...... \u «om WW CGEEI IIIIII .‘ h. PW maxsflz ...... .. ... e comczom.|lllll I u o coco - N u To .. \., T :8 a c s o T: .. . / I, . c a \ 2 ~ . u M u s // x n W o \ \ In \ a VA .a \ .60... soxoo o r O? .L. o .\ oco c.0000 - m . .oo \l/ {6‘ .. \\ \. I a / .. 0.5.0:... \ . // ooXopoo {1K \ co 0. \‘l \ ‘ .0... O . \< Ill. I.‘ 1),, /x - .._.. x 2 68 g ‘1' || \ s\ I‘ 1‘ Iv.\\ flaw \ . s SUI ARV, C KCIUSIONS, AND RECOMMENDATIONS Summary This study was undertaken to determine the most signif- icant variables in performing the double-back somersault. Five subjects performed the trick and a mathematical and geometrical analysis of motion was made of motion pictures taken during the performance. In the total analysis, the most important parameters turned out to be time and distance of flight, lift of the center of gravity, initial velocity, forces at the kick and at the initial part of the flight, flight angle of the center of gravity, angle of kick, acceler— ation at the kick, hip and knee flexions, angle of body rotation, and angular velocity of body rotation. The trick could be totally described using these terms. Conclusions Subject to the limitations inherent in cinematographic analysis and the other limitations given for this study, the following conclusions have been drawn: 1. In the subjects studied, it would be impossible to perform a double-back somersault without a preceding back—hands ring. 2. The bodv angle of take-off should be less than 9O degrees J _, l and the initial flight angle of the center of gravity should be approximately SO degrees. The body tuck during the latter half of the flight in all subjects was found to be more than that of the first half. The angular velocity of body rotation should be greater in the latter half of the stunt than in the first. The first somersault in all subjects required 55 per cent or more of the flight time. The height attained by the center of gravity in flight should be at least lOOcm above the center of gravity at take—off. Recommendations Upon the cinematographic study of mechanics of the double-back somersault, the following recommendations have been drawn: 1. Necessary height of flight and enough initial velocity are needed in order to obtain a double—back somersault. A tight tuck should be maintained at the latter half of the flight. In spotting a double-back somersault, these suggestions will be helpful to coaches: a. To lift performer, the use of a spotting belt is helpful. b. Spotting should be given at the latter half of the flight. A more significant analysis could be achieved with more subjects. kwfiterrmfiion ‘ z r - '- r~ a r - v \ C(‘LIJL (lilflll'fg “rt” pictu.e ‘ ) 1— ‘ C} could be resolution. aeiiieiM2d L» 3 LA ‘. 13:87 *0‘ _ JRAPHY BIBIZ GRAPHK Books Bunn, John W. Scientific Principles of Coaching. En ngle wood Cliffs, New Jersey: Prentice—Hall, Inc., 1360 Aleshi 7uka, Tetsuo. Taiiku o Keisan Suru (Scientific Principles of Coaching). Tokayo: Fumai—do, 1953. Hiyahata, Torahiko (ed.). Shintai Undo no Kagaku (Kinesi— 01L gy). Tokyo: Gakugei Shuppan-sha, 1960. Ono, Katsuji. Pikujo W yogi no Fikigaku (Kinesiology of Truck and rield AlhI—tlc ). Tokyo: Dobun-shoin. 1957. Scott, M. Gladys. Analysis of Human Motion. Nev York: Appleton-Century-Crofts, Inc., 1942. Szypula, George. Tumbling and Balancing for All. Dubuque, Iowa: Wm. C. Brown, Co., 1957. Periodicals Brodeur, John and Johns on, Steve. "Study of Back Somersault,” The Aodern Gymnast, 4:19, December, 1962. Cureton, Thomas K. HElementary Principles and Techniques of Cinematographic Analysis as Aids in Athletic R search, Research Quarterly, 10:3-24, May, 1939. H . HAiechanics of Broad Jump,H Scholastic Coach, 4:13-21, way, 193A. . HMechanics of High Jump," Scholastic Coach, 4:9-12, April, 1935. Mitakis, Steve. HSpotting the Back Double,” The Modern Gymnast, 1:24-25, December, 1956. Unpul lislled Material Austin, Jeffrey H. HCinematog graphic Analysis of Double Backward Somersault. Unpublished Master' s thesis, Department of Physical Education, University of Illinois, 1959. Interviews Holmes, Hal. Personal interview, Hyman, Lewis. Johnson, Steve. Personal interview, Personal interview, March 1, F'fianCh l , 1962. 1962. March 1, 1962. LA j APPEN C! l X m LU APPENDZX A DEFINITION OF TERMS Double—back somersault. A gymnastic trick done by one person, taking-off into the air with both feet together spin backward twice before landing on the floor or on the mat with both feet; the spin is in the same direc- tion as the post-anterior line of the performer. Preceding velocity. Velocity of the center of gravity of the performer by a back—handspring, until the center of gravity temporarily stops moving at the latter stage of take-off. Take-off. The beginning stage of double—back somersault from when the feet touch the mat until the feet leave it. Body angle of take-off. The angle made by the mat and the line between the center of the shoulder and the center of the soles of the feet, at the last moment of take-off. The height of the center of gravity at the take—off. The vertical height of the center of gravity at the last moment before take-off. The flight. The main part of the performance, when the center of gravity moves as a projectile on a parabola. The latter part of take—off is also considered to be the beginning part of the flight, when the center of gravity movement is in the parabola, even though the 10. 11. feet are still on the mat. Since the height of the center of gravity of landing is lower than that of take—off, due to the hip flexion, the flight finishes at the time when the center of gravity comes down to the same height as that of the beginning of the flight. The flight until the feet come down on the mat-should be called "whole flight.H The distance and the time of flight. The distance and the time length of which the center of gravity moves in the’flight." Lift of the center of gravity. The margin between the height of the center of gravity of the earliest stage of flight and the highest height of the center of gravity. The former is lower than the normal height of the center of gravity at the standing position, due to his body leaning forward. Flight angle of the center of gravity. The angle between the mat and the direction of the center of gravity flight into the air at the beginning stage of the flight. Initial velocity. The velocity of the center of gravity at the beginning stage of the flight. Theoretical initial velocity. The velocity of the center of gravity movement at the beginning stage of the flight given theoretically by the following formula: . p V0 = theoretical initial velocity V0 7v/ we a 2 distance of flight 2sine 0088 g = gravity 9 2 flight angle of the center of gravity 12. 13. 14. 16. A0 This calculation stands on an assumption that the center of gravity flies as a parabolic projectile. Angle of preceding movement of the center of gravity. The angle between the mat and the direction of the movement of the center of gravity at the preceding back—handspring. If the center of gravity is declining at this moment, the angle is explained as ”plus,” if climbing, Hminus.H Initial force. The force involved at the beginning stage of flight, which is calculated as follows: Vo P 2 n1-————— F 2 force t V0: change of velocity from zero m — mass t = time Force at the take—off. The force involved at the take- off, which is found as the other element of the initial force which is the component of the forces of take-off and preceding movement. It is given geometrflzally as seen in Figure 9. Acceleration at the take-off. The acceleration which the performer receives at the take-off as changes of velocity and direction of movement of the center of gravity. It is given by the same principle as the force at take—off. Hipfiangle. Degree of hip flexion at the performance. If the hips are kept straight, the hip angel equals zero. in Figure 10, Q is the hip angle. 18. 20. Al Knee angle. Degree of knee flexion at the performance. :f the knees are kept straight, the knee angle equals zero. In Figure 11, Q is the knee angle. Angle of body rotation. Degree of body rotation is counted from the vertical line and the upper body which is defined as the line drawn between the middle parts of the shoulder and the hip. In Figure 12, Q is the angle of body rotation. Angular velocity_of body rotation. Speed of body rotation, given as follows: 13v : :”l _ [BO 13v 2 angular velocity of t body rotation [BO 2 initial body angle [31 = body angle at the next frame t = time interval between [BO and 431 Degree of tuck. The length of radius of body rotation which indicates the degree of tuck of the whole body. It is expressed as a percentage of a half of the normal height of the performer, and is measured as the distance from the center of gravity of the performer when in tuck position to his outermost extremity. 12 force at the preceding movement initial force (result) force of take—off Figure 9. Forces at the Take-off Figure 10. Hip Angle ; Figure 11. Knee Angle Figure 12. Angle of Body Rotation eewwww mace; we ego one wH oe was. ow mo pgwwaw we use om: ewe me :e om». 0H ofie ea we mos. mg mew wee om ow wee. ea eme me eofi ewe. ea emoa com Hm egg men. ea we: em egg wee. :w AR©.mmv pflsemLeEom pmwwe we use oowa sem owe :wH com. ma wem we owe men. we zpeooflo> wmwzwsm we Ezefixae eewfl omw Hod egg ewe. Ha hee>egm go Lopceo we eewwme emezwwe mew oofi HHH mew. OH wefia owe em on mmm. m - eefl ee em www. e nee mow me .oH omw. e we ee Hofl mow. e ooe em ww em New. e m we we mww. : omm\owm: : - we :: mmo. m ego-exmp wo wee ma- ma ow weo. w eewwflw we ewaew . we- -- -- .omm 000. 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