Hill’s Equation in the Arm Push of Shot Put

This is an electronic reprint of the original article. This reprint may differ from the original in pagination and typographic detail. Author(s): Title: Year: Version: Please cite the original version: All material supplied via JYX is protected by copyright and other intellectual property rights, and duplication or sale of all or part of any of the repository collections is not permitted, except that material may be duplicated by you for your research use or educational purposes in electronic or print form. You must obtain permission for any other use. Electronic or print copies may not be offered, whether for sale or otherwise to anyone who is not an authorised user. Hill’s Equation in the Arm Push of Shot Put Rahikainen, Ahti; Virmavirta, Mikko; Ranta, Matti A. Rahikainen, A., Virmavirta, M., & Ranta, M. A. (2016). Hill’s Equation in the Arm Push of Shot Put. British Journal of Applied Science and Technology, 18(5), Article no.BJAST.30766. https://doi.org/10.9734/BJAST/2016/30766 2016 _____________________________________________________________________________________________________ *Corresponding author: E-mail: [email protected]; British Journal of Applied Science & Technology 18(5): 1-12, 2016; Article no.BJAST.30766 ISSN: 2231-0843, NLM ID: 101664541 SCIENCEDOMAIN international www.sciencedomain.org Hill’s Equation in the Arm Push of Shot Put Ahti Rahikainen 1* , Mikko Virmavirta 1 and Matti A. Ranta 2 1 Department of Biology of Physical Activity, Neuromuscular Research Center, University of Jyväskylä, Finland. 2 Aalto University, Finland. Authors’ contributions This work was carried out in collaboration between the authors. Author AR designed the study, created the theory, performed the measurements and analysis and wrote the first draft of the manuscript. Author MV read, edited, and approved the final manuscript. Author MAR did the solution of Eq. (7) – Eq. (9) and approved the final manuscript. Article Information DOI: 10.9734/BJAST/2016/30766 Editor(s): (1) Ya-mei Gao, College of Life Science and Technology, Heilongjiang Bayi Agariculture University, Daqing, Heilongjiang, China. Reviewers: (1) Adil Loya, PAF Karachi Institute of Economics and Technology, Pakistan. (2) Danúbia da Cunha Sá-Caputo, Universidade do Estado do Rio de Janeiro, Rio de Janeiro, RJ, Brazil. (3) Yüksel Savucu, Firat University, Elaziğ, Turkey. Complete Peer review History: http://www.sciencedomain.org/review-history/17615 Received 29 th November 2016 Accepted 19 th January 2017 Published 26 th January 2017 ABSTRACT Aims: The purpose of this paper was to continue the previous study of arm rotation movement where A.V. Hill’s force-velocity relationship was transformed into a constant maximum power model consisting of three different components of power. Methodology: In the present study a new model of Hill’s equation was applied for accelerated motions. This theoretically derived model of further development of Hill’s force-velocity relationship was tested by fitting it into two arm push measurements of shot put experiments. The results of the further development of Hill’s equation for accelerated motions were compared with the mechanics of the constant power model of the previous study. Results: The analyses of the present study verified that this theoretically derived equation for accelerated motions was in agreement with the measured data of shot put experiments. The fittings succeeded and they coincided with the velocity curves of the measured shot put experiments and the constant power model of the previous study. Conclusion: In the present study the progress of movement was concluded to be as follows: 1) the Original Research Article Rahikainen et al.; BJAST, 18(5): 1-12, 2016; Article no.BJAST.30766 2 state of low speed, maximal acceleration which applies to the hypothesis of constant force, 2) the state of high speed, maximal power which applies to the hypothesis of constant power, where the constant power model of previous study and the present development of Hill’s equation for accelerated motion were acting. This is a new approach to Hill’s equation. Keywords: Muscle mechanics; muscle power; force-velocity relationship; Hill’s equation; arm movement; arm push in shot put. 1. INTRODUCTION British Nobel laureate A.V. Hill invented the famous model of muscle mechanics which describes the force-velocity relationship of skeletal muscle contraction. The equation of this model is (F + a)(v + b) = b(F0 + a), where F is maximum force in muscle contraction, a is constant force and b is constant velocity, F0 is isometric force of muscle or the constant maximum force generated by muscle with zero velocity and v is velocity ([1,2], Fig. 1). This equation was based on the laboratory measurements in which the force (F) of activated muscle was measured as the muscle was contracting at a constant speed in an isolated condition. In the equation the vectors of forces and velocities have the same direction and therefore Hill’s equation can be presented in a scalar form. Other early experiments of forcevelocity relationship of skeletal muscle were done by e.g. Fenn and Marsh [3] and good reviews are also available (e.g. [4,5]). Fig. 1. Hill’s force-velocity curve with the corresponding power P The arm rotation experiments of Rahikainen et al. [9] followed the theory, where movement was described to have four (4) different phases: 1) start of motion 2) movement proceeds at constant maximum rotational moment during the first part of the movement 3) movement proceeds at constant maximum muscular power during the second part of the movement 4) stopping of motion. For validation of these assumptions the equation was solved for angular velocity-time function:       C P Td d I This theoretically derived equation with constant maximum power (phase 3 above) was in good agreement with the experimentally measured results. The study of Rahikainen and Virmavirta [6] continued the experiments of the previous study [7]) and further developed its theory of mechanics resulting in further solution of Hill’s equation. The results were based on the assumption that in muscle mechanics there is a constant maximum power which the muscle is able to generate within a certain range of velocity. The principle of constant maximum power is also in Hill’s equation and in this respect the two models can be considered the same. In the left side of Hill’s equation the term (F + a)(v + b) is muscles’ total power including Fv, which is the power of moving the external load. The right side of the equation, b (F0 + a), includes only constants and thus the equation can be considered as a constant power model. However, the constant maximum power in the study of Rahikainen and Virmavirta [6] is a characteristic of whole muscle group instead of separate muscle fibers as in the Hill’s equation. The model was based on the muscular system’s ability to transfer chemical energy and, therefore, it is not necessary to know the contribution of the individual muscles involved. The constant power of Hill’s equation presented by Rahikainen and Virmavirta [6] is not the power of Hill’s original curve as it is usually considered in biomechanics, but it is the sum of three different power components. It was inferred that the constant F/P v F0 v0 a b P0 Rahikainen et al.; BJAST, 18(5): 1-12, 2016; Article no.BJAST.30766 3 power model of the study of this paper acts during high speed movements with no external load, where Hill’s equation does not seem to fit the experimental points ([2], p. 32, Fig. 3.2) very well. As an explanation for this mismatch Hill mentioned that “sharp rise at the end of the curve in the region of very low tension was due to the presence of a limited number of fibers of high intrinsic speed and no such equation could fit the observed points below P/P0 = 0.05”. Because Hill’s equation is also a constant power model, it is acting only in a certain state of motion, which is constant power movement at low speed of motion decelerated by counter force. Therefore it is not a model of motion which is suitable for every state of muscle motion. Although Hill’s force-velocity relationship has been an important part of muscular mechanics models, it has deficiencies which to a great extent restrict its application for the real muscular mechanics of human motion. Hill’s equation is a constant power model and it has no term for the power of acceleration and therefore it cannot be applied within accelerated motions. Also at the point of maximum speed, where force is zero, Hill’s equation does not seem to fit the experimental points well. The third reason for the deficiencies of the models based on Hill’s equation is that they take no account of the effect of the elastic properties of muscle-tendon unit. The purpose of the present study was to further develop Hill’s equation and to create a model, which could be applied to accelerated motions, and to test its function in arm push of the Olympic winner in shot put. 2. METHODS 2.1 Hill’s Equation in Accelerated Motion In order to find out the function of Hill’s forcevelocity relationship in accelerated motion, equation of motion was derived from Hill’s equation. In the present study Hill’s equation is presented in a form      baFbvaF  0H (1) where F is the muscular force in Hill’s equation. The force F must be constant during the whole muscle contraction. If it is not, the equation of motion of muscle contraction will be much more complex. vH is constant velocity in Hill’s equation, and during the muscle contraction the muscular force F corresponds to the velocity vH in Hill’s equation. The results of Hill’s experiments could be transformed into hyperbola equation describing force-velocity dependence of the movement. The left side of Hill’s equation represents the maximum total power consumed into muscle contraction, and the right side of Hill’s equation indicates that this maximum total power is constant. In the following this maximum total power is divided into three power components. Hill’s equation, the motion with maximum constant power is the second state of motion. The first state of motion is with the maximum constant force, and in that state of motion acceleration is constant. Fig. 2 represents a further development of Hill’s force-velocity relationship. Hill’s equation, (F + a) (vH + b) = constant, implies that the area of the rectangle (F + a) (vH + b) is constant. The total power of the muscle is comprised of three different components represented by rectangles A, B and C. The area of rectangle A = FvH represents the power needed from muscle against an external load (see the power curve in Fig. 1). If there is no external load, this power is consumed by acceleration. The area of rectangle B = (F + a) b represents the power of muscle’s internal loss of energy. This power creates a counter force against an external load. As the velocity is zero, this power B is highest and, therefore, it is not related to external movement. When velocity increases, this power decreases rapidly initially, then slowly at higher velocities. The area of rectangle C = vH a represents the power of friction due to the motion of the muscle – load system. Because power is force multiplied by velocity, the force of friction is a. This is not force directly proportional to velocity, generally known as liquid friction (which is the friction used in the present study in paragraph 2.3), but constant force of friction which is known as glide friction. Now we can see that there are three different states of motion: 1) at the beginning of motion characterized by a state of low speed constant maximal acceleration, then 2) as the motion continues a state of high speed, constant maximal power, which applies to motion of Eq. (21) and to Hill’s equation. The maximum power is due to the fact that the transfer of energy within the muscle system must have a maximum rate and, therefore, muscle’s power generation must also have a certain maximum rate. Rahikainen et al.; BJAST, 18(5): 1-12, 2016; Article no.BJAST.30766 4 Fig. 2. Hill’s force-velocity relationship presented with asymptotes (broken lines) and three rectangles of power. In traditional presentation of hyperbola a and b are negative, but here they refer to the positive constant terms of Hill’ equation Application of Hill’s equation into human movement is problematic. Hill’s force-velocity relationship has no power term for acceleration, and therefore it is not valid in accelerated motions. Hill’s force - velocity relationship was measured with a measuring device in which the muscle force is measured at constant velocity. Its applications must also be constant velocity movements. Herein theoretical experiment is performed: A mass m is accelerated by a muscle contraction. The force generated by the muscle is F and its counter force is – F. In the beginning velocity is zero, and the movement is at a state of high acceleration. As the movement continues, velocity v increases and acceleration decreases, and if the movement continues sufficient long distance at some point the movement can be regarded as constant. Then there is force F corresponding to velocity vH as it is in Hill’s force – velocity relationship. The total power of Hill’s equation can be divided into three separate power components (Fig. 2): the power of the work done against counter force FvH, the power of friction avH and the power consumed within generation of muscle force (F + a) b. Because the muscle force F is constant and a and b are also constants, the power (F + a) b is also constant. The total power at the phase of constant velocity is the sum of the three rectangles A, B and C (Fig. 2) which is F v H + av H + (F + a) b = (F + a)(v H + b) = (F 0 + a) b (2) At the phase of acceleration the power consumption into acceleration is v dt dv mP  acc (3) where v is general velocity in movement containing also accelerated phase of motion. Because at the phase of acceleration the velocity v is less than the constant velocity vH, the power of the work done against counter force Fv and the power of friction av are less than that at the velocity vH. The difference of these powers is equal to the power into acceleration. We obtain the equation of motion         vvavvFbaFbaF avavFvFvv dt dv m   HH HH (4) Solution   vv m aF v dt dv    H (5)           H H 1 vv vv m aF dt dv (6) dt m aF dv vv vv             H H 1 11 (7) dt m aF dv vv             H 1 1 1 (8)                t dt m aF dv v vvv v v dv 000 HH H 1 1 ln 1 (9)       Ct m aF vvvv    1ln1ln HH (10) F F0 vH0 a b vH F B A C Rahikainen et al.; BJAST, 18(5): 1-12, 2016; Article no.BJAST.30766 5 constant C = 0 ln (1) = 0     HH 1ln vvvv aF m t   (11) This is the equation of motion as mass m is accelerated by muscle contraction. Hill’s velocity vH corresponds to muscle force F in Hill’s equation. Hill’s velocity is the velocity after the phase of acceleration as the motion can be regarded as constant velocity movement. Calculation of the values of Hill’s velocity vH and muscular force F (Eq. 1) are      baFbvaF  0H   bv avbF vF    H H0 H (12) Substituting F = 0 into Hill’s equation bFva 0H0  (13) 0 H0 F av b (14) 2.2 Numerical Calculations of Hill’s Equation in Accelerated Motion Theoretical velocity functions of the mass lifted against gravity force by muscle contraction are determined by selecting constant values F0 = 1 N and vH0 = 1 m/s (for convenience), and a / F0 = 0.27 ([8], p. 194). Moving mass m is equal to force divided by gravitational coefficient m = F / g. First Hill’s velocities are chosen for the curves of contraction equations which will be calculated (vH = 0.2 m/s, 0.4 m/s, 0,6 m/s, 0.8 m/s, 0.9 m/s, Fig. 3). Then the values of constant force a = 0.27 F0 and constant velocity b using Eq. (14) are chosen (a = 0.27, b = 0.27). After that the corresponding values of force F using Eq. (12) are calculated (F = 0.460 N, 0.242 N, 0.124 N, 0.051 N, 0.023 N, Fig. 3). Finally corresponding Fig. 3. Velocity curves of moving mass accelerated in muscle contraction resisted by counter force which is equal to the force of gravitation. Hill’s velocity vH is the limit velocity that the velocity of the moving mass approaches. Hill’s velocity vH corresponds to the velocity in the Hill’s force velocity relationship and the force F corresponds to the force of Hill’s force velocity relationship 0.0 0.2 0.4 0.6 0.8 1.0 0.000 0.010 0.020 0.030 0.040 0.050 0.060 V(m/s) t(s) V H 0.9 m/s F0.023 N m0.0023 kg V H 0.8 m/s F0.050 N m0.0051 kg V H 0.6 m/s F0.124 N m0.0126 kg V H 0.4 m/s F0.242 N m0.0246 kg V H 0.2 m/s F0.460 N m0.0468 kg Rahikainen et al.; BJAST, 18(5): 1-12, 2016; Article no.BJAST.30766 6 values of force F and Hill’s velocity vH are substituted into equation of muscle contraction (Eq. 11), and the velocity curves of mass accelerated in muscle contraction resisted by constant counter force F are obtained, (Fig. 3). 2.3 Constant Power - Liquid Friction Model of Muscle Contraction The model used in the present study is constructed according to Newton’s II law, which was first used in linear motion of arm push in shot put [9] and then applied to rotational motion by Rahikainen et al. [7] and Rahikainen and Virmavirta [6]. The theory of arm movement is as follows: At the beginning of the movement, velocity is naturally zero and it takes some time to generate force. At that phase of motion, passive elements of muscle-tendon unit have influence on the motion, but after reaching the full state of tension, they have no further dynamic effect. After that it can be assumed that a maximum muscle force takes action and at that phase of motion constant value glide friction acts. Because the muscle system is able to transfer only a certain quantity of chemical energy during the time of contraction, there must be a constant maximum power, which the muscle is able to generate within a certain range of velocity. As the velocity increases the motion reaches the point where the maximum power takes action and acting force is less than the maximum force. This way power remains constant as the velocity increases and the force decreases. At high velocity phase of motion, liquid friction, directly proportional to velocity, acts. The constant value glide friction decreases as forces at the joint decrease and it becomes indifferent. The model of arm movement during constant power phase in shot put study was constructed as follows: accelerating force is mass multiplied by acceleration which equals muscle force minus the force generated by inner friction of muscle. The effect of gravitational force is added afterwards. VC V P Td Vd m (15) where The weight of the shot 7.27 kg and the weight of the arm approximately 3.5 kg, or total weight 10.8 kg (Table 1). Mass of shot and arm m Velocity of shot V Power generated by arm P Time of arm push T Pushing force P / V Internal friction in arm C V Internal friction of muscle is liquid friction inside muscle, which is directly proportional to velocity. The same liquid friction was also used in the study of Rahikainen et al. [7] which was initially adopted from Alonso and Finn [10]. Table 1. Body segment masses [11] and estimated moving mass of the shot putter (140 kg) in the present study % of total mass mass (kg) moving mass (kg) 1 trunk 34.70 48.6 2 upper arm L 2.65 3.7 3 forearm L 1.82 2.5 4 hand L 0.50 0.7 5 upper arm R 2.65 3.7 ¼ = 0.925 6 forearm R 1.82 2.5 ¾ = 1.875 7 hand R 0.50 0.7 0.7 8 shot 7.3 7.3 9 head 6.72 9.4 1 2 3 4 5 6 7 8 9 Rahikainen et al.; BJAST, 18(5): 1-12, 2016; Article no.BJAST.30766 7 Solution of velocity in Eq. (15) TddV CVP V m  2 (16)     TV TddV VCP VC C m 0 2 0 1 2 2 (17)     T m C PCVP 2 lnln 2  (18) T m C P VCP 2 ln 2           (19) T m C eV P C 2 1 2   (20)           T m C e C P V 2 1 (21) 2.4 Effect of Gravitational Force on the Movement in Shot Put The force which is induced by gravity was omitted from the motion model. The power generated by this gravity force is Pgr = mg·sin(41º)·V = 69.5 N·V, where mg is gravitational force of moving mass (Fig. 4), V is velocity of arm movement. In Fig. 6, velocity (V in Eq. 21) coincides the measured velocity curve between 4 and 6 m/s and the best fit for power (4750 W, Fig. 6) is in the middle of these velocities, at 5 m/s. As the force of gravity is relatively small, the power induced by gravity was calculated in this study as a constant factor. It is included in the power P as follows; 1) P0 = Pacc + Pfr + Pgr (22) P = P0 ‒ Pgr = Pacc + Pfr P is power in Eq. (21), P0 is muscle power, acc is acceleration, fr is friction, gr is gravity, force of gravity is F. At the point B in Fig. 5 velocity is 4 m/s and the real power can be calculated as follows; 2) Preal = P + (5 – 4) m/s ·F = 4750 W + 69.6 W = 4819.6 W The real velocity can be solved from the power ratio of Eq. (21); 3) 0073.147506.4819 real VV (23) m/s029.40073.14 real V which is within the accuracy of this study 4 m/s In constant acceleration phase of movement F0 = Facc + Ffr + F Fig. 4. Sideview of shot’s path during the arm push (distance of the put 19.47 m) Beginning of arm push x (m) y (m) Release 41 41 2.1 2.0 1.9 1.8 1.7 1.6 1.5 1.4 2.12.01.91.81.71.61.51.4 2.2 2.3 mg Rahikainen et al.; BJAST, 18(5): 1-12, 2016; Article no.BJAST.30766 8 3. RESULTS 3.1 Analysis of 19.47 m Put Using Eq. (21) If the time of arm push is known, it is possible to determine the speed of shot during the arm push using speed curves from Rahikainen and Luhtanen [9], (Figs. 5 and 6). Thereby, the part corresponding to the time of arm push is separated from the end of the speed curve (e.g. Fig. 5). In the path of the shot (Fig. 6) it can be seen that in section A - B the arm push continues to generate speed with the maximal pushing force and the inclination of the speed curve is almost constant. This is because the maximal generation of speed is limited by the shot putter’s maximal arm-pushing force. As the arm push continues, in section B - C - D, the pushing force accelerating the shot decreases and the inclination of the speed curve decreases as well. There are three different factors that cause the decrease in acceleration. First: as the speed of the shot increases, the rate of increase is not limited by a maximal pushing force, but by a maximal propulsive power, in which case force is power divided by velocity. Second: The internal friction of the pushing arm, which can be considered to be directly proportional to the velocity, decreases the velocity of the shot. Third: as the shot putter in the rotational motion turns sideways in respect to the direction of the arm push, the pushing force of the arm decreases and disappears and the arm just follows the shot without accelerating it. In Fig. 6 the broken line describes the effect of the first and second factor mentioned above. In section B – C, the measured speed curve and the broken line coincides. In this phase of the arm push, the two above-mentioned factors are the principal factors influencing the speed of the shot. In section C – D, the measured speed curve travels under the broken line. In this phase of the arm push, the shot putter turns so much sideways in respect to the direction of the arm push that the acceleration of the shot decreases further. If the shot putter would not turn (or rotate) during the arm push, the measured speed curve would combine with the broken line in section C - E. By fitting Eq. (21) into the measured speed curve in Fig. 6 values of internal friction and power are obtained C = 64.8 kg/s and P = 4750 W. 3.2 Analysis of Arm Push in Shot Put Using Hill’s Equation In Hill’s equation the velocity of muscle contraction vH is measured, as the force -F is resisting the motion. The muscle force is then F. In the beginning of movement the velocity of muscle contraction is zero, then the muscle force accelerates the motion, and the velocity increases. At some point it reaches maximum value, and at this constant speed phase of movement, velocity vH in Hill’s equation corresponds to the muscle force F. Theoretically the time of motion for the constant maximum velocity vH is indefinite, but if the counter force -F is strong enough, the movement decelerates and the constant maximum phase really exists in Fig. 5. The measured speed of the shot and the length of arm push (shaded area AD) in 19.47 m put Length of arm push = area below speed of shot 42 cm Speed of shot at the beginning of arm push v 0 = 6.5 m/s 12 10 14 8 6 4 2 0 m/s Speed of shot Time of arm push T k = 0.112 s A D Speed of right shoulder t4 t5 frame rate 125 fr/s