Hill’s Equation in Arm Push of Shot Put and in Braking of Arm Rotation

This chapter consists of the earlier study of shot put where A.V. Hill’s force-velocity relationship was transformed into a constant maximum power model consisting of three different components of power. In addition, the braking phase of the arm rotation movement was examined where Hill’s equation was applied for accelerated motions. Hill’s force-velocity relationship was tested by fitting it into two arm push measurements of shot put experiments and one braking phase of whole arm rotation. Theoretically derived equation for accelerated motions was in agreement with the measured data of shot put experiments and the braking phase of the whole arm rotation experiment. Maximum power in these experiments was also tested by three different equations and two of them seemed to function well. The progress of movement in the studied experiments was concluded to be as follows: 1) the state of low speed and maximal acceleration which applies to the hypothesis of constant force, 2) the state of high speed and maximal power which applies to the hypothesis of constant power.


INTRODUCTION
British Nobel laureate A.V. Hill invented the famous model of muscle mechanics which describes the force-velocity relationship of skeletal muscle contraction (Fig. 1). The equation of this model is (F + a)(v + b) = b(F 0 + a), where F is maximum force in muscle contraction, a is constant force and b is constant velocity, F 0 is isometric force of muscle or the constant maximum force generated by muscle with zero velocity and v is velocity [1,2]. 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 force-velocity relationship of skeletal muscle were done by e.g. Fenn and Marsh [3] and good reviews are also available [4,5].
The arm rotation experiments of Rahikainen et al. [6] 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: This theoretically derived equation with constant maximum power (phase 3 above) was in good agreement with the experimentally measured results.

Fig. 1. Hill's force-velocity curve with the corresponding power P
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 (F 0 + 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 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/P 0 = 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 present chapter consists of the earlier study of the arm push of the Olympic shot put winner where the further develop of Hill's equation was applied. As an example of the accelerated motion, the braking phase of the whole arm rotation was also examined. Maximum power in these experiments was also tested by three different equations.
Advances in Applied Science and Technology Vol. 6

Hill's Equation in Accelerated Motion
In order to find out the function of Hill's force-velocity 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: 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.
v H is constant velocity in Hill's equation, and during the muscle contraction the muscular force F corresponds to the velocity v H in Hill's equation.
The results of Hill's experiments could be transformed into hyperbola equation describing forcevelocity 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) (v H + b) = constant, implies that the area of the rectangle (F + a) (v H + 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 = Fv H 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 = v H 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.
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 forcevelocity 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 v H 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 Fv H , the power of friction av H 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: At the phase of acceleration the power consumption into acceleration is: 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 v H , the power of the work done against counter force Fv and the power of friction av are less than that at the velocity v H . The difference of these powers is equal to the power into acceleration. We obtain the equation of motion.
This is the equation of motion as mass m is accelerated by muscle contraction. Hill's velocity v H 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 v H and muscular force F (Eq. 1) are:

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 F 0 = 1 N and v H0 = 1 m/s (for convenience), and a / F 0 = 0.27 (MacIntosh and Holash [8], p. 194). Moving mass m is equal to force divided by gravitational coefficient m = F / g. Finally corresponding values of force F and Hill's velocity v H 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).

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 (Rahikainen and Luhtanen [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.  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]. (21)

Effect of Gravitational Force on the Movement in Shot Put
The force that is induced by gravity was omitted from the motion model. The power generated by this gravity force is P gr = 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) P 0 = P acc + P fr + P gr (22) P = P 0 -P gr = P acc + P fr P is power in Eq. (21), P 0 is muscle power, acc is acceleration, fr is friction, gr is gravity, force of gravity is F. At the point B in Fig

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 abovementioned 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.

Analysis of Arm Push in Shot Put using Hill's Equation
In Hill's equation the velocity of muscle contraction v H 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 v H in Hill's equation corresponds to the muscle force F. Theoretically the time of motion for the constant maximum velocity v H is indefinite, but if the counter force -F is strong enough, the movement decelerates and the constant maximum phase really exists in muscle mechanics. All the velocities v from zero to maximum velocity correspond to muscle forces greater than F (muscle force velocity = constant power) and therefore in this study the maximum velocity corresponding F is marked as Hill's velocity vH. In arm push of shot put the corresponding progress of velocity does not reach the velocity value v H because the range of movement is too short for that. In that movement all the velocity values are calculated from Eq. (11).

Determining the Constants in Eq. (11)
The moving mass m is presented in Table 1. Gravitational force of the moving mass is mg = 10.8 · 9.82 = 106 N and the force resisting the acceleration of the shot is -F = -mg · sin(41º) = -69.6 N, and the muscle force of Hill's equation is F = 69.6 N.
The maximum muscle force F 0 can be determined by two different ways. Shot putter's maximum bench press with two hands has been measured 260 kg and thus the estimated result for one hand is 130 kg. The maximum muscle force is then F 01 = 130 kg · 9.82 m/s2 = 1280 N. The other way to determine the maximum muscle force F 02 is to use Eq.

Analysis of 20.90 m Put
Another shot put performance (20.90 m, Rahikainen and Luhtanen [9]) was analyzed in order to be able to compare the two puts and to learn more about the characteristics of the arm push. Another analysis was also needed to confirm the validity of the equation of motion. Measured speed curve of this put (dots) is presented in Fig. 7. The speed which is calculated with Eq. (21) coincides with the speed of further development of Hill's equation Eq. (11) in section B -C (Fig. 7). The propulsive power of the arm push is approximately P = 4520 W. The arm push in Fig. 6 has a greater pushing force 1260 N than in Fig. 7 1200 N, but at the end of the push the decrease in acceleration in Fig. 6 is so great that the total velocities generated during both arm pushes are almost equally high. The reason for the large decrease in acceleration is probably due to muscle's mechanics, maybe a pressure decrease in muscles as the turning of the body is getting larger. In the optimal arm push, this large acceleration decrease must be eliminated.

Model of Muscle Contraction in the Braking Phase of Arm Rotation
Two models of muscle contraction have been used for the braking phase of arm rotation. The first model is constructed according to Newton's II law, which was first used in linear motion by Rahikainen and Luhtanen [9] and then applied to rotational motion by Rahikainen et al. [7] and Rahikainen and Virmavirta [6]. The second model is Hill's equation, which has been solved for accelerated movement by Rahikainen et al. [12].
The theory of arm rotation in the braking phase is as follows: In the beginning of the braking phase, angular velocity has reached the maximum value. At this phase elasticity of the muscle-tendon unit has influence on the motion, which can be seen as a wave motion between points B -C in Fig. 8, but thereafter the muscle-tendon unit have full state of tension, and it has no further dynamic effect in the braking phase of movement between points C -E. The phase of braking movement between points C -D is the phase of maximum constant power.

50). For convenience, the time for Eq. (50) is presented from right to left
Because the muscle system is able to transfer only a certain quantity of chemical energy during the time of contraction, it is obvious that arm rotation must have maximum power that cannot be exceeded. It can also be assumed that the maximum power acts within a certain range of velocity and it is a constant maximum power. As the braking of the movement starts, the maximum power takes action and the rotational moment is less than the maximum moment. This way power remains constant as the angular velocity decreases and rotational moment increases, until rotational moment is the maximum rotational moment at point D. Cϕ &

Coefficient of friction C
Inner friction of muscle is fluid friction which is directly proportional to velocity. It can be seen in Fig. 8 that initially movement accelerates at Maximum constant moment M 0 between points A -B, which implies that the moment generated by muscle force is constant. After that the movement has reached maximum velocity between points B -C. As the velocity decreases, the movement decelerates at maximum constant power within the range of movement between points C -D in Fig. 8 and the power P is constant.

Constant Velocity Movement
Experiments of Rahikainen et al. [7] and Rahikainen and Virmavirta [6] verified that the conclusions of the theoretically derived equation with constant maximum power Eq. (21) were in agreement with experimentally measured results. As it is compared to Eq. (34), it can be seen that it is the same save the negative power. As Hill's equation is a constant power model, it is same as the model of Eq. (34) and Eq. (21) in that respect. Hill's force -velocity relationship was derived from the experiments in which the velocity of muscle contraction was measured against a certain constant force. The experiments of Hill's equation naturally started at zero velocity and continued in the same manner as the experiment of the present study through all the phases. Hill's equation was derived from muscle contraction experiments, in which moving masses were very small. The scale difference has a great effect on muscle forces and inertial forces of moving masses. For instance 10 times larger muscle has 100 times stronger muscle forces (related to cross-section area of the muscle) and 1000 times stronger inertial forces of moving masses (related to the volume of moving mass). Therefore Hill's equation is valid within the muscular mechanics of mice and mosquito, but within muscular mechanics of human being it must be solved into accelerated motion [13,14] and [15]. The measurement of the this study was made without external load and it did not reach the maximum theoretical velocity of Eq. (21).  Fig. 2). This power creates a counter force against external load. As the velocity is zero, the force is highest. Thereafter, as velocity increases, initially this power decreases rapidly, then slowly at higher velocities. This power has no velocity variable, and therefore it is not related to movement. The power H ro ϕ& a represents the power of friction due to the motion of muscle's internal structures (corresponding the area C in Fig. 2). Because power is force multiplied by velocity, the force of friction is a ro . This is not the force directly proportional to velocity, generally known as fluid friction (which is the friction used in the upper model of muscle contraction), but constant force of friction which is known as sliding friction.

Model of Hill's Equation in the Braking Phase of Arm Rotation
The Hill's equation Eq. (36) for decelerating movement in Fig. 8 was solved between points C -D. The solution has been calculated from the theoretical maximum velocity phase of movement, in which Hill's equation Eq. (36) is valid.

Decelerated movement (braking phase)
At the phase of deceleration the power consumption into deceleration is and solution of rotational motion corresponding to Eq.(11) is

Analysis of the Braking Phase of Whole Arm Rotation Movement
At the section of braking movement C -D deceleration decreases, and the movement decelerates at constant power movement. This is followed by stopping of the movement.
The theory of muscle contraction in arm rotation movement in Fig. 8 is as follows:

Maximum rotational moment between points A -B.
Maximum power acceleration the motion after point B (very short).

The phase of Maximum constant velocity
Arm rotation experiment presented in Fig. 8 was recorded by the camera system of Rahikainen [16]. The measuring technique has been described in papers [6,7,17,18].

Braking Phase of the Arm Rotation Movement
Calculation of the angular velocity ϕ & at point D (Fig. 8) where the measured velocity points and the theoretical velocity curve (Eq. 50) coincide is checked first below.
After the velocity v = 3.9 m/s Hill's equation acts in the motion. In Fig. 6 it can be seen that the measured velocity curve begins to bend approximately at that point and therefore, it can be inferred that Hill's equation functions right in accelerated motion. The check for the shot put of 20.90 m (Fig. 7) Fig. 7 it can be seen that this point is not a match to the point at which the measured velocity curve begins to bend.

DISCUSSION AND CONCLUSIONS
Rahikainen et al. [7] showed that in the first part of the arm rotation movement acceleration was constant and during the second part of movement equation of constant power (Eq. 34) fitted the measured velocity curve. The following study [6] continued the arm rotation experiments and a new approach to Hill's equation was presented.

Shot Put
The shot put experiments of the present study have the similar progress of movement with the previous ones, and the above mentioned constant acceleration and constant power phases have been proved to be true as well. The correct functioning of Eq. (21) is also proven as the two forces F 01 = 1260 N and F 02 = 1200 N are close to each other.
In Figs. 6 and 7 the measured velocity curve has a phase of constant acceleration between velocity values ~1.5 -3.5 m/s and it can be inferred that within this range of velocity constant maximum muscle force is produced presuming that the force of friction is constant value glide friction. The constant friction a in the phase of constant acceleration and in the phase of constant power in Eq. (11) is assumed to be the same and mechanics in the equation seems to function properly with this assumption. In MacIntosh and Holash [8] the values of friction coefficient a/P 0 for human elbow flexors are given, 0.45, 0.4 and 0.39, in which P 0 is maximum moment for values 0.4 and 0.45 and maximum force for value 0.39 and a corresponds to the force of friction. Values of friction coefficient a/P 0 0.2-0.3 are also used. Proper functioning of Hill's equation implies the action of counter force F and if the counter force is weak, velocity increases, and the friction coefficient a/P 0 may be bigger 0.6 -0.7.
The original purpose of this study was to find out how well the models of Eq. (21) and the further development of Hill's equation, Eq. (11), match the measured velocity in the arm pushes of two shot put performances. The fittings of these two constant power equations succeeded and they did function very much the same manner. Both of power equations fitted the measured velocity curve between velocity values 4 -6 m/s and thus the constant power model proved to be true within this range of velocity. However, these two models of constant power, Eq. (11) and Eq. (21), are different and they differ at higher velocities significantly. The highest velocity value of Hill's equation in accelerated movement (Eq. 11) in Fig. 6 is v H = 13.2 m/s and the highest velocity of Eq. (21) is C P V / = = 8.56 m/s. It was verified that in the phase of constant acceleration three different constant forces are acting: the force of acceleration, the force of friction and the force of gravity and added together they represent the maximum muscle force. The range of correspondence between the measured and theoretical velocities of the shot put experiments was long enough to confirm the existence of constant power models. Kinetic friction was assumed to be directly proportional to velocity at the beginning of the movement. It is possible that kinetic friction at small velocities is constant and at high velocities is directly proportional to velocity. This leads to a constant torque accelerating the movement at the beginning of movement. It is also possible that the constant acceleration phase of movement is rather a matter of human ability to learn effective modes of motion than a direct cause of natural laws. It may be that human nervous system controls the rotational moment accelerating the arm movement.
The present application of Hill's equation in accelerated motion could be worth to apply in the accelerated rotational movement as well. The analysis of additional subjects with different performance level would also help to understand better the function of Hill's equation in accelerated movement.
After checking the shot put of 20.90 m experiment, it can be concluded that further analysis is needed. The friction term of Hill's equation a = 288 N must be bigger.

Arm Rotation Movement
The braking phase of whole arm rotation of this study was initially analysed by Rahikainen and Virmavirta [6]. However, in that analysis an error was made, so that the whole arm rotation was analysed as downwards rotation, but actually it was upwards rotation. Therefore the time had wrong direction. The error was found, and there was no doubt about the direction of the movement, because maximum velocity of downward movement is much higher than upward movement. Because of this error the very important finding was made that the muscle's braking movement, which is called eccentric movement, is much the same as accelerating movement, which is called concentric movement. The actual Hill's equation at maximum constant velocity values is valid only within concentric movement, but not within eccentric movements. However, the arm rotation movement in Fig. 8 shows that Hill's equation is valid both in accelerated and decelerated movements. Another important finding was that as the equation that fitted the decelerated arm rotation movement was known, the corresponding equation of motion, Eq. Comparing the two equations of motion in accelerating and decelerating movement, it can be seen that the term of friction C ϕ & is always against the muscle force P / ϕ & , not against the direction of motion as it usually is. Therefore, it must be related to muscle's force and power generation process decreasing muscle's force generation.