Swimming propulsion and muscle force moments

Abstract

Based on 3D video analysis of swimming movements hypotheses on the mechanisms of propulsion are deduced. Body point coordinates and their first and second derivatives are computed. The limb environment where water particle displacement occurs is estimated. We apply the Navier–Stokes equation to compute the total force for those particles. The shoulder torque is calculated by summation over environments of hand, forearm and upper arm of infinitesimal torques of displaced water particles. Similarly, hip torques are computed by summation over environments of foot, shank and thigh. Our aim is to determine individual shoulder and hip torques over one movement cycle. These muscle force moments are related to the velocity of the mass centre as a measure for propulsion. Simultaneously they serve as controlling data for dry land strength training. Recommendations for best propulsion techniques are derived.

Keywords:

• 3D analysis
• Navier–Stokes equation
• swimming
• muscle force moment
• propulsion

Introduction

For more than 40 years researchers have invested much work into a better understanding of propulsion in swimming. In the past, explanations for movement in the water run from the mechanics of the paddle steamer [1], or that of the ship's propeller [2] up to undulations, such as vibrations of the fins of fishes. A very comprehensive overview on the discussion whether drag or lift primarily causes propulsion is given by Sanders [3]. Counsilman [4] was the first who proposed the theory of hydrodynamic lift as the main contributor to propulsion in swimming. However, in the last two decades of the last century Wood and Holt [5], Cappaert [6] and Schleihauf [1] presented facts that drag is the dominant force. These studies indicated that drag forces are more important than lift forces in all strokes other than breaststroke. Rushall et al. [7] gave convincing arguments in favour of drag as the dominant propulsive force in freestyle swimming. Sanders [8,9] obtained lift and drag coefficients of the hands. The greatest forces are obtained when the hand plane is almost orthogonal to the flow. At this orientation the force is due to drag. Lift makes its greatest contribution to propulsion at angles close to 45°. However, even at these angles, the contribution because of drag is as great as the contribution because of lift. It was found that drag made a larger contribution than lift throughout the propulsive part of the pull. Currently the vortex theory of Matsuuchi et al. [10] is most popular. According to Colwin [11] vortices can generate something like firm resistance from which one may repel. He distinguished between starting, bound and shed vortices. The majority of the vortices, however, are immediately left behind the swimmer and it is impossible to regain their energy.

In our studies we assume the flow to be incompressible without shearing forces. Our approach uses the Navier–Stokes equation F = ρ ∂ v/∂t + ρ (v grad) v + grad p, in this form also known as Euler equation. It has three different hydrodynamic forces on the right: inertial force, convective force and pressure gradient. In international scientific literature there are no publications paying attention to this distinction of three forces; no one applies the Navier–Stokes equation to calculate propulsion forces. Sanders [8], for example, applied the following formula to compute the force on the hand F = ρ A (C _X , C _Y , C _Z )∙v ² + ρ A (D _X , D _Y , D _Z )∙a, where C is a constant vector referring to drag and lift and D is a vector including the effective mass accelerated by the hand with acceleration a. The acceleration term is similar to the first term in the Navier–Stokes equation, the velocity term is different.

In swimming propulsion is reflected by the velocity of the mass centre. Such calculations can be performed after a 3D video analysis. For training methodology it is interesting to know in which way propulsion is related to swimming technique. To understand this relation the drag forces acting on the limbs as well as the total drag acting on the body have to be known. The last question is an especially difficult issue. This article presents the results of a 6-year study that consisted of the following elements: Design of a feasible measuring system, determination of the limb velocity by 3D video analysis, explanation of propulsion from a phenomenological point of view based on measuring data, quantification of the net muscle force moments for propulsion by using the Navier–Stokes equation and finally deduction of recommendations for the improvement of the individually best propulsion techniques.

Our main aim was to determine individual shoulder and hip torques at any state of one movement cycle and to find out how propulsion technique and swimming velocity are related to different torque patterns.

Phenomenology of propulsion

To move quickly in water we use our relatively large hands and feet to find water resistance and we repel ourselves like from a soft solid. In crawl stroke this happens directly with the hands when they are moved against the swimming direction. Difficulties in the interpretation of swimming movement stem not only from the high degree of complexity, but also from the two different perspectives. One viewpoint is that of cyclic movement like shovels of a paddle steamer (Figure 1, right), another one is the absolute movement of a ship's propeller (Figure 1, left).

Figure 1. Two pictures from animation of breaststroke swimming. Left: trajectory of the hand in an external reference frame (black curve through feet, mass centre and head); right: the same movement relative to the mass centre (closed black curve).

When swimming crawl stroke or dolphin with high speed no part of the legs can push the water against the swimming direction. The feet are moved forwards all the time [12]. In the downstroke of the legs the streaming water generates an acceleration force. Because the legs are bent in the knees and the hip joints, the muscle force moments cause the body to stretch, pushing the mass centre while the joints are opening. The upstroke is even more complicated. The feet are coming up close to the water surface. One can see the formation of a water peak at the highest point of the toes. Muscle forces cause a lift of the legs, which in turn induces a negative drag. In other words, if the upstroke is strong enough the inflowing water sliding upwards along the legs will be accelerated. By Bernoulli's equation a slipstream arises at the lower legs, which can be transformed during the subsequent downward movement into propulsion. This tricky technique must be learned. In addition, vortexes appear in the water in a more complicated way than one could expect from the dynamics of the mass points. When they occur, vortexes generate a slipstream at the back of the hand. We include this effect into our formulas (see Assumption 1 below).

Data acquisition

For our experiments the swimming flume at the Olympic Training Centre Hamburg was used. Once a year that flume was examined on laminar flows. Up to a flow speed of 2 m/s no standing wave was produced on the surface. Up to 1.8 m/s no water bubbles appeared. Technology and software were elaborated by Drenk et al. [13] at the Institute for Applied Training Science Leipzig. The performances of five elite swimmers have been recorded by two 50 Hz video cameras. One camera was located transversal in front and one transversal behind the subject. The optical axes formed almost a right angle and were oriented into the water. Two additional cameras were used to locate hidden points, one camera above the water and another one on the floor of the flume. The flow speed ranged from 1.15 to 1.65 m/s. Several methods have been developed and tested to correct the spherical barrel-like distortion (see Figure 2).

Figure 2. Lateral view of the flume.

We have chosen the following experimental procedure: The entire movement area in and above the water was divided into cubes with an edge length of 20 cm. This calibration frame of 1.40 m times 2.00 m was captured (see Figure 3). For each body landmark the computer program had to detect the cube of the grid that contained the body landmark. This yielded the projective mapping to reconstruct the correct spatial coordinates [13]. Displaying epipolar lines in the video images was quite useful in determining the correct location of body landmarks in particular of those points close to the water line. The accuracy of the method was established by calculating the lattice points of a canted grid implemented into the flume. The mean error over 100 lattice points was zero whereas the maximum error was 3 mm. The procedural error was equal to the pixel resolution. Computing the joint centres the mean error was about 3 cm [14].

Figure 3. One position of the calibration frame during calibration recordings.

Method

Position, velocity and acceleration

Twenty-two body point coordinates for each frame obtained by 3D video analysis are the input to our model. We use their first and second derivatives to calculate two of the three forces. The position of the mass centre was determined according to the human body model of Zatsiorsky and Seluyanov [15]. Via cubic splines the velocities and accelerations of the mass centre and those of all body landmarks have been calculated. Exactly one entire movement cycle of 50–70 frames was analysed. Because we assume optimal movement of the athletes the movement can be thought to be periodic. Therefore, accelerations at the beginning and at the end of a cycle are almost equal. These values of accelerations have been determined by a search algorithm.

Forces on particles and simplifying assumptions

The Navier–Stokes equation involving force F on a volume element and its velocity v is

(1)

where p denotes the pressure, ρ the fluid density, ρ ∂v/∂t the inertial force and ρ(v grad)v the convective force. In coordinates this reads as follows:

where (x,y,z) are the coordinates of a particle and v = (u,v,w) is the flow velocity of the particle at position (x,y,z). In our convention y denotes the longitudinal axis in swimming direction, z corresponds to the bottom-top axis and x is the left-right axis or lateral axis.

Because velocity distribution and pressure close to a moving body and other boundary conditions are not known, the boundary value problem cannot be solved.

We rather use these equations as a definition of the force F. We do not consider the whole swimmer, but only limbs that produce propulsion. Thus we can assume that water particle displacement occurs only close to the propulsive areas. The hands move on an S-shaped curve in a way that the hands catch into still water all the time. The resulting velocity, acceleration and velocity gradient are put into the right-hand side of the Navier–Stokes equation. On the left-hand side we then obtain the force acting on a single mass element. Finally we can compute the torques through cross product of force and distance vector to the joints. We are aware that our assumption that limbs always catch in still water is quite strong in particular in the case of lower limbs. For upper limbs it is less critical.

The mass of displaced water particles, their velocity and acceleration cannot be determined directly from the given data. One crucial point is to estimate the neighbourhood of the limb where water displacement occurs. We can assume that water is moved only in a close neighbourhood of the limb. To determine this neighbourhood we placed a nine-point grid into the flow. At a distance of 40 cm cotton yarn of length 30 cm was knotted in a square. A freestyle swimmer was asked to swim with the forearms between the threads. The threads were moving only in a distance less than 10 cm to the arms. Consequently, in our model the integration domain is a box where the limb surface is representing the bottom and which has a height of Δ = 10 cm. Outside this box no water displacement occurs and all forces are zero.

We apply the following three simplifying assumptions:

1. To calculate the force at the edge of limbs we take the effect of the vortices into account by designing an experimental form factor equal to two [16].

2. The velocity at some inner point of the limb is the linear interpolation of the velocities at the end points. The velocity in a small area close to the limb is constant in any plane orthogonal to the vector from one end point to the other. The same applies to accelerations.

3. Instead of the hydrodynamic pressure we use the hydrostatic pressure p = p _o + ρgz, z being the depth of the water, p _o the atmospheric pressure and g the gravitational acceleration. Consequently, 1/ρ grad p = g = (0,0,–g).

By Assumption 3 system (1) now reads as follows:

(2)

where f = F/ρ or in vector form

(3)

The mathematical model

We assume that for all body landmarks position r, velocity v and acceleration c = ∂ v/∂t are computed. For any segment, hand, forearm, upper arm, foot, shank and thigh, we know the two end points P ₁ and P ₂ of the segment and its width. As an example, we consider the hand, where P ₁ is the wrist and P ₂ the finger tip (see Figure 4 below). This direction is denoted by a´= P ₂  – P ₁. The length of the hand is a = |a´|, the hand width is denoted by b and is assumed to be known. Let a = a´/ a be the normalization of vector a´. We introduce a new system of coordinates with centre P ₁ and orthonormal axes a, n and b. Here n is the normalized vector that is orthogonal to the hand plane, in particular to a. By efficiency of the swimming movement we assume that the velocity vector of the finger tip P ₂ is orthogonal to the hand plane. Hence n has the same direction as velocity v(P ₂). We have chosen v(P ₂) rather than v(P ₁) because the distal point P ₂ has a greater impact on the torque than the proximal point P ₁. Hence, n = v(P ₂)/|v(P ₂)| and finally, b = a × n.

Figure 4. The box model of the area where water particle displacement occurs. P ₁ is a wrist point and P ₂ the finger tip. The hand moves in the direction n.

In our model water particle displacement occurs only in the box Q = [0,a] × [0,b] × [0,Δ] with edges of lengths a, b and Δ. The top face of the box is the inner hand and water particles are displaced only within a distance of Δ downwards from the inner hand.

Our next objective is to calculate v and ∂ v/∂t for all points of Q. Let k = k(x,y,z) be a function with given values k(P ₁) = k ₁ and k(P ₂) = k ₂. Later we will specify k to be a velocity or an acceleration. We want to approximate k within the box Q knowing k ₁ and k ₂ only. To do so we use Assumption 2, namely the following interpolation: k is linear between P ₁ and P ₂ and k is constant in any plane r = r ₀, which is orthogonal to vector a. Let ǩ(r,s,h) = k(x,y,z) be the same function k written in new coordinates corresponding to the box (r,s,h)ϵQ. For example, r = 0, s = 0, h = 0 corresponds to the point P ₁. According to our assumption we have

(4)

Now we want to compute grad k. Clearly, grad ǩ = ((k₂–k₁ )/a, 0, 0). Knowing the transformation matrix for the change of coordinates one gets grad k = [(k₂ – k₁ )/a]a. This coincides with the geometric picture that k changes only in direction a with constant slope (k₂ – k₁ )/a. Knowing k and grad k we can now compute the total acceleration f in Equation (3)(3), where c = ∂ v/∂t:

(5)

Inserting k = v and k = c into Equation (4)(4) and putting the result into Equation (5)(5) we obtain

(6)

The infinitesimal torque at some water particle at position r = r ₀ + r a + s b + h n is then

(7)

Torques

The shoulder torque is the sum of three individual torques generated by the hand, the forearm and the upper arm:

Integrating Equation (7)(7) over the box Q, one gets

where x = (r,s,h)ϵQ and the position vector r _hand points from the shoulder joint to the particle position x in Q. In the same way we get T _forearm and T _{upper arm}. Similarly, the hip torque is a sum of three parts T _hip = T _foot  + T _shank + T _thigh. Note that Equation (7)(7) is a quadratic function in variables r, s and h. The details of the integration are omitted here. They are carried out in a FORTRAN program. The numerical results are presented below in 5–10.

Figure 5. Left shoulder torque of two elite crawl stroke swimmers. Left figure: deep arm pull, v-flow = 1.6 m/s, t _cycle  = 1.1 s. Right figure: perpendicular lower and upper arm and pressure on forearm until the arm is leaving the water, v-flow = 1.55 m/s, t _cycle  = 1.3 s.

Figure 6. Dolphin stroke at v-flow = 1.78 m/s, t _cycle  = 1.1 s; shoulder and hip torques. Right figure: The velocity of the mass centre in swimming direction is added as a black solid line.

Figure 7. Dolphin stroke at v-flow = 1.78 m/s; contribution of foot, shank and thigh to the total hip torque. The shank part of the hip torque is dominant.

Figure 8. Different breaststroke techniques, shoulder torques. Left figure: swimming technique at v-flow = 1.15 m/s. Right figure: swimming technique at v-flow = 1.65 m/s.

Figure 9. Breaststroke, left hip torques. Left figure: swimming technique at v-flow = 1.15 m/s. Right figure: swimming technique at v-flow = 1.65 m/s.

Figure 10. Breaststroke, left shoulder torques. Contribution of left hand (left diagram) and left lower arm (right diagram) to the shoulder torque at v-flow = 1.65 m/s. The left upper arm contribution to the shoulder torque (T_max <5 Nm) can be neglected.

Results

The principle that is used to produce propulsion with the arms is a little bit different from the principle for the legs. In breaststroke initially, after diving into the water, the arms move in swimming direction and against the water. The water is flowing upwards to the shoulder. When the arms are taken to the body there is one moment when the inner palm, the forearms and parts of the upper arms are pushing against the swimming direction on the unmoved water resulting in a reverse of the streaming direction of the water. Obviously, just in these phases the drag force can be completely transformed into propulsion. It does not matter if the hands are led exclusively against the swimming direction, because the resulting force is always created against the local streaming direction. In crawl stroke the hands are mainly led backwards towards the hip. Particularly, in breaststroke swimming they are also led laterally. The drag that the hands are faced with is used with the help of the muscle force moment in shoulders and arms in such a way that the trunk pulls forwards [17]. Referring to crawl stroke one could imagine a hold at an anchor in the water from which one is pushing off. In breaststroke swimming the hands are pushed together against the imaginary anchor to pull the body forwards over the shoulders.

Crawl stroke

The principles are implemented individually: In case the stretched arm catches deeply into the water a big torque results from the long hand lever (Figure 5, left). In case the forearm is quickly moved into a perpendicular position towards the swimming direction drag is created at the whole forearm. This force lasts longer (Figure 5, right).

The thick solid line represents the shoulder torque T _x around the transverse or lateral axis on the left with a maximum of 91 Nm and on the right with a maximum of 69 Nm. The dotted line represents the torque component T _y around the longitudinal axis and finally the thin solid line with the smallest amplitude represents the component T _z around the vertical axis. The technique shown on the left diagram requires both higher force values and higher joint performances. These two swimming techniques require different dry land training.

Dolphin stroke

For the dolphin technique the lateral component T _x dominates for both the shoulder and hip torques. The thick solid line has its maximum at T _x  = 100 Nm (shoulder torque) and T _x  = 443 Nm (hip torque). The other two components are of minor significance. Even the upwards stroke (right diagram in Figure 6) can have a propulsion effect that is seen in the last negative T _x peak at t = 0.8 s, where the velocity v _y attains its maximum v _y,max  = 2.1 m/s.

Comparing the contributions of foot, shank and thigh to the hip torque, the dominating torque is generated by the shank. Although foot and thigh show similar torque pattern and similar amplitudes, the lateral component T _x of the shank is up to five times larger (Figure 7).

Breaststroke

In breaststroke the movements of the arms are different for women and men. Men swim about 0.5 m/s faster than women. For men, by lack of time, a backward movement of the hands against the water is impossible (Figure 1, left). The very first technique of women breaststroke, in which hands create pressure against the swimming direction, becomes more and more ineffective with increasing swimming speed. Men develop the following strategy: when the outward movement of the hands is completed they are immediately put together below the breast. This move generates a drag transversal to the swimming direction. The result is a much bigger torque about the body axis, as shown in Figure 8. This technique requires different control of arm and shoulder muscles and therefore training that is different from the one for the classical technique.

We compare shoulder and hip torque of these two different techniques. When swimming with a velocity of 1.65 m/s arm propulsion is dominating (T _max,shoulder > T _max,hip), but when swimming with 1.15 m/s leg propulsion is dominating (Figures 8 and 9).

We compare the different contributions of hand, forearm and upper arm to the shoulder torque in case of fast breaststroke. Figure 10 shows that the hand has almost double the effect than the forearm. The upper arm contribution is smaller than 5 Nm.

The three hydrodynamic forces

Finally we compare the effect of the three hydrodynamic forces on the hip torque for a dolphin stroke (Figure 11, left) and on the shoulder torque for a breaststroke (Figure 11, right). To do so we split the torque T _x from Figure 6, right diagram, into three summands corresponding to convective, inertial and pressure forces (Figure 11, left). The same is done with the shoulder torque T _y from Figure 8, right diagram. In both cases the convective force is dominant. Its contribution to the torque is more than twice compared with that of the inertial force (dolphin stroke) and about 50% bigger than the contribution of the inertial force to the shoulder torque (breaststroke). In both cases the pressure force has the smallest contribution to the torques.

Figure 11. Convective, inertial and pressure force contribution to torques. Right: dolphin stroke (tru) with total hip torque T _x , v = 1.78 m/s, Left: breaststroke (war50-6) with total shoulder torque T _y , v-flow = 1.65 m/s.

Conclusion

The quantification of the individual propulsion moment improved our understanding of the propulsion processes. We can prove that there are different ways of creating propulsion depending on swimming velocity. This has to be taken into account especially in long and short breaststroke swimming as well as when learning swimming techniques. As expected, the contribution of the torques arising from the upper arm and the thigh are small compared with the torques resulting from hand, foot and lower leg movements (see Figure 7 for the contribution of lower limbs). Compared to earlier estimates [12] the torque in the shoulder joint is about 50% bigger. The significance of the three terms in the Navier–Stokes equation is of great interest. The biggest gain is produced by the convective force (Figure 11), whereas the effect of the pressure gradient is significantly smaller (that could affirm Assumption 2). Inertial force has only the same significance for long breaststroke events. The dolphin stroke in butterfly swimming and crawl stroke swimming has more than double the effect compared with the typical breaststroke leg stroke (compare the hip torques in Figure 6, right, and in Figure 9). In this way the great importance of the dolphin stroke for the performance trend in swimming has been proved.

Our studies have not yet been completed. We think that the hydrostatic pressure condition might be critical (Assumption 3). Therefore, measuring the pressure in a close neighbourhood of the limbs will become indispensable in future studies to obtain more precise estimates of the velocities. Moreover, thigh and shank do not act against still water as in the case of hand and arm. This will be the topic of subsequent studies.