Journal of Fluid Science and Technology

Journal of Fluid Science and

Technology

Vol. 7, No. 1, 2012

11

Measurement and Modeling of Unsteady Fluid Force Acting on the Trunk of a Swimmer

Using a Swimmer Mannequin Robot*

Motomu NAKASHIMA** and Yusuke EJIRI*** **Graduate School of Science and Engineering, Tokyo Institute of Technology

2-12-2 Oookayama, Meguro-ku, Tokyo 152-8552, Japan E-mail: [email protected]

***Graduate School of Information Science and Engineering, Tokyo Institute of Technology Abstract The objectives of this study were to measure the unsteady fluid force acting on the trunk of a swimmer using the ‘swimmer mannequin robot’ and to model the fluid forces based on the formulation of the swimming human simulation model SWUM, which was developed by the authors’ group. The swimmer mannequin robot consisted of a swimmer mannequin and a driving mechanism. The scale of the swimmer mannequin was 1/2 (half scale) and the three-dimensional shape of an athlete swimmer taking the gliding position was reproduced in detail. The driving mechanism could move the mannequin in the pitching, heaving and rolling motions. Using the swimmer mannequin robot, the trunk motions of four strokes (crawl, breast, back and butterfly) were reproduced, and the unsteady fluid forces acting on the mannequin were measured by the dynamometers installed in the robot. On the other hand, the swimmer mannequin was modeled using the swimming human simulation model SWUM. The fluid force coefficients in the model were identified so that the simulated fluid forces became as consistent as possible with the experimental ones for each stroke case. The identified coefficients were then unified into ones which can be used for all cases. It was found that the precision of the model almost did not decrease as a result of the unification. It was also found that the overall performance of the simulation using the determined fluid force coefficients to predict the time variation of the fluid forces was satisfactory.

Key words: Swimming, Sport Engineering, Fluid Force, Bio-Fluid Mechanics, Simulation

1. Introduction

In competitive swimming, the thrust produced by the four limbs is balanced with the drag acting on the other parts of the body, such as the trunk. Therefore, the characteristics of the drag acting on the trunk is important as well as the thrust by the limbs. The characteristics of the drag acting on a swimmer’s whole body, including the trunk, have been studied by many researchers. In the early stage, the stationary drag acting on a swimmer who takes the gliding position was investigated. This drag was called “passive drag.” At the next stage, the dynamic drag acting on a swimmer during swimming drew the attention of researchers. This drag has been called “active drag.” To date, the various attempts to measure active drag have been made. Kolmogorov and Duplishcheva1) estimated it by an experiment in which an additional hydrodynamic body was towed by a swimmer. Toussaint et al.2) utilized the ‘MAD system’ for this purpose, in which a swimmer

*Received 23 Aug., 2011 (No. 11-0528) [DOI: 10.1299/jfst.7.11]

Copyright © 2012 by JSME

Journal of Fluid Science and Technology

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pushed the fixed paddles in water to propel instead of stroking. Takagi et al.3) developed an innovative method in which a swimmer fixed to a circulating tank by a harness swam at various flow speeds. In these experimental methods, the subjects of actual swimmers were basically employed. In such experiments, the dispersion due to individuals and the bias due to the physical limitations of the subjects sometimes caused problems. Therefore, the problem of the active drag still has room for the development of alternative experimental methods and further examination from various different viewpoints. As an alternative experimental method, the robotic approach can be cited. Takahashi and Nakashima4) measured the unsteady fluid forces acting on the limbs during swimming using a robot arm. When a robot is used for an experiment, the above-mentioned problems according to using actual subjects are not likely. However, there is no example to measure the dynamic drag acting on the whole body including the trunk using a robot. Therefore, the first objective of this study was to measure the unsteady fluid forces acting on the trunk of a swimmer using the ‘swimmer mannequin robot.’ This robot consisted of a swimmer mannequin and its driving mechanism. In the swimmer mannequin, the 3D shape of an actual swimmer was reproduced in detail, although the mannequin did not have any joints in the body, that is, the arms and legs could not be moved. The driving mechanism could move the mannequin as a whole in the pitching, heaving and rolling motions.

However, it is not sufficient to merely measure the fluid forces in order to fully clarify its characteristics. For better understanding of the characteristics, modeling based on a formulation is necessary. The authors’ group has already modeled the fluid forces acting on the limbs, which was measured using a robot arm5). Therefore, the second objective of this study was to model the unsteady fluid forces measured in the experiment using the swimmer mannequin robot. For the modeling, the swimming human simulation model “SWUM,” which has been developed by the authors’ group6) 7), was utilized. To date, various analyses using SWUM have been conducted and its validity has been confirmed as well8)-14). The modeling of the present study also contributes to the improvement of the accuracy of SWUM. Such a highly accurate model will be useful for various mechanical analyses of swimming in the future.

In this paper, the experimental method including the swimmer mannequin robot is described in §2. The modeling method including the formulation of the fluid forces in SWUM is explained in §3. The results and discussion are presented in §4. The conclusions are summarized in §5.

Note that the effect of moving upper and lower limbs on the flow field around the trunk was neglected in the present study since the upper and lower limbs of the swimmer mannequin could not move unlike the actual swimmer. Although this effect is considered as secondary, the detailed investigation for this effect will be the next step.

2. Experimental Method

2.1. Swimmer mannequin robot 2.1.1. Swimmer mannequin

A swimmer mannequin was manufactured for the experiment. The scale of the mannequin was selected as 1/2 (half scale) since the full scale was too large due to the limitation of the power of the driving mechanism, which is described in the next section. To begin with, the 3D shape of a swimmer was measured. The subject swimmer was a top-ranking Japanese male swimmer. His stature and weight were 1.79m and 72kg, respectively. The swimmer took a gliding position, and his upper and lower halves of his body were scanned respectively by a Whole Body 3D Scanner (WB4 Scanner, Cyberware Inc., USA). After the scanning, these data were composed into one whole body. The obtained 3D shape of the swimmer is shown in Fig.1. Using Rapid Prototyping technology, plastic shells of the swimmer were constructed from the obtained 3D shape. The inside of


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the shells were filled with plaster, as well as several steel shafts for reinforcement, and styrene foam for buoyancy. The constructed swimmer mannequin is shown in Fig.2. Note that two horizontal pipes were embedded in order to pierce shafts for mounting on the driving mechanism. The total length (from the hand tip to the toe), weight, and the specific gravity of the mannequin were 1.20m, 10.6kg and 1.19, respectively. 2.1.2. Driving mechanism

A driving mechanism for the swimmer mannequin was developed for the present study. The photographed driving mechanism and its drawing are shown in Fig.3 and Fig.4, respectively. As shown in Fig.3, four struts were connected to the two shafts which were fixed to the swimmer mannequin. When the mannequin performed the rolling motion, the right two (anterior and posterior) struts and left ones move upward and downward in opposite phase. When the mannequin performed the combination motion of heaving and pitching, the anterior two (right and left) struts and posterior ones move upward and downward in different phases. These motions were driven by three AC servo motors. The minimum/maximum angle of the rolling was ± 30 degrees. The minimum/maximum heave (up and down) displacement is ± 100mm. The time curves for these angle and

(a) Top view

(b) Side view

(c) Diagonal view

Fig.2 Photographed swimmer mannequin

(a) Diagonal view (b) Bottom view (c) Side view Fig.1 Obtained 3D shape of the swimmer


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displacements can be fully programmable at a sampling frequency of 200Hz. In order to measure the forces acting on the swimmer mannequin, four dynamometers were embedded at the tip of the struts, as shown in Fig.5. Each dynamometer detected the forces in the x (flow) and z (vertical) directions. The maximum ranges of the forces were ± 9.8N for x and

± 98N for z.

Struts

Fig.3 Photographed driving mechanism

Heave drive motor 1 Heave drive motor 2

Roll drive motor 1

4 two-axis dynamometers

Screws for adjusting vertical position

Fig.4 Drawing of the driving mechanism

Dynamometers

Fig.5 Four dynamometers embedded at the tip of the struts


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2.2. Experimental conditions The swimming motions for the four strokes (crawl, back, breast and butterfly) were

reproduced in the experiment. For crawl and back strokes, the rolling motion was performed. For the breast and butterfly strokes, the combined motions of heaving and pitching were performed. These aimed to reproduce the main components of the trunk motion in the respective swimming strokes. The input angles of these motions were created based on movies of an actual swimmer 15). In the movies, the scenes shot from the front of the swimmer were used for the roll, while the scenes shot from the side were used for the heave and pitch. The displacements and angles for each time frame of the movie were computed from the coordinates of the points which corresponded to the locations of the shafts of the mannequin. These points were determined by manually digitizing, that is, manually clicking the points on the images. On the computation of the heave displacements, the known stature of the swimmer was used for the reference with respect to length. The time curves of the input roll angles for the crawl and back strokes are shown in Fig.6. The dark blue dots and lines represent the measured angles which were calculated from the movies. The pink dots, on the other hand, represent the actual angles input to the driving mechanism, in which the unnecessary fluctuations due to the measurement error were eliminated and simplified. In Fig.6(a), it can be seen that the measured angles in the latter half became larger than those in the first half because of the breathing motion. Therefore, the amplitude in the first half (non-breathing side) was taken for the input angles. The time curves of the heave displacements and the pitch angles for the breast and butterfly strokes are respectively shown in Fig.7 and Fig.8. Note that the heave displacements at the anterior shaft of the swimmer mannequin are depicted in Fig.7. The heave displacement was defined as zero at the water surface. The pitch angle was defined as zero when the two shafts of the mannequin were horizontal. The actual depths of the swimmer mannequin in the experiment were determined so that the projecting magnitudes of the back of the swimmer mannequin

(a) Crawl (b) Backstroke Fig.6 The time curves of roll angles for crawl and back strokes

(Original; measured angles, Simplified; simplified actual input angles)

(a) Breaststroke (b) Butterfly Fig.7 The time curves of heave displacements for breast and butterfly strokes (Original; measured angles, Simplified; simplified actual input displacements)


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from the water surface became the same as those in the movies of an actual swimmer as possible.

In the experiment, the amplitudes of these motions were variously changed. For this purpose, the ‘amplitude ratio’ was defined so that it becomes 1.0 when the amplitude of the motion input to the driving mechanism is equal to the simplified one, which is shown in Figs.6 to 8. For example, if the amplitude ratio is 0.5 for the roll in the crawl, all the input angles are half of the ones shown in Fig.6(a).

Two conditions were considered for the motion cycle and the flow velocity of the circulating water tank. The first one is called ‘half velocity.’ Under this condition, the flow velocity was determined as simply the halved value of the swimming speed for the actual (full scale) swimming, which was also calculated from the movie of an actual swimmer. The motion cycle, on the other hand, was equal to the actual swimming in this condition. The second condition is called ‘Froude similarity.’ Under this condition, the flow velocity and motion cycle were determined so that the Froude number of the experiment became equal to that of the actual swimming. Therefore, the flow velocity as well as the motion cycle could be obtained as those of actual swimming divided by 2 .

Ten stroke cycles were performed by the mannequin in each experiment. By eliminating the beginning two and the last two cycles, the intermediate six cycles were used for the evaluation. The six curves of the measured forces for the six cycles were averaged and low-pass filtered to eliminate the unnecessary noise.

3. Modeling Method

3.1. Formulation of fluid forces In SWUM, the swimmer’s body is modeled as a series of body segments. Each body

segment is represented as a truncated elliptic cone. The inertial force due to added mass of the fluid, the drag force normal and tangential to the longitudinal direction, and buoyancy are taken into account. Each truncated elliptic cone is divided into thin elliptic plates along the longitudinal axis, as shown in Fig.9(a), and all fluid force components except buoyancy are assumed to act at the center of each plate. Buoyancy, on the other hand, is calculated by integrating the pressure due to the gravitational force acting on each tiny quadrangle, into which the edge of the thin elliptic plate is again divided in the circumferential direction, as shown in Fig.9(b). The unit vectors in the directions of the ellipse’s two axes, their half-lengths, and the thickness of the plate are respectively denoted as e1, e2, r1, r2 and dl as shown in Fig.9(c). In addition, the normal component of the acceleration vector of the center of the ellipse in inertial space, that is the component which is perpendicular to the longitudinal axis, is denoted as an. Thus the inertial force, Fa , due to the added mass is assumed to be given by:

(a) Breaststroke (b) Butterfly Fig.8 The time curves of pitch angles for breast and butterfly strokes (Original; measured angles, Simplified; simplified actual input angles)


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( ) ( ){ }22n2

111n2

2faa rrdlC eeaeeaF ⋅+⋅−= περ (1)

where Ca is the coefficient for the added mass effect which becomes 1.0 in a two-dimensional ideal fluid. This coefficient is used to correct the fluid forces for a three-dimensional real fluid. The coefficient εf , defined below, represents the ‘submerged fraction’ of the plate.

If the absolute velocity component of the elliptic plate’s center normal to the longitudinal axis is denoted by vn , the drag force in the normal direction, Fn , is assumed to be given by:

(2)

where Cn is the drag coefficient for the normal direction. In Eq.(2), the normal absolute velocity component, vn , is decomposed into components along the plate’s two axes, respectively. The fluid forces proportional to the squares of the velocities in both directions are then computed. Therefore, the resultant fluid force corresponds to the sum of the ‘lift’ and ‘drag’ components. Note that the coefficient p represents the effect of ellipse flatness on the fluid forces. The coefficient p was determined to be 1.0 in a previous study7).

Next, if the absolute velocity component of the elliptic plate’s center tangential to the longitudinal axis is denoted by vt , and the ellipse’s circumference by c, the drag force in the tangential direction, Ft , is assumed to be given by:

ttftt cdlC21 vvF ερ−= (3)

where Ct is the drag coefficient for the tangential direction. This equation represents the

( ) ( )

⋅

+⋅

−= 22nn

p

2

1111nn

p

1

22fnn r

rrrrrdlC eevveevvF ερ

(a) Acting points of fluid forces (b) Acting point and direction of buoyancy

(c) Divided quadrangle (d) Classification of submerged quadrangles

Fig.9 Analytical models of the fluid force in SWUM


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tangential force proportional to the elliptic plate’s side surface area acting in the longitudinal direction.

Buoyancy is calculated by integrating the pressure, Fb , due to gravity acting on the tiny quadrangles, as shown in Fig.9(b). When the area of the quadrangle, the vector normal to the quadrangle, and z coordinate of the quadrangle center are respectively denoted as ds, en , and zq , the pressure force, Fb , is given by:

nqb gdsz eF ρ−= (4)

where g represents the gravitational acceleration. Since the pressure force Fb does not act above the water surface, each tiny quadrangle is classified as to whether it is below the water surface or not, i.e., whether zq< 0 is true or not, where z = 0 represents the water surface. The pressure force Fb is assumed to act only on the quadrangles classified as being below the water surface, as schematically shown in Fig.9(d). The black points represent the centers of the quadrangles. The shaded quadrangles are classified to be below the water surface and the nonzero force Fb on them is thus calculated. Since this classification is performed in the simulation program by dividing the elliptic plate into tiny quadrangles in the circumferential direction, the ratio (number of quadrangles where zq < 0 holds)/(number of all quadrangles) can be calculated for each elliptic plate. In the present model, this ratio is called the ‘submerged fraction,’ εf . This is a factor in the inertial force of the added mass, and the drag force in the normal and tangential directions, as already shown in Eqs.(1),(2) and (3). 3.2. Modeling of swimmer mannequin

The simulation model of the swimmer mannequin was constructed on SWUM. The body geometry in the simulation was represented by the 21 truncated elliptic cones in SWUM, whose dimensions were determined as consistently as possible with the actual mannequin. The constructed model is shown in Fig.10. Note that the two horizontal shafts were also modeled, as shown in Fig.10(b), since the fluid forces acting on them could not be

(a) Mannequin only

(b) Mannequin with two horizontal shafts Fig.10 Simulation model of the swimmer mannequin


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neglected. The same movements as the experiments were performed by the swimmer mannequin in the simulation as well. That is, the vertical displacements and the roll angle of the two shafts were input to the simulation. Note that the effect of surface material of the mannequin on the fluid force was represented by the fluid force coefficient Ct in the simulation, which had to be determined using the experimental results. 3.3. Method of identification of fluid force coefficients

Since the fluid force coefficients of Cn (normal drag force) and Ca (inertial force due to added mass) for the upper and lower limbs have been identified in the previous study using a robot arm5), the coefficients of Ct (tangential drag force) for the whole body as well as Cn and Ca for the trunk (including head) were identified in this study. In addition to this, since only the roll motions were given in the crawl and backstroke, the tangential drag force (the term of Ct) was considered to be dominant, while the normal drag force (the term of Cn) as well as the inertial force due to added mass (the term of Ca) were considered to be small in this case. Therefore, the coefficient of Ct was identified using the experimental results of the crawl and backstroke. In the breast and butterfly stokes, on the other hand, the normal drag force and the inertial force due to added mass were considered to be large since the heave and pitch motions were large. Therefore, the coefficients of Cn and Ca were identified using the experimental results of the breast and butterfly strokes. These fluid force coefficients were determined using the optimizing calculation so that the simulated values became as consistent as possible with the experimental ones. For example, the objective function for the identification of Ct was defined as:

( )2

1simexp)( ∑

=

−=n

iixixt FFCf (5)

The symbols ixF exp and

ixF sim represent the fluid forces in the x direction obtained in the

experiment and simulation, respectively. The symbol n represents the number of the time-series data for one stroke cycle. This function was minimized by the optimizing calculation. The objective function for the identifications of Cn and Ca was defined as:

( )2

1simexp),( ∑

=

−=n

iizizan FFCCf (6)

The symbols izF exp and

izF sim represent the fluid forces in the z direction obtained in the

experiment and simulation, respectively. This function was minimized as well. The

Downhill Simplex method was used for the optimizing algorithm. The fluid forces expxF

and expzF in the experiment were calculated by firstly summing up the four values detected

by the four dynamometers, and next deducting the inertial force acting on the mannequin. For this procedure, an experiment to measure the inertial force acting on the mannequin was carried out. In this experiment, the swimmer mannequin was driven on land in the same motions as those in the water, and the measured values were regarded as the inertial forces. This procedure was necessary because the inertial force became comparable to the fluid force in the experiment.


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4. Results and Discussion

4.1. Results of identification of fluid force coefficients The results of the identification for the fluid force coefficient, Ct , are shown in Table 1.

Note that the amplitude ratio of 1.0 could not be performed due to the limitation of the power of the driving mechanism. In most cases except for Ba4, it was found that the identified values were within 0.025 – 0.035. Therefore, the unified value of 0.0285 was obtained as the average of the values for all cases. The results of identification for the fluid force coefficient Cn and Ca are shown in Table 2. Note that the unified value of Ct = 0.0285 was used for all cases of the simulations for identification. Although the values varied widely both for Cn and Ca (Cn : 1.341–6.439, Ca : 0.121–0.641), the unified values of Cn = 3.11 and Ca = 0.393 were calculated as the average of the values for all cases.

By this unification, there was a possibility for increase in the modeling error since the unified coefficients were different from the identified ones optimum for each case. Therefore, the precision of the fluid forces calculated with the unified coefficients as well as those identified for each case were examined. For this purpose, the index of the precision, P, was defined as:

( )MinzMaxz

n

iizisimz

FFn

FFP

−

−=∑=1

exp (7)

The above equation is an example with respect to the fluid forces in the z direction. The

symbols MaxzF and MinzF represent the maximum and minimum of the fluid forces within

one case, respectively. The results of comparison with respect to P are shown in Table 3. As

Table 2 Results of identification of the fluid force coefficients Cn and Ca

ID Stroke Condition Flow velocity[m/s]

Cycle [s]

Amplitude ratio Cn Ca

Br1 0.80 1.885 0.563Br2

Half velocity 0.50 2.153

0.98 1.341 0.407Br3 0.60 2.494 0.121Br4

BreaststrokeFroude

similarity 0.75 1.444 0.80 1.601 0.318

Bu1 0.60 4.957 0.641Bu2

Half velocity 0.75 1.324

0.80 3.228 0.641Bu3 0.20 6.439 0.175Bu4

Butterfly Froude

similarity 1.10 0.912 0.40 2.954 0.279

Table 1 Results of identification of the fluid force coefficient Ct

ID Stroke Condition Flow velocity[m/s]

Cycle [s]

Amplituderatio Ct

Cr1 0.60 0.0306 Cr2

Half velocity 0.70 1.866 0.80 0.0281

Cr3 0.20 0.0278 Cr4

Crawl Froude similarity 0.95 1.379

0.40 0.0258 Ba1 0.40 0.0354 Ba2

Half velocity 0.65 1.828 0.60 0.0322

Ba3 0.20 0.0332 Ba4

BackstrokeFroude similarity 0.90 1.315

0.40 0.0153


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shown in this table, it was confirmed that the precision almost did not decrease as a result of the unification. Therefore, the fluid force coefficients were finally determined in the present study as: Ct = 0.0285 (for the whole body), Cn = 3.11 and Ca = 0.393 (for the trunk). The reason why the precision did not decrease despite the large variation of Cn and Ca as shown in Table2 may be because the contribution of the upper and lower limbs to the fluid force was larger than that of the trunk.

The fluid force coefficient of Ct = 0.0285 is supposed to be within the reasonable range since it is only 20% lower than the previous value of 0.036 (7). However, the coefficient Cn = 3.11 is considered to be significantly large. It is much larger than the generally known values, such as 1.2 for a two-dimensional cylinder and 2.0 for a two-dimensional plate. One possible reason is that the wave-making drag affects the fluid forces not only in the horizontal direction, but also in the vertical one. The coefficient Ca = 0.393, on the other hand, is considered to be smaller than 1.0 for the two-dimensional ideal fluid around a cylinder. A possible reason is the complexity of the flow around the trunk due to its complicated shape. 4.2. Results of time variations of the fluid forces

Using the fluid force coefficients determined in the previous section, comparison of the time variations for the fluid forces between the simulated and experimental values were carried out. Examples of the results are shown in Fig.11 for Fx as well as Figs.12 and 13 for Fx and Fz , respectively. These are the cases where the amplitude of the motion was maximum in each stroke. The abscissas represent the nondimensional time normalized by the stroke cycle, that is, where t* = 0 – 1.0 corresponds to one stroke cycle. The roll or pitch angles are also drawn in the figures. As described in Section 2.3, the experimental curves

Table 3 Comparison of precision, P, between simulations with un-unified and unified fluid force coefficients Cn and Ca

ID Un-unified Unified Br1 0.09 0.09 Br2 0.08 0.09 Br3 0.10 0.09 Br4 0.08 0.08 Bu1 0.08 0.09 Bu2 0.07 0.08 Bu3 0.13 0.14 Bu4 0.13 0.13

Average 0.10 0.10

0 0.2 0.4 0.6 0.8 10

2

4

6

8

-40

-20

0

20

40

Nondimensional time t*

Fx [N

]

Simulation Experiment Roll angle

Rol

l ang

le [d

eg]

0 0.2 0.4 0.6 0.8 10

2

4

6

8

-40

-20

0

20

40

Nondimensionl time t*

Fx [N

]

Simulation Experiment Roll angle

Rol

l ang

le [d

eg]

(a) Crawl (Cr2) (b) Backstroke (Ba2) Fig.11 The unsteady fluid forces of Fx for crawl and backstroke


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were the average for six stroke cycles. The significant variation among six cycles could not be seen for all the cases. From Fig.11, it can be found that the simulated fluid forces in the x direction for the roll motions could reproduce the overall characteristics of the time variations, such as the temporal increases when the roll velocity becoming maximum (t* is around 0 and 0.5). The discrepancy between the simulation and experiment was larger for the backstroke than the crawl. A possible reason for this is the flow around the swimmer mannequin in the backstroke became more complicated than that in the crawl.

For the breaststroke, it was found that both Fx and Fz of the simulation were sufficiently consistent with those of the experiment, as shown in Fig.12. It can be seen that both Fx and Fz reached their maximums at t* = 0.25, which is the late phase of the breathing motion. Note that the reason why the simulated values do not become smooth curves is that not the smoothed reference values, but the actual measured ones of the swimming motion, were used for the simulations. For the butterfly stroke, as shown in Fig.13, it can be found that Fz of the simulation seems consistent with that of the experiment, while the discrepancy becomes somewhat larger with respect to Fx , especially at t* = 0.7–0.9, although its absolute value was small (2–3N). This small discrepancy occurred although Fz could be predicted relatively accurately by the simulation, as shown in Fig.13(b). The good prediction in Fz means that the reason for the discrepancy in Fx was not due to the inappropriate fluid force coefficients. The possible reason for the discrepancy, therefore, is mechanical error of the driving mechanism inducing error in pitch angle. Since the horizontal component of the vertical fluid force is proportional to the sine of the pitch angle, a small error in the pitch angle can induce a relatively large error in the fluid force in the horizontal direction.

From the above discussion, the overall performance of the simulation to predict the time variation of the fluid forces can be said to be satisfactory.

0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

-20

-10

0

10

20

Nondimensionl time t*Fx

[N]

Simulation Experiment Pitch angle

Pitc

h an

gle

[deg

]

0 0.2 0.4 0.6 0.8 10

50

100

-20

-10

0

10

20


Fz [N

]


Pitc

h an

gle

[deg

]

(a) Fx (b) Fz Fig.12 The unsteady fluid forces of Fx and Fz for breaststroke (Br2)

0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

-20

-10

0

10

20


Fx [N

]


Pitc

h an

gle

[deg

]0 0.2 0.4 0.6 0.8 10

50

100

-20

-10

0

10

20


Fz [N

]


Pitc

h an

gle

[deg

]

(a) Fx (b) Fz Fig.13 The unsteady fluid forces of Fx and Fz for butterfly stroke (Bu2)


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5. Conclusion

In this study, the unsteady fluid forces acting on a swimmer was measured using a swimmer mannequin robot. The fluid forces were then modeled based on the formulation of the swimming human simulation model SWUM. Findings are summarized as follows:

(1) The unified fluid force coefficients in the model could be determined by averaging

those identified for each case. It was confirmed that the precision of the model almost did not decrease as a result of the unification.

(2) The coefficient for the tangential drag force was determined to be 0.0285. This value is 20% lower than the previous value used in SWUM.

(3) The coefficient for the normal drag force was determined to be 3.11. This value is significantly larger than the conventional value.

(4) The coefficient for the inertial force due to added mass of the fluid was determined to be 0.393. This value is smaller than the conventional value.

(5) The overall performance of the simulation using the determined fluid force coefficients to predict the time variation of the fluid forces was satisfactory.

With respect to (3) and (4), the deviations of the coefficients from the conventional values do not necessarily mean the failure of the identification since the experimental fluid forces can be reproduced well using these values, as described in (5). However, the detailed explanation for these values will be the important future task.

The fluid force model developed in the present study can be utilized for the estimation of the fluid force acting on the swimmer. Such estimation will be utilized for various purposes. For example, the evaluation of individual swimming form with respect to the thrust generation can be possible. Such evaluation will provide the useful feedback to athlete swimmers for the improvement of their performance.

Acknowledgments

The authors wish to thank Hisayoshi Sato, Junichi Sato and Yohei Sato for their cooperation in the 3D body scanning. This work was partly supported by JSPS Grants-in-Aid for Scientific Research (No.20300208).

References

(1) Kolmogorov, S.V. and Duplishcheva, O.A. Active Drag, Useful Mechanical Power Output and Hydrodynamic Force Coefficient in Different Swimming Strokes at Maximal Velocity, Journal of Biomechanics, Vol. 25, No. 3, (1992), pp.311-318.

(2) Toussaint, H.M., de Groot, H.H., Saveberg, C.M., Vervoorn, K., Hollander, A.P. and van Ingen Schenau, G.J. Active Drag Related to Velocity in Male and Female Swimmers, Journal of Biomechanics, Vol. 21, No. 5, (1988), pp.435-438.

(3) Takagi, H., Shimizu, Y. and Kodan, N. A Hydrodynamic Study of Active Drag in Swimming, JSME International Journal, Series B, Vol. 42, No. 2, (1999), pp.171-177.

(4) Takahashi, A. and Nakashima, M. Clarification of Unsteady Fluid Force Acting on Limbs in Swimming Using an Underwater Robot Arm (Development of an Underwater Robot Arm and Measurement of Fluid Force), Transactions of the Japan Society of Mechanical Engineers, Series B, Vol.75, No.750, (2009), pp.284-293.

(5) Takahashi, A. and Nakashima, M. Clarification of Unsteady Fluid Force Acting on Limbs in Swimming Using an Underwater Robot Arm (2nd Report, Modeling of Fluid Force Using Experimental Results), Transactions of the Japan Society of Mechanical Engineers, Series B, Vol.76, No.761, (2010), pp.66-75.

(6) SWUM Developer Team, “Swimming Human Simulation Model SWUM”. SWUM Website. (online). available from <http://www.swum.org/>, (accessed 2011-8-18).

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Whole Body, Journal of Fluid Science and Technology, Vol. 2, No. 1, (2007), pp.56-67.

(8) Nakashima, M. Mechanical Study of Standard Six Beat Front Crawl Swimming by Using Swimming Human Simulation Model, Journal of Fluid Science and Technology, Vol. 2, No. 1, (2007), pp.290-301.

(9) Nakashima, M. Analysis of Breast, Back, and Butterfly Strokes By The Swimming Human Simulation Model SWUM, Bio-mechanisms of Swimming and Flying - Fluid Dynamics, Biomimetic Robots, and Sports Science - (eds. Naomi Kato and Shinji Kamimura), Springer, Tokyo, (2007), pp.361-372.

(10) Nakashima, M. Simulation Analysis of the Effect of Trunk Undulation on Swimming Performance in Underwater Dolphin Kick of Human, Journal of Biomechanical Science and Engineering, Vol. 4, No. 1, (2009), pp.94-104.

(11) Kiuchi, H., Nakashima, M., Cheng, K.B. and Hubbard, M. Modeling Fluid Forces in the Dive Start of Competitive Swimming, Journal of Biomechanical Science and Engineering, Vol.5, No.4, (2010), pp.314-328.

(12) Nakashima, M., Kiuchi, H. and Nakajima, K. Multi Agent/Object Simulation in Human Swimming, Journal of Biomechanical Science and Engineering, Vol. 5, No. 4, (2010), pp.380-387.

(13) Nakashima, M., Suzuki, S. and Nakajima, K. Development of a Simulation Model for Monofin Swimming, Journal of Biomechanical Science and Engineering, Vol. 5, No. 4, (2010), pp.408-420

(14) Nakashima, M. Modeling and Simulation of Human Swimming, Journal of Aero Aqua Bio-mechanisms, Vol.1, No.1, (2010), pp.11-17.

(15) IPA (Information-technology Promotion Agency, Japan. “Physical Education (Swimming)” (in Japanese). Picture Materials for Education (in Japanese). (online), available from <http://www2.edu.ipa.go.jp/gz2/list.html#suiei>, (accessed 2011-8-18).

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