- Email: [email protected]
Results of Single Sculling Technique Analysis Using 1D Mathematical Model
Proceedings of the 9th Vienna International Conference on Proceedings ofModelling the 9th Vienna International Conference on Mathematical Proceedings ofModelling the 9th Vienna International Conference on at www.sciencedirect.com Available online Mathematical Proceedings of the 9th Vienna International Conference on Vienna, Austria, February 21-23, 2018 Mathematical Modelling Mathematical Modelling Vienna, Austria, February 21-23, 2018 Vienna, Austria, February 21-23, 2018 Vienna, Austria, February 21-23, 2018
ScienceDirect
IFAC PapersOnLine 51-2 (2018) 879–883 Results of Single Sculling Technique Analysis Using 1D Mathematical Model Results of Single Sculling Technique Analysis Using 1D Mathematical Model Results Analysis Using 1D Mathematical Model Results of of Single Single Sculling Sculling Technique Technique Analysis Using 1D Mathematical Model * ** ***
M. Wychowanski*, G. Slugocki**, G. Orzechowski***, M. Wychowanski**,, G. Slugocki** Orzechowski *** ***** **,, G. ***,, M. Z. Staniak****, D. Radomski ***** M. Wychowanski Wychowanski , G. G. Slugocki Slugocki , G. G. Orzechowski Orzechowski , Z. Staniak****, D. Radomski ***** Z. Z. Staniak****, Staniak****, D. D. Radomski Radomski***** *Faculty of Rehabilitation, Joseph Pilsudski University of Physical Education
Rehabilitation, Joseph Pilsudski University of Physical Education in*Faculty Warsaw,of Poland (Tel:+48 600-331-303; e-mail: [email protected]), *Faculty of Rehabilitation, Joseph University of *Faculty of Rehabilitation, Joseph Pilsudski Pilsudski University of Physical Physical Education Education in Warsaw, Poland (Tel:+48 600-331-303; e-mail: [email protected]), **Faculty of Power and (Tel:+48 Aerospace600-331-303; Engineering e-mail: [email protected]), University of Technology, Poland in Warsaw, Poland in Warsaw, Poland e-mail: [email protected]), **Faculty of Power and (Tel:+48 Aerospace600-331-303; Engineering Warsaw University of Technology, Poland [email protected]), **Faculty of of Power Power and and Aerospace Aerospace(e-mail: Engineering Warsaw University University of of Technology, Technology, Poland Poland **Faculty Engineering Warsaw (e-mail: [email protected]), *** Department of Mechanical (e-mail: Engineering, Lappeenranta University of Technology, [email protected]), (e-mail: [email protected]), *** Department of Mechanical Engineering, Lappeenranta University of Technology, Finland (e-mail: [email protected]), *** Engineering, Lappeenranta *** Department Department of of Mechanical Mechanical Engineering, Lappeenranta University University of of Technology, Technology, Finland (e-mail: [email protected]), ****National Research Institute of Sport, [email protected]), Warsaw, Poland (e-mail: [email protected]), Finland (e-mail: Finland (e-mail: [email protected]), ****National*****Institute Research Institute of Sport, Warsaw, (e-mail: [email protected]), of Radioelectronics andPoland Multimedia Techniques, ****National Research of Poland (e-mail: [email protected]), ****National*****Institute Research Institute Institute of Sport, Sport, Warsaw, Warsaw, Poland (e-mail: [email protected]), of Radioelectronics and Multimedia Techniques, Warsaw*****Institute University of Technology, Poland and (e-mail: [email protected]) of Radioelectronics Multimedia Techniques, *****Institute of Radioelectronics and Multimedia Techniques, Warsaw University of Technology, Poland (e-mail: [email protected]) Warsaw Warsaw University University of of Technology, Technology, Poland Poland (e-mail: (e-mail: [email protected]) [email protected])
Abstract: Sports results in rowing depend on two the most important factors: the athlete physical features Abstract: Sports results in rowing depend on two theoptimisation most important factors:techniques the athlete are physical features and the motion’s techniques. Bothdepend assessment andthe of rowing possible only Abstract: Sports in on most factors: the physical features Abstract: Sports results results in rowing rowing depend on two two theoptimisation most important important factors:techniques the athlete athlete are physical features and the motion’s techniques. Both assessment and of rowing possible only whenthe onemotion’s disposestechniques. the reliableBoth mathematical model predicting theof regatta results, that is are the possible time to cover and assessment and optimisation rowing techniques only and the motion’s techniques. Both assessment and optimisation of rowing techniques are possible only when one disposes theAreliable mathematical model predicting the regatta results, that isisthe time to cover the assumed distance. single mathematical scull participating in predicting the 2000 meters distance’s regatta ourtime subject. The when one the reliable model the results, that to when one disposes disposes theA reliable mathematical model predicting the regatta regatta results, that is isisthe the time to cover cover the assumed distance. single scull participating in the 2000 meters distance’s regatta our subject. The purpose of this study is to create a simplified mathematical model to simulate the rowing boat dynamics. the assumed distance. A single scull participating in the 2000 meters distance’s regatta is our subject. The the assumed distance. single scull participating in the 2000model meterstodistance’s regatta is our subject. The purpose of this study isAisto create a simplified mathematical simulate the rowing boat dynamics. The boat-rower systemis treated here as a material point here. The oar has a prescribed angular motion vs. purpose of this study to create aa simplified mathematical model to simulate the rowing boat dynamics. purpose of this study is to create simplified mathematical model to simulate the rowing boat dynamics. The boat-rower system is treated here as away material point here. The oar developed has a prescribed angular motion vs. oarlock depending upon the time. The of hydrodynamic force on theangular oar’s motion blade was The boat-rower system is treated here as a material point here. The oar has a prescribed vs. The boat-rower system is treated here as away material point here. The oar developed has a prescribed motion vs. oarlock depending upon the motion, time. The of hydrodynamic force on theangular oar’s blade was modelleddepending here. Thenupon the boat with theof influence both of wind and water current, was described by oarlock the time. The way hydrodynamic force developed on the oar’s blade was oarlock depending upon the time. The way of hydrodynamic force developed on the oar’s blade was modelled here. Then the boat motion, with the influence both of wind and water current, was described by a single nonlinear ordinary differential equation (NODE).both Theof used simple modelcurrent, gives the possibilities of modelled here. Then the boat motion, with the influence wind and water was described by modelled here. Then the boat motion, with the influence both of wind and water current, was described by a single nonlinear ordinary differential equation (NODE). The used simple model gives of the possibilities of and nonlinear reliable simulation of the singleequation sculling(NODE). techniqueThe andused of forecasting the result rowing regattas. aafast single ordinary differential simple model gives the possibilities of single nonlinear ordinary differential equation (NODE). The simple model gives of therowing possibilities of fast and reliablesensitivity simulation of the single sculling technique andused of forecasting the result regattas. The and parametric coefficient wassculling definedtechnique here, as well the efficiency coefficient ofrowing sculling. fast reliable simulation of and of the regattas. fast reliablesensitivity simulationcoefficient of the the single single and the of forecasting forecasting the result result of ofof rowing regattas. The and parametric wassculling definedtechnique here, as well efficiency coefficient sculling. The parametric sensitivity coefficient was defined as the coefficient of The parametric sensitivity coefficient was defined here, here, as well well the efficiency efficiency coefficient of sculling. sculling. © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. reserved. Keywords: sculling, mathematical model, motion’s equation, hydrodynamic force onAll therights oar. Keywords: sculling, mathematical model, motion’s equation, hydrodynamic force on the oar. Keywords: Keywords: sculling, sculling, mathematical mathematical model, model, motion’s motion’s equation, equation, hydrodynamic hydrodynamic force force on on the the oar. oar.
1. INTRODUCTION 1. INTRODUCTION 1. INTRODUCTION 1. A INTRODUCTION single sculling is a rowing with two oars, one in one hand. A single scullinginvolves is a rowing with atwo oars, one in oneinhand. This discipline passing specific distance the A single sculling is with oars, in A single scullinginvolves is aa rowing rowing with atwo two oars, one one in one oneinhand. hand. This discipline passing specific distance the shortest possibleinvolves time and therefore the efficient sculling This discipline passing a specific distance in This discipline involves passing a specific distancesculling in the the shortest possible time and therefore the efficient techniquepossible is of a paramount importance. This paper attempts shortest time and therefore the efficient sculling shortest possible time and therefore the efficient sculling technique of mathematical a paramount importance. This paper attempts to developis the model of a rowing to be useful technique is of aa paramount This attempts technique isthe of mathematical paramount importance. importance. This paper paper attempts to develop model of a rowing to be useful fordevelop analysis,thelearning, improvement and motiontotechniques to mathematical model of a rowing be to develop thelearning, mathematical model ofand a rowing totechniques be useful useful for analysis, improvement motion optimization. Accordingimprovement to Zatsiorsky Yakunin (1991) for analysis, and motion techniques for analysis, learning, learning, improvement andand motion techniques optimization. According to Zatsiorsky and Yakunin (1991) "Despite relatively numerous studies, the of optimization. According to and Yakunin optimization. According to Zatsiorsky Zatsiorsky and biomechanics Yakunin (1991) (1991) "Despite relatively numerous studies, the biomechanics of rowing remains poorly understood". These authors were "Despite relatively numerous studies, the biomechanics of "Despite relativelypoorly numerous studies, the biomechanics of rowing remains understood". These authors were among the first who hadunderstood". attempted rowing mathematical rowing remains poorly These authors were rowing remains poorly understood". These authors were among the A first had attempted mathematical description. lot who of authors have dealt rowing with similar subjects, among first who had rowing mathematical among the the A first who had attempted attempted rowing mathematical description. lot of authors have dealt with similar subjects, Pulman, C. A (access 2014), among others, gives a thorough description. lot of authors have dealt with similar subjects, description. A lot of 2014), authorsamong have dealt withgives similar subjects, Pulman, C. (access others, a thorough description of(access the phenomena concerning different aspects of Pulman, C. 2014), among others, gives a thorough Pulman, C. of(access 2014), among others,different gives aaspects thorough description the phenomena concerning of a rowing. Some authors such as: Findlay, M. and Turno, S. R. description of the phenomena concerning different aspects of description of the phenomena concerning different aspects of a(2010), rowing. Some authors such as: Findlay, M. and Turno, S. R. Kinoshita et al. (2008), and MillarM. S.and K. et al. (2015) aa rowing. Some authors such as: Findlay, Turno, S. R. rowing. Some authors such as: Findlay, M. and Turno, S. R. (2010), Kinoshita et al. (2008), and Millar S. K. et al. (2015) have with et similar and problems, the (2010), Kinoshita al. Millar et (2010),dealt Kinoshita etthe al. (2008), (2008), and Millar S. S.aa K. K.specially et al. al. (2015) (2015) have dealt with the similar problems, specially the hydrodynamics phenomena related to the flow past the oar. have dealt with the similar problems, a specially the have dealt with the similar problems, a specially the hydrodynamics phenomena related to the flow past the oar. The main aim ofphenomena the present related work istotothe carry outpast the analysis hydrodynamics flow the oar. hydrodynamics phenomena related to the flow past the oar. Thea main of the present workWe is to carry out analysis of singleaim sculling technique. proposed thethe following The aim of work is carry the analysis Thea main main aim of the the present present workWe is to to carry out out thefollowing analysis of single sculling technique. proposed the criteria of technique assessment We for proposed this aim. These criteria of a single sculling technique. the following of a single sculling technique. We proposed the following criteria of technique assessment for this aim. These criteria were sculling efficiency and parametrical sensitivity regatta criteria of technique assessment for this aim. These criteria of technique assessment for this aim. These criteria criteria were sculling efficiency and parametrical sensitivity regatta result on chosen parameters. were sculling efficiency and parametrical sensitivity were sculling efficiency and parametrical sensitivity regatta regatta result on chosen parameters. result result on on chosen chosen parameters. parameters.
Fig. 1. Physical model of the sculler-boat-oars system. Fig. 1. Physical model of the sculler-boat-oars system. Fig. Fig. 1. 1. Physical Physical model model of of the the sculler-boat-oars sculler-boat-oars system. system. 2. MATHEMATICAL MODEL 2. MATHEMATICAL MODEL 2. MATHEMATICAL MODEL 2. MATHEMATICAL MODEL The mathematical model of a sculling technique has been The mathematical model of a sculling technique derived from the second Newton’s dynamics law andhas rulesbeen the The mathematical model of technique has been The mathematical modelNewton’s of aa sculling sculling technique has been derived from the second dynamics law and rules the describing hydrodynamic forces and the aerodynamic forces, derived from the second Newton’s dynamics law and rules the derived from the second Newton’s dynamics law and rules the describing hydrodynamic forces and the aerodynamic forces, for the physical model of forces rowingand shown in the Figure forces, 1. The describing hydrodynamic the aerodynamic describing hydrodynamic forces and the aerodynamic forces, for the emphasises physical model of rowing shown in generation the Figure 1. model the description of force on The the for the physical of in the The for the emphasises physical model model of rowing rowing shown shown in generation the Figure Figure 1. 1. The model the description of force onboat the oars phenomenon. Thedescription equation ofofmotion of one scull model emphasises the force generation on the model emphasises the force generation onboat the oars phenomenon. Thedescription equation ofof motion of one scull takes the wind and water current in consideration. oars phenomenon. The equation of motion of one scull boat oars The equation motion of one scull boat takesphenomenon. the wind and water current inofconsideration. takes takes the the wind wind and and water water current current in in consideration. consideration.
2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2018, 2018 IFAC 1 Hosting by Elsevier Ltd. All rights reserved. Peer review©under of International Federation of Automatic Copyright 2018 responsibility IFAC 1 Control. Copyright © 2018 IFAC 1 10.1016/j.ifacol.2018.04.025 Copyright © 2018 IFAC 1
Proceedings of the 9th MATHMOD 880 Vienna, Austria, February 21-23, 2018
M. Wychowanski et al. / IFAC PapersOnLine 51-2 (2018) 879–883
with the oar cycle period: 2.1 The equation of boat motion
T 0.5(T1 T2 )
The equation of motion of the sculler-boat-oars system is shown below:
and the oar angular velocity:
m x(t ) 2 F (t ) sin OA (t ) 0.5 H O SVAcD H O [ x(t ) xVS (t )]2 sgn[ x(t ) xVS (t )] 2
2
0.5 AIR S LV cD AIR[ x(t ) xLS (t )]2 sgn[ x(t ) xLS (t )]
(7)
k (t ) 2 (t iT) k 2 (t ) 2 (t 0.5T1 iT) OA (t ) ( OAMAX OAMIN) 1 sin sin T2 T1 T1 T2
(1)
(8)
where: TIN oar blade immersion time,
where: m mass,
TOUT oar blade emersion time,
x ( t ) boat position,
k1(t) and k2(t) are the switching functions.
xVS ( t ) water current velocity,
2.3 Definition of the Efficiency Coefficient We define it for one cycle i:
xLV (t ) wind velocity,
( i 1) T
F( t ) force developed by oar, Ei
AIR air density,
F(t ) sin
OA
( t )dt
iT
( i 1) T
0i n 1
F(t ) dt
H O water density,
(9)
iT
2
and for the entire motion period:
cDAIR aerodynamic drag coefficient,
nT
cDH O hydrodynamic drag coefficient, 2
E
SLV reference surface flown by air,
F(t ) sin
OA
( t )dt
0
(10)
nT
F(t ) dt
SVA reference surface wetted by water,
0
OA oar angle
where n is the total number of oar motion cycles.
l OA distance from oarlock on the oar trunk and blade, 2.4 Definition of the sensitivity result from the given parameter The regatta result TREG sensitivity coefficient on the given parameter pi is defined as follows: TREG T TREG p i (11) S T REG p i TREG p i p pi
b OA oar blade width as a function of l OA . 2.2 Derivation of oar force formula The force developed by oar has the following form: F (t ) 0.25cDOA H O [1 sgn OA (t )] (t ) sgn g (t ) 2
REG
(2)
lOAM AX
{lOAOA (t ) [ x(t ) xVS (t )] sin OA (t )} bOAdlOA 2
i
lOAM IN
where Δpi=0,01pi.
where the oar’s blade immersion-emersion function is as follows: (t)
t iT TIN
iT t iT TIN
(t) 1
iT TIN t iT 0.5 T1 TOUT
t (iT 0.5T1 TOUT ) (t) 1 TOUT
iT 0.5 T1 TOUT t iT 0.5 T1
(t) 0
iT 0.5 T1 t ( i 1) T
3. RESULTS
0 i n 1
(3)
The following values given below are the listed parameters such as: the system total mass of (the rower, the boat, the oars) m=100kg, the boat hydrodynamic drag coefficient cDH20=0.07 with the reference surface SVA=0.1075m2 and water density ρH2O=1000kg/m3, aerodynamic drag coefficient cDAIR=0.19 of the boat-rower system with the reference surface SLV=0.3231m2 and the air density ρAIR=1.225kg/m3, dimensionless drag coefficient of the oar cDOA=1.13, the distance between an oar lock and the oar blade tip lOAmax=2.04m, the distance between the oar lock and the oar blade root lOAmin=1.6m, the oar blade width bOA=0.23m, the total rowing cycle period is T=2s, the cycle active phase period TA=0.5T1=0.7s, the cycle period passive phase TB=0.5T2=1.3s. The system is modeled using MATLAB® package. The resulting equation of motion is solved with ode45 code (Shampine and Reichelt, 1997) to be based on the explicit Runge-Kutta formula of orders 4 and 5.
with an adjusting term: lOAM AX
(4) sgn g (t ) sgn{ {lOAOA (t ) [ x(t ) x VS (t )] sin OA (t )}bOAdlOA} lOAM IN
The oar angle vs. time: OA ( t ) 0.5( OAMAX OAMIN ) 0.5( OAMAX OAMIN ) 2 ( t 0.5T1 iT) 2 ( t iT) k 1 ( t ) cos k 2 ( t ) cos T1 T2
[k 1 ( t ) 1] [k 2 ( t ) 0]
[k 1 ( t ) 0] [k 2 ( t ) 1] 0i n iT 0.5 T t ( i 1) T iT t iT 0.5 T1
1
0i n
(5)
(6)
2
Proceedings of the 9th MATHMOD Vienna, Austria, February 21-23, 2018
M. Wychowanski et al. / IFAC PapersOnLine 51-2 (2018) 879–883
The equation is solved with relatively tight error tolerances of 10-10 for absolute and 10-8 for relative tolerances. The use of the default tolerance values results in wrong solution. In the beginning we have carried out the regatta simulation for three ranges oar rotation. The assumed ranges of the oar rotation γOA=45°÷135º, γOA=60°÷150º. were: γOA=30°÷120º, The calculation result in sensitivity parameters, rowing efficiency and regatta time for oar rotation all ranges are presented in the Table 1. The most efficient sculling technique was for γOA=45°÷135º range of the oar rotation. The best regatta result was observed for the γOA=60°÷150º. The sensitivity coefficients computation of regatta result in wind and water current were performed for the most efficient oar rotation range. Result of computing of efficiency, sensitivity coefficients and regatta results for different wind velocities are presented in the Table 2 and the Table 3 and the Figure 2. Efficiency computing, sensitivity coefficients results and regatta results for different water current velocities are presented in the Table 4 and the Table 5 and the Figure 3. The best time to cover the distance 2000m was observed for favorable water current velocity of 2m/s. The worst regatta time was observed for opposite water current of 2m/s. The efficiency coefficient for this oar rotation range did not change. The greatest influence on the regatta result has the time of active phase of rowing as shown on the Figure 4. The sensitivity of regatta results on active phase time vs. the wind velocity is constant as shows the Figure 5.
881
Table 3. Regatta result sensitivity on parameter [%], E - efficiency of rowing [%] and regatta time [s] for favorable wind velocities [m/s] Favorable wind CDAIR CDH2O ρH2O TA TB TIN TOUT E Regatta time
Table 1. Regatta result sensitivity on parameter [%], E - efficiency of rowing [%] and regatta time [s] for ranges of oars angles [deg] Range of oars 30÷120 45÷135 60÷150 CDAIR 0 0 0 CDH2O 16 18 16 ρH2O 5 4 2 TA 86 88 85 TB 11 12 11 TIN 1 1 0 TOUT -3 -1 1 E 69 84 83 Regatta time 491.5 488.8 471.6
0 0 18 4 88 12 1 -1 84 488.8
0.5 0 19 4 88 12 1 -1 84 488.6
1 0 18 4 87 12 1 -1 84 488.4
2.5 0 17 4 87 13 1 -1 84 488.1
5 10 0 0 18 17 4 4 87 86 12 13 1 1 -1 -1 84 84 487.9 486.3
Fig 2. Regatta time TREG vs. wind velocity VLV.
Table 4. Regatta result sensitivity on parameter [%], E - efficiency of rowing [%] and regatta time [s] for opposite water flow velocities [m/s]
Table 2. Regatta result sensitivity on parameter [%], E - efficiency of rowing [%] and regatta time [s] for opposite wind velocities [m/s] Opposite wind -10 -5 -2.5 -1 -0.5 CDAIR 2 1 0 0 0 CDH2O 15 17 17 16 17 ρH2O 2 3 4 4 4 TA 88 87 87 87 88 TB 12 13 13 11 11 TIN 1 1 1 1 1 TOUT 0 -1 -1 -1 -1 E 85 84 84 84 84 Regatta time 497.7 492.0 490.1 489.3 489.0 3
Opposite flow
-2
-1.5
-1
-0.5
CDAIR
0
0
0
0
CDH2O
33
27
23
21
ρH2O
7
6
5
5
TA
171
138
115
100
TB
23
18
15
15
TIN
1
1
1
1
TOUT
-1
-1
-1
-1
E
84
84
84
84
Regatta time
951.6
769.4
645.8
556.4
Proceedings of the 9th MATHMOD 882 Vienna, Austria, February 21-23, 2018
M. Wychowanski et al. / IFAC PapersOnLine 51-2 (2018) 879–883
Table 5. Regatta result sensitivity on parameter [%], E - efficiency of rowing [%] and regatta time [s] for favorable water flow velocities [m/s] 0 0.5 1 1.5 2 Favorable flow CDAIR CDH2O ρH2O TA TB TIN TOUT E Regatta time
0 0 0 0 0 18 15 13 13 12 4 3 3 3 3 88 78 69 65 58 12 11 9 9 8 1 1 1 0 0 -1 -1 -1 0 0 84 84 84 84 84 488.8 435.9 393.4 358.3 329.1 Fig 5. Comparison of the sensitivities of regatta time STA on oar active phase time TA vs. water current VVS and vs. wind velocity VLV.
6. CONCLUSIONS i) This model provides the opportunity to study the effect of different rowing technique elements and wind and water currents on the regatta outcome. ii) The mean angle range of oar rotation should be close to angle of 90°. Fig 3 Regatta time TREG vs. water current velocity VVS . iii) The time of active phase of oar work has the greatest influence on regatta result. iv) The opposite wind is more aggravated by the result of the regatta than by the wind at the same velocity which improves it. v) The water currents have a very large effect on the regatta result. The opposite current of water results in the worse regatta result than the water current at the same velocity which improves it. vi) The best regatta result was obtained for the rotation range of the oars located maximally in front of the boat. Fig 4. Sensitivity of regatta time STA on oar active phase time TA vs water current VVS.
4
Proceedings of the 9th MATHMOD Vienna, Austria, February 21-23, 2018
M. Wychowanski et al. / IFAC PapersOnLine 51-2 (2018) 879–883
883
Acknowledgement: Project "Mathematical modelling of scullling on a one-man sport boat in order to computer simulation, optimization and teaching movement techniques" financed by Ministry of Science and Higher Education Republic of Poland, within the program “Development of Academic Sport” contract: No.0030/RS4/2016/54 dated 03.31.2016.
REFERENCES Day, A. H., Campbell, I., Clelland, D., Cichowicz J. (2011) An study of unsteady hydrodynamics of a single scull. Proceedings of The Institution of Mechanical Engineers Part M-Journal Of Engineering For The Maritime Environment, Vol. 225 Issue: M3 pp. 282-294. Findlay, M. and Turno, S.R. (2010) Mechanics of a rowing stroke: surge speed variations of a single scull. Proceedings of the institution of mechanical engineers part p-Journal of Sports Engineering and Technology, Vol. 224 Issue: P1 pp: 89-100 Special Issue: SI Formaggia, L., Miglio, E., Mola, A., Montano, A. (2008) Fluid structure interaction problems in free surface flows: application to boat dynamics. International Journal for Numerical Methods in Fluids, 56, pp. 965-978. Formaggia, L., Miglio, E., Mola, A., Montano, A. (2009) A model for the dynamics of rowing boats. International Journal for Numerical Methods in Fluids, 61, pp. 119143 Kinoshita, T., Miyashita, M., Kobayashi, H., Hino, T. (2008) Rowing velocity prediction program with estimating hydrodynamic load acting on an oar blade. In: Kato N., Kamimura S. (eds) Bio-mechanisms of Swimming and Flying. Springer, Tokyo Millar, S.K., Oldham, A.R.H., Hume P.A. and Renshaw I. (2015). Using rowers’ perceptions of on-water stroke success to evaluate sculling catch efficiency variables via a boat. Instrumentation System Sports, 3(4), 335-345 Pulman, C. The Physics of rowing. http://eodg.atm.ox.ac.uk/user/dudhia/rowing/physics/rowing.p df (access 2017). Shampine, L. F. and Reichelt, M. W. (1997) The MATLAB ODE Suite. SIAM Journal on Scientific Computing, Vol. 18 Issue: 1 pp. 1-22. Sliasas, A. and Tullis, S. (2010) The dynamic flow behavior an oar blade in motion using a hydrodynamic-based shallvelocity-cuppled of the rowing stroke. Proceedings of The Institution of Mechanical Engineers Part p-Journal of Sport International Journal for Numerical Methods in Fluids Engineering and Technology Vol. 224, Issue: P1 Special Issue, pp. 9-24. Zatsiorsky, V.M. and Yakunin, N. (1991) Mechanics and biomechanics of rowing - a review. International Journal of Sport Biomechanics, Vol. 7, Issue: 3, pp. 229-281.
5
ScienceDirect
IFAC PapersOnLine 51-2 (2018) 879–883 Results of Single Sculling Technique Analysis Using 1D Mathematical Model Results of Single Sculling Technique Analysis Using 1D Mathematical Model Results Analysis Using 1D Mathematical Model Results of of Single Single Sculling Sculling Technique Technique Analysis Using 1D Mathematical Model * ** ***
M. Wychowanski*, G. Slugocki**, G. Orzechowski***, M. Wychowanski**,, G. Slugocki** Orzechowski *** ***** **,, G. ***,, M. Z. Staniak****, D. Radomski ***** M. Wychowanski Wychowanski , G. G. Slugocki Slugocki , G. G. Orzechowski Orzechowski , Z. Staniak****, D. Radomski ***** Z. Z. Staniak****, Staniak****, D. D. Radomski Radomski***** *Faculty of Rehabilitation, Joseph Pilsudski University of Physical Education
Rehabilitation, Joseph Pilsudski University of Physical Education in*Faculty Warsaw,of Poland (Tel:+48 600-331-303; e-mail: [email protected]), *Faculty of Rehabilitation, Joseph University of *Faculty of Rehabilitation, Joseph Pilsudski Pilsudski University of Physical Physical Education Education in Warsaw, Poland (Tel:+48 600-331-303; e-mail: [email protected]), **Faculty of Power and (Tel:+48 Aerospace600-331-303; Engineering e-mail: [email protected]), University of Technology, Poland in Warsaw, Poland in Warsaw, Poland e-mail: [email protected]), **Faculty of Power and (Tel:+48 Aerospace600-331-303; Engineering Warsaw University of Technology, Poland [email protected]), **Faculty of of Power Power and and Aerospace Aerospace(e-mail: Engineering Warsaw University University of of Technology, Technology, Poland Poland **Faculty Engineering Warsaw (e-mail: [email protected]), *** Department of Mechanical (e-mail: Engineering, Lappeenranta University of Technology, [email protected]), (e-mail: [email protected]), *** Department of Mechanical Engineering, Lappeenranta University of Technology, Finland (e-mail: [email protected]), *** Engineering, Lappeenranta *** Department Department of of Mechanical Mechanical Engineering, Lappeenranta University University of of Technology, Technology, Finland (e-mail: [email protected]), ****National Research Institute of Sport, [email protected]), Warsaw, Poland (e-mail: [email protected]), Finland (e-mail: Finland (e-mail: [email protected]), ****National*****Institute Research Institute of Sport, Warsaw, (e-mail: [email protected]), of Radioelectronics andPoland Multimedia Techniques, ****National Research of Poland (e-mail: [email protected]), ****National*****Institute Research Institute Institute of Sport, Sport, Warsaw, Warsaw, Poland (e-mail: [email protected]), of Radioelectronics and Multimedia Techniques, Warsaw*****Institute University of Technology, Poland and (e-mail: [email protected]) of Radioelectronics Multimedia Techniques, *****Institute of Radioelectronics and Multimedia Techniques, Warsaw University of Technology, Poland (e-mail: [email protected]) Warsaw Warsaw University University of of Technology, Technology, Poland Poland (e-mail: (e-mail: [email protected]) [email protected])
Abstract: Sports results in rowing depend on two the most important factors: the athlete physical features Abstract: Sports results in rowing depend on two theoptimisation most important factors:techniques the athlete are physical features and the motion’s techniques. Bothdepend assessment andthe of rowing possible only Abstract: Sports in on most factors: the physical features Abstract: Sports results results in rowing rowing depend on two two theoptimisation most important important factors:techniques the athlete athlete are physical features and the motion’s techniques. Both assessment and of rowing possible only whenthe onemotion’s disposestechniques. the reliableBoth mathematical model predicting theof regatta results, that is are the possible time to cover and assessment and optimisation rowing techniques only and the motion’s techniques. Both assessment and optimisation of rowing techniques are possible only when one disposes theAreliable mathematical model predicting the regatta results, that isisthe time to cover the assumed distance. single mathematical scull participating in predicting the 2000 meters distance’s regatta ourtime subject. The when one the reliable model the results, that to when one disposes disposes theA reliable mathematical model predicting the regatta regatta results, that is isisthe the time to cover cover the assumed distance. single scull participating in the 2000 meters distance’s regatta our subject. The purpose of this study is to create a simplified mathematical model to simulate the rowing boat dynamics. the assumed distance. A single scull participating in the 2000 meters distance’s regatta is our subject. The the assumed distance. single scull participating in the 2000model meterstodistance’s regatta is our subject. The purpose of this study isAisto create a simplified mathematical simulate the rowing boat dynamics. The boat-rower systemis treated here as a material point here. The oar has a prescribed angular motion vs. purpose of this study to create aa simplified mathematical model to simulate the rowing boat dynamics. purpose of this study is to create simplified mathematical model to simulate the rowing boat dynamics. The boat-rower system is treated here as away material point here. The oar developed has a prescribed angular motion vs. oarlock depending upon the time. The of hydrodynamic force on theangular oar’s motion blade was The boat-rower system is treated here as a material point here. The oar has a prescribed vs. The boat-rower system is treated here as away material point here. The oar developed has a prescribed motion vs. oarlock depending upon the motion, time. The of hydrodynamic force on theangular oar’s blade was modelleddepending here. Thenupon the boat with theof influence both of wind and water current, was described by oarlock the time. The way hydrodynamic force developed on the oar’s blade was oarlock depending upon the time. The way of hydrodynamic force developed on the oar’s blade was modelled here. Then the boat motion, with the influence both of wind and water current, was described by a single nonlinear ordinary differential equation (NODE).both Theof used simple modelcurrent, gives the possibilities of modelled here. Then the boat motion, with the influence wind and water was described by modelled here. Then the boat motion, with the influence both of wind and water current, was described by a single nonlinear ordinary differential equation (NODE). The used simple model gives of the possibilities of and nonlinear reliable simulation of the singleequation sculling(NODE). techniqueThe andused of forecasting the result rowing regattas. aafast single ordinary differential simple model gives the possibilities of single nonlinear ordinary differential equation (NODE). The simple model gives of therowing possibilities of fast and reliablesensitivity simulation of the single sculling technique andused of forecasting the result regattas. The and parametric coefficient wassculling definedtechnique here, as well the efficiency coefficient ofrowing sculling. fast reliable simulation of and of the regattas. fast reliablesensitivity simulationcoefficient of the the single single and the of forecasting forecasting the result result of ofof rowing regattas. The and parametric wassculling definedtechnique here, as well efficiency coefficient sculling. The parametric sensitivity coefficient was defined as the coefficient of The parametric sensitivity coefficient was defined here, here, as well well the efficiency efficiency coefficient of sculling. sculling. © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. reserved. Keywords: sculling, mathematical model, motion’s equation, hydrodynamic force onAll therights oar. Keywords: sculling, mathematical model, motion’s equation, hydrodynamic force on the oar. Keywords: Keywords: sculling, sculling, mathematical mathematical model, model, motion’s motion’s equation, equation, hydrodynamic hydrodynamic force force on on the the oar. oar.
1. INTRODUCTION 1. INTRODUCTION 1. INTRODUCTION 1. A INTRODUCTION single sculling is a rowing with two oars, one in one hand. A single scullinginvolves is a rowing with atwo oars, one in oneinhand. This discipline passing specific distance the A single sculling is with oars, in A single scullinginvolves is aa rowing rowing with atwo two oars, one one in one oneinhand. hand. This discipline passing specific distance the shortest possibleinvolves time and therefore the efficient sculling This discipline passing a specific distance in This discipline involves passing a specific distancesculling in the the shortest possible time and therefore the efficient techniquepossible is of a paramount importance. This paper attempts shortest time and therefore the efficient sculling shortest possible time and therefore the efficient sculling technique of mathematical a paramount importance. This paper attempts to developis the model of a rowing to be useful technique is of aa paramount This attempts technique isthe of mathematical paramount importance. importance. This paper paper attempts to develop model of a rowing to be useful fordevelop analysis,thelearning, improvement and motiontotechniques to mathematical model of a rowing be to develop thelearning, mathematical model ofand a rowing totechniques be useful useful for analysis, improvement motion optimization. Accordingimprovement to Zatsiorsky Yakunin (1991) for analysis, and motion techniques for analysis, learning, learning, improvement andand motion techniques optimization. According to Zatsiorsky and Yakunin (1991) "Despite relatively numerous studies, the of optimization. According to and Yakunin optimization. According to Zatsiorsky Zatsiorsky and biomechanics Yakunin (1991) (1991) "Despite relatively numerous studies, the biomechanics of rowing remains poorly understood". These authors were "Despite relatively numerous studies, the biomechanics of "Despite relativelypoorly numerous studies, the biomechanics of rowing remains understood". These authors were among the first who hadunderstood". attempted rowing mathematical rowing remains poorly These authors were rowing remains poorly understood". These authors were among the A first had attempted mathematical description. lot who of authors have dealt rowing with similar subjects, among first who had rowing mathematical among the the A first who had attempted attempted rowing mathematical description. lot of authors have dealt with similar subjects, Pulman, C. A (access 2014), among others, gives a thorough description. lot of authors have dealt with similar subjects, description. A lot of 2014), authorsamong have dealt withgives similar subjects, Pulman, C. (access others, a thorough description of(access the phenomena concerning different aspects of Pulman, C. 2014), among others, gives a thorough Pulman, C. of(access 2014), among others,different gives aaspects thorough description the phenomena concerning of a rowing. Some authors such as: Findlay, M. and Turno, S. R. description of the phenomena concerning different aspects of description of the phenomena concerning different aspects of a(2010), rowing. Some authors such as: Findlay, M. and Turno, S. R. Kinoshita et al. (2008), and MillarM. S.and K. et al. (2015) aa rowing. Some authors such as: Findlay, Turno, S. R. rowing. Some authors such as: Findlay, M. and Turno, S. R. (2010), Kinoshita et al. (2008), and Millar S. K. et al. (2015) have with et similar and problems, the (2010), Kinoshita al. Millar et (2010),dealt Kinoshita etthe al. (2008), (2008), and Millar S. S.aa K. K.specially et al. al. (2015) (2015) have dealt with the similar problems, specially the hydrodynamics phenomena related to the flow past the oar. have dealt with the similar problems, a specially the have dealt with the similar problems, a specially the hydrodynamics phenomena related to the flow past the oar. The main aim ofphenomena the present related work istotothe carry outpast the analysis hydrodynamics flow the oar. hydrodynamics phenomena related to the flow past the oar. Thea main of the present workWe is to carry out analysis of singleaim sculling technique. proposed thethe following The aim of work is carry the analysis Thea main main aim of the the present present workWe is to to carry out out thefollowing analysis of single sculling technique. proposed the criteria of technique assessment We for proposed this aim. These criteria of a single sculling technique. the following of a single sculling technique. We proposed the following criteria of technique assessment for this aim. These criteria were sculling efficiency and parametrical sensitivity regatta criteria of technique assessment for this aim. These criteria of technique assessment for this aim. These criteria criteria were sculling efficiency and parametrical sensitivity regatta result on chosen parameters. were sculling efficiency and parametrical sensitivity were sculling efficiency and parametrical sensitivity regatta regatta result on chosen parameters. result result on on chosen chosen parameters. parameters.
Fig. 1. Physical model of the sculler-boat-oars system. Fig. 1. Physical model of the sculler-boat-oars system. Fig. Fig. 1. 1. Physical Physical model model of of the the sculler-boat-oars sculler-boat-oars system. system. 2. MATHEMATICAL MODEL 2. MATHEMATICAL MODEL 2. MATHEMATICAL MODEL 2. MATHEMATICAL MODEL The mathematical model of a sculling technique has been The mathematical model of a sculling technique derived from the second Newton’s dynamics law andhas rulesbeen the The mathematical model of technique has been The mathematical modelNewton’s of aa sculling sculling technique has been derived from the second dynamics law and rules the describing hydrodynamic forces and the aerodynamic forces, derived from the second Newton’s dynamics law and rules the derived from the second Newton’s dynamics law and rules the describing hydrodynamic forces and the aerodynamic forces, for the physical model of forces rowingand shown in the Figure forces, 1. The describing hydrodynamic the aerodynamic describing hydrodynamic forces and the aerodynamic forces, for the emphasises physical model of rowing shown in generation the Figure 1. model the description of force on The the for the physical of in the The for the emphasises physical model model of rowing rowing shown shown in generation the Figure Figure 1. 1. The model the description of force onboat the oars phenomenon. Thedescription equation ofofmotion of one scull model emphasises the force generation on the model emphasises the force generation onboat the oars phenomenon. Thedescription equation ofof motion of one scull takes the wind and water current in consideration. oars phenomenon. The equation of motion of one scull boat oars The equation motion of one scull boat takesphenomenon. the wind and water current inofconsideration. takes takes the the wind wind and and water water current current in in consideration. consideration.
2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2018, 2018 IFAC 1 Hosting by Elsevier Ltd. All rights reserved. Peer review©under of International Federation of Automatic Copyright 2018 responsibility IFAC 1 Control. Copyright © 2018 IFAC 1 10.1016/j.ifacol.2018.04.025 Copyright © 2018 IFAC 1
Proceedings of the 9th MATHMOD 880 Vienna, Austria, February 21-23, 2018
M. Wychowanski et al. / IFAC PapersOnLine 51-2 (2018) 879–883
with the oar cycle period: 2.1 The equation of boat motion
T 0.5(T1 T2 )
The equation of motion of the sculler-boat-oars system is shown below:
and the oar angular velocity:
m x(t ) 2 F (t ) sin OA (t ) 0.5 H O SVAcD H O [ x(t ) xVS (t )]2 sgn[ x(t ) xVS (t )] 2
2
0.5 AIR S LV cD AIR[ x(t ) xLS (t )]2 sgn[ x(t ) xLS (t )]
(7)
k (t ) 2 (t iT) k 2 (t ) 2 (t 0.5T1 iT) OA (t ) ( OAMAX OAMIN) 1 sin sin T2 T1 T1 T2
(1)
(8)
where: TIN oar blade immersion time,
where: m mass,
TOUT oar blade emersion time,
x ( t ) boat position,
k1(t) and k2(t) are the switching functions.
xVS ( t ) water current velocity,
2.3 Definition of the Efficiency Coefficient We define it for one cycle i:
xLV (t ) wind velocity,
( i 1) T
F( t ) force developed by oar, Ei
AIR air density,
F(t ) sin
OA
( t )dt
iT
( i 1) T
0i n 1
F(t ) dt
H O water density,
(9)
iT
2
and for the entire motion period:
cDAIR aerodynamic drag coefficient,
nT
cDH O hydrodynamic drag coefficient, 2
E
SLV reference surface flown by air,
F(t ) sin
OA
( t )dt
0
(10)
nT
F(t ) dt
SVA reference surface wetted by water,
0
OA oar angle
where n is the total number of oar motion cycles.
l OA distance from oarlock on the oar trunk and blade, 2.4 Definition of the sensitivity result from the given parameter The regatta result TREG sensitivity coefficient on the given parameter pi is defined as follows: TREG T TREG p i (11) S T REG p i TREG p i p pi
b OA oar blade width as a function of l OA . 2.2 Derivation of oar force formula The force developed by oar has the following form: F (t ) 0.25cDOA H O [1 sgn OA (t )] (t ) sgn g (t ) 2
REG
(2)
lOAM AX
{lOAOA (t ) [ x(t ) xVS (t )] sin OA (t )} bOAdlOA 2
i
lOAM IN
where Δpi=0,01pi.
where the oar’s blade immersion-emersion function is as follows: (t)
t iT TIN
iT t iT TIN
(t) 1
iT TIN t iT 0.5 T1 TOUT
t (iT 0.5T1 TOUT ) (t) 1 TOUT
iT 0.5 T1 TOUT t iT 0.5 T1
(t) 0
iT 0.5 T1 t ( i 1) T
3. RESULTS
0 i n 1
(3)
The following values given below are the listed parameters such as: the system total mass of (the rower, the boat, the oars) m=100kg, the boat hydrodynamic drag coefficient cDH20=0.07 with the reference surface SVA=0.1075m2 and water density ρH2O=1000kg/m3, aerodynamic drag coefficient cDAIR=0.19 of the boat-rower system with the reference surface SLV=0.3231m2 and the air density ρAIR=1.225kg/m3, dimensionless drag coefficient of the oar cDOA=1.13, the distance between an oar lock and the oar blade tip lOAmax=2.04m, the distance between the oar lock and the oar blade root lOAmin=1.6m, the oar blade width bOA=0.23m, the total rowing cycle period is T=2s, the cycle active phase period TA=0.5T1=0.7s, the cycle period passive phase TB=0.5T2=1.3s. The system is modeled using MATLAB® package. The resulting equation of motion is solved with ode45 code (Shampine and Reichelt, 1997) to be based on the explicit Runge-Kutta formula of orders 4 and 5.
with an adjusting term: lOAM AX
(4) sgn g (t ) sgn{ {lOAOA (t ) [ x(t ) x VS (t )] sin OA (t )}bOAdlOA} lOAM IN
The oar angle vs. time: OA ( t ) 0.5( OAMAX OAMIN ) 0.5( OAMAX OAMIN ) 2 ( t 0.5T1 iT) 2 ( t iT) k 1 ( t ) cos k 2 ( t ) cos T1 T2
[k 1 ( t ) 1] [k 2 ( t ) 0]
[k 1 ( t ) 0] [k 2 ( t ) 1] 0i n iT 0.5 T t ( i 1) T iT t iT 0.5 T1
1
0i n
(5)
(6)
2
Proceedings of the 9th MATHMOD Vienna, Austria, February 21-23, 2018
M. Wychowanski et al. / IFAC PapersOnLine 51-2 (2018) 879–883
The equation is solved with relatively tight error tolerances of 10-10 for absolute and 10-8 for relative tolerances. The use of the default tolerance values results in wrong solution. In the beginning we have carried out the regatta simulation for three ranges oar rotation. The assumed ranges of the oar rotation γOA=45°÷135º, γOA=60°÷150º. were: γOA=30°÷120º, The calculation result in sensitivity parameters, rowing efficiency and regatta time for oar rotation all ranges are presented in the Table 1. The most efficient sculling technique was for γOA=45°÷135º range of the oar rotation. The best regatta result was observed for the γOA=60°÷150º. The sensitivity coefficients computation of regatta result in wind and water current were performed for the most efficient oar rotation range. Result of computing of efficiency, sensitivity coefficients and regatta results for different wind velocities are presented in the Table 2 and the Table 3 and the Figure 2. Efficiency computing, sensitivity coefficients results and regatta results for different water current velocities are presented in the Table 4 and the Table 5 and the Figure 3. The best time to cover the distance 2000m was observed for favorable water current velocity of 2m/s. The worst regatta time was observed for opposite water current of 2m/s. The efficiency coefficient for this oar rotation range did not change. The greatest influence on the regatta result has the time of active phase of rowing as shown on the Figure 4. The sensitivity of regatta results on active phase time vs. the wind velocity is constant as shows the Figure 5.
881
Table 3. Regatta result sensitivity on parameter [%], E - efficiency of rowing [%] and regatta time [s] for favorable wind velocities [m/s] Favorable wind CDAIR CDH2O ρH2O TA TB TIN TOUT E Regatta time
Table 1. Regatta result sensitivity on parameter [%], E - efficiency of rowing [%] and regatta time [s] for ranges of oars angles [deg] Range of oars 30÷120 45÷135 60÷150 CDAIR 0 0 0 CDH2O 16 18 16 ρH2O 5 4 2 TA 86 88 85 TB 11 12 11 TIN 1 1 0 TOUT -3 -1 1 E 69 84 83 Regatta time 491.5 488.8 471.6
0 0 18 4 88 12 1 -1 84 488.8
0.5 0 19 4 88 12 1 -1 84 488.6
1 0 18 4 87 12 1 -1 84 488.4
2.5 0 17 4 87 13 1 -1 84 488.1
5 10 0 0 18 17 4 4 87 86 12 13 1 1 -1 -1 84 84 487.9 486.3
Fig 2. Regatta time TREG vs. wind velocity VLV.
Table 4. Regatta result sensitivity on parameter [%], E - efficiency of rowing [%] and regatta time [s] for opposite water flow velocities [m/s]
Table 2. Regatta result sensitivity on parameter [%], E - efficiency of rowing [%] and regatta time [s] for opposite wind velocities [m/s] Opposite wind -10 -5 -2.5 -1 -0.5 CDAIR 2 1 0 0 0 CDH2O 15 17 17 16 17 ρH2O 2 3 4 4 4 TA 88 87 87 87 88 TB 12 13 13 11 11 TIN 1 1 1 1 1 TOUT 0 -1 -1 -1 -1 E 85 84 84 84 84 Regatta time 497.7 492.0 490.1 489.3 489.0 3
Opposite flow
-2
-1.5
-1
-0.5
CDAIR
0
0
0
0
CDH2O
33
27
23
21
ρH2O
7
6
5
5
TA
171
138
115
100
TB
23
18
15
15
TIN
1
1
1
1
TOUT
-1
-1
-1
-1
E
84
84
84
84
Regatta time
951.6
769.4
645.8
556.4
Proceedings of the 9th MATHMOD 882 Vienna, Austria, February 21-23, 2018
M. Wychowanski et al. / IFAC PapersOnLine 51-2 (2018) 879–883
Table 5. Regatta result sensitivity on parameter [%], E - efficiency of rowing [%] and regatta time [s] for favorable water flow velocities [m/s] 0 0.5 1 1.5 2 Favorable flow CDAIR CDH2O ρH2O TA TB TIN TOUT E Regatta time
0 0 0 0 0 18 15 13 13 12 4 3 3 3 3 88 78 69 65 58 12 11 9 9 8 1 1 1 0 0 -1 -1 -1 0 0 84 84 84 84 84 488.8 435.9 393.4 358.3 329.1 Fig 5. Comparison of the sensitivities of regatta time STA on oar active phase time TA vs. water current VVS and vs. wind velocity VLV.
6. CONCLUSIONS i) This model provides the opportunity to study the effect of different rowing technique elements and wind and water currents on the regatta outcome. ii) The mean angle range of oar rotation should be close to angle of 90°. Fig 3 Regatta time TREG vs. water current velocity VVS . iii) The time of active phase of oar work has the greatest influence on regatta result. iv) The opposite wind is more aggravated by the result of the regatta than by the wind at the same velocity which improves it. v) The water currents have a very large effect on the regatta result. The opposite current of water results in the worse regatta result than the water current at the same velocity which improves it. vi) The best regatta result was obtained for the rotation range of the oars located maximally in front of the boat. Fig 4. Sensitivity of regatta time STA on oar active phase time TA vs water current VVS.
4
Proceedings of the 9th MATHMOD Vienna, Austria, February 21-23, 2018
M. Wychowanski et al. / IFAC PapersOnLine 51-2 (2018) 879–883
883
Acknowledgement: Project "Mathematical modelling of scullling on a one-man sport boat in order to computer simulation, optimization and teaching movement techniques" financed by Ministry of Science and Higher Education Republic of Poland, within the program “Development of Academic Sport” contract: No.0030/RS4/2016/54 dated 03.31.2016.
REFERENCES Day, A. H., Campbell, I., Clelland, D., Cichowicz J. (2011) An study of unsteady hydrodynamics of a single scull. Proceedings of The Institution of Mechanical Engineers Part M-Journal Of Engineering For The Maritime Environment, Vol. 225 Issue: M3 pp. 282-294. Findlay, M. and Turno, S.R. (2010) Mechanics of a rowing stroke: surge speed variations of a single scull. Proceedings of the institution of mechanical engineers part p-Journal of Sports Engineering and Technology, Vol. 224 Issue: P1 pp: 89-100 Special Issue: SI Formaggia, L., Miglio, E., Mola, A., Montano, A. (2008) Fluid structure interaction problems in free surface flows: application to boat dynamics. International Journal for Numerical Methods in Fluids, 56, pp. 965-978. Formaggia, L., Miglio, E., Mola, A., Montano, A. (2009) A model for the dynamics of rowing boats. International Journal for Numerical Methods in Fluids, 61, pp. 119143 Kinoshita, T., Miyashita, M., Kobayashi, H., Hino, T. (2008) Rowing velocity prediction program with estimating hydrodynamic load acting on an oar blade. In: Kato N., Kamimura S. (eds) Bio-mechanisms of Swimming and Flying. Springer, Tokyo Millar, S.K., Oldham, A.R.H., Hume P.A. and Renshaw I. (2015). Using rowers’ perceptions of on-water stroke success to evaluate sculling catch efficiency variables via a boat. Instrumentation System Sports, 3(4), 335-345 Pulman, C. The Physics of rowing. http://eodg.atm.ox.ac.uk/user/dudhia/rowing/physics/rowing.p df (access 2017). Shampine, L. F. and Reichelt, M. W. (1997) The MATLAB ODE Suite. SIAM Journal on Scientific Computing, Vol. 18 Issue: 1 pp. 1-22. Sliasas, A. and Tullis, S. (2010) The dynamic flow behavior an oar blade in motion using a hydrodynamic-based shallvelocity-cuppled of the rowing stroke. Proceedings of The Institution of Mechanical Engineers Part p-Journal of Sport International Journal for Numerical Methods in Fluids Engineering and Technology Vol. 224, Issue: P1 Special Issue, pp. 9-24. Zatsiorsky, V.M. and Yakunin, N. (1991) Mechanics and biomechanics of rowing - a review. International Journal of Sport Biomechanics, Vol. 7, Issue: 3, pp. 229-281.
5