A Model for the Simulation of Rowing Boats Dynamics in Race Conditions Andrea Mola Edie Miglio Claudia Giannini Antonio DeSimone 9/12/2013
Physics of Rowing Rowing is a sport with a long tradition in which races can be decided by the smallest particular: Boats are very narrow and light, their shape is designed to reduce drag to a minimum Oars are connected to the boat at oarlocks mounted on outriggers extending from the boat Rowers sit facing backwards and pull oars towards them to generate thrust. To exploit the leg power and extend the length of each stroke, their seat slides on rails
Motivations The motion of the rowers on the boat induces a set of secondary motions superimposing to the mean surge motion of the hull, so: a hull shape optimized for steady state motion might not be optimal in presence of secondary motions the rower with highest performance on a rowing machine, might not be the best in race conditions For this reason, in the last years both CONI, the Italian Olympic committee, and Filippi Lido s.r.l., Italian manifacturer of rowing boat hulls, funded research programs with Politecnico di Milano.
An Interdisciplinary Problem In the study of rowing boat dynamics, instruments from several fields are needed: Rigid body dynamics Fluid dynamics Musculoskeletal mechanics Numerical analysis Models that account for the coupled complex dynamics are needed for the prediction of rowing boats performance
A single scull Oar Blade Bow Outrigger Foot Stretcher Sliding Seat Oarlock Stern
Rowing Boat Dynamical Model: Assumptions The rowing boat is a rigid body with a given geometry and hull shape. It also has a given mass M and matrix of inertia I G in principal axes Each rower s body is decomposed into a set of 12 body parts, which are assumed to be point wise masses m ij moving with prescribed motion laws x ij in the boat non inertial reference frame The oars are assumed rigid and massless, hence the oarlock forces F oj and hand forces F hj are proportional, with scaling coefficients depending on the boat geometry The aim is to compute for any time t the hull center of mass position (GX h (t), G Y h (t), G Z h (t)) and its orientation (ψ(t), θ(t), φ(t))
Rigid Body Dynamics: Governing Equations Introducing the values of oars, hands, seat and footboard forces obtained from the rowers and oar models yields M Tot Gh + ω m ij R T x ij = i,j r h (F olj + F orj ) m ij R T ẍ ij L j i,j m ij ω ω R T x ij m ij 2ω R T ẋ ij + M Tot g + F w i,j i,j i,j m ij R T x ij G h + RI G R 1 ω + m ij R T x ij ω R T x ij = ω RI G R 1 ω i,j m ij R T x ij R T ẍ ij m ij R T x ij ω ω R T x ij i,j i,j m ij R T x ij 2ω R T ẋ ij + R T x ij m ij g + M w i,j i,j + + [ ( X olj j [ ( X orj j G h) L r h L G h) L r h L ( X hlj G h)] F olj ( X hrj G h)] F orj to close the problem the water forces F w and moments M w have to be modeled, as well as the rowers parts motion laws x ij and the oarlock forces F olj and F orj
Musculoskeletal Dynamics: Motion Capture Data The rowers kinematics has been reconstructed starting from experimental data coming from the motion capture of actual athletes Light reflecting markers are placed on the rowers, their is and position tracked by means of synchronized digital cameras
Musculoskeletal Dynamics: Oarlock Forces The oarlock force F oj (t) can be measured by means of sensors placed in the oarlock F X [N] 1200 1000 800 600 400 Oarlock Force F Xmax Active Phase Passive Phase The horizontal component of the oarlock force, F x, is modeled by means of the polynomial equations 200 F x=c 3X t 3 +c 2X t 2 +c 1X t+c 0X for 0 < t < τ a 0 200 F Xmin 400 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 t [s] F x=k 2X t 2 + k 1X t + k 0X for τ a < t < T where τ a is the time length of stroke active phase and coefficients c ix, k ix are chosen so as to fit experimental data
The Rowers Proportional Control The system is unstable on the roll degree of freedom and indifferently stable on the yaw degree of freedom: we modeled in the following way the control exerted by the rowers to stabilize the system φ Foz Foz MTotg Foz F oz = k Roll φ, Foz F w r F w Fox Fox F max ox = k Yaw ψ 0 ψ F w r Fox Fox
Reduced Fluid Dynamic Model: Force Splitting The hydrostatic and hydrodynamic forces and moments are decomposed in the following way F w = F h + F m + F s M w = M h + M m + M s F h and M h are the hydrostatic force and moment respectively F m = 1 2 ρs ref C dx (G h X )2 e X is the drag due to mean motion F s and M s are the forces and moments due to the secondary motions of the hull F h, M h and S ref are computed run time by means of a surface mesh displaced in the instantaneous boat position
Reduced Fluid Dynamic Model: Radiation Problem The secondary motion hydrodynamic forces F s and M s are computed by solving six elliptic partial differential problems for the complex potentials given by ψ α (t, X) = Re (Ψ α (X)e iωt ) Ψ α = 0 on Ω Γ h Γ fs Ψ α ω2 z g ψα = 0 on Γw Ψ α n i ω 2 g ψα = 0 on Γ Ψ α = 0 on Γ bot n Ψ α = N α on Γ b n α = 1,..., 6 Γ b Γ where N = [n x, n y, n z, yn x xn y, xn z + zn x, yn z + zn y ] T is the generalized normal vector
Reduced Model: Added Mass and Damping Matrices It turns out that the pressure integral on the boat surface gives rise to two terms the damping matrix S αβ = ρω Im(ψ α )N β dγ Γ c the added mass matrix M αβ = ρ Re(ψ α )N β dγ Γ c that are computed offline and introduced in the system for the boat dynamics to close the problem. The reduced model has been implemented in a stand alone C++ software and is currently employed in the boat design process
Very first results Here is an old comparison between (very rare) experimental data and numerical predictions of horizontal velocity for a single scull pushed by a woman rowing at 20 strokes per minute. 5 Horizontal velocity 4 3 V x [m/s] 2 1 0-1 10 11 12 13 14 15 16 17 18 19 20 The numerical model seemed to be able to correctly reproduce the physics of a stroke, although some differences were present, in particular during the catch phase. t [s]
Model uncertainties Several parameters and inputs had to be completely guessed in this test: The rower is female, but her weight and height had to be guessed The weight distribution among body parts was not specific to the athlete considered The motion of the body parts was also not specific to the rower considered The hand force exerted by the rower was not known, and had to be guessed More experimental results were needed to better validate the numerical model!
CONI Experimental Campaign Thus, the model seemed somehow suitable for the evaluation of different hulls, as it was able reproduce the race conditions under which a hull is utilized. Yet, the Italian Olympic Committee (CONI) shown interest in the software for rowing boat dynamics simulation developed. In fact, the model can in principle address several trainers questions like: Is a strong and heavy rower better then a weaker but lighter one? What is the trade off between of stroke frequency and strength? Given a set of rowers, what is the best crew disposition? So, CONI carried out an experimental campaign to gather data for validating the software and increase its accuracy.
Insean CNR Experimental Setup The most interesting experiment has been carried out by CONI in the towing tank facility of Insean CNR, Rome, on a female double scull. A set of light reflecting markers were placed on the body joints of each rower and on the boat shell; Several HD digital cameras were installed around the basin (for about 20 m length) to capture the rowers motions; A 6 dofs accelerometer was placed on the shell, and force sensors were mounted on the oarlocks.
Validation: Longitudinal Motion The software was fed with the measured oarlock forces and body parts motions. The body part weights specific to the rowers were also employed. 120 Spazio percorso Computed Experimental 6 Velocity Computed Experimental 100 5 80 4 Distance (m) 60 Velocity (m/s) 3 40 2 20 1 0 0 5 10 15 20 25 30 0 0 5 10 15 20 25 30 Time (s) Time (s) The predicted longitudinal displacement and velocity curves are in close agreement with the experimental ones.
Validation: Sink and Pitch The vertical and angular positions were obtained through markers located on bow and stern. Thus, they are available for only 4 seconds. 0.1 Sinking Computed Experimental 0.01 Beccheggio Computed Experimental 0.08 0.005 0 0.06 Sinking (m) Angle (rad) -0.005 0.04-0.01 0.02-0.015 0 0 5 10 15 20 25 30 Time (s) -0.02 0 5 10 15 20 25 30 Time (s) Both sink and trim frequencies seem correct. Yet, the amplitudes especially the sink motion one are not recovered accurately.
Validation: Roll and Yaw Yaw and control experimental data were used to tune the gain coefficients of the rowers active control model. 0.015 Rollio Computed Experimental 0.006 Yaw Computed Experimental 0.01 0.004 0.005 0.002 Angle (rad) 0 Angle (rad) 0-0.005-0.002-0.01-0.004-0.015 0 5 10 15 20 25 30-0.006 0 5 10 15 20 25 30 Time (s) Time (s) The super-harmonic behavior of roll motion and sub-harmonic behavior of yaw motion are both correctly reproduced. The quantitative result still needs to be improved.
Conclusions The validation campaign confirmed that the model developed can be effectively used both in the hull design process and for training design purposes. Further improvements of the model can be obtained through work in the following areas: The oars model has been overlooked so far. A suitable study is needed for more accurate results; The rowers body is split into point wise masses. Introducing the tensor of inertia of each body part would improve the software predictions; Identifying a suitable high fidelity fluid dynamic model and validating its results through the experimental data is mandatory for evaluating effects in small changes of hull shape.
Further Work In the last years, a model for ship hydrodynamics based on potential flow theory has been developed at SISSA, in the framework of the Open- SHIP project, funded by Regione Friuli Venezia Giulia. In the next year, this model will be interfaced with the rowing boat dynamical model developed so far.