# Multi-physics modelling and sensitivity analysis of olympic rowing boat dynamics

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## Abstract

A multidisciplinary approach is implemented to model and analyse the performance of Olympic rowing boats. A reduced-order model that couples rowers motions with the hull, oars and hydrodynamic and hydrostatic forces is detailed. This model is complemented with a sensitivity analysis carried out by means of a non-intrusive polynomial chaos expansion. Sensitivity analysis results for two different boat classes, namely, a lightweight single scull and a coxless four are presented and discussed. This analysis contrasts, for both classes, the effects of varying forces exerted by the rowers, weights of rowers and cadence of motions on the boat performance.

## Keywords

Rowing Sensitivity analysis Water forces Rigid body motions Rowers biomechanics## List of symbols

*A*Wave amplitude

*a*_{kX}Coefficients of the polynomial approximation of the active phase of the longitudinal oarlock force;

*k*= 0, 1, 2, 3*a*_{kZ}Coefficients of the polynomial approximation of the active phase of the vertical oarlock force;

*k*= 0, 1, 2, 3*b*_{kX}Coefficients of the polynomial approximation of the passive phase of the longitudinal oarlock force;

*k*= 0, 1, 2*b*_{kZ}Coefficients of the polynomial approximation of the passive phase of the vertical oarlock force;

*k*= 0, 1, 2*C*_{dw}Hull wave drag coefficient

*C*_{v}Hull viscous drag coefficient

*F*_{Xmax}Maximum value of the longitudinal oarlock force

*F*_{Xmin}Minimum value of the longitudinal oarlock force

*F*_{Zmax}Maximum value of the vertical oarlock force

*F*_{Zmin}Minimum value of the vertical oarlock force

- \(\user2{F}_{{\rm ol}_j}\)
Left oarlock force of

*j*th rower- \(\user2{F}_{{\rm or}_j}\)
Right oarlock force of

*j*th rower- \({\bf F}^{w}\)
Water force on the hull

- \({\bf G}^{h}\)
Coordinates of the hull centre of gravity in the global reference frame

- \(\dot{\bf G}_{X}^{h}\)
Surge velocity

- \(\dot{\bf G}^{h}\)
Velocity of the hull centre of gravity in the global reference frame

*H*_{p}Hermite polynomial of order

*p**i*Index running on body parts of each rower in the equations of motion

*I*_{G}^{h}Inertia tensor of the boat centered at the hull centre of gravity

*j*Index running on the number of each rower in the equations of motion

*L*Oar length

*L*_{b}Boat length

*m*_{ij}Mass of the

*i*th body part of*j*th rower- \(m_{r_{j}}\)
Mass of the

*j*th rower*M*_{Tot}Total mass of rowers and hull

- \({\bf M}^{w}\)
Water moment on the hull

*n*Number of random parameters

*N*Number of samples used for polynomial chaos expansion analysis

*n*_{r}Number of rowers on the boat

*P*Number of random modes considered in the truncated expansion

*p*Maximum degree of polynomial bases considered in the expansion

*p*_{d}Dynamic pressure due to secondary motions

*q*Number of body parts of the rowers

- \(\mathcal{R}\)
Rotation matrix, obtained from the Euler angles values

*R*Total resistance due to the mean motion

*r*Rowing cadence, i.e., number of strokes per minute

*r*_{h}Distance between each rower’s hand and the oarlock

- \(r_{gi}\)
Boat hull radii of gyration

*T*Duration of the stroke

*T*_{2}−*T*_{1}Duration of 15 strokes

*t*Generic deterministic time variable

- \(\eta\)
Energy ratio

*V*_{X}Mean surge velocity

- \(\user2{x}\)
Generic deterministic space variable vector

- \(\user2{x}_{ij}\)
Position of the

*i*th body part of*j*th rower in the boat reference frame- \(\alpha^{*}\)
Generic variable representing stochastic boat motions

- \(\mathcal{M}^{s}\)
Radiative potential added mass matrix

- \(\mathcal{S}^{s}\)
Radiative potential damping mass matrix

- \(\varvec{\omega}\)
Angular velocity of the boat with respect to the global reference frame axes

- \(\user2{X}_{{\rm ol}_j}\)
Position of the left oarlock of

*j*-th rower in the global reference frame- \(\user2{X}_{{\rm or}_j}\)
Position of the right oarlock of

*j*th rower in the global reference frame- \(\user2{X}_{{\rm hl}_j}\)
Position of the left hand of

*j*th rower in the global reference frame- \(\user2{X}_{{\rm hr}_j}\)
Position of the right hand of

*j*th rower in the global reference frame- \(\Upgamma^{b}\)
Boat surface area

- \(\Upgamma^{b}_{X}\)
Projection of \(\Upgamma^{b}\) on the plane with a normal vector in the

*X*-axis- \(\Upgamma^{b}_{Z}\)
Projection of \(\Upgamma^{b}\) on the plane with a normal vector in the

*Z*-axis- \(\rho_{f}\)
Fluid density

- \(\lambda\)
Wave length

- \(\mu_{\alpha^{*}}\)
Vector of the mean values of each of the random parameters

- \(\mathcal N\)
Generalized normal vector

- \(\phi\)
Roll Euler angle

- \(\Upphi_{r}\)
Total radiative potential due to secondary motions

- \(\Upphi_{s}\)
Radiative potential due to a small harmonic motions onto the

*s*th degree of freedom of the boat hull- \(\psi\)
Yaw Euler angle

- \(\Uppsi_{i}\)
*i*th random basis function- \(\rho\)
Probability density function

- \(\sigma_{\alpha^{*}}\)
Vector of the standard deviations values of each of the random parameters considered in the dynamical system

- \(\Upgamma_{0}^{h}\)
Hull hydrostatic wet surface

- \(\tau_{a}\)
Time length of the stroke active phase

- \(\theta\)
Pitch Euler angle

- \({\bf \xi}\)
*n*-Dimensional random variable representing the varying parameters in the stochastic problem

## Notes

### Acknowledgments

The authors would like to thank Alessandro Placido of Filippi Lido s.r.l., maker of competition rowing boats, for providing data of the boats geometry. They would also like to thank Laura Milani, lightweight single scull vice World Champion, who contributed to the validation of the rowing boat model in the framework of her Bachelor Degree final project at Politecnico di Milano. This validation helped in setting up simulations using realistic values of world class athletes.

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