Water walking as a new mode of free surface skipping

Deformable elastomeric spheres are evaluated experimentally as they skip multiple times over a lake surface. Some spheres are embedded with small inertial measurement units to measure the acceleration experienced during water surface impact. A model for multiple impact events shows good agreement between measured acceleration, number of skipping events and distanced traveled. The experiment reveals a new mode of skipping, “water walking”, which is observed for relatively soft spheres impacting at low impact angles. The mode occurs when the sphere gains significant angular velocity over the first several impacts, causing the sphere to maintain a deformed, oblong shape. The behavior is characterized by the sphere moving nearly parallel to the water surface with the major axis tips dipping below the water surface with each rotation while the shorter sides pass just above, giving the impression that the sphere is walking across the water surface.


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A rigid rotation Q through an angle ψ about the e 3 axis (this does not change the shape of the solid).
3. A volume preserving stretch V that deforms the sphere into an ellipsoid. By symmetry, one of the principal directions of stretch must be parallel to the e 3 direction. The other two principal stretch directions are parallel to unit vectors m 1 , m 2 , which lie in the vertical plane (see Fig. 5 and Fig. A1). We let α be the angle between the m 1 and e 1 directions (positive α corresponds to rotation of the principal stretch directions about the m 3 axis); thus α describes the attack angle of the ellipsoid. We introduce the principal stretches λ 1 , λ 2 and λ 3 = 1/λ 1 λ 2 . The coordinates of a material particle in the sphere before deformation are given by x, and y defines the coordinates after deformation.
The following identities are useful for further calculations. Elementary trigonometry shows that the unit vectors e i and m i are related by where P is a proper orthogonal tensor whose components in both the basis {e 1 , e 2 , e 3 } and {m 1 , m 2 , m 3 } can be expressed as a matrix Since e i are independent of time, derivatives of the vectors m i can be calculated aṡ The sphere undergoes a volume preserving stretch V that deforms the sphere into an ellipsoid. The principal stretches are defined as λ 1 , λ 2 and λ 3 = 1/λ 1 λ 2 and we let α be the angle between the m 1 and e 1 directions, positive α corresponds to rotation of the principal stretch directions about the m 3 axis.
Since P is orthogonal it follows that Substituting this result into equation 3 and making use of equation 1 yieldṡ Evaluating this expression shows thaṫ The standard definition of principal stretches implies that the volume preserving stretch V can be expressed as where a ⊗ b denotes the tensor product of two vectors; i.e the operator with the property that [a ⊗ b] · c = (b · c) a for all vectors c. Taking the time derivative of this expression and using equation 6 then shows thaṫ The following identities are useful for further calculations: The coordinates d 1 , d 2 and ψ describe the rigid body motion and λ 1 , λ 2 and α describe the deformation of the sphere. Our goal is to calculate equations of motion for these generalized coordinates. With this description we can write the deformation mapping as The velocity and acceleration fields follow as We introduce a virtual velocity field where the kinematic variables are associated with virtual rates of change δḋ 1 , δḋ 2 , δψ, δλ 1 , δλ 2 , δα about the current (deformed) state. The governing equations for d 1 , d 2 , ψ, λ 1 , λ 2 and α are obtained from the principle of virtual work (i.e., weak form of the momentum conservation equation, 1 ).
where b i represents body forces, t i are traction forces (i.e., applied to the sphere boundary) and V and A denote integration over the volume and surface of the deformed solid, respectively. The stretch rate in the solid is given by The first term in equation 16 is the virtual rate of change of strain energy in the sphere, which can be calculated directly as To evaluate the remaining terms, the following identities are useful where x i denote the coordinates of a material particle with respect to the center of the sphere and the integrals are evaluated over the undeformed sphere. Thus, where we have made use of incompressibility to convert the integral over the volume of the deformed sphere (V ) to an integral over the volume of the undeformed sphere (V 0 ). Also, we have noted that b i = −ρ s gδ i2 , have substituted equation 14 for δ v i and have made use of the first two integrals in equation 19. Using equations 13-14 and again imposing incompressibility, the inertia term can be expressed as Expanding the terms on the right hand side yields The nonzero terms can be interpreted physically as rates of change of translational and vibrational kinetic energies. Finally, consider the term involving the external traction, which represents the pressure applied by the fluid on the elastomer surface,

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It is preferable to express the integrand in terms of spatial coordinates. Note that from equation 11 we have x = Here, F represents the resultant hydrodynamic force acting on the solid, the second term on the right hand side represents the virtual power associated with a force dipole tending to distort the elastomer and the third term on the right hand side represents a generalized moment (tending to make the body rotate).
Equation 16 can now be expressed in terms of the generalized coordinates. Substituting the expressions of equations 7-10 into equations 18,20-22 and 24, then setting each of δḋ 1 , δḋ 2 , δψ, δλ 1 , δλ 2 , δα to be nonzero in turn will yield a set of coupled second order nonlinear ODEs. Working through this procedure yields the following governing equations: where F h and F v are the horizontal and vertical force components in {e 1 , e 2 , e 3 } coordinates, respectively; t i and y i are the components of the traction vector and position vector on the surface of the ellipsoid in {m 1 , m 2 , m 3 } coordinates, respectively. Note that the translational degrees-of-freedom (DOF) decouple; they are coupled to vibration through the fluid.

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A.2. The water walking mode From our experimental observations, the water walking mode appears to consist of a rigid body rotation with the sphere holding steady, deformed principal stretches; we also allow for the possibility that the principal axes rotate with steady rotation rateα. Therefore, we look for a steady-state solution in which the sphere is undergoing a rigid body rotation with constantψ andα, with all other time derivatives and second time derivatives vanishing and all tractions vanishing. In this case, equations 25-28 are satisfied trivially and equations 29-30 reduce to Adding these equations results in which requires (λ 1 λ 2 ) 3 ≥ 1 or λ 1 ≥ 1/λ 2 for real solutions. Equation 32 thus describes a sphere in steady-state translation and rotation that holds a constant deformed shape described by λ 1 λ 2 . Equation 32 can be rearranged to

A.3. Tracking the lowest point on the sphere with deformation model
It was shown in Belden et al. 2016 that the governing equations 25-30 permit a different steady state mode than that described above in whichψ = 0, but in which the principal axes {m 1 , m 2 , m 3 } rotate with constant angular rateα = ± 5G 2ρ s R 2 , which is proportional to the term in the denominator of equation 33. They also showed experimentally thatα ∝ G ρ s R 2 for single impact events. Furthermore, we note that in the steady-state rotation case derived in section A.2, with constant λ 1 λ 2 , equations 29-30 actually permit a steady state value ofα, in the case that all other derivatives and tractions vanish. Thus, the termψ/ 5G ρ s R 2 in equation 33 can be thought of as the ratio of the steady state rigid body rotation rate to the steady state angular rotation rate of the principal axes.
It seems likely that what is actually observed in the field experiments is the sphere in a state with both nearly steady state and non-zeroψ andα. We also observe that the sphere tends to be oblong with λ 1 > λ 2 , but with each nearly steady. To analyze the sphere in this state, we consider the deformation for different values ofψ/ 5G ρ s R 2 . Consider tracking a single point on the sphere perimeter x, and that the sphere experiences a rigid rotation Q and a volume preserving stretch V (see Fig. 1). The final position of the point x, in the {e 1 , e 2 , e 3 } coordinate frame, is defined by For a point in the e 1 − e 2 plane, x is defined as

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Following rotation and deformation, the position of this point relative to the COM is Letting sin () = s () and cos () = c () , equation 36 becomes To find the lowest point on the sphere, we seek the of γ that minimizes the e 2 component of y − d. Thus we find a solution for the expression The derivative of equation 38 is We then solve for γ to get

Appendix B. Maximum skipping distance
Finally, we look for any correlation between number of skips N and distance traveled d in figure 2(a). There seems to be little correlation, which is supported by qualitative observations. For example, the largest value for distance recorded was d = 164 m; though this sphere was difficult to see when it landed so far away, and may have skipped an extra time prior to entry, the ball clearly skipped less than 10 times, which is at the low end of N. On the other hand, the event with the greatest number of skips N = 124, traveled just under 90 m before entering the water. Even when only considering the traditional skipping events in the figure 2(b), there is not an observable trend between N and d. However, one clear conclusion from figure 2(a) is that when striving to maximize number of skips N, one should strive for a water walking type 2 skipping event.