# A Numerical Study for the Effect of Ski Vibration on Friction

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## Abstract

In ski jump, minimizing the energy loss when sliding down the track is crucial for improving jump performance. In particular, it is essential to reduce the friction between the ice track and the jumping ski, which is typically achieved by properly waxing the ski base according to weather and ice conditions. Additionally, it might be possible to reduce the friction by controlling the vibration of jumping skis as many experiments and numerical analyses have reported that vibration can reduce the friction in both dry and wet conditions. However, they have been done mostly for small, rigid specimens and therefore the results may not be directly applicable to jumping skis because they have unique vibrational modes due to their slenderness and structural flexibilities. Here, we investigate a potential effect of ski vibration on friction using finite element analysis. Finite element beam models for jumping skis are constructed by measuring the geometry and bending stiffness experimentally. We employ a pressure-dependent friction model between the ski base and the ice track derived from the reported experimental data. Various vibration conditions are tested for four jumping ski models with different bending stiffness profiles. Results demonstrate the possibility of friction reduction by designing the bending stiffness profile of a jumping ski that controls its vibrational modes.

## Keywords

Vibration Friction model Friction reduction Ski jump Finite element method## Introduction

Ski jump is a dynamic winter sports game involving various interactions among athletes, skis, and environment (ice slope and air) during four different phases including in-run, takeoff, flight and landing. Most studies [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11] related to ski jump have been focused on aerodynamic analysis of an athlete with jumping ski during flight to investigate an optimal posture maximizing the lift-to-drag ratio for flying longer. This is mainly due to the fact that the flying speed is almost 100 km/h and the aerodynamic resistance must be effectively overcome to improve the performance. However, little attention has been paid to the in-run (sliding) state where the friction between the track and the ski base in addition to aerodynamic drag force must be minimized in order to gain the take-off speed as high as possible.

Most common approaches for the friction reduction during in-run are to apply wax layers and groove patterns on the ski base taking the weather and icy track conditions into consideration. It might be possible, in addition to these approaches, to reduce the frictional force by controlling the vibration that jumping skis experience while sliding down the track. Friction reduction due to vibration has been reported in many papers [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24]. But they have been limited to small and rigid specimens where the externally applied vibration force is transmitted to contact surfaces as it is. However, for slender structures like jumping skis, their inherent vibration modes need to be considered as they vary the location and condition of contact. In this study, we investigate the effect of ski vibration on friction numerically using the finite element method. The geometric and structural properties of commercial jumping skis are measured experimentally and corresponding finite element beam models are constructed. We employ a pressure-dependent friction model on ice derived from the reported experimental data. The degree of friction reduction is calculated by performing transient dynamic analysis for jumping skis under various vibration conditions. Results are expected to provide an important insight into the interaction between the ski and the ice track which might offer an additional way of reducing friction by controlling the dynamic properties of jumping skis.

## Method

In this section, we present our finite element model for a jumping ski, the friction model used to describe the interaction between ski plate and icy running slope, and analysis procedure to investigate the effect of ski vibration on friction.

### Finite Element Model for Jumping Ski

Geometric data of the jumping ski used to generate the finite element model

Ski parts | A | B | C | D | E | F | G | H | I | J | K | L |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Length (mm) | 230 | 230 | 230 | 230 | 172.5 | 172.5 | 172.5 | 172.5 | 140 | 140 | 140 | 140 |

Average width (mm) | 111.2 | 109.2 | 107.3 | 105.4 | 105.4 | 107.3 | 108.9 | 110.3 | 111.6 | 113.0 | 114.3 | 114.7 |

Average thickness (mm) | 10.3 | 15.7 | 21.0 | 24.1 | 23.6 | 20.8 | 17.1 | 13.7 | 11.2 | 9.4 | 7.6 | 7.0 |

### Friction Model in Icy Condition

Jumping ski in winter games runs on an ice track. The friction between the ski base and the track develops through complicated physical mechanisms during in-run state because the track may exist in multiple phases from solid (ice) to liquid (water) exerting a strong influence on friction. In the beginning of sliding, the track is mostly in solid phase and hence the dry friction due to solid-to-solid asperity contacts dominates the early stage of in-run resulting in a relatively high frictional force. Heat generated by the frictional force melts the top surface of icy track and a thin water layer is, in consequence, formed at the interface. As the water layer becomes thicker, the friction force decreases because water bridges are formed between two solid surfaces and their capillary drag determines the friction force. If the ice melts down even more, the friction increases again with the film thickness because the hydrodynamic friction by viscous shearing of the film becomes the principal friction mechanism. These three friction mechanisms on ice are usually referred to as boundary, mixed, and hydrodynamic friction (or lubrication) regimes, in the order described.

Here, \(\mu_{dry}\) is the dry friction coefficient without water layer, \(v_{r}\) is the relative velocity of a solid body, \(\eta\) is the viscosity of water layer, \(h_{w}\) is the thickness of water layer, and \(p_{r}\) is the pressure. In this model, the contribution of dry friction with respect to the hydrodynamic friction is controlled by \(\alpha\) ranging from 0 to 1.

^{2}as follows.

Here, \(A_{c}\) is the contact area and we substitute the contact width, \(w_{c}\), with the one of the specimen used in experiments (0.5 cm). We used this friction model to describe the interaction between the ski base and the in-run track in this study.

While it is almost impossible to achieve a perfect dry condition without any water layer generated when both a solid body and the ice come in contact, Bowden et al. could obtain a generally acceptable value of dry friction coefficient (\(\mu_{dry} = 0.3\)) between the ski base material and the ice with very thin water layer being formed [26, 28]. It can be deduced from this value and Eq. (3) that the dry friction contributes only 7.6% (or \(\alpha = 0.076\)) and therefore, the hydrodynamic friction dominates (92.4%).

### Analysis Procedure

To simulate the ski response during in-run state and investigate the effect of vibration on friction, transient dynamic analysis is performed using an implicit time integration method. First, we apply the normal force from the gravity only so that the jumping ski becomes fully contacted with the icy track. Then, sinusoidal forces are imposed at both ends to induce normal vibrations over the ski. Finally, we apply the tangential force from the gravity and allow the ski to slide along the track. Time-variant frictional forces are acting on the contacting nodes according to the friction coefficient in Eq. (3). Since this analysis is highly nonlinear due largely to frictional contact, Bathe method is used for time integration, which is a composite implicit time integration method proven to be more stable and accurate than conventional Newmark method [29, 30, 31]. Here, the time interval (\(\Delta t\)) is set to 0.000125 s considering the frequencies of the applied vibration and the natural frequencies of the jumping ski beam model. Analysis is performed for 6 s, usual elapsed time to slide down a large hill track, using the commercial finite element analysis software, ADINA version 9.0.5 [32].

## Numerical Studies

### Effect of Vibration Frequency on Friction

### Effect of Vibration Amplitude on Friction

### Effect of Bending Stiffness Distribution on Friction

## Conclusions

- (1)
In icy condition, the contact area between the ski base and the track governs the friction.

- (2)
In general, the vibration decreases the friction force. However, it does not monotonically decrease with the frequency and amplitude of applied loads. It is rather mode/amplitude specific probably because of inherent nonlinearity in ski deformation and its contact with the track.

- (3)
Smoother bending stiffness distribution seems more beneficial to friction reduction.

## Notes

### Acknowledgements

This research was supported by the Convergent Research Program for Sports Scientification (grant number 2014M3C1B1033983) and the EDucation-research Integration through Simulation On the Net (EDISON) Program (grant number 2014M3C1A6038842) through the National Research Foundation of Korea (NRF) Grant funded by the Ministry of Science and ICT.

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